FUNDAMENTALS
OF MACHINE DESIGN

MIR PUBLISHERS MOSCOW

P. ORLOV

OF MACHINE DESIGN

TRANSLATED FROM THE RUSSIAN
BY YU. TRAYNIGHEV

MIR PUBLISHERS . MOSCOW

First published 1976

THE RUSSIAN ALPHABET AND TRANSLITERATION

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© English translation, Mir Publishers, 1976

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Contents

Preface .. 7

Chapter i. Principles of Machine Design . 9

1.1. Objectives of Machine Design. 9

1.2. Economic Factors of Design. 10

1.3. Durability . 28

1.4. Operational Reliability . 52

1.5. Machine Cost . .. 57

1.6. Building up Machines Derivatives on the Basis of Unification 61

1.7. Reduction of Product Range. 70

1.8. Preferred Numbers and Their Use in Designing. 78

*1.9. General Design Rules ..' 84

Chapter 2. Design Methods .. 88

2.1. Design Succession .. 89

2.2. Study of Machine Application Field ......... 91

2.3. Choice of Design . 92

2.4. Development of Design Versions . 93

2.5. Method of Inversion . .. 98

2.6. Composition Methods . 103

2.7. Composition Procedures .. 106

2.8. Design Example . 107

Chapter 3. Weight and Metal Content ..' 131

3.1. Rational Sections ... 133

3.2. Lightening of Parts . .. 148.

3.3. Rational Design Schemes. 170

3.4. Correction of Design Stresses . 184./ ■>.»

3.5. Materials of Improved Strength . 211

3.6. Light Alloys .. 226

3.7. Non-Metallic Materials . 235

3.8. Specific Indices of Strength of Materials. 246

Chapter 4, Rigidity of Structures .. 252

4.1. Rigidity Criteria .. . 253

4.2. Specific Rigidity Indices of Materials . .. . 260

4.3. Enhancing Rigidity at the Design Stage . 272

4.4. Improving the Rigidity of Machine Constructions ... 305

Chapter 5. Cyclic Strength . 348

5.1. Improvement of Fatigue Strength . 391

5.2. Design of Cyclically Loaded Components. 397

5.3. Cylindrical Joints Operating under Alternating Loads 412

6

Contents

Chapter 6 . Contact Strength . 41 g

V y- 6.1. Spherical Joints . 424

6 . 2 . Cylindrical Connections .. . ! 428

^ ^ Chapter 7. Thermal Stresses and Strains .. 439

7.1. Thermal Stresses . 430

7.2. Thermal Strains . ■ ' \ 4 g-j

7.3. Temperature-Independent Centring ....!!!!! 472

7.4. Heat Removal.j 431

w y Chapter 8 . Strengthening of Structures. 486

8.1. Elastic Strengthening ... 435

8.2. Plastic Strengthening ..] j 439

Chapter 9. Surface Finish . 493

9.1. Classes of Surface Finish . 500

9.2. Selection of Surface Finish Classes . 510

Index . 516

Preface

The purpose of the present book is to offer the reader an attempt
at a systematic exposition of rules for rational designing.

With all the diversity of the modern, machine-building the tasks
facing the designer are similar in many respects. It is the reduction
of the weight and specific metalwork weight of the machine, the
improved suitability for industrial production, greater durability
and reliability that are of importance for the design of any machi¬
ne, the difference lying only in the relative significance of these
factors. All this enables one to formulate the principles of rational
designing as a code of general rules for machine building.

The prime intention of the book is to make the designer learn to
work creatively. To design imaginatively means:

to abstain from blindly copying the existing prototypes and to
design meaningfully, selecting from the entire store of the design
solutions offered by the present-day mechanical engineering the
ones that are most suitable under given conditions;

to be able to combine various solutions and find new, better
ones, i. e., display initiative and put vim in the work;

to continually improve the machines’ characteristics and to
contribute to the progress in the given branch of mechanical engi¬
neering;

to follow the dynamic development of the industry and devise
versatile machines of long life, amenable to further modernization
and capable of meeting the ever-growing demands of the national
economy without running the risks of obsolescence for a long time
to come.

Particular attention in the book is attached to the problems of
durability and reliability. The author endeavoured to strongly
emphasis the leading role of the designer in tackling these problems.

8

Preface

In presenting the material the author followed the principle
“qui vidit—bis legit” (the one who sees reads twice). Most of the
designers are individuals of visual thinking and visual memory.
For them a drawing or even a simple sketch means much more than
many pages of explanatory notes. For this reason, each point in
the text is accompanied by design examples.

To better the understanding most of the illustrations are arranged
in such a way as to enable it to compare wrong and correct, inexpe¬
dient and expedient design versions.

The solutions given as correct are not the only possible ones.
They should be regarded not as precepts, suitable for use in all
cases, but rather as examples. In particular conditions other ver¬
sions may prove more advisable.

Chapier I

Principles of machine design

1.1. Objectives of Machine Design

The chief aim of the designer is to develop a machine that would 1
satisfy most fully the needs of the national economy, would be most
economic, and would have the best technical and operational charac¬
teristics.

The most important characteristics of machines are their produc¬
tivity, efficiency, strength, reliability, weight, specific metalwork
weight, size, power intensiveness, scope and cost of repairs, labour
costs, service life, in-between repair times, degree of automation,
simplicity and safety of maintenance, and convenience of operation,
assembly and dismantling.

Any machine must meet the industrial design requirements, i.e., it
must have a plain but attractive finish.

The priority of each of the above characteristics depends on the-
purpose of the given machine, namely:

for generators and energy converters the main characteristic is-
their efficiency which is indicative of the degree of useful energy
conversion;

for power tools—productivity, precision and reliability of opera¬
tion, and degree of automation;

for metal-cutting machines—productivity, accuracy of machining,-
and range of operations;

for control and measuring instruments—sensitivity, accuracy, and.'
stability of readings;

for transport machines (particularly for air and spacecraft-
weight and engine efficiency which determine the amount of on¬
board fuel.

Economical considerations are of tremendous importance in;
engineering.

When designing a machine, the designer must do his best to
make the machine as economical as possible throughout its service-
life.

This aim is achieved by way of enhancing the efficiency of the-
machine, increasing its service life, and cutting operational expen¬
ditures.

.10

Chapter 1. Principles of Machine Design

At the same time the designer must minimize laborious manufac¬
turing operations, lower production costs, and reduce the time
;spent on designing, making, and running-in the machine.

A vast number of technological, organizational, processing, eco¬
nomic, and other factors affect the total cost of engineering products.

This book deals only with design methods which enhance effi¬
ciency and reduce production costs.

1.2. Economic Faetors of Design

Economic factors must be made the basis of designing. Designing
particulars should never overshadow the main aim—-increase of
machine efficiency.

Many designers consider that to design economically means cutting 1
production costs, avoiding complex and expensive solutions, using
•cheapest materials and applying simplest processing methods.

This is but a part of the problem. The economic effect is determi¬
ned by machine output and the total operational expenditures during
■service life. The cost of the machine is not the only and not always
. the main part of the expenditures.

Economy-oriented designing means'consideration of all the factors
•determining the efficiency of the machine and a correct evaluation
•of the relative importance of each of these factors.

This principle is often ignored. In an attempt to obtain cheaper
products the designer often achieves economy in one way only,
while missing others and more effective ones. Moreover, such a one-
■sided economy, which disregards the totality of the essential factors,
often results in a lower overall economy of the machine.

(a) Profitability (Commercial Value) of Machine

Machine profitability q is determined by the ratio between output
(production) Ot over a certain period of time, expressed in terms
■of money, and the total operational expenditures Ex over the same
period

The term “output” implies the cost of products made on the machi¬
ne (the cost of finished and semi-finished products, and the cost
•of intermediate operations and useful work performed by the machi¬
ne).

Generally, the total expenditures Ex cover the following: De —
■depreciation charges for the machine; Pr —cost of power consumed;
Mr —cost of materials consumed; Lbr —cost of labour force; Mntce —
•cost of maintenance; Ovhd —overhead costs; Rpr— cost of repairs;

1.2. Economic Factors of Design

11

general depreciation charges for the plant, i.e.,

Ex — De + Pr -f- Mr + Lbr -j- Mntce -j- Ovhd -\-Rpr-\-Gd

The value of q must always be greater than unity, otherwise the
machine will operate unprofitably, in other words, its existence
•will become commercially useless.

( b) Economic Effect

The annual economic effect (annual profit) Q from the machine
:'is the difference between the annual output and expenditures

Q = Ot-Ex = Ot (1.2)

where q is the profitability.

The total profit 2$ for the entire service life of the machine is
•equal to the difference between the total output 2 Ot and total
•expenditures ^Ex

2 <2 ^HOt-^Ex

•or

2 Q= 2 Ot — 2 jPr-f 2 Mr- f- 2 Lbr+ 2 Mntce +

4- 2 Ovhd -{- 2 Rp r + 2 Gd) (1.3)

The quantity 2 Q depends on the duration of the machine opera¬
tion. Let us introduce the following more precise definitions: H—
service life, i.e., the total period (in years) of the machine’s being
in operation; h—actual running time (in years) for the entire service
period. If we assume that the machine will run until its physical
resources are fully exhausted, then, obviously, h is the durability
of the machine, i.e., its potential running time.

The relation

Pase— ~jg~ (1.4)

is the use factor characterizing the operational intensity of the
machine.

In Eq. (1.3) some terms (2 Rpr> 2 Gd) are proportional to the
service life, i.e., 2 Rp r — HRpr ; 2<3d = HfGd, while the others
&t, 2 Pt, '2jLbr,'2jMntce, 2-/lTr, ^Ovhd), to the actual running time
(i.e., to the machine durability, given the above assumption is
valid) and are equal to hOt, hPr, etc., respectively.

The depreciation expenditures for the entire service life are equal
to the cost of the machine

2 De=*C

( 1 . 5 )

12

Chapter 1. Principles of Machine Design

Substituting tbe above values for the respective terms of Eq. (1.3)
we will have

2 Q = hOt -[C-\~h ( Pr 4* Mr -f- Lbr -j- Mntce + Ovhd) 4-

+ H(Rpr + Gd)}

Let us designate the expenditures proportional to the durability h
as Ex' and those proportional to the service life H as Ex". Then

2 Q^hOt~(C + hEx' + HEx") = hOt~[c + h (Ex' + ^-Ex"} ]

E 1

Since, according to Eq. (1.4), -r =» —> then

^Q^hiot-Ex' - |~)~C (1.6)

In terms of service life H the total profit (economic effect) is

2 Q = H [T] use (Ot- Ex ') - Ex" I —C (1.7)

The recoupment term T r of the machine is the service period for which
the aggregate economic effect equals the cost of the machine (2(? —
= C). Substituting this expression in Eq. (1.7), we get

Tr W {Ot-Ex')~Ex“

When determining the recoupment term of the machine, the
repair costs can be ignored because at the initial stages of operation
they are insignificant.

(c) Coefficient of Operational Expenditures

The ratio between the total expenditures for the entire service
period of the machine and its cost is called coefficient of operational
expenditures:

k

H Ex

C+h (ex'+JeL)
' Pase /

Ex”

r\v.se

)

(1.9)

Equation (1.6) can now be given in the following form

2 Q = hOt~kC (1.10>

The percentage ratio of the machine cost to the total expenditures
i s 6 . e< I ua l to the reciprocal of the coefficient of operational expenditu¬
res

2 * * 4

• 100 %

c

( 1 . 11 )

1.2. Economic Factors of Design

13

Coefficient k is usually much larger than unity and may be as
great as 10-100.

As is seen from Eq. (1.9), the coefficient of operational expendi¬
tures increases with an increase in durability h of the machine.
Correspondingly, the proportion of the machine cost in the total
amount of expenditures decreases.

(d) The Influence of Operational Factors
upon Economic Effect

Equation (1.6) shows that the overall economic effect, i.e., the
total gain for the entire machine service life,, is proportional to
durability h. This gain will be the greater, the higher the annual
output Ot and the less the machine cost C and expenditures Ex'
and Ex".

Let us consider the relative importance of each of these factors
by analyzing the operation of an exemplary metal-cutting machine
tool.

In this case it is best to determine the net economic effect
comprising the total profit less the cost of materials and consumable
tools. Furthermore, we will ignore the general factory overhead
which is difficult to consider and limit ourselves here to the overhead
expenditures directly related to the operation of the machine (main¬
tenance. expenses are included in labour costs).

Let the machine cost C be 1500 roubles (rbl), power consumption
of the machine drive electric motor, 10 kW. The machine operates
on a double-shift basis with a load factor of 0.8. Taking into consi¬
deration holidays and Sundays (75 days per year), the machine use
factor will be

% se = 0.8 • -

365 - 75
365

The actual machine running time per year will be
365-24-0.4 « 3500 h/year

Assuming that on the average the machine operates at 0.75 of its
rated power, the annual electrical power consumption is

0.75 • 10 • 3500 = 26 250 kWh/year

With an industrial tariff for power of 2.5 kopeks per 1 kWh, the
annual cost of the power consumed is

Pr — 26 250-0.025 650 rbl/year

If the annual operator’s pay is 1500 rbl, then the cost of labour
on a double-shift basis will be

Lbr — 2 • 1500 — 3000 rbl/year

14

Chapter 1. Principles of Machine Design

Let the overhead rate be equal to 25% of the labour cost
Ovhd — 0.25*3000 = 750 rbl/year

Assume that the total cost of repairs by the end of the service
life of the machine is equal to its cost, i.e.,

Then, the overall economic effect in terms' of the service life is
2 Q = H [ Ot — (Pr + Lbr + Ovhd)} — 2 Rpr—C =

— H [Ot — (650 + 3000 + 750)] -1500 -1500 =

— H(Ot— 4400) — 3000 (1.12)

In order to calculate the output, assume that the profitability
of the machine, related to the sum of expenditures Pr, Lbr and Ovhd >
is

__ Ot _ , „

^ P r-j- Lbr + Ovhd

Then

Ot -1.6 (650 + 3000+750) « 7050 rbl/year
and Eq. (1.12) becomes

2 Q' = H (7050 - 4400) - 3000 = f?2650 - 3000

On the basis of Eq. (1.12), let us analyze variations in the profit
with an increase in the machine’s service life and output, and also
with changes in the cost of the machine, labour and power. Let the
initial duration of seiwice life H be equal to 2.5 years, which with
the adopted use factor corresponds to the machine durability h
of 1 year. The results of estimates for service lives of 2.5 to 25 years
are given in Fig. 1 and Table 1.

From Table 1 and Fig. 1 the following conclusions can be
drawn.

The profit sharply rises with the increase of h, i.e., with the
increase of H , provided that r| us<3 — const. Taking the profit for
If — 2.5 years to be equal to unity, then with H — 10 years the
profit increases by 6.5 times and with H — 25 years, by 17.5 times..

The coefficient of operational expenditures increases from its
initial value of k = 9 up to k = 73 (with H = 25 years) with the
increase in the machine’s service life. This correspondingly lowers
the ratio between the machine cost and the total operational expen¬
ditures. This ratio, equal to 11 % with the initial service life figure
(2.5 years), decreases to a negligible value (3-1.5%) when the service
life is increased to more than 10 years.

1.2. Economic Factors of Design

15 -

Reduction of the machine cost appreciably influences the profit
only when short service lives are involved. For example, reducing
the cost in half (which is quite a sizable value) results in a 20.5-per

Fig. 1. Relation between overall economic effect and machine's service life H

l — ratio lorlnltial output and labour cost; 2 — coefficient of operational expen¬

ditures; S — ratio of machine cost to operational expenditures; 4 — increase of economic
effect with machine cost halved; s — decrease of economic effect with machine cost increased
1.6 times; e — increase of economic effect with 10-percent increase in machine efficiency;

7—ratio wittl !aiour cost reduced by 30%; a— ratio with out P ut

increased 1.5 times; 9— ratio 2^S^ 2 " 8 wltl1 out P ut doubled; 10 — ratio 2^S® 2 ' 8 11,11:11
labour cost reduced by 30% and output doubled

cent increase in the profit when Ii = 2.5 years, but with a service
life of over 10 years the increase comes to 3.5-1% only.

Conversely, the rise of the machine cost has a very slight effect
upon the profit when the service life is long. Thus, increasing the
machine cost by one half lessens the profit by 21% when H — 2.5
years and only by 3 to 1 % with service lives longer than 10 years.

Chapter 1. Principles of Machine Design

16

Table 1

Economic Effect as a Function of Service Life and Operational Factors

Service life H, years

Economic Indicators

2.5

1 5

1 10

1 15

| 20

| 25

Durability (n U5e = 0- 4 ), years

i |

I 2

!. 4

t 6

| 8

1 10

Profit rbl

3625

10 250

23500

36 750

50 000

63 250

increase as compared with

2*23.5

1

2.82

6.48

10.2

13.75

17.4

Coefficient of operational expen¬
ditures k

9

16.2

30.4

44.3

58.8

73

Ratio of machine cost to opera¬
tional expenditures, %

Profit increase, %:

11

6.15

3.3

2.25

i •

: 1.7

1.4

j

with machine cost reduced by
half

: 20.5

j

7.5

3.8

2

1.5

1.25

with machine efficiency increa¬
sed by 10%

4.

3

2.5

2.45;

2.4

2.35

Profit decrease (in per cent) with
machine cost increased by 1.5 times
Profit increase C^jQl^Qz. 5 ):

21

i

7.3

3.2

2

1.5

1.2

with labour cost decreased by
30%

with output increased by:

1.62

4.05

9

13.9

18.8

23.7

1.5 times

3.3

7.4

15.7

24

32.2

40.5

2 times

5.9

12.5

26

39.2

52.5

66

with output increased by 2
times and labour cost decreased
by 30%

9.6

18

36.5

54.5

71.5

90

Consequently, making the machine more costly but of greater
durability is quite justifiable economically since the gain from the
enhanced durability by far exceeds the drop in the profit caused by
the rise of the machine cost. Thus, increasing the initial durability
by 6 times entailing even a two-fold rise of the cost increases the

profit by 10 times.

Raising die efficiency of the machine (lowering the power costs)
in our example has no significant effect. For instance, a 10-per
cent increase in the efficiency raises the profit only by 4% when
H — 2.5 years, and on the average by 2.5% when H > 10 years.

1.2. Economic Factors of Design

17

Profit is greatly increased by lowering the labour costs through
automation, attendance of many machines by one operator, etc.
Thus, decreasing expenditures on labour and maintenance and asso¬
ciated overhead charges by 30% enhances the profit by 9 times when
jj = 10 years and by 23.7 times when H — 25 years.

Increasing the machine output has a great effect. An output increase
of 50% increases the profit by 15.7 times when H = 10 years and by
40.5 times when H — 25 years, these figures in the case of a doub¬
led output being 26 and 66 times, respectively.

A sharp rise in the profit is obtained by simultaneously increasing
the durability and output of the machine and reducing the labour costs.
For instance, with the durability increased by 6 times, output
doubled, and labour costs cut down by 30% the profit rises by 36.5
times when H — 10 years and by 90 times when H — 25 years.

The effect of the increase in the durability and output in this
case is so great that it nullifies the influence of the other factors,
e.g., the cost of the machine and power.

The above calculation example is sketchy since, apart from the
assumptions that simplify computations, it does not consider the
dynamics of operational changes (e.g., possible future alterations
of the power cost, a fall in the output due to the gradual wear of the
machine, etc.). Nevertheless, for machine tools the approximation
clearly shows how operational expenditures influence the profit.

Naturally, for other machines, and with a different structure of
operational expenditures, the influence of various factors ou the
profit will vary.

Let us take, for example, the cost of labour. There is an extensive
range of machines (non-automatic machine tools; automobiles; road¬
building, constuction and agricultural machines, etc.), which
cannot function without the constant attendance of an operator.
Here the labour cost is comparatively high and there is no way to
substantially reduce it. Correspondingly, as proved earlier, the
relative importance of the machine cost within the totality of ope¬
rational expenditures is hot very great.

For machines that can run for long periods without any attendan¬
ce (electric motors, electric generators, pumps, compressors, etc.)
the labour costs only consist of the costs of periodic maintenance
and inspection.

Most economic from the standpoint of labour costs are automatic
and semiautomatic machines. For these machines the relative
importance of the machine cost is much greater.

Power requirements of various machines differ considerably. For
heat engines the fuel cost is of far greater importance than the cost
of the machine and, occasionally, the cost of labour.

There are also machines which have an insignificant power con¬
sumption thanks to their high efficiency (electric generators, reduc-

2-01395

18

Chapter 1. Principles of Machine Design

tion gears, etc.). If, in addition, the labour costs are also small*
the machine cost will then be the dominant factor.

All other things being equal, the machine cost to a decisive degree
depends upon the number of the machines produced. When machines-
are produced on a mass scale their cost is not very high and its
relative importance in . the totality of operational expenditures is
much slighter than in the case of machines produced on a small-lot
basis or, the more so, to an individual order.

For some classes of machines the costs of depreciation, mainten¬
ance and-repairs of industrial premises and installations carry great
weight. These, expenditures may by many times exceed those con¬
nected with running the machine.

An economic calculation similar to the above makes it possible
in each .particular case to determine the structure of operational-
costs and the relative importance of their constituents, and to estab¬
lish an economically rational basis for designing the machine. .

Generally, profit depends to the greatest degree on .the output
and durability of the machine. Therefore when designing a machine
the designer must focus his attention on these factors. Equally
important is the reliability which determines not only the durabili¬
ty but also the scope and cost of repairs carried out during the machi¬
ne’s being in use. In the previous example the consequence of the-
repair costs is somewhat overshadowed because in the calculation
these costs for the; service life of the machine have been taken at
a moderate value equal to the machine cost. In other words, the-
repair costs are taken to be such as they must be for a correctly
designed and reasonably run machine.

In practice repair costs may reach very large values exceeding:
in some cases the machine cost several times. Sometimes repair
expenditures absorb a major part of the profit gained from operating,
the machine, thus making its further use unprofitable.

Nowadays the transfer of machines to repairless operation is
a problem of precedence.

The term repairless operation implies:
exclusion of capital repairs;

exclusion of repairs to worn parts, meaning instead complete
replacement by new parts, units and assemblies;

exclusion of emergency repairs, caused by the breaking or wearing
down of parts, by way of planned maintenance.

The transfer of machines to repairless operation is a complex of
problems the prerequisites for the Solution of which are as follows:
prolongation of the service life of wearing parts;
construction of machines on a unit assembly principle enabling
independent replacement of worn parts and units;

provision in machines of non-wearing datum surfaces for locating
changeable parts.

1.2. Economic Factors of Design

19

These design measures must be accompanied by technical and
organizational measures, the most important of which is the organi¬
zation of a centralized production of replacement parts and units.

The above said by no means implies that the designer may pay
less attention to the problem of decreasing the machine cost. Such
a conclusion would be wholly wrong. As noted earlier, the value
of the machine cost depends on the type of the machine and may be
great for machines having low energy and labour requirements, as
well as for those with a comparatively short service life. It is only
necessary to correctly evaluate the relative importance of this factor
among other factors influencing the economy of the machine and be
able to sacrifice it in cases when the reduction of the machine cost
contradicts the requirements for enhanced output, durability and
reliability.

It is noteworthy that along with decreasing the cost of individual
machines, there is another, more effective way of lowering the cost
of machine products as a whole, namely, reducing the list of machi¬
ne products by selecting optimal types of machines, and satisfying
the needs of the national economy while minimizing the number
of the machine type-sizes (see p. 70).

Successful solution of the above listed problems should be the
main activity of the designer, who must, first, set the policy in the
machine building field and, second, develop progressive designs,
providing for high economic efficiency of machines, reduced opera¬
tional expenditures and low cost of machine products,

(e) Influence of Durability on the Size of Machine Fleet

Increasing the durability is an effective and economical means
to increase the number of machines being used at one time.

The number N of machines running at a given time is proportional
to the product of their durability h and the number n of machines
produced per year in the previous time. ■ •

As an example, consider a case when the annual production n is
constant and equals 100 machines per year. Let the machine’s
durability h be 3 years; the machines operate continuously, i.e.,
their service life is equal to the durability.

The diagram in Fig. 2 a pictures the utilization of the machines
by the years. The number of the machines manufactured yearly is
shown by blackened rectangles. The sum of the rectangles along the
horizontal shows the duration of the machines’ being in service,
equal in this case to three years; the sum of the rectangles along the
vertical represents the number of the groups of machines produced
in different years and being in operation at one time. This quantity
is equal numerically to the durability {h — 3), provided the annual
production of the machines and their durability remain constant.

1667 1968 1969 1970 1971 1972 1973 1979 1975

(C)

Fig. 2. Machine utilization diagrams
(a) with rinse *“ 1; with ’''use = °- 5 ’ <c) wittl 1 W “ °- 3

1.2. Economic Factors of Design

21

Hence* the total number of the machines in use every year is
N^hn = S -100 = 300

Now assume that starting from 1966 the manufacturer doubles-
the durability (hatched rectangles). The machine’s service duration,,
represented by the horizontal sum of the blackened and hatched
rectangles, becomes ti — 6 years. From 1970 onward the size of the
machine fleet increases, reaching in 1972 a stable value which for
all the subsequent years remains constant and equal to 600 machines,
i.e., to the product of the new durability and the annual production

N' = h'n

Thus, with the production of the machines being the same, a two¬
fold increase in the durability raises by as many times the size of
the machine fleet and, cosequently, the annual output of products
(and; the total output for the entire service period of the machines
as well).

Let us examine a case when a machine is not fully used, i.e.,
when the machine’s service period is lengthened in comparison with
its durability reserve (Fig. 2b).

The service life H is equal to the quotient of durability h by the
use factor r| ttSe accounting for all forms of forced or scheduled stop-
pages

Let h — 3 as before, and r] use = 0.5, then the service life H =
3 ft

= 03 = 6 years.

The reduction of the degree of utilization of machines is equiva¬
lent to reducing the number of the machines operating at one time.
In our case (%;.„ — 0.5) this reduction is shown on the diagram by
halving the height of the blackened rectangles. The number of the
machines of the same year’s manufacture, operating simultaneously
in the course of one year becomes equal to 50.

The number of the groups of machines of different years’ manu¬
facture, running at the same time is equal to the vertical sum of the’
rectangles. With the initial assumptions (n — const, h — const)
being valid, this quantity for any year is numerically equal to the
machine’s service life (H — 6).

The total number TV of machines in the fleet for any year is the
product of the service life and the actual number of the machines
cf each group operating at the same time ( n act = m ) use )

N ~ JFInx\ use

22

Chapter 1. Principles of Machine Design

But

Hence

Puse

N — hn

In the case being considered N = 3 *100 = 300.

The result can easily be checked by simply finding the total
number of machines in any one of the vertical columns of the diag¬
ram (N = H *50 = 6 *50 = 300).

Thus, the annual size of the machine fleet in operation does not
depend on the use factor or the service life, but only on the durabi¬
lity and the number of machines manufactured yearly.

This conclusion, naturally, holds true only if the total service
life of the machine is equal to its mechanical life. However, when
the service life is limited because of obsolescence the picture sharply
changes: the machine is rendered obsolete before its mechanical
life has expired and one has to prematurely discard it, thus losing
products which could have been manufactured if the durability
reserve of the machine had been fully utilized during a shorter
period of time.

Let us illustrate this (Fig. 2c). Let the machine’s durability be
6 years. Assume that, due to a low shift factor, prolonged downtimes,
etc., the machine utilization at any given moment is equal to 30%.
With the service life corresponding to the full utilization of the

ft

durability reserve, i.e., with H — ^ = 20 years, the machine

(as is obvious from the above) will yield production equal to 6A,
where 4 is the annual output. With an annual manufacture of
machines n — 100, the total output of the machines of one year’s
manufacture will be 6004, which for 10 years (1966-1975) becomes
60004.

Now assume that the obsolescence period is 10 years. Then, in
1976, i.e., 10 years after their manufacture, the machines must be
discarded. Up to this time the 1966 machines will yield only 50%
of their potential production (0.5 *600.4 = 3004); 1967 machines
45%; 1968 machines*, 40%, etc. The total output of all the machines

manufactured in 1966-1975 will be 16504, i.e., —~*100% =

= 27.5% of the production they would have yielded if their dur¬
ability reserve had been fully utilized. Thus, the limits imposed
because of obsolescence sharply reduce the total output. In our
example the manufacture of machines, which in the near future
would become obsolete, would inevitably lead to huge losses.

The analysed cases belong to the simplest ones. The picture beco¬
mes much more complicated with annual changes in the number

1.2. Economic Factors of Design

23

of the machines manufactured and their durability. Yet, the general
regularity remains valid: increasing the durability (within the
limits of the obsolescence period) is always accompanied, in the years
to come, by an increase in the actual size of the machine fleet, the
increase being proportional to the size of the annual manufacture
of the machines and their durability.

(/) Influence of Durability on Output

Let us now examine the problem of the output obtained from
a group of machines working simultaneously.

The total output (in terms of money) obtained from a machine
throughout its entire service life H is equal to the product of the
annual output Ot and actual running time of the machine

2 S — Ot'H "TltMe

Assuming that the machine fully exhausts its durability reserve

\U*}us e ~ n '),

%S = Ot-k (1.13)

The annual output from a group of machines will equal the product
of the annual output from each machine, use factor r] use , and number N
of the machines being in operation at one time

2 Sannl — Ot-X] use 'N (1.14)

The number N of machines running in any given period of time
is equal to the product of the number n of machines produced yearly
und their service life H

N = nH

■Substituting this relation into Eq. (1.13), we obtain

^Sanni—Ot-n-H’^e^Ot-n-h (1.15)

Hence, the total production delivered by a machine throughout
its service life [Eq. (1.13)] and the annual output from a group of
■simultaneously running machines [(Eq. (1.15)3 are proportional to
the product of the annual output and the durability of the machine.

Doubled durability will double the annual production. If, simul¬
taneously, the output is also doubled, the total output will then
be increased four times. If the annual output is specified, then the
increase of the durability and output will allow the number of the
machines produced yearly to be reduced in proportion to the product
nh. In this instance the total expenditures on the machine manufac¬
ture and labour will also be reduced, which means added financial
gain.

24

Chapter 1. Principles of Machine Design

Using the numerical values from the above example, we may com¬
pute the increase in financial gain resulting from the reduction of
the number of machines manufactured yearly made on account of
increased durability and output of the machines.

Assume that the required annual production is provided for when
the number of machines manufactured yearly is 100, the machines
having service life H — 5 years and annual output Ot = 7050 rbl/year.

With the doubled service life and output ( H —10 years, Ot
= 14 100 rbl) the number of machines to be made each year to
provide for the specified production will be reduced to 25 machines.

According to Eq. (1.12), the total output of the machine for
the entire service life

2 Ot = H-Ot

The total gain from 100 machines with service life H — 5 years and
annual output Ot = 7050 rbl is

2 Oti oo = 100 • 500 • 7050 = 3 525 000 rbl

The expenditures over the entire service life of the machine
[Eq. (1.12)3 are

2 Ex = H ( Pr -J- Lbr -j- Ovhd) -j- 2 Apr -j- C

Substituting the numerical values from the above example (Pr =
— 650 rbl/year, Lbr — 3000 rbl/year, Ovhd — 750 rbl/year, and
BRpr — C — 1500 rbl), we obtain

2 Ex = 5 (650 + 3000 + 750) +1500 +1500 = 25 000 rbl

The total expenditures for 100 machines

2 Ex m = 100-25 000 = 2 500 000 rbl

The total profit from 100 machines

2 <?100 - 2 Ot m - 2 Ex m = 3 525 000 - 2 500 000 = 1025 000 rbl

Assume that the cost of machines with service life H' = 10 years
and output Ot’ — 14 100 rbl/year is equal to the doubled cost of the
original machines (C — 2 C — 3000 rbl). Consider also that the
difficulty of repairs increases proportionally to the machine cost,
that is, 2Apr' = 2C = 3000 rbl.

Then, the total expenditures over the entire service life of the
machine

JjEx' = IP (Pr + Lbr -f Evhd) 4- 2 Rpr’ + C' =

= 10 (650 + 3000 + 750) + 3000 + 3000 = 50 000 rbl

1.2. Economic Factors of Design

2S

The total expenditures for 25 machines

2 #4 = 25-50 000 = 1 250 000 rbl

The total output from 25 machines possessing the higher durabi¬
lity and output -will, according to the original equation, be the
same as for 100 machines

2 25-10-14 100 = 3 525 000 rbl

The total gain from 25 machines will be

2<?; s = 2 04 - 2 £*» = 3 525 000 -1 250 000 = 2 275 000 rbl

Consequently, when replacing 100 machines with 25 machines of

higher durability and output, the profits rise by 1 025 C;06 ~ ^-2-'

times in spite of the fact that the machine cost is doubled.

The picture will only slightly differ, if the power consumption-
is taken to be proportional to the output, i.e., when Pr' — 2Pr —
= 1300 rbl/year. In this case

2 Ex' = 10 (1300 + 3000 + 750) + 6000 = 56 500 rbl

For 25 machines

2 Ex ' t8 = 25-56 500 = 1 412 500 rbl

The total financial gain

2 <?; 6 = 3 525 000 -1412 500= 2 112 500 rbl

i.e., compared with the original estimated profit for 100 machines,.

the gam rises by JQ25000 = 2,05 ^mes.

Conclusions. The increase of the machine’s output and durabi¬
lity is a most effective and profitable means to raise the production
and financial gain.

The durability increase allows a proportional reduction in the
number of the machines produced annually to be made without
decreasing the total industrial output, and thus provides for cutting-
down the cost of the machines manufacture, materially reducing-
the operational expenditures, and raising the overall economic
effect.

On the other hand, increasing the machine’s durability without
any changes in the annual number of the machines produced, their
output and the cost of their manufacture will ensure a larger machi¬
ne fleet, raising accordingly the total industrial output.

It is obvious that enhancing the machine’s durability as a means-
of increasing the size of the machine fleet, their total output, and
the power intensiveness of the national economy is by far more-

26

Chapter 1. Principles of Machine Design

profitable than simply increasing the number of the machines
■without raising their durability.

Increased machines manufacture is achieved by commissioning
new plants, expanding the production areas and equipment of the
■existing .enterprises, or by increasing the productivity of the existing
■equipment through intensification of production processes (which
is most advisable economically).

In the first and second cases subject to increase are the costs of
the machines manufacture, while the operational expenditures rise
in all of the cases because of the increased number of the machines
in operation.

As a rule, an increase in the output and durability of machines
is only accompanied by a relatively small rise of their cost, and at
the same time it leads to a reduction in the operational expenditures.
The final result is the same: a larger actual size of the machine
"fleet and greater overall industrial output, but with incomparably
lesser expenditures and materially greater overall economic effect.

Thus, we may conclude that in mechanical engineering it is
■a good policy to combine the increase of the machines manufacture
with the increase of their output and durability and, in certain
■cases, to moderate the machines manufacture, giving preference
to the more advantageous way of increasing their output and dura¬
bility.

Also, from the above we may conclude that increasing the annual
'•machines manufacture does not necessarily mean an increase in the
number of actually working machines and the total industrial
output. The figures of a yearly rise of the machines manufacture
have no value as an economic growth indicator if they are not supple¬
mented with objective data on the durability of the machines and
the quality of production they yield.

These figures may mean: progress, if the durability remains at
■a constant level or increases; stagnation, if the durability lowers
in the same proportion as the output grows; and regression, if the
.purabulity falls to a greater proportion than does the output grow.

(g) Output

Output is expressed by the cost of products or of effective work
-done by the machine in unit time. The output depends on the machi¬
ne’s productivity, i.e., on the number of operations (or work units)
performed in unit time, and also on the cost of operations (work
units).

To increase the output is a complex problem, the solution of
which depends to a large extent upon the correct operation of the
machine. For motor vehicles, for example, operational means of
increasing the output include the elimination of empty runs, aug-

1.2. Economic Factors of Design

27

mentation of running speeds, increase of the load-carrying capacity
{use of trailers), etc. The productivity of machine tools is improved
by intensifying the technological procedures and by using special
.attachments and fixtures.

But in the main this problem must be solved at the design stage.
The machine must be made of the highest possible productivity
according with the actual requirements of the production and pros¬
pects of its development. The business end of the machine should
be designed with a view to maximizing the scope of operations by
appropriately selecting the kinematic scheme, power capacity,
.strength and rigidity.

The main ways to enhance the productivity of machines are as
follows:

increasing the number of simultaneous operations performed on
the work;

increasing the number of workpieces being simultaneously machi¬
ned;

automating the production process.

The first method is most fully expressed in the design of multi-
head metal-cutting machines enabling workpieces to be machined
-on several or all surfaces simultaneously. An automatic multiple-
tool lathe is another example.

The second trend is represented by rotary machines on which
several components are processed at the same time. Another example
is a batch-type processing, in which several workpieces in multiseat
fixtures are worked upon at a time.

Rotary machines are, in fact, roundabout-type machines provided
with rotating tables. Several workpieces which are to be subjected
to one or several operations are mounted on the rotary table and
allotted to an individual operative member (block, chuck or spindle),
also fixed on the table, and are fully machined as the table makes
■one complete i*evolution.

Rotary machines possess a high productivity (e.g., rotary filling
machines with 40-60 spindles perform up to 1500 filling operations
per minute).

The high productivity of rotary machines is due to their conti¬
nuous operation and the simultaneous processing of many compo¬
nents. The number of components processed simultaneously is

z = a i

where i — number of operative units positioned along the periphery
of the rotary table;

a — circumference portion over which the processing opera¬
tion takes place

The productivity of a rotary machine

Q — i-n parts/min

28

Chapter 1. Principles of Machine Design

where n — rotary table speed, rpm;

i — number of operative units positioned on the rotary
table

The speed of the rotary table is dependent upon the time during
which the operation is being performed (so-called phase time)

where t phase — phase time, min;

a = circumference portion over which the processing-
operation takes place (generally a == 0.65-0.75)

Consequently,

Q — i — parts/min

‘'phase

From this equation it is obvious that the productivity of a rotary
machine can be increased by:

decreasing the phase time (i.e., speeding up the operation);

increasing the number of operative units.

The possibilities of decreasing the phase time are limited, whereas
increasing the number of operative units enables the productivity
to be raised, practically, as much as desired. The limiting factor
in the latter case is the speed of loading and unloading the workpieces.

Rotary machines can be easily arranged to form automatic lines
offering high productivity, compactness and economic effect (thanks
to the reduction of the amount of power consumed in transferring
workpieces from one machine to another).

1.3. Durability

Similarly to the output the durability of a machine is heavily
dependent upon the conditions and technical level of its use: care
of the machine, timely maintenance, and prevention of overloads—all
these measures may greatly add to the machine’s durability. Conver¬
sely, low-skilled, poor attendance shortens the machine’s service
life.

But it is the correct design of the machine that determines its
durability.

(a) Durability Criteria

The durability of a machine is the total time during which the
machine can . work under normal conditions and still retain its
principal ratings, due account being taken of repairs, provided
their total cost is within reasonable limits.

Occasionally, application also finds the notion of durability
reserve, i.e., the time in hours of machine running until the first
major overhaul.

1.3. Durability

29

In many cases, particularly for machines of aperiodic action,
■durability is measured in terms of the total output for the entire
service life. In this case the durability is the total number of ope¬
rations (units of work) that the machine or unit is capable of per¬
forming until complete wear out. For instance, the durability of
motor vehicles and railway rolling stock is determined as the maxi¬
mum distance travelled; that of instruments and testers, as the
total number of switchings; of foundry mills, as the total number
of melts; and of soil-cultivating machines, as the total area culti¬
vated.

Depending on operating conditions, the actual durability may
substantially differ from the rated one. It falls due to systematic
overloads, speeds higher than rated, increased working forces,
adverse climatic conditions, etc. Conversely, the durability increa¬
ses when the machine operates under lighter conditions.

The influence of operating conditions upon the durability may
be taken account of by introducing a duty factor r\dv.ty Thus, the
actual durability will be

Jl' Crated
r}duty

(1.16)

where h Tate d is the rated durability.

The magnitude of this factor can be reliably determined on the
basis of a differentiated investigation into the influence of operating
■conditions and duty of the machine upon its durability. This is
a problem pertaining to the statistical theory of durability. If pre¬
cise data is not available, one may take the following approximate
values: for light duty — rj duty — 0.7-0.8; medium duty —
r\duty = 1; heavy duty — r\ dlLty = 1.2-1.5.

(b) Service Life

The service life of a machine is the total period it remains in
operation up to complete exhaustion of its durability reserve.

For machines of aperiodic action the service life is determined as
the quotient of durability h, expressed in terms of the number of
operations (work units), by the average number of such operations
per year.

For instance, the service life of an automobile rated for a total
run of L kilometres is

where i\d u ty — duty factor;

l — average distance (in km) travelled by the automo¬
bile per year

30

Chapter 1.-Principles of Machine Design

When dealing with machines whose durability is expressed in
units of time the total service life is the quotient of the durability h
by the use factor T] use , the latter showing the average degree of actual
machine utilization throughout the entire period of its service.
Considering also the duty factor

H

h

In a general case the use factor is

7]«se — hspos '7) ofj-d ' "^rep’^shlft ’haec. br

(1.17)

(1.18)

The seasonal work factor q seQS is the ratio between the duration
h seas of the working season of the machine (in days) and the total
number of days in the year

„ hseas

TWas — 3 g 5

Included in the category of machines whose work is dependent,
upon climatic and seasonal conditions are most of agricultural
machines (combine harvesters, cultivators, croppers, etc.), as well
as the machines used for processing perishable agricultural products
(equipment for canneries), road-building machines, snow-clearing
vehicles and river transport vessels which have a limited period
of navigation.

For special-purpose agricultural machines, e.g., potato diggers,,
cotton pickers, etc., ii seas = 0.05-0.02.

For machines used the whole year round rj sects — 1.

The off-day factor r\ of f-d is the ratio of the number of working
days to the total number of days in the year (365). Off-days include
Sundays (52 days per year) and national holidays (on the average
7 days per year). Besides, the off-day factor takes account of the
shortening of the working days preceding the national holidays,
which on the average, amounts to 4% of the total annual working
time.

Thus, the off-day factor averages to

365—52—7 n na A Q
^1 off~d — li 1 ggg • 0*96 0.8

This factor is valid for machinery working to a calendar schedule.
For. machines run continuously throughout the year (e.g., equipment
of power stations) Tjo/f-d — 1-

The shift factor is the ratio of the duration h shi jt of the working
shifts of the machine in hours to the 24-hour day

hshijt
24

lf Uhijt

1.8. Durability

3t

For single-shift operation r\ shi ft & 0,3; for double-shift operation
r\ sh ift » 0.6; for triple-shift operation « 0.9; and for 24-hour
operation r\ shin — 1.

The repair downtime factor y\ rep is the ratio between the duration
h act of the actual operation of the machine and the sum. of the actual
operation duration h act and repair downtime duration h rep

^ _ h ac t

re ^ h-act "h hrep

The magnitude of this factor depends primarily on the machine’s-
reliability, which defines the length of time between repairs and the
scope of the latter, and on the level of organization of repair work.

Factor r[ Tep depends also on the time of the machine’s being in use-
Being quite negligible initially, repair downtimes lengthen progres¬
sively as the machine wears down and may reach by the -end of the
service life quite a great value.

For processing machines operating -to a calendar schedule the-
average value of T| rep equals 0.85-0.95. For other machine categories
this factor varies very widely.

For machines of seasonal and 'sharply aperiodical operation q rep
is equal to unity because such machines are almost always repaired
during their off periods.

The machining time factor is determined as the ratio of

the machining time h maC h (actual time of machining) to the sum
of machining time h maC h and auxiliary time h aux (time spent in
mounting and removing the work, setting up, adjusting and- servicing
the machine)

... hmach

'i mach — ~Z , r~L ~
n rnach"ir n a ux

This factor is valid for manually-controlled machine tools (e.g,,.
metal-cutting machines, metal-forming equipment). The vaule of.
if mach depends on the type of equipment, degree of perfection of the
working process, work batch size. For a metal-cutting machine
tool this factor on the average equals 0.8-0.9. As the degree of auto¬
mation goes up r\ mach approaches unity. For automatic transfer
machines r ] mac h equals unity.

The load factor rj Iocc d is the ratio of the duration h aci of the actual
operation to the sum of the duration of the actual operation and
the duration hat of the downtimes for the same period, the downti¬
mes being due to the impossibility of loading the machine fully

’ 1 ‘” i ”TC+fcr

For machines running without any hourly schedule, as well as
machines of periodic action whose loading is not regulated (auxiliary,,
emergency, repair), q ;oa(J is low.

32

Chapter 1. Principles of Machine Design

Special-purpose machines may be underloaded in a continuous
production process when they perform a limited number of opera¬
tions on a narrow component range. The same is true of machines
whose output capacities exceed those of the neighbouring machinery.

As a rule, a low value of factor p Ioat j in processing machines is due
to' planning defects and incorrect choice of equipment as far as its
•quantity, type and capacity are concerned.

In production processes with changeable, objects of production
the magnitude of this factor depends upon the component being-
machined and upon the type of machining, and hence may vary
with time. For example, if turning operations are predominant,
‘then the lathes will be the most occupied machines, whereas machi¬
nes of other types (milling, boring, etc.) will either be underloaded
•or idle.

Under the conditions of small-batch production the average value
of rjzoad equals 0.7-0.75, in batch production—0.8-0.85, aud in
multiple production—0.9-0.95. In stable continuous large-lot pro¬
duction T]i oad = 1.

The accidental breakdown factor "n acc .j, r is the ratio of the duration
'■h act of the actual operation to the sum of the duration h ar j and idle
time h acc , br caused by breakdowns eliminated on the spot

1]acc. br —

_ hqqt _

itqct "b hacc. br

For well designed and carefully operated machines this factor
is close to unity. For machines having design defects, or for those
■operated by unskilled personnel, r) flCC . 6r . may be much lower than
■unity.

Let us prove Equation (1.18).

In a generalized form the use factor is

dttse —

hact

F

where h aci — total number of machine’s running hours per year;
F — 365-24 = 8760 = annual time fund, hours

Hence,

rjitsp —

. 8760-2

8760

8760

(1.19)

where I s the total number of downtime hours per year.

Downtime for non-working- seasons

hseaa ^ 8760 (1 —p seas)

Downtime for off-days

h<>f}-d — 8760r] seas (1 r\off-d)

1.3. Durability

33

Downtime for repairs

hreP " 8760l'] se (i S T|o^.. I 2 (1 “ Tpep)

Downtime for the non-working part of the day (24 hours)
kshift — 8760p seaa p o _fy. rf p re p (1
Downtime for the non-machining period

fo-mach — 8760 q se( j 5 T) o /^.. c p r ) re p 7 | s / l j/j (1 broach)
Downtime because of incomplete machine loading
hload. = 8760'p S g as p o ^_d'>'}rep , 'bhi/i , Hmc!cft (1 hJoad)
Downtime because of operating troubles

hacc.br — 8760Tj se a s bo//-d'*Vep'nsh2/thmac?ih2oad (1 • Pace, hr)
Summing up all the above downtimes gives

2 hdt ~ 8760 {1 — bseashoif-abrepbshs/fhmacft'lkoadhacc.fcr)
Substituting this expression in Equation (1.19) we obtain
2 h *t

'fuse — 1'

8760

bseasbo//-d''l7'ep , *1shl/;(hroacfth2oc!dhacc.6r

(c) Design Durability. Design Service Life

For general-purpose machines running to a yearly time-schedule
with preset repair downtimes the degree of utilization, and hence
the difference between service life H and durability h, will depend
mostly on the shift factor r\ S hut-
Figure 3 shows the relation between H and h for different condi¬
tions of work. The diagram is plotted under the following assump¬
tions: T\ duty = 1 (Eq. (1.16)]; T] seas — 1; x) 0 ff~d — 0-81 (except for
all-the-year-round continuous operation when (\ 0 ff-d — 1);

^mach?\^oad'^acc. ftrhrep ~ 0.8.

On these assumption the service life is

tt _ h __ __jjj_

0-81-0.8T| s Ai/t 0.648q s ftj/f

where r\ shift = 0.3; 0.6; 0.9 (for single-, double- and triple-shift
operation, respectively).

For all-the-year-round continuous operation

h

H = -

0.95

where factor is

introduced to account for accidental
repair downtimes. .

and

3-01S95

34

Chapter 1. Principles of Machine Design

Considering the graph, one may conclude that the durability
reserve of a machine, sought for at the design stage, must accord
with the use factor r) uss and, in the first place, with the shift factor
r) shift- Raising the durability of machines which would be used
not so very intensively will result in an increase in their service

hours years

sooooo

mooo

300000
250000
200000
ISO 000

120 000
100000
90000
80000
70000
60000
soooo
40000
35000
30000
25000
20000

15 000
12 000
10 000
9 000
8000
7000
6000
5000
4000
3000

2000

1500

Fig. 3. Durability h versus service life H

l — single-shift operation; 2 — double-shift operation; 3 — triple-shift operation; 4 — all-

the-year-round operation

life which cannot be practically used to the full because of obsoles¬
cence. For instance, when durability h = 10 years, the service life
under the conditions of a double-shift work will be 28 years, and
under those of a single-shift operation, 50 years, which surpasses
all conceivable obsolescence limits.

It is advisable to use high design durability values for machines
which would be intensively operated. Thus, machines with a rated
10 -year durability, operating to a triple-shift schedule have their
service life equal to 17 years, and in the event of a continuous yearly

1.3. Durability

35

operation, to 11 years, a value which for most machine categories
falls within obsolescence limits.

Table 2 gives rounded-off values of design durability (determined
on the basis of the above diagram) for machines of different opera¬
tive intensities, with the machines’ service lives being preset.
This data can be used for determining approximately the design
durability of machines of different classes.

Table 2

Design Durability as a Function of Service Life and Operative Intensity

Service life*
years

Design durability, thous. hours

single-shift

operation

double-shift

operation

triple-shift

operation

continuous
(yearly) operation

i

1.8

3.5

5.2

8

2

3.5

7

10

16

3

5.2

10

16

24

5

9

18

27

40

10

18

35

55

80

15

27

55

80

120

20

35 •

70

105

160

25

45

90

135

200

In a most common case of a double-shift operation of machines
with a service life of 10-15 years the durability rating of the machines
ranges from 40 000 to 60 000 hours. These figures may be taken as
the basis for determining the durability of most of processing machi¬
nes. For intensively operated machines (i.e., for those operating
to a triple-shift schedule or continuously all the year .round) the
durability values should be taken at 60 000 to 100 000 hours, the
machines’ service lives being the same.

( d) Theory of Durability

The theory of durability is now being developed. Its main subjects
are as follows:

determination of technologically and economically reasonable
limits of durability,

development of methods for studying the operation of machines
(statistical processing of actual operational data);

study of operating conditions and their influence upon the machi¬
nes’ durability; standardization of ranges of operating conditions;

3 *

36

Chapter 1. Principles of Machine Design

determination of the degree of utilization of machines, and corre¬
lation between the machines’ durability and service life;
diagnosis of the causes of failures;

revelation of parts limiting the durability; study of the effect
of the components’ durability upon the durability of the machines
as a whole;

development of methods of bench and field tests of machines,
units and elements for durability; prediction'of the machine’s dura¬
bility on the basis of the bench test data;

development of reliable durability indices for machines being
produced.

When defining the durability, the multiplicity and heterogeneity
of factors influencing it (technological level of operation, fluctuations

Fig. 4. Graph of probable durability

1 — probable service life (survival percentage); 2 — probability of destruction; s — densitv

of service life probability

of operating conditions, quality of manufacture, etc.) and indeter¬
minacy of many other factors (scattering of the strength characte¬
ristics of materials, effects of different regional and climatic condi¬
tions, etc.) necessitate the use of the probability theory and mathe¬
matical statistics. Because of this, the theory cannot provide an
unambiguous solution for the problem of anticipated durability,
restricting itself to the determination of a functional relation between
the probability of failure and the duration and conditions of opera¬
tion (Fig. 4). The theory can only define that probable duration of
the machine’s operation under given conditions will be, say, 7200,
10 500 and 15 000 hours at a failure probability of 90, 80 and 60%,
respectively, or define probable number of machines remaining in
operation (survival percentage) after certain periods of work.

Further, it is necessary to take account of the type and degree of
failures, i.e., to determine with a certain confidence whether vital

1.8. Durability

37

or less important parts or units have failed, whether the machine
remains repairable, what are probable scope and cost of repairs.

From these positions the durability of a machine may be defined
as probable duration of the machine’s operation under specified
conditions, with which the probability of the machine’s failure
does not exceed a certain preset limit (say, 10%), the machine
remains repairable, and probable cost of repairs does not exceed
a definite value expressed, say, as a percentage of the machine cost.

The formulation of durability standards is a difficult problem
and requires the collection and processing of vast information.

The study of machine durability would become much easier if
every new machine were equipped with a “workmeter”, i.e., with
an hour-counter or operation-counter (similar to the odometers on
automobiles). All new machines should be fitted with such devices.

Conclusions drawn from the studies of actually operating machi¬
nes refer to machines manufactured in the past years and are thus
inevitably out-of-date. As a result, such conclusions in essence are
inapplicable to new machines incorporating latest design and tech¬
nological improvements. Therefore predictions of the durability of new
machines, which are of vital practical importance, have to be based
upon bench tests of the machines (or new units incorporated into them).

Thus, one of the chief aims of the theory of durability is the
development of accelerated test methods and correlation of such
tests with actual operation.

The theory of durability, based on statistical data, is essentially
applicable to articles of mass production and in a much lesser degree,
to small-lot, and still less, individual products. Generally spea¬
king, it should be noted that the theory of durability, as formulated
earlier, is based on phenomenological grounds and operates with
figures of the already achieved durability. Of much greater impor¬
tance is the development of methods for enhancing the durability.
Here the study of the physical regularities of failure, wear and
damage of machine components (depending on the type of loading,
properties of materials, condition of surfaces, etc.) comes to the
fore. The problems listed above are so differentiated and specific
that it is hardly possible to confine them within the framework of
a general theory of durability. Such problems are solved by the
methods of the theories of strength and wear, and particularly through
concentrating the activities of designers and production engineers
upon the enhancement of the durability of machines.

ii(c) Means to Enhance Durability

The main factors limiting tne'durability and'Teliability of machi¬
nes are as follows: breakage of parts; wear of friction surfaces;
surface damages caused by contact stresses, work hardening and

38

Chapter 1. Principles of Machine Design

corrosion; plastic deformations occurring because of local or general
stresses exceeding the yield limit, or creep (at high temperatures).

Strength in most cases is not an insurmountable limit. In general-
purpose machines breakage can be fully excluded. With the available
range of modern machine-building materials, the existing methods
of manufacture, and the present state of the theory of strength, the
machines of this class have no parts that cannot be made of virtually
limitless durability.

The problem is more difficult in the case of highly-stressed machi¬
nes, such as, for example, transport vehicles. Size and weight requi¬
rements make it necessary to increase the design stresses, which
results in a greater probability of failure. Nevertheless, the conti¬
nuous improvement of strengthening technology and refinement of
computation methods make it possible, even in this case, to elemi-
nate or significantly widen the strength limits of durability.

Many chance factors can be minimized: manufacturing factors
(fluctuations in the mechanical properties of materials, technolo¬
gical defects)—through careful product quality control; operation
factors (overloads, wrong handling of machines)—by purely design
methods (by introducing protective systems, safety devices, inter¬
locks, etc.).

Heat engines are in the worst position. Their durability depends
primarily on the endurance of parts operating under high tempera¬
tures (pistons, piston rings, and valves in internal combustion
engines; rotor and stator blades in steam and gas turbines; combus¬
tion chambers in gas turbines).

The strength of materials sharply drops with an increase in tem¬
perature. Furthermore, high temperatures give rise to the creep
phenomenon (i.e., plastic flow of material under the effect of com¬
paratively low stresses) which results in a change in the dimensions
of parts, and hence, in the loss of their efficiency.

Machine parts operating under high temperatures have limited
durability ratings. The service life of such parts can only be increa¬
sed by design methods (reducing stresses, proper cooling) and,
mainly, by using heat-resistant materials (high-alloyed chrome-
molvbdenum, chrome-vanadium-molybdenum, chrome-tungsten-mo-
lybdenym steels, titanium alloys, nickel-base alloys). For manufac¬
turing thermally stressed parts nowadays wide use is made of sin¬
tered metal-ceramic materials (cermets). Cermet materials are made
with oxides, nitrides and borides of Ti, Cr and Al, and carbides
and nitrides of B and Si as the base, the bonding material being
metallic nickel, cobalt and molybdenum.

Practically, the durability of machines is mostly dependent on
the wear of their components. Gradually developing wear worsens
the machine characteristics, lowers the accuracy of the operations
done, and results in a poorer efficiency of the machine, greater

1.3. Durability

39

energy consumption and lower profits. Wear may eventually become
ruinous. The progressive destruction of surfaces causes failures
(breakage of antifriction bearings, pitting of gear teeth, etc.).

The main kind of wear in machines is mechanical wear which
is subdivided into abrasive, sliding friction, rolling friction, and
contact wear. Some parts are subject to chemical wear (corrosion),
thermal, and cavitation-erosion wear. The multiplicity of kinds

30~to~5Q 60 70Rc

Fig. 5. Wear resistance versus surface hardness (after Goodwin)

of wear and the differences in their physical-mechanical nature
require a differentiated studies into wear and special methods for
its prevention.

The main methods for improving the resistance to mechanical
wear are as follows: increasing the hardness of rubbing surfaces,
selecting properly the material for friction pairs, decreasing the
unit pressure on friction surfaces, improving surface finish, and
ensuring correct lubrication.

Figure 5 shows the effect of surface hardness on the resistance to
wear, the graph being plotted on the basis of experiments on the
wear of surfaces under the action of an abrasive material (corundum).

The wear resistance of a surface with a Vickers pyramid hardness
(VPH) of 500 (« 48 Rc) is taken as the measurement unit. As evident
from the diagram, each 500-VPH increment of the surface hardness
gives a '10-fold increase in the resistance to wear.

The experimental conditions (abrasion wear) differ from the
actual conditions of operation of lubricated surfaces in machines.

40

Chapter 1, Principles of Machine Design

Nevertheless, these experiments give an idea of the huge influence
that the surface hardness has on the resistance to wear.

Modern technology has at its disposal effective methods for raising
the surface hardness: case-hardening and induction-hardening
(500-600 VPH), nitriding (800-1200 VPH), beryllizing
(1000-1200 VPH),diffusion chromizing (1200-1400 VPH), plasma car¬
bide facing (1400-1600 VPH), boronizing (1500-1800 VPH),
boron-cyaniding (1800-2000 VPH).

Another trend is to improve the antifriction properties of surfaces
by depositing phosphate films (phosphating), by saturating the
surface layer with sulphur (sulphidizing), graphite (graphitization),
molybdenum disulphide, etc. Though having a moderate hardness,
such surfaces exhibit a greater slipperiness, low friction coefficient,
and high resistance to tearing, jamming and seizure.

The above methods (especially sulphidizing and saturation with
molybdenum disulphide) increase the wear resistance of steel parts
by 10-20 times. Combined methods also find application. This can
be exemplified by the method of sulphocyaniding which simulta¬
neously enhances the hardness and slipperiness of surfaces.

Of vital importance is the correct combination of the hardnesses
of friction surfaces. Maximum hardness of both surfaces in contact
is advisable when they operate at low relative speeds and with
high loads imposed thereon, whereas operation at high relative
speeds and in the presence of a lubricant requires a combination of
a hard and a soft surfaces, the latter preferably possessing better
antifriction properties.

An efficient way to improve the wear resistance of friction joints
is to decrease the unit pressure therein. Occasionally this can be
achieved by reducing the loads (rational distribution of forces) or
by minimizing the cyclic and impact character of loads. But most
simply this can be effected by increasing the area of the friction
surfaces, often accomplished without any appreciable increase in
the overall dimensions of machines.

To illustrate, let us examine a machine tool guideway acted upon
by a unilateral load (Fig. 6a). An appropriate change of the guideway
profile (Fig. 66) makes it possible to double the bearing surface
area, halve the unit pressure and, hence, improve the durability,
with the overall guideway dimensions remaining the same. Ridged
guideways offer still greater durability (Fig. 6c). In this case the
unit pressure is reduced four times, the original overall dimensions
being only doubled.

If the design permits, it is strongly recommended that point
contact be changed to line contact, line contact, to surface contact,
and sliding friction, to rolling friction. Point contact gears are to
be avoided; these include transmissions with skew axes, spiral
bevel gears, helical wheels with large spiral angles, and screw gears.

1.3. Durability

41

The latter have a further disadvantage as their contact spot moves
rapidly along the tooth under sliding friction conditions, while
in conventional involute gears rolling friction occurs at a rather
low speed.

A special method is wear compensation accomplished either auto¬
matically or periodically. Units with a periodic wear compensation
include sliding (plain) bearings in which radial or axial clearances,
can be adjusted (bearings with tapered trunnions or seating surfaces,
or with inserts or bushes that can be periodically tightened up).

Fig. 6. Reduction of unit pressure on friction surfaces (case of machine-tool

guideways)

Another example of periodic wear compensation is the axial tighte¬
ning of antifriction bearings (radial thrust or tapered) or adjustment
of clearances in rectilinear guides by means of tightening wedges
and wear strips.

More perfect are systems with an automatic wear compensation
(self-grinding-in conical plugs of taps, face- and lip-type seals,
spring-preloaded antifriction bearing units, hydraulic clearance
compensation systems in lever-type mechanisms, etc.).

Proper lubrication of friction units is of decisive importance.
Wherever possible, fluid friction should be ensured, thus elimina¬
ting semifluid and semidry friction.

Open mechanisms lubricated by way of a periodic packing should
be avoided. Open gear transmissions must never be used, and chain
drives should better be avoided.

All rubbing parts should be enclosed in housings and reliably
protected against the ingress of dust, dirt and atmospheric moisture.

The best solution is to use hermetically sealed systems provided
with a continuous forced oil feed to all the lubrication points.

Excessive lubrication is not recommended for units operating
in conditions of high periodic contact loads and speeds (antifric¬
tion bearings, gear teeth, etc.). It is advisable to lubricate such units
with a metered stream of lubricant, while at higher rotational speeds
atomized (oil-mist) lubrication is preferable.

The viscosity and thermal characteristics of oils musx be compa¬
tible with the operating conditions of the given unit or machine.

42

Chapter 1. Principles of Machine Design

The efficiency of lubricants can be enhanced considerably by ad¬
ding to them special substances which improve their lubricity (col¬
loidal graphite and sulphur, molybdenum disulphide) and oiliness
(oleic acid, palmitic acid, and some other organic acids), preclude
oxidation (organic and metal-organic compounds containing sul¬
phur, phosphorus, and nitrogen) and prevent seizing (organosilicon
compounds).

When the application of liquid oils is either impossible (operation
under high temperatures, in chemically aggressive media, in high
vacuum) or inefficient (under the conditions of high-frequency con¬
tact loads) use is made of dry-film lubricants: graphite, molybde¬
num disulphide (MoS a ), lead monoxide (PbO), cadmium oxide (CdO),
lead iodide (Pbl 2 ), cadmium iodide (Cd-Ig), lead sulphide (PbS),
etc. Dry-film lubricants are usually employed in the form of films
applied to metal surfaces. To improve their lubricity and endurance
properties, such films are reinforced with binders (powdered metallic
nickel, silver, and gold).

An ideal alternative, from the standpoint of wear resistance, is
complete elimination of any metal-to-metal contact between the
working surfaces. Examples of wear-free units are electromagnetic
clutches and brakes in which torque is produced on account of
electromagnetic forces developing in the gap between their working
surfaces.

Approximating, in principle, to a wear-free operation are sliding
bearings with a hydrodynamic lubrication. This is achieved than k s
to a continuous delivery of oil and provision of a wedge-shaped oil
gap making for the delivery of the oil into the loaded zone; under
steady operating conditions the metallic' surfaces in such bearings
are fully separated, thus making, theoretically, the bearing unit
wear-free. The vulnerable point in this case is the disturbance of
fluid friction in non-stationary conditions, particularly during
starts and stops, when the delivery of oil is discontinued due to the
reduced rotational speeds, causing a metal-to-metal contact to
develop between the trunnion and the bearing and thereby intensi¬
fying their wear.

Recently use has been made of hydrostatic bearings in which
oil is fed at high pressures into the gap from an independent source.
In these bearings the rubbing surfaces are separated by an oil film
even before the machine is actually started; the variation of the
machine speed in no way affects the lubrication of the bearing or,
in other words, Its serviceability.

An example of a hydrostatic support (step bearing) is shown sche¬
matically in Fig. 7. Oil from a pump flows through choke 1 into
pocket 2 with confining edge 8. The pressure in the pocket depends
on the ratio of the cross-sectional area of the choke to the changing
areas of the cross section between the confining edge and the verti-

1.8. Durability

43

cal journal of the bearing. As the load on the beairng increases this
cross section becomes smaller and the pressure in the pocket rises,
reaching at its maximum the pressure built up by the pump. In the
case of impact loads the pressure in

the pocket may by far exceed the
pump pressure because of the blocking
up of the choke as a result of a sharp
increase in its hydraulic resistance.

In full-journal bearings loaded by
alternating forces use is made of
a system of radially arranged pockets
(Fig. 8). In Fig. 8 the lower pocket is
the supporting one. There is no pres¬
sure in the upper'pocket because of an
enlarged clearance along the upper
arc of the bearing. The pressures in
the lateral pockets are mutually
equalized and so these pockets do not

take up any load. The oil flowing out of yjg, 7 Hydrostatic step bearing
the upper and lateral pockets func¬

tions as a conventional coolant.

Apart from the hydrostatic taking up of the load, a certain hydro-
dynamic effect also occurs. The oil flowing through the upper and
lateral pockets is entrained by the rotating shaft and forced into

Fig. 8. Four-chamber cylindrical hydrostatic bearing

the converging wedge-shaped gap along the lower arc a of the bearing,
building up an increased pressure on the surface of the confining
edges and also in the supporting pocket (because of the hydraulic
blocking up of the choke).

With the load direction altered through 180°, the upper pocket
becomes the supporting one and the lower pocket, the feeding one.
Similar effects occur when changing the load direction through 90°.

44

Chapter 1. Principles of Machine Design

Thus, responding to the shaft movements, the bearing automati¬
cally adjusts itself to the taking up of the load in the direction
of the acting force vector.

Recently for taking up alternating loads use has been made of
honeycombed and porous hydrostatic bearings. These bearings
operate similarly to the one above; in this case the honeycomb or
pores act as pockets.

In certain cases (e.g., when dealing with high-speed spindles or
guideways of metal-cutting machine tools) it is advantageous to
employ aerostatic or gas-static lubrication, in which case the rubbing
surfaces move on an air (gas) cushion built up by the air (gas) forced
into the clearance between the surfaces.

In view of the development of hydrostatic bearings there occurs
a revaluation of the comparative merits of sliding and rolling'bearings,
a certain preference being up to now given to the bearings of the
latter type. The sliding bearings, when provided with proper lubri¬
cation, are in principle more advantageous since they totally exclude
metal-to-metal contact and assure wear-free operation, whereas in
rolling bearings such contact and wear are inevitable.

The employment of hydrostatic bearings, however, is limited
because of their more elaborate lubrication systems and, in parti¬
cular, because of the necessity for separately driven oil pumps during
start and stop periods.

One of the most frequent causes of premature machine failures
is corrosion. To preclude this, highly effective means of protection
must be provided, especially for machines working in the open,
under conditions of high humidity, or in chemically active media.
Such means include electroplating (chrome, nickel or copper pla¬
ting), deposition of chemical films (phosphating, oxidation), depo¬
sition of polymer films (eapronization, polythenization).

Yet, the most advantageous solution is the use of corrosion-resis¬
tant structural materials (stainless steels, titanium alloys). Slightly
loaded machine components operating in contact with chemically
active agents should be made of chemically stable.plastics (polyole¬
fins, polyfluoroethylene resins).

The application of the above described technological and con¬
structional means may allow the service life of the majority of
components in general-purpose machines to be increased to practi¬
cally any value required to ensure the necessary durability of the
machines as a whole.

When working on a new project, designers often do not plan the
durability of components but only select their shapes, sizes and
machining methods, following the traditions and standards existing
in a certain branch of engineering, which in new conditions of con¬
tinuously intensified operation of machines and in the light of
new ideas about the import of durability must be revised. In most

1.3. Durability

45

cases it is sufficient to clearly outline the task and apply general
rational design processes so as to solve already at this stage many
a durability problem that otherwise would have to be solved by
way of refining the design of the already built machine, which is
waste of time and effort.

Like in the aircraft industry where each component at its design
stage is subjected to a careful check for weight, in the general machi¬
ne building it would undoubtfully be a good practice to subject
all parts to a systematic durability control.

Naturally, there are exceptions from the general rule. It is very
difficult to impart durability to parts running in direct contact
with abrasive media (impellers of pumps handling slurries, soil-
cultivating implements, cutters in mining machines, caterpillar
tracks, jaws of stone-crushers, chains and drives of countinuous
conveyers for handling cement, coal, etc.).

The service life of such components in some instances (e.g., rock
hits) is only tens of hours and can be prolonged only by selecting
the most wear-resistant materials and applying the most effective
strengthening processes.

Of course, measures taken for improving the durability make the
design more expensive. They will mean the use of high-grade mate¬
rials, introduction of new technological processes and, occasionally,
establishment of new sections in a factory. The additional capital
investments not uncommonly daunt managers who consider costs
in the enterprise and not the effect of the machine’s durability and
reliability on the national economy. Such expenditures are justified.
The cost of producing components determining the durability of
a machine is negligible in comparison -with the machine cost and
the latter, generally, is rather small when compared to the total
sum of operational expenses. Inappreciable in the total cost balance,
the above-said additional expenditures on improving the durability
will eventually bring huge financial gains because the downtimes
and cost of repairs are reduced.

Hence an important practical conclusion’, trying, as a general
rule, to make the machine cheaper, one should not economize on
the manufacture of components determining the durability and
dependability of the machine. One should never spare expenses on
researches associated with the development of new, better materials
and processing methods that would enable a greater durability to
be obtained.

Many manuals on mechanical engineering recommend the use of
the cheapest materials and simplest manufacturing methods fit for
the function of a given part. This recommendation cannot be accep¬
ted without reserve. The selection of materials and production
techniques must always he based on the assessment of the relative

46

Chapter 1. Principles of Machine Design

weight of the additional manufacturing costs in the totality of ope¬
rational expenditures.

All components on which the durability and reliability of machi¬
nes depend should be made of the highest-grade materials and by
the most perfect processing methods.

Let us consider, as an example, the manufacture of piston rings
used in internal combustion engines and in other piston-type machi¬
nes. The quality of these rings largely predetermines the engine
maintenance cycle. As the rings wear down the engine efficiency
falls and the fuel and oil consumption increases. At present the ser¬
vice life of such rings amounts to 500-1000 hours. The use of the
latest technological procedures which increase the wear resistance
of the ring-cylinder pair (porous chrome plating and sulphocyani-
ding of the rings and nitriding of the cylinder faces, etc.) makes it
possible to prolong the service life up to 5000-10 000 hours. The
insignificant rise of the engine cost, due to the more costly rings,
is repaid many times because of the increased wear resistance of the
engines which are now so common (in automobiles, tractors, diesel
locomotives, ships). As a result, a great economic effect is provided
for the national economy.

Another example are antifriction bearings. Most authors recom¬
mend the use of less accurate bearings, arguing that the higher the
bearing accuracy, the greater their cost.

Let us assume that the manufacturing cost of the class H bearings
(ordinary accuracy) is one unit. Then, the manufacturing costs of
bearings can be expressed by the following proportionate figures
(in the order of rising accuracy): improved (class II)—1.3; extra-
improved (class BII)—1.7); high (class B)—2; extra-high (class AB)—
3; precision (class A)—4; extra-precision (class CA)—7; and super¬
precision (class C)—10.

At first sight these figures rather convincingly speak in favour
of the use of the less accurate bearings.

However, such a conclusion is short-sighted. As the wear and
damage of antifriction bearings is one of the commonest causes of
machine failures, which to a significant degree predetermines the
in-between repair periods, one must acknowledge that it is more
sensible and economically profitable to employ bearings of higher
accuracy in spite of their higher cost. Of course, this does not mean
that precision bearings must always be used, and it does not libe¬
rate the designer from his responsibility for ensuring the durability
of the bearings by their correct installation and lubrication.

Limits to increasing the durability. The effectiveness of increa¬
sing the durability, as a means of enlarging the actual size of the
machine fleet, lowers with the increase of the absolute durability
values. With a successive rise of the durability, each year added
to the absolute durability value gives an ever-decreasing gain in

1.8. Durability

47

the size of the operating machine fleet as compared to the previous
year.

The graph in Fig. 9 shows the change in the relative size of the
machine fleet with the rise of the durability of the model being
manufactured. The durability of the original model is assumed
to be equal to one year. Increasing the durability by one year doubles

Fig. 9. Relative growth of machine fleet with increase of machine durability

(size of machine fleet with initial durability h— 1 year is taken as unity)

the size of the machine fleet. With the durability increased by as
much again, the gain relative to the previous model comes to 1.5 times
(though in reference to the original model it is 3 times). Increasing
the durability by another year brings the size of the machine fleet
up to 1.33 times that for the previous model (though in reference
to the original model the gain is 4 times). With each yearly increase
in the durability the size of the machine fleet grows less and less.
Hence, it is important to set the most reasonable limit to the dura¬
bility increase, which will give a significant gain in the size of the
machine fleet without causing an excessive rise of the model cost.
In the case illustrated by Fig. 9 the growth of the machine fleet
practically stops when the durability increase reaches 5-6 times.

The magnitude of technically obtainable durability largely
depends on the loading rate of the machine.

For transport machines the durability totals 10 000 to 20 000 hours,
and the service life, 5 to 8 years. For stationary machines, e.g.,
machine tools, the durability comes to 50 000-100 000 hours, which
corresponds to 15-25 years of service to a double-shift schedule and
10-20 years with a triple-shift operation. At such service life periods
the problem of obsolescence becomes very urgent.

48

Chapter 1. Principles of Machine Design

The mechanical life of machines can be artificially prolonged
by way of restoration.

However, from the standpoint of economy this way is inadvisable,
as the cost of repairs often exceeds by many times that of the original
machine.

At the initial stage of operation the repair expenditures are usually
small but later on they sharply grow as the need for routine and
medium repairs arises. Eventually, when the machine is subjected
to a capital repair the total repair costs reach a considerable value
comparable to the machine cost. The question of whether a machine
is worth further use should be settled just before submitting the
machine to an overhaul. Putting aside for the present the problem
of obsolescence, it is economically advisable to consider the limit
to the machine use to be the moment when the impending capital
repair expenditures approximate the cost of a new machine. Then
it is unquestionably more advantageous to buy a new machine than
to repair the old one. The more so as new machines are of higher
quality than repaired ones and have better characteristics thanks to
continual technical improvements, Moreover, there is a regular
lowering of the cost of new machines due to the undeviating intensi¬
fication and improvement of industrial processes.

The total sum spent on all repairs should also be considered when
deciding the question of the termination of a machine use: as a gene¬
ral rule it may be said that the total sum spent on repairs during
the machine life should not exceed the machine cost.

Attempts have been made to find optimum durability, i.e., the
durability with which the cost of the products turned out by the
machine is minimal. One proceeds from the following prerequisites:
the cost of production is equal to the sum of fixed expenditures
independent of the duration of the machine use (expenditures on
power, materials, labour, etc.) and variable expenditures dependent
on the duration of the machine use (depreciation expenditures which
are inversely proportional to the machine use duration, and repair
expenditures which grow larger with time because of the wear of
the machine).

The variation of production costs Cp as a function of the duration
of the machine service is expressed by the following formula

Cp = Ef + -^ + R P r=y(H) (1.20)

where Ef = fixed expenditures;

C — machine cost;

H — duration of service;

Rpr — cost of repairs

Summing up the above constituents gives the costs of production
as a function of D (thick line in Fig. 10a). The cost curve has its

1.3. Durability

49

minimum, and it is suggested that the durability corresponding to
this minimum be considered optimum.

This interpretation is too simple to be used in practice. Firstly,
in most cases the variable expenditures are negligible as compared
to the fixed ones, so that even when the cost curve has a minimum,

(a) (b)

Fig. 10. Machine cost (heavy lines) as a function of machine’s service life H

(a) with Rpr = 4C; (6) with Rpr ~ C; 1 — annual cost of repairs; 2 — depreciation expen¬
ditures

the latter is too weakly expressed. The machines which have exce¬
eded the optimum durability term are still capable of turning out
products for a long time, though at a somewhat less profitable rate.

Secondly, the total cost of repairs ^Rpr throughout the entire
period of the machine service is not accounted for. Thus, for the
case depicted in Fig. 10a, where the repair expenditures for the last
year of service are assumed to be equal to the cost of the machine
(lines cd), the total repair expenditures (the area between the repair
cost curve and fixed expenditures line) equals four times the machine
cost, which is certainly an overestimated figure.

The picture becomes different after setting sensible limits on the
total cost of repairs. Thus, if one takes these costs not to exceed the
machine cost, then there will be a definite minimum product cost
(thick curves in Fig. 10b) for any given length of the machine service.
These minimums decline and become less clear as the service life
increases. As the service life grows longer, the envelope of these
minimnms continuously falls down. Thus, with the total repair
costs confined within certain limits, the concept of optimum durabi¬
lity vanishes; the costs of production continuously decrease.

Finally, it should be mentioned that the discussion on the concept
of optimum durability neglects the dynamics of changes in the fixed
expenditures which, generally, tend to reduction (decrease of the

4-01395

50

Chapter 1. Principles of Machine Design

costs of power,'materials and labour thanks to automation and impro¬
vement of processes). This reduction may still more change the
picture in favour of longer service lives^

(/) Durability and Obsolescence

The problems of increased durability and obsolescence are inti¬
mately associated. A machine becomes obsolete when its performance
characteristics are no longer fit to satisfy the production needs becau¬
se of higher requirements, or of more perfect machines having made
their appearance, although the machine itself still retains its mecha¬
nical capabilities.

The signs of obsolescence are as follows: degradation (compared to
the mean value) of reliability, product quality, and productivity;
increase of power consumption per unit product, costs of labour,
maintenance and repairs, and, as a general result, reduction of the
machine’s profitability.

Obsolescence is not necessarily connected with physical wear
(although, physical ageing impairs the machine performance, thereby
expediting obsolescence). A machine may become obsolete while
being quite sound mechanically, and even new.

The main effect of obsolescence is reduced productivity per unit
of labour force, which is the main indicator of economic progress.

Undoubtedly obsolescence will ensue in two cases: when chan¬
ging over to a new product (a complete change of the technological
process), and with the development of new processes or new designs
allowing machines better than those of the old type to be made.

The latter kind of obsolescence is exemplified by the development
of turbojet engines which in aircraft ousted almost fully the older
internal combustion engines (piston engines).

It must be noted, however, that such radical and prompt changes
are rather uncommon. Under the conditions of gradually improving
technology the problem of obsolescence is otherwise.

To begin with, in most cases, particularly in heavy-duty machines,
physical wear by far outpaces obsolescence. For example, the physi¬
cal reserves of automobile trucks operated under heavy conditions
are exhausted in 5-6 years, although their technical and economic
ratings provide for a much longer service life.

Secondly, efficient methods of preventing machine obsolescence
exist.

The most important of these methods is a prospective machine
design, i.e.; the design considering the dynamics of changes in that
particular branch of industry for which the machine is intended.
The design of the original model must be capable of further develop¬
ment in respect to its production capacities, power, output and ver¬
satility. Such a provision will in due time allow the necessary modi-

1,3. Durability

51

fications to be made, thus keeping the machine ratings on'a par^witb
the growing demands without retirement of the]basic models and,
consequently, avoiding the construction of new production lines
which is inevitable when switching over to a new model.

With machines being already in use, the presence of reserves makes
it possible to intensify the operation of the machines as the demands
of production grow higher.

Another method of preventing obsolescence is to use the machine
more fully in actual service. The shorter the period during which
it will exhaust its durability reserves, i.e., the closer its service life
to the mechanical life, the higher is the guarantee that the machine
will not become obsolete. The reduction of the service life of machi¬
nes to 3-4 years will practically guarantee them from the risk of
obsolescence.

The reduction of the machine’s service life by no means implies a
decline in the production output. As shown earlier, the total output
of a machine depends not on its service life, but on its durability.

The reduction of the service life with the durability remaining
constant is essentially effected by intensifyingjthe use of thejmachine.

With regard to machines operated to a calendar-based schedule the
problem is solved by increasing the number of working shifts and
enhancing the loading level.

In design this problem is solved by making the machine as univer¬
sal as possible, i.e., by extending its range of operation, and, pri¬
marily, by enhancing its reliability, which results in shorter emer¬
gency and repair downtimes.

The degree of utilization of machines of aperiodic action (e.g.,
seasonal machines) can be raised by applying changeable trailed or
mounted equipment, which increases their working period in a year.

The rate and degree of obsolescence depend on the scale and techni¬
cal level of production. At large industrial enterprises which are
capable of accelerating the production and continuously improving
the working processes machines become obsolete earlier, in contrast
to those employed at medium and small enterprises developing more
slowly. .

Machines which are considered obsolete in conditions of advanced
production can be used to advantage either at less important sections
of the same plant or by smaller enterprises having insufficient equip¬
ment where these machines will successfully contribute to the out¬
put. Although such machines will somewhat lower the rate of the
total productivity rise they will keep delivering products until their
mechanical life is fully exhausted.

Proceeding from the above one may say that obsolescence is not
the absolute limit to the durability increase. This limit can either
be moved farther away by rationally selecting the design parameters
of the machine or entirely eliminated by intensifying its use. Conse-

4 *

52

Chapter 1. Principles of Machine Design

quently, obsolescence cannot be regarded as a serious argument
against enhancing durability. Naturally, this by no means implies
that the designer may neglect the obsolescence factor; on the con¬
trary, he must do everything to avoid it.

1.4. Operational Reliability

The following are the reliability features of a machine: high dura¬
bility, trouble-free operation, stability of action (ability to main¬
tain original parameters over long periods), endurance (ability to
sustain overloads), simplicity of maintenance, survivability (capa¬
bility of operating for some time after partial failures, even if with
poorer characteristics), repairability, long periods between repairs
and, finally, small scope of repair work.

Because of the many factors influencing the reliability it is
rather difficult to formulate a single reliability criterion. When defi¬
ning reliability one most often proceeds from the concept of machine
failure, i.e., any forced stoppage of the machine.

The reliability^ a machine may be characterized by:

frequency of failure;

duration of an uninterrupted operation of the machine between
failures;

regularity of changes in the frequency of failure during the machi¬
ne’s being in operation;

severity of failures and the scope, cost, and duration of repair
work necessary to eliminate them.

The duration of the forced stoppages of a machine is characterized
by the downtime factor rj di (in other words, faultiness factor) which
is the ratio of the total duration of downtimes h dt over a certain
period to the sum of the actual working time h act and downtimes h dt
over the same period

„ _ hat _ 1

^ h a ct + hdt a i h aa t

The duration of a trouble-free operation of a machine is characteri-
zedibv the^faultlessness factor

h fault

ha a fr hdt

= 1

— *1<B

According to their severity all failures are subdivided into light,
medium|and heavy.

Light failures are those which can be eliminated on the spot by
the operating personnel and by means of the available hand tools.

Medium;failures require that the machine be stopped for a period
of time sufficient for a partial disassembly and change (or recondi¬
tion) of faulty parts, this being accomplished by repair services.

1.4. Operational Reliability

53

Heavy failures axe such breakages as affect vitaljmachine parts,
necessitating a prolonged stoppage of the machine for the purpose
of repairs. General wear of the machine can also be regarded as a
heavy failure for at a certain stage it requires the complete overhaul
of the machine and replacement of badly worn parts.

As to the oi’igin of failures, there are distinguished failures due to
design and manufacturing defects, failures due to maloperation, and
incidental failures.

Maloperation implies negligent attendance, violation of operating
instructions, disregard of prescribed operating conditions (over¬
loads), mistakes in the order of executing control operations (wrong
engagements), noncompliance with safety regulations, etc.

Most failures attributed to maloperation may be, with good reason,
put down to design defects. A well-designed machine must incorpo¬
rate features which preclude its operation with dangerous overloads,
prevent conflicting engagements and reduce to a minimum the effect
of the quality of maintenance on the serviceability of the machine.

It would be possible to characterize the reliability of a machine
by the scope of work necessary to eliminate failures, i.e., in the final
analysis, by a repair cost index which characterizes in an aggrega-
tory manner the frequency and degree of failures and the repairabi-
lity of machines. But this index is relative for the’ fc following reasons.
Firstly, the in-between repair intervals and cost|of repairs depend
not only on the machine’s reliability, but also on the quality of
maintenance and repair. Secondly, the cost of repairs is influenced
not only by the scope of the repair work required, but also by the
organizational level of the repair services. Thirdly, the total cost
of repairs depends on the policy of increasing the machine’s service
life. When following a wrong policy of restoring machines instead of
increasing the manufacture of new machines and, particularly, their
durability, the repair expenditures may reach formidable values.

The theory of reliability is at present being developed. Its objecti¬
ves are as follows: determination of requirements for reliability from
the standpoints of technology and economy; study of statistical laws
governing failures; investigation into causes of failure (failure diagno¬
sis); detection of parts and units most frequently causing failures;
prediction of failures; assessment of the severity of failures and the
difficulty of their elimination; study of failure effects upon the eco¬
nomical aspects of machine operation; development of reliable indi¬
ces characterizing the reliability of machines.

Special sections of the theory deal with the joint reliability of
complex sets of machines (flow and continuous production, semi- and
fully-automatic lines, transfer machines, etc.). These sections treat
the following problems: control of such complex sets of machines and
monitoring of their interaction; redundancy (introduction of stand-by
machines and circuits); accumulation (introduction of magazines

54

Chapter 1. Principles of Machine Design

which assure an uninterrupted operation of a group of machines in
the event of a short-time failure of some machine in the group), etc.

The|theory of reliability in many respects is closely associated
with the theory of durability, as far as their essence and methods are
concerned. Like the (theory of durability, the theory of reliability
is^based upon the methods of the theory of chances and mathematical
statistics and its conclusions are formulated as probability rela¬
tions.

The reliability of a machine is characterized by the mean-pro¬
bable time of an uninterrupted operation (mean-probable output of
the machine per failure) as a function of the duration of service, or
by the mean-probable frequency of failure, and also by the density
of distribution of failures throughout the service life of the machine.

The theoryj of reliability which predicts failures most frequently
occurring in practice may serve as a valuable aid to a mechanical
engineer. The main active role of the latter is the elimination of weak
parts in the design, thus raising the reliability of the machine as a
whole. By applying all modern design methods and technological
procedures, the designer can, in principle, (at least in many catego¬
ries of machines) eliminate all failures, except those occurring by
pure chance.

(a) Means for Improving Reliability

The^reliability of a machine in the first place is dependent on the
strength and rigidity of its structure.

The following methods improve the strength of the machine with¬
out increasing its weight: application of the most advantageous sec¬
tions an deforms, maximum utilization of the strength of material^- and
uniform, as far as possible, loading of all the elements of the itruc-
■ ire.

Rigidity can be increased by correctly selecting the loading pat¬
terns, rationally arranging the supports, and giving a stiff shape to
the structure.

The faultlessness of operation and the duration of in-between repair
periods depend to a large extent on the correct use and careful main¬
tenance of the machine, timely preventive repairs and prevention of
overloads. Yet, it would bejwrong to wholly depend on the quality
of maintenance. The prerequisites for correct operation must be
built into the machine design. Reliable operation must be assured
even in conditions of unsatisfactory maintenance. If the machine
fails because of unskilled hands it means that the construction was
not designed correctly from the standpoint of reliability.

The human factor in the maintenance and operation of a machine
should be eliminated as far as possible and the maintenance opera¬
tions should be minimized.

1.4. Operational Reliability

55

Subject to elimination are periodic operations, such as readjust¬
ments, tightening-up operations, lubrication, etc., which in the case
of unskilled servicing may lead to an excessive wear and premature
failure of the machine.

For instance, in internal combustion engines the necessity for the
adjustment of clearances in the valve mechanism can be eliminated
by means of automatic devices compensating for wear and thermal
expansion (hydraulic or some other devices). Apart from simplifying
the maintenance, such compensating devices practically eliminate
all the clearances in the valve mechanism, thus materially enhanc¬
ing its durability.

Periodic adjustments of the main and crankpin bearings can also
be obviated. The modern level of lubrication techniques allows
such bearings to be designed as will operate practically infinitely
and with a minimal wear.

The necessity for a periodic tightening up of nuts and bolts, which
work loose in service, can be dispensed with by using modern self¬
locking screw connections.

Machine operation can be heavily complicated by an inadequate
lubrication system requiring much attention of the maintenance
personnel. Intermittent point lubrication should be avoided. When,
because of design features, this is not feasible, it is advisable to use
self-lubricating bearings or introduce a centralized system which
feeds oil to all the lubrication points from one station.

As far as dependability and convenience of operation are con¬
cerned, the best solution is to introduce a fully automated lubrication
system which does not require periodic oil changes. This will be
feasible, if measures are taken to prevent the oxidation and thermal
degeneration of oil and if means are provided for continuous oil
refining and regeneration.

Lubrication systems must incorporate emergency devices which
ensure at least a minimum delivery of oil in the event of a failure
of the main system.

One of the ways to enhance operational reliability is to duplicate
servicing devices which most frequently fail in the course of work.
This can be exemplified by a dual ignition system of petrol engines
or a dual automatic control system. Multiple duplication of control
systems is exercised where a failure may endanger human lives.
Of great significance is automatic protection against accidental or
preconceived overloads. Such a protection is effected by safety and
limiting devices operating in a tracking mode and coming into action
as soon as an overload occurs..

Fully automated control is most expedient since it turns the ma¬
chine into a self-servicing and self-regulating, unit which adjusts
itself to optimum operating conditions.

56

Chapter 1. Principles of Machine Design

This can be illustrated by self-changing gearboxes and automatic
transmission systems of automobiles which provide for a stepless
control of the transmission ratio from the engine to the road wheels.
Such a system adjusts automatically to the optimum transmission
ratio which suits the given road and running conditions, thus increas¬
ing the engine efficiency and life.

High reliability can only be achieved by. implementing a whole
complex of design, technological, and organizational measures.
The increasing of reliability requires a prolonged, daily, scrupulous,
purposeful, combined work by designers, process engineers, metallur¬
gists, researchers and practical engineers, done to a carefully deve¬
loped programme.

The prerequisites for the manufacture of high-quality products
are: progressive manufacturing procedures, high level of production,
rigorous adherence to the prescribed processing conditions, and care¬
ful control of products at all stages of manufacture, beginning with
the manufacture of component parts and ending with the assembly
of the finished products.

Objective evaluation of the indices of reliability, durability and
operational costs presents the task of great difficulty. These indices
can only be determined for certain upon expiration of a long period
of time, with the products being sent far away from the parent facto¬
ry and dispersed among various remote operational points.

Under such circumstances the accelerated test methods for deter¬
mining the durability of parts, units, assemblies and machines as a
whole assume great importance. Great assistance in this respect can
be rendered by durability research laboratories carrying out systema¬
tic tests on the wear and service life of finished products.

A wider use should be made of the method of simulating actual
working conditions, which consists in the bench and operational
testing of machines under conditions much more severe than those
of the normal service. Thus, the machine in a very short period of
time undergoes a test which under normal conditions would take
several years. Tests are continued until the wear limit of the machine
is reached or even until its full or partial breakage occurs. At times
they are stopped for the degree of wear to be measured, the state of
parts registered, and tokens of impending failure determined.

Similar vigorous tests enable structural defects to be found and
the necessary precautions for their exclusion taken. Accelerated tests
also provide sufficiently reliable starting information for assessing
the actual machine’s durability.

Refinement of machines in operation. To develop reliable and
durable machines it is necessary to carefully study the experience
gained in their operation. The work of designers on a machine must
not terminate in the State tests of the prototype or in putting the
machine into serial production.

1.5. Machine 'Cost

57

The final refinement of the machine begins, in fact, after it has
been commissioned. Testing the machine in actual service is the
best way to find and then eliminate the weak spots in its design.

The disadvantages of a machine are especially clearly revealed in
repairs. Therefore close and continuous connections must exist be¬
tween the. designers and maintenance departments. For factories
engaged in mass or multiple production it is convenient to have their
own repair sections functioning as study laboratories and schools for
better design. . e

Of interest is the system of repair organization which is in practice
at engineering plants in the United States. According to the data
obtained during a survey of 356 American firms by the Institute of
Economy of the USSR Academy of Sciences, in 46% of cases the
chiefs of design departments are simultaneously in charge of repair
shops; in 18% of cases the repair shops are subordinated to the chiefs
of design departments; in 24% of cases—to other services, and only
in 12% of cases are repair shops independent sections. Thus, in 64%
of cases repair shops are, in one form or another, directly linked with
design departments. This tendency is not accidental but the result of
a definite policy of linking the design and repair services closer toge¬
ther so that the quality of products can be enhanced,

When examining defects it is absolutely necessary to discriminate
between random and systematic defects. Random defects usually
result from unsatisfactory control or technological inadequacies at
the manufacturing plant, while systematic defects are an evidence
of poor design requiring immediate rectification.

Operational observation of a machine should be included into
the work plan of design offices along with the design work and occupy
a significant part of the designer’s time. When isolated from the
actual operation of the machine, the designer cannot advance and
will never come to the top in his profession.

1.5. Machine Cost

Lowering the cost of mechanical-engineering products is a complex
task, both design and manufacturing. Sharp decreases in the cost of
manufacturing the machines can be obtained through rationalization
of production (mechanization and automation of production proces¬
ses, concentration of processing operations, specialization of plants,
industrial cooperation, etc.).

These measures are practicable and have the greatest effect in the
case of a stable large scale production. First and foremost here is
the designer. He must incorporate in the design prerequisites for the
production of one model over a long period of time and on the largest
scale possible, i.e., develop a design possessing wide applicability
and capable of future development and modernization.

58

Chapter 1. Principles of Machine Design

SOf great importance is the reduction of the number of the machine
type-sizes through a rational selection of the machine types and
parameters, as this makes it possible to manufacture the ma¬
chines on a multiple production basis and thus cut down their
cost.

The suitability of the design for industrial production is no less
important. Under the suitability for industrial production we under¬
stand the combination of features which ensure the most economical,
prompt and efficient manufacture of machine's by advanced techni¬
ques, parallel with the higher quality, accuracy and interchangeabi¬
lity of parts.

This concept should also cover the features providing for the most
efficient assembly (suitability for assembly) and the most convenient
and economical repairs (suitability for repairs).

The suitability for industrial production depends on the scale and
kind of the production. The piece and small-lot production require¬
ments as to the suitability of the design differ greatly from the
multiple and mass production requirements. The features characteri¬
zing the suitability of the design for industrial production are not
the same for different machine components.

Large economic gains are obtained from the unification and stand¬
ardization of parts, units and assemblies.

(a) Unification

Unification consists in a repeated use of the same elements in the
design, which enables the range of parts to be reduced, cuts down
the cost of manufacture and facilitates the operation and repairs of
machines.

The unification of constructional elements makes it possible to
curtail the number of processing, measuring and mounting tools.
Subject to unification are connection joints (by diameters, classes of
fits and accuracy), threaded connections (by diameters, types of
threads, classes of fits and accuracy, sizes of jaw), key and spline
connections (by diameters, shapes of keys and splines, classes of fits
and accuracy), gears (by modules, teeth types, classes of accuracy),
chamfers and fillets (by sizes and types), etc.

Unification of originally designed^ parts and units can be internal
(for a given product only) or external (adoption of components of
other machines produced at the same or adjacent factories).

The greatest economic gain is obtained when components already
being produced on a mass scale are used in new machines. The
adoption of components of individually produced machines, or of
old machines already discarded or about to be discarded, and also
of machines produced by other enterprises wherefrom the procure¬
ment of the components is difficult or even impossible has only one

1.5. Machine Cost

59

advantage: the successful use of such parts has been practically veri¬
fied. Not uncommonly this is enough to justify unification. j£|
The unification of grades and assortment of materials, electrodes,
sizes and types of fasteners and other standardized parts, antifriction
bearings, 1 ’[etc., eases the supply of materials, standards and purchased
parts to the manufacturing plant and repair departments.

The degree of unification is estimated by factor r\ unif which is
the ratio of the number of the unified components to the total num¬
ber of components in a given product

— ■~™"’ 100 %

or the ratio of the weight of the unified components to the total
weight of the product

100 %

or else the ratio of the cost of the unified components to the total
cost of the product

The disadvantage of the first index is in that it does not take
account of the specific value of the unified components in the machi¬
ne construction. The second index allows only for the weight of the
unified components in the total weight of the machine. The third
index is most correct, but it is more difficult to determine than the
former ones.

The degree of internal unification can be evaluated by the coeffi¬
cient of recurrence

Pt-ot = (1 —) • 100%

where N n — number of differing names of parts in the product;

N p = total number of parts in the product

This coefficient is readily determined from the summary parts list
and characterizes the degree of perfection of the design in respect to
the reduction of the range of parts. In well-designed machines r\ re( . —
= 40-60%.

For a differential assessment, the following indices are applied.

The degree of unification of original components

where orig

/V ■

' tV OTlg

T| unif.orig —

N unif.arlg
Nq rig

■ 100 %

= number of the originally designed components
unified;

— total number of the originally designed com¬

ponents

60

Chapter 1. Principles oj Machine Design

The degree of unification of constructional elements
‘nelera = ( 1 Nglem ) ' 100%

where N t . s — number of type-sizes of given elements;

N e iem — total number of given elements in the product

For example:

the degree of unification of threads

where N thrd . t . $ = number of thread type-sizes;

Nfhrd — total number of threaded connections in the pro¬
duct

the degree of unification of fasteners

where Nf St . t . s — number of fastener type-sizes;

N fsi = total number of fasteners in the product

( b ) Standardization

Standardization is the regulation of the designs and type-sizes of
components (fasteners, bushes, pipe fittings,, unions, nipples, oilers,
seals, control elements, such as handles, levers, knobs, wheels, etc.)
and units and devices (couplings, cocks, shutters and gates, lubri¬
cating devices, pumps, filters, pressure reducing valves, air- and
fluid-actuated units, etc.) which are widely used in mechanical
engineering.

All existing industrial standards are classified as National, branch
(pertaining to individual industries) and departmental. Nearly every
specialized design organization standardizes parts and units which
are typical of a given branch of mechanical engineering.

Standardization hastens the design work, facilitates the production,
operation and repairs of machines. A correct choice of the design of
the standardized parts increases the machine’s reliability.

Standardization produces the greatest effect when the number of
normally applicable standard sizes and types is reduced, in other
words, in the case of their unification. The design organizations
practically solve this problem by issuing the so-called limiting
lists covering the minimum range of standard items, which satisfies
the needs of the given class of machines.

The advantages of standardization are fully realized when the
standard parts are manufactured in a centralized fashion at speciali¬
zed factories. This relieves machine-building plants of the laborious

1.6. Building up Machines on the Basis of Unification

61

work of manufacturing standard parts and simplifies the supply of
spares to repair enterprises.

The degree of standardization is evaluated by the factor

f*A00%

where N stand = number of standard parts;

Nt — total number of parts in the product
For the successful accomplishment of standardization the stan¬
dard parts must be of high quality. Furthermore, the application
of standard parts should in no way restrict the creative initiative of
the designer and prevent him from finding better, more rational
solutions. The designer should not refrain from the application of
new solutions in the fields already covered by standardization, pro¬
vided that these solutions have obvious advantages over the existing
standards.

1.6. Building up Machines Derivatives
on the Basis of Unification

Unification is an effective and economical means for building
up on the basis of an original model a number of machines of the
same purpose, but having different characteristics as to their power
capacity, productivity, etc., or machines of different purposes,
performing qualitatively different operations, and also those inten¬
ded to turn out other products.

At present this problem is solved in several ways not all of which
are universal. In most cases each method is applicable only to cer¬
tain machines categories, the economic effects of these methods being
also different.

The classification of the methods for building up unified machi¬
nes derivatives, given below, is arbitrary. Some of the listed methods
are so close together that it is rather difficult to draw a clear demar¬
cation line between them.

A combined and parallel use of two or more methods are also
possible.

(a) Sectionalization

This method consists in dividing a machine into identical sections
and building up machines derivatives by stacking the required num¬
ber of the unified sections.

Sectionalization is applicable to many types of materials handling
machinery (belt, scraper and chain conveyers). Sectionalization in
this case comes to building up machine frameworks from unified
sections and making machines of various lengths with a new carrying
belt.

62

Chapter 1. Principles of Machine Design

Particularly easy to sectionalize are machines with a link-type
carrying belt (bucket elevators, platform conveyers with roller
chains) in which the length of the belt can be varied by removing or
adding links.

The economic efficiency of this method is little affected by the
need for introducing some non-standard sections which may become
necessary to make the machine length suit local conditions.

Suitable for sectionalization are also plate filters, plate heat-
exchangers, and centrifugal, vortex and axial hydraulic pumps.
In the latter case, by stacking different numbers of sections one can
obtain a series of multi-stage pumps rated for different pressures,
with their main working members being unified.

(b) Method of Changing the Linear Dimensions

This method enables the productive capacity of machines and
units to be changed by varying their length without changing the
cross-sectional shape. The method is applicable to a limited class
of machines (mostly of the rotary type) whose capacity is propor¬
tional to the length of their rotors (gear, rotary and vane pumps,
Ruth’s compressors, agitators, etc.).

With this method the degree of unification is rather moderate
because subject to unification are only the end plates of housings
and some auxiliaries. The main economic gain is due to the fact that
the basic equipment for machining the rotors and interiors of the
housings remains unchanged.

Increasing the capacity of gear trains, speed reducers and gear¬
boxes by extending the tooth length of the gears without changing
their module is a-particular case of the application of this method.

(c) Method of Basic Unit

This method enables machines of various purposes to be obtained
from one basic unit to which special equipment is attached to suit a
particular task. It is most popular in the construction of road-build¬
ing machines, mobile cranes, loaders, pilers, snow-removing machi¬
nes and also special-purpose self-propelled vehicles. In this case the
basic imit is a mass-produced automobile or tractor chassis. Machi¬
nes for different purposes are obtained by mounting auxiliary equip¬
ment on the chassi .

The basic unit method is extensively used in the manufacture of
agricultural machinery.

The mounting of auxiliary equipment necessitates the develop¬
ment of additional devices and attachment mechanisms (power take¬
off boxes, hoisting and swivelling mechanisms, winches, reversing

1.6. Building up Machines on the Basis of Unification

63

and friction gears, brakes, control mechanisms, cabs, etc.). These
additional mechanisms may, in their turn, also be unified consi¬
derably.

( d ) Conversion

With the conversion method the basic machine or its principal
elements are used to make different-purpose units, sometimes alike
and sometimes different as to their working process.

As an example may serve the conversion of internal combustion
engines from one kind of fuel to another, or from one type of the
ignition process to another (from spark-ignition to compression-
ignition).

Petrol engines are comparatively easy to convert into gas engines.
This is achieved by replacing the carburettor with a mixer, changing
the compression ratio (this is most readily done by changing the
height of the pistons), and making some minor constructional alte¬
rations. The engine by and large remains the same.

Converting a petrol or gas engine to a diesel is a much more diffi¬
cult job, chiefly because of the higher operating forces inherent in
the diesel engines due to their high compression ratio and enormous
ignition pressure. Consequently, the engine to be converted must
possess significant safety factors. Conversion in this case consists in
replacing the carburettor by an injection pump and injectors (or
pump and injector units) and changing the compression ratio (repla¬
cing cylinder heads, increasing piston height or altering the confi¬
guration of piston heads).

Still another example of conversion is the change-over of piston
air compressors to some other working fluid (gas, ammonia, Freon).
Here account should be taken of the different physical and chemical
properties of the new working fluids and select the materials for the
working members accordingly.

A radical conversion is illustrated by the change of an internal
combustion engine to a piston compressor. In this case the cylinder
heads are replaced by valve boxes, which requires corresponding
changes in the timing gear, thereby involving many alterations.

( e) Compounding

The method of compounding (parallel combination) consists in
combining machines or units in parallel with a view to increasing the
total power or capacity of the installation.

The machines to be combined can either.be placed side by side,
as self-contained units, or linked by synchronizing, transporting or
other devices, or else designed as one integral unit.

An example of the combination of the first type is a twin-engine
ship propulsion system with each engine driving its own screw, or

64

Chapter 1. Principles of Machine Design

a twin- or multiple-mounted engines in aircraft wings. Apart from
increasing the total power output (when it is difficult to build one
engine of a sufficient power capacity), this method occasionally helps
successively solve other problems. Thus, parallel-mounted engines
in a ship add to its manoeuvrebility, particularly at low speeds.

Installation of several engines on aircraft contributes to better
banking and steering. Several engines also enhance aircarft reliabi¬
lity: if an engine should fail the aircraft will'still be capable of fly¬
ing, although at a lower speed.

An example of the combination of the second type is a parallel
(bank) installation of machine tools in groups, 2-3 pieces in each.
Such installation is practised in transfer machines (lines), when the
productive capacity of one machine in the flow line is considerably
inferior to that of the whole line. In this case the component flow
is divided into two or more streams (to suit the number of the machi¬
nes installed in parallel) which are then again combined into a sin¬
gle flow.

An example of the combination of the third type is the doubling
or trebling of the line-type machine tools, i.e., combining of several
working routes on a single stand. As a result, there is a multi-line
parallel-flow machine with its productivity increased in direct
proportion to the number of the routes.

(/) Modification

Modification implies an alteration of a machine with a view to
adapting it to other operating conditions, or other operations or
kinds of products without changing its basic design*.

An example of modification is the adaptation of a machine for
work in various climatic conditions. In this case required modifi¬
cation comes mainly to a change of materials. Corrosion-resistant
materials are used in machines operating in humid tropical condi¬
tions while the machines intended for use in regions of severe cold
are made of cold-resistant materials, with their lubrication systems
being specially designed to assure trouble-free operation.

Another example is the modification of engines for use on marine
transport. Here the task is to make the engine as lightweight as
possible by replacing heavy alloys (east iron) by light ones (alumi¬
nium alloys) and by introducing materials that will resist the corro¬
sive action of sea water and air.

The modification of the machines intended for work in close con¬
tact with active chemical agents consists in protecting them against
the adverse effects by introducing heavy-duty seals and using chemi-
cally-resistant materials.

* Occasionally the term modification may imply modernization of machines
and improvement of their characteristics.

1.6. Building up Machines on the Basis of Unification

65

Modification is much more difficult when the machine has to be
adapted to different operations or kinds of products. In this case
the modification method comes very close to the multi-station ma¬
chine system.

( g ) Multi-Station Machine System :

This method consists in building machines out of a varying number
of standard self-contained units, or combinations of such units,
which are mounted on a common base.

This principle is most fully embodied in the design of multihead'
metal-cutting machines. Such machines are built up of individual
unified assemblies and elements (machining units, combination
housings, synchronizing mechanisms, rotary tables, general-purpose
housings, beds, pedestals, auxiliary units, coolant systems, electri¬
cal controls, hydraulic drives, etc.).

In the course of machining the workpiece generally remains sta¬
tionary and the machining units are brought to it so that the machi¬
ning takes place simultaneously on all sides, thus considerably speed¬
ing up the entire working process.

The chief advantages of the multi-station machine system consist
in that it reduces the time and cost of the design and manufacture of
machines, simplifies their servicing and repairs, and allows the pi
to be easily reset for processing various components.

The method is rather promising and applicable to other machine
tools.

A partial implementation of this method comprises the use of
mass-produced standardized units (speed reducers, pumps, compres¬
sors), or the adoption of units and assemblies of serially-manufactu¬
red machines (gearboxes, differential gears, shifting mechanisms,
clutches, friction couplings, etc.). ;

( h ) Integrated Standardization

The method of integrated standardization is close to the previous
one. It is successfully applied to simple units (vessels, settling tanks,
evaporators, and mixture-preparing units which are extensively used
in the chemical and food industries).

The structural simplicity of the above listed units enables all or
nearly all of the elements of their design to be standardized. Suitable
for standardization as to their sizes and types are the shells of tanks,
bottoms, covers, manholes, hatches, fittings (valves, gates and shut¬
ters), fastening feet, stands. Subject to standardization are also
whole assemblies (heat exchangers, agitator drives, metering devi¬
ces, etc.).

5 - 01.395

Ref.

No.

Engine type

Number of
cylinders

Ref.

No.

Engine type

Number of
cylinders

I

Single-row

2

XIV

Four-row, twin crankshaft

24 (16)

II

Same

4

XV

Same

24 (16)

III

Same

6

XVI

Six-row, twin crankshaft

36(24)

IV

Same

8

: XVII

Single-row, radial cylinder

3

V

Double-row, V-type

8

XVIII

Same

5

VI

Broad-arrow, W-type

12

XIX

Same

7

VII

Four-row, X-type

16

XX

Same

9

VIII

Double-row, opposed cylinder

12(8)

XXI

Double-row, radial, staggered cylin¬
der

14

IX

Double-row, V-type

12

XXII

Same

18

X

Broad-arrow, W-type

18

XXIII

Row, cross-type

24 (16)

XI

Same

00

XXIV

Row, radial cylinder

. 36(24)

XII

Four-row, X-type

24(16)

XIII

Double-row, twin crankshaft

12(8)

' •

68

Chapter 1. Principles of Machine Design

A specific feature of such, apparatus is the wide use therein of
purchased auxiliary equipment (pumps, vacuum pumps, filters,
condensate traps, instruments and controls, automation means).

By using standard parts, unified assemblies, and purchased equip¬
ment one can build up:

apparatus with the same working process, but differing in size
and productive capacity; ,

apparatus of the same purpose, but rated for different parameters
of the working process (pressure, vacuum, temperature);

apparatus differing both in the purpose and working process.

(i) Unified Series of Machines

In some instances it is possible to form a series of machines deriva¬
tives of different power or productive capacities by changing the
number of their main working members and using them in various
combinations. Such a series is called a family, a range or just a series
of machines. The method is applicable to machines whose power or
productive capacity is directly proportional to the number of the
working members.

The method provides the following technological and operational
advantages:

simplification, acceleration and cheapening of the machine design
and manufacturing processes;

possibility of applying highly efficient methods of machining
unified parts;

shortening of the time required for testing and refining experi¬
mental models (because the main components have already been
tested);

simplification of operation;

reduction of the training periods for specialists, shortening of
repair times, and simplification of spares problem.

A classical example of the development of unified machines is the
formation of series of four-stroke internal combustion engines on the
basis of a unified cylinder assembly and partially unified rod-and-
piston assembly.

The number of possible cylinder combinations is limited because
of the need to have balanced inertia forces due to reciprocating mas¬
ses-and a regular firing. The combinations that meet these require¬
ments are presented in Table 3.

A greater degree of unification is characteristic of the twin crank¬
shaft engines (XII1-XVI) which have their rod-and-piston assembly
and crankshafts fully unified, as well as the cylinder assembly.

Sincqfhe power output of an engine is proportional to the number
of its cylinders, the illustrated series of engines enables, in prin¬
ciple, a family of engines to be obtained which offers a very wide

1.6. Building up Machines on the Basis of Unification

69

range of power outputs. For instance, if the power output of one
cylinder is 100 hp, the obtainable range of the series then will be
200-3000 hp.

However, out of the large number of schemes shown in Table 3
comparatively few are employed in practice.

Engines with a limited number of cylinders (<4) are notable for
their irregular torque and bad balance.

Engines with a large number of cylinders (>24) are rarely used
because of the difficulty of maintenance and greater liability to
troubles.

Row engines with a small angle between cylinders (VII) are inac-
ceptable because of the difficulty of arranging the inlet and outlet
pipes between the cylinders.

In the category of low- and medium-power engines (automobile,
tractor and other transport engines) use is mostly made of schemes 11,
III, IV and V, while in that of high-power engines (ship engines)—
schemes IX, XI, and more rarely, XXIII and XXIV.

The radial schemes (XVII-XXII) were extensively used for air¬
cooled aero-engines.

Another application field for the method of unified machines
series is represented by rotary machine tools. Since the productivity
of rotary machines is directly proportional to the number of opera¬
tional units installed thereon, it is possible to form a series of machi¬
nes of different productive capacities by using various combinations
of unified operational units. In contrast to piston engines the number
of operational units which can be installed on a rotary machine is
practically limitless and depends only on the required productive
capacity of the machine.

Besides changing the number of the operating units on rotary
machines, it is also possible to change the units themselves, thus
adapting the machines for performing various functions. This is an
example of combining the unified series method with conversion or
multi-station machine system.

Limitation of the method. The methods of developing machines
derivatives on the basis of unification cannot be regarded as univer¬
sal and fully comprehensive. Each of these methods is applicable only
to a limited category of machines. Many machines (steam and gas
turbines) do not permit the development of machines derivatives be¬
cause of their design features. Also, it is not possible or advisable to
develop derivative series for special-purpose machines, high-power
machines, etc., which remain in the category of individually pro¬
jected devices.

Not uncommonly unification worsens the product quality, parti¬
cularly in the case of derivative series having extensive ranges. The
extreme members of these series are generally inferior to specially
made machines as to their size, specific metalwork weight, specific

70

Chapter 1. Principles of Machine Design

gravity and operational characteristics. Such an impairment is tole¬
rable when unification provides for a greater economy and the size
and weight are only of secondary importance.

The method is readily applicable to general-purpose machines, but
has a limited application, and often inapplicable at all, to machines
whose overall dimensions and weight must be kept to a minimum.
In the category of high-class machines one often has to refuse unifi¬
cation and follow the path of individual design.

In this connection it is necessary to say a few words about the
technological trend in designing wherein emphasis is placed upon
processing aspects and particular attention is paid to the methods of
unification and development of machines derivatives series, which
are considered almost the main principle of rational design.

The chief merit of this trend is that it establishes close links be¬
tween design and production. The suitability of a design for industri¬
al production should be achieved not by a series of subsequent amend¬
ments, but through planned designing.

However, the suitability of the design for industrial production
cannot be the main aim of designing.

The main trend in the machine design is the enhancement of the
quality, reliability, durability and efficiency of machines. Production
engineering must use all its means to help solve all these basic pro¬
blems, but not determine the design trend.

Also, one must not overestimate the role of the development of
machines derivatives and their series as a means for reducing the
cost of machines. These methods have a limited application and
their efficiency is inferior to that of other methods such as the auto¬
mation and mechanization of production, reduction of the number of
type-sizes of machines, etc.

_ It is wrong to consider the suitability of a machine for theforma-
tioxi on its basis of machines derivatives and their series to be an
indication of the suitability of its design for industrial production
even because this method cannot be applied to all machines. It would
be very strange to regard, for instance, the design of a huge heat
machine (e.g., a powerful steam turbine) as being unsuitable for
industrial production on the grounds that no series of machines deri¬
vatives can be built up on the basis of it.

1.7. Reduction of Product Range

Decreasing the range of products on the basis of a rational selec¬
tion of their types facilitates their serial manufacture, mechaniza¬
tion and automation of the production and introduction of progressi¬
ve techniques, thus increasing the products output, cutting down
their cost and improving their quality. In this way the scattering
of capital in producing machines in small lots is eliminated, the

1.7. Redaction of Product Range

71

operation, maintenance and repairs of the machines are made easier,
and good opportunities for a profitable manufacture of spare parts
are provided.

The problem of decreasing the range of products is solved in the
following three ways:

by developing parametric series of machines with rationally
chosen design parameter intervals between the adjacent machines
in the series;

by improving the versatility of machines, i.e., by widening their
operational range;

by providing the machine designs with reserves for further deve¬
lopment and subsequently using these reserves as the demands of the
national economy grow higher.

All these methods can be combined one with another and with
unification methods as well. For instance, it is possible to simulta¬
neously develop unified and parametric series of piston engines.
In this case the unified series comprise engines using the same cy¬
linders, hut differing in the cylinder number and arrangement, where¬
as the parametric series include engines having the same number
and arrangement of cylinders, but differing in the cylinder diameter.

(a) Parametric Series

Parametric series are series of machines of the same purpose, with
their design and characteristics being regulated as well as the diffe¬
rences in the characteristics (gradations) between the adjacent machi¬
nes in the series.

In many cases it is advisable to base a series upon a single machine
type, establishing the required differences in the characteristics of
the machines in the series by changing the dimensions of the original
machine while preserving the geometric similarity of the machines.
Such series are called size-similar or simply size series.

In other instances it is advisable to establish for each range of
characteristics its own machine type with its own dimensions. Such
series are called type-size series.

This can he exemplified by ship engines. For low power ratings it
is advisable to employ four-stroke internal combustion engines,
while for medium and high power ratings it is more advantageous to
use two-stroke engines as they feature smaller dimensions and weight
for the same power output, or gas-turbine engines which are capable
of a still greater concentration of power.

Combined series are occasionally used: some of the machine modi¬
fications in the series are of the same type and have similar geomet¬
ries, while others are developed on the basis of other machine types.

The use of different machine types (as in the case of type-size and
combined series) in no way affects the efficiency of the parametric

72

Chapter L Principles of Machine Design

series method since the economic advantage of such series results
from the reduction of the number of models. The technological gain
is a centralized and hence, highly productive machine manufacture
stemming from the increase in the number of each machine model
produced.

The parametric series method has its maximum effect when pro¬
ducing machines for mass application, for such machines generally

Fig. 11. Frequency of use

I — arithmetic series ; II — geometric series; Ill — series correlated with frequency of use

have extensively ranging characteristics (internal combustion engi¬
nes, electric motors, machine tools, pumps, compressors, speed redu¬
cers, etc.).

The most important factor when developing a parametric series is a
correct selection of the machine types, the number of the members
in the series, and the intervals between the members. When deciding
upon these matters it is necessary to consider the frequency of use of
various members within the series, operating conditions likely to be
met with in actual service, the flexibility and adaptability of the
machines of the given class (i.e., feasibility of varying their opera¬
tional parameters), and possibilities for their modification and deve¬
loping additional machines derivatives on their basis.

It is advisable to increase the number of the members of the series
in the range of the most frequently used parameters and have larger
intervals between the members in the range of parameters which are
seldom required.

1.7. Reduction of Product Range

73

Let us consider the case of three-phase electric motors. Let the
curve expressing the frequency of use of these motors he such as
shown in Fig. 11. The scales in the lower part of the graph show sche¬
matically the motor power rating gradations obtained when deve¬
loping parametric series of the motors to an arithmetic (1) and geo¬
metric (II) progressions. It is quite obvious that both these series do
not accord with, the curve of the frequency of use. The density of the
distribution of members in the arithmetic series is the same both in
the range of low and high frequency of use, which is clearly irration¬
al. The density of the distribution of members in the geometric,
series is too high in the range of small power ratings and too low in.
the range of the maximum frequency of use.

A good compromise is presented by a rationally developed se¬
ries (III). Here the members are arranged more densely in the regi¬
on of the highest frequency of use and more sparsely in the region,
of the lowest frequency of use. Such a distribution makes it possible
to satisfy more fully the requirements of the majority of consu¬
mers.

To realize fully the economic effect of parametric series, it is
necessary that they be in use for a sufficiently long period of time.
Therefore when developing a parametric series of machines account
must be taken not only of the contemporaneous conditions, but also
of the prospects of the development of the branches of industry for
which the series is intended.

(b) Size-Similar Series

The designing of size-similar machines has its own particular-
features, the main one being that the output parameters of such
machines depend not only on their geometrical dimensions, but
also on the parameters of their working processes.

To maintain full similarity of the machines of different dimensions
it is necessary to ensure, firstly, their geometrical similitude and,
secondly, the similarity of their working processes, i.e., to make the.
thermal and mechanical stresses developing in the machines as a
whole and in their individual parts equal for all the machines in a
given series.

Similarity criteria are formulated for most machines and working-
processes. For instance, internal combustion engines (Fig. 12) have
the following two similarity criteria:

(1) equality of mean effective pressure p e depending on the pres¬
sure and temperature of the mixture at the inlet;

(2) equality of mean piston speed v p — ~ ($ — piston stroke,..
n- engine revolutions per minute), or equality of the product

74

Chapter 1. Principles of Machine Design

D -n (D —cylinder diameter which, in geometrically similar engines
is related to the piston stroke by the relationconst).

In a general form

f(jp e , Dh) = const (1.21)

Should this criterion be the same, then all.the geometrically simi¬
lar engines will have the following identical parameters: thermody¬
namic efficiency, mechanical efficiency, actual efficiency (hence, the

Fig. 12. Size-similar series of internal combustion engines

same specific fuel consumption in terms of g/e.h.p -h), thermal load
(heat transfer per unit cooling surface), specific power, stresses due
to gas pressure and inertia forces, specific bearing loads and con¬
structional weight (weight related to the sum of the squares of the
cylinder diameters).

It is clear from Eq. (1.21) that to ensure the constancy of the above
listed parameters whenever the cylinder diameter is increased it is
necessary to reduce either the engine revolutions per minute or the
mean effective pressure. Therefore, the effective engine power is
proportional to the square and not to the cube of the cylinder diame¬
ter. The litre power (power referred to the swept volume) decreases
proportionally to the cylinder diameter, but the engine weight per
effective horsepower rises in direct proportion to the cylinder diame¬
ter. The greater the cylinder diameter, the weaker the resistance to
•bending of the engine parts and the engine as a whole.

It is necessary to say that strict adherence to geometrical simili¬
tude in the region of small cylinder diameters is impracticable be¬
cause of manufacturing difficulties. The minimum cross-sectional di-

1.7. Redaction of Product Range

75

mensions of the engine components are limited because of the need to
make them sufficiently rigid to withstand machining (e.g., turning),
assembly and transportation. This is why quite a large number of
components of small-size machines are made larger than required
if the conditions of geometric similitude are met. Therefore, small-
cylinder engines have a greater weight per effective horsepower, but
are more reliable, strong and rigid, and are capable of being super¬
charged and operated at increased speeds.

The above illustrated example of internal combustion engines is
a particular case of an extensive category of machines in which the
stresses developing in the parts depend upon the operating pressures
and speeds. A general law for the machines of this class may be for¬
mulated as follows: the stresses developing in geometrically similar
constructions working under similar pressures and speeds are iden¬
tical.

From the above one can draw the following conclusions.

Size-similar series should be developed on the basis of the output
characteristics (power, productivity, etc.) of the machines and not
on their geometrical parameters (displacement, cylinder diameter,
size of rotors in rotary machines) because, due to the inherent laws of
similarity, the output characteristics follow a regularity which
differs from that of the change of the geometrical characteristics;
the latter are obtained as derivatives.

It is necessary to consider the changes in specific parameters (e.g.,
weight per horsepower, litre power) and mechanical properties (e.g., :
flexural rigidity) which are inevitable in geometrically similar
machines.

(c) Universalizing of Machines

The universalizing of machines is aimed at expanding their func¬
tional potentialities and operational range, and widening the range
of components machined with them. It improves the adaptability of
the machines to the production requirements and increases their use
factor. The chief economic gain from universalizing is that it enables
the number of manufacturing units to be reduced: one universal
machine is equivalent to a number of specialized machines which
perform specific operations.

Widening the functions and application areas of machines can be
attained by the following methods: introducing additional opera¬
ting members and replacement equipment, providing built-in adjust¬
ments with a view to widening the range of the products machined,
and controlling the output characteristics (speed, power, producti¬
vity).

Universalizing may be illustrated by piano-milling machines,
which combine planing and milling operations, or by slabbing-bloo-

76

Chapter 1. Principles of Machine Design

ming mills designed for turning out both blooms (billets for rolling
section steel) and slabs (billets for rolling sheet steel).

Many agricultural machines are readily adaptable to universali¬
zing. By providing a basic machine -with auxiliary equipment, trailed
or mounted, a multi-function machine with an extended seasonal
employment interval can be readily obtained.

Universalizing techniques may be illustrated by using as an exam¬
ple automatic rotary filling machines intended for filling up contai¬
ners of various capacities.

The first condition for universalizing a piston-type filling machine
is the provision of a metering mechanism which can meter out bat¬
ches within broad limits. Such a mechanism can be made in the form
of a swash plate whose inclination can be changed by means of an
adjusting screw. On the template of the machine there are several
metering cylinders with their piston rods sliding over the plate as
the turntable rotates. The inclination of the swash plate determines
the length of the piston stroke and, consequently, the size of the
batch being metered out. The adjustment of the swash plate inclina¬
tion -is stepless.

The problem of handling containers of various diameters is solved
by using adjustable container guideways or changeable container-
feed mechanisms which deliver empty containers to the turntable
for filling and then remove the filled ones. For handling containers
of different heights, either the vertical position of the turnplate with
the metering cylinders is changed, or that of the container-carrying
turntable.

The time needed to fill a container is proportional to its capacity.
Therefore, a gearbox, or a stepless speed variator is added to the
filling machine to adjust the turntable rotary speed.

It is important to set reasonable limits to the degree of univer¬
sality. Universal machines designed to suit too extensive a list of
products or operational range are complex, heavy, cumbersome and
inconvenient in use. It is sometimes more advisable to design a series
of moderately universalized machines so that the complete series
will provide for the, required degree of universality.

In other cases universal machines may be supplemented with two
or three special machines designed for parts sharply differing in size
or configuration from the basic component type.

{d) Consecutive Machine Development

The resources for future development, incorporated into a machi¬
ne’s design, allow the machine to be systematically improved so as to
keep its characteristics in line with the ever-growing demands of the
national economy. Such a development method eliminates the perio¬
dic replacement of obsolete machines, assures a stable production of

1.7. Reduction of Product Range _ 77

machines of the same basic design over many years, and, being one of
the main ways of lowering the cost of machine products, it yields
a great economic gain.

The resources embodied in the design of a machine depend on the
purpose of the latter. Thus, in heat engines the original model must
possess ample working volume (displacement) reserves and allow
increases in rotational speeds and improvements in the thermal
processes to be effected. Machine tools, for which productive capa¬
city is of prime importance, must permit increasing their speed and
Tange of operations.

In all instances the initial model must have adequate strength and
rigidity margins. This, however, does not mean that the basic model
must be overweight. It is important to make stronger the most heavi¬
ly stressed parts and units which may become deterrents to the inten¬
sification of the machine operation.

Immensely important is the efficiency of the kinematic scheme of
the machine which predetermines the general capabilities of inten¬
sified operation inherent in the machine’s design.

Not uncommonly the improvement of machines involves subse¬
quent introduction of additional units-(speed reducers, gearboxes,
■automation means). Their installation should not interfere with the
basic design of the machine, therefore the latter must be provided
with already machined seating surfaces and fastening points.

Along with the use of the original resources, the machine should be
■continuously improved by utilizing newly developed production
and design techniques with a view to lowering the machine weight,
increasing its power capacity and degree of automation, and impro¬
ving its durability, reliability and maintainability.

A striking example of the above trend is the history of the Soviet
•aircraft engine model AM-34 which lasted for 15 years in all and,
thanks to the continuous development, remained in each of its
stages the best in the world of aviation. During this period its power
was raised from 800 to 1800 hp through supercharging, increasing the
speed and using a high-octane antiknock fuel. Engine service life
rose from 200 to 1000 h. Thanks to the improvement, the engine of
the last model maintained its power rating at high altitudes (up to
■6000 m). The efficiency of the propeller unit was increased by using
a speed reducer and a variable-pitch propeller. Though the total
weight of the engine had somewhat increased (because of the addi¬
tional built-in units, namely, the supercharger and reducer), its
weight per horsepower was almost halved (from 0.9 to 0.5 kg/e.h.p.).

This progress- was made owing to the built-in displacement reserves
of the original model and systematic development of the engine with¬
out changing the basic, design and initial geometry.

Machine tools may serve as another example of the advisability of
providing the initial design with future development reserves.

78

Chapter 1. Principles of Machine Design

Machine tools with increased strength, rigidity and resistance to
vibration proved to he capable of being used (without alterations)
for high-speed high-force cutting, while those of lower rigidity had
to be reconstructed to suit the new, heavier operating conditions.

It should be emphasized that unlike other methods of cutting down
the cost of machine products, which have been described above, the
method of consecutive machine development is universal and appli¬
cable to all categories and classes of machines, including unique
machines.

1.8. Preferred Numbers and Their Use In Designing

Standardization in mechanical engineering is based on series of
numbers obeying definite regularities. Up to recent years broad
use has been made of arithmetic series in which each term, after
the first, is derived by adding to the preceding one a constant quan¬
tity (the common difference, or constant). With the co m mon diffe¬
rence being equal to 10, an arithmetic series of diameters (from 10
to 200 mm), most popular in mechanical engineering, is as follows

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120,

130, 140, 150, 160, 170, 180, 190, 200

Arithmetic series are noted for their relative non-uniformity, Their
higher regions are more densely filled with dimensional gradations
than the lower ones.

The ratio of each member of an arithmetic series to the preceding
one is

*-‘+4

where A — common difference (in the case under consideration
A = 10);

n = numerical value of the preceding'term

This ratio sharply decreases with the growth of the number of
terms in the series. Thus, in the above series the ratio of the second
term to the first one is cp = 2; fifth to fourth, cp = 1.25; tenth to
ninth, cp = 1.1; twentieth to nineteenth, <p = 1.05,

The nonuniformity may be partially corrected by changing the
value of A for different regions of the series. Thus, for the case under
consideration, within the regions D < 50, D — 50-100 and D >
> 100, A can be taken at 5, 10 and 20, respectively.

As a result, the following series is obtained

10, 15, 20, 25, 30, 35, 40, .45, 50, 60, 70, 80, 90, 100,

120, 140, 160, 180, 200

1.8. Preferred Numbers and Their Use in Designing

79

This series is more uniform, though -with a step-wise change of the
size gradations.

More rational are series developed on the principle of the geometric
progression in which each term, after the first, is derived by multi¬
plying the preceding one by a constant value cp (the constant ratio).

(a) Basic Preferred Numbers Series

The USSR State Standard TOCT 8032-56 stipulates four series of
preferred numbers with different values of

<P“ Vt ( L22 >

where a and l are the first and the last terms of a series, respective¬
ly. In the standard series — = 10. Then, Eq. (1.22) becomes

<p = /l0

The index n of the radical is taken at 5, 10, 20 and 40. These figu¬
res preceded by the letter R make up the designation of a series.

In this way series i?5, i?10, i?20 and i?40 are obtained, for which
the value of cp is 10 sm 1.6, y^lO « 1.25,10 » 1.12 and y^lO «
fv 1.06, respectively.

Any term in a series is

a h = aq h

where k — serial number of the term;

a — the first term of the series, to which the zero serial num¬
ber is assigned

With a decrease in the value of <p the intervals between the terms
in a series become shorter, the total number of the terms grows lar¬
ger, and the series itself becomes more fractional. As an exception,
the use of the very fractional series R 80 with cp — 8 10 « 1.03 is
permissible.

According to the USSR State Standard TOCT 8032-56 the basic
series of preferred numbers in the range of 1-10 are as follows

125: T; 1.6; 2.5; 4; 6.3; 10
JR10: 1; 1.25; 1.6; 2; 2.5; 3.15; 4; 5; 6.3; 8; 10
1220: 1; 1.12; 1.25; 1.4; 1.6; 1.8; 2; 2.24; 2.5; 2.8; 3.15; 4;
4.5; 5; 5.6; 6.3; 7.1; 8; 9; 10

JR40: 1; 1.06; 1.12; 1.18; 1.25; 1.32; 1.4; 1.5; 1.6; 1.7; 1.8;
1.9; 2; 2.12; 2.24; 2.36; 2.5; 2.65; 2.8; 3; 3.15; 3.35; 3.55; 4;
4.25; 4.5; 4.75; 5; 5.3; 5.6; 6; 6.3; 6.7; 7.1; 7.5; 8; 8.5; 9; 9.5; 10

.80

Chapter 1. Principles of Machine Design

The numerical -values of the terms of all series are rounded off to
within ±1%. Each subsequent lower series is obtained by deleting
oven terms from the preceding higher series.

An additional series with the common ratio cp — 1.03 is

B80: 4; 1.03; 1.06; 1.12; 1.15; etc.

(b) Derived Series ( Subseries )

From the basic series one can obtain geometric series for any range
of numbers, i.e., with any value of the first and the last terms. In
accordance with the basic law governing the development of geo¬
metric progressions, derived series are obtained by multiplying
successively the first term of a new series by the terms of any basic
series (i?5, RIO, etc.) until the quantity 10 a. is obtained, which, in
its turn, is multiplied by the terms of the same basic series, etc.

Let us cite as an example a series of numbers ranging from 1 to
1000, derived on the basis of,the f?5 series

1; 1.6; 2.5; 4; 6.3; 10; 16; 25; 40; 63; 100; 160; 250;

400; 630; 1000

Series developed on the basis of geometric progression can be made
sparser by selecting the m%h terms (m —a serial number multiple
of any integer). As a result, a new series with a common ratio <p m is
obtained. Examples of such a thinning out are, in fact, the basic se¬
ries of preferred numbers.

The series R20 (<p m = 1.06 2 = 1.12), i?10 (cp m = 1.06 4 — 1.25)
and R5 (<p m = 1.06 s = 1.6) are obtained by selecting from the RAO
series (<p — 1.06) all the terms whose serial numbers are multiples
of 2, 4 and 8 , respectively. By selecting from the i?40 series the
terms whose serial numbers are multiples of 3, 6 and 9 it is possible
to obtain accordingly series with the following common ratios

<p m = l.Q 6 3 = 1.19; cp m = 1.06 s = 1.41; and 9 ™ = 1.06® = 1.68

Derived series can also be obtained by other methods. Raising
the terms of a geometric progression to any power gives a new pro¬
gression with a new common ratio. Thus, when terms of the R 5
series are squared, the following progression with a common ratio
of 2.56 is obtained

1; 2.56; 6.25; 16; 39.7; 1000

Thus, if the linear dimensions of a series of parts from a geo¬
metric progression, the magnitudes of their cross sections, volumes,
weights, resisting moments, and moments of inertia of sections'then
also form geometric progressions, but with other common ratios any
different first and last terms.

1.8. Preferred Numbers and Their Use in Designing

81

From the ability of the cross sections, resisting moments and mo¬
ments of inertia to form geometric progressions it cannot be conclu¬
ded that all the parts whose linear dimensions follow the order of a
geometric progression display the properties of equal strength and
equal rigidity. For this to be true the acting loads should be propor¬
tional to the square of the linear dimensions of the parts, a case very
seldom met with in practice.

(c) Standard Linear Dimensions

On the grounds of the basic series of preferred numbers there have
been developed series of standard linear dimensions (USSR State
Standard FOGT 6636-60) differing from the basic series in that their
numbers are rounded off more roughly. The series of standard linear
dimensions are designated as i?a5, BalO, i?a20, and f?a40.

Bob: 0.1; 0.16; 0.25; 0.4; 0.6; 1; 1.16; 2.5; 4; 6; 10; 16; 25;
40; 60

Ba 10: 0.1; 0.12; 0.16; 0.2; 0.25; 0.32; 0.4; 0.5; 0.6; 0.8; 1;
1.2; 1.6; 2; 2.5; 3; 4; 5; 6; 8; 10; 12; 16; 20; 25; 32; 40; 50;
60; 80

Ra2Q: 0.1; 0.11; 0.12; 0.14; 0.16; 0.18; 0.2; 0.22; 0.25; 0.28;
0.32; 0.36; 0.4; 0.45; 0.5; 0.55; 0.6; 0.7; 0.8; 0.9; 1; 1.1; 1.2;
1.4; 1.6; 1.8; 2; 2.2; 2.5; 2.8; 3; 3.6; 4; 4.5; 5; 5.5; 6; 7; 8; 9;
10; 11; 12; 14; 16; 18; 20; 22; 25; 28; 32; 36; 40; 45; 50; 55;
60; 70; 80; 90

B«4Q: 0.1; 0.105; 0.11; 0.115; 0.12; 0.13; 0.14; 0.15; 0.16;
0.17; 0.18; 0.19; 0.20; 0.21; 0.22; 0.24; 0.25; 0.26; 0.28; 0.3;
0.32; 0.34; 0.36; 0.38; 0.4; 0.42; 0.45; 0.48; 0.5; 0.52; 0.55; 0.6;
0.63; 0.65; 0.7; 0.75; 0.8; 0.85; 0.9; 0.95; 1; 1.05; 1.1; 1.15; 1.2;
1.3; 1.4; 1.5; 1.6; 1.7; 1.8; 1.9; 2; 2.1; 2.2; 2.4; 2.5; 2.6; 2.8; 3;

3.2; 3.4; 3.6; 3.8; 4; 4.2; 4.5; 4.8; 5; 5.2; 6; 6.3; 6.5; 7; 7.5; 8;

8.5; 9; 9.5; 10; 10.5; 11; 11.5; 12; 13; 14; 15; 16; 17; 18; 19;

20; 21; 22; 24; 25; 26; 28; 30; 32; 34; 36; 38; 40; 42; 45; 48;

50; 52; 55; 60; 63; 65; 70; 75; 80; 85; 90; 95

The TOCT 6636-60 Standard covers dimensions up to 95 mm. When
necessary, these series can be continued by following the principle
of geometric progression while keeping the common ratio unchanged.
For instance, for the range of 100-250 we have

B«5: 100; 160; 250
BalO: 100; 120; 160; 200; 250
JS«20: 100; 110; 120; 140; 160; 180; 200; 220; 250
Bam: 100; 105; 110; 115; 120; 130; 140; 150; 160; 170; 180;
190; 200; 210; 220; 240; 250

S—01395

82

Chapter 1. Principles of Machine Design

The application of the standard linear dimensions is advisable
for surfaces subject to an accurate machining, especially for connec¬
tion joints*. This helps standardize cutting, checking and measuring
tools, and facilitates the setting up of machine tools.

The greatest economic gain is obtained when reducing the number
of the terms in a series, i.e., when using in each individual case the
lowest series providing for the necessary size range.

Standard dimensions which are referred to surfaces not involving
precise co-ordination are of less importance.

The series of the diameters of wire and bar-steel rolled stock,
thickness of sheet-steel rolled stock, linear cross-sectional dimen¬
sions of shape-steel rolled stock—all are based on the standard linear
dimensions.

The series of round rolled stock must be consistent with the series
of machined diameters, so that adequate machining allowances are
given, providing for the minimum waste of metal in the form of
chips.

If the RaiO series containing dimensions 10, 12, 16, 20, 25, . . .,
is taken as the basic one for machining purposes, then for the blanks
it is more advisable to take a modified Ra 20 series obtained by
selecting from the latter series only those dimensions which are
shifted by one serial number relative to the dimensions of the basic
RaiO series. Thus, one may obtain the following series:

Diameters of machined work-

pieces, mm ..i0 12 16 20 25 32 40 50 60 80

Diameters of blanks, mm . . . . 11 14 18 22 28 36 45 55 70 90

The setting up of the diameters series for blanks and machined
workpieces by following the same principle and with the same com¬
mon ratio can lead to a greater waste of metal in the form of chips.

It is irrational to use the series of preferred dimensions for surfa¬
ces uot subject to machining (castings, forgings). In such instances
even a partial standardization of dimensions has no actual advanta¬
ges and only makes the process of design and manufacture more dif¬
ficult.

Series of preferred numbers used in designiug. The significance of
preferred numbers used in designing should not be overestimated.
Some authors are of the opinion that the series of preferred numbers
must he used not only for the purposes of standardization, hut also
for all the spheres of designing. They are wrong.

* The USSR State Standards pertaining to the tolerances and fits for general-
purpose joints, as well as antifriction bearings, are not yet brought in line with
the preferred numbers series.

1.8. Preferred Numbers and Their Vie in Designing

83

The use of preferred numbers series is advisable when it is neces¬
sary to establish a sequence of gradations of some parameter, the
gradations being uniformly distributed throughout the entire range
of the series (e.g., gear ratios in speed and feed boxes of metal-cut¬
ting machine tools).

However, such a uniform distribution of gradations is not always
the most rational. It would be more correct, in principle, to pro¬
ceed from the density of distribution of the frequency of use of a
given parameter.

As an example, the graph in Fig. 13 shows the frequency of use of
gear modules in general mechanical engineering. It is obvious that

Fig. 13. Frequency of use of gear modules in general mechaniclial engineering

90% of all the gears used have their modules lying within the range
of m — 1-5. The maximum application find gears with modules
m = 2-3. Thus, it is reasonable to increase the number of gradations
in the range of the highest frequency of use and reduce it in the re¬
gion of the least popular modules. In other mechanical engineering
branches (instrument-making, heavy machine-building) relations
may he different. In each of such branches it is possible to determine
the density of distribution of the frequency of use and then select
accordingly the standard module gradations. In essence, such a diffe¬
rentiated approach to standardization is also necessary in the case
of other mechanical engineering parameters (sizes of fitting diame¬
ters, threads, etc.).

The series of preferred numbers are inapplicable when developing
unified series of machines with repeatedly used working members.
Parameters of unified series follow other laws, which are dependent
on the feasibility of combining unified members and conditions of
the technical application of the members of the series, and cannot he
arranged in the order of a geometric progression.

6*

84

Chapter 1. Principles of Machine Design

When setting up a parametric series due account must be taken of
the frequency of use of different machine categories, their flexibili¬
ty, etc. Formal application of geometric progressions may lead to
serious errors.

The series of preferred numbers are also inapplicable to the deter¬
mination of the parameters of progressively developed and moder¬
nized machines whose characteristics at each development stage
depend on the technical capabilities and requirements of the corres¬
ponding branches of industry. Thus, the power capacity of heat engines
is dependent on their original parameters (pressure and temperature)
and speed. Not one of these parameters can be arbitrarily increased.
In some cases they have an optimum value (e.g., the degree of com¬
pression in gas turbines), whose changes worsen the machine chara¬
cteristics. Increasing the temperature and speed can only be done on
the basis of technical improvements (increasing the heat resistance
of materials, improving the cooling of thermally stressed parts).
The results of such investigations cannot be kept within preferred
numbers series.

1.9. General Design Rules

When designing a machine, it is recommended to keep to the follo¬
wing basic rules:

design the machine with a view to enhancing the economic effect
which is primarily dependent on the efficiency and durability of the
machine and its maintenance costs throughout the entire service life;

achieve a maximum output by increasing the productive capacity
of the machine and extending its range of operations;

take all measures to lower operational expenditures by decreasing
the power consumption and costs of servicing and repairs;

automate the machine as fully as possible with the aim of increa¬
sing its productivity and the product quality and reducing labour
costs;

increase the machine’s durability as much as possible in order to
enlarge the actual fleet of machines and raise their total output;

minimize the risks of obsolescence by endowing the machine with
high initial parameters and reserves for future development and
improvement;

make the machine capable of being intensively operated by impro¬
ving its reliability and. versatility;

provide possibilities for manufacturing derivatives on the basis
of the original model;

reduce the number of the type-sizes of machines as much as possib¬
le and improve their operational flexibility;

strive to satisfy the needs of the national economy not by produ¬
cing more machines but by increasing their output and durability;

1.9. General Design Rules

85

design the machine so as to assure its repairless operation with
the complete exclusion of capital repairs and reduction of all other
repairs to unit replacement;

avoid the design of rubbing surfaces directly on housings or machi¬
ne bodies; to facilitate repairs of such rubbing surfaces design them
on individual and easily changeable parts;

follow consistently the unit design principle (multi-station machi¬
ne system); construct machine assemblies as self-contained units;

exclude special selection and matching of parts during their assem¬
bly; make them fully interchangeable;

exclude the alignment and adjustment of parts and units during
assembly; provide the necessary locating elements that enable cor¬
rect mounting of the parts and units;

impart, high strength to machine parts and to the machine as a
whole without increasing their weight (use the most rational shapes
providingfor the best utilizationof the properties of materials, employ
high-strength materials, introduce hardening treatment, etc.);

pay particular attention to increasing the cyclic strengths of parts;
use shapes providing for a high fatigue strength; eliminate stress
concentrations; introduce treatment improving the fatigue strength
(induction hardening, nitriding, shot-blast work hardening,
etc.);

in machines, units and mechanisms, operating under cyclic or
dynamic loads, introduce resilient elements which dampen impacts
and load fluctuations;

make constructions highly rigid without increasing their weight
(use hollow and shell-type constructions; avoid deformation with
the aid of diagonal and transverse braces; select rational positions
for supports and stiffening elements);

increase operational reliability in every way possible, thus making
the operation of the machine wholly trouble-free;

make machines as simple to maintain as possible; reduce the scope
of maintenance; eliminate periodic adjustments; make mechanisms
as self-servicing units;

prevent the overloading of the machine during operation; intro¬
duce automatic monitors, protection and limiting devices that should
prevent the machine from working under dangerous conditions;

exclude failures and accidents ensuing from unskilled and careless
attendance;, introduce interlocking devices which should exclude
wrong manipulation of controls; make the machine control as auto¬
matic as possible;

exclude any possibility for an incorrect assembly of machine parts
and units requiring precise coordination with respect to one another;
introduce interlocks which should only permit correct assembly;

eliminate periodic lubrication; ensure an automatic and conti¬
nuous feed of the lubricant to rubbing surfaces;

86

Chapter 1. Principles of Machine Design

. avoid open mechanisms and devices; enclose all mechanisms in
housings which should prevent dust, dirt and moisture from getting
onto rubbing surfaces and allow a continuous lubrication to be ef¬
fected;

. assure a reliable locking of screwed connections to prevent them
from self-loosening, lock internal connections by positive methods
(apply cotters, keys, tab washers, etc.);

prevent corrosion of parts, particularly in machines operating in
the open air or coming into close contact with chemically aggressive
media, by applying chemically stable coatings, in the form of either
paints and varnishes or electroplates, and also by manufacturing
the parts of corrosion-resistant materials;

cut down the cost of the machine by making its design suitable for
industrial production, and also through unification, standardiza¬
tion, decreasing the specific metalwork weight and reducing the num¬
ber of the machine type-sizes;

reduce the weight of the machine by making it more compact and
also by applying rational kinematic and loading schemes, elimina¬
ting unfavourable kinds of loading, changing bending stresses to
tensile or compressive stresses and using light alloys and non-me-
tallic materials;

make the machine design as simple as possible; avoid complex
multi-component constructions;

wherever possible change mechanisms with a rectilinear reciprocal
motion to the more advantageous rotary mechanisms;

assure the maximum suitability of parts, units and the machine
as a whole for industrial production by building into its design
preconditions for the most efficient manufacture and assembly;

reduce machining operations; anticipate the manufacture of parts
from blanks whose shape is close to'that of the finished parts; change
the machining to other more productive methods which do not invol¬
ve metal cutting;

implement the maximum unification of constructional elements
with a view to reducing the cost of the machine, shortening the time
of its manufacture and facilitating its operation and repairs;

broaden the use of standard parts; adhere to the national and
local standards in force;

never use specially designed parts or units when it is possible to
employ standard, unified, adopted and purchased parts and units;

economize on expensive materials and materials in short supply by
replacing them with equally good substitutes; if the use of the mate¬
rials in short supply cannot be avoided, keep their consumption to
the minimum;

cheapen the production but do not limit expenses on the manufac¬
ture of vital parts determining the machine’s durability and reliabi¬
lity; produce such parts from high-grade materials and use modern

1.9, General Design Rales

87

production processes providing for increased reliability and service
life;

impart to tlie machine simple and smooth shapes which should
facilitate keeping it in neat and tidy condition;

pay proper attention to the industrial design requirements, make
the machine well-proportioned and provide good external finishes;

wherever possible concentrate controls in a single place convenen t
for manipulation and observation;

make the machine units and mechanism which require periodic
inspection easily accessible;

ensure safety of attending personnel; preclude accidents by making
the operation of the machine as fully automatic as possible, by intro¬
ducing interlocks, enclosing mechanisms and by providing guards;

in machine-tools and automatic machines provide devices that
should enable the adjustment and setting-up to he carried out while
driving the machine manually; provide the possibility for the machi¬
ne to be slowly driven from the drive motor (with reverse, when
required for the machine setting-up);

in machines having electric motor drives consider the possibilities
of wrong switchings, while in those driven from internal combustion
engines take precautions to avoid the back-firing of the engine; pro¬
vide reversible drives or introduce safety devices (e.g., overrunning
■clutches);

carefully study the experience gained in running the machines;
eliminate without delay all defects discovered during use; remember,
that studies into the operation of machines are the best means of
improving and refining them and an effective method of improving
the skill of the designer;

continuously improve the design of machines which are being
produced on a mass scale so as to keep them in line with the ever¬
growing industrial demands;

. anticipate new designs, thus preparing in advance for the produc¬
tion of new, more perfect machines to replace obsolete models;

when developing new designs, as well as machines intended for new
technological processes, check all new elements by experiment, simu¬
lated processes, test runs, etc.;

make wider use of known designs; use the experience gained in
the allied and, if necessary, more remote branches of mechanical
engineering.

Chapter 2

Design methods

The starting material for designing may be the following:

a technical assignment worked out by a planning organization
or customer; such an assignment generally specifies the required
parameters of the machine, and the field and conditions of its use;

a technical suggestion initiated by a designing organization or a
group of designers;

a research work or an experimental model based on the re¬
search;

an invention proposition or an invention prototype;

a machine prototype of foreign make which has to be reproduced
with modifications and alterations.

The first case is more general and most convenient to following
the design processes.

It is necessary to approach any technical assignment critically.
The designer must well know the branch of industry for which the
machine is intended. He must verify the assignment and, when ne¬
cessary, prove the need for corrections.

A critical approach is particularly necessary when the customer is
an individual factory or branch of industry. In the latter case the
designer must uot only satisfy the customer’s requirements but also
consider that it is possible for the machine to be used at other facto¬
ries and in the allied branches of industry.

A circumstance not always considered is the fact that from the
moment of the inception of designing to the introduction of the
machine into production a time period elapses, which is the longer
the more complex the machine. This period includes the following
stages: design, manufacture, finishing and adjustment of the proto¬
type, industrial testing, introduction of necessary alterations to
eliminate defects found when testing, State Acceptance Tests and,
finally, acceptance of the prototype. This is followed by the prepa¬
ration of technical documents for the initial production series, its
manufacture, and industrial tests, after which documents for mass
production are prepared, the production processes modified to suit
mass production, and then the production is started.

2.1. Design Succession

89

Under the most favourable conditions when there are no large
troubles or complications this process takes about a year and a half
or two years. Sometimes from the beginning of designing to the
beginning of mass production two or three, or even more years elap¬
se. With the modern tempo of the technological progress in the me¬
chanical engineering this seems a long time.

Machines with incorrectly chosen, underestimated parameters,
based on trite solutions, not assuring technical progress and incom¬
patible with new ideas about the import of quality, reliability and
durability, become obsolete even before the initiation of mass pro¬
duction. Then, the whole time spent on designing, manufacture and
tests will be vain and the industry will still lack the machine it
needs.

2.1. Design Succession

Succession in design implies the use during designing of the pre¬
vious experience gained in the given branch of mechanical engine¬
ering, as well as in the allied branches, and introduction into the
projected unit all the useful features of the existing machines*.

Nearly every modern machine is the result of the work of several
generations of designers. The original model of a machine is gradual¬
ly improved, equipped with new units and mechanisms, and made
more perfect as a result of new constructional solutions, fruits of the
creative efforts of many people. Some design solutions die due to the
development of more rational ones, new technological procedures or
higher operational demands, while some, proved exceptional, remain
in force for very long periods, although at times being slightly modi¬
fied.

In the course of time, machine specifications, both technical and
economic, are made, higher, their power and productive capacities
are enhanced, automation is more extensively introduced, reliability
and durability are bettered. Alongside the improvements to existing
machines new machines of identical purpose are devised, which are
principally based on other design schemes. In competition more pro¬
gressive and viable designs win.

When reviewing the history of mechanical engineering it is seen
that a huge number of miscellaneous design schemes were tried be¬
fore. Many of them disappeared and became completely forgotten,
were revived after tens of years on a new technological basis, and
again returned to use. Thus historical reviews help the designer to
avoid errors and the repetitions of the stages traversed, and allow
him to choose more prospective methods of machine development.

Some authors use the design succession concept to denote the general ten¬
dency of typification, unification, development of standard series, etc. Such,
use of the term does not correspond to the sense of its content.

50

Chapter 2. Design Methods

It is useful to plot graphs showing the changes of basic machine
parameters over the years. Tendencies of constructional forms are
well expressed by graphs showing as percentages the occurrence of
various designs. An analysis of such graphs and their extrapolation
make it possible for a clear picture to be drawn of future machine
parameters and design trends.

Of particular importance is the study of the starting data when
working on a new design. Here the main task is to choose machine
parameters correctly. Partial design errors can be eliminated during
machine manufacture and tests, but errors in basic parameters are
often irrepairable and may lead to failure of the design. Therefore,
at this stage neither time nor effort on investigations should be
.spared. Here, more than anywhere, the proverb “look before you
leap” is valid.

The selection of parameters must be preceded by a comprehensive
study of all factors determining the machine’s life. It is necessary to
•carefully study the experience gained in the operation of foreign
and domestic machines, correctly analyse their advantages and disad¬
vantages, choose a correct prototype, and make clear the tendencies
•of the development and requirements of the branch of engineering
for which the machine is intended.

An important prerequisite for correct design is the availability
of pertinent references on constructional materials. Apart from archi¬
ves of their own products, design organizations should have at their
•disposal design catalogues from neighbouring organizations.

It is necessary to undertake systematic studies of patents and
periodicals, both foreign and domestic.

Owing to the general tendency of capitalist firms to keep their
novel technicalities secret, publications are often camouflaged.
Therefore the designer must be able to read between the lines because
sometimes a brief information may conceal an important innovation
in a given branch of engineering.

In his particular sphere the designer during the course of his
searches must process the scientific investigations of research insti¬
tutions.

In addition to the researches carried out in its own branch, the
design organization should make wide use of the experiences of other,
sometimes rather alien branches of mechanical engineering. This
widens the outlook of the designer and enriches his store of con¬
structional means and devices.

It is especially useful to study the experience of advanced branches
of engineering, in which the design and technological concepts,
forced by the high demands for the product quality (aviation) and
mass production (automotive industry), continuously develop new
designs, methods of increasing strength, reliability and durability,
and better production techniques.

2.2. Study of Machine Application Field

91

The use of the experience already gained often helps solve partial
problems arising during designing. Sometimes the designer wrestles
■over the development of a specialized unit or assembly new for
the given machine, but which has long ago been developed and
proved in other branches of engineering.

The concept of design succession does not mean limitation of
initiative. The designing of any machine presents an unlimited field
•of activity to the designer. He must not invent anything already
invented and must not forget the rule formulated early in the 20th
century by Giildner: “weniger erfinden , mehr konstruirer.i' (invent less,
design more).

The process of continuous machine improvement, caused by the
ever-growing demands of the industry and national economy, results
in the development of the design school and the cast of the design
thought. The striving for a perfect design penetrates the flesh and
blood of the designer, becomes his life. A genuine designer is per¬
petually charged with vigour to wrestle with any difficulties. He
will feel fully satisfied only when his numerous relentless efforts,
failures and errors are finally crowned with the finding of a most
perfect solution promoting the progress in a given branch of engi¬
neering.

The designer must constantly improve and enrich his store of
design solutions. An experienced designer will invariably note and
mentally “snapshot” interesting solutions, even in machines “foreign”
to him.

The designer must always be aware of all the latest technological
processes, including physical, electrophysical and electrochemical
methods of processing (spark-erosion, electron-beam, laser, ultrason¬
ic, dimensional electrochemical etching, processing with blast,
■electrohydraulic impact, electromagnetic impulses, etc.). Otherwise
he will he rather restricted in his choice of the most rational shapes
for parts and unable to build into the design the conditions for
productive manufacture.

2.2. Study of Machine Application Field

The progress in mechanical engineering is inextricably linked
with the development of the different branches of industry and
national economy, appearing as the machine users. Industrial
development is accomplished by continuous improvements, includ¬
ing: growth of productivity; decrease of operational cycle; develop¬
ment of new technological processes; rearrangement of production
lines; change in equipment layout; continuous expansion of mechan¬
ization and automation of production. This accordingly results in
higher requirements for machine parameters, their productivity
and level of automation. As new technological processes come into

92

Chapter 2. Design Methods

being, some machines become obsolete. The necessity arises for new
machines or for modification of the old.

Occasionally such modifications may be very extensive and affect
large categories of machines. Thus, the introduction of the continu¬
ous steel casting process means the death or, in any case, reduction
in the manufacture of such complex and large machines as blooming
and slabbing mills. Production of steel in oxygen blast converters
will reduce the use of open-hearth furnaces unless they are radically
improved. The introduction of magnetogasdynamic generators,
enabling thermal energy to be directly converted into electrical
energy, will cause die-out of electric generators and considerable
reductions in the employment of heat engines.

Before designing machines intended for a specific branch of indu¬
stry, it is necessary to examine carefully the dynamics of the quan¬
titative and qualitative development of that particular branch,
the requirements for the given machine category and the possibility
of the development of new technological processes and new methods
of production.

The designer must be well acquainted with the particular charac¬
teristics of that branch of industry and the working conditions of
the machine. The best designers, from the author’s observations,
are those who have been practical engineers and combine their design
aptitudes with knowledge of running conditions for the machines
being designed.

When choosing machine parameters it is necessary to consider
actual working conditions. For instance, the productivity of a ma¬
chine must not be arbitrarily increased without considering production
requirements for which the machine is intended and without con¬
sidering the needs of adjacent machines. Under certain conditions
machines with higher productivity may operate underloaded and
be more idle than at work. This lowers the degree of their use and
lessens the economic advantages.

2.3. Choice of Design

When choosing the parameters of a machine, its basic scheme and
type of construction, attention must be given to the factors deter¬
mining its economic efficiency, i.e., high output, low energy con¬
sumption, low maintenance and operation expenditures, long service.

Generally, the final machine version is chosen from several, which
have been carefully evaluated and compared from all sides of con¬
structional expediences, perfection of kinematic and force schemes,
cost of manufacture, energy consumption, cost of labour, reliable ope¬
ration, size, metal content, weight, suitability for industrial produc¬
tion, unitizing, servicing, assembly-disassembly, inspection, sett¬
ing and adjustment.

2.4, Development of Design Versions

93

It must be made clear to what degree the original design assures
future development, forced working and subsequent improvement,
forming on its basis of machines derivatives and modified versions.

Not always, even during the most careful search, is a successful
solution found, which fully answers the posed demands. Exceptional
in constructional practice is a variant being a success in all res¬
pects. The matter is sometimes not the lack of inventiveness, but
rather conflicting requirements being imposed. Under such cir¬
cumstances it is necessary to find a compromise solution, waiving
some requirements which are not of prime importance in the given
conditions of the machine application. Often a variant is chosen not
because it incorporates more advantages, but rather because it has
less disadvantages in comparison with others.

After selecting the scheme and basic parameters of the machine,
a sketch is made, and then a general arrangement drawing, on the
basis of which are composed the initial, technical and working
designs.

2.4. Development of Design Versions

The development of design versions is not a matter of individual
habits or leanings of the designer, but a regular design method aimed
at seeking the most rational solution.

As an example of the development and comparative analysis of
design versions, let us consider a bevel reduction gear unit (Fig. 14)
which is very often used in mechanical engineering.

To simplify the example we will not consider constructional
variants of power input and take-off, types of supports, methods of
fixing the axial position of gears. We shall examine only the working
arrangement of the drives, casing construction and spacing of sup¬
ports.

Widely used in practice are reduction gear units which have can¬
tilevered gears mounted inside a common casing (Fig. 14a). The
popularity of this design is due to its undoubted advantages. Gear
shafts are located inside a single casing enabling a highly accurate
mutual positioning of gears to be attained during manufacture. Free
access is effected through an inspection hole provided with a detach¬
able cover. The entire mechanism can easily be inspected when
assembled.

Gear mesh is controlled by spacing washers m whose installation
requires the disassembly of the gear mounting unit.

Generally, the reduction gear unit is held in position by lugs.
For assembly purposes the outer diameter of the pinion must be less
than that of the holes holding the shaft bearings.

In another alternative (Fig. 14 b) the shaft bearings are mounted
in intermediate sleeves, thus enabling the pinion diameter, admis¬
sible from the standpoint of assembly conditions, to be slightly

Fig. 14. Angle drive design versions

•96

Chapter 2. Design Methods

increased. This facilitates the adjustment of the mesh, as in this
case it is not necessary to dismantle each gear, but merely change
spacing washers m placed under the flanges of the intermediate
sleeves. This is particularly easy if the spacer washers mare made
as half-rings locked in place by bolts (Fig. 14b, lower part). To
•change the washers the bolts are unscrewed and the intermediate
sleeve shifted to suit the new washer.

In the construction illustrated in Fig. 14c the gears are installed
inside detachable casings. This version preserves the advantages
of Fig. 14b but the rigidity of the main casing is worse. This means
that the alignment of centering shoulders and seating surfaces of
bearings must be kept to extremely close tolerances when manufac¬
turing the casing. According to its design features this reduction
unit is more suitable for suspension-type mounting, although it
can also be mounted on feet (cast integral with the lower
cover).

The design is worse, if the gear shank is extended upward
(Fig. 14 d). Now the mechanism cannot be inspected when assembled
and the removal of the gear casing disturbs the entire unit.

Gear mesh can be controlled only by blacking-in, which means
dismantling and refitting the gear many times. In order to inspect
the reduction unit interior it is necessary to disconnect the gear
drive shaft.

In the design shown in Fig. 14e the gears are installed inside
a split casing (the split line runs along the pinion axis). The design
is simple to assemble and easy to inspect. Inspection of gear engage¬
ment can only be complete when the pinion shaft in assembly with
the bearings is pressed against the lower bearing seatings.

Machining of a split casing is more difficult. First, it is necessary
to finely machine the joint surfaces, connect two halves by locating
dowels and machine the bearing seats after assembly. The joint
surfaces must be lapped. Joint seals cannot be used, as their use
spoils the true cylindrical shape of the bearing seats.

The remaining variants have the aim of reducing overall size
by bringing the supports into the casing interior.

In the design in Fig. 14/ the gear shaft is mounted in two sup¬
ports, one of which is inside the casing, and the other, in the cover.
Here the distance between the supports is increased and bearing
loads are less than in the cantilever design. The chief disadvantage
of this construction is the difficulty of inspection and adjustment
of the mechanism. When casing cover is removed, the gear shaft
is held by the lower support only and because of this freedom the
mesh accuracy cannot be checked. Furthermore, since the bearings
are located in different casing sections, the shaft alignment is worse.
The necessity of machining the bearing seats in the assembly of the
casing and cover complicates the process technology.

2.4. Development of Design Versions

97

The more compact design illustrated in Fig. 14 g proves unsatis¬
factory when examined closely. Here it is practically impossible
to inspect the mechanism. The removal of the gear is only possible
after the cover and pinion have been dismantled, consequently,
the mechanism must be fully disassembled.

A more preferable design is shown in Fig. 14 h, in which the gear
shaft is mounted inside the casing. Here the entire mechanism is
accessible through the bottom cover. The design is suitable only
when suspension-mounted and cannot be used if installed upon
a stand by its bottom, as it would be necessary to detach the entire
reduction unit in order to inspect it. The advantage of this design
is the possibility of taking power from the upper end of the gear
shaft for which it is sufficient to extend the shaft through the upper
cover.

A similar design is given in Fig. 14i. To inspect the mechanism
it is necessary to disconnect the gear drive. The meshing portion
is viewed from the end face.

The size of the transmission can be decreased by bringing one
of the pinion shaft bearings into the casing (Fig. 14;). The bearing
is installed in partition n cast integral with the casing sides. This
design is suitable for both suspension and installation by its base.
The mechanism is inspected through the non-bearing cover; the
drive shafts remain connected. The disadvantage of this design
is in that the inspection of the meshing portion, screened by par¬
tition n, is rather difficult.

In the alternative illustrated in Fig. 14 k the internal pinion
shaft bearing is transferred to tbe opposite casing wall. The design
features well-spaced supports and convenience of the mechanism
inspection obviating the necessity to disconnect the drive shafts.
Power take-off from the pinion shaft is possible.

The disadvantage: the gears cannot be dismantled separately;
to dismantle the gear it is necessary first to dismantle the
pinion.

From the dimensional viewpoint the most preferable designs are
those in which the gear and pinion bearings are accommodated inside
the casing (Fig. 14Z, o).

In the design shown in Fig. 14 1 the bearings of the gear shaft
are mounted in an integrally cast hub. The inside pinion bearing
is accommodated in the side hole of the hub. The gear is mounted
through the lower non-bearing cover, which is also an access for
inspecting the mechanism. Gear mesh is viewed from the end face.
This design is only suitable for suspension mounting.

Figure 14m shows another type of reduction unit with the output
end of the gear shaft upward. The unit is mounted by its lower sur¬
face.

7-01395

98

Chapter 2. Design Methods

The disadvantage of the alternatives in Fig. 14Z, m is that the gear
drive shaft must be disconnected before the inspection cover can
be removed.

The variant illustrated in Fig. I4n enables inspection without
the disconnection of the drive.

Another alternative, featuring reduced casing height and a large-
size moulded lid, contributing to easier inspection is shown in
Fig. 14o.

The final choice of a reduction unit design depends on its installa¬
tion and application. For general use the best designs are those in
Fig. 14a, c. If it is necessary to reduce size and weight, the compact
designs shown in Fig, 14 l-o are preferable.

2.5. Method of Inversion

Among the large number of methods devised to facilitate the
design there is the method of inversion. Its essence consists in invert¬
ing the function, shape and position of parts. In assemblies it is
occasionally preferable to invert the functions of parts, e.g., to
turn a driving element into a driven one, to turn a leader into a fol¬
lower, to make a female component out of a male one, to convert
a stationary part into a movable element, etc.

Sometimes it becomes advisable to invert shapes of parts, e.g.,
to change a female taper to a male one, a convex spherical surface
to a concave one. In other eases it may be more suitable to shift
some elements from one part to another, for instance, to move a key
from a shaft to a boss, or a striker from a lever to a pusher.

Each time the design will acquire new qualities. The designer
must carefully evaluate all the advantages and drawbacks of the
original and inverted alternatives, taking into account durability,
size, suitability for industrial production and convenience of ope¬
ration, and then choose the most suitable version.

The method of inversion is an inalienable instrument of thought
for every experienced designer. It significantly lightens the search
for solutions as a result of which a rational construction is found.

Some typical examples of inversion in mechanical engineering
are shown in Fig. 15.

Rod drive (Fig. 15a). In sketch I the rod is actuated by a fork
lever acting through the agency of the rod pin. In the inverted design
II the pin is transferred to the lever. This results in a reduced trans¬
verse force acting on the rod. However, the design shown in sketch I
has smaller sizes.

Tappet drive (Fig. 15b). In design I the striker of the rocker arm
is flat and the tappet plate—spherical. In the inverted scheme II
the rocker arm is spherical and the tappet plate—flat. The result
is reduced transverse forces acting upon the tappet. Moreover, the

2.5. Method of Inversion

99

striker can be made cylindrical, thus assuring a linear contact in
the joint, whereas in design I the contact is point.

Connecting rod and gudgeon pin (Fig. 15c). In scheme I the gud¬
geon pin is secured to the connecting rod and runs in the fork bear¬
ings; in the inverted design II the pin is fixed to the fork and the
bearing is in the connecting rod. In this case the inversion reduces
overall dimensions and lightens the work of the bearing acquiring
greater rigidity.

Nipple unions. In sketch I (Fig. 15 d) the nipple is tightened by
an internal captive nut, but in the inverted version II an external
nut is used. As a result, axial dimensions tare decreased, and the
radial ones increased. The alternative shown in sketch II is more
convenient to tighten.

In the nipple union presented in Fig. 15e the female taper on
the nipple (version I) has been changed to a male taper (version II),
thus reducing the overall dimensions.

Spherical pipe joint (Fig. 15/). The spherical surface of the joint
(version I) can be substituted by two such surfaces (version II);
advantages: smaller dimensions and lighter weight. However, the
manufacture of such a joint is more complicated, because it is neces¬
sary to make the two spherical surfaces truly concentric.

Fixing a turbine blade (Fig. 15g). Changing the female fork blade
fixing on the T-shaped male tenon of the rotor (scheme I) to a female
tenon in the rotor and male tenon on the blade (scheme II) lessens
the weight, gives a greater rigidity and eases the manufacture of
the blade base.

By-pass valve (Fig. Ibh). Better precision is obtained if the valve
is piloted not by a rod press-fitted into the valve body (scheme I),
but By a shank sliding in the valve body hole (scheme II). This is
because the pilot hole and valve seat can be machined in one setting
and with better concentricity.

Self-aligning thrust bearing (Fig. 15i). In the bearing 1 shown in
sketch I the thrust journal runs on a spherical surface, and in
sketch II —on flat surface, which is more advisable. The two schemes
differ also in the position of self-alignment centres.

Rocker drive-(Fig. 15/). Transferring the spherical surface from
the rod (schemb I) to the striker (scheme II) improves lubrication,,
as the oil from the mechanism’s interior will collect in the cup¬
shaped tip of the rod. In version I penetration of oil into the joint
is almost excluded.

Vee way (Fig. ibk). The design given in sketch II is more advan¬
tageous as far as lubrication is concerned. The groove will retain
lubricant, thus adding to the machine’s durability and accuracy
of movement.

Spring (Fig. 151). In scheme II the tension spring is changed
to a compression spring with a reverse. The compression spring is

7*

102

Chapter 2. Design Methods

Pig. 15. Inversion o! units (cont'd)

stronger and longer lasting than the tension spring in which
the tug ends are often stretched. The inverted design is more reli¬
able, but is more complex and heavy than the conventional design

' S °r«nihuckle (Fig. 15m). This can he made in the form of an exter¬
nally threaded rod (version I) or as a threaded sleeve (version n )-
In the latter case the overall dimensions are reduced by the hexa¬
hedron length which in scheme I is longer.

Stud securing boss (Fig. I5n). Version II is more advantageous
in terms of the screwed joint strength since the boss of enhanced,
compliance distributes thread loads more uniformly.

Guide key (Fig. l5o). When secured in a hub (scheme II), the
key will make the design more practical. The difficulty of mounting
a loug key is avoided since it is easier to make a precision slot m
the shaft than fit a long key to the same tolerances.

Lifting screw (Fig. I5p). It is more advantageous technically
to cut a long thread on a rod (version II) than in a nut (version I).
With the same thread diameter and connection length, the rod shown
in the second version will be stronger.

Idle gear (Fig. 15§). The arrangement presented in sketch 11
is better as far as dimensions and manufacturing conditions are

3.6. Composition Methods

103

■Concerned. Thus, bearing operating conditions are improved as the
.supports are more rigid. Furthermore, when the gear is installed
according to mode I, the shaft, affected by the drive, will be sub¬
jected to a cyclic load, while in mode II this will be a static load.

Should the gear be mounted in antifriction bearings (Fig. 15r)
then one must bear in mind that the bearings’ durability in scheme II
(in which the outer race is rotating) will be worse than that of the
bearings in scheme I (in which the inner race runs).

Slider (Fig. 15s). In the first version slider 1 slides along a sta¬
tionary rod, whereas in the inverted scheme II the slider is fixed
to the rod which moves in its guideways. This inversion gives improv¬
ed directional movement.

Splined coupling (Fig. 15 1). Scheme II is better than scheme I
in terms of axial dimensions and manufacture (internal splines are
usually broached).

Actuation of rod by a disk master-form (Fig. 15w). In version I
the rod is actuated by the two rollers 1 mounted on the rod, which
are rolling over master-form 2; in version II —by one roller 3 posit¬
ioned between two master-forms 4. The latter construction is more
compact and eliminates the drive backlash as the clearance between
the master-forms and roller can be adjusted.

Suspended hook trolley (Fig. 15u). In scheme I the trolley has
two pairs of rollers (the Figure shows only one pair) rolling on a
monorail. In the inverted scheme II one pair of rollers travel bet¬
ween two rails. The second scheme is more compact, but requires
two rails.

Hydraulic cylinder (Fig. 15 w). In design I the cylinder is station¬
ary, and its piston and drive rod move. In the inverted design II
the piston and rod remain stationary, and the cylinder fitted
with a driving fork moves over them. The advantage of scheme .//:
•drive can be effected from any point along tbe cylinder.

2.6. Composition Methods

Generally composition has two stages: draft and operational.
During the draft composition the main arrangement and general
■design of a given unit are elaborated (sometimes several versions).
After discussing the draft the working scheme is composed, defining
more accurately the design of the unit and serving as the initial
material for furthering the project.

During composition it is important to separate the main items
from the secondary and, accordingly, establish the correct order
of development and construction.

An attempt to compose all elements of a design at a time is a
typical error characteristic of young designers. Having received
the assignment presenting fully the purposes and parameters of

104

Chapter 2. Design Methods

the projected design, the designer often begins at once to draw the
complete design in all its details, depicts all the design elements
to compose a picture as if it were an assembly drawing, a technical
or working specification of the project. Such a method makes the
design an irrational one, as it forms a mechanical string of construc¬
tional elements and units, poorly arranged as is well known.

It is necessary to begin the composition with the solution of the
main design problems, i.e., selection of rational kinematic and
power schemes and correct sizes and shapes of parts and determin¬
ation of their most suitable relative positions. Any attempt to fully
describe parts at this stage of design is not only useless, but harmful
as it draws the attention from the main problems of the composition
and confuses the logical course of developing the design.

Another fundamental rule of composition—parallel development
of several design alternatives, their analysis and selection of the
best version. It is a mistake if the designer at once sets the direction
of designing by choosing the first type of design which occurred to
him, or a commonplace alternative. The designer must analyse
carefully all feasible versions and choose the one most suitable for
the given conditions. This requires much work and the problem is
not at once solved, and sometimes only after long investiga¬
tions.

Full development of each alternative is not necessary. Usually
hand pencil sketches are sufficient to get an idea of the prospects
of the variant and decide on the question of whether it is advisable
to continue with this particular alternative.

Composition must always be accompanied with calculations, if
only tentative approximations. Main design components should
be calculated not only for strength, but also for rigidity.

Never rely on the eye when selecting dimensions and shape of
parts. Of course, there are very skillful designers who almost with¬
out mistakes can establish sizes and cross-sections assuring stress
levels acceptable for the given branch of engineering, but this is
a doubtful virtue. Copying trite shapes and keeping to traditional
stress levels, one cannot create progressive designs.

To wholly depend upon calculations is wrong. In the first place
the existing techniques of strength calculations do not consider
many factors influencing the fitness of the design and secondly,
there are some parts (e.g., housings) which cannot be calculated
at all. Thirdly, other factors besides strength affect the sizes of
parts. For example, the design of cast parts in the first place depends
on casting technology. For parts being mechanically machined it
is necessary to consider their resistance to cutting forces and give
them a certain rigidity. Heat treated parts should be large enough
to avoid buckling. Dimensions of controls components should be
chosen so as to ensure manipulation convenience.

2.6. Composition Methods

105

Thus, alongside calculations, the designer must refer to existing
designs and introduce, if necessary, well-founded corrections.

Another prerequisite for correct designing is a continuous con¬
sideration of the manufacturing problems; at the very beginning
the component should be given a technologically rational shape.
A skilled designer composing a part at once makes it easy to produce.
Young designers should constantly consult production engineers.

Composition must be carried out on the basis of standard dimen¬
sions (fitting diameters, sizes of keyed and splined connections,
diameters of threads, etc.). This is particularly important when
composing units with several concentric fitting surfaces, as well
as stepped parts, whose shape to a large extent depends on the gra¬
dation of diameters.

At the same time maximum unification of standard elements
should be sought for. Chosen elements unavoidable in the design
of main parts and units should be utilized in the remaining parts
of the design as much as possible.

In the process of composition the designer must take into account
all the conditions defining the operability of the machine, develop
the systems of lubrication and cooling, assembly and disassembly,
and attachment of adjacent parts (drive shafts, pipings, electric
wires, etc.); provide for convenient maintenance, inspection and
adjustment of the mechanism; choose correct materials for the main
components; think out methods for improving the machine's dura¬
bility and wear-resistance of rubbing surfaces, and methods of
corrosion protection; investigate and determine the limits of the
machine when operating under forced conditions.

Composition does not always proceed smoothly. Often during
designing some small defects, overlooked in the first estimates, are
revealed. For their elimination it sometimes turns out necessary
to return to schemes rejected earlier or to develop new ones.

Some units are not always successfully designed from the first.
This should not discourage the designer—he has to devise some
“tentative” alternatives and raise the design to the required level
in the process of further activities. In such cases it is useful, as
Italians say “dare al tempo il tempo ” (to give time to time), i.e.,
to take a breathing space, after which, as a result of subconscious
thought, the problem is often solved. After a while the designer
looks at the outline drawing in another light, and sees the mistakes
made at the first stage of the development of the main design idea.

Sometimes the designer unintentionally' loses his objectiveness
and does not see the drawbacks of his favourite variant or the poten¬
tialities of other versions. In such cases impartial opinions of out¬
siders, the advice of elders and co-workers, and even their fault¬
finding criticism turn out to be most opportune. Moreover, the sharper
the criticism, the greater is the benefit derived by the designer.

106

Chapter 2. Design Methods

At all stages of the composition manufacturers and operators
should be consulted.

A general rule is such: the wider the discussion of the composi¬
tion and the more attention the designer pays to useful advice, the
better becomes the composition and the more perfect the design.

Do not spare time or effort on working up the design-. The cost
of designing is only a small portion of the machine manufacturing
expenditures (excluding machines produced- individually or on
a small-batch production basis). In the final analysis, the greater
the development work on the design, the more are the savings in the
machine cost, time of manufacture and finishing, the better its
-quality and the greater the economic gains over the machine’s
.service life.

2.7, Composition Procedures

If the overall dimensions of a project allow, its composition is
best of all drawn to a 1 : 1 scale. This facilitates selection of required
dimensions and sections and enables a realistic presentation of machine
part proportions, their strength and rigidity, to he individually
and as a whole obtained. In addition such a scale obviates the neces¬
sity for a large number of dimensional specifications and simplifies
subsequent stages of design, in particular detail drawings, since
such dimensions can be taken directly from the drawing,

Tracing to a reduced scale, .particularly at reductions exceeding
1 : 2, strongly obstructs the composition process, distorts proportions
and clarity of the drawing representation.

If the dimensions of design deny the use of a 1 : 1 scale, then at
least individual units and groups should be composed to natural size.

Composition of simple components may be developed in one
projection if the drawing explains it sufficiently well and clearly.
The form of the construction in a cross-sectional plane will be filled
by a three-dimensional imagination. However, when applied to
more sophisticated compositions, this method may cause serious
errors; therefore, the design must be developed in all necessary
projections and cross sections.

The fulfilment of an arrangement drawing is, a continuous process
of research, trial, approximation, development of alternatives, their
comparison and rejection of unsuitable versions. Sketches should
be lightly traced with a pencil during the composition corrections,
follow one another, which means that a rubber is applied more
often than a pencil.

Gross sections can be left unhatched, but if hatched, then only
by hand. Do not waste time on drawing parts. Typical components
and units (fasteners, packing, springs, antifriction bearings, etc.)
should be simply depicted, examples of this are shown in Fig. 16.

2.8. Design Example

107

Contour outlining, hatching, provision of a legend and particulars
of small parts are made at the final composition stage, when the
arrangement is ready for discussion.

There is a school for making hand-drawn compositions, the design
is drawn with a pencil on a sheet of squared paper. The author
always used this method, and considered that such composition
had great advantages as to capacity, flexibility and easiness of

Fig. 16.' Simplified sketches of common parts as used on general arrangement

drawings

introducing corrections. The method almost practically excludes
any errors in connecting dimensions and ensures comprehension
•of the part sizes.

This method is especially useful for showing smooth outlines
■characteristic of modern designs.

The method is convenient for designers having certain aptitudes
for drawing. Some designers are capable, when applying this method,
of preparing in a few hours complete arrangements, which can be
handed over for detailing.

2.8. Design Example

To illustrate the technique of composition let us consider the
design of a centrifugal water pump. This device has certain specific
features, which influence the techniques and sequence of compo¬
sition. In the case under consideration there will be rather a stable
original basis in the form of a sketch, prepared by the draft design
department, describing the hydraulic parts of the pump. It remains

108

Chapter 2. Design Methods

for the designer to embody the draft design in metal. Very often
the designer is given only a most general scheme of the machine
without basic sizes. Sometimes the designer begins his work with
knowledge of only the purpose and specifications of the machine
without even a scheme of its future construction. Under such cm
cumstances he has to begin with thinking over the main concept
of the design and investigate versions of design schemes. Only after
this does the composition take shape.

The technique of composition described below is not the only
one possible. Like any other creative process the process of arran¬
gement composition is subject to and in many instances is largely
dependent on the experience, skill and aptitudes of the designer.
There may be different ways in composition, sequence of developing
the design and also solutions to the problems occurring during
design. The technique, illustrated below, is for example only
and is intended to illustrate the basic regularities to this or that
degree inherent in any composition process. These are:

sequence of development, revelation at the first design stage
of the basic constructional elements ignoring details of construc¬
tion;

consideration of several alternatives, choosing the best on the
basis of comparison of technology and operational practice;

parallel with designing, tentative calculations for strength, rigi¬
dity and durability;

building-in at the first composition stages machine reserves for
future development and the limits of its operation in a forced regime;

making the design suitable for industrial production, implementing
successively unification and standardization;

developing practicable assembly and disassembly schemes;
careful examination of machine parts and units for durability
and operational reliability.

In the next example the results of each composition stage are
given as separate drawings. A young designer may have the impres¬
sion that the composition process consists of the successive prepa¬
ration of such drawings. In actual fact, it is one and the very same
composition drawing which is continuously being amplified and
clarified as the development proceeds until it gains finality.

In the interests of better representation in the drawings below
small details are presented (not always) in full. In reality in the
composition process they are represented simply by conventional
symbols and sometimes not at all.

In the book a full account of the variants has to be drawn so
that detailed explanation may be given when comparing the advan¬
tages and disadvantages of the different constructional solutions.
During design the greater part of the variants are thoughtfully
compared by the designer who at once rejects unsuitable solutions

2.8. Design Example

109

and only sometimes sketches by hand other variants without even
observing scales.

Sometimes the designer cannot explain why he chose or rejected
one of the design solutions and only limits his remarks by laconically
saying: “I like (or dislike) it”. Thus what appears to be tasteful
justification to an experienced designer, in fact, conceals an intui¬
tively correct appraisal of the instructional, operational and other
complications, which he carries out when rejecting certain lines
of thought.

However, the process of comparison and selection of alternate
designs progresses much more rapidly than it may seem from the
descriptions and illustrations given below.

Much time has to be spent on the solution of complex or new design
problems which may occur in a designer’s work requiring creative
work searches for similar supporting examples from various engi¬
neering branches and sometimes the organization of experiments,
which, depending on the allotted design time, are carried out
quickly or to the dictated circumstances.

(a) Initial Data

The starting material for designing the pump is a sketch of the
pump hydraulic cavity, with basic dimensions (Fig. 17). This must

be a single-stage pump with an axial inlet and cantilevered impeller.
In the intake pipe is fitted a guiding device assuring axial flow of
the water stream to the impeller.

The pump is driven by an induction motor (n = 2950 rpm).
The impeller peripheral speed is 35.5 m/s, the design head, 50 m

110

Chapter 2. Design Methods

WG, and the pump delivery, 40 1/s. The pump has two symmetri¬
cally arranged discharge pipes, each 40 cm 2 in cross-sectional
area.

The calculations supplied with the sketch give the number and
arrangement of vanes (eight vanes curved to suit direction of impeller
rotation), the profile of the impeller flow portion, cross sections
of discharge volutes at circumferential angular positions. The pump-
service life is preset 10 years of operation to a double-shift schedule).

The calculated pump durability is equal to the product of the
pump service life in hours and the shift and off-days factors (no
factor for repair downtimes is introduced as a repairless operation
is assumed). Hence,

~ shift ' off-d ,' H

where H — 10 -365 -24 = 87 600 h is the nominal service life;

Ps/ii/t — shift factor (»0.6 for double-shift operation);

— 0.8 is the off-days factor

The calculated durability

ft —0.6-0.8-87600 » 40 000 h

(b) Shaft Supports

It is advisable to begin with selection of the type, dimensions
and arrangement of impeller shaft supports. The latter are assumed
to be ball bearings which advantageously differ from plain bearings,
as the lubrication of the former is much simpler.

The radial load upon the bearings is established from the weights
of the impeller and shaft and the centrifugal load due to incomplete
static balance of the impeller. In addition, the supports withstand
the axial pressure from the operating fluid on the impeller.

On the basis of the preliminary rough estimates assume that
impeller weight Gi mp — 4 kgf (with a margin), the weight of shaft
and connected parts (hearing inner races, drive flange, clamping
nuts) G shajt - 2 kgf.

The unbalanced centrifugal force of the impeller may be found
from the magnitude of the static unbalance. Assume that the value
of the static unbalance at the impeller periphery is 5 kgf. Then, the
maximum possible unbalanced centrifugal force will be

dig :

0.005co 2 i?

0.005-3102.0.115

9.81

9.81

-5.5 kgf

The maximum radial force acting upon the impeller in the plane
of its centre of gravity will be

P = Gimp + Pctfg ==4 + 5.5 ==9.5 kgf

2.8. Design Example

ill

The load Ri on the bearing nearest to the impeller is

Ri = P( l+x) (2- 1 )

■where l — distance from the impeller centre of gravity to the front
support;

L = distance between the supports

The load i? a on the second bearing will be

i? 2 = i? 1 _p = pJ_ (2.2)

As is well known the best ratio Lil lies within the range of 1.5-2
(see Fig. 125). Below these values the forces Rj and i? a sharply in¬
crease; increasing the LIl ratio in excess of 2 does not significantly
reduce these forces, but only leads to increases in the axial dimen¬
sions of the unit.

Assume L/l = 1.5.

Then, according to Eqs. (2.1) and (2.2)

f? t = 1.66P = l.66-9.5 = 16 kgf
j? 2 = 0.66P = 0.66-9.5= 6.3 kgf

To these values it is necessary to add the shaft weight G shaft —
2 kgf which is distributed almost equally between the two bearings.
Thus,

7?! = 16+1 = 17 kgf
i? 2 = 6.3+ 1 = 7.3 kgf

For the sake of unification assume the two bearings to be of the
same type. As the rear bearing bears a lighter load, it is better to
make this bearing fixed, so that it may take up axial forces.

(c) Counterbalancing of Impeller Axial Force

The rear faces of open impellers receive the full force of the hyd¬
rostatic pressure, being built up at the discharge outlet (in our case
p — 5 kgf/cm 2 ). The force, acting in the opposite direction, is sub¬
stantially less, because the pressure upon the impeller disk from the
vane side changes to a square law, beginning from vacuum in the
intake pipe up to 5 kgf/cm 2 at the impeller discharge point. As
a result, an axial suction force, directed to suction, occurs, which
in the considered case reaches approximately 1000 kgf. This force
can be eliminated by employing an enclosed double-disk impeller
with a bilateral seal, and by introducing bleed holes in-between
suction and pressure cavities (Fig. 18). Such a system fully counter¬
balances the hydrostatic pressure upon the impeller as it is subjected
to identical pressures (5 kgf/cm 2 ) on both sides.

112

Chapter 2. Design Methods

Apart from hydrostatic forces the impeller is affected also by the
reactive force of the jet, which, at the intake, is directed against the
suction. However, this force is not large and can therefore be neglected.

The condition for hydrostatic balance requires that both seal
■diameters be the same and the total area of bleed holes be at least

equal to the circular clearance in the
seal.

Assuming that seal diameters D sea i =
130 mm, radial clearance s — 0.1 mm,
number of the bleed holes n = 8 (equal
to the number of the vanes), we obtain

n^-^0AnD se ai

hence

V0.05D s ,,az^2.5 mm

With some margin we take the value
of d at 5 mm.

The seals are made in the form of cylin¬
drical bosses on the impeller disks and
fit, with clearance, into rings press-fitted
into the pump casing. To allow for pos¬
sible touching of sealing surfaces the
rings are made of an antifriction mate¬
rial (soft bronze, grade Bp. OH,G).

The seals can also be made of fluoro¬
carbon or silicone plastics, which do not
swell in water. However, the high cost
of these materials should be kept in
mind and their high coefficient of linear expansion makes the
problem of ring fixture in the casing much more complex.

(d) Durability of Supports

Tentatively assume that the impeller shaft diameter d = 40 mm
and that the supports are, in effect, single-row ball bearings (light
series) whose work capacity coefficient C — 39 000.

The work capacity coefficient providing for the required pump
durability

C = Rk a {nhf‘ z

where R = load on bearing (in this case the greatest bearing load
R = 17 kgf)

k a = bearing duty factor (assume k a — 1.5)
n — shaft speed, rpm ( n = 2950 rpm)
h = given durability (h = 40 000 h)

Fig. 18. Impeller with cou¬
nterbalanced axial force of
hydrostatic pressure

2.8. Design Example

113

Consequently, C = 17-1.5 (2950-40 000) °' 3 = 6800. Thus, the
selected bearings will satisfy (with an ample safety factor) the pump
durability period and enable larger loads and increased speeds to
be used in the event, of forced working.

To consider bearings for durability alone is not sufficient, as the
calculations assumed normal working conditions for the bearings.
Mistakes in assembly, too little
or too much lubrication may
bring all calculations to
nought and cause premature
wear or even failure long before
their service life has expired.

( e) Spacing of Supports

Whith the chosen ratio LU—

1.5 the spacing between the
supports will entirely depend
on the overhang value l of
the centre of gravity of the
impeller relative to the front
support. The latter value is de¬
termined by the arrangement
of the seal between the front bearing and the pump hydraulic cavity.

On the basis of the preliminary estimates assume that the seal
length is 45 mm and the distance between the front seal face and
the impeller centre of gravity, 10 mm. The bearing width is 18 mm.
Hence, the total length of the overhang

l — 45 -j-10 -f- 9 = 64 mm
and the distance between the supports

L = 1.51 -l & 100 mm

Fig. 19. Arrangement of impeller shaft
supports

This stage of the pump design ends with the drawing of a sketch
of the impeller shaft, which also illustrates the support spacing
(Fig. 19).

(/) Discharge Volutes

The cross sections of volutes can be positioned so that the extreme
internal points of the sections are equispaced about the impeller
circumference.

Then the centres of the cross-sections are arranged along a spiral
described by the equation

D

imp

<p0°

180°

8-01395

2

114

Chapter 2. Design Methods

and the extreme external points of the cross-sections -will follow
the spiral described by the equation

p' = _j_ s + d Q j/'-jggs-

where D imp — diameter of impeller;

s = distance at which the internal points of the cross-
sections are located from the impeller circumference;
d 0 — diameter of the volute discharge cross-section;
p, p' and cp = current coordinates

Fig. 20. Design variants of pump discharge volutes

Such a spiral volute with a split symmetrically along the cross-
section (Fig. 20a) ensures simple coreless moulding and enables
easy cleaning of the volute cavity.

Disadvantages:

the split plane intersects the volute discharge pipes; as a result,
a T-joint very difficult to seal is formed in the discharge pipe flanges
and in outlet pipe joints associated;

2J8. Design Example

115

the volute halves can be fixed relative to each other by fitted
dowel pins only; centring with the aid of a cylindrical spigot is
excluded;

the volute radial dimensions are too large (when s = 20 mm,
the maximum size, without discharge pipes, will be 470 mm)...

In the design shown in Fig. 20 b the volute is a complete casting.
The impeller is assembled through detachable cover. The discharge
pipes are one piece. The cover is centred relative to the casing by
a locating spigot diameter. The volute size is somewhat reduced due
to elimination of the peripheral flange (maximum size is now 422 mm).
The impeller hydraulic cavity is enclosed and shaped, during
pouring, by moulding cores. The cavity can be cleaned only
by hydropolishing (water jet containing suspended abrasive
particles).

In the design in Fig. 20c the external points of cross-sections are
spaced over the circumference whose radius is equal to the minimum
initial radius of the volute. Outwardly the centres of cross-sections
are gradually shifted towards the pump axis, following a spiral
pattern described by the equation

P^T (A>—^o]/")

where D 0 — volute outer diameter;

d 0 = diameter of the volute discharge cross-section

In the last portions of the volute the impeller, together with
the walls binding it to the casing, penetrate into the volute section.
The overall size of volute is sharply decreased (370 mm). The joint
line is along the volute cross-section plane of symmetry. The volute
halves are positioned by a locating spigot diameter (interrupted
in the discharge pipe areas). The discharge pipes are intersected by
the split plane.

The design disadvantage is that the water, as it flows to the out¬
put from the impeller, bifurcates forming in the exit part of the
volute two spiral vortices causing greater hydraulic losses.

The discharge pipes can be made in one piece if the volute section
is offset from the axis of impeller symmetry (Fig. 20 d). In this case
the impeller mounting is accomplished through a cover. The eli¬
mination of the peripheral flange decreases the volute size still more
(the maximum size is 330 mm).

Offsetting the volute sections will cause a water vortex but
hydraulic losses will be less here than in the design displayed in
Fig. 20c.

Finally, comparison of the schemes preliminarily establishes that
the best design is given in Fig. 206, as it possesses significantly few^r
drawbacks and a comparatively larger number of advantages.

8 *

416

Chapter 2. Design Methods

(g) Hydraulic Cavity

An arrangement drawing of the pump hydraulic cavity consisting
of volutes, cover and intake pipe guiding device, is presented in
Fig. 21. The guiding device consists of radial vanes cast on to the

pipe walls and to a streamlined

shaped central boss which assures'
a smooth water approach to the
impeller.

The cover-to-volute joint is
lined with rubber cord a placed
into a ring-shaped recess cut in
the centring spigot. Cover dis-

Fig. 21. General arrangement drawing Fig. 22. Guide device variants
of pump hydraulic cavity

mantling is simplified by recesses b, provided for insertion of dis¬
mantling -instruments, which are disposed in-between stud bosses of
the casing.

During development of the prototype pumps with detachable
guiding device can be made to help manufacture (Fig. 22a), but for
mass production the simpler version, shown in Fig. 22b, is preferred.

A filter should he provided when the pump is expected to work
in contaminated waters (Fig. 22b). A tapered threaded drainage plug
should, be located below the volute in the longitudinal plane of
volute symmetry (Fig. 21).

2.8. Design Example

117

Water drainage can be automated by closing the drain hole with
a spring-loaded valve. As the pump is started, the valve is closed
by the pressure-of water in the volute; as soon as the pump is stop¬
ped, water pressure drops and the
spring again opens the valve,
thus connecting the volute direct¬
ly to drainage piping. The possi¬
bility of introducing such a de¬
vice should be noted and drawn
(Fig. 23) and later discussed at
the final development stage.

The remaining pump elements
(design of discharge pipes, lugs
separating discharge pipes from
volutes, etc.) which seem rather
simple will be considered at the
stage of their design. Fig. 23. Automatic drainage device

(h) Hydraulic Cavity Sealing

The seal separating the hydraulic cavity from the bearing is
a very important unit on which to a significant degree depends the

operational reliability and dura¬
bility of the pump.

To fully exclude ingress of
water from the hydraulic cavity
into the oil space, it is better to
seal in two stages, positioning
one seal on the water side and
one on the oil side with an open
space between, having bleed
holes open to atmosphere..

The water stage, for the most
effective sealing, must be packed
with a face seal since it possesses
the property of self-running-in in
contrast to conventional packing
glands which require periodic
retightening. The oil side should
be packed with a sevanite cup
seal held in position by a bra¬
celet spring (Fig. 24).

In the first sketch (Fig. 25a) the
face seal is disk 1 carrying seva¬
nite seal 8. The disk face is, in effect, the sealing surface. The
seal movable part comprises washer 2 rotated by a toothed rim

118

Chapter 2. Design Methods

cut on the inside of the impeller relief packing ring. The washer is
constantly pressed to the non-rotating disk by the action of a spring
resting against the impeller face. The secondary seal, which prevents
water from leaking along the impeller shaft space bush 3 , is made

Fig. 25. Design versions o! end face seals

in the form of rubber cup 4 tightly encircling the surface of the
distance piece. The cup collar is pressed to washer 2 by the
same spring acting through steel sleeve 5. Infiltration of water
through the joint between the impeller and distance piece is preclu¬
ded by ring gasket 6 mounted at the joint.

.The intermediate space is formed by cavity 7 between the sevanite
, seal '8 and wall of disk 1 and is connected by radial hole 9 in the disk

2.8. Design Example _ 119

flange with the radially drilled hole 10 in the casing which, in
its turn, communicates with atmosphere through hole 11. For con¬
venience of inspections, viewing condition of packing (water leak¬
age), hole 11 is extended sideways by a pipe fitted in the casing wall.

Disadvantage of the design: when dismantling the impeller the
spring forces sealing washer 2 out of engagement with the impeller
and pushes rubber cup 4, consequently the seal disintegrates. Reas¬
sembly of the impeller and seal is rendered difficult for the same
reasons.

In the design shown in Fig. 256 sealing washer 2 is fixed axially
in the impeller with the aid of split spring ring 12 mounted on the
toothed rim of the impeller with clearance n assuring axial displa¬
cement of washer 2, thus compensating wear of sealing surfaces.
The secondary cup seal is fixed on the cylindrical extension of
washer 2. During dismantling the movable sealing unit slips off,
as a whole, together with the impeller. Assembly is also easier
since the above units can be freely slipped onto the shaft.

In another design (Fig. 25c) the secondary cup seal is installed
on the cylindrical extension of the impeller hub. The design is
better in that the impeller centring on the shaft is more precise.

In both of the cases (Fig. 256 and c) it is not necessary to provide
an additional sealing gasket between the impeller hub and distance
piece (Fig. 25a, gasket 6).

The final alternative is shown in Fig. 25 d. Here in the rotating
seal washer 2 is made to move by splines cut on the impeller hub,
making also a more compact design.

The possibility of water penetration along the splines is averted
by tightening the impeller on the shaft by a captive nut with a
sealing gasket between the nut and the impeller hub end faces.

With a unit pressure of 2 kgf/mm 2 on the seal working surfaces
the axial force developed by the spring is negligible and can be
disregarded when calculating acceptable loads £ on the fixed
bearing.

( i) Mounting of Bearings and Impeller on the Shaft

The first design sketch of the shaft with bearings, impeller and
drive flange assembled is shown in Fig. 26.

The main requirement for reliable mounting of bearings on .a
shaft is their adequate tightening in the axial direction. Prelimi¬
narily we apply the following procedure for fastening the bearings
on the shaft: the front (right-hand) bearing is tightened by the cap¬
tive nut which secures the impeller, pressing the bearing against
the shaft shoulder through a distance piece. The rear bearing is
tightened through the drive flange hub by the nut which secures
the flange.

120

Chapter 2. Design Methods

The drive flange hnb shoud be long enough to accommodate the
external shaft seal. For the sake of unification we also use a sevanite
seal here similar to that in the shaft face assembly. The length of
the drive flange hub should also suit dismantling tool lugs to be put
behind the flange. An acceptable hub length in the first instance
appears to be 25 mm.

The impeller and drive flange are spline-mounted. For the sake,
of unification the splines of the impeller and. drive flange, as well

Fig. 26. Shaft, impeller and bearings assembled (general arrangement drawing)

as the threads of the fastening nuts, are similar. The splined con¬
nection is assembled by a wringing fit and is centred on the external

spline diameter and the side faces of the splines

±]h_\

W S 2 w) ■

The torque from the electric motor shaft is transmitted to the
drive flange by involute splines cut in the flange periphery (spline
ring width 15 mm, diameter 80 mm).

A similar flange is mounted on the electric motor drive shaft.
The flanges are then joined by splined sleeve 1 which is loosely fit
on the splines of the two flanges and held in place axially with
a split ring. This coupling transmits high torques with small axial
dimensions and enables shaft misalignments to. be compensated.

We now introduce improvements: an internal thread 4 is cut in
the impeller hub to receive an extractor and washer 2 is installed
between the impeller hub and the distance piece for axial adjust¬
ment of the impeller inside the casing.

2.8. Design Example

121

The captive nut is locked by tab washer S, whose tabs on one side
are bent into slots in the impeller hub and on the other side into
slots cut in the edge of the captive nut. The tab washer is made of
annealled stainless steel, grade 1X18H9, which also allows the
washer to be used as a sealing gasket, preventing infiltration of
water into the splines, as well as the nut and extractor threads.

(/) Assembly and Disassembly

The order of assembly and disassembly is closely linked with
the system of installing the bearings on the shaft and in the casing.
In principle, two assembly-dismantling techniques are fea¬
sible

When applying the first technique, the bearings are installed in
the casing by an interference fit, and on the shaft, by a centring
or wringing fit. The dismantling procedure will be as follows. First,
remove the drive flange, then, moving the shaft to the right, extract
it together with the impeller out of the bearings bores (Fig. 21a).

Another order of dismantling is possible: first remove the impeller
and then, extract the shaft out of the bearings by moving it to the
left by the drive flange (Fig. 276).

The above-described procedure avoids the tightening of bearings
against the shaft shoulders but requires installation of a distance
piece 1 between the bearings. Under such circumstances the impeller
must be fixed axially on the shaft by pushing it against the spline
step 2. Both hearings are tightened against the impeller end face
by means of the drive flange fastening nut. The tightening force in
this case is transmitted to the first (right-hand) bearing through
the distance piece.

Disadvantages of the scheme are the following:

after the shaft extraction, the distance piece remains in the pump
casing; this makes assembly of the shaft more difficult;

the shaft fitting surface under one of the bearings, as the shaft
is being extracted, may he damaged by the inner race of the
other bearing.

The worst feature of the arrangement is that the inner races are
not interference fitted to the shaft. During long-time operation the
fitting surfaces may become worn under the action of radial forces.
In principle it is more advisable for slide fits to be used on the outer
race diameter where the unit pressures are much less (in this case
half).

In the second assembly scheme (Fig. 27c) the bearings are fitted
on the shaft with interference and are extracted together with the
shaft during dismantling. In this event bearings can be tightened
against shoulders machined on the shaft. The bearings are indivi¬
dually secured: the front bearing is tightened through the distance

122

Chapter 2. Design Methods

piece by the impeller captive nut; the rear bearing is held in place
by the drive flange locking nut.

£(a) and (6) interference fit of bearings in casing; (c) interference fit of bearings on shaft

It is better to mount the bearings inside the casing in reducing
sleeves; in doing so, the rear, fixed bearing should be installed in
the sleeve with an interference, but the sleeve in the pump casing,

2.8. Design Example

123

by a slide fit. The front bearing also should be installed in a reducing
sleeve by a slide fit. The sleeve, integral with the front seal hous¬
ing, is installed inside the pump casing by a wringing fit and bolted.

The dismantling procedure is as follows. Detach the impeller
from the shaft, undo the bolts fastening the rear seal housing and
with the movement to the left extract the shaft together with bearings.
The rear bearing is removed out of the casing together with its
sleeve and sevanite seal housing. The front bearing seal remains
in the pump casing. As the shaft is being extracted, the front bearing
freely passes through the enlarged fitting hole of the rear bearing.

When completely dismantling, the bearings are pressed off the
shaft, this being easier than extracting bearings from the casing
(as in the first scheme).

Comparative analysis of the two schemes shows that the second
one is preferable. Therefore the second scheme is taken as the basic
one.

( k ) Lubrication System

Generally, the pump bearings run under light loads and com¬
paratively high speeds. The walls of oil retaining housings are
very well cooled owing to the proximity of the water flow inside the
hydraulic cavity. Under such circumstances splash lubrication is
reasonable, employing a low viscosity oil with a gently sloping
viscosity-temperature characteristic. A suitable grade is industrial
oil grade 12, 12 cSt at 5G°C.

When composing the lubricating system the following problems
have to be solved:

prevent bubbling and frothing of oil which causes excessive heat¬
ing and accelerates thermal degeneration;

provide ample oil reserves for prolonged running;

assure a moderate and regular supply of oil to bearings;

avoid excessive bearing lubrication and prevent balls and cages
from oil splashes;

provide ventilation of the oil cavity to avoid cavity pressure rise
and forcing of oil through seals during warming (starting) periods
and formation of vacuum when cooling (stoppage periods);

provide oil drainage and filling facilities;

provide a convenient means for controlling oil level.

The first two of the above-listed problems may in the main be
solved by providing a large oil sump in the lower part of the pump
casing (Fig. 28). If the sump capacity is insufficient and its size
cannot be increased in the axial direction then it may be increased
by expanding it laterally.

Protection of bearings from too much oil is achieved by special
anti-splash disks 2 installed inside the oil cavity at the bearing
side faces.

124

Chapter 2. Design Methods

Fig. 28. Pump with oil cavity (general arrange ment drawing)

to the level of the lower balls does not solve the problem, as the oil
level lowers (due to evaporation of volatiles) and the bearings lack

lubrication long before the oil
becomes fully exhausted compel¬
ling also frequent addition of
oil.

A conventional oil ring, loosely
riding on the shaft, cannot be
used due to mounting conditions
and such a ring would hamper
shaft extraction from the casing.
Introduction of a simply driven
oil pump is connected with the
appearance of excessive rubbing
parts. Besides, the pump drive
Fig. 29. Oil atomizer would hamper the dismantling

procedure.

A reasonable way out is mounting upon the shaft a folding spring-
loaded oil atomizer.

The atomizer (Fig. 29) is the lever 3 made of thin-sheet steel
and fixed to the front bearing anti-splash disk. Spring 1 holds the
lever against the shaft. The action of centrifugal force on the lever

2.8. Design Example

125

overcomes spring tension, the lever swings out radially and dips
into the oil sump. As the pump stops, the spring returns the lever
to its initial position back to stop 2. The shaft may then be easily
extracted from the casing.

The slight outbalance arising from the swinging lever is easily
eliminated with the aid of a small counterweight 4, secured on the
anti-splash disk.

Since the oil atomizer is a self-adjustable device, the amount of
oil is automatically supplied at approximately the same rate regard¬
less of sump oil level. When striking the oil surface, the atomizer
is deflected in a direction opposite to shaft rotation, catching each
time a small portion of oil, thus precluding excessive bubbling.

The upper oil level in the sump is in line with the lowest points
of the ball bearings. With the selected dimensions the total quantity
of oil held by the oil sump is approximately 1.3 litre and the work¬
ing volume, being determined from the atomizer immersion depth
(its most advanced position), is about 1 litre. This enables the pump
to be run for a long period of time without the addition of fresh
oil.

To ventilate the oil cavity a breather is installed, which is also
used for oil filling. The most rational place to mount the breather
is close to the rear bearing in plane A-A (see Fig. 28), far away from
the operation zone of the atomizer. For stable readings an oil gauge
should also be installed in the same zone.

The breather consists of housing 10 with extension sleeve 3,
which protects the breather from oil splashes. In the housing a long
cylindrical gauze filter 4 is placed so that oil can be poured in through
a large-size, funnel.

The filter is secured to the housing shoulder by washer 8 sliding
along rod 7 which, in its turn, is fixed inside breather cap 9 and
loaded by spring 6. On the breather housing the cap is fastened by
a bayonet lock and held in position by the same spring 6.

An assembly of washers 5 mounted on extension of the rod 7,
preclude the throw-out of oil droplets through the breather. When
the cap is removed, washer 8 sits on the shoulder of rod 7 and at
the same time the washer assembly 5 is taken out, exposing the filter
ready for filling.

An oil gauge glass is set on the same side as the breather. This
enables the oil level to be controlled during filling. Directly behind
the glass with a small space between a white plastic screen is fitted,
having holes top and bottom which communicate with the oil sump.
The screen has two functions: it shows the oil level and protects the
oil gauge glass from splashes, when filling or running.

For the oil cavity to be viewed from the opposite side to the oil
gauge a hatch provided with an easily-detachable lid should be
made.

126

Chapter 2. Design Methods

To fasten the pump to its mounting frame four screwed holes
should be provided: two (a) in the breather location plane A~A,
and the other two—near the discharge volute.

Thus from this stage of design the general arrangement of the
pump is drawn (Fig. 30).

The oil drain hole a must be positioned in the sloping channel
of the oil cavity. To avoid mixing of oil sediments we separate the.
channel from the atomizing zone by a baffle plate. The drainage

Fig. 30. General view of pump (general arrangement drawing)

tube should be on the same side as the breather and oil gauge glass.
The seal drainage tube should also be brought to the same side.
Drainage is effected through tube b screwed into the front bearing
boss. The opposite end of the tube is flared to fit the casing wall.

( l ) Alternative Design. Smaller-Size Volute

Sketch the pump with the smallest radial sizes, according to
Fig. 20d. This time the impeller is given a conical form (Fig. 31)
and the volute is offset to bring it closer to the pump casing.

The drainage channel of the face seal must be inclined and also
shifted to the side to bypass the volutes.

In this version it is better to fasten the pump directly to the hous¬
ing of the flange-mounted drive electric motor using adapter 1.

2.8. Design Example

127

With such a fastening it is no longer necessary to install the pump
on a frame. In addition, the drive cluteh will be totally enclosed by
the adapter casing, thus the installation as a whole will be more
compact and lighter.

The design is quite suitable for fitting to a flange-mounted elec¬
tric motor. The addition of fastening holes in the pump base for

Pig. 31. Pump with smaller volutes (general arrangement drawing)

attachment to a frame, if necessary, will make the fastening still
more universal.

If so, the front fastening holes should be made in the pump casing
flange (plane a) leaving the back holes (2) in place.

Considering the smaller-size volutes as a positive advantage of
the construction we establish this as the main variant. The initial
version (see Fig, 30) is also brought forward for discussion, as well
as the individual pump units sketched during the composition
(e.g., automatic water discharge unit, Fig. 23, which may not be
necessary).

(m) Durability

In addition to the measures accepted earlier, assuring bearing
durability, we introduce induction hardening of the shaft surfaces
where the bearings-are fitted, making the hardness of these surfaces-

128

Chapter 2. Design Methods

not less than 50 Re. Finally rolling with hardened rolls to still
further improve the hardness. The shaft is made of steel grade 45.

To prolong service life of the lubricant and hearings, oil stipulated
in specifications and containing stabilizing additives is used (e.g.,
complex additives UHATHM-330, A3HHII-8).

Pump durability depends primarily on the service life of face
seals and the corrosion-resistance of the impeller, the pump casing
and other parts in contact with water.

Seal durability is determined by the material of the rubbing
surfaces. The stationary casing seals are made of stainless steel
grade 4X13, improved by nitriding (700-800 VPH). The rotating
disk seal is made from the same steel but its working surface is
coated with a composite layer of cermet bronze-graphite, impreg¬
nated also with silicone plastic.

The following materials are suitable for manufacture of the impel¬
ler and casing:

high-strength grey cast iron grade C *7 28-48. The strength (in modi¬
fied state) Gt, == 26-30 kgf/mm a , hardness, 180-250 BH, specific
weight, 7.2 kgf/dm 3 . The material casts well. Its disadvantages are
brittleness (elongation 6 < 0.3 per cent) and comparatively low
corrosion resistance in water;

corrosion-resistant cast Iron grade IIX 15-7-2 (Ni-resist).
Tensile strength or & = 25 kgf/mm 2 , hardness, 150-170 BH, specific
weight, 7.6 kgf/dm 3 . This material advantageously differs from grey
iron by a higher plasticity (8 — 3 to 4 per cent) and its corrosion
resistance in water is 15-20 times higher;

silumin grade A JI4 (8-10 % Si; 0,4% Mn; 0.25% Mg; balance—Al).
Its strength (in modified state) cr & = 15-25 kgf/ mm 3 , hardness,
70-80 BH, elongation 6 — 2-3 per cent, specific weight, 2.65 kgf/dm 3 .
The material has good casting properties. Its corrosion resistance in
fresh water lies between those of grey cast iron and Ni-resist.

The main advantage of silumin is its low specific weight which
offers (with equal sized sections) sharp decrease in stresses (almost
three times) when under the action of centrifugal forces in compa¬
rison with the other listed materials. However, one should always
remember of its lower resistance to abrasion, which is the result
of its low hardness. This disadvantage is critical for an impeller
subjected to the action of rapid water flows and moving with
even greater speed relative to the water layers in the gaps between
the casing walls and impeller disks.

From the comparative analysis of the listed materials it is clear
that it is better to choose silumin for the casing and Ni-resist for
the impeller. The higher price of the latter is repaid in higher dura¬
bility and reliability.

When building the pump casing of silumin, the softness and plas¬
ticity of this material should be kept in mind. Studs should be

2,8. Design Example

129

used as fasteners and nuts provided with, washers. The holes for
a drainage plug and pump fastening bolts must have reinforcing
threaded steel bushes. Owing to the low rigidity of silumin, casing
walls must be at least 8 mm
thick and be properly ribbed.

To further corrosion protection
of the water cavity walls a zinc
protector 1 (Fig. 32) is introduced
at the hub of the stationary vane
apparatus.

All other parts coming into
contact with water (some sealing
details, impeller captive nut,
drainage plug, etc.) are made of
stainless steels: the sealing
spring, fastening elements, nuts
and drainage plug — from heat-
treated steel grade 4X13 and
stoppers — from mild stainless
steel grade 1X18H9.

Amidst other methods applied
to enhance durability and reli¬
ability is the heat-treatment of shaft splines, as well as all fastening
elements. The splined rim of the drive flange should have a hardness
of not below 55 Rc which can be achieved by induction hardening.
The surfaces on which the sevanite sealing cups work must have
a hardness of not lower than 45 Rc and a surface finish not worse
than class 10.

Nuts of internal fastening elements must be locked positively
(e.g., by tab washers).

Fig. 32. Installation of zinc protector

( n) Working Arrangement

After discussion the final design is chosen and the working, arran¬
gement drawn, which is then used for designing the compo¬
nents. ■' •

The drawing of the working arrangement of the pump, (see
Fig. 33) should contain the main coordination, coupling and overall
dimensions, as well as the dimensions of connection and locating
joints, classes of fits and grades of accuracy, and code numbers of
bearings. The maximum and minimum sump oil levels and sump
capacity should also be specified.

The drawing should also indicate the main specifications of the
pump (capacity, head, speed, rotation direction, power consumption,,

9-01395

type of electric motor) and technical requirements (hydraulic pres¬
sure testing of -water cavities, overspeed test of impeller). .

On the basis of the working arrangement check calculations oi
strength are carried out.

Chapter 3

Weight and metal content

Weight is an important machine parameter, especially in trans¬
port and, even the more so, in aviation, where each kilogramme of
excess weight lessens the payload, speed and cruising range. In
general engineering a reduced weight of machines means metal eco¬
nomy and cheaper production.

The reduction of weight becomes a particularly important factor
under mass production conditions, as it enables metal to be saved
on a national scale. This in no way relieves us of the necessity for
reducing the weight of individually produced machines or those
produced in small quantities as their total output makes up in
general a considerable fraction of all machine production.

Of course, weight reduction is by no means the end in itself. The
cost of material is an inappreciable part of the total sum of expen¬
ditures spent during machine exploitation, as expenditure depends
primarily on machine reliability and durability. The tendency to
reduced weight should be rejected without hesitation if it degrades
such machine parameters as strength, rigidity or reliability, espe¬
cially so in general engineering. It is better to have a somewhat
heavier machine than a lighter one possessing worse reliability
characteristics and durability.

Comparative weight characteristics of machines of similar pur¬
poses are assessed by their weight factor which is the quotient of
the machine weight G by the main machine parameter.

For machine power generators such a main parameter is their
power N. The weight-to-power ratio of such a machine will' be

This characteristic accounts for the design perfection of a machine,
as well as the use of light alloys and non-metallies.

For various types of internal combustion engines the weight-
to-power ratios are as follows:

stationary, 8-15; ship, 3-8; automobile, 2-5; aircraft, 0.5-
0.8 kgf/e.h.p.

Chapter 3. Weight and Metal Content

.132

In transport the weight factor is expressed, as the ratio between
the total machine weight and the payload, and has the following
(values: for ship transport, 20-30; railway transport, 10-20; automo¬
bile transport, 3-5; aircraft, 1.2-2.5.

The quality of metal-cutting machines in these terms is assessed
as the ratio between the machine weight and the nominal power
of its drive motor (this is not expressive as -it neglects the degree
to which the nominal power is utilized and, also, the machine pro¬
ductivity).

The design perfection of speed reduction units is evaluated by
the weight-to-torque ratio or by the ratio between their weight
and the product of the transmitted power and the gear ratio (reduc¬
tion ratio).

The notion of the metal content must be differentiated from that
-of the weight. 1 These are not equivalent.

Let us explain this by an example. Assume that two machines
.have identical dimensions and parameters but one of them is made
mostly of heavy metals (steel and cast iron) and the other, of light
.alloys (e.g., aluminium). It is clear that the weight of the second
machine is less than the weight of the first, as the specific weight
of the heavier material is approximately twice that of the light alloy,
but the metal content, which represents the amount of the metal
used, is the same for both machines.

Metal content can best of all be expressed by the volume of metallic
parts making up the machine. Then, along with the weight factor,
it is possible to introduce an index of the specific metal content
(specific volume), expressed as the quotient of the volume of the
metal parts by the machine main parameter.

This index will allow the evaluation of first, the economy of metal
achieved in a given machine, and second, the quality of the design,
i.e., the rationality of the design scheme and the form perfection
of the components regardless of the specific weights of the materials
■used. ■, •

As machines are generally manufactured of metals with different
.specific weights, then, in a general case, the index of thejspecific
metal’ content will be

, , h G ™

v ~^r + '^r' + ••• + ~^r

where. 2<?i, . . ., TiG m = total weight of components made from

materials with respective specific weights

' '■ , Tn • • -, : 7m

N — machine main parameter

3.1. Rational Sections

133

The reciprocal N/V can be defined as the volume utilization
factor.

A reduction in weight along with a decrease in the metal content
may be obtained by giving parts rational cross-sections and shapes,
by reasonably utilizing the strength characteristics of materials,
by making use of stronger materials and rational design schemes,
by eliminating excessive safety margins and substituting non-
metallics for metals.

3.1. Rational Sections

Maximum reduction in weight can be achieved by giving machine
elements complete equal strengths. The ideal case is when the longi¬
tudinal stresses are distributed equally in every cross-section and
at every point in the cross-section. This is the condition when equal
strength is most fully expressed. Here, the material is used to its
best possible advantage, and the weight of a part, for the designed
stress level, is the minimum obtainable. In principle, such a case
is possible only for a few types of loading when the load is taken by
all cross-sections of a part, that is in tension and compression and,
partially, in shear.

In flexure, torsion and in complex states of stress the stresses
along the section are distributed unevenly. They achieve their
maximum at the extreme points of the section and at other positions
may fall to zero (e.g., on the neutral axis of a section subjected to
flexure). In these cases it is only possible to approximate conditions
of equal strength by equalizing the stresses over the entire section,
removing metal from the less loaded portions and adding it at the
most loaded points, i.e., at the periphery.

Let us take, for example, a cylindrical component subjected to
flexure or torsion (Fig. 34).

The stresses in a solid component part having a continuous round
section (implied here are ordinary bending stresses and torsional
shear stresses) are distributed according to the law of a straight line
(Fig. 34 a) passing through the section centre (in the Figure the
stress diagram pertaining to the case of flexure is turned through
90° from its true direction).

Removal of metal from the weakly loaded centre zone, i.e., making
it tube-shaped, assures a more equal stress distribution in the remain¬
ing zones (Fig. 34&). The thinner the wall, i.e., the greater the
d!D ratio, the more regular is the stress distribution. Keeping
the external diameter constant causes the wall stress level to rise)
However, increasing the outside diameter easily reduces the stress
level to the previous values (Fig. 34c) or even significantly decreases
it (Fig. 34 d).

! -This principle,, which can be defined as the principle of equally
stressed sections, may be applied to sections of any shape.

134

Chapter 8. Weight and Metal Content

The weight savings obtainable under such circumstances are
illustrated in Fig. 35. The latter shows a number of profiles with
identical cross-sectional areas and, consequently, with equal weight

Fig. 34. Distribution of stresses over sections of solid and hollow cylindrical
components (in flexure and torsion)

per unit metre, in the order of decreasing metal thickness over the
periphery. The most rational forms of sections (hollow and rolled
steel sections) are noted for high strength and rigidity. The advant-

i

rpzazza mzsEzizzzsa

W=4.6

/ = s.s

w=52

Fig. 35. Resisting moment W and moment of inertia I for solid and hollow stock
having identical cross-sectional areas (in flexure)

ages increase with symmetrical shapes proportionally to the increase
in radial dimensions entailing under the given conditions (F =
; == const) wall thinning, but with asymmetric profiles more
according to the metal mass concentration in the area of the greatest
acting normal stresses (e.g., in rolled section profiles with an
increase in the thicknesses and widths of flanges).

3.1. Rational Sections

135

(a) Strength and Rigidity Indices of Profiles
The relative weight advantage of a profile subjected to bending
is characterized by the quantities ~~ and — ( reduced profile strength

F F

and rigidity). The reciprocals ^ and -j- are defined accordingly as
the reduced profile weight in terms of strength and rigidity.

These indices have linear dimensions cm,, y- cm 2 j and totally
define the profile’s advantage in shape and linear dimensions.

If the denominator in the expression cm 3 /cm 2 ) is raised to

/

the power 8 / a —■■ 1 j,, and in the expression cm 4 /cm 2 |,

1 j,, we shall obtain the following dimen-

cm 4

L (cm 2 ) 2
sionless expressions

to the power 2

w-

W

jg.3/2

/

- F z

(3.1)

(3.2)

which define the degree of rationality of the profile shape regardless
of its absolute dimensions. Although rather approximate repre¬
sentations, the expressions (3.1) and<(3.2)|enable one to have a suf¬
ficiently clear impression of the weight advantage.

The dimensionless reciprocals characterize the profile weight

p3/2,

Sw :

W
F 2
I

With

indices

equal success we

may use

(3.3)

(3.4)

the following dimensionless

w

or

, tf 2/3

F

(3.5)

V~I

1 F

(3.6)

, F

7T

(3.7)

r ^

II

•>

tut)

(3.8)

Some actual values for the indices w and i t calculated from for¬
mulae (3.1) and (3.2) for the most widely used profiles (flexure in
the vertical plane), are given in Table 4.

Table 4

Sketches of
profiles

Specific Indices of Strength and Rigidity of Profiles (in Flexure)

p

W

1

w

W ~F*li

I

1 '~ J?2

0.7857) 2

0.17)3

0.057)4

0.14

0.08

£2

£3/6

£4/12

0.166

0.083

B'ic (c=H/B)

£ 3 c 2/6

£4c s /12

0.1661/1

0.083c

0.785 Z) 2 (l— a 2 )

(a — d/D)

0.17)3 (l_ a 4)

0.05Z) 4 (1 — a 4 )

0.14

(1— a2)»/a

0.08 i ~~ ai
(1—«2)2

£ 2 (1— e)

(e = b/B)

£3

12- f 1 —

1—e 4

1 — e 4

6(1— e 2 ) 8 /a

12 (1— e 2 )3

Continued

138

Chapter 3. Weight and Metal Content

( b ) Strength and Rigidity of Round Hollow Profiles

Round profiles (shafts, axles, etc.) are of particular importance
■in mechanical engineering. Let us consider some typical cases which
•illustrate the advantage of hollow profiles under conditions of flexure
.and torsion.

'Fig. 36. Change of resisting moment W, inertia moment I and weight G of

d

■cylindrical components as a function of a = with D — const (in flexure and

torsion)

Case 1. Given: outside diameter of a component part (D = const).
For such a case the following relations are valid:
relative strength and rigidity

relative weight

W

W 0

a 4

In these formulae the subscript 0 refers to a solid round section,
and the value a is the ratio between the hole diameter d and the

outer diameter D of the part [a = .

Assume W 0 , I Q and G 0 of a solid part to be equal to unity. Then
in Fig. 36 we can show changes in the resisting moment,, moment
•of inertia and weight of the part with the increase of the ratio a.
From this figure we see that:

small-diameter holes (d < 0.2D) hardly affect the strength,
rigidity and weight of the part;

when a — 0.3-0.6 a significant reduction in the weight with
a simultaneous reduction of the strength and rigidity are observed

3.1. Rational Sections

139

(when a = 0.6 the weight of the part is almost 40% less, while the
strength and rigidity are only worse by approximately 10%).

Thus, in the case being considered it is possible to introduce
holes with diameter d — 0.6Z) and obtain a large weight reduction
with an inappreciable reduction in the strength. Increasing 'the
values of d in excess of 0.6D

considerably lowers the strength.

Case 2. Given: the shaft
strength (W = const). The outside
diameter of the part changes.

For this case the following
relationships are valid

D _ 1

Do f

G F D2-i v 1— a 2
Go ” F 0 “ Dl ~ (l _ a 4)2/3

_/_ 1—a* _1

h ~ (l _ a i ) 4 / 3 ” fjCZaF

Fig. 37. Change of inertia moment I,
outside diameter D and weight G of
cylindrical components as a function

of a = 4; with W = const (in flexure

D

and torsion)

Figure 37, plotted on the basis
<of these relations, shows the rigi¬
dity and weight indices of a
part as a function of a = d/D.

With d/D and D increasing
simultaneously, the weight and
rigidity characteristics of a part continuously improve. The enlarge¬
ment of the outside diameter, a condition for equistrength, is hardly
noticeable at the beginning. Even when d/D — 0.7, the O.D. of
a part must only be increased by 10%, then the weight is lowered
by 40%. The moment of inertia will increase with the same regu¬
larity as the outside diameter.

The regularities shown in Fig. 37 are further expressed in Fig. 38,.
where cylindrical components having equal strength in flexure and
torsion with progressively ascending d/D ratios are shown.

After a certain point the d/D ratio cannot be increased as the
wall thinning may result in local deformation, particularly in
areas where loads are applied, making difficult also the use of such
design elements as keyways, recesses, threads, etc. Actually d/D >
> 0.7 is very rarely used. Parts in which d/D = 0.8-0.95 refer to
pipes, tubes and shells.

Thin-walled tubes, having walls 1-2 mm thick, can he used to
good advantage for torque transmission, provided significant longi¬
tudinal and transverse loads are absent. The design elements which
take up torque are welded to gimbal drives of the tubes. Auto¬
mobile cardan shafts are such examples.

140

Chapter 3. Weight and Metal Content

At large d/D ratios the saving in weight may be considerable;
e.g., when dlD = 0.95 the weight of a tube is as small as 20% of
an equally strong solid shaft and its torsional rigidity is almost
twice as high as that of the solid shaft.

i~0.7

--a-o-

\z3ZZZMZZZZ32skzzk.

G-0.61

6=0.51

G=039

Fig. 38. Cylindrical

components of equal strength in flexure and torsion,
■ having different ratios a =~~

Case 3. Given: the shaft weight (G = const).
The calculation formulae for this case are

D

1

D q

Y 1 —

W

l—a «

W 0

(1 _a2)3/2

I

1— a*

h ~

(p_ a 2)2

The values of D/D calculated by these formulae,,, are illustrated
in Fig. 39 as a function of a = d/D.

The graph illustrates the advantages of hollow thin-walled sec¬
tions. When d/D — 0.9 the resisting moment and the moment of
inertia are increased 4.5 and 10 times respectively % and when d/D =
= 0.95—6 and 20 times respectively when compared with a solid
part of the same weight.

Increasing the relative sizes of the outside diameter in parallel
with the simultaneous introduction of hollow shell sections and
holes leads to a sharp increase in the strength and rigidity indices,
which is also accompanied by reduction of weight, giving improved
operational characteristics to shafts and parts associated with them
as well as providing safe margins which allow machine power and
speed to be enhanced.

3.1. Rational Sections

141

la modern high class machines solid shafts are almost always
replaced by hollow ones.

The regularities considered in this section underlie the modern
mechanical engineering trend to applying,, whenever possible, thin-
walled, tubular and shell constructions for components when extre¬
mely high strength and rigidity in conjunction with minimum

meter D of cylindrical components as
c?

a function of a — ^ with G — const
(in flexure)

weight are required. Dangerous loss of local stability due to the
influence of working loads is prevented by increasing local rigidity
at critical points by using tension-compression braces.

Examples of shell-like structures incorporating tubular elements
are shown in Fig. 40a and b. These structures are welded to the
solid elements.

(c) Designs of Equal Strength

In the case of torsion, bending and complex state of stress,
when the equality of stresses throughout the cross-section is, in
principle, unobtainable, components considered equally strong
are those which have identical maximum stresses in each section
along the axis.

In bending, a design is considered to be equistrong if the ratios
of the bending moment acting at each given section to the
resisting moment of the given * section are identical. For torsion
this condition consists in the equality of the torsional resisting mo-

142

Chapter 3. Weight and Metal Content

ments of resistance across each given section, and in cases of com¬
plex states of stress, in the equality of the safety factors.

The concept of an equistrong design is applicable both to some
parts and to a machine as a whole. Of equal strength are those con¬
structions in which the parts have identical safety factors with
respect to the loads acting on them. This rule can be applied to parts
made of different materials. Thus, a steel part with stress 20kgf/mm 2:
and yield limit cr 0 . 2 = 60 kgf/mm 2 will be equistrong with an alu¬
minium part having stress level equal to 10 kgf/mm 2 and yield:
limit cr 0 . 2 — 30 kgf/mm 2 . In both these cases the safety factor is 3.
This implies that the two parts will succumb to plastic deformation
as soon as the loads acting upon them are tripled. Independent
of this, each of the parts being compared may still possess equal'
strength in the above-said meaning, i.e., have identical stress, levels
along their entire length.

The value of the working loads and stress levels at various sections
of a part are found by calculation. A part already calculated as
equistrong, will indeed be equistrong if the calculations correctly
determine the true values and stress distribution along the part
axis, which is not often the case.

Shapes required by equal strength conditions are sometimes tech¬
nically difficult to obtain and must be simplified. Additional ele¬
ments on almost all parts are unavoidable (trunnions, shoulders,,
grooves, recesses, threads, etc.); sometimes they cause local streng¬
thening, but more often, stress concentrations and weakening of the
part which also means the introduction of corrections for a true-
stress distribution along the part.

For all these reasons the concept of equal strength is rather rela¬
tive. In practice design of an equistrong part amounts to an appro¬
ximate shape reproduction, dictated by equistrength requirements,
with necessary measures taken to reduce sources of local stress con¬
centrations.

Weight savings from the application of the equal strength prin¬
ciple mostly depend on the type of loading and the method of accom¬
plishing equal strength. Some idea of the weight saving can be
conceive from the example below where a cylindrical part, to be of
equal strength when supported at its ends, is subjected to flexure by
a centrally applied transversely acting load (Fig. 41).

Case 1. The part is made equistrong by changing its external shape..

Maximum normal stress in the central section of the original cylin¬
drical part 1 (Fig. 41a)

where M 0 is the bending moment at the beam centre, equal to the
product of the end reaction and the distance LI2 from the central
section to the reaction plane.

\

3.1. Rational Sections

143 ;

Stress in any arbitrary section

M

G.1D*

2i

where M — M 0 -jr is the bending moment in the given section;,

l = distance from the given section to the end reaction
Consequently

, 2Mpl
LOAD*

The maximum stress in any section of an equistrong designed part

Fig. 41. Methods of imparting equal strength properties to cylindrical parts (in.

lateral flexure)

(a) original shapes; (b) shapes of equally strong parts; (e) design versions of equally strong

parts

must be constant and equal to

hence, the diameter for an equistrong^part should he

The profile of the equistrong part 1 is shown in Fig. 41 b. Figu¬
re 41c illustrates a design of equistrong part 1 for a geaif made inte¬
gral with a shaft mounted in antifriction bearings.

444

Chapter 3. Weight and Metal Content

The equistrong shapes are more simple. The simplified shaft has
bearing journals at its ends.

Case 2. Part 2 is made equistrong by removing internal metal
Irom the shaft while keeping the outside diameter unchanged.

The equistrength requirement

a _„ ..... M q _ 2M 0 l.

0 O.lDg L0.W§{l—a*)

const

where a is the ratio between the varying diameter d of the inter¬
nal lightening recess and the constant outside diameter D 0 .

Hence,, the current internal diameter will be

d=D a yi-%

The profile of the equistrong part 2 in this case is as shown in
Fig. 416,, and its design, in Fig. 41c.

The high weight saving (weight of the equistrong part is one
third of that of the original) is the result of applying not only
the equistrength principle, but also the principle of equal cross-
sectional stress.

However, it should be noted that this method of imparting equal
strength means enlargement of the bearing support diameters which
naturally decreases somewhat the resultant weight saving.

Case 3. Equal strength of a hollow part 3 (Fig. 41) is obtained by
■changing its outer configuration.

The equistrength requirements for this case are expressed by the
following formula for determining the varying outer diameter:

D

■D„

1—a4

2l_
' L

where a 0 — d d /D 0 is the initial ratio between the hole and outside
part diameters;

a — current value of d 0 /D for the equistrong part
With moderate values of a 0 the weight saved in this case will be
close to that in the case of part 1.

Case 4. Equal strength of hollow part 4 (Fig. 41) is achieved by
changing its internal shape.

The value of the varying diameter of the internal cavity
satisfying the equal strength requirements is

d ^D 0 y l-^-(i-aj)

where a 0 — d 0 /D 0 is the ratio between the hole size and the outside
diameter of the original part

3.1 Rational Sections

145

The weight saved in this case is similar to that in the Case 2.

It should be emphasized that other conditions being equal the
rigidity of a part having equal strength is inferior to the rigidity of
parts which have even locally improved safety factors.

In those cases when the rigidity of a part is important lowering
of its value can be averted by lowering the stresses (which, natural¬
ly, lowers the weight saving) or by applying in each separate case
a rational method of achieving equal strength.

Thus, the equistrong component 2 (Fig. 41b) achieved by remov¬
ing internal metal, is much more rigid than part 1 (Fig. 41b), al-

:^^ZZZZZZZZZZZZZZZ^0 zzzzzz ^

.i f n t

I 2 (a) II

Fig. 42. Imparting equal strength, properties to parts

though it is lower in rigidity than the original solid cylindrical part
2 (Fig. 41a).

Some examples of imparting equal strength properties to parts are
illustrated in Fig. 42.

Flanged shaft / (Fig. 42a) loaded by a constant torque has unequal
strength between the splines and flange. The maximum stresses occur
on the splined portion, while between the splines and flange, where
the shaft diameter is enlarged, the stresses are considerably less. The
equistrong (stress-balanced) design, II, given in Fig. 42a, has been
obtained by calculations, based on the constancy of the torsional
resisting moment across any section of the shaft.

HO—01395

146

Chapter 3. Weight and Metal Content

The design of pinion-shaf t I (Fig. 42 b) with a through hole of con¬
stant diameter is of unequal strength in spite of its simplicity and
technological effectiveness. Shaft II with stepped-turned portions
is approximately of equal strength. Design III is a carefully consi¬
dered design aimed at improving fatigue strength and having a
smooth shape of the internal recesses.

Shafts II and, particularly, III are much more expensive to manu¬
facture. Nevertheless, the necessity for lightness and higher fatigue
strength often justifies more complicated and expensive production.

A lighter design is exemplified by the terminal gear made integral
with the shaft (Fig. 42c).

Here, in order to obtain the shape, which approximately meets
the equal strength requirements, upsetting is first used followed
by machining (variant IV),

It is especially important to observe equal strength requirements
in discs which rotate at high speeds (turbine rotors, centrifugal
and axial compressors). The centrifugal forces developing in such
components cause stresses increasing in the inward radial direction
(towards the hub) as a result of the summation of centrifugal forces
occurring in each radial layer of metal from the periphery to the disc
centre. The conditions for equal strength in this case require the disc
to be tapered becoming thinner towards the periphery. This measure
lightens the disk as a result of removing metal from the periphery
and contribute to lesser hub stresses.

Equal strength calculations for rapidly rotating disks are complex
since in a number of cases thermal stresses must be considered.
These stresses appear as a result of temperature variations over
the disk body. In many cases the picture is complicated by the effect
of thermal shock produced under certain operating conditions by
unstable heat fluxes flowing inwardly or outwardly.

( d ) Equal Strength of Units and Connections

Implementation of the equal strength principle, as applied to
units and joints, we consider by examples.

Figure 43a shows a turnbuckle tightening two screwed rods.
Design I cannot be regarded as of equal strength, since elementary
calculations prove that the tensile stresses in the tubular section
are three times less than those in the rods.

In the equal strength design II the turnbuckle cross-section has
been reduced.

As a general remark to this example, we should say that circular
sections are deceptive when their strength is evaluated visually.
The sectional strength of such components is proportional to the
square of their diameter the resisting moment and torsion, to the cube,
and the moment of inertia, to the fourth power. These conditions are

3.1 Rational Sections

147

not always taken into account when designing such components.
When evaluating tensile or compressive strengths and also rigidity
of round parts the designer generally makes the mistake of exagge¬
rating their sizes.

The design (Fig. 43 b) of chain conveyer link joints with lugs of
the same width is not equistrong for the following three reasons:

the tensile safety margin at the base of the lugs in the upper
link is 3/2 times less than that in the lower link (the figure shows
the ratio between the numbers of lugs in the upper and lower links);

II

Fig. 43. Imparting equal strength properties to units

the shear safety margin for the pin stud is half the tensile safety
margin of the lower link lugs;

the tensile safety margin of the lug heads across section a-a,
with head wall thickness equal to the lug base thickness, is twice
that of the lug bases.

In other words, the lugs of the upper link are weaker than their
lower counterparts, and the pin strength is less than that of the
whole link. The lug head walls are also too thick.

In the equistrong design II the total lug width in the lower and
upper links is the same, which assures the given equality of stresses!n
the lugs. The pin diameter is accordingly enlarged; and the lug
wall heads thinned to suit. “ r

In the construction III equal strength of the pin has been obtained
by increasing the number of lugs (seven in place of five in the
previous exa mple). In this way the pin diameter can be decrea¬
sed bv V 2/3 times as compared to that in version II.

10 *

148

Chapter 3. Weight and Metal Content

3,2. Lightening of Parts

Very often equal strength requirements cannot be met either due
to the complex configuration of a part or because of the indetermi¬
nancy of the acting stresses which develop in it. Under such circum¬
stances the weight of a part is reduced by removing metal from the
less stressed areas.

Examples of part lightening are presented, in Figs. 44-47 and in
Table 5.

Figure 44 a shows how the weight of the crankshaft webs can be
reduced. The outside corners of webs / are not involved in transmit-

Fig, 44. Examples of lightening parts and Joints

(a) crankshaft! (6) flanged shaft; (c) clamp connection; <d) bevel gear; l ~ original design;
11-111 — reduced-weight designs

ting forces from crank necks to the crankshaft. Elimination of these
web corners (version II) substantially reduces the weight without
reducing the strength.

, The bevel gear (Fig. 44 d) can be lightened by removing all non-
usable teeth portions over the minor diameter. Apart from the weight
saved, the shortened teeth assure a more uniform pressure distribu¬
tion over the tooth length and lessens tooth loads due to the in¬
crease in the average pitch diameter.

In flange-type components a substantial reduction of weight can
be achieved by changing the flange outside contour (Fig. 45). The
radius of the fastening holes centres circle is assumed to be the same
in all cases. The weight of the round flange is assumed to be equal
to unity.

1

3.2. Lightening of Parts

149

The possibility of lightening a part by removing surplus metal
must never be neglected, no matter how small the amounts may be
(Fig. 48). Although weight savings in each separate case may be
small the total effect from many similar components may be quite
considerable.

Particular attention should be paid to decreasing the weight of
fastening elements, as this results in an appreciable reduction
of the total machine weight. Besides, the use of rationally shaped

Fig. 46. Methods of lightening a grooved drive plate

fastening elements is accompanied further by strength and technolo¬
gical advantages. Let us take, for example, the tie bolt shown in
Fig. 48/. The reduced-weight design, given in sketch II, is not only
lighter in weight, but also possesses higher resistance to cyclic loads,

Methods of Lightening a PI anetaryTCarrler

Table 5

Method of
lightening

Original design

Method of
lightening

Thinning

disk

Method of
lightening

Fixing flange
strengthened
by two annular
ribs

Round and elon¬
gated through
holes

Thinning as
above and sha¬
ped recesses be¬
tween fixing
holes

Stud hubs and
satellite holder
connected by
radial ribs

Annular recess
between hub
and stud fixing
holes

To increase ri¬
gidity fixing
hole hubs have
strengthening
rib

As above and
shaped recess
between stud
hubs

Annular recess
as above and
extra through
holes between
fixing holes

Annular recess
as above and
shaped recess
between fixing
holes

Fixing hole
hubs have rein¬
forced with
strengthening
internal annular
rib

As above and
shaped recess
between fixing
holes

As above and
lightening holes
in disk

Stud hubs
strengthened
with peripheral
ribs

Peripheral ribs
and lightening
holes in disk

M (a)

Fig. 47. Lightening of levers

(a), (c) original designs; (6), ( d) reduced-weight versions

Fig. 48. Methods of lightening component elements

) a ) shaft collar; (6) thrust shoulder in a hole; (e) flange; (d) disk- and hush-type components;
(<?) threaded shaft end; y) nut; ( g ) ring nut; (h), (i) screwed piugs and stoppers; 0) holt;
(fel locating bolt; I — original designs; II and III — reduced-weight versions

3.2. Lightening of Parts

153

particularly if the thread is rolled and the stem coined on a rota¬
ry-forging machine.

For the fitted bolt (Fig. 48&) the reduced stem diameter will mean
less precise machining.

In light-weight machines, where lowered weight plays an impor¬
tant role, the use of reduced-
weight nuts and bolts is obli¬
gatory.

(a) Effect of Diameter on the

Effectiveness of Lightening

When lightening cylindrical
components, disks, wheels, rings,
as well as parts with figured
outlines, e.g\, polygons, it should
be borne in mind that the great¬
est effect is achieved when remo¬
ving material from the periphery
and comparatively lesser effect,
from parts close to the centre.

In the general case the weight
saving is proportional to the
square of the diameter.

Let us compare the effective¬
ness of the weight reduction
when decreasing axial dimensions on different diameters.

Fig. 49a depicts a disk with a rim and a hub. Let us determine
the saving in weight, obtained by removing metal of indentical
thickness b from the rim and hub (black portions).

The volume of metal removed from the rim

The volume of metal removed from the hub

Vi =JL {d Z„ d *)b = JLMt[i~(lL) 2 ]

The ratio

V ^2

obeys a square relation, relaxed by the effect of the ratio between
the outside and inside diameters of the rim and hub.

Assume that the wall thicknesses of the rim and hub are the same
and D 1 /D z — 0.8. With the relations given in Fig. 49a the value of

(3.9)

Fig. 49. The weight of part as in¬
fluenced by:

(a) redaction of axial dimensions in centre
and on periphery; (6) removal of annular
volumes in centre and on periphery

154 Chapter 3. Weight and Metal Content

djd 2 — 0.5 and. Eq. (3.9) becomes

D\_ 1-0.64 p , ilhVi

V z ~~ d\‘ 1-0.25 \ d 2 I

With D 2 /d 2 = 3.

1^45

f 2

In. a particular case, when — p- a perfect s

(3.10)

a perfect square relation is

obtained

Figure 495 illustrates weight reduction by removing annular volu¬
mes of the same thickness a from different diameters.

In this case the volume of metal removed from the rim will be

Vi m nDal

The volume removed from the hub

The relation

V 2 ~ ft dal

Vj_ __ D_
V z ~ d '

(3.11)

i.e., it is directly proportional to the ratio of the diameters.

Thus, the comparative saving in weight, obtained by removing
metal from different diameters, depends on the mode of relieving
and the part shape. Its dependence on the diameter fluctuates within
the limits from D/d to (D/d) 2 .

Radial thinning towards the periphery of disks, flanges, covers,
etc., is widely applied, the more so as this shape often conforms
to the law of radial stress variation (covers loaded centrally by a
transverse force; flanges loaded with a torsional or tilting moment;
disks subjected to centrifugal forces, etc.).

The weight saving obtained by thinning disks in the peripheral
direction can be evaluated for a most simple case of changing a
disk of a rectangular profile (in the meridional section) to most
a trapezoidal profile (Fig. 50).

The volume of the square-section disk

(3.12)

The volume of the trapezoidal-section disk

3

( 3 . 13 )

3.2. Lightening of Parts

155

The relation between weight G of the trapezoidal-section disk
and weight G 0 of the rectangular-section disk will be

i.e ., it is totally dependent on the ratio b/B.

Fig. 50. Relative weight of tapering
disks as a function of b/B

Fig. 51. Tapering of flange towards
periphery

When b/B — 0 (the case of a triangular profile) the disk will be
three times lighter than the rectangular-section disk. For the most
widely used ratio b/B = 0.3-0.5,

trapezoidal-section disk weights ' \ Af/.AA

will be 0.5-0.65 times less ii£||

than those of rectangular disks. B a"

A flange thinned towards the
periphery is shown in Fig. 51a, b. ( a >

To impart better rigidity and rf/y' f .1"

stability in cross-sectional pla- ! gaTT

nes, the .lightened flanges are ■ Te3

often made conical (Fig. 51c).

Removal of metal from the lar- wT'

ger [diameters when lightening

even small components should be lg ‘ 5 "' Centnn 2 oi spring

kept in mind. (a) from outside d %Tetlr r ’ W * rora mslde

156

Chapter 3. Weight and Metal Content

For instance, springs should be centred from their inner diameter
(Fig. 52 b) and not from their outside diameter (Fig. 52a): this auto¬
matically saves weight.

Spacer bushes are better lightened by an external recess (Fig. 535)
and not by an internal one (Fig. 53a). The weight ratio of the former
to the latter (with identical wall thicknesses) is:

j. Fig. 53. Lightening of bushes
(a) on outside diameter; (b) on inside diameter

for the external spacer bush

Gj _ i?2~H

Gz &2 + ^3

for the internal spacer bush

Si _ ^a-Mi

g 2 ^2 + ^3

and for the relationships given in the Figure, this ratio for the exter¬
nal spacer bush is -i- = 0.92 and for the internal spacer bush:

g 2

— 0.88. However, although this is a comparatively small weight

g2

saving (8-12%), it should not be neglected, if one considers the large
numbers of such parts used in mechanical engineering.

(b) Effect of Fillets , Chamfers and Tapers

The weight of parts can significantly be lowered by increasing
wall conjugation radii, i.e., by giving the walls more smooth outli¬
nes. The weight savings obtained when changing a right-angle con¬
jugation of walls to a smooth conjugation of radius R and when
increasing this radius can be evaluated by the following figures.

3.2. Lightening of Parts

157

Case 1. Conjugation of two flat walls at an angle (Fig. 54 a). The
weight saved by enlarging the fillet radius is easily deduced from
Fig. 54a by the relation

G n 1

where r and R = are respectively the original and increased fillet
radii;

G and G 0 — respective weights of conjugated portions
From this expression it is clear that the weight saving increases
with the increase of R. For a right-angle conjugation (r = 0).

G o.785,

<? 0 4

i.e., the weight saved with respect to any right-angle conjugation
of the same length approximately equals 20%.

Fig. 54. Lightening o£ conjugations

Economy of the same order can be obtained by making the wall
conjugation skew (Fig. 54 b). The ratio between weight G of the
skew conjugation and right-angle joint weight G 0 is

G . 1

G 0 sin os-j-cos a

This expression has its minimum when a — 45°

Go sin 45°-j-cos 45°

Thus, the weight saved, relative to the right-angle joint, is appro¬
ximately 30%.

158

Chapter S. Weight and Metal Content

Case 2. Conjugation of three mutually perpendicular flat walls.
Increasing the spherical fillet radius from r to R gives

For a conjugation at a right solid angle (r = 0)

^ = 'f = 0 - 52 ’

i.e., the weight reduction is 48%.

For the spatial conjugation of three mutually perpendicular walls,
skewed at 45°

G _ sin60°

G 0 “ 3 sin2 45° ~ U ‘ 0/ ’

i.e., the weight saving equals 43%.

It should be emphasized, however, that considered here is only
the weight reduction of the conjugated section. The weight saved
in terms of the part as a whole will naturally depend on the correla¬
tion between the weight of the relevant conjugation and that of the
complete part.

The examples illustrated in Fig. 54 c-e show methods of lightening
cylindrical bodies consisting of a flat wall and a rim by introducing
fillets and tapers in the conjugation section and by replacing
the flat wall with a tapered one.

The weight saving achieved with the fillet (Fig. 54c)

G n(t — R/D)

G 0 ~ 2(2 —RID)

where D is the diameter of the rim.

For a right-angle conjugation (R = 0)

ir—r - 0 - 785

Consequently the weight saving obtained with the filletted
conjugation is approximately 20%.

The weight reduction due to the tapers (Fig. 54 d and e) is given by

_G_ _1_

G ° coaa+ T+W siaa

The G/G o ratio is graphically shown in Fig. 55_as a function of
angle a for different d/D values.

It is evident from the graph that an angle of 60° gives the most
substantial weight saving.

3.2. Lightening of Parts

159

The lowering of the weight in the considered case is in the main due
to the shortening of the rim length by m (see Fig. 54e). In the event
of a preset rim length, the G

conical form will increase the T

wall weight in the ratio

The increase in the weight °- s --

is insignificant, approaching >

4% when a — 15° and 6% oj --

when a — 20°. This is why

weight is often reconciled to_ '

the fact that conical walls • — 'ST'" '

strongly improve rigidity of , -■—^

a part. 0.51 - J - 1 - - I — . -Z-.

Conical shapes are not re¬
commended for high-speed Fig. 55. Relation between the weight of
rotating parts since for the conically conjugated cylindrical parts

- and angle a

given examples the centrifugal

forces will cause a complex three dimensional flexure, tending
to reduce the cone to a flat disk.

(c) Constructions from Pressed Sheet Metal

One effective way to save weight is to use pressed metal com¬
ponents (Fig. 56).

The parts, made in the form of solids of rotation and shown in
Fig. 56, are manufactured by spinning on lathes (individual or

Fig. 56. Substitution of stampings for cast parts
(a-b) bearing unit cover; (c-ft) v-belt pulley

160

Chapter 3. Weight and Metal Content

small-lot production) or by pressing. In mass production, when
the output scale repays the cost of press tool manufacture, it is cer¬
tainly economical to produce
large-size sheet metal components
by pressing (dashes, panels, cases,
enclosures, diaphragms, fairings,
linings, etc.).

The lower strength and rigidity
of thin-sheet parts are compensa¬
ted for by making them of a
shell or arched form, by corru¬
gation and flanging, by introdu¬
cing ties, braces, and welded stif¬
feners, etc.

Parts made from plastic metals
(low-carbon steels, duraluminum
in its annealed or freshly har¬
dened state) with sheet thickness
no greater than 3-4 mm are pro¬
duced by cold stamping, and in
Fig. 57. Gears (shell-type design) the case of deep drawing, in

several operations, with inter¬
mediate annealing stages. Hot stamping is employed when dealing
with sheets thicker than 4 mm.

Some examples of welded thin-walled components are illustrated
in Fig. 57.

Fig. 58. Table of a housing component
(a) one-piece cast structure; (6-d) skeleton structures with a thin-sheet skin

In a series of cases substantial reductions in the weight of casings
is achieved by using skeleton-constructions (Fig. 58). Elements of
parts which must have accurate mutual positions are cast and later
connected by lightweight ties. The formed skeleton is then faced
with sheet materials.

3.2. Lightening of Parts

161

The original fully-cast structure is shown in Fig. 58a. The alterna¬
tive depicted in Fig. 585 shows the first lightening step: the casting
is made smaller and the outer, functionally necessary surface, is
formed by the thin-sheet facing 1.

Fig. 60. Reduced-weight shapes

In Fig. 58c, the skeleton casting has an open lattice form, which
interconnects the central and peripheral bosses; the lattice is cove¬
red with a thin-sheet facing.

11—01395

162

Chapter 3. Weight and Metal Content

In another form of skeleton construction, Fig. 58d, the central
and peripheral bosses are linked together by T-section ribs.

A method of securing steel facings without external fastening
elements is shown in Fig. 59a. Internally threaded bosses 1 are spot-
welded to the facing and afterwards tightly bolted to the casting
skeleton.

If the design uses removable bushes, the facing can be secured by
tightening the bush flange (Fig. 595).

In particular instances casing components can advantageously
be manufactured by welding steel sheets. Such fabrication technique
can be applied to simple box-section parts, e.g., gearbox casings.
Sheet welded components are much stronger than similar parts made
of cast iron. It is not profitable to make intricate casings this way
because very many blanks are necessary and a] large volume of
welding operations.

To save weight frame and girder structures can be replaced to
a large extent by sheet steel cold-formed profiles (Fig. 60). Such
profiles are generally produced on roll-forming and bending
machines.

(d) Extrusion

A very good way of reducing weight is given by the extrusion pro¬
cess (forcing of a metal, heated to a plastic state, through a die)
used these days not only for light alloys but also for steels.

Extruded products can be made of virtually any complex shape
to suit a component’s function by introducing form mandrels into
the die holes. In particular it is possible to obtain profiles rationali¬
zed in strength and rigidity with, for instance, internal ribs (Fig. 61a,
/), partitions (Fig. 616, e ), diagonal ties (Fig. 61c), cellular and
honeycomb sections (Fig. 61 d, g , h).

Of interest is the possibility of obtaining components which have
changing profiles along their length. Such shapes are formed by
a programmed displacement of the form mandrels relative to the
dies due to which the size and contour of the profiting section,
through which metal flows out, changes.

When pressing tubes it is possible, by using a stepped mandrel
reciprocating to a preset programme, to obtain extended tubes with
changing wall thicknesses (Fig. 61i), with thicker ends (Fig. 61/, k ),
with internal circular ribs (Fig. 61Z), with wafer ribs (Fig. 61m) and
even partitions (Fig. 61 n). Internal helical ribs can be obtained by
making the mandrel rotate during extrusion process.

The method is suitable for producing plates with varying thick¬
nesses and rib depths (Fig.. 61a). '

164

Chapter 3. Weight and Metal Content

(e) Effect of Type of Loading

Rational loading of parts, with the greatest use of the material,
is one of the ways of reducing the weight of construction.

Fig. 62 illustrates various ways of loading a round bar. Stress
values are shown conventionally by the thicknesses of the full black
lines.

In bending (Fig. 62a) the most affected portions of the section are
found at the section extreme points in the acting force plane. As the
stresses approach the neutral axis they become proportionally less

Fig. 62. Stress distribution over the cross-section of a cylindrical part under diffe¬
rent loading conditions
(a) flexure; 0>) torsion; (c) tension-compression

and are zero at the centre. In torsion (Fig. 62f>) all peripheral points
have identical loads. But the annular stress values decrease and at
the centre they are zero.

The most advantageous case is when the material is in tension-
compression (Fig. 62c) when all points in the cross-section have iden¬
tical stresses and thus the material is used to its utmost.

Where possible, avoid bending and change to tension-compression
stress. The most advantageous constructions as to weight and rigidity
are those whose elements work mainly in tension-compression (e.g.,
girder and rod systems). .

If flexure is still unavoidable, for instance, on functional grounds,
its negative effects should be suppressed by the following measures;

use rational sections in which the material flow lines are distri¬
buted along the lines of maximum stresses (i.e., sections with uniform¬
ly distributed stresses); _

reduce the bending moment by shortening its bending force arm,
i.e., decrease spans, position supports rationally and eliminate
cantilever loads which give higher stresses and strains.

Often bending stresses, occurring in a tensile-compression system,
are due to asymmetry of sections, off-centre loads or curvilinearity
in the shape of the part.

3.2. TAehteningoj Parts

165

Let us consider an example showing how an eccentrically applied
load affects the stresses in a part.

Figure 63a depicts a rectangular beam (width a and thickness b)
subjected to tensile force P. On one side of the beam there is a recess
of width an {n = 0-1). The influence of the recess causes a bending
moment equal to the product of force P and the arm 0.5 an.

Fig. 63. Determination of stresses for off-centre tension

The maximum tensile stress a in the beam centre section will
equal the sum of the tensile stresses caused by the action of force P
and moment 0.5 Pan

P . 0.5-6 Pan

ab(i — n) ' & a 2(i_„)2

(3.15)

Assume that the force P is applied at the middle section centre
(Fig. 63 b). In this case the tensile stress in the mid-section will be

o. ^hL.' (3.16)

Dividing Eq. (3.15) by (3.16), we get

I^n

In Fig. 64 the a/a 1 ratio is shown as a function of n. From the cur¬
ve it is clear that the eccentric application of force P increases ten¬
sile stresses, their value being directly proportional to the eccentri¬
city. Thus, when n — 0.25 the stress is twice as high as in the case
when the load is applied at the centre. Hence, such a simple measure
as shifting the application point of force P to the centre (in this case
by 0.125a) will reduce the beam stress by half.

166

Chapter 3. Weight and Metal Content

■ Another method of strengthening is the addition of a symmetri¬
cally made recess on the opposite side of the beam (Fig. 63c). Despite
the.reduced cross section, the stresses are less due to the elimination
■of the bending moment. The stress in this case is

o 2

P

ah (1-2 n)

(3.17)

Dividing Eq. (3.15) by (3.17) we obtain

a 1—2 n , 3«(1 —2re)

1 -n + (1 — n) 2

(3.18)

The curve in Fig. 64 plotted on the basis of Eq. (3.18) shows the
o/a 2 values as a function of n. The introduction of a symmetric bila¬
teral recess with an n value of 0-0.4 ensures a certain gain in

H

■

|

■

■

■

■

■

■

1

■

■

■

■

n

■

i

■

1

n

■

■

i

■

■

■

m

■

i

s

SB

g

■

m

■

g

■

■

K

■

■

n

■

■

■

m

■

■

■

■

■

a

B

■

■

■

■

■

K(

■

■

■

■

■

S!

a

■

1

■

■

s

an i

■■

■

■

■

6_

C'2

U

1.2

1

0.8
0.6
0.4
0.2 '

0.1 0.2 0.3 0.4 0.5 0.S 0.7 08 n

□3323333

*********

Fig. 64. Stress ratios as a function of
recess width n

m

tzzm

(c)

Fig. 65. Relieving connecting rod from
bending stresses

strength. When «=0.25 and o/o 2 = max, the gain will equal 25%.
When n — 0.4 the strengths of unilaterally and bilaterally recessed
beams are equal.

A connecting rod subjected to a compressive load is shown'in Fig. 65.
An eccentrically applied load (Fig. 65a) produces additional flexu¬
ral stresses in the rod stem. To sustain these additional stresses the
designer has to enlarge the stem section, which, consequently, in¬
creases the weight of the unit; 1 '

3.2. Lightening of Parts

167

The same drawback, though somewhat less, is encountered in
the alternative presented in Fig. 656, where an out-of-centre flexure
occurs as a result of the connecting rod stem sectional asymmetry
with reference to the direction of the loading force.

A more rational design is given in Fig. 65c, in which sections are
symmetric in respect to the applied loads. In this case the load is
purely compressive. Other conditions being equal, the latter design
ensures minimum weight.

In a part subjected to flexure asymmetrical sections cause torsion
(Fig. 66) and excessive shear stresses which are summed up with
those of flexure.

Another example shows a lever (Fig. 67) with forces applied to
its ends in the plane A-A . Owing to the displacement of the acting
force plane relative to the stem of the lever, the latter is subjected
to torsion (Fig. 67 a, b). The correctly designed version, in which
the sections are arranged symmetrically relative to the acting for¬
ces, is shown in Fig. 67c.

When parts are subjected to pure bending, it is advisable to make
their sections slightly asymmetric to reduce the tensile stresses
on the account of increased compressive stresses.

Most structural materials have better resistance to compression
than to tension. Failures almost always begin in those parts of a com¬
ponent which are in tension and not in compression, since the tensile
stresses are the first to reveal the internal material defects (micro¬
cracks, microflaws, microvoids, etc.). The compressive stresses,
on the other hand, make for the closure of microdefects.

This property is very sharply expressed in plastic metals. Figu¬
re 68 illustrates tension and compression curves of low carbon steels.
Steels under tension pass through the well-known stages: after elas¬
tic deformation the metal begins to yield (portion m) and, as a result
of cold working, hardens (portion n). Upon reaching the ultimate

168

Chapter 3. Weight and Metal Content

stress limit, a neck begins to form, which terminates in brittle
failure of the specimen.

The picture is quite different when the material is subjected to
compression. After the elastic deformation period the material
continually becomes stronger as a result of both the cold working

Fig. 67. Elimination of torsion in a Fig. 68. Loading curves for a cons-
lever tant-diameter specimen made of

plastic steel (grade 20) in tension
' and compression

and transverse expansion of the specimen (cambering). No plastic
material can be brought to failure.

In brittle materials (e.g., cast iron) compression results in brittle
failure, starting with formation of cracks and terminating in fis¬
sion. However, such materials display sharp anisotropy of mechani¬
cal properties under the effect of tensile and compressive stresses.
For instance, the ultimate compressive strength of cast iron is
2.5-4 times greater than tensile strength.

Metals having plasticity somewhere midway between the above
described extremes, as a rule offer, better resistance to compressive
stresses than to tensile ones. Thus, the ultimate compressive strength
of steel grade 45 (hardened and tempered at 100°G), duralumin grade
J3.16 (after hardening and ageing) and hard brass grade JI-O70-1
exceeds their tensile strength by 1.3-1.5; 1.6-1.8 and 2-2.2 times,
respectively.

Exceptions from this general rule are magnesium alloys which,
contrary to the above, show better resistance to tensile stresses.

3.2. Lightening of Parts

169

Examples of good and bad loading methods for materials in bend¬
ing are given in Fig. 69. The lowered level of tensile stresses (Fig. 696,
d) improves part strength (despite the simultaneous increase of com¬
pressive stresses).

Fig. 69. Loading schemes for asymmetric shapes (stress diagrams are given in

the plane of the Figure)

(a), (c) inadvisable; (b), (d) advisable

In the constructions presented in Fig. 69a, b the correlation be¬
tween maximum compressive and tensile stresses is predetermined
by the profile shape and is not always the optimal one.

Fig. 70. Reinforcement of sections
subjected to tensile stresses

Fig. 71. Design versions of a cast iron
bracket

a) poor; (6) satisfactory

On the average, the ratio between the admissible compressive and
tensile stresses lies in the range of (1.2-1.5) : 1. To utilize this rela¬
tionship it is better to use slightly asymmetrical profiles, similar
to that shown in Fig. 70a. The segments subjected to tensile stres¬
ses are preferably reinforced with straps made from a material
that is stronger than that of the basic part (Fig. 706).

For materials having highly asymmetric strength properties (e.g.,
grey iron and plastics) with a better resistance to compression, the

170

Chapter 3. Weight and Metal Content

relationship between the maximum compressive and tensile stresses
is advisably increased in the ratio of their ultimate strengths.

Figure 71 shows irrational and rational designs of a grey iron
casting subjected to bending.

3.3. Rational Design Schemes

The greatest possibilities for reducing weight lie in the application.

of rational design schemes having

the least number of parts and
the smoothest power arrange¬
ment assuring compactness
and small constructional sizes.

(a) Reduction of the Number
of Links

Elimination of unnecessary
mechanism links significantly
helps to lower a device’s
weight. An instance of this is
exemplified by the elimination
of a crosshead (Fig. 72 a) in
piston engines formerly instal¬
led to relieve cylinder walls
from lateral forces caused by
the connecting rod inclination
during the crank rotation. 'It
was proved that the crosshead
function could be fulfilled by
the piston, if its height were
increased and lubrication were
better. The height of the
trunk piston engines (Fig. 726)
has been reduced nearly by
half, thanks to the elimination
of the crosshead.

Fig. 72. Elimination of superfluous links A cam drive mechanism is
in a piston engine another example (Fig. 73a, b).

In the design shown in Fig. 73a
the cam actuates the rocker arm through a tappet. In a number of
cases it is more rational to let the cam act directly on the rocker arm
(Fig. 13b). Apart from the lesser number of parts and smaller overall
sizes, this scheme brings the acting forces closer together. In the
first version the forces are balanced over the casing section h, which
means that the casing must be strong enough to withstand the
operating forces. In the second version the length of the loaded
section h x is much less, this enabling a further reduction of the weight.

172

Chapter 3. Weight and Metal Content

In the unit driving two shafts through a system of bevel gears
(Fig. 74) the elimination of superfluous members will lessen the
weight of the construction and reduce the number of the gear types
from 4 (Fig. 74a) to 1 (Fig. 746).

(6) Compactness of Constructions

Substantial reduction of weight can be reached through rational
arrangement of parts and mechanisms which decreases overall size.
This is demonstrated by the double-reduction unit in Fig. 75.

Fig. 75. Reduction of the weight of a double reduction gear

The original design containing a conventional compounded train
of gears (Fig. 75a) may be made more compact and lighter by moun¬
ting the final gear 4 of the train coaxially with gear 1 (Fig. 756).

(a) (b) (c) (d.)

Fig. 78. Rational use of material in a centrifugal friction clutch

174

Chapter 3. Weight and Metal Content

Apart from the reduction of the weight and size, such an arrange¬
ment of gears 1 and 4 will considerably lower the forces acting upon
intermediate gears 2 and 3 and determining the hearing loads, and
the loads on the casing walls. In Fig. 75a the forces P 1 and P 2 of
the drive from the first and the last gears are directed to one and the
same side so that the resultant R has rather a large magnitude. In
the design shown in Fig. 756 the forces act in opposite directions,
as a result of which the resultant R' has a much lesser value.

A further reduction of weight and size are obtained by reducing
gear diameters (Fig. 75c), Increased peripheral forces can be counter¬
acted by increasing the tooth face width, substitution of helical
or herring-bone gears for spur gears, use of harder and stronger mate¬
rials and application of a rationalized lubrication system.

The overall casing size should be utilized to its maximum so that
it will accommodate as many operative elements as possible. This
principle, which can be described as the principle of close packing,
allows significant weight savings to be obtained.

The flexible coupling illustrated in Fig. 76a transmits torque
through six stacks of washers made from an elastic material. Within
the same overall sizes eight stacks of washers can be accommodated
(Fig. 766) thus increasing the transmitted torque 1.33 times. Conse¬
quently, for a given torque the coupling can also be reduced in weight
and size.

A free-wheeling (overrunning) clutch of simple design (Fig. 77a)
consists of three balls placed into angular cutouts in the driving plate
and forced by springs into a jammed position.

The carrying capacity of the clutch can be substantially raised
by replacing the balls by rollers (Fig. 776) and increasing their num¬
ber.

In a still more closely packed construction (Fig. 77c) the elements
transmitting the torque are in the form of prisms inclined relative
to the radial direction at an angle smaller that the friction angle.
The tightening split spring ring 1 continuously turns the prisms to
a jammed position. In this case the torque is being transmitted
by practically the entire periphery of the driving plate. The carrying
capacity of such a design is tens times that of the original ver¬
sion.

An example illustrating a rational material utilization is given
in Fig. 78 which depicts a centrifugal friction clutch. According
to the' design alternative shown in Fig. 78a, the driving element is
a set of annular bronze segments 1 connected by pins 2 with a driver
(not shown in the Figure).

The driving force is equal to the.product of the total segment
centrifugal force P c j and the coefficient / of friction between the
segments and friction surfaces of the driven plate, and is also pro¬
portional to the square of the driver’s rotational speed (in rpm).

3.3. Rational Design Schemes

175

The torque transmitted by the clutch

M=:P C ' f fR = -yW Z pfR

where G — total weight of segments, kgf;

g — gravitational constant (g = 981 cm/s 2 );
o> = angular velocity, rad/s;

p = distance from the axis of rotation to the centre of
gravity of segments, cm;

R — radius, of friction surface (friction radius), cm
In the design presented in Fig. 78 b the bronze pieces have a trape¬
zoidal section and are placed in a biconical recess of the driven com¬
ponent. In contrast to the previous version, which has only one
friction surface (cylindrical), the second design has two conical
friction surfaces.

Owing to the wedge shape of the bronze pieces, the torque trans¬
mitted by the clutch, other conditions being equal (i.e., the mass of the

pieces'and radius p remaining the same), will be increased

times, where i? x is the new value of the friction radius.

Let us designate the torques transmitted by the friction clutches
shown in Fig. 78a, b as Mj and Mu, respectively.

For the relations presented in Fig. 78 b

M ii _ 1 7? _ 1 a q_n y

Mi “sin a/2 ‘ 7?j ~ sin 20° **“ *•'

Thus, the torque, transmitted by the clutch, with the weight
of the bronze pieces remaining the same, will be increased nearly 3
times as compared with the original design presented in Fig. 78a.

In the design displayed in Fig. 78c each piece is divided into three
parts: one internal (trapezoidal) and two lateral (triangular). The
centrifugal force of such a composite piece, acting on the cylindrical
friction surface of the driven component is equal (provided the total
mass of the pieces and radius p are the same) to the centrifugal force
developing in the first design (Fig. 78a). At the same time, the inter¬
nal trapezoidal element acts upon the lateral elements in a wedge¬

like manner, thus causing additional transverse

forces

Po.f
tan a/2

(P c .f — centrifugal force of the internal element) which are taken
up by the cheeks of the driven component.

In this case there are three surfaces of friction: one cylindrical and
two plane.

The additional torque

G'

R.

1

176

Chapter 3. Weight and Metal Content

where G' IG — weight ratio between the trapezoidal internal ele¬
ments and complete pieces;

F {2 = friction radius on the driven component cheeks
The relation between the total torque Mm transmitted by the
clutch and the torque Mi of the initial design will be

Mm A | dr' 1 R 2

Mi ' G tan a/2 "

For the relations given in Fig. 78c

%i

Mj

— 1 + 0.6

1

tan 20°

• 0.9 = 2.5

Thus, the separation of insert pieees without increasing their
weight will increase the transmitted torque 2,5 times as compared
with the original design.

If the design is changed to two rows of trapezoidal-section insert
pieces (Fig. 78d), then each of them will act upon two surfaces, na¬
mely: conical (cheek) and plane (central disk of the driven compo¬
nent). All in all there will be four frictional surfaces in the clutch.
Provided the total weight of the insert pieces and distance p are the
same, the torque transmitted by this clutch will exceed that of the
original one in the ratio

jtfiv

-( 1

t 1

Mi

\ sia a/2

! tan i

relations given

in Fig.

78 d

Jtfiv /

i

Mi

sin 20° 1

tan 20°

0.9

7?s

R,

— 5.2

The increased torque in this case has been achieved mostly by
halving the wedge angle of the insert pieces in comparison to the
design shown in Fig. 78 b. However, a similar result can be achieved
also in the design of Fig. 78 b by decreasing the insert piece wedge
angle from a — 40° to a — 20°, but the specific loads on the fric¬
tion surfaces in this case will be twice those in Fig. 78 d.

Thus, for the same weight of the driving elements the transmitted
torque can be increased five times by imparting to the clutch a more
rational design, or, conversely, for the same initial value of the tor¬
que the weight and overall dimensions of the clutch can be substan¬
tially reduced by the same design measures.

(c) Effect of Power Schemes

The weight of a unit to a large extent is dependent upon the power
scheme, i.e., on how the main acting forces in the construction
are taken up and balanced. The power arrangement is considered ra¬
tional if the acting forces are balanced over a short section by means

178

Chapter 3. Weight and Metal Content

of elements which operate preferably in tension or compression. To use
the existing elements of a structure is better since the introduction
of special elements is accompanied by an increase in the weight.

It is unreasonable to effect a machine drive by means of a chain
drive from an electric motor through a reduction gear unit (Fig. 79 a),
as this causes transversely acting forces on the machine and reducer
driving sprockets and produces additional loads upon the shafts and
casings. The installation is distinguished by its large overall sizes..
A more preferable drive is obtained from a flange-mounted motor
through a co-axial reducer directly connected to the machine (Fig.
795). In this case the reaction forces of the drive will be counterba¬
lanced over the shortest path in the reducer casing, not bringing
additional loads onto the system’s elements. The overall dimensions
of the entire installation are sharply decreased. In addition, all the
driving mechanisms are enclosed, which facilitates lubrication.

Illustrated in Fig. 79 c-e are typical schemes of power distribution
applied in modern designs of internal combustion engines with de¬
tachable cylinder blocks.

Three ways of transferring the flash impacts to the crankcase are
possible: through cylinder jackets (Fig. 79e), via cylinders (Fig. 79 d)
or through tie studs (Fig. 79e), named bearing jackets, bearing
cylinders and bearing studs, respectively.

The disadvantage of the first system is that flash impacts are taken
up by the cast jacket walls possessing lowered strength. Conse¬
quently, wall sections have to be thickened.

In Fig. 79 d, the flash impacts are taking up by steel cylinder
walls. The latter, due to manufacturing reasons, must have a cer¬
tain minimum thickness. Therefore they generally have an ample
safety margin, which allows them to withstand the action of gas
pressure forces. Hence, they can be loaded by tightening without
increasing their sections. This makes the bearing cylinder system
principally the lightest one.

The version illustrated in Fig. 79e is heavier than the previous two
because of the tie studs, the role of which in the first two cases
were fulfilled by the existing elements: jacket (Fig. 79e) and cylin¬
ders (Fig. 79 d).

Figure 80 shows power arrangements of turbine rotors fitted with
a series of blades.

The one-piece forged massive rotor I is quite unsuitable because
of its weight. The second design 2 is somewhat better as it has lighten¬
ing recesses at the ends.

The drum-like hollow rotor 3 has a small weight, but its strength
and rigidity are insufficient to withstand the action of the blade
centrifugal forces.

In the versions 4, 5 and 6 the drum-shaped rotor is strengthened
internally against bow by ribs.

3.3. Rational Design Schemes

Fig. 80. Power schemes of turbine rotors with insert blades

The disks can be united into a single unit on a central shaft
(rotors 7-9), clamped by means of peripheral bolts (rotor 10) or wel¬
ded (rotors 11-12).

12 *

180

Chapter 3. Weight and Metal Content

In design 7 the disks are tightened on the central shaft against
a hub, which causes undesirable bending stresses. This draw¬
back is eliminated in design 8 where the disks are tightened at the
rims.

In an original rotor design 9 the disks are positioned between
the blades, this facilitating manufacture of slots and blade assembly.

Fig. 81. Diagrams of gear drives with an intermediate gear

Welded constructions have two forms: in the first (rotor IT) the
disks during welding are centred one relative to another with the
aid of a false-arbor inserted through centring disk bores which lower
the disk strength.

In version 12 the disks are centred on spigot diameters near the
rims. This enables the disks to be made as a continuous assembly.

An example of improved power distribution scheme in a gearing
unit incorporating an intermediate gear is depicted in Fig. 81.
The arrangement of the intermediate gear greatly affects the amount
of load acting upon the gear supports.

Let the small gear 1 be the driving pinion running clockwise. It
is not wise to position the intermediate gear to the right of the drive
axis (scheme I), because the drive forces P acting upon intermediate
gear 2 add vectorially and produce a substantial force R which
loads the gear supports.

It is more advantageous to position the intermediate gear to the
left (scheme II). In this case forces P when vectorially added will to
a significant degree neutralize each other and the resultant force R
loading the supports of gear 2 will be considerably less.

3.3. Rational Design Schemes

181

The magnitude of the resultant force R will depend, in both cases,
upon angle cp between the axes 2-3 and 2-1 oTthe gear centres.

For scheme /

R — 2P sin (cp/2 + a) (3.19)

For scheme II

R — 2P sin (cp/2 — a) (3.20)

where P — driving pinion peripheral force;

a = pressure angle (in standard gears a = 20 °)

Figure 81 shows variation of force R as a function of angle cp. The
value of force R for the most disadvantageous case, i.e., when the
intermediate gear is on to the right and <p — 140° [i? = 2 P sin
(70° + 20°) =2 P] is taken as a unit force.

From the graph, the magnitude of force R in scheme II with the
same values of angle cp is much less than that in scheme I. For exam¬
ple, when cp = 100° the resultant force in scheme II (R — 0.5) is
nearly half that in scheme I (R — 0.95).

Hence, such a simple constructional measure as the repositioning
of the intermediate gear from one side to the other .can assure
a significant advantage in the power arrangement and considerably
lower the level of the acting forces.

When cp — 40° in scheme II R — 0. Actually this is an unreal con¬
dition since the intermediate gear would require an enormous diame¬
ter. The least real values of angle 9 are 80-60°.

However, should the rotational direction of pinion 1 be changed
from that in the Figure it will be more profitable to position the
intermediate gear to the right. The same will be true if, with the
given direction of running, the driving gear is the large
gear 3.

The position of the intermediate gear may be decided by a general
rule: the best position for the intermediate gear is on the side where
the force of the driving gear seems to push it into engagement.
The most disadvantageous position is when the driver tends to force
the intermediate gear out of engagement.

(d) Multi-Limb Power Schemes

Considerable weight savings can be achieved by the application
of multi-limb power schemes, i.e., by distributing the power through
several parallel channels. The transmission of torque through
the agency of several, parallel working gears (stage gearings, multi¬
satellite planetary gearings, etc.) is an example.

Figure 82 illustrates the overall dimensions of two different gear
systems: a single-limb (Fig. 82a) and a four-satellite planetary mul-

182

Chapter 3. Weight and Metal Content

ti-limb system (Fig. 8 2b) with identical gear ratios. A comparison
of the overall dimensions clearly shows the advantage of the multi¬
limb version.

Fig. 82. Comparative sizes of (a) single- and (b) milti-limb transmissions

(e) Rational Selection of Machine Parameters

Significant savings may also be attained by a rational selection
of machine parameters. As an example Fig. 83 shows two internal
combustion engines of equal power with stroke-to-diameter ratios
of S/D = 1.5 (Fig. 83a) and S/D = 1 (Fig. 836).

The small height, typical of short-stroke machines can be reduced
still more by reducing the height H of the piston (in Fig. 836, H —
= 0.75D; in Fig. 83a, H = D), or by decreasing the ratio of the con¬
necting rod length L to the crank radius R (in Fig. 836, L — 3.2jR;
in Fig. 83a, L — 3.8 R). Combination of these two measures enable
the sizes and weight of the machine to be substantially reduced.

3.3. Rational Design Schemes

183

It is necessary to stipulate the increase in piston force carrying capa¬
city because reducing piston height and the LIR ratio leads to a spe¬
cific pressure increase on cylinder walls.

In some categories of fluid- or gas-operated machines (hydraulic
presses, air- and steam-operated hammers, air- and hydraulically-
operated drives) a significant
reduction in size and weight
can be accomplished by increas¬
ing working pressures of
liquids (gas). Up to certain
limits the same is applicable
to internal combustion engi¬
nes (supercharging and higher
compression ratios), which
allows the displacement to be
decreased or, for the same
displacement, the power to be
raised.

Occasionally, for example,
in some machine power gene¬
rators, it is possible to reduce
the weight by increasing the
rotational speeds.

This method has its limita¬
tions. In internal combustion
engines increasing the number
of revolutions per minute is limited by gas suction speeds accom¬
panied, in turn, by insufficient filling of cylinders and less litre
power. In steam and gas turbines increased revolutions mean
a corresponding rise of fluid through rates, causing a rise in inter¬
nal losses.

Furthermore, higher speeds increase dynamic loads and wear.

It is also necessary to consider the characteristics of machine-
consumers of energy. If the revolutions of a machine-energy consu¬
mer are stipulated, then, increasing the revolutions of a machine-
generator will require the use of a reduction unit or a greater degree
of reduction in the existing reducer which increases the weight of
the installation.

Thus, careful consideration must be given to the positive and nega¬
tive sides of the argument when higher rotational speeds are used as
a means of saving weight.

High weight reductions can be obtained by the transfer to princi¬
pally new designs and processes. Thus, steam engines were replaced
by steam turbines as the latter have much higher power/weight
ratios. High-power internal combustion piston engines are being
ousted by gas turbines. Steam turbines apparently in time will give

184

Chapter 3. Weight and Metal Content

way to gas turbines which do not require large auxiliaries (boilers,
condensers). In the field of electrical energy installations the root
of the change will be brought about by MHD generators which
directly convert thermal energy into electrical energy.

3.4. Correction of Design Stresses

General calculation methods allow stress determinations to a satis-'
lying degree of accuracy only for a comparatively small number of
simple loading schemes. In several cases the magnitude and distri¬
bution of stresses inside the body of a component part cannot be
calculated. The majority of casings, mounting frames, beds, crank¬
cases belong to the incalculable category.

The effectiveness of the method of correcting design stresses and
lowering strength reserves as a means of lowering the total machine
weight depends on the weight correlation of the calculable and in¬
calculable parts.

Returning to the calculable parts, it must be pointed out that
calculations are based on simplifications not always holding true
in real conditions.

The most important factors determining the deviations of the
actual values of stresses and strength reserve from the calculated
ones are as follows:

scatter of material strength characteristics (ultimate strength,
yield and fatigue) from their rated values which are averaged sta¬
tistically from a series of tested specimens;

heterogeneity of material; scatter of strength characteristics in
different parts and different points of sections;

changes of materials strength as a function of the loading chara¬
cter (rate of loading);

deviation of the design loading scheme from actual loading con¬
ditions;

deviation of actual acting force magnitudes from their nominal
values;

deviation of actual stresses from nominal stresses caused by the
system’s resilience;

disregard of strength and rigidity characteristics of parts adjacent
to the part being calculated;

formation of local stresses at the component fixing points and the
points of force application;

formation of additional forces and stresses, caused by inaccurate
manufacture, assembly and installation (e.g., high edge pressure due
to a misalignment or distortion of supports);

overloads in operation;

internal stresses originating during the manufacture owing to
macro- and micro-heterogeneities of the material.

8.4. Correction of Design Stresses

185

(a) Local Stresses

Machine elements are usually rather short and the loads are applied
relatively close to each other.

Local stresses occurring at the spots where forces are applied, as
well as in supports and fixing points, propagate into the depth of
material, at times throughout the entire length of the part. Con¬
sequently, the actual magnitudes of stresses and character of their
distribution differ strongly from the calculated values.

Another feature typical of engineering components is the comple¬
xity of their forms and change in sections. At section transition
points stress concentrations occur.

Thus, in machine components a very large role is played by local
stresses which at times are the deciding strength criteria.

For example: in the conrod of a piston engine (a compressor or
internal combustion engine) its stresses formally determined by
calculation from the action of gases and inertia forces have a value
close to the actual stress value only at the section midway along the
length of the conrod quite far from the big and small ends. The stres¬
ses prevailing in the conrod ends and in the points where the ends
are mated with the rod, are very complicated, particularly, in the
conrod split big end or when knuckle conrods are involved (V- and
W-type machines). The type of stressed state, as well as the mag¬
nitude and distribution of stresses in the conrod end are influenced
by many factors, in particular by the tightening forces and resilien¬
ce of conrod bolts, configuration and rigidity of the end, and by the
crankshaft rigidity where the crank neck joins the conrod.

To account for all these factors is difficult. True stress values are
often only defined by analyzing fractures, cracks and breaks which
appear after the conrod has run in the machine.

When a hollow cylindrical shaft supported at the ends is subjec¬
ted to bending, a simplified calculation scheme is used assuming that
the load is concentrated in the centre, halfway between the bearing
surfaces (Fig. 84a), or that it is uniformly distributed throughout
their length in the plane of acting forces (Fig. 846), and the stresses
are determined from a simply supported beam. Thehchemes do not
consider the actual force distribution along the length and circum¬
ference of the supports or the influence of transverse load components
on the strength and rigidity of the part, nor the influence of support
rigidities on the load distribution or even the values of edge pressures
and local stresses in. the loaded areas.

If a part is interference mounted in its supports, then additional
compressive and crushing stresses occur in the seating areas. If the
mounting has a loose fit, shocks will accompany any alternating or
pulsating loads, causing additional stresses.

186

Chapter 3. Weight and Metal Content

A true loading picture can be imagined (with a sufficient degree
of confidence) for a particular case if we assume that the part is
mounted in plain liquid-friction bearings and that the transversely
acting force is taken up by a plain bearing (e.g., a floating gudgeon
pin of a crank and conrod assembly).

It is well known that the pressure distributed along the bearing
oil layer changes in accord with a parabolic curve whose maximum

Fig. 84. Loading schemes for an axle supported at two points

ordinate is 2.5-3 times greater than the mean specific pressure
k ik = = j 2 )- In the transverse section the pressure is distributed over
an arc of 90-120° (Fig. 84c).

From a comparison of formal loading schemes (see Fig. 84a, b )
it becomes obvious that the first scheme overestimates the stresses
occurring in the critical section of the part, and the second scheme
underestimates these stresses; neither of these schemes considers the
transversely acting components of the load and the stresses and
strains produced by these components.

The load distribution picture over the oil bearing layer depicted
in Fig. 84c is closer to the actual condition. However this picture
may substantially change owing to the elastic deformations of the
shaft and bearings, the emergence of increased edge pressures, etc.

The local stresses and strains, in the areas of force application,
can reach significant values and decisively affect the serviceability
of a part.

Let us consider a typical practical case.

The end of a crankshaft mounted in plain bearings was loaded with
a comparatively modest load P from a gear positioned between the
supports. Usual strength calculations for the crankshaft with per¬
mitted stress values of 20 kgf/mnU led to the shaft end configuration
shown in Fig. 85a. The recurring front bearing failures made the

3.4. Correction of Design Stresses

187

designers examine the unit more carefully. When the crankshaft
was bench-tested and subjected to an acting force equal to the design¬
ed one, it was] proved that the shaft end deformed, acquiring an
elliptical cross section whose major axis exceeded the bearing diame¬
ter by 0.2 mm. Meanwhile, with a diametral clearance of 0.1 mm an
ellipticity amount of only 0.05 mm eliminates completely the
wedge shape of the oil layer in the zone of the maximum closeness
between the shaft and bearing, which is indispensible for the
correct functioning of the bearing.

mm

wwwwwwWv ■

///AiW//////////// ■

!

■

■1

eeaeeeei

,BBB]

(a)

Fig. 85. Increasing the rigidity of a crankshaft by pressed-in plug

This defect was rectified very simply without serious weight
penalty. A plug press-fitted into the shaft end (Fig. 856) greatly
increased the rigidity of the part and enabled the true cylindricity
of the shaft to be kept.

The arbitrariness of strength calculations can be illustrated by an
example of pure shear. With usual stress calculations it is generally
assumed that a part operates over its full section and that shearing
stresses are the same throughout the entire section and equal to the
quotient obtained from the division of shear loads by the area of the
section.

However, the actual picture of the stress distribution sharply dif¬
fers from the described scheme. The mechanism of shear has been well
studied on the punching of sheet materials. Along its contour the
sheared section is subjected to high crushing and shearing stresses
under the action of which in the material there occur, at first not
deep, microflaws and plastic shear before the bulk of the material
begins to respond.

With reference to common engineering parts this means that the
serviceability of parts is disturbed long before the shear stresses in
the section attain dangerous values. In fact, the part fails as a result
of stress concentrations in the surface layer, accompanied by local
crushing and plastic deformation at the place where the shear force
is applied. This phenomenon is sharply expressed in the shear of
cylindrical parts, when the stresses are concentrated over a small arc

188

Chapter 3. Weight and Metal Content

of the surface close to the acting force. The softer the material of the
part being sheared in comparison to the punch and the higher the
punch rigidity, the greater the crushing.

After crushing the surface layer of a part becomes stronger due to
work hardening and the bulk of the metal entering the action. The
joint, however, is proved spoilt due, in the first instance, to the
sharp drop in strength from local cracks and- flaws which become
stress concentrators with successive load applications and, secondly,
to the disturbed joint geometry caused by the onset of crushing.

From the above discussion we may derive the following important
rule for designing connections working in shear: the hardness of
the cutting material and the material being cut must be as equal as
possible; the greater the surface hardness of the parts, the more
reliable the insurance of the connections against fracture.

The complexity of shear stress distributions cannot be accounted
for by elementary calculations. Such an important factor as the sur¬
face hardness of a material, which determines the connection’s servi¬
ceability, is also neglected.

( b ) Effect of the Resilience of System

Formal calculations do not usually account for resilience charac¬
teristics of a constructional system, which in practice have a great
effect on the true stress values.

As an example we consider a shaft, a component widely used in
machine building, supported by its ends and subjected to a bending
load in the centre from a connecting rod. For the sake of simplifi¬
cation we neglect the influence of transverse load components,
as well as support reactions, keeping during the calculations to the
standard schemes of load distribution along the axis of the part.

If the rigid sections are assumed to be at the connecting rod centre
and at the edges of the supports (scheme 1, Table 6), then it is possi¬
ble to consider that the shaft is acted upon by concentrated force P
at the centre and that the bearing reactions are applied at the extreme
shaft points separated by span I.

With this loading scheme the stresses in the critical section in the
shaft are

PI

where W is the resisting moment of the shaft section

The maximum flexure of the shaft

, _ Pl *
h 4 8BI

where I — moment of inertia of the shaft section;

E — Young’s modulus for the shaft material

Load Distribution Schemes

Table 6

Continued

3.4. Correction of Design Stresses _ 191

Assuming these values of o and / to be equal to unity, we consider
how the strength and rigidity of the system are changed with the
loading schemes.

In the designs where the rigid sections of the supports are at their
centre it is possible to assume the following probable schemes of
the acting forces: bending by the point load P with a span equal to
0.75Z (scheme 2), and bending by a load distributed parabolically
(scheme 3).

Under such circumstances we shall obtain respectively

02 = 0.7505; / 2 = 0 . 42/5

and

o 3 — O.SOoq; / s = 0 . 37/5

The uniformity of the load distribution is improved with an
increase in the rigidity of the connecting rod and supports (scheme 4).

In this case

04 = 0 . 505 ; / 4 = 0.27/j

The strength and rigidity of the design can further be enhanced
by transferring the rigid sections to the supports inner ends (schemes
5, 6 and 7). Depending on the connecting rod rigidity and on the
assumed scheme of load distribution, we obtain the following stress
and flexure values

O5-7 = (0.5 to 0.25) 05
/s -7 = (0.125 to 0.015) /j

The strength and rigidity will grow still more (schemes 8 to 11)
with encastre ends (without clearance or with an interference).
Depending on the connecting rod rigidity and the law of load distri¬
bution, the values of a and / will vary within the following limits:

< 78—11 ~ (0.25 to 0.04)05
/s-u = (0.031 to 0.0035 )/1

With fixed mounting in the rod (also without any clearance
or with an interference), when the shaft ends are cantilevered
(illustrated in schemes 12, 13, and 14) the gains in strength and
rigidity are less. This is attributable to the inferior strength and
rigidity characteristics inherent in cantilevered systems in general.
In this case the magnitudes of a and / are

O 52-14 = (0.5 to 0.08) G t

/i 2 -u = (0.125 to 0.012) / t

The general conclusion, that can be drawn from the above review,
says that the resilience of a system and the manner in which loads
are distributed have immense influence upon strength and rigidity.

192

Chapter S. Weight and Metal Content

Within the range of the above described schemes, the stress value
can be 25 times and flexure, approximately 300 times (scheme 11)
less than those in the original scheme 1.

The gain due to the system’s resilience can be realized by making
the design as rational in shape as possible. However, it should be
noted that any assessment of the system characteristics, particularly,
the law of 'load distribution along the axis of a part, is unavoidably
somewhat arbitrary. Hence, the above-listed relationships are rather

Aid

F

fP-

— %|r-

(A )

Fig. 86. Loading schemes for a pin supported at two points

construction recommendations. Their value as to the calculation
accuracy is relative because they only indicate probabilities for the
given form of load distribution.

It should also be noted that the loading scheme and law of the
load distribution are dependent not only upon the design of the unit,
but also on its deformability which is determined by the level of the
acting stresses, the material of the part (Young’s modulus), etc. With
the given design of the unit the loading scheme sets in on its own
accord as a result of the interaction of the loads and deformation
developing in the unit.

We can explain this by the same example of bending a simply sup¬
ported shaft which has its fixed section at the centre between the
supports (Fig. 86). The loading scheme illustrated in Fig. 86a is
possible with small loads or with a high rigidity of the system. As
the force grows (or section rigidity lowers) the system is deformed^
as shown in somewhat exaggerated manner in Fig. 86& (for the sake
of simplicity only the shaft deformation is shown). The deformation
acts as a strengthening";agent causing load concentration on the edges
of the support surfaces. As a result a new system of acting forces
arises, which follows a triangular law or (as shown in the Figure)
a parabolic one. In this case the stresses are sharply lowered, and
still more sharply decreases the deformation.

As the loads increase (or the section rigidity falls) the scheme appro¬
aches that of pure shear (Fig. 86c), which lowers stresses and deforma¬
tion even more.

Thus, the load increase is accompanied by a self-strengthening
process caused by the development of deformation which contributes

193

3d. Correction of- Design Stresses

to a more favourable load distribution. However, the deformation
simultaneously increases the system rigidity acting reversely and
at a certain stage equilibrium is reached, fixing the load distribution
and defining the true values of stresses and strains occurring under
the loads. The actual instantaneous strength and rigidity of the sys¬
tem are fully dependent upon the magnitudes of the loads and rigi¬
dity of the areas which transfer and take up the loads. In most cases

Fig. 87. Load distribution along a gear tooth lor differently designed gears

the fashion of loading cannot be established by calculation. Thus
it is clear that a system, self-adapting to load conditions, arrives
at a state intermediate between the extremes depicted in Fig. 86a, c.

Let us consider one more example of the effect of rigidity on stress
values: the load distribution over the tooth length of two gears in
mesh (Fig. 87). The character of load distribution and its maximum
value depend on the relative position of gear webs. If the webs are
located in the same plane close to the end face of the gear (Fig. 87a),
then the load is concentrated mostly at the rigidity node, i.e.,
in the plane of the webs. The remaining parts of teeth positioned on
a relatively elastic rim are loaded less. The probable load distribu¬
tion in this case is shown by the triangle, with its apex found in the
plane of the webs. Maximum load on a unit length tooth equals 2 p,
where p —average load, with the usual assumption that the load is
uniformly distributed over, the entire tooth length.

If the webs are positioned in the plane of engagement symmetry
(Fig. 8Tb), then the picture of load distribution is expressed by the
triangle having its apex in the plane of symmetry. Maximum load
as.in the previous case is equal also to 2 p.

The load on the teeth will, he distributed more evenly when the
webs are arranged on different sides relative to the plane of engage¬
ment symmetry (Fig., 87c).

13-01395

194

Chapter 3. Weight and Metal Content

(c) Effect of the Strength of Mating Parts

In practical calculations the strength of parts (hubs, bushes, sup¬
ports, etc.) mating with the designed part is often neglected. The part
being designed is generally treated in isolation and the effects of
adjacent parts, which transmit or accept the load, are considered
(and not always) in the design scheme of the load distribution along
the part. This is admissible only when the length of a mated piece
is small in comparison with the length of a part being designed, or

if these two parts are mated
with clearance. If the length
of the mated part is com¬
mensurable with that of
the part being designed
(especially when the fit is
without clearance or with
interference) any disregard
of the mating part can result
in quite sizable errors.

Let us again consider the
same case of a simply sup¬
ported axle loaded with
transverse force P. In
Fig. 88 (in a force-deflec¬
tion coordinate system) are
shown the results of testing
three specimens made from
steel grade V8A heat-treated
to a hardness of 45 Rc.
Specimen 1 is a rod 10 min
diameter and 80 mm long.
Specimen 2 is a similar
rod upon which three bushes are slide-fitted, each bush being 18 mm
in diameter. The two bushes at the rod ends imitate support bushes,
while the middle one imitates the bush of a conrod. Specimen S
(a reference specimen) is a rod 18 mm in diameter with two annular
grooves corresponding to the end clearances between the bushes in
specimen 2.

The specimens were tested to failure. Specimen 1 failed at an
800-kgf load and a deflection of 1 mm; specimen 2, at 2900 kgf and
2.2 mm. The composite specimen 2 proved stronger than the large
solid specimen S having the same configuration, which failed at a
load of 1700 kgf and 1.5-mm deflection. This can be explained by
stress concentrations at the undercuts.

With similar strains the composite specimen proved to be appro¬
ximately twice as strong, and with similar loads, 3-5 times more rigid

3.4, Correction of Design Stresses

195

than the plain one and as to the failure load, 3.7 times stronger.

The obtained values refer to the region of plastic deformations.
Nevertheless they illustrate the fact that calculations which disreg¬
ard the effect of mating parts are only very approximate.

(d) Deviation of Acting Forces from Nominal Values

Another reason for calculation inaccuracy is the difficulty of
defining in a number of instances the actual values of the acting
loads. This is particularly true of changing, pulsating and impact
loads.

Let us take for example, a well investigated mechanism, acrank-
and-connecting-rod assembly. In internal combustion engines the
initial data for the strength calculations is the maximum gas pres¬
sure forces acting on the piston. It would seem that in determining
of these forces there can be no mistake. However in practice the
value of these forces and the stresses caused by them in the links
of the mechanism depend upon many factors, but most of all, on
the resilience and mass of the links.

Some of the flash energy is spent on the work of the resilient ex¬
tension of cylinder walls, fixing studs and crankcase (within the
limits of elastic strains). Another part of the energy is consumed in
compressive deformations of the piston and connecting rod and also
in the bending of the gudgeon pin, bending and torsion of the crank¬
shaft, and expulsion of the oil layer from the clearances between the
mating parts. A substantial portion of energy is spent on accelerat¬
ion of reciprocating and rotating members. The largest part of the
energy is returned and recovered at later stages in the cycle but ener¬
gy is wasted on work of viscous shear, expulsion of oil layers from
clearance gaps and hysteresis during elastic strain.

The greater the resilience of the system (i.e., the longer are the
fastening studs, the more yielding is the crankshaft, the lesser are
the cross sections and inertia moments of the parts, as well as the
modulus of elasticity of the material), the smaller is the actual force
stressing the parts and the weaker the forces reaching the final links
of the mechanism. The introduction of resilient connections into
the system, e.g., provision of spring couplings between the shaft and
the final member (flywheel, screw propeller, generator, reducer),
elastic torsion-bar suspension of the motor, etc., will sharply decrease
maximum stresses in the system.

Any increase in the mass of intermediate links increases the in¬
stantaneous value of maximum forces acting upon preceding links,
and lessens the forces acting on the subsequent links. The action of
the enlarged mass of intermediate links is similar to^that of an anvil,
which absorbs the energy of an impact and protects”other parts from
higher stresses.

13 *

196

Chapter 3. Weight and Metal Content

Of certain importance is the rate at which the pressure of operating
gases is built up at the moment of ignition. The greater this rate,
i.e., the closer the load approximates an impact one, the greater are
the stresses in the system. However, the strength of the material
increases with the increase in the loading rate.

( e) Internal Stresses

Internal stresses, which are inherent in any material, arise during
manufacture and operation. The actual strength of a part depends on
the interaction between internal stresses and stresses caused by exter¬
nal loads.

When establishing admissible stress values the previous history
of the part is often neglected, as well as its future history, i.e., the
gradual changes of mechanical properties of the part material
during machine operation. Such changes can both weaken and streng¬
then the material. The weakening factors include corrosion, wear
and damage of part surfaces; accumulation of microdamages as a
result of multiple-repetitive loads; local tempering, which follows
the heating brought about by cyclic loads.

The strengthening factors include the so-called material “training”
processes caused by short-time stresses which exceed the yield
limit; deformation strengthening caused by the structural changes
in the stressed microvolumes of the material; spontaneously deve¬
loping ageing processes accompanied by recrystallization and spread
of internal stresses. Positive effects are felt from the design’s adap¬
tability, i.e., total dr local plastic deformations,' arising under the
effect of overloads and causing load redistribution. A definite strengt¬
hening effect is obtained at the first wear stage (smoothing of sur¬
face microirregularities), favouring an increase in actual contacting
surface areas, lowering the pressure peaks and levelling out the load
on the surfaces.

It is generally accepted that defects appearing during the manufac¬
ture and use of a part are mostly random ones. This circumstance
partly explained by the dispersion in the strengths characteristics
of parts, a fact which is well known. Amongst components in one
and the same batch, all of which were subjected to similar processing,
some have high durability, while others will fail within quite a
short period of use as a result of unnoticed initial defects or defects
occurring in use.

Internal stresses are generally divided into the following three
categories:

the first —those developing in a substantial volume of the part
due to the heterogeneity of the metal’s macrostructure (some¬
times arbitrarily called macrostresses);

3.4. Correction of Design Stresses

197

the second—those developing in the volume of the crystallites or
in a group of crystallites due to the heterogeneity of the crystallite
metal structure (sometimes arbitrarily called microstresses);

the third —those developing in the submicroscopic volumes due
to defects and distortions in the atomic-crystalline lattice (sometimes
arbitrarily called submicrostresses).

Stresses of the first group are often the result of technological
processes which the component undergoes during forming operations.
Since such technological treatment is a many-stage process, the
stresses existing in the finished component are the result of accumulat¬
ed and interacting stresses arising at each stage of the process. Hete¬
rogeneities of an ingot pass to the forging (or rolled stock); hot-pres-
sure processing introduces new heterogeneities. Machining opera¬
tions although eliminating the nonuniformity present in the cut-off
metal, causes redistribution of the stresses which had been intro¬
duced at the earlier stages, leading to new additional stresses at
surface layers. Heat treatment partially eliminates stresses, from
the earlier stages, but at the same time it creates stresses of its
own.

The internal stresses appearing in castings are generally the
result of a non-uniform crystallization in foundry and shrinkage of
material when cooling. As a rule, the stresses are concentrated around
extrusion defects, shrink voids, pores, etc., often obtaining large
values and inviting fractures, local, cracks and general fissuredness
of the casting. Other defects, often found in castings, are burnt spots,
entrapped slag, spalled particles of the mould, oxide inclusions,
sulfides and silicides, local liquation and local dendricity.

The main source of internal stresses when hot working by pressure
is the uneven metal flow through sections oriented differently to
the action of the deforming tool. Especially often irregularities
occur at points of discontinuity, at conjugation zones of different
section thickness, at exterior and interior corners. Other defects in¬
clude shearing of material, folding, spalling, overlaps, hair cracks.

Lemon spots or flakes as they are usually termed in the parlance
of iron and steel manufacture (which are in fact non-welded hydro¬
gen bubbles) are rather common defects in alloy steel.

High residual stresses occur during heat treatment, particularly
when hardening is followed by rapid quenching. Because of the
non-uniform removal of heat from surfaces and inner layers of metal,
as well as from changing sections, zones of high stresses are formed,
some limes leading to quenching cracks. In materials with low har-
denability the phenomenon is aggravated by the interaction between
hardened and soft zones. The martensite zones which have the larg¬
est specific volume are subjected to compression by the action of the
adjacent denser layers of troostite, sorbite or perlite structures in
which reactive tension stresses arise.

198

Chapter 3. Weight and Metal Content

Substantial stresses occur in surface layers in the process of machin¬
ing. Plastic shear and destruction of metal in the cutting process are
accompanied by residual tensile stresses arising in neighbouring
layers. The rougher the machining (i.e., the thicker the cut-off layer
and the greater the applied cutting force), the higher the residual
stresses (e.g., in rough-turning steel residual tensile stresses reach
80-100 kgf/mm 3 ). The mechanical stresses are added to thermal stres¬
ses appearing in the zone of cutting as a result of heat liberation,
as well as to stresses forming from the structural and phase trans¬
formations in the zones of intensive heat liberation.

Even fine machine operations, such as, for example, grinding, are
accompanied by residual tensile stresses reaching 20-40 kgf/mm 2 .
The greatest particular defect in grinding is a burnt spot, which
causes local tempering in chilled steels and the development of soft
spots of troostite or sox*bite. Conversely, normalized or structurally
improved steels, when subjected to a high temperature and emulsion
cooling, can harden and form hard spots of martensite.

High local stresses can also originate during welding from the
local heating up to a melt-point temperature and subsequent cooling
accompanied by material elongation in the weld seam. Local
stresses also arise in the zones of weld defects (incomplete penetra¬
tion, poor fusion, loose areas, oxide and slag inclusions, etc.).

Stresses of the second group arise in the main because of the metal
crystalline structure heterogeneity and due to differencies in phy¬
sical and mechanical properties of the alloys phases and structures.
Phases (e.g., ferrite, austenite, cementite and graphite in ferrous
metals) have different crystal lattices, thereby their density, strength
and resilience, as well as thermal conductivity, thermal capacity
and thermal expansion characteristics are also different. Structures
appearing as phase mixtures (e.g., perlite in steels) or hardened
structures possess, in turn, such properties which make them differ¬
ent from neighbouring structures. Variations in crystalline orienta¬
tion of metal grains cause anisotropy of the physical and mechanical
properties in the microvolumes of metal. As a result of the combined
action of these factors both intra- and inter-granular stresses come
into existence, starting from original crystallization and following
subsequent transformations in the process of cooling. At high tem¬
peratures the stresses are counterbalanced by the plasticity of mate¬
rial. However, they develop in low-temperature areas as a result of
phase recrystallization and drop out of secondary phases (phase cold-
hardening); with every total or local rise of temperature (due to the
difference between heat conductivity and coefficient of linear expan¬
sion of structural components); due to the application of external
loads (because of difference and anisotropy of mechanical properties);
and also as a result of cold hardening, which follows total or local
transition of stresses beyond the material yield limits.

3.4. Correction of Design Stresses

199

Amidst other sources of the second group stresses are intra- and
inter-granular foreign inclusions, microporosity, segregates, residual
austenite (in chilled steels).

Stresses of the third group are due to numerous submicrodefects
(dislocations) inherent in atomic-crystalline metal lattices. The
fields of elastic stresses, forming around dislocations, can break the
interatomic bonds, i.e., cause plastic deformation.

The third group includes also stresses which develop at the phase
boundaries of phases with different crystal lattices (e.g., cementite
and ferrite in Fe — C alloys, cuprous, magnesian and ferrous phases
in A1 alloys). Belonging to the same group are also the stresses occurr¬
ing at the boundaries of subgrains (crystalline blocks) as a result of
reorientation of the latter either due to heat treatment, or under
the action of external loads, or else caused by cold hardening.

Occasionally submicrostresses can encompass extensive regions
and turn into microstresses (for example, this is the case, when such
stresses develop at grains boundaries owing to the difference
between crystal lattices of grains and interlayers of the mate¬
rial).

Multiple distortions of crystal lattices, developing throughout
extensive areas of material, can cause macrostresses, which will
encompass a number of layers or the entire thickness of material
(such are, for instance, the stresses occurring in macrovolumes due
to the total plastic deformation of metal).

Numerous microstresses eventually turn into macrostresses co¬
vering extensive areas of metal or the whole metal volume. This
can be illustrated by phase cold hardening, but it should he emphasi¬
zed that phase cold hardening will lead to a higher dislocation den¬
sity, as well as to distortions of crystal lattices and crystal blocks
boundaries, thus producing submicrostresses throughout the entire
metal volume.

It is obvious that classification of internal stresses into the three
groups is rather arbitrary since all of these stresses are intermixed
and may be of a local, regional or general nature.

Of significance from the practical point of view is that internal
stresses can act either as weakening or as strengthening factors. One
must bear in mind that internal stresses, whose sign coincides with
that of the working stresses, are hazardous (e.g., tension-tension pair
of stresses). Conversely, should the sign of internal stresses he oppos¬
ite to that of the working stresses, the situation is favourable (e.g.,
tension-compression pair of stresses). Furthermore, it is noteworthy
that internal stresses of the same sign are always accompanied by
counterbalancing stresses of opposite sign in the neighbouring volu¬
mes. The relative value of oppositely-signed stresses depends on
the extent of the material volumes in which these stresses are
predominant. Thus, we may conclude that the critical factors deter-

200

Chapter 3. Weight and Metal Content

mining the strength parameter are: first, the position and orientation
of the stressed volumes relative to the acting working stresses; and,
second, the magnitude of the internal stresses whose sign and direct¬
ion coincide with those of the working stresses. The heterogeneities
which produce areas of high tensile stresses, break the continuity of
metal, and cause flaws and local plastic shear are, in effect, metal
defects. The heterogeneities, which produce extensive zones of com¬
pressive stresses, contribute to metal consolidation and prevent the
inception and propagation of plastic shear are, in fact, strengt¬
hening factors. They can be utilized to enhance the resistance of
material to working loads.

Out of the weakening factors the most dangerous are macrodefects,
as they create regions in which tensile stresses of the first group
occur. Moreover, when working loads are applied, the macrodefects
behave as stress concentrators, thus increasing still more the high
level stresses.

Modern technology possesses efficient means of preventing and
rectifying macrodefects.

Thus, the defects forming at the primary stage, during melting,
to a significant degree are eliminated by melting in vacuum, in elec¬
tric or electron-beam furnaces, by refining of steel, by electroslag
remelting, etc. Ingot defects are lessened by vacuum casting, by
assuring uniform crystallization of ingots, and also by using conti¬
nuous-casting techniques.

Casting defects are eliminated by imparting rational shapes to
castings, which assure more uniform crystallization and paralyze
shrinkage effects; by properly chosen sand; by vacuum and die
casting, etc.

Many types of microstresses are successfully eliminated by stabi¬
lizing heat-treatment. Crystal-structural defects of blanks, produced
by hot plastic working, are eliminated by recrystallization annealing.
Internal stresses in castings can be removed by low-temperature
annealing (ageing).

Hardening stresses are reduced by special processes (step-by-
step hardening, isothermal hardening, etc.). The. stresses liable
to appear because of limited hardenability are eradicated by the
introduction of alloying elements (nickel, chrome, molybdenum
and, particularly, boron).

The stresses, being caused by machining, are eliminated by proper
selection of cutting conditions and removal of damaged layers by
finishing operations (fine grinding, honing, super-finishing, power
polishing, etc.).

In general the main task is to choose the most optimum production
technique and observe the established production processes.

Macrodefects stealing in despite all the precautions should be
discovered through careful inspection of all blanks at all stages of

3.4. Correction of Design Stresses

20 i

their production by the application of high-sensitive methods (X-ray,
magnetic and ultrasonic defectoscopy).

Deep crystal-structural defects are detected only by cutting micro¬
sections, i.e., through destruction of a part. In practice this method
is applied on the basis of random sample testing.

The stresses of the second and particularly of the third group are,,
in effect, unavoidable. Here, the primary aim is not to eradicate
the stresses (which would be impracticable), but rather to control
their behaviour and utilize them for strengthening the material.
Thus is the subject of the so-called strengthening technique which,
has extreme practical value.

(/) Experimental Determination of Stresses

Limitations and drawbacks of calculations compel the designer
to determine stress values through experiments.

For the investigation of stresses the optical-polarization technique
is of value. The method is based on the ability of certain transparent
elastic materials to change their optical properties when acted upon
by stresses.

The stresses in flat specimens are easier to determine. A specimen
made of an optically-active material (generally organic glass), is.
placed in a beam of monochromatic polarized light and viewed thro¬
ugh a second polarizer, whose light path intersects that of the first
polarizer. Should the specimen he free of stresses, the second pola¬
rizer will extinguish the light beams that have passed through the
first polarizer, and the specimen appears dark.

However, when subjected to a load, the specimen material will
acquire a quality to divert the plane of polarization through an
angle which is proportional to the magnitude of existing stresses.
A part of polarized light passes through the specimen, which, owing
to the interference of light beams, will he seen as streaks of altern¬
ate light and dark bands. The intensity and position o! these bands
will characterize the magnitude and direction of the existing stress¬
es. When exposed to a beam of white light, the specimen will show
coloured bands with continuously changing colours.

Such a method is applied when examining the distribution of
stresses in regions with concentrated loads or in areas with stress
concentrators (local weaknesses, sharp transitions, etc.).

Employment of this method for determining spatial distribution
of stresses is very complex, which limits its value.

In recent years for experimental researches of surface stresses wide
use has been made of strain gauges.

A strain gauge is a wire element which registers change of distance
between two points on the specimen when the latter is affected by
some load. The magnitude of the stress is determined indirectly from

202

Chapter 3. Weight and Metal Content

the deformation on the basis of Hooke’s law. Of the numerous types
•of strain gauges those most convenient and mostly used in practice
are electrical resistance ones made as loops of wire or foil (Fig. 89a, b)
0.01 to 0.03 mm thick, glued to a strip of tough paper. The strain
gauge is then glued upon the surface of the part to be tested in such
a way that the length of its loops coincides with the direction of
the expected deformation.

The measuring device consists of a current source, a galvanometer
and a bridge comprising four equibalanced resistors, one of

which is the strain gauge
(Fig. 89c).

The equation of balance is

R^_Rs_

Rz R/,,

Usually the R 2 resistor is
also a sensor similar to R x ,
and the R s and i? 4 resistors
are equalized, and the gal¬
vanometer set to zero. Any
length change in the speci¬
men changes the length of
the strain gauge wire, thus
altering its electrical resi¬
stance and causing a cur¬
rent, which is proportional to the strain, to flow through the
galvanometer circuit. To exclude the effects of temperature the
gauges should be made of elinvar or constantan.

If the deformation direction is unknown, it is advisable to position
a number of strain gauges at 45° to one another. This will allow
both the direction and value of deformation to be determined.

The base length of a strain gauge (i.e., the length of loops) is gene¬
rally 20 mm. Yet, strain gauges with 3-5-mm bases are made to enab¬
le differential investigations of strains on intricately shaped com¬
ponents over tiny areas of surface to be carried out.

When studying quickly changing deformations (e.g., in the event
■of cyclic loads), an amplifier is introduced into the circuit and the
deformations recorded by means of an oscillograph. This method
is applied when testing pulsations. The specimen is installed on a
test bench and subjected to vibrations which simulate working loads.
The same method can be used to study deformation of parts on a
running machine.

The strain gauge indications reveal the surfaces withstanding the
highest tensile stresses, which consequently need strengthening.
■Small values or complete absence of stresses shows it is possible to
lighten the part at these sections.

Transducer

•O 0>

(a)

Fig. 89. Electric tensometers

•(a) wire strain gauge; ( b) foil strain gauge;
(c) measuring bridge

3.4. Correction of Design Stresses

203

Internal stresses in the material are exposed in the following way:
a strain gauge (sensor) of a measuring instrument is glued upon the
surface area to be examined and the instrument set to zero; the sec¬
tion of metal to be tested is cut out and the internal stresses deduced
from its size change.

Recently strain gauges with a base length of 0.5 mm have been
manufactured. Also produced are semiconductor (silicon) strain
gauges, whose sensitivity factor is 100-200 times higher than that of
the constantan sensors and which have an elastic-plastic measuring
range of up to 20%.

Fatigue tests are conducted with the aid of multichannel in¬
struments which enable measurements to be taken simultaneously
at many (up to 200) points. The cyclic stresses occurring at these
points and varying between 50 to 50 000 Hz are recorded in digital
or coded form on a film or tape. The obtained data can also be trans¬
mitted over a distance to an illuminated display, where the data
is presented in the form of stress curves.

Temperature-compensated strain gauges employed for strain mea¬
surements under high temperatures exclude the effects of the apparent
strains due to the thermal surface expansion. Such compensated
constantan wire gauges allow temperatures up to 300°C to be mea¬
sured, those of nichrome wire, up to 750°G and. of platinum wire, up
to 1100°G. High-temperature strain gauges are fixed on the tested
surface by means of a ceramic cement.

The lacquer-coating method has the advantages of simplicity and
visuality. The surface to be tested is coated with a thin film of lac¬
quer. After a load has been applied to the specimen, in the zones of
high strain in the lacquer coating there will form cracks perpendicular
to the direction of tensile stresses. This enables stress directions to
be defined. If loads are applied gradually and the ultimate strength
of the lacquer is a strain function known, then from the appearance
of the first cracks the deformation (and stresses) may be deter¬
mined.

As the load increases, the cracks widen and, at the same time, new
cracks appear in the regions where stresses begin to exceed the
ultimate strength of the lacquer film. The final pattern of the cracks
show the tensile stress distribution over the area being researched.

In compression zones the orientation and magnitude of compres¬
sive stresses can also be deduced from the wrinkles and creases
occurring in the lacquer film. Further compressive stress increases
turn these wrinkles and creases into cracks.

One of the simplest formulae of the lacquer composition reads:
60 g resin and 10 g celluloid dissolved in 100 g acetone. By varying
the composition a spectrum of lacquers with different strength cha¬
racteristics may be obtained, thus increasing the application and
accuracy of the method.

204

Chapter 3. Weight and Metal Content

Surface strains occurring under high temperatures can be deter¬
mined through the agency of brittle ceramic coatings deposited by
hotpulverization upon the surface to be tested.

The low sensitivity and the impossibility of quantitatively measur¬
ing stress values are disadvantages of the film technique: films crack
only under sizable strains, which belong to the category of plastic
deformations. To observe elastic deformation by the method is dif¬
ficult. The film-coating method by no means can replace other,
more exact methods of stress measurement, for example, strain
gauges. However, this simple method does allow the general charac¬
ter of the stress distribution to be quickly seen and, with some skill,
weak and nonrigid sections to be faultlessly located and, if necessary,
strengthened.

Moreover, it is a valuable aid when applying and positioning
strain gauges, since it allows the stressed areas to be preliminarily
selected.

The film method is used for studying stresses on a working ma¬
chine. Like strain gauges, it only defines the amount of stresses on
the specimen surface, which, in most cases, are of decisive impor¬
tance for the component’s strength.

The most simplest method of checking parts for strength and
rigidity is by bench tests under a static load and conditions appro¬
ximating most closely the working ones. The resultant deformations
are measured by indicators or strain gauges.

Bench tests provide best results when checking such parts as
high-speed rotors, e.g., rotating discs' of centrifugal or axial com¬
pressors undergoing mostly centrifugal loads. The speed of the
part being tested is gradually increased to a value, which ex¬
ceeds by 20-40% its working speed {correspondingly increasing
stress values 40-100% in excess of design stresses). Such tests
enable actual loads to be simulated (except thermal stresses which
develop in heat engine rotors).

To determine safety merging tests are sometimes carrier to
overspeed limits, i.e., to complete failure.

The most trustworthy, although the most expensive way of test¬
ing, is a complex check of the machine as a whole. Such a testing
includes a long-time trial running of the machine under severe work¬
ing conditions, either on a bench or in the field. At definite inter¬
vals of time the machine is partially or completely dismantled and
the conditions of parts evaluated with the aim of forecasting forth¬
coming failures. This method jointly reveals the machine elements
which are inferior not only in respect of their strength, but also in
wear-resistance. Reliability of parts is only established indirectly,
namely, on their good condition after prolonged operation.

3.4. Correction of Design Stresses

205

( g ) Raising the Design Stresses

Decreases in the weight of a part can only be achieved by increas¬
ing the design stresses and decreasing the safety margins.

First an essential stipulation. We speak here about an actual safety
margin decrease consisting in the increase of the actual stresses and
reduction of the part’s cross section. A formal decrease in the safety
margin, obtained only as a result of correcting the stress values, is
another matter.

Let us explain it by an example.

The safety margin

n

<y

(3.21)

where o & = breaking stress;
a = design stress

For the simplest case of a cylindrical part in flexure the design
stress will be

0

M

0 . 1 £»

where M — bending moment acting upon the part;

D = diameter of the part
, Substituting this expression into Eq. (3.21), we get

0 6 O.1Z)3

n ~ M

The actual lowering of the safety margin will only occur if the
part diameter is decreased. Let the shaft diameter be decreased by
10%. Then the new safety margin will be

n' — n- 0.9 3 = 0.74n

and the new value of the design stress

0 ' =cr (o ) 3 = 1 - 35 °

With the formal correction of the design stress value Eq. (3.21)
will have the form

where, a is a factor accounting for higher stresses when correcting
the design stresses (for example, when considering stress concen¬
trations).

Let a be. equal to 2. Then, the safety margin is halved, but this,
however, does not indicate , any actual drop in the strength or rise
in the stresses. Qne and the same part, calculated in different ways,
may have different safety margins, but nevertheless fail under the
same breaking load, regardless, of the design safety, margin.

206

Chapter 3. Weight and Metal Content

Reduction of weight through a decrease in the safety margin de¬
pends on the character of loading. The largest weight savings are
obtained in the simple tension-compression case. The dependence
of weight on stress is expressed by the formula

G __ op
G 0 o

where G 0

and o 0 = original weight and stress, respectively;

G — weight at the increased design stress a
In flexure and torsion the relationship between weight and stress
is less strong

G __ / Co \ 2/3

V*T j

It should be borne in mind that increasing design stresses wit¬
hout changing the part shape is always accompanied by lower rigi¬
dity, which in many cases defines the serviceability of the part.

In terms of effectiveness, the method of increasing the stress
level concedes to other ways of decreasing weight.

Let us consider it by an example. Assume that a shaft is loaded
with a transverse flexural force or torsional moment. If the design

stress is significantly increased (for instance, 1.5 times) the shaft

i

outside diameter can be decreased in the ratio of 1.5"^ = 0.87. In

2

doing so, the shaft weight is reduced in the ratio of 1.5 3 = 0.75
(i.e., it is decreased by 25% only), and its rigidity, in the ratio of

1.5"^ — 0.57 (by 43%).

Such a weight reduction can be obtained' by drilling in the shaft
an axial hole whose bore is 0.5 that of the outside diameter. In
this case the stresses will rise by only 6% and the rigidity, fall also
by 6%.

A weight saving of 25% at the same stress level with rigidity in¬
creased by 5% can be obtained by enlarging the shaft outside dia¬
meter by as little as 5% and simultaneously increasing the bore
diameter by 10%.

Thus, we conclude that; giving rational forms to components
is a much more effective and advisable way of reducing weight than
enhancement of stress levels.

In most engineering constructions increasing stress has an insigni¬
ficant effect because of the limited number of the calculable parts,
whose weight is generally only a small fraction of the total con¬
struction’s weight. The overwhelming part of constructions are
incalculable parts. For the majority of machine classes (piston
engines, compressors, turbines, pumps, machine tools, etc.) the

3.4. Correction of Design Stresses

207

weight of the housings (generally castings) generally makes up
60-80% of the total machine’s weight, and the calculable parts share
10-20%. If we take into account the fact that housings have high
safety factors (due to manufacturing technology), then it will he
quite obvious that the main potentialities for weight reduction are
in these parts.

Assume that in a 5-t machine the cast housing components make
up 70% of the weight and that the average wall thickness is 12 mm.
Even if we reduce the wall thickness by 1 mm only, it will give us
a weight-saving of approximately 300 kg. This result is unattainable
even through the most scrupulous design calculations aimed at
saving weight.

Naturally, lowering the weight of housings must not lower their
strength, rigidity and stability. Reduction of cross-sections must
be compensated for by improved casting technology, increased wall
strength, elimination of casting defects, etc. Purely design techni¬
ques, which enable housing components to be lightened without
deteriorating their strength and rigidity, include:

smoothing outlines; rounding corners and removing sharp edges;
using shell-shaped and arched structures; proper ribbing pentroding;
braces, struts, trusses, stays and other ties, binding the elements of
a structure; using rational power schemes.

There are, however, some structures, in which calculable parts
account for a comparatively large portion of the total weight. These
include: machines, in which metal frameworks are predominant
(jibs, gantry cranes, derricks); aircraft hull and wing structures
with extensive carrying framework elements; frame constructions
for various applications (support structures, uprights, towers, poles,
etc.). For machines and structures of these types, refined calcula¬
tions and reasonable safety factors give substantial weight savings.

(h) Design Stresses and Safety Margins

There are two main ways of selecting the design stresses and
safety margins.

The first method (now rather obsolete) consists in the preliminary
choice of the safety factor and the establishment of the design stres¬
ses on its bases, determining cross-sections and inertia moments
from strength of material and theory of elasticity formulae account¬
ing for the main loads in the design scheme (usually, maximum power
or maximum revolutions).

The method is also applied in the reverse sequence: first the part
sizes are roughly specified, then the stresses acting at critical sec¬
tions are calculated, and finally the safety factor found. If the
latter corresponds with established traditional values, the de¬
sign is considered complete; if no, the part dimensions are corrected.

208

Chapter 3. Weight and Metal Content

In this method, all factors determining deviations of true stress
values from the designed ones are summarily inserted into the sa¬
fety factor which consequently acquires a large value (5-10).

The second, modern way is aimed at complete and exact explana¬
tion for the actual stresses existing in the part. To help the analytical
stress determination experimental methods are enlisted'. Combina¬
tion of the analytical and experimental methods establishes a more
accurate stress distribution and determines stress values closer to
the true maximum ones. With the improvement and refinements in
the calculation methods the number of unknown factors reduces and
the number of defined factors increases.

Indeterminate factors include internal stresses caused by macro-
and micro-structural defects, and also the stresses due to . imperfect
manufacture and assembly. It is necessary to consider these factors
when defining safety margins.

Furthermore, the safety margin must reflect the importance of the
component part and possible consequence of its failure. If the fail¬
ure of the part is likely to cause complete machine failure, then the
safety margins must be increased.

The method of ascertaining stress values and putting into the
safety factor a few random and unaccounted for factors is most cor¬
rect. Naturally with such a refined method the safety factor may
be lowered on average 1.5-3 times.

However, accurate computing methods are drawn up only for a
limited number of loading cases and types of parts.

The third intermediate way tries to make up for deficiencies of mo¬
dern design techniques by transferring unknown values into the mar¬
gin of safety, but only in a differentiated form.

The margin of safety is generally expressed as the product of a num¬
ber of individual factors, each representing some indeterminancy. The
safety factor is often determined as

n —

where re 1 = factor accounting for the difference between the values
of the actual and theoretical loads, as well as the diffe¬
rence between the actual and theoretical stresses (due
to errors in formulae);

n 2 — factor accounting for the heterogeneity of material, as
well as for the influence of macro- and micro-defects
and residual stresses in the material;
n 3 — factor accounting for the importance of the part and
the reliability requirements for the part during opera¬
tion

Some of the authors differentiate still more, asserting that the
safety margin should be expressed as the product of many indivi¬
dual factors (ten and even more), thus encompassing- all or-nearly all

3.4, Correction of Design Stresses

209

the indeterminate factors listed above (see p. 184). Then they give
recommendations on the choice of numerical values of each of these
factors, depending on the authenticity of calculations, quality
of manufacture, complexity of the parts shape, etc.

It is evident that this system is similar in principle to the old sys¬
tem of gross safety factor. The difference is that when applying the
former system the designer commits one gross error in the choice
of the safety margin but applying the differentiated system he makes
a few small mistakes which superimpose one upon another.

Obviously, the evaluation of the indeterminate factors by this
method is hypothetical, causing, for example, a doubt factor as to
the degree of calculation accuracy. The numerical evaluation of
this factor presumes the existence of an absolutely true calculation
allowing true stress values to be defined. Then there is no need for
a correction factor: it is sufficient to introduce the true stress values
into the calculations.

Furthermore, it is a well established fact that numerical values
of correction factors, belonging to sharply different categories, as, for
example, accuracy of calculation and perfection of technological
processes, are incompatible.

In practice, however, with such calculations the use of the differ¬
entiated factor system often leads to adjustment of their numerical
values in order to obtain admissible values of the total safety factor
in the former conception of the term.

Thus, one may state that at present this problem has yet to be
solved. The old methods are obsolete, and the new methods are not
everywhere fully developed. Whenever exact experimentally proved
methods of stress calculations are available, it is better to use the
second method, and introduce into the safety factor only actual in 1
determinate factors.

When it is essential to nse simplified methods of calculation
they must be supported by experience of similar constructions al¬
ready in operation.

Long trouble-free running of a machine is the best indicator that
the stresses in the machine parts are acceptable (although it does
not imply that these stresses cannot be decreased). Preservation of
geometrical similarity between the part being designed and its pro¬
totype and selection of its absolute sizes from conditions of stress
similarity from the main acting loads and perhaps of slightly high¬
er stresses almost faultlessly lead to the development of a du¬
rable component part.

As a general rule, great care must be taken when decreasing
safety factors. In most cases, the saving of weight at the expense of
increased stresses is not great because of the relatively low weight
of calculable parts in the structure of the majority of machines.
The risk is great as in the first place the rigidity of parts is lowered

14-01395

210

Chapter 3. Weight and Metal Content

-which in many cases determines the machines durability. Reduced
rigidity may cause the appearance of supplementary, difficult to
consider loads, which worsen working conditions for the parts.
That is why when increasing design stresses it is obligatory to ana¬
lytically or experimentally check the degree of reduced rigidity.
It is better to combine increased design stresses with constructive
methods of improving rigidity by giving parts rational forms.

An indispensible condition for the immediate comparison of safe¬
ty factors accepted in different mechanical engineering branches
is that identical calculation methods and identical strength of mate¬
rial theories must be assumed as the calculation base for complex
stressed states.

It is also necessary to consider the specific branch of engineering.
High-class machines manufactured under conditions of a strict
technological discipline with carefully formulated quality control
eliminating the possibility of defective material being passed for
assembly may have lower safety factors. To mechanically give
such low safety factors to other machines produced in less strict
conditions would be a mistake.

In aircraft constructions the margins of safety related to the
stresses which have been carefully computed by special methods
and tested experimentally come to 20-30%. Obviously, such safe¬
ty factor values are unacceptable for parts designed by routine
simplified methods, manufactured and inspected less rigorously,
and intended for a much longer working life than that in aviation.

(i) Design Operating Conditions

A mandatory condition for a correct design is the refining of the
design operating conditions by a careful study of possible overloads
which may occur during operation. The design operating conditions
do not always coincide with those of the maximum power or speed.
They may, for instance, include starting conditions, when some
machines (ac squirrel-cage motors or d-c series-field motors) develop
increased torques, or the conditions of braking, stopping, reversing,
speed or load changing, or, finally, those of an abrupt load drop
when the machine begins to overspeed.

In a machine whose drive incorporates an irreversible mechanism
(worm gearings) higher stresses occur during stopping, when the ro¬
tating and rectilinearly moving driven links, owing to the stored
energy, become driving elements in respect to the irreversible
mechanism.

In crankshafts of internal combustion engines maximum stresses
arise during torsional vibrations, while in flexible turbine
shafts they occur when passing through the critical rotational
range. •

3.5. Materials of Improved Strength

211

In many cases overloads can be eliminated or materially lessened
through appropriate design modifications, e.g., by introducing
regulators or speed limiters, limit speed monitors or clutches, vi¬
bration dampers, etc.

Under certain conditions overloads during machine operation are
inevitable. This is the case, for example, in road-building machines,
used on heavy or rocky terrains, damp soils, slopes, side banks, etc.
For processing machines, variations in the quality of raw materials
being handled is of great importance.

All these factors should be carefully studied from the standpoint
of their influence upon strength before finally selecting the design
scheme.

3.5. Materials of Improved Strength

An effective means of lowering a construction’s weight is to use
stronger materials. In contrast to the increased stress method, ac¬
complished by lowering the safety factor with the risk of weakening
the part, reliability in this instance is not impaired (if the same
value for the safety factor is preserved). Another difference is that
the improved material strength method can be applied to all parts
without exception while the increased stress method is valid only
for calculable parts.

The principal methods which enable materials to be strengthen¬
ed include: hot working, alloying, thermal and thermal-chemical
treatment, cold working.

Hot working under pressure. S trengthening of metals through hot
working occurs due to the conversion of a loose ingot structure into
a compact one with oriented crystallites. As the liquid metal cools,
voids appear at the boundaries of the forming crystals. The voids are
caused by metal contraction in the transfer from the liquid to the
solid state, by the evolution of gas bubbles as the solubility of gases
in metal decreases with decreasing temperature, etc.

As crystals are growing, the impurities, which are inevitably pre¬
sent in metal, are ejected and accumulated at the grain boundaries,
where crystallization is completed. Because of this cast structure,
insufficient bond between grains is inherent causing the lowered
strength and toughness of cast metals. In the course of hot work¬
ing (by pressure) the voids amidst crystallites are reduced and welded,
while interlayers of impurities between crystals are crushed and,
under the effect of high temperature and pressure, pass into metal.

For improving metal strength the ,recrystallization process is of
great value. Recrystallization takes place during metal cooling
within definite temperature interval (450-700°C for steels). From the
crystals fragments, which are crushed and deformed during reduction,
are formed new finely-sized grains and the impurities in the growing
recrystallized grains remain absorbed. For this reason the structure

14*

212

Chapter 3 Weight and Metal Content

of forged metals consists typically of fine rounded grains closely asso¬
ciated with one another, which accounts for the increased strength
and .toughness of forged metals.

Forged and, particularly, rolled metals show anisotropy of mechan¬
ical properties along and across the grain. Toughness is especially
influenced by the direction of metal fibres (Fig. 90).

The direction of metal fibres in forged and. stamped parts should
agree with the configuration of the part and the direction of acting

loads. Crankshafts, forged or
stamped in several passes, with
fibres following the throw con¬
figuration (Fig. 916), possess much
higher strength than crankshafts,
made of prismatic blanks, with
sheared fibres (Fig. 91a). The
gear produced by hot rolling plus

fa)

d%

oc«

Fig. 90. Ultimate tensile strength 05 , Fig. 91, Orientation of fibres

elongation 6 and specific toughness a h
of steel as influenced by orientation
of fibres

cold) sizing will have correct grain orientation with respect to the
loads acting on the gear teeth (Fig. 91d).

Alloying has various aims: to improve corrosion resistance and
heat resistance; to give better welding properties and special phys¬
ical characteristics. The main purpose of the alloying process is to
improve strength with a differentiated improvement in other par¬
ticular characteristics of the strength, toughness, plasticity, elasti¬
city and wear resistance. Addition of certain elements may improve
hardenability of steels, thus allowing better mechanical properties
throughout the entire cross section of a part to be obtained.

3.5. Materials of Improved Strength

213

The best all-in-all strength characteristics are possessed by chrome-
nickel steels, particularly polyalloy chrome-nickel-tungsten and
chrome-nickel-vanadium steels.

To acquire good mechanical properties it is necessary to comple¬
ment alloying with thermal treatment.

The comparative characteristics of alloy and carbon steels are
given in Table 7.

Table 7

Average Strength Characteristics of Carbon and Alloy Steels
(with optimum heat-treatment)

Steels

Tensile

strength,

o b ,

kgf/mm 2

Yield

limit,

°0 2'
kgf/mm 2

Elongation
per unit
length, 6, %

Fatigue
limit, c-i,
kgf/mms

Impact
strength a k ,
kgf.in/cnia

Low carbon

35-50

25

25

20

3-6

Medium carbon

60-80

40-50

12

25-30

4-8

High-strength,

alloy

100-180 ,

100-150

6-8

60-100

6-10

Strengthening thermal treatment (hardening with high, medium
and low tempering, isothermal hardening) causes the formation of
unbalanced structures with strongly deformed crystal lattices (sor¬
bite, troostite, martensite, bainite).

The composition of these structures, the sizes and shapes of grains,
as well as different mechanical properties are obtainable by regulating
the conditions of heat treatment. For constructional steels the ap¬
plication of quenching and tempering heat treatment (hardening
with high tempering) assures the most favourable combination of
strength, toughness and plasticity.

In recent years the induction hardening process has become very
popular, which consists in heating a component’s surface layers with
high-frequency currents. Besides its purely technological advantages
(economy, high productivity) this kind of heat treatment has a sig¬
nificant strengthening effect resulting from residual compression
stresses in the hardened surface layer.

Chemical-thermal treatment consists in saturating the surface
layer with carbon (carbonizing), or nitrogen (nitriding, cyaniding),
with the formation in the latter case of Fe-nitrides and alloy ele¬
ments. These kinds of heat-treatment are aimed at making the sur¬
face hard and wear-resistant. At the same time they increase strength
(particularly under the conditions of cyclic loads) owing to the for¬
mation of a compression stressed state in the surface layer.

214

Chapter 3. Weight and Metal Content

Cold working (shot blasting, rolling, coining, cold reduction and
sizing) promotes the development of compressive stresses in the
surface layer and enhances the fatigue strength.

(a) High-Strength Cast Irons

Grey cast irons are one of the most widely used constructional
materials. Cheapness, good castability, high resistance to cyclic
loads—these qualities ensure wide application of grey cast irons
in the manufacture of housing components (castings) of stationary
and transport machines when weight considerations are of secondary
importance.

Disadvantages of grey cast irons are their low strength, low im¬
pact toughness and brittleness.

Modification is the first measure in improving the strength of
cast irons. This means adding small amounts of inoculants (cal¬
cium-silicon, ferrosilicon, graphite powder) to the liquid iron before
casting. These inoculants (additives) improve cast iron properties
and promote homogeneity of the cast structure in all sections. Gra-
phitizing modification stops the formation of graphite flakes so
that the latter are present as nodules preventing chilling and giving
a pearlite structure, this being more favourable for the strength of
this material. The inoculated irons have a strength level 30-50%
higher than that of grey irons.

In recent years methods have been developed which produce
high-strength irons, i.e., alloy cast irons (containing Mg, Mn, Cr
and some other elements), heat-treated to give granular pearlite
with globular-shaped graphite inclusions. The process is called sphe-
roidizing modification (inoculants: Mg, Gr or alloys of these metals
with Cu and Ni). A typical chemical composition of a high-strength
alloy cast iron is: 3.4-3.6% G; 2-2.2% Si; 0.03-0.06% Mg; 0.15-
0.25% Cr; 1.15-1.3% Mn; not more than 0.005% S and 0.12% P.

The heat treatment includes: normalizing at 950°G for 6-8 h
with subsequent cooling at a rate of 30-60°G/min, then temper¬
ing by heating to 700-720°C for 8 h with subsequent cooling
in air.

High-strength cast irons have better mechanical properties than
grey cast irons (see Table 8), approaching, in this respect, steels.
These high-strength cast irons are used for manufacturing heavily
loaded housings with complicated configurations. They can be
processed by induction hardening, shot-blasting and nitriding.
High-strength nitrided cast irons (with admixture of Al) possess
a hardness of about 900 VPH.

Nowadays, important heavily loaded parts are made from high-
strength cast irons, e.g., crankshafts, which successfully compete
in strength with forged and moulded shafts of carbon and low-alloy

3.5. Materials of Improved Strength

215

Table 8

Characteristics of High-Strength and Grey Cast Irons

Characteristic

.

Irons

grey

high-strength

Ultimate strength, kgf/mm 2 :

tensile

15-30

45-80

flexural 0/

30-45

50-90

Yield strength cr 0 .2, kgf/mm 2

10-20

40-50

Fatigue strength o_i, kgf/mm 2

6-15

15-25

Elongation per unit length 5, %

<0.3

2-10

Impact strength a kgf-m/cm 2

0.2-0.4

1.5-3

Elastic modulus E, kgf/mm 2

8000

15000

steels and even exceed the latter in respect of wear-resistance. Fur¬
thermore, the manufacturing costs of cast iron shafts are many
times less than those of forged steel shafts.

Castability (casting properties) of high-strength cast irons is
worse than that of grey cast irons (shrinkage of grey cast irons is
0.8-1.2% and of high-strength cast irons, 1.3-1.8%). However high-
strength cast irons cast better than casting steels. Particular atten¬
tion should be paid to desulfurization of cast irons, otherwise mag¬
nesium sulfides will occur in castings (in the form of the so-called
black spots), which will cause local softening of castings.

It is noteworthy that in respect of cyclic toughness the high-
strength cast irons are much more inferior to the grey cast
irons.

By cyclic toughness is meant the property of metals to partially
transform the energy of elastic strain into heat as a result of inter¬
nal losses due to friction. The higher the cyclic toughness, the bet¬
ter the metal property to dampen vibrations when subjected to cyclic
loads. Grey cast irons possess the highest cyclic toughness.

The value of cyclic toughness is given by the hysteresis coef¬
ficient ¥ (percentage ratio of energy loss v per strain cycle to the
full strain energy w):

'F = — • 100%

W

Figure 92 gives hysteresis coefficient values for cast irons and
steels as a function of amplitude x of stress variations per strain
eycle^ As evident from the diagram, the cyclic toughness of grey
cast irons is 5-6 times that of carbon steels and 10-20 times that of

216

Chapter 8. Weight and Metal Content

alloy steels. Higli-strength cast irons approximate steels in terms
of cyclic toughness and inoculated cast irons are midway between

the grey and high-strength

Fig. 92. Hysteresis coefficient f as a
function of stress amplitude % per defor¬
mation cycle

3 — grey east iron; $ — inoculated cast iron;
3 — steel grade 20; 4 — steel grade 45; 8 —
high-strength oast iron; s — steel casting;
7 — steel grade 40X

cast irons.

The cyclic toughness of
non-ferrous metals is extre¬
mely low, except magnesium
alloys which approximate car¬
bon steels in their cyclic tou¬
ghness.

( b ) Extra-High Tensile Steels

The development of super¬
strong materials is based on the
modern concept of disloca¬
tions (i.e., local distortions of
atomic-crystalline space lat¬
tices,), which are considered to
be the original cause of dis¬
crepancy between the actual
strength of metals and the
theoretical value, predicted on
the basis of atomic bonds in

crystalline lattices. The theo¬
retically inferred strength value amounts approximately to
(0.1-0.15) E, where E is the Young’s, modulus. Yet, the actual
strength is tens, and occasionally hundreds times less than theoret¬
ical values. In other words, in modern metals only a very small por¬
tion of their potential strength is utilized.

Up till recently it has been a generally accepted consideration
that the process of plastic strain consists in simultaneous shear of
crystal planes one relative to another. However, such a concept is
inconsistent with the magnitude of those forces which are needed to
overcome the strength of atomic bonds at the slip planes. Now it is
generally considered that the shear occurs not at once, but in sub¬

sequent stages.

In the areas of dislocations, as a result of crystal lattices distortion,
there are formed plateaus of easy slip, so that a comparatively small
shearing force will suffice to displace crystal planes in the area over
an interatomic distance.

Such a. displacement will result in a corresponding shift of the
plateau along or opposite to the direction of the acting force. In
the new location of the plateau another displacement over an inter¬
atomic distance will occur, this being, in turn, followed by a new
shift of the plateau.

3.5. Materials of Improved Strength

217

In this way the plateau will successively move in the direction
of the acting force, making, in doing so, the entire crystal plane shift
over an interatomic distance. Should the force be continued, the
action will keep on repeating many times so that a macroshift of
crystal planes will take place.

Obviously, successive shearing, which requires only a local rup¬
ture of atomic bonds, will be actuated by such a force that is many
times less than that necessary to simultaneously displace the entire
crystal plane.

It is the mechanics of the described shift which is mostly re¬
sponsible for the lowered actual metal strength when compared with
theoretical values.

The displacement of ready-to-slip plateau will continue until the
dislocation has] emerged onto the surface of a crystal block or
encountered some obstacle (e.g., foreign interstitial solute, perpendi¬
cular dislocation, similar dislocation but of the opposite sign,
etc.). Dislocations of opposite signs mutually cancel each other.

Thus, we may conclude that increasing heterogeneities, i.e.,
enlarging the amount of admixtures and the number of crystal lat¬
tice distortions, as well as comminuting crystal blocks, will make me¬
tals stronger owing to impeded and locked dislocations.

Dislocations inevitably occur in huge quantities in all metals.
For instance, the average density of their distribution in steels
reaches 10M0 10 cm -2 .

Dislocations arise due to many reasons: superfluous interstitial
crystal layers,, the so-called extraplanes (line dislocation), spiral
displacement of crystal planes relative to each other (screw dislo¬
cation). An alternative of dislocations is a vacancy, i.e., absence of
atoms in the nodes of crystal lattices, as well as foreign atoms inocu¬
lated into interstices. Local distortions in lattices occur under appli¬
cation of external loads and also in the zones where internal stres¬
ses act.

The existing dislocation may cause new dislocations in adjacent
areas. There are sources of spontaneous dislocations: two superim¬
posed line dislocations will form a continuously acting generator of
dislocations (Frank-Read sources).

There are two main methods of increasing strength of metals:

(1) elimination or reduction of the number of dislocations (pro¬
duction of metals possessing homogeneous and regular crystalline-
structure);

(2) increasing the number of heterogeneities (creation of obstacles-
that will check the development and propagation of disloca¬
tions).

The potentialities of the first method are rather limited, since a
defectless structure can only be obtained for extremely pure mate¬
rials and in very small volumes, which would exclude the inception

218

Chapter 3. Weight and Metal Content

and development of dislocations. Exceedingly thin whisker crystals
have been prepared under laboratory conditions. Such a whisker crys¬
tal (several millimetres long and 0.05-2 p,m thick) has a tensile
strength of 1350 kgf/mm 3 , this being approximately 100 times that of
•conventional commercial iron and 10 times that of high-grade alloy
steels. Alongside such a high tensile strength the whiskers pos¬
sess good elasticity characteristics.
Thus, the elastic elongation of whis¬
kers reaches 5%, while the same para¬
meter offered by conventional iron
does not exceed 0.01 %.

Enhanced strength and elasticity
of single crystal whiskers are obtain¬
ed owing to extreme purity of their
materials and regularity of crystal
structure. The development of dislo¬
cations in whiskers is practically
impossible, since the diameter of
a whisker is less than the average
extension of a dislocation. With an
increase in diameter the strength of
crystal whiskers sharply falls (Fig.93)
because of dislocations.

The whiskers are also produced
from non-metallics (graphite alumi-
0 4 8 12 16 jam n j um oxide A1 3 0 3 , silicon oxide Si0 2 ,

Fig. 93. Strength of iron whis- silicon Carbide SiC); the strength of

kers (kgf/mm 2 ) as a function of non-motallic wiiiskors is higher than
diameter (pm) that of their metallic counterparts

(Fig. 94).

The actual strength of whiskers is 50-60% of the theoretical values.
The tiny dimensions hamper, however, their technical application.

Perhaps, the only real way of whiskers utilization is their employ¬
ment in the manufacture of composite materials, comprising single
crystal whiskers, laid in a definitely oriented pattern within a me¬
tallic (e.g., aluminium) or plastic matrix. If the whisker has suffi¬
cient length to ensure a good bond with the matrix (along its
side surfaces) its strength will be successfully used. It is
noteworthy that the strength of composite materials, which contain
40-50% (by weight) of whiskers, attain approximately 30% of the
latter’s strength. Thus, a composition of sapphire single crystal
whiskers (Al 2 O a ) and metallic aluminium has a strength 500-
■800 kgf/mm 3 .

Naturally, such materials are very expensive (their cost nearly
equals that of platinum); for this reason, their application is limited
to only special constructions.

3.5. Materials of Improved Strength

219

The second method seems more promising, tending to increase the
degree and number of heterogeneities. The first step is alloying
and heat treatment, whose strengthening effect, in essence, amounts
to an increase in the density of dislocations.

Further achievements in the development of high-tensile steels are
based on the fact that in some multi-component alloy steels (com¬
prising a relatively low percentage of alloying elements) the process

aj, kgf/mm 2

Fig. 94. Theoretical strength of materials (unshaded rectangles), strength of
whiskers (shaded rectangles) and actual technical strength (black rectangles)

of cooling, starting from the temperature of austenitic transforma¬
tion within a certain range of temperatures (450-550X), is not accom¬
panied by decomposition of austenite (it should be pointed out that
such a decomposition is inevitably followed by the formation of
hard ferrite-cementite mixtures). Consequently, within this range
of temperatures steels can remain plastic for an unlimited period of
time and be ready for forging, stamping and rolling.

Thus begins the thermomechanical processing method combining,
in fact, the processes of thermal treatment and plastic deformation.

Low-temperature thermomechanical processing (LTTP) is an inten¬
sive plastic deformation of steel within the temperature range of
the stable austenitic state.

Temperature Temperature Temperature

Fig. 95. Schematic representation of thermomechanical treatment (diagrams of
isothermal austenite decomposition: temperature — time)

3.5. Materials of Improved Strength

221

The process (Fig. 95a): heating to 900-1000°C; rapid cooling down
to 450°-550°C; multiple plastic deformation at this temperature
with a high degree of deformation (90%); martempering (to produce
a fully martensitic structure); and final tempering at 250~400°C.

The low-temperature thermomechanical treatment can readily
be applied to steels having the following approximate chemical com¬
position: 0.4-0.6% C; 1-1.5% Ni; 0.74.5% Mn; 1-1.5% Si; 1-3% Cr
and 0.54.5% Mo. The steels of such a composition have the stable
austenitic state within the above-mentioned range of tempera¬
tures.

The LTTP assures a significant increase in the material strength
(tensile strength cr & = 320-350 kgf/mm 2 , yield limit ovs —
= 280-300 kgf/mrn 2 with elongation 6 = 8-12%). This is
about twice the strength indices of the best modern alloy steels. Of
great significance is the fact that the LTTP sharply increases fatigue
strength.

The enhancement of strength attained by LTTP is attributed, in
the main, to the substantial destruction of the crystalline structure as
a result of semi-plastic deformation, which is accompanied by com¬
minution of crystalline groups (blocks) down to a fourth-fifth frac¬
tion of crystalline groups obtainable by conventional heat treat¬
ment.

The shortcoming of the LTTP lies in that the treated parts cannot
be subjected to high temperatures because the steel will lose in this
event that hardness which it has acquired before. Hence, the LTTP-
treated parts forbid welding.

The process is applicable for rolled stock and parts of simple form.
Intricately-shaped components will not yield good results when sub¬
jected to the LTTP, as it is impossible to obtain uniform deformation
and homogeneity of metal properties throughout the entire compo¬
nent.

Another disadvantage is the necessity for increased forces which
are needed to deform the material in its semiplastic state.

To avoid this disadvantage use is made of the high-temperature
thermomechanical processing (HTTP), by means of which (Fig. 95Z?)
the material is deformed in the range of temperatures between 800-
900°C, with the deformation degree reaching 20-30%. Afterwards
the component is martempered and after-hardened. Occasionally
heat treatment for bainite takes place (Fig. 95c).

The HTTP method brings about a lesser accretion In strength as
compared with the above described method (LTTP): the strength
level is raised to 220-280 kgf/mm 2 , yet it is still 1.5-2 Aimes that
obtained by applying separately pressure working and heat
treatment. Furthermore, the HTTP technique betters plasticity and
impact toughness in parallel with reducing the sensitivity of steel
to stress concentrations.

222

Chapter 3 , Weight and Metal Content

The HTTP treatment may be applied, though with poorer strength
results, to conventional medium carbon steels. Thus, high tempera¬
ture thermomechanical processing increases the ultimate strength
of steel grade 45 to 180-200 kgf/mm 2 .

When combined, the LTTP and HTTP treatments (Fig. 95d)
will produce a tensile strength increment of 15-20%.

Still another method of material hardening is based on martensite
strain ageing (MSA). In this case (Fig. 95e) the steel is first subjected
to the usual heat treatment (hardening plus tempering at 250-400°G)
and afterwards deformed in a cold state, the deformation degree
being 1-3%. Then follows ageing for 1-2 h at a temperature which ex¬
ceeds by approximately 100°C the tempering temperature. In the
process of ageing the steel tensile strength rises to 200-250 kgf/mm 2 .
It is significant that the ratio between the yield limit and ultimate

tensile strength becomes approximately equal to 1. Owing to

this the strain-aged steels, as regards their yield limit value which
is the main strength characteristic of the material, are close to steels
which have been strengthened by more complicated methods (such
as those described above).

Deformation can be performed by any method: reduction, exten¬
sion, torsion, stamping, rolling, etc. Intricately-shaped components
are deformed by applying loads, which simulate the working ones.
Thus, storage vessels are hardened by applying high internal pres¬
sure with subsequent ageing.

Increased hardness in the course of strain ageing is obtained on
account of two simultaneously acting factors: cold working (i.e., higher
density of dislocations) plus comminution of martensite blocks.

An alternative to this method is bainite hardening followed by
strain ageing (Fig. 95/). Satisfactory results are obtained also by
combining strain ageing with LTTP (Fig. 95g) and HTTP
(Fig. 95 /j, i).

In recent times a hardening process has been suggested which is
based on the ageing of carbonless alloy martensite. This method can
be applied to carbonless (<0.01% C) alloys Fe — Ni — Co — Mo,
containing 18-20% Ni; 7-10% Co and 3-5% Mo with obligatory ad¬
mixtures of Ti (0.3-1.5%) and A1 (0.1-0.3%), which are, in fact, the
primary hardening elements. ,;i

These alloys are heat-treated by martempering, which, in con¬
trast to conventional tempering of alloy steels, does not require high
cooling rates and proceeds as the steel is being cooled down in open
still air from 800-l000°C (generally alloys are hardened from forging
temperature). This hardening will result in soft martensite (10-15 Rc),
which readily yields to deformation in the cold state.

The material is then subjected to ageing, keeping it at 400-500°C
for approximately 3 h. After ageing the ultimate tensile strength

3.5. Materials of Improved Strength

223

increases to 210-250 kgf/mm 2 (with — « 1); the martensite hard¬
ness is raised to 50 Rc, keeping its high plasticity (8 = 10-12%)
and toughness (a k = 8-12 kgf/cm 3 ). The strengthening effect is
attained mainly due to segregation of intermetallics, such as
Ni (Ti, Al) and Ni 3 (Ti, Al, Mo).

Martensitic-ageing alloys have good technological characteristics.
Ageing does not distort the part, this treatment can be used at the
final manufacture stage. Such alloys can be worked through hot
plastic deformation of any kind (forging, rolling). In a hardened
state (prior to ageing) these alloys can be worked by pressure
(deep drawing, spinning operations, etc.). The alloys have good
machining and satisfactory welding qualities after hardening and
after ageing. Softening in the weld zone (when welding in aged
state) is eliminated by reageing.

The disadvantage of the martensitic-ageing alloys is the increased
content of expensive Ni and Mo. Ample tensile strength and good
toughness can be obtained by introducing 1.5-2% Mn, the Ni content
not exceeding 8-12%.

Soviet scientists have developed a method which enables low-
carbon steels to be strengthened by multiple thermomechanical
processing (MTMP). It is in deforming a specimen 5-6 times, each
deformation stage corresponding to the yield plateau length on the
stress-relative elongation diagram (total deformation reaching 6-8
per cent) until the yield plateau fully vanishes. This is followed by
ageing at 100-200°C for 10-20 h. The processing allows the yield
limit to be raised by 25-30% (nearly approaching the ultimate
strength) and the fatigue limit by 30-50%.

Recently, along with the thermomechanical processing, additional
strength is obtained by applying a magnetic field which on the
strength of the well-known magnetostriction phenomenon changes
crystal sizes. The stresses caused by magnetostriction will be added
to the stresses produced during the previous thermomechanical pro¬
cess, this strengthening the steel still further (approximately by
10-15% when compared with the original strength). This is called
the thermo-mechano-magnetic processing method (TMMP).

Samples of super-high strength steels have been produced under
laboratory conditions. These samples have ultimate tensile strength
as high as 400-500 kgf/mm 2 , i.e., ten times that of carbon steels and
3-4 times that of modern alloy steels.

With the appearance of high-strength steels arises a series of new
design problems since parts made from the stronger metals are less
rigid. This is because the modulus of elasticity of each metal has
a stable value and only slightly depends on heat treatment and con¬
tents (in normal quantities) of alloying elements. Since elastic
strains are proportional to the ratio between the stress and elasti-

224

Chapter S. Weight and Metal Content

city modulus, then with increased stress values (which is the reason
for the application of high-tensile materials) the strains will also
increase proportionally and the rigidity will fall inversely propor¬
tionally.

This will be true only if the length of parts is assumed unchanged
(the case of most applications). The linear dimensions of a con¬
struction are generally chosen to suit machine operational condi¬
tions. Thus, in power generators and converters these dimensions
depend on the working capacities and specifications, e.g., in internal
•combustion engines these dimensions depend upon cylinder size,
which is, in turn, dependent upon the gas operating pressure; in
machine tools, on the workpiece size to be machined; in metallic
frameworks, upon the length and the height of the construction.
In all these cases the use of high-strength materials will affect only
the cross section and not the length of a part.

A series of machines exist whose linear dimensions are dependent
•only on the strength of materials. To such machines are referred
reduction gear units. In this instance the application of high-strength
materials allows not only cross sections to be decreased, but also
a proportional reduction in the length of separate parts and the
overall unit dimensions as a whole to be obtained. In these cases
the introduction of high-tensile materials does not lower construc¬
tion rigidity.

Let us consider a case when the linear dimensions of a part are
unchanged. Assume, there are two equallystrong (in tension) bars
of the same length—one from carbon steel grade 45, with on ultimate
tensile strength 50 of kgf/mm 2 , and the other from a high-strength
steel with an ultimate strength 500 kgf/mm 2 . The rigidity of the
latter bar is obviously 10 times less than that of the former.

Now let us evaluate the absolute deformation values.

Take as an example a connecting rod, length L — 400 mm, of an
internal combustion engine. If the compressive stress developed in
a conventional steel rod is 20 kgf/mm 2 , the elastic deformation will
be

X== -|~ L== 2IW* 400 ^ °* 4 mm

Now, if we compare it with the compressive deformation in a
connecting rod from the high-strength steel with a proportionally
less cross-sectional area (for the equistrong conditions), we shall
find that the deformation is much greater

X = 4 mm

In the tension-compression case no means exist of fighting rigidity
reduction because with the given values of a and E the deformation
depends only on the cross-sectional area and is quite independent

3.5. Materials of Improved Strength

225

of its shape. In flexure, torsion and longitudinal bending the rigid¬
ity reduction is greater, but in the given case there are some means
of fighting this phenomenon.

Assume two equistrong bars of equal length and similar cross-
sectional profiles made from the same steels as in the above example
and subjected to flexure or torsion. Then, for the above case the
rigidity of the bar made from high-strength steel will be less by
10V3 J 21.5 times.

Let us consider a numerical example. Assume a shaft 60 mm in
diameter and 400 mm long, supported at its ends and loaded in the
centre with a force P.

The maximum deflection of the shaft under the action of flexural
moment ( M f i ex — PLU)

* PIA Mnex&

J ~~ 482 ?/ “ 12 El

As I = W -y- , then

t M fUs L* g
1 ~ 6 EDW ~ E ' GD

If the bending stress in the carbon steel is 20 kgf/mm 2 , then

, _ 20 4002

21000 ' 6-60

0.45 mm

The deflection of the high-strength steel shaft with its proportion¬
ally less section will have a very large value

/' = 21.5-0.45 « 10 mm

Thus, the application of high-strength materials with full use
of their strength reserves and reduction of cross section without a
corresponding decrease in length may lead to a catastrophic drop in
rigidity.

Generally the rigidity characteristic is improved by increasing
diametral sizes of the part and making simultaneously its walls
thinner. However, in the case under consideration, this would be
useless: with the increased moments of inertia the resisting moments
of the part are simultaneously increased and this is accompanied
by decreased stresses. Consequently, this method lowers the level
of stresses which conceals the main advantage of high-strength ma¬
terials, namely, the possibility of raising design stress to obtain
a corresponding weight saving. The advantage is only partly realiz¬
ed with a very large reduction of wall thickness (in general mechanic¬
al engineering the wall thickness would only be 1-2 mm), i.e., this
means a switch over to shell-like components.

15-01395

226

Chapter 3. Weight and Metal Content

For some engineering components (disks, containers, gears, con¬
necting rods, levers, shafts, etc.) the shell-like shapes are practicable,
although they require radical changes in design and manufacturing
processes. Therefore, along with the increase in inertia moments
it is necessary to apply other methods of lessening deformations,
such as shortening the length of parts, closer arrangement of sup¬
ports, etc.

In all cases the employment of high-strength materials puts new
problems before designers and technologists, the solving of which
requires a lot of creative and original thought.

A positive feature of- parts made from high-strength steels is
their extraordinary high capability of withstanding impact loads
being accounted for by the large elastic deformation values. The
resistance to impact loads is approximately proportional to the

ratio [see Eq. (3.33)], where cr 0 . 2 is the yield limit and E, Young’s

modulus. Should the yield limit be considered as proportional to
ultimate strength, then the resistance of high-strength steels to im¬
pact loads will be greater than that of conventional steels in the
ratio

where cr& and a b are the ultimate tensile strengths of the high-strength
and conventional steels, respectively.

When — == 10 the resistance of high-strength steels to impact

loads will be 100 times that of conventional steels.

3.6. Light Alloys

The use of materials with low specific weights is a significant way
of lowering the weight of constructions. However, real weight
savings not only depend on the value of the specific weight; they also
depend on the material strength. The lower strength and rigidity of
the light materials may in a series of cases reduce to tought the
weight savings.

Additional factors limiting the application of light-weight mate-
rials are: low hardness, insufficient resistance to corrosion, low heat-
and cold-resistance, high sensitivity to stress concentrations. Finally,
it is necessary to consider cost and availability of these materials
or components made of them.

Some of the light-weight materials being applied in general engi¬
neering are:

3.6. Light Alloys

227

V in itgf/dias

aluminium alloys .... 2.6-3.2
magnesium alloys . . . .1.8

titanium alloys.4.5

plastics.1-1.8

improved wood.1.3-1.6

ceramic materials .... 2.2-3.2

(a) Aluminium Alloys

Aluminium alloys are the most widely used light alloys. They
are noted for their low specific weight (y 3 kgf/dm 3 ), high heat-
conductivity (X = 100-150 cal/m-h-°C), satisfactory strength and
plasticity and are easily machined. Many of them can be welded
by the argon-arc method when using non-consumable tungsten, elec¬
trodes or by hydrogen-arc welding. Use can be made of gas submerged
welding (LiCl, NaCl, KG!, KF). Sheet materials are resistance-
welded.

Aluminium alloys withstand corrosion in dry atmospheres, are
resistant to the influence of alkalies and weak acid solutions, hut
corrode in humid (especially sea) air and are non-resistant to strong
acids and rather soft (their hardness varies within 60-130 BH). Coef¬
ficient of linear expansion a — (20 to 26) .10“ 8 *C -1 . Young’s modu¬
lus E = 7000-7500 kgf/mm 3 .

The strength of aluminium alloys rapidly decreases with the
increase of temperature. There are, however, some alloys which
retain satisfactory mechanical properties up to 250-300°C,

Aluminium alloys are divided into two main categories: castable
and wrought (i.e., worked by forging, stamping, rolling).

Castable alloys (Table 9) by their chemical composition are sub¬
divided into aluminium-copper, aluminium-magnesium, aluminium-
zinc, and also aluminium-zinc-silicon, aluminium-copper-silicon,
aluminium-silicon and composite (with admixtures of Ni, Ce,
etc.).

The highest combined characteristics belong to the aluminium-
silicon alloys ( silumins ), distinguished for their low specific weight
(y = 2.6-2 .J kgf/dm 3 ), good casting and welding qualities and
high corrosion resistance. Silumins can be used to particularly good
advantage for casting thin-walled intricately shaped components.
To enhance their mechanical properties silumins are generally , inoc¬
ulated before casting (with metallic sodium, Na and K fluorides),
owing to which the silicon inclusions acquire a grain-like structure
favourable to strength.

Aluminium alloys are often used for casting bases and housings.
The low strength and rigidity of aluminium alloys are compensated
for by increasing cross-sections, moments of inertia and resisting.

15 *

Table 9

Chemical Composition and Mechanical Properties of the Main Castable Aluminium Alloys

Chemical tomposition.

%

Mechanical properties

Alloys

Alloy

grade

Cu

Mg

Mn

si

Zn

Ultimate ten¬
sile strength,

<J 6 , kgf/mmS

Elongation, 6,

%

Al—Cu

I

■

4-5

!

20

!

6

VfTTTT^I

4.5-5.3

.6-1.0

—

—

30

8

1

Al— Mg

AJI8

—

—

9.5-11.5

1

—

29

9

Al—Zn— Si

AJI11

—

—

1

0.1-0.3

6-8

7-12

18-25

1.5-2

Al—Cu—Si

AJ15

1-1.5

0.35-0.6

4.5-5.5

16-23

0.5-1

AJI6

2-3

—

—“•

4.5-6

-.

15

1

Al — Si

m

■M

10-13

14-16

mm

H

■

0.2-0.5

0.17-0.3

1

8-10.5

15-24

mm

Table 10

Chemical Composition and Mechanical Properties of Some Wrought Aluminium Alloys

Chemical composition, %

Mechanical properties

Alloys

Cu

Mg

Mn

Ni

Fe

Si

Other

elements

Ultimate
tensile
strength,
a b •

ltgf/mm2

Yield

limit,

^0.2’

kgf/mm2

Endurance

limit,

a ~lb>

kgf/mm2

Elonga¬
tion, 8,

%

fll

3.8-4.9

0.4-0.8

0.4-0.8

—

—

—

—

40-45

25-30

10-12

10-12

3.8-4.9

0.3-0.1

1-2-1.8

—

—

—

—

45-50

25-35

12-15

8-10

B95

1.4-2.0

0.2-0.6

1.8-2.8

—

—

5-7Zn
0.1-0.25Cr

50-60

40-50

12-15

5-7

AK2

3.5-4.5

0.4-0.8

—

0.8-1.3

0.8-1.3

0.5-1.2

—

40-45

25-30

10-12

4-5

AK4

1.9-2.5

—

1.4-1.8

1-1.5

1.2-1.5

0.5-1

—

35-40

20-25

8-10

6-8

AK6

1.8-2.6

0.4-0.8

—

0.7-1.2

—

35-40

20-25

8-10

5-6

230

Chapter 3. Weight and Metal Content

Despite this the employment of aluminium alloys gives signifi¬
cant weight savings.

Components made from aluminium alloys which require sealing
(e.g., sumps, oil-casings, boxes, etc.) are impregnated with synthetic
thermosetting compounds (often with bakelite) and are then heated
to the bakelite setting temperature (140-160°C).

The most popular in the family of wrought alloys (Table 10) is
duralumin, which is, in fact, an A1 — Cu — Mg alloy. Also widely
applied are alloys with admixtures of Mn, Si, Fe and Cr.

The duralumin-type alloys (fll, J3.16, B95) are heat-treated to ob¬
tain the highest mechanical properties. The heat treatment con¬
sists in water-quenching from a temperature of 500-520°C with sub¬
sequent ageing at room temperature for 75-100 h (natural ageing) or
at 175-150°C for 1-2 h (artificial ageing). Duralumins are most com¬
monly used in production of sheet and rolled shapes.

To prevent corrosion the rolled stock of aluminium alloys is an¬
odized. The process comprises an electrolytic treatment in a bath
of a 20% solution of H 2 S0 4 at a current density of 1-2 A/dm 2 and
tension of 10-12 V. The component serves as an anode and the cathodes
are made in the form of lead plates. As a result of the process the
component surface is coated with a film of aluminium oxide A1 3 0 3 , :
which effectively protects the metal from corrosion and adds better
hardness and abrasion-resistance to the surface. To enhance the
stability of the coating it is then treated with a hot 10% solution
of potassium bichromate (K 2 O,0 7 ).

The sheet rolled stock of aluminium alloys is protected by clad¬
ding, i.e., applying upon the surface some layers of pure aluminium.

The AK-type alloys are utilized for forging and stamping of parts
(connecting rods of high-speed engines, disks of centrifugal and axial
compressors, blades of axial compressors, etc.). The AK4 heat-resi¬
stant alloy is used for the manufacture of internal combustion engine
pistons and cylinder heads of air-cooled engines.

The wrought aluminium alloys possess satisfactory antifriction
properties. The Ni-doped alloys are used in production of plain
bearing bushes. An ample circulating supply of lubricating oil for
such bearings must be assured for their good serviceability. As for
shafts, they must possess increased hardness (45 > Rc).

( b) Magnesium Alloys

The magnesium alloys consist of Mg (90 % or more) and of alloying
elements (Al, Zn, Mn, Ti, etc.) They have low specific weight (y
« 1.8 kgf/dm 3 ), low elasticity values (E — 4200-4500 kgf/mm 2 )
and low hardness (60-80 BH). The coefficient of linear expansion of
these alloys is very large: a = (27 to 30) -1O -0O G“ 1 , and the heat con¬
ductivity varies between 60 and 70 cal/m -h *°C.

3.6. Light Alloys

231

The strength of magnesium alloys is lower than that of aluminium
ones and rapidly drops with rise in temperature. The magnesium
alloys are sensitive to stress concentrations. They are easily machined
hut precautions must be taken to avoid ignition of chips.

The magnesium alloys are subdivided into castable and wrought
(Table 11).

The worst disadvantage of magnesium alloys is their low corrosion
resistance, particularly in humid atmosphere. Therefore, all parts
made of magnesium alloys should be adequately protected against
corrosion.

Generally this is achieved by dichromizing—a process during
which a stable anticorrosion film is formed on the metal surface
(the film is composed of magnesium chrome salts).

Table 11

Chemical Composition and Mechanical Properties of Basic Magnesium Alloys

Chemical composition, % j

Mechanical properties

Alloys

Alloy-

grade

A1

Zn

Mn

Ultimate
tensile :
strength

Yield

limit,

Endurance

limit,

Elonga¬
tion, 6,

°b’

icgf/mm a

a 0.2'

kgf/mm2

°-i 6’
kg?/mm a

%

Cast-

able

MJI2

—

—

1-2

8-9

5-6

2.5-3

MJI4

5-7

2-3

0.15-0.5

14-16

9-12

.2.5-3

MJ15

7.5-9

0.2-0.8

0.15-0.5

12-15

9-12

1.5-2

Wrou-

MAI

—

—

1.3-2.5

16-18

10-12

6-9

1.5-2

MA2

3-4

0.2-0.8

0.15-0.5

24-26

14-18

10-12

3-5

ght

MA5

7.8-9.2

0.2-0.8

0.15-0.5

28-30

18-20

12-14

6-8

The dichromizing process comprises several stages. First of all
the part is treated with a cold 20% solution of chromic anhydride
Cr0 3 to remove oxide films. Then an electrolytic treatment follows.
This is carried out in a bath filled with an acidified aqueous solution
of potassium bichromate (K 2 Cr 2 0 7 ) and ammonium persulphate
(NH 4 ) 2 S0 4 . Finally the surface is treated with a hot 10% solution
of chromium anhydride.

Recently selenium treatment has been applied in which the part
is treated with a 20% solution of selenium acid (H a Se0 3 ) containing
a small addition of potassium bichromate.

The part should at least he treated twice: first in the as-cast
(as-stamped) condition and then after machining.

Direct contact should be avoided between parts made of magnesi¬
um alloys and those made of metals, whose electrochemical poten¬
tial is higher than that of magnesium (steel, copper alloys, nickel
alloys). In this case the parts must be either zinc- or cadmium-plat-

232

Chapter 3. Weight and Metal Content

ed. To protect components operating in humid atmosphere (par¬
ticularly in sea air) the use of zinc or cadmium protectors is recom¬
mended.

Magnesium alloys are cast in a protective atmosphere (e.g., in an
atmosphere of sulphur dioxide gas produced by powdering the
mould interior with sublimed flower of sulphur). However, it is
still rather difficult to produce a sound casting with uniform mechan¬
ical properties, especially when the casting is large.

Cast magnesium alloys (MJI4, MJI5) are strengthened by means
of a proper heat treatment (heating to 380-410°C for 10-18 h, cooling
in air and ageing at 175°C for 16-18 h).

In general, magnesium alloys are used for non-power components
(non-hearing casings, covers, engine sumps). In particular instances,

(a) split spring ring seal; { b ) installation
of antifriction bearing in intermediate bush
in housing

however, these alloys are used to manufacture some important hous¬
ings. Wrought magnesium alloys are often used for parts subjected
to centrifugal loads.

The disadvantages of magnesium alloys, particularly their poor
corrosion-resistance, restrict their application to the cases when eight
weight saving is the chief factor.

Particular features of parts made from light alloys. When produc¬
ing parts from aluminium and magnesium alloys it is necessary to
consider their characteristics. Thus, it is possible to compensate for
the low strength and rigidity of these alloys by increasing their cross
sections and of resisting and inertia moments, by giving the construc¬
tion a rational form which assures maximum strength and rigidity,
and also by suitable ribbing.

The softness and low tensile strength of light alloys forbid the
employment of screwed-in fixing bolts (Fig. 96a). Should the latter
he absolutely necessary for some design reasons, then the holes for
tapping must be reinforced with steel bushings (Fig. 96fc). The best
fastenings are studs (Fig. 96c) or bolts (Fig. 96d) with large steel

3.6. Light Alloys

233

washers placed in-between the bolt head and/or nut and the part
surface, otherwise, the supporting surfaces are crushed and wear
down as the nut is screwed tight.

Friction surfaces in parts made from light alloys should be rein¬
forced with bushes of some hard metal (Fig. 97a); antifriction bear¬
ings must be mounted in intermediate steel sleeves (Fig. 97 b).

Light alloy surfaces should never be used for supporting springs,
especially when the latter operate under cyclic loads. In such cases-

Pig. 98. Spline-fitting of a light-alloy
part

Pig. 99. Composite structures

it is necessary to apply bearing washers, made of some hard metal*
so that abrasion of bearing surfaces under the action of multiple
repetitive loads is prevented.

Transmission of torque through the agency of keyed or splined
connections made directly in a light alloy component is not recom¬
mended (Fig. 98a). It is. better to reinforce the fitting surfaces
with steel bushes or transmit torque with the aid of locating screws
or pins, spacing them over the radius as much as possible (Fig. 98 b).

When matching light alloy parts to steel parts make due account
for the difference between the linear expansion coefficients of these-
materials. High thermal stresses can occur in fixed joints, in which
the expansion of parts made of light alloys is restricted (locked up)
by steel parts. In movable joints, where the male part is made of
some light alloy,.and its female—from steel (e.g., a cylinder of an
internal combustion engine with an aluminium piston) increased
clearances should be given so that piston seizure at high temperatures
is avoided.

H a part must possess certain qualities (e.g., high hardness, wear-
resistance,, etc.), which light alloys cannot provide, then to make
the part lighter use is made of composite constructions. The non-

234

Chapter 3. Weight and Metal Content

■working portion of a part can be made of light alloy and to this
correctly fastened the operative portions which are manufactured
from materials having the necessary qualities (Fig. 99). Figure 99a
shows a composite construction, a cam plate, whose body is made
•of light alloy whereas the cam rim and the internal drive gear are
made of hardened steel. The rim is riveted to the body. Shown in
Fig. 99 b, is an aluminium alloy impeller of a centrifugal compressor;
the impeller is reinforced with a steel bush, which has internal driv¬
ing splines.

(c) Titanium Alloys

Titanium alloyed with Al, Cr, Mn, Mo, Fe and Si is often used in
■general engineering. The average specific weight of these alloys
is about 4.5 kgf/dm 3 , coefficient of linear expansion a — 8.5 •10 -sO C -1 ,
thermal conductivity, cal/m -h .°G.

The main advantages of titanium alloys are: combination of high
.strength with low specific weight; high heat and corrosion resistance.
The strength of titanium alloys rivals that of alloy steels, and their

Table 12

Mechanical Properties of Titanium Alloys

Alloy grade

Ultimate strength, kgf/mra 2

Elongation

Brinnel

tensile, cr^

yield, a 0 2

fatigue, o_ 1&

per unit
length, % j

hardness

BT-3

95-115

85-105

40-55

:

10-16

285-320

BT-4

80-90

70-80

35-40

15-20

285

BT-5

80-95

70-85

35-45

12-25

285-340

BT-6

90-100

80-90

40-50

8-13

320-360

BT-8

105-118

95-110

45-55

6-12

320-380

-anticorrosion qualities are better than those of stainless steels. The
titanium alloys (Table 12) maintain their high strength within a
.broad range of temperatures (from minus 200 to plus 600°C). They
possess excellent punching and forging characteristics. These alloys
-are more difficult to machine than steels and require more powerful
tools for the purpose. It should be emphasized that under high-speed
cutting conditions titanium chips can ignite and the dust is explosive.

3.7. Non-Metallic Materials

235

To cast titanium is difficult because of the high chemical reactivity
of this metal, and for it easily interacts with the moulding materials,,
as well as the gases given out during casting.

Many titanium alloys can be resistance- and argon-arc welded.
Titanium alloys can be subjected to heat treatment (hardening, tem¬
pering), chemical-thermal treatment (case hardening, nitriding) and
thermal-mechanical treatment.

They can also be improved by \kgf/mm z
cold working.

Antifriction properties of tita¬
nium alloys are not high. Tita¬
nium parts, operating under high
friction conditions are nitrided
and hardened to 900-1000, VPH. 1f 0

Wear-resistance of titanium parts
can also be enhanced through go

diffusional saturation with cop¬
per, tellurium and selenium.

Titanium alloys are exten¬
sively used in the aircraft and
rocket industries, when the com¬
bination of high strength and
low specific weight is important. Fig.100. Ultimate tensile strength of
These alloys are indispensable titanium alloy grade T12 versus tem-

for the manufacture of parts perature

subjected to high inertia loads,

in particular, high-speed rotors, in which the stresses are directly
proportional to the material’s specific weight.

High heat resistance (Fig. 100) and stability against hot corrosion
make titanium alloys suitable for the manufacture of parts working
under high temperatures and loads (gas turbine blades). The good
corrosion-resistance of titanium ensures its application in the chemi¬
cal industries.

3.7. Non-Metallic Materials

(a) Plastics

Plastics (polymers) are, in fact, synthetic high-molecular com¬
pounds produced by polymerization or condensation-polymerization of
monomers—substances comprising of simple molecules having small
molecular weights. Nowadays, we have a wide assortment of plas¬
tics, possessing different physical and mechanical properties. The
most important features of plastics, when they are used as structural
materials* are:

236

Chapter 3. Weight and Metal Content

low strength. (10-30 times less than that of steels);
low rigidity (20-210 times less than that of steels);
low impact strength (20-50 times less than that of steels);
low hardness (10-100 times less than that of steels);
low heat resistance (100-250°C);

low heat conductivity (100-400 times less than that of steels);
low stability of form, owing to low rigidity, hygroscopicity, creep
(inherent in many plastics) and high coefficient of linear expansion
(5-20 times that of steel);

low stability of properties; embrittlement when subjected to the
prolonged influence of changing temperatures.

Plastics possess excellent dielectric properties and high chemical
stability.

Plastics are most commonly applied in electrical engineering,
electrical- and radio-instrument industry and chemical engineering.
In mechanical engineering plastics are used generally for the pro¬
duction of light-weight housings, covers, panels, controls, decorative
elements. Elastic plastics (e.g., PVC, polyolefines, etc.) are widely
applied for the manufacture of flexible hoses, sleeves, collars and
sealing elements.

Certain plastics (such as, polyamides and fluoroplastics) possess
high wear resistance and a low friction coefficient, making them
valuable materials for the production of plain bearing sleeves and
silent gears.

For load-carrying structures plastics reinforced with fibre-glass
and glass-fabric are often applied. Glass-fibre moulding materials
are employed for the manufacture of boat hulls, fairings, car bodies
and other components of shell-type constructions which successfully

7 Table 13

Values of C for Metals and Plasties and the Ratio Cpiast/^metai

Materials

Amino

plastics,

C=l25

Vini

plastics,

C=170

Capron,

C=235

Fibreglass
reinforced
plasties,
C=680

Epoxies,

C=*130G

Fluoro¬
plastics,
C=Z 2 000

C -plast/C metal

Steels:

carbon
((7 =--13.5)

9.3

12.6

17.5

50

95

1600

ft?

1 —|

O

«<2

O

1 !

to

10.5

14

19.5

56

110

1850

stainless
((7 = 50)

2.5

3.4

4.7

13.5

26

440

Aluminium alloys

3-3

4.5

6.5

18

35

600

(C — 37)

Bronzes ((7 = 110)

1

1.15

1.5

2.2

6

12

200

3.7. Non-Metallic Materials

237

compete in strength with similar metallic parts. Insufficient rigidity
is compensated for by increased thickness and sections.

It should be noted that plastics constructions for a time remain
more expensive than metal ones.

Relative costs of materials are given in terms of specific cost in
Table 13, which shows the cost of equal-strength components made
from different materials

where P = price per ton of material in roubles;
y — specific weight of material, lcgf/dm 3 ;
a b = ultimate tensile strength of material,; kgf/mm 2

(b) Reinforced Wood

Wood materials applied in mechanical engineering are generally
impregnated with synthetic resins and pressed under high temperatur¬
es. The most used are wood-laminated plastics made from the best
birch veneer 0.3-1.5 mm thick. The veneer is impregnated with raw
bakelite (resol or phenolformaldehyde resin), placed into metal
moulds and subjected to a hydraulic pressure of 300-500 kgf/cm a at
the bakelite setting temperature (160-180°C).

Reinforced wood ( delta-wood , laminated birchwood) has ultimate
tensile strength (along layers) cr & = 15-20 kgf/mm 2 , and compres¬
sive strength (across layers) a Mmpr = 25-35 kgf/mm 2 specific
weight 1.2-1.4 kgf/dm 8 ; the mechanical properties, shown for tension
across the layers and compression with the layers are 30-40 per
cent lower.

Balinite (a wood-resin laminate) is prepared by the same method
except that the wood is treated with a 5% NaOH solution prior to
impregnation. The mechanical properties of balinite are somewhat
higher than those of delta-wood.

Sheets and plates of wood-laminated plastics are widely used for
manufacture of panels and various facing pieces. Products from
wood-laminated plastics can be moulded into the required shapes.

Lignostone (a kind of laminated wood) is birchwood, bakelite-
impregnated and pressed into blocks. This material is used mostly
for the manufacture of segment-shaped bearings intended to be oper¬
ated with water lubrication.

Bushes, gears and other parts are made of birchwood sawdust
impregnated with bakelite and then formed under pressure into
the required shapes. Wooden gears work well under smooth (impact-
free) loads at pressures not exceeding 30-50 kgf per 1 cm of gear
tooth width. Wooden gears show high wear resistance when meshed
with metal gears.

238

Chapter 3. Weight and Metal Content

(c) Glassceramics ( Sitalls )

Sitalls (glassceramics) are a silicate glass, possessing a fine-crystal¬
line structure changing radically material properties. They possess
improved strength, lack of blittleness and thermofragility inherent in
glass and can withstand impact loads successfully.

In contrast to usual glass, which becomes softer with the rise of
temperature, sitalls keep their hardness and strength up to 600°G.
Like metals, glassceramic materials have a definite melting point*
which varies between 1200 and 1400°C depending on the grade.

Their ultimate tensile strength o 6 = 40-80 kgf/mm 2 , approximates
that of carbon steels and high-strength cast irons. In the laboratory
sitalls have been obtained with ultimate tensile and compressive
strengths of 100 kgf/mm 2 and 150 kgf/mm 2 , respectively.

Glassceramics are excellent dielectrics and display high resistance
to aggressive chemicals surpassing in this respect plastics, stainless
steel and titanium alloys. They also successfully resist the attack of
the strongest acids and alkali (except hydrofluoric acid).

Specifications: specific weight 2.2-2.3 kgf/dm 3 ; thermal capacity
0.2 cal/kg°G; average thermal conductivity 2-4 cal/m -h *°C. Modulus
of normal elasticity from 10 000 to 15 000 kgf/mm 2 .

An interesting feature of sitalls is possibility of regulating within
wide limits their linear expansion coefficient. Depending on the
chemical composition and structure of a sitall, its linear expansion
coefficient can vary from 20-10'" 6O C“ 1 to zero. Thus, the possibility
is provided of making parts which will not change their linear di¬
mensions despite temperature variations and, hence, will be free of
thermal strains. Some sitalls have even negative coefficients of linear
extension (a = — 2 -lO^G -1 ), i.e., their sizes reduce with the
rise of temperature.

Sitalls with low linear expansion coefficients are noted for their
high thermomechanical stability (products of such sitalls even when
heated up to 800~900°C can safely be immersed into cold water). This
feature makes sitalls particularly valuable for the manufacture of
parts subjected to thermal shocks.

Outwardly, sitalls are glass materials, whose colouring, depend¬
ing on the chemical composition and structure, can be white, cream,
grey, yellow-brownish, brown and dark, down to black. Some sitalls
are transparent, others—semitransparent with yellowish or brownish
hues.

The main advantage of the glassceramic material is its cheapness
and practically inexhaustable raw material reserves. Sitalls are
produced from rock minerals: magnesium-alumosilicates, calcium-
alumosilicates, calcium-magnesium-alumosilicates (petrositalls ) or
metallurgical slags and cinder ( slagsitalls).

The process of manufacturing products from sitalls is as follows.

8.7. Non-Metallic Materials

239 >

From a charge of the required composition a glass is made from
which in the liquid or plastic state the product is formed either by-
casting or extruding. The obtained products are heat-treated in¬
steps (first step, at 500-700°C and the second, at 900-1100°C) after
which they acquire a crystalline structure.

Nucleators are introduced into the glass composition which are
substances forming centres of crystallization. Earlier, colloidal
particles of Cu, Ag and Au were used as nucleators, which became-
actually nuclei (seeds) of crystallization as a result of product
irradiation by penetrating radiation (photocerams ).

Nowadays the expensive photochemical process is excluded; as-
the nucleators now used are iron sulfides, titanium oxides, fluorides-
and phosphides of alkali and alkali-earth metals (Na, Ca, Li).

At the last step of heat treatment products are uniformly crys¬
tallized. The content of a crystalline phase reaches 95% and crystal
sizes are rather fine (up to 0.05 pm), i.e., hundreds of times less than
those in fine-grained steels. Dimensional changes of a product in the
course of crystallization does not exceed 2%.

Crystallized products can be machined by carbide, boron and dia¬
mond tools, as well as by ultrasonic technique.

The combination of high strength, toughness, hardness, thermal
and chemical resistance, low specific weight and good formability
with the use of the most efficient forming techniques make sitalls a.
promising construction material.

From sitalls are prepared parts for chemical equipment, pumps,,
heat-exchangers, pipes, vessels, reservoirs, male and female parts
of dies, draw plates, parts for radio receivers, electrical machines and
instruments.

In civil and industrial construction work sitalls are extensively
used as a facing material possessing high strength, durability, ab¬
rasion resistance, good heat-insulating properties and complete
moisture resistance; furthermore, they resist well the influence of
high temperatures, thermal shocks and gas erosion.

Sitalls are also used in the manufacture of thermally stressed parts,
Sitall slide bearings can run without lubricant under moderate loads
and rotational speeds at temperatures up to 500°C.

Many structural components in general machines may be made of
sitalls.

(d) Reinforced Concrete

In some branches of machine building the use of reinforced con¬
crete structures is promising. It is good practice to prepare from rein¬
forced concrete large-sized housings and basic parts of heavy machines
(e.g., beds of unique metal-cutting machines, presses, hammer an¬
vils, etc.). In this case the metal volume is sharply reduced with
large savings in manufacturing costs.

240

Chapter 3. Weight and Metal Content

For reinforced concrete structures only first-class portland-cement
is used, which is a finely powdered silicate mixture roasted before at
1500°G. The mixture is prepared from limestone, clay and quartz
sand. Generally, the composition of roasted cement includes: 65-70%
CaO; 20-25% S.iO 2 ;8-10% Al a 0 3 and 2-5% Fe 2 0 3 . During interact¬
ion with water the cement hardens and after a certain interval of
time turns into strong monolithic mass. Best hardening conditions
require a temperature not lower than 15-20°G and high humidity of
the ambient air. Hardening is slower at low temperatures and dis¬
continues at minus temperatures; it accelerates when subjected to
heat and moisture treatment (wet steam heating).

The quality of portland-cement is dependent on its mineralogical
composition and the fineness of grinding: the finer the cement the
more quickly and fully it interacts with water and the higher its
strength. Portland-cement usually sets in 1-1.5 h and completely
hardens in 10-12 h. With successive maturing the strength of cement
increases but after approximately 30 days the hardening process
becomes slower.

Portland-cement is produced in the following grades: 200 , 250,
300, 400, 500 and 600. The figures indicate ultimate strength in
kgf/cm 2 (cube strength), from compression testing a standard cub¬
ical sample measuring 20-20*20 cm 3 , prepared from a mixture of
cement and quartz sand (ratio 1:3). Testing is carried out after a
28-day hardening period at 15-20°C and 90% relative humidity of
the ambient air. The volume weight of portland-cement is
3-3.2 kgf/dm®.

Concrete is a hardening mass comprising a mixture of cement, fine
aggregate (quartz sand) and rough aggregate (gravel). The strength
of concrete depends on the quality of cement, as well as on the pro¬
perties and granulometric composition of aggregates, percentage ratio
of cement and aggregates, hardening conditions (ambient tempera¬
ture and humidity) and also on the method of placement and compac¬
tion degree of the mixture.

The weight ratio of concrete constituents is expressed by the
following formula:

, . W
1: x : y :

where 1 = weight of cement, assumed equal to unity;

x — number of parts of sand by weight;

y — number of parts of gravel by weight;
yty

-jT- = water-cement modulus, i.e., water-cement ratio

The lower the water-cement modulus, the stronger the concrete.
For normal hydration it is enough to introduce water in quantities

amounting to 20% of the cement weight (~r = 0.2). However, in
practice, this ratio is generally taken as equal to 0.3-0.5, because

3.7. Non-Metallic Materials

241

reduced water content impairs “liveliness” of the concrete mixture.
The usual concrete mix is: 1 : 1 : 2 : 0.5.

To produce a strong concrete use is made of a quartz or granite sand,
whose particles measure within 0.2-0.4 mm; chippings of approx¬
imate size 20-30 mm from hard crystalline rocks (granite, syenite, dia¬
base, basalt). Thin-walled concrete products, walls 30-40 mm thick,
are made of cement-sand or cement-chippings mixtures. The chipping
size in this case should not exceed 0.25 of the wall thickness.

The strength of concretes is taken as equal to the ultimate compres¬
sive strength of the standard cubical sample. As a rule, this cube
strength reaches 500-600 kgf/cm 2 . When steel chips are utilized as
reinforcement (steel reinforced concrete) the cube strength can be
as high as 1000 kgf/cm 2 .

The volumetric weight of concrete depends on its composition and
aggregates. Concrete of the above compositions have volume weights
which vary within 2.2-2.7 kgf/dm 3 .

Light concretes (volume weight -< 1.5 kgf/dm 3 ) are obtained by
employing light sedimentary rocks as aggregates (pumica, tuff,
shell rock), as well as cinder or metallurgical slags. Although inferior
in strength, light concretes have excellent heat- and sound-proof pro¬
perties.

Honeycomb and foamy concretes with a volume weight of about
0.2 kgf/dm 3 also possess good heat- and sound-proof qualities.

From the viewpoint of a structural material, concretes display
brittleness and sharp anisotropy of mechanical properties. Con¬
crete strength in tension is worse than that in compression and it
shows propensity for brittle cracking even when subjected to slight
tensile stresses. Its ultimate tensile strength is 10-20 times less
than its ultimate compressive strength.

Concretes possess yielding properties. Under compressive loads,
exceeding 0.3-0.5 of the cube strength, concretes acquire a yield
state and their dimensions spontaneously change which means limit¬
ing design compressive stresses to rather low values (150-250 kgf/cm 3
for concretes with a cube strength of 500-600 kgf/cm 2 ).

Another feature typical of concretes is their low elasticity modul¬
us stipulating the poor rigidity of parts. For concretes the elasticity
modulus E = 1500-4000 kgf/mm 2 (average value 3000 kgf/mm 2 ),
being approximately three times less than that of cast iron and
seven times less than that of steel. The shear elasticity modulus
G = 1400-1600 kgf/mm 2 .

Concrete has poor resistance to acids, alkali, machine oils and
cutting fluids. To protect concrete from the attack of such compounds
it is better employ components covered with a sheet metal skin.
Concrete resistance against aggressive chemical substances can be
effectively improved by introducing silicone-type polymers into the
mixture (polymerconcretes).

16—01395

242

Chapter 3. Weight and Metal Content

The low shrinkage when hardening is a positive feature of con¬
crete as a construction material. The linear shrinkage coefficient
of concrete averages 0.03%. This assures the dimensional stability
of concrete castings and accuracy of relative position of metallic
elements imbedded in the concrete mass. It also reduces machining
of metallic base parts of a manufactured product. There are concretes
with practically no shrinkage (containing admixtures of gypsum,
etc.).

Reinforced concrete, i.e., a concrete with imbedded steel rods,
network or lattices, is almost exclusively employed for structures
undergoing tensile stresses, as well as dynamic and alternating
loads.

The concrete coefficient of linear expansion (a = 12-iO -6 ^" 1 )
approaches that of steel, thus ensuring good bond between the con¬
crete and reinforcing elements during temperature fluctuations.

Prestressed reinforced concrete is obtained when the reinforcing
elements are subjected to tension' during forming process (pretension
by jacks or by induction heating), imparting to the concretes
higher tensile strength. The weight of steel reinforcement is general¬
ly 15-30% of the total weight of reinforced concrete.

Preliminary tensile stresses in the steel reinforcement amount to
150-250 kgf/cm 3 . Admissible tensile stresses in the prestressed rein¬
forced concrete average 100-150 kgf/cm 2 and admissible compressive
stresses from 300 to 500 kgf/cm 2 .

Reinforced concrete possesses an extremely high cyclic toughness,
namely, approximately double that of grey cast iron. This feature
contributes to the high anti vibration properties of concrete products.

It is obvious from the above that reinforced concrete, when used
as a structural material, will be inferior to metals in relative strength
and rigidity. Permitted tensile and compressive stresses in reinforced
concrete are approximately three times less than in grey cast iron.
To obtain reinforced concrete structures, equal in strength to the
cast iron ones, it is necessary to increase accordingly cross sections
and resisting moments. A rule, kept in practice, says that the rein¬
forced concrete products cross sections must be three times greater
than those of the cast iron counterparts for the same strength to be
achieved. Since the elasticity modulus for reinforced concrete is
approximately 1/3 that of cast iron, then increasing reinforced con¬
crete products cross section in the same ratio will bring the rigidity
of the latter, when under tension-compression loading, up to the
level of the cast iron products.

In practice the rigidity of reinforced concrete structures subjected
to tensile-compressive stresses is calculated in terms of the reduced
cross section

F Te <l = F c + F T ^-

3.7. Non-Metallic Materials

243

where F c and F r — areas occupied in the calculated cross-section
by concrete and reinforcement, respectively
E c and E r = Young’s modulus for concrete and reinforcement,
respectively

Assuming Young’s modulus of concrete E c — 3000 kgf/mm 2 and
that of reinforcement E r = 21 000 kgf/mm 2 , we obtain

Frrt&Fe + IF,

simplifying

^=^( 1 + 7 -^)

V r.sec 1

where F sec is the total cross-sectional area subjected to tensile and
compressive stresses.

Similarly, when calculating reinforced concrete components subj¬
ected to bending, use is made of the reduced resisting moment

w red = W c + Wr « W c 4 - 1W V

where W c

and W r are the resisting moments across cross-sections
occupied respectively by concrete and reinforcement
relative to the neutral axis of the cross-section.

Or

W nd -W„'( 1 + 7 -^-)

where W sec is the moment of inertia across the entire section of part.

In respect of the weight reinforced concrete constructions give
way to cast iron ones (of the same strength). The specific weight
of reinforced concrete varies within.3-4 kgf/dm 3 depending on the
weight of reinforcement. For the increase in the cross-section be only
thrice in comparison with the equistrong cast iron part (cast iron
specific weight 7.2 kgf/dm 3 ), the resultant weight of the reinforced

concrete components will he —- = 1.3-1.7 times greater than
that of cast iron parts.

Thus, the chief gain obtained from the employment of reinforced
concrete components is due to decreased metal volume (3-4 times on
average) and cheaper production cost. Additional advantages are,
obtained from simpler production techniques as requirements for
pattern-making, moulding and heat treatment of castings are obvi¬
ated. With correct management of the pouring and hardening pro-
ceses waste is practically eliminated. ...... ;

16 *

244

Chapter 3. Weight and Metal Content

However,*the manufacturing process of machine engineering
reinforced concrete components means more manual labour (assembly
of moulds, particularly those of metal, positioning and alignment of
metallic part locations, placement and tensioning of reinforcement).
Another shortcoming is the long production cycle and the necessity
to keep the reinforced concrete castings under controlled conditions
of temperature and humidity for 15-20 days. The latter disadvantage
is eliminated by wet-heat treatment, after which the reinforced con¬
crete attains 70 % of its designed strength within 6-8 h.

Use of reinforced concrete is justified in the manufacture of spe¬
cially designed and individually produced large-size machines and
assemblies. Casting base parts for such machines from cast iron
presents formidable difficulties. In some cases (when lacking suf¬
ficiently powerful casting machines) the employment of reinforced
concrete structures is the only practical solution.

In general machine building, as a means of increasing strength
and rigidity, the concrete material can be used to fill hollow spaces
in hollow parts. In other instances it is advisable to fill tubular or
box-shaped components (uprights, columns, pedestal legs, brackets,
beams, v etc.) with concrete.

Machine building structures of reinforced concrete. Reinforced
structures applied in machine building are essentially frame-
reinforced castings, into which steel or cast iron elements are imbed¬
ded (pilot bushes, inserts, locating plates, pedestals, uprights, bra¬
ckets, etc.) to suit the expected purpose of a structure.

There are two principal manufacturing techniques.

The first method: reinforced concrete components are cast in
wooden frames which are removed after the concrete has set. The
second method: concrete components are cast in welded thin-sheet
skins (moulds) 1.5-2 mm thick, internally reinforced with longitudi¬
nal and transverse braces (named permanent metallic framings). To
avoid bulging from the hydrostatic action of the liquid concrete
when pouring the shell is additionally strengthened on the outside
with removable wooden sections. Holes provided in the shells for
risers and gates are welded after the casting has set.

The second method is more perfect, as the metal shell isolates the
concrete from the influence of external medii (namely, lubricating
oils, lubricating and cooling fluids) and protects it from chance
mechanical damage, spalling and shear. However, this method is
much more expensive and requires more labour than the first one.

Of utmost importance for the casting strength is the uniformity
and compactness of the mould filling. It is obligatory to vibrate the
mould at a frequency of 1000-3000 oscillations per minute for 5-10
minutes when filling.

The best results are obtained when the mould is installed and
shaken in a vibrostand. When manufacturing large-size components

3.7. Non-Metalllc Materials

245

consolidation is effected from the surface with the aid of plate and
rake vibrators. When compacting difficult-to-reach areas depth vi¬
brators are used.

If casting is performed in wooden or open metallic moulds, the
humidity of ambient air in the shop should be maintained within
80-90 %. To avoid too quick drying open sections should be moistened.
After the casting has been freed from its mould (usually in 10-12 days)
it must be covered with damp sawdust. These requirements are
less rigorous when casting is done in metallic moulds as sufficient
moisture is inside the mould.

The density of finished castings is checked by X-ray and ultra¬
sonic techniques.

After seasoning the casting for 15-20 days, its metallic bearing
surfaces can be machined. This interval can be cuCdown to 1-2 days
if the casting has been subjected to steaming.

Design rules. When designing reinforced concrete castings the
following rules should be observed:

simplify in every possible way the shape of castings, particularly
when pouring into metallic shells (moulds), make casting elements
in the form of simple geometrical bodies (cylinders, tubes, cones,
prisms, etc.);

make wall thicknesses not less than 30-40 mm;

assure smooth transitions from section to section; avoid, whenever
possible, difficult-to-fill cavities and pockets in which pits and
voids may form; to fill such spaces properly, provide additional
risers and gates;

for complex configurations use a livelier concrete with an increas¬
ed water-cement ratio (0.7-0.8);

in the direction of tensioning forces arrange the load-carrying
metallic reinforcement; in zones subjected to bending concentrate
the reinforcement at the places where maximum tensile stresses act.

Large-size irregularly shaped components are better made in
separate parts. At butt jointing surfaces of parts connected' by
bolting or welding, suitable metallic fasteners should be cast-
imbedded into the concrete. It should be taken into account,
however, that jointing lessens the rigidity of the construction, im¬
pairs the accuracy of relative position of basic structural surfaces,
furthermore, also means more machining.

Constructions, as a whole, must possess a rigidity sufficient not
only for normal functioning in stationary conditions but also for
transportation and installation in its place. Extensive constructions
must include sturdy longitudinal reinforcement elements made of
high-grade rolled stock.

Basic metallic components, being cast into the concrete, must
be reliably secured and relieved of stresses as much as possible. Such
components should not be employed as stiffening elements. Stiff-

246

Chapter 3. Weight and Metal Content

ness must be assured by internal reinforcement and suitably shaped
sections.

The adhesion strength of the metallic shell and concrete can be
enhanced by welding wire or plate anchors to the inside of facing
sheets.

3.8. Specific Indices of Strength of Materials

The -weight advantages of materials can be evaluated by specific-
indices characteristic of every type of loading;

Tension-compression. The weight of parts, subjected to tension
or compression, other conditions being equal (identical lengths and
loads) is calculated by the formula

G = const Fy (3.22)

where F = cross-sectional area of the part;
y — specific weight of the material

The cross-sectional area is inversely proportional to the acting
stress

„ const

For parts of equal strength (i.e., parts with equal safety
margins) the stress value a can be changed by the ultimate strength
value o 6 . In this case

rp const
r = —--

Inserting this expression into Eq. (3.22), we obtain

G = const —

Ob

Consequently, the weight of equistrong parts subjected to tensile
or compressive stresses under other equal conditions is determined
by the ratio between the material’s. ultimate strength and specific
weight

(3.23)

This factor, termed specific strength, characterizes the weight
advantage of the material under tension or compression.

Modern design practice diverts from the evaluation of strength in
terms of o b) i.e., according to failure stresses. Much more often the
design calculations pertaining to statically loaded parts are based
either on the proportionality limit a p (when this value is sharply ex¬
pressed in the material) or on the yield limit g 0 .2 (i.e., the stress at
which the residual strain comprises 0.2 %).

When dealing with parts operating under cyclic loads the initial
value for the calculation is the fatigue limit o D .

3.8. Specific Indices of Strength of Materials

247

The yield limit is not proportional to the ultimate strength. The
a 0 . a value for different materials varies between 0.5 and O.950 b .
Therefore any calculations based on the ultimate strength, even
with large safety margins, may lead to gross
errors.

As the ultimate strength belongs to the
most readily definable characteristics and
since a certain proportionality exists be¬
tween o 03 and o b for some groups of mate¬
rials, the o b parameter is often taken as
the basic strength characteristic.

When calculating in terms of the yield
limit o 0 . 2 , the specific strength factor will
be expressed as

(3.24) ^

and when calculating in terms of the fa¬
tigue limit

f- . ( 3 - 25 )

This factor is called the specific fatigue
limit when under tension or compression.

The specific strength indices can he il¬
lustrated by the following..

Imagine a freely hanging bar of an arbitrary section. One end of
the bar is secured (Fig. 101) and the bar is, thus, loaded with its own
weight. The most dangerous section is a-a, upon which acts the full
weight

... G = FLy (3.26)

where F = area of cross-section;

L — length of the bar;
y — specific weight of the bar material

The tensile stress across this section will be

—J-

! 's

Fig. 101. Freely hanging
bar

or, taking into account Eq. (3,26),

o = Ly (3.27)

Let the tensile stress attain the ultimate strength value (a = <y b ).
This will occur when the bar has a certain length L br (i.e., breaking
length) which, according to Eq. (3.27), is

248

Chapter 3. Weight and Metal Content

This value coincides with the specific strength of the material,
Eq. (3.23).

Should Gb be expressed in kgf/mm 2 , and y, in kgf/dm 3 , then

Lbr

Gb kgf dm 3

y mm 2 kgf

— • 10 6 mm = — km (3.28)
'y y ' 7

i.e., in this case the length L br will be expressed in kilometers.
Similarly

r Qq. 8 kgf dm 3
y y mm 2 kgf

(3.29)

expresses the length (in km) of a freely hanging bar, at which the
stresses across the critical section reach the yield limit.

The displacement of the bar free end (complete extension) as is
well known equals

f — ^

1 ~ 2EF

Since G = FLy and L — — , then, when L — L y and o

u 0.2
2 Ey

km —10 3 ■

° 0.2
2 Ey

m

= tfo.a
(3.30)

where cr 0 . 2 and E are in kgf/mm 2 , and y, in kgf/dm 3 .

The f y value defines the resilience and impact resistance of the
material.

Flexure (bending) and torsion. For bending and torsion the weight
advantage of the material is computed by the proportion

0 2/ 3

V

(3.31)

where a is the rupture stress for the given type of load (<?& for flex¬
ure and % b for torsion).

This factor is known as the specific flexural (or torsional) strength.
If calculations are carried out in terms of yield or fatigue limit, then
the numerical values of corresponding limits are substituted into the
numerator of Eq. (3.31).

The weight advantage evaluation is only an approximation.
This is why the comparison of all types of loading is generally ac¬
complished with the aid of such structurally simple factors as —

and , pertinent to compressive-tensile loads.

Impact loads. The ability to resist impact loads is defined through
the work of recoverable strain. When tensioning a bar of constant
cross section F and length L

U

P*L

o z FL
2 E

2 EF

S.S. Specific Indices of Strength of Materials

249

The value of U at stress a equal to proportionality limit cr p shows
the material’s ability to withstand impact loads within the region
obeying Hooke’s law

Having divided this value by weight G = FLy, we obtain

,t2

r J-

u

''FLy

2Ey

(3.32)

This factor is the specific impact strength, which characterizes
the material weight advantage under impact loads.

When approximate comparison will suffice, the limit of propor¬
tionality o p can be changed to the yield limit cr 0 . 2 (which is rather
close to it). Then

V

° 0.2
2 Ey

(3.33)

This Equation is similar to Eq. (3.30) which pertains to the com¬
plete extension f y of a freely hanging bar of length L y , at which
the stresses in the critical cross section reach the yield limit.

(a) Comparative Weight Evaluation
of Structural Materials

Table 14 cites average values of a b , cr 0 .2 and E, as well as specific
strength characteristics of structural materials. For certain specific
strength indices the upper values of o 6 and o 0-2 are given.

High-strength steels, whose specific strength, expressed in terms
of a breaking length (Fig. 102a) reaching 45 km, have the best'weight
advantage. Next to these are glass-fiber anisotropic materials
(GFAM) with L br = 37 km (this value is valid only for those cases
when fibers are favourably oriented with respect to the loads). The
last place is occupied by grey cast irons ( L br — 5 km).

Approximately the same order is preserved when classifying ma¬
terials in terms of their specific strength ^ (Fig. 102&) and specific
2 *

impact strength (Fig. 102c). In the latter case the extra-high-

strength steels are well ahead of the rest, of materials, because their
specific impact strength is 2, 5 and 40 times better than that of titani¬
um alloys, alloy and carbon steels, respectively.

It is necessary to say that the'choice of. a material is governed not
only by strength-weight characteristics, but also by the purpose and
operational conditions. When selecting a suitable material attention
is paid to such characteristics as rigidity, hardness, toughness, plas-

Table 14

Specific Strength Characteristics of Structural Materials

Specific indices

Cl>

►a

to

C*

►O

£

Material

. w>

>0

' § A

+»s

11

’3

b i a

bo

fi

b

*? \P-

o ^

B'S

o , ~ s *
u

p, -

Ultimate

strength,

Itgf/mmz

■a -

Modulus

ticity,

kgf/mm 2

A

t I

I *

» £

si 2

■fj p!

bo

P

g a>

» 6

** * ltd

“«eps

b .

2* «
a s <*

Carbon-steels

m

35-80

21-48

6

7

Alloy steels

100-180

80-145

21000

18.5

64

High-strength steels

■

250-350

225-315

40

300

Grey cast irons

bi

20-35

15-25

8 000

5

3.5

5.5

High-strength cast irons

m

45-80

32-56 •

15 000

11

7.7

13

Aluminium alloys

cast

2.8

18-25

■ 13-17.5

9

6.5

wrought

40-60

28-42

21.5

. 15

Magnesium alloys

cast

1.8

12-20

8-13

4 500

11

25-30

16-20

wrought

16.5

11

25

Structural bronze

8.8

40-60

32-48

11000

7

5.5

12

Titanium alloys

4.5

80-150

70-135

12 000

33

30

170

delta-wood

1.4

15-20 (along

5 000.

13

_

_

layers)

Structural plastics

mmm

1.6

25-30

--

5 000

19

—

—

GFAM

1.9

1tiitei

|

6 000

37

—

—

Sitalls

3

50-80

45-72

15 000

27

24

58

3.8. Specific Indices of Strength of Materials

251

ticity, technological characteristics (machinability, punchability,
weldability), wear, corrosion and heat resistance, thermal stabil¬
ity (the two latter terms are for parts working under high temperatu¬
res).

Of no less importance is the cost of a material, its short supply and
expensive and/or difficult-to-obtain components.

Fig. 102. Specific strength indices of materials
l — extra-high-strength steels; a — alloy steels; 3 — caTbon steels; 4 — grey cast irons;
S — high-strength cast irons; fi — structural bronzes; 7 — wrought aluminium alloys;
3 _ cast aluminium alloys; 9 — wrought magnesium alloys; 10 — cast magnesium alloys;
11 — titanium alloys; ra — delta-wood; 13 — GFAM; U — glass-fibre materials; is —

sitalls

QThe best universal properties with high strength-weight indices
belong to alloy steels. By introducing alloying elements and apply¬
ing special heat-treatment processes, it is possible to change their
properties within wide limits adding, depending on the requirements,
hardness, heat and corrosion resistance, etc. This makes steel become
the most widely used and universal material for the manufacture
of load-carrying structures.

The same properties of rigidity and high strength-weight indices
are possessed by titanium alloys.

Chapter 4

Rigidity of structures

Rigidity is one of the basic factors, which determine the work-
capability of a design, and has the same, if not greater, effect on
reliability as the strength.

Increased strains can disturb normal functioning of a construction
long before stresses dangerous for strength appear. Disturbing the
uniform load distribution, they cause a concentration of forces at
separate parts of the part, the result being local stress concentrations
far exceeding the nominal stress values.

Non-rigidity of housings spoils the arrangements of mechanisms
installed inside and causes increased friction and wear of movable
joints; non-rigidity of shafts and gear-carrying supports disrupts
gear engagement leading to quicker gear tooth wear; non-rigidity of
trunnions and sliding bearing supports causes higher edge pressures,
appearance of local semi-liquid and semi-dry friction spots, overheat¬
ing, seizure or shorter bearing life; non-rigidity of fixed joints sub¬
jected to dynamic loads brings about fretting, galling and sticking.

In machines performing accurate operations, for example, metal¬
cutting machine tools, the rigidity of operative members as well
as their supports determine the dimensional accuracy of the finished
workpieces.

Rigidity parameter is of the greatest importance for the lighter-
class machines (transport vehicles, aircraft and rockets). In pur¬
suit of lighter constructions and maximum utilization of materials’
strength reserves the designer in given instance is compelled to raise
stress levels which leads to larger strains. Extensive employment of
equally strong constructions with the most favourable use of weight
in its turn causes greater strains as the equistrong structures possess
less rigidity.

Particular attention is given to rigidity problems especially with
the appearance of high- and super-high tensile strength materials
whose application sharply increases the deformation of constructions.

Not uncommon are the cases when the magnitude of forces, affect¬
ing the design, is underestimated. Very often during calculations
very low values of working forces are obtained while the actual loads
unexpectedly reach very high levels leading to partial or complete

4.1. Rigidity Criteria

253

failure of a part. This may be caused by imperfect assembly, flexure
of insufficiently rigid constructional elements, residual strains, over-
tightening of fasteners, higher friction, distortion and seizure of
rubbing parts, loads occurring during transportation and installa¬
tion of a machine and other factors not considered at the design stage.

Strain values can be calculated only for the simplest cases, i.e.,
for ones, which can be solved through routine solutions based on the
strength of materials and elasticity theories. However, in practice,
the designer has to deal with “incalculable” parts for which it is
actually impossible to even approximate the amount of future strains.

Under such circumstances the designer has to recourse to simulat¬
ing techniques, experiments, including data on practical use of
similarly designed machines in the field and sometimes depend on
his experience gained over many years. An experienced designer
knowing the direction and magnitude of acting forces evaluates
more or less correctly their direction and value, amount of future de¬
formation, finds the weakest links and, applying various techniques,
increases rigidity, thus creating a rational design.

Conversely, constructions designed by beginners usually have
insufficient rigidity.

4.1. Rigidity Criteria

Rigidity is the capability of a system to resist the action of exter¬
nal loads with the least deformation. In mechanical engineering
rigidity can be defined thus: rigidity is the capability of a system to
resist the action of external loads with permitted deformations which
do not destroy the work-capability of the system. The inverse of
rigidity is elasticity, i.e., the property of a system to acquire com¬
paratively large deformations under the action of externally applied
loads. In respect to machine-building the most important factor is
rigidity, however, in certain cases elasticity does play the major
role (springs, shock absorbers and other elastic parts).

The rigidity characteristic is evaluated by the coefficient of
rigidity (or stiffness), which is the ratio between the force P applied
to a system and the maximum deformation / produced by the force.

In the simplest case of tension-compression of a beam of constant
cross-section within elastic limits the coefficient of rigidity, in
accordance with Hooke’s law, will be

where E = elasticity modulus of material;

F = cross-section of beam;
l — beam length measured along action of force

254

Chapter 4. Rigidity of Structures

The inverse value

= TT ( 4 - 2 >

characterizing the elastic yielding capacity of the beam is called the
coefficient of elasticity.

Determined according to relative strain e = ///, the coefficient
of rigidity

X'tens=EF kgf

indicates, in fact, the load in kgf causing relative strain e — i.
The corresponding coefficient of elasticity

p'-^kgf- 1

gives the relative strain under application of 1-kgf load.

For a beam of constant cross section subjected to torsion, the
coefficient of rigidity is the ratio between the applied torque moment
M tr „ e and the angle <p through which the beam sections twist over
the length l

■j, ' Mtrde t/.

Mors - - -— (4-d)

where G — the shear modulus of material;

I p = polar moment of inertia of beam section
The coefficient of. rigidity for a beam of constant cross-section
upon which a flexural load acts, is given by

hfiex — -j-==a-jf (4.4)

where I — moment of inertia of beam section
l — length of beam

a — coefficient depending upon loading conditions
Figure 103 illustrates values of the rigidity coefficient for some
forms of flexural loads. Taken as unity is the % value which corres¬
ponds to the flexure of a simply supported beam loaded by a concen¬
trated force P at its midspan.

It is clear that the rigidity of the system depends on the loading
conditions. Thus, a beam having a uniformly distributed load
(Fig. 103b) has a rigidity 1.52 times greater than that of a beam
loaded with a concentrated force of the same total value (Fig. 103a).
The rigidity is affected still more by the type and arrangement of
supports. For instance, the rigidity of a simply supported beam with
encastre ends (Fig. 103c, d) is 5-8 times greater than that of a beam
freely supported at its ends (Fig. 103a,' b).

4.1. Rigidity Criteria

255

The rigidity of a cantilevered beam loaded with a concentrated
load (Fig. 103e) is only 0.063 of the rigidity of a simply supported
beam of the same length loaded with the same force at the midspan.

A-i

(a) (a-m

q= p /l

(b)

M33..

(•)

A - 1.52
(a = 77)

2.=0.053
(a-3)

-t-

q = p /L

A=4
(a =192)

■ (C)

'/////a

K

(f)

A ='0.77
fe-A)

Fig. 103. Rigidity coefficient y for different flexure schemes

For given load values and linear dimensions.of a system the rigi¬
dity is determined by the maximum deformation value /. The latter
is often employed for the practical evaluation of deformation in
geometrically similar systems.

(a) Factors Determining the Rigidity of Constructions

The rigidity of constructions is governed by the following factors:
modulus of elasticity of the material (modulus of normal elasticity
E under tension-compression in bending and shear modulus (? in
torsion);

the cross-section’s geometrical characteristics of the deformed body
(section F for tension-compression and moment of inertia 1 in fle¬
xure, the polar moment of inertia I p in torsion);

linear dimensions of the body undergoing deformation (length Z);
type of loading and type of supports [factor a in Eq. (4.4)}.

The modulus of elasticity is an extremely stable characteristic
of metals and depends solely on the density (packing) of the atomic-
crystalline lattice, i.e., on the magnitude of the mean interatomic
distance. From all, the commercial metals only three possess high
moduli of elasticity, namely: tungsten, molybdenum and beryllium
(E — 40 000, 35 000 and 31 000 kgf/mm 2 , respectively).

The modulus of elasticity of practically all structural materials
being used varies from 22 000 (steel) to 4500 kgf/mm 2 (magnesium
alloys). However, actual application of a material will mostly depend
on operating conditions for the part. Therefore the main practical

256

Chapter 4. Rigidity of Structures

means of improving rigidity is by maneuvering the geometrical
parameters of the system.

The dimensional and sectional characteristics of a part also have
a strong bearing on rigidity. In the case of tensile-compressive loads
rigidity is proportional to the square of the cross-sectional sizes,
and for flexural loads, to the fourth power (in the direction of the
acting bending moment).

The effect of linear dimensions of a part is rather moderate for
tension-compression cases (rigidity is inversely proportional to the
first power of length), but very significant in flexure (rigidity is
inversely proportional to the third power of length).
e**The design factors influencing the rigidity can be combined in
one factor (for a constant acting force P)r
in tension-compression

*•-4

(4.5)

in flexure

= IT

(4.6)

For a round beam:
in tension-compression

?

>4 = 0.785 -y

(4.7)

in flexure

X'f

— 6.25-10 -4 y-

(4.8)

Equal rigidity conditions for beams with different l and d values
and loaded with an identical force P are
in tension-compression

-y- = const

in flexure

ii!8

= consi

The strength of the material indirectly influences the rigidity of
the construction.

The maximum deformation of a part can be expressed in the follow¬
ing way:

in tension-compression

PI a

Jef e

where or is the tensile-compressive stress acting in the part;

(4.9)

4.1 rapidity Criteria

257

in flexure

/

PI 3

aEI

Mflexl Z ... Gflexl 2
aEW ~ abE

(4.10)

where Oj iex — maximum flexural stress acting in the system;

a — coefficient accounting for loading conditions;

b = yx is a ratio constant for the given form of section

Thus, it is quite obvious that, other conditions being equal, defor¬
mations are proportional to stresses. In practice the magnitude of the
stresses are generally assumed to be proportional to the strength
characteristics of the material; the stresses are, in effect, the ratio
between the ultimate stress (or ultimate yield) and the safety factor
coefficient. Hence, the higher the strength of the material, the great¬
er the values of stresses admitted and, other conditions being
equal, the larger the deformation of the system. Conversely, the
smaller the safety factor and the closer the stresses acting in the
system to the ultimate strength, the larger the deformation and
the less the rigidity of the system.

The simplest method of reducing deformation is to minimize
stresses. However, this way is not rational as it entails penalty in
weight. In the case of flexure the rational way to decrease deforma¬
tion is to select the best section shapes, loading conditions, types
and positioning of supports. As the influence of the linear parameters
in the system during flexure is rather great [Eq. (4.8)], then for the
given case the designer uses a number of different ways to enhance
rigidity which allow deformation to be lowered tens of times as
compared with the original design and sometimes to eliminate flex¬
ure altogether.

In the torsional case the rigidity parameter can be efficiently
improved by decreasing the length of the part over the portion subjec¬
ted to torsion and particularly by increasing the polar moment of
inertia.

The improvement of rigidity is much more difficult in the tension-
compression case since the shape of the section does not play a large
role and the deformation depends only on the sectional area, which
is determined by the strength requirements. Here, the only way of
increasing rigidity is to shorten the length of the system. If the
length cannot be changed, there is no possibility of manoeuvring
further.

In accordance with Hooke’s law, the deformations of a system
subjected to tensile-compressive loads are found by Eq. (4.9)

/

_£

E

l

and at given values of stress cr and length l, the deformations depend
only on the elasticity modulus.

17—01 395

258

Chapter 4. Rigidity of Structures _

The deformation value depends not only on the maximum 'stress
acting within the system (across the critical cross-section), but also
on the law of stress distribution throughout all other sections, i.e.,
on the form along the length of the part. Equally strong parts (in
which stresses are identical across all sections and equal to the maxi¬
mum) possess the least rigidity.

(b) Rigidity Beyond Margins of Elastic Strains

In practice it is necessary to concede the possible occurrence of
plastic deformations. Even in systems designed to operate within
the margins of elasticity it is not unusual to encounter local plastic

Fig. 104. Loading curve of a plastic metal specimen in tension (a) and effect
of the strength of steels on plastic deformations of parts (6)

deformations in the weakest links of the constructions, at stress con¬
centration spots and in elements unfavourably positioned relative
to the acting forces, etc. General or zonal plastic deformations may
also occur when operating at overloads.

The behaviour of material in these conditions can be followed from
the graph of force and relative extension for the simple tension case
(Fig. 104a). As long as the part operates within the elastic deforma¬
tion region (Fig. 104a, with loads not exceeding 6 tf), the deformation
has an insignificant value (s = 0.2% on average); loading and unload¬
ing take place along line ab\ and when the load is removed, the
system each time returns to its original state.

4.1.'Rigidity Criteria ■.

259

If the acting force rises to a value, surpassing the elastic limit,
the deformation is sharply increased- due to' the appearance of resi¬
dual strains. For example, with the load increased to 9 tf (point b’),
the relative deformation reaches 2.5%. While removing the. load,
the unloading takes place along line b'a '. Even with complete load
relief, the system will not return fully to its initial state but will
acquire some residual deformation, in this case equal to 2%. At
the same time the system attains greater hardness owing .to cold
working occuring during the plastic flow of the material.

With repeated force applications loading occurs along line a'b'
and the system can sustain loads up to 9 tf without any additional
residual deformation. However, this lessens the reserves of plastic
load carrying capacity (the latter is the difference between the ulti¬
mate force and the force corresponding to the elastic limit). If before
application of the force which caused the residual deformation the
reserves of load carrying capacity amounted to 9.5 — 6 = 3.5 tf,
now it is reduced to 9.5 — 9 — 0.5 tf.

It is clear that the rigidity drop beyond the elasticity limit is
temporary (provided the overloading stress does not exceed the
ultimate strength of the material). Having sustained the residual
deformation, the system regains its initial state of resilience. The
behaviour of the system under repetitive loads is determined by
laws of elastic deformation but only with new values of elastic limits
and new initial coordinates.

The appearance of moderate residual deformations does not present
danger if the load is static and the deformations do not affect the
work of the unit and adjacent components. On the contrary, they
will help to strengthen the part.-The degree of strengthening depends
on the ratio between ultimate strength o & and ultimate elasticity
of the.material (or, close to the latter, the yield limit o 0 . 3 ). The
o 0 . 2 /n& ratio is small for soft and plastic materials (e.g., for low-
carbon steels o 0 . 2 /o & = 0.5-0.6) and rises with the increase of
ultimate strength, reaching as much as 0.85-0.95 in high-strength
steels. Thus, the degree of strengthening can be substantial only
for plastic materials; the possibility of strengthening high-tensile
steels by plastic deformation is not great.

Should residual strains impair the efficiency of a unit (such a case
may be encountered in accurate joints), then they must either be
eliminated completely, or reduced to very small values.

As evident from the above, the value of the deformation beyond the
elastic limits depends primarily on the material’s strength and the
character of its changes in the plastic deformation area,- i.e., on the
kind of loading curve.

The illustration in Fig. 1045 shows comparative values of plastic
deformation in parts made from three steels of. different strengths.
Assume, a part is acted upon by a 7.5-tf tensile force causing stresses

17 *

260

Chapter 4. Rigidity of Structures

surpassing the elastic limit of all steels. The relative deformation e
under the action of this force (line aa ) for steels I to 3 will respective¬
ly be equal to 0.5, 1 and 2.5%. Thereby, the deformation of a part
made from the strongest steel is 1/0.5 — 2 times less than that in
the case of steel 2 and 2.5/0.5 — 5 times less than that for steel 3.

The advantages of high-strength steels in the considered case can
be illustrated in some other way. Let the given limit relative defor¬
mation s = 1% (line bb). The part made from'the strongest steel 1
will have this deformation under a 9.5-tf load; from steel 2 — under
a 7.5-tf load, and from steel 3 — under a 6-tf load.

Thus, from the above it is clear that the rigidity of a system in the
plastic deformation area is determined mostly by the strength
factors.

(c) Rigidity of Thin-Walled and Composite Structures

In thin-walled and particularly shell structures stability of the
system is of great importance. Constructions of such a nature are in
certain conditions at stresses which are safe from the viewpoint
of nominal strength and rigidity calculations prone to abrupt local
oiygeneral deformations bracing the character of collapse.

The main means in the battle against the loss of stability (apart
from improvement in material strength) is the reinforcement of easi¬
ly deformable sections in the system by the introduction of stiffen¬
ing elements, or braces, between the deforming sections and rigid
units.

In composite structures (i.e., in systems, composed of several parts
by tight fits) the rigidity will depend also upon a factor, sel¬
dom considered/but having great practical significance, namely,
the rigidity of matched units. Presence of gaps or clearances in such
matched units often results in deformations which surpass many times
the natural (inherent) recoverable strains of the structural elements.
When dealing with such units special attention must be paid to
the rigidity of fastening and built-up parts.

Other efficient ways at enhancing the rigidity of composite systems
are dead tightening of joint units, interference fits, larger bearing
surfaces and providing greater rigidity at connection points.

4.2. Specific Rigidity Indices of Materials

'' ; To compare the indices of rigidity, strength and weight of parts
made from different materials, four main cases must be considered:

1. Parts identical in shapes (with equal loads the parts have the
same stresses).

■ '2. Parts of equal stiffness (have equal deformations with different
sections and stresses). .

4.2. Specific Rigidity Indices of. Materials

261

3. Parts of equal strength (have identical safety factors, different
sections and stresses proportional to the ultimate strength of the
material).

4. Parts having identical weight.

The first case (changing the material of a part for another material
but leaving its geometrical dimensions the same) is encountered in
practice when the sectional sizes of the part are suited to the manu¬
facturing conditions (e.g., cast housing components). It is the same
for parts not specially designed, with small or indeterminate stres¬
ses.

The second and the third cases occur when not only the material
of a part but also its sectional sizes are changed (designed parts in
which stresses and deformations are determined rather closely and
specified so as to make the maximum use of the material’s
strength and rigidity).

The fourth case is when the weight of a structure is determined
by its function and conditions of use.

During the comparison of strength, weight and rigidity indices
of parts made from different materials, we assume that the parts
have equal length and their sections (for the last three cases) are
changed in a geometrically similar manner.

1. Parts of similar shapes (a — const). For tension-compression,
the rigidity coefficient

where P and l = cross-sectional area and length, respectively;

E — Young’s modulus
Given: l — const, and F — const, hence

X — const E (4.11)

i.e., the rigidity of parts in this case is dependent only on the value
of Young’s modulus.

Safety factor

_

where o b = ultimate tensile strength;
a = acting stress in a part
Given: a = const, hence

n — const (4.12)

The value n defines the maximum load which the part can sustain

Pmax —

262

Chapter 4. Rigidity of Structures

In the case under consideration, the maximum load is determined
by the metal tensile strength a b , and the .weight of a part, only by
the material’s specific weight, i.e.,

£? = const y (d.13)

The correlations are absolutely similar for flexure and torsion, the
only difference being that in torsion the rigidity of a part is depen¬
dent on the value of the shear modulus.

2. Equally rigid parts {X — const). The conditions for equal
rigidity in tension and compression

With identical lengths l

EF

l.

■■ const

F .

const

~~E

Consequently, the weight of equally rigid parts

G = Fy — const •

Stresses - •

: const

X

E

(4.14)

(4.15)

Taking into account Eq. (4.14)

a — const- E

The safety factor (design redundancy)

n = -2k = const —jF
o E

In flexure the weight of equally rigid parts

G — const

V'
£ 1/2

The safety factor (design redundancy)

n = const

06

£ 3 / 4

(4.16)

(4.17)

(4.18)

3. Equistrong parts (n
tension and compression

const). Conditions for equal strength in

n —

Ob

■■ const

As a

const

n — const <JbF — const

4.2. Specific Rigidity Indices of Materials

263

Hence, for equistrong parts

const

Ob

and their weight

G — yFl ~ const —.
* Ob

(4.19)

(4.20)

The coefficient of rigidity is tension-compression

EF

Taking into account Eq. (4.19)

X — const

E

Ob

In flexure the relations will be

G — const -

X= const•

.213

E

«V3

(4.21)

(4.22)

(4.23)

4. Parts of equal weight (G — const). The conditions for equal
weight in tension-compression

G ~ Fyl — const

leads to the relation

Stresses in a part

F =

const

(4.24)

const

o — —=— — const y

The safety factor (design redundancy)

n- ^ - const ^ (4.25)

ay

The coefficient of rigidity

X — = const— (4.26)

h y v

In flexure

(4.27)

(4.28)

264

Chapter 4. Rigidity of Structures

For the purpose of comparison use is made of the simplest equa¬
tions pertaining to tension-compression.

For all the above analyzed cases, in tension and compression,
the indices of weight, rigidity and strength of parts made from diffe¬
rent materials are cited in Table 15. The last two bottom- rows give

Table 15

Characteristics of Weight, Rigidity and Strength

Parameter

Parts

similar

shapes

equal

rigidity

equai

strength

equal

weight

Weight, G

■v

V

E

JL

const

Rigidity, X

E

const

E

06

E

y

Strength (n;

P max)

0&

06

E

const

' .Hh

y

X

E .

G

y

P max

Ob

G

y

relative values of K/G and jPmax/G; the first of these indicates the
weight advantage in terms of rigidity and the second — in terms
of strength.

From the Table it is seen that factor %/G, of the weight advantage
in terms of rigidity, is the same for all cases and equal to Ely. This
is called the specific rigidity for tension-compression; this factor
is the basic weight-rigidity characteristic of materials.

The specific rigidity in flexure

in torsion

£ 1/2

y

G m

y

where G is the modulus of shear.

In all cases the weight advantage in terms of strength is defined
by the specific strength factor (failure length o b /y).

Table 16

Strength and Rigidity Characteristics of Structural Materials

Materials

£

o

o

CO

Ultimate ten¬
sile strength,
<s b , Xgt/mm 2

1 .«

” £

« - s
-3

PS’S 1

Modulus ol
elasticity,

E, kgf/mm2

O

| Rigidity characteristics

iO

1 S
Suw

eg p~

Ss °

CO

»

O

W|r-

Cl

t

o

iM

\

O

tH

03

« £

Carbon steels

7.85

35-80

21-48

21000

2.67

1

1.56

1.3

Alloy steels

100-180

80-145

mm

0.94

|

High-strength steels

250-350

225-315

0.6

0.54

Grey cast irons

7.2

20-35

14-25

8000

|

1.1

2.3

1.6

0.3

High-strength cast irons

7.4

45-80

32-56 | 15 000

7000

2

1.9

1.3

1.1

Aluminium

alloys

cast

18-25

13-17.5

7 200

2500

2.67

2.9

2.04

0.45

wrought

40-60

28-42

1.2

0.83

1.1

Magnesium

alloys

cast

1.8

12-20

8-13

4500

1500

2.3

2.1

1.35

0.32

wrought

25-30

18-21

1.4

1.2

0.52

Structural bronzes

8.8

40-60

32-48

11000

4200

1.25

1.85

2.3

0.6

Titanium alloys

4.5

80-150

70-135

12 000

4200

2,66

0.8

0.72 | 3.6

Structural

plastics

delta-wood

1.4

15-20 (along
layers)

—

4000

—

2.9

2

—

—

glass-fibres

1.6

25-30

—

5000

—

3.1

1.67

—

GFAM

1.9

40-70 (along
fibres)

—

[6000

—

3.1

0.86

— ■

—

Sitalls

50-80

45-72

15 000

—

5

1.87

1.68

3.6

266

Chapter 4. Rigidity of Structures

The rigidity of equally strong parts is defined by the ratio E/a b
of the elasticity modulus to the ultimate tensile strength, termed the
rigidity factor of equistrong parts under tension-compression.

In flexure the rigidity factor is expressed as

Table 16 and Pig. 105 describe the specific rigidity characteristics
of the principal structural materials, defined in the Table according to
the maximum ultimate tensile strength values for the given material.

Fig. 105. Specific rigidity factors of materials

I — superhigh-strengtli steels; 2 — alloy steels; 3 — carbon steels; -i — grey cast irons;
S — high-strength, cast irons; <3 — structural bronzes; 7 — wrought aluminium alloys;
S — cast aluminium alloys; o — wrought magnesium alloys; 10 — cast magnesium alloys;

II — titanium alloys; 12 — delta-wood; is — GFAM; li — glass-fibre materials; is —

sitails

By the magnitude of the — factor (Fig. 105a) the first place is

occupied by sitalls; these are followed by composite plastics, tita¬
nium alloys and steels. By tbe magnitude of tbe Eh b factor (Fig. 1056)
the strongest materials are found to occupy the last place.

The highest specific rigidity is shown by the beryllium-aluminium
alloys (24-38% Al, balance—Be)-, being- applied as yet on a reduced
scale. The specific weight of these alloys y — 2-2.1 kgf/dm 3 , ultimate
tensile strength a b = 45-60 kgf/mm 2 , Young’s modulus E =
= (20-22) *10 3 kgf/mm 2 , relative elongation 6 — 5-8%. Specific

rigidity of these alloys — 11-10 8 , i.e., approximately

4 times that of steels. The failure length L fail — 60 ; 2 = 30 km.

In the case of parts having similar shapes (Fig. 106a) the most
advantageous are steels for their rigidity (determined by E) and
strength (determined by <j 6 ), but in respect of weight (determined
by the specific weight of materials) steels, as well as bronzes and
cast irons are worse.

Fig. 106. Influence of material on the weight, strength and rigidity of construc¬
tions

(a) of similar configurations; (b) equirigid; (c) equistrong; ( d ) of equal weight; I —
superhigh-strength steels; 2 - alloy steels; 3— carbon steels; 4 — grey cast irons; s —
hign-strengtii cast irons; 6 —structural bronzes; 1 - wrought aluminium alloys; « —
cast aluminium alloys; 9 — wrought magnesium alloys; If—cast magnesium alloys;
11 — titanium alloys; 12 — delta-wood; 13 — GFAM; 14 — glass-fihre materials; is— si-

268

Chapter 4. Rigidity of Structures

Let us compare the rigidity, strength and -weight of structures
when changing over from castings of pig iron to castings of alumi¬
nium and carbon steels without altering the parts’ shape. Assuming
the rigidity, strength and weight of grey cast iron structures as
equal to unity, we obtain:

Aluminium

Carbon

alloys

. steel

Rigidity (E) .

. . 0.9

. 2.6

Strength (a&)

. . 0.72

2.3

Weight (y) . .

. . 0.39

1.1

Thus, the transfer to aluminium alloy castings hardly affects
rigidity, somewhat lowers strength (approximately 30%) and sig¬
nificantly reduces the weight of the structure (2.5 times). Changing
to steel castings increases rigidity and strength approximately
2.5 times and the weight is practically the same.

The above listed relations are based on the fact that Young’s
modulus has a constant value and only slightly depends on the pre¬
sence (in normal amounts) of alloying elements, heat treatment and
strength characteristics of alloys of the given metal. For example:
in respect to steels, beginning from low carbon and up to high-alloy
ones, Young’s modulus varies from 19 000 to 22 000 kgf/mm 2 , and
the shear modulus, from 7900 to 8200 kgf/mm a . With regard to
aluminium alloys £'—7000-7500 kgf/mm a and G— 2400-2700 kgf/mm 2 .

Consequently, for the manufacture of identically shaped compo¬
nents when rigidity is the first requirement and the stress level is
low, the use of cheapest materials is recommended (low carbon steels
instead of alloyed ones; simple aluminium alloys instead of complex
ones). This will not affect the rigidity of the construction but will
enable the cost of the latter to be reduced.

The above does not hold when the strength of the construction is as
important as the rigidity. Thus, for instance, identically shaped
structures, one made from low-carbon steel, and the other, from alloy
steel, will have the same rigidity but the load capacity of the first
structure will be as many times less as the tensile strength of carbon
steel is less than that of alloy steel.

For equally rigid parts (Fig. 1065) the greatest strength advantage
is with materials having the highest a b /E ratio (high-strength
steels, titanium alloys, GFAM and wrought aluminium alloys). In
terms of weight advantage (this is, in our case, proportional to the
y/E factor) the above-listed materials possess approximately equi¬
valent values. Worse weight characteristics are held by bronzes aud
grey cast irons.

Now, let us compare the strength and weight characteristics of
equally rigid parts when changing from carbon steel to wrought
aluminium alloys, alloy steel and titanium alloys. Assuming the

4.2. Specific Rigidity Indices of Materials

269

strength and weight of the carbon steel structures to be equal to

unity, we obtain:

Aluminium-

Alloy

Titanium

alloys

stesl

alloys

Strength .

. . 2.15

2.3

3.3

Weight (-J) .

. . . 1-05

0.43

1

Hence, for the equally rigid parts, the change from carbon steel
to alloy steel, titanium and aluminium alloys is accompanied by
a substantial strength increase (from 2.15 to 3.3 times). The construc¬
tion’s weight is lowered 2.3 times when using alloy steels and practi¬
cally. remains the same with aluminium and titanium alloys.

In the case of equally strong parts (Fig. 106c) the weight of the
structure depends on the magnitude of factor y/c r&: in terms of weight
the advantages lie with the materials which have a low y/a b factor
(sitails, GFAM, titanium alloys and high-tensile steels). Grey cast
irons and bronzes have lower weight characteristics (advantages).

The rigidity of constructions in the considered case depends on the
rigidity factor Eh b . The greatest rigidity advantages are possessed
by materials which have the highest values of this factor, i.e., the
less- strong materials (cast aluminium and magnesium alloys, car¬
bon steels, grey cast irons) and the worst rigidity by the high-strength
materials (alloy steels, titanium alloys, extra-high tensile steels).

The meaning of these circumstances must be evaluated correctly.
All the above is true only in assuming that the design stresses are
chosen proportionally to the material’s ultimate strength. In this
case the use of high-strength materials for equal-strength construc¬
tions definitely lowers their rigidity.

Thus, with the same safety factor (ft— 5) and admissible stresses
of 30 kgf/mm 2 , the rigidity of alloy steel parts will be one third of
the rigidity of equal-strength parts made from carbon steels having
admissible stresses of 10 kgf/mm 2 . The rigidity of parts made from
extra-high-strength steels, admissible stresses 70 kgf/mm 2 , will be
2.5 times less than that of the rigidity of equal-strength alloy steel
parts and 7 times less than that of equally strong carbon steel parts.

The rigidity of structures made of high-strength materials may
be practically unreservedly increased by lowering design stresses,
but with detriment to the weight advantage of the structure and
incomplete utilization of the materials strength reserves.

Thus, we conclude that] when using high-strength materials in
equal-strength constructions it is necessary t oaccount for the fall in
rigidity and compensate for this effect by other design measures.

Let us compare .the rigidity and weight of equally strong structures
when carbon steel is changed by wrought aluminium alloys, alloy steel

270

Chapter 4. Rigidity of Structures

and titanium alloys. Having assumed that the rigidity and weight
of carbon steel are equal to unity, we obtain:

Aluminium

alloys

Alloy

steel

Titanium

alloys

Rigidity

(!)■

. . 0.53

0.51

0.35

Weight ^

i) •

. . 0.48

0.45 ■

0.3

Thus, the transfer, in equally strong parts, to aluminium alloys,
alloy steel and titanium alloys causes an abrupt (2-3 times) drop of
rigidity. Simultaneously, approximately in the same proportion,
the weight of constructions is reduced.

With respect to parts of equal weight (Fig. 106<2) the strength of
a structure is determined by the ajy specific strength factor. Here
it is more advantageous to use materials with a high a b /y ratio
(high-tensile steels, titanium alloys, GFAM and sitalls). In terms
of structural rigidity the above-listed materials are approximately
the same, except sitalls which have sharply increased rigidity.

Now, let us compare the strength and rigidity of structures hav¬
ing the same weight, when changing from carbon steel to wrought
aluminium alloys, alloy steel and titanium alloys. Assuming the stre¬
ngth and rigidity of carbon steel to be equal to unity, we obtain:

Aluminium Alloy Titanium

alloys steel alloys

Strength ... 2.1 2.25 3.23

Rigidity (f) . . . 1 1 1

Hence, the change to aluminium and titanium alloys and alloy
steels leads to increased strength of equal-weight parts by 2-3 times
with rigidity unaffected.

(a) Generalized Index

As shown above the weight advantage of a material in terms of
strength is described by the factor a b ly (or a 0 , 2 /y), and the rigidity
advantage is expressed by the factor Ely. Uniting these we obtain

a generalized strength-rigidity index ^ (See Table 16), which

characterizes the capability of a material to sustain the highest
loads with the least deformations and least weight of a construction.

Finally, this is very important. Rigidity and strength are insepa¬
rable. Rigidity, if taken solely by itself, has no practical value unless
the material is capable of sustaining high loads. A slender rod made
of soft carbon steel can easily be bent by hands. The rigidity of such

4.2. Specific Rigidity Indices of Materials

271

a rod is insignificant' and its constructional value is zero. The same
rod produced from high-tensile heat-treated steel can carry consi¬
derable loads. The rod will be “rigid” even though the elasticity
modulus of its material is the same.

Here is an example from mechanical engineering. A hoisting
crane manufactured from low-carbon raw steel will have the same
rigidity as a crane made from high-grade heat-treated steel, when

Fig. 107..- Generalized strength-rigidity indices
1 —superhigh-strengtli steels; s —alloy steels; a—carbon steels; '4—grey cast irons; 5— hi¬
gh-strength cast irons; a—structural bronzes; t —wrought aluminium alloys; S —cast alu¬
minium alloys; 9— wrought magnesium alloys; 10 — cast magnesium alloys; 11— titanium
alloys: IB— sitalls

judged in terms of the material’s characteristics. However, the former
crane will deform and settle under the action of high loads, which
in the latter crane only cause insignificant elastic deformations.

According to the value of the generalized weight and strength-rigi¬
dity factor (Fig. 107), the materials are divided into four rather pre¬
cise groups: (1) extra-high-strength steels 8 -10 5 j ; (2) alloy

steels, titanium alloys and sitalls — (3.5-4)-10 5 j ; (3) carbon

steels, high-strength cast irons and wrought aluminium alloys

( —1-10 5 ) ; (4) structural bronzes, wrought magnesium alloys,
\ y i

272

Chapter 4. Rigidity of Structures

cast aluminium and magnesium alloys, grey cast irons ( a -~- <.
< 0.5 -10 5 ).

In practice, the choice of material is determined not only by the
strength-rigidity characteristics, hut also by some other properties.
That is why preference is given to those design measures which
enable strength and rigidity to be obtained, even when applying
materials of low strength and rigidity.

4.3. Enhancing Rigidity at the Design Stage

The main constructional methods of improving rigidity are:

do all that is possible to eliminate flexure which is unremunerative
from the viewpoint of rigidity and strength, changing it to compres¬
sion and tension;

for parts working in flexure, position supports rationally, exclude
loads not advantageous to rigidity;

rationally increase section inertia moments, reinforce joints and
the parts transferring forces from one section to another, all without
causing weight increases;

for thin-walled box-shaped parts, use shell, arched, vaulted,
spherical and oval (egg-shaped) forms.

(a) The Change of Flexure to Tension-Compression

The higher rigidity of parts, working in tension-compression is
due to the better use of the material with this form of loading. In
the case of flexure and torsion tile most loaded fibres are at the cross-
sections extremes. Thus, the loading limit begins when the stresses
in these fibres reach dangerous values and the core remains under¬
loaded. In tension-compression conditions the stresses are equally
distributed throughout- the entire cross-section and the material is
fully utilized. The ultimate level of loading arrives only when the
stresses in all points of a section simultaneously approach, at least
theoretically, some critical level. In addition, the action of loading
under tensile-compressive forces is independent of the length of
a part; the deformations are proportional to the part length, whereas
in the case of flexure the action of load is dependent on the distance
between the plane of acting flexural forces and the critical section;
deformations in this case are, proportional to the third power of the
part length.

Let us compare a cantilevered beam of circular cross section
(Fig. 108a), loaded with a flexural force P, and an equivalent truss
system, comprised of bars having the same section as the beam. In
the latter case the upper bar under the action of force P works in
tension and the lower one in compression. From the relations

4.3. Enhancing Pdgidity at the Design Stage

273

presented in the figure, the stresses in the truss bars are 550 times
less than the maximum stresses in the beam and the maximum defor¬
mation at the application point of force P is 9000 times less.

If the maximum bending stresses in the beam (Fig. 1086) equal
the tensile-compressive stresses in the bars (equal-strength case of
both systems), the cross-sectional area of the beam must be increased

• (c)

Pig. 108. Rigidity, strength and weight comparison of a truss system and cantile¬
vered beam

<a) with equal cross-sections of truss bars and beam; (fc) with equal stresses; (c) with equal

deflections

60 times. In this case the weight of the beam, excluding the fixed
end, will be 20 times that of the truss system and the flexure of the
beam in the plane of the applied load will be 3.3 times that of the
truss system.

For equal maximum deformations in both systems (Fig. 108c)
the beam cross-section must be increased 110 times and its weight
will exceed that of the truss system 40 times. .

The connection between flexure f b of the round cantilevered beam
(in the plane of applied force) and flexure f tr of the truss system with
equal sections (Fig. 108a) may be-, expressed by the approximate

1 8-01395

274

Chapter 4. Rigidity of Structures

relation

—-=10.5 sin 2 a cos a

where l — span length;

d — beam and truss bars diameter;

a = half-angle taken at the apex of the truss-formed triangle
The relation p- as a function of a angle is plotted to a semi-log,

Jtr

scale for different values of— (Fig. 109). From this graph, with equal

sections, it is seen that the flexure of the cantilevered beam can be
hundreds and thousands times greater than that of the truss system,
the difference sharply increasing with the enhancement of the Ud
ratio, i.e., with relative thinning of the beam and bars. Even for
the most rigid bars (lid = 10) the difference remains in favour of
the trussed bar system.

The fblft r ratio reaches its highest value when a = 45-60°. This
means that the truss shown in Fig. 108 possesses its maximum rigid¬
ity when a = 45-60°.

Having plotted the dependence of the ratio of maximum flexural
stress G b in the beam to the stresses of tension-compression a tr in
the truss bar system on angle a (for various Ud values) (Fig. 110)

4.3. Enhancing Rigidity at the Design Stage _275

it is possible to establish that the stresses in the beam are many
times those in the truss bars (e.g., when a = 45°, 100-1000 times).

A cast bracket (Fig. 111a, b) is a structure similar to the systems
depicted in Fig. 108. The rigidity of the bar connecting units at
the bracket stays changes their operating conditions as compared
to the pure truss in which the bars are articulated; nevertheless, for

Fig. 111. Designs of cast brackets

the truss-type bracket (Fig. Ill b) the bars work mainly in tension-
compression, but the cantilevered form of bracket (Fig. 111a) will
be subjected to flexure.

The structure becomes stronger and more rigid if the bars are joined
together by a solid web converting the entire unit into completely
rigid system (Fig. Ulc).

The truss-type bracket with the horizontal upper bar shown in
Fig. 111c?, is much less rigid than the bracket shown in Fig. 1116,
because the end of the horizontal bar, when subjected to loads,
is displaced approximately along the direction of the acting force
and its rigidity is not used for limiting deformations.

Design calculations show that the best value of angle a, when
considering rigidity and weight, lies between 90 and 120°.

In the short thin-walled cylinder carrying transverse load P
(Fig. 112a) all the sections, positioned along generatrices undergo
flexure. The loads are mainly taken up by the side walls (Fig. 112 b)
which are parallel to the bending moment action (blackened in the
Figure), since their rigidity in this direction is many times more than
the rigidity of the walls positioned perpendicularly to the plane
of the acting moment.

In the conical form (Fig. 112c) approximating a truss-type con¬
struction (Fig. 108) the walls positioned in the plane of the acting
bending moment work as follows: the upper wall in tension, the
lower wall resembling a strut in compression. The side walls mainly
undergo flexure and their'rigidity is commensurable with the rigidity
of the upper and lower walls. Consequently, all walls of a conically
shaped component take part in the work and the strength and rigidity
of the construction is increased.

18 *

276

Chapter 4. Rigidity of Structures

Connection between tbe tensioned and compressed walls is accom¬
plished by a ring of stiffness m, set at the end of the unit. Apart from
acting as a load-closing element, this ring prevents the cone becom¬
ing oval under the action of the load. Such rings are indispensable
for thin-wall hollow components to operate correctly.

Close to the conical components in terms of rigidity are spherical
(Fig. 112 d) and other convex-shaped parts.

Another example of eliminating flexural stresses is given in
Fig. 113. Here, a simply-sup ported beam, subjected to flexure
(Fig. 113a) is changed to a much more advantageous bar structure,

Fig. 113. Schemes

<a) freely supported beam; (&) truss system; (a) arched beam

whose elements work in compression (Fig. 113b). Close to this case
is the' arched beam (Fig. 113c) working also mainly in compression.

: The relationship between the flexure /& of the simply supported
beam in the plane of the acting bending forces and the flexure f tr
of the truss system3(Fig. 113b) is given approximately by

• = 1.3 ^"^ 2 sin 2 acosa

where l — span of'the beam;

d = diameter of the beam (and bars of the truss system);
a — side angle of the truss triangle
The fi)lftr ratio is plotted in the graph (Fig.-114) as a function of
angle a for different values of l/d.

4.3. Enhancing Rigidity at the Design Stage

277

The graph indicates the large advantages of the system working
in compression over the system subjected to flexure. In the acting
load plane the beam flexure is hundreds and thousands of times grea¬
ter than that noted for truss-type bar structures. Even with small
values of angle a («*15°), the flexure of the truss-type structure, for
example, when lid — 50, is 200 times less than that of the'^beam
system subjected to flexure.

As in the preceding case (see Fig. 108) the trussed bar system has
the highest rigidity when a — 45-60°.

From the graph (Fig. 115), showing the relation between the maxi¬
mum flexural stress in a beam and compressive stress o tr in a
trussed bar system as a function of angle a for various l/d values,
the stresses in the beam on average are 30-300 times larger than
those in the truss system.

In rods having large l/d ratios working in compression the danger
of buckling occurs. This feature must be considered during design.

Figure 116« shows loading of a cylinder. The force applied along
the cylinder axis causes flexure of the end faces, transferred to
the coarse through the connecting flange (the deformations are)shown
by dashed lines). The system is not rigid.

On changing the cylinder to a cone (Fig. 1165), by the main scheme
of taken-up forces, it approximates the case of the truss bar system
in Fig. 1135. The cone walls work mainly in compression, the role

278

Chapter 4. Rigidity of Structures _

of the sides withstanding the thrust in this case is fulfilled by the
stiff circular sections of the body, thus limiting radial deformations
of the walls. The highest rigidity with lowest weight is possessed
by cones having vertex angles from 60 to 90°.

Improved rigidity is displayed by other components when their
shapes are similar to a cone, for example, spherical (Fig. 116c),
egg-shaped (Fig. 116 d) constructions, etc.

Fig. 116. Constructions operating in compression

As for the cone working in flexure, in order to improve the strength
and rigidity of thin-walled components it is essential to use stif¬
fening collars (Fig. 116e, /), of which the upper collar m takes up
compressive loads and the lower one n —tensile loads.

(b) Blocking the Deformations

When expressed in most general terms, the problem of improving
rigidity means finding the points of greatest deflection under the

Fig. 117. Operation of diagonal struts

action of acting loads in the system and avoid these displacements
by introducing tension-compression elements arranged in the line
of the displacements.

A classic example of solving this problem is the rigidity improve¬
ment in frames and truss constructions by using diagonal struts.

Figure 117a, shows a bar frame subjected to shear forces P. The
insignificant rigidity of this frame system is determined only by
the flexural rigidity of vertical bars and the rigidity of the corner

4.3. Enhancing Rigidity at the. Design,Stage

279

joints. Afterj some stiffening elements, for example, gussets, have
been introduced into the system (Fig. 117b), the working system
of the frame approximates that of the constrained beams and some¬
what lessens the deformations.

(0 (j) Ik) (l)

Fig. 118. Flat trusses

The most efficient way, however, is the introduction of diagonal
struts, which work in tension or compression. A tensile strut
(Fig. 117c), diagonally linking the frame elements, must extend
by,.-the value A l as shown in the Figure. Owing to the inherent fea¬
tures of this form of loading, the small deformation values efficiently

Fig. 119. Arresting deformations

stop frame distortion. A compression strut (Fig. 117d) will act simi¬
larly, however, in this particulas case it is necessary to consider
possible buckling of the compressed bar, which in this case makes
this system less desirable.

If the loads act alternately in both directions, Fig. 117c, the
struts are either diagonally crossed or, Fig, 117/, in alternating
directions.

Figure 118 shows various design schemes of simple trusses (girders)
-in order of rising rigidity (Fig. 118 a-h), and also composite simple

280

Chapter 4, Rigidity of Structures

trusses with stiffening elements preventing buckling and loss of bar
stability (Fig. 118i-Z).

The wall deformation of a cylindrical vessel under the action of
internal pressure is shown in Fig. 119a. Parts m having the greatest
deformations are advisably connected by tension elements
(Fig. 119&): around the course by ring 1 and the end plates by anchor¬
ing bolt 2.

(c) Cantilevered and Double-Support Systems

Should flexure for constructional and functional reasons be un¬
avoidable, the decrease of deformations and flexural stresses must
be the first task.

Fig. 120. Loading schemes

Beam flexures, most commonly encountered in practice, are given
in Fig. 120: a cantilevered (overhanging) beam (Fig. 120a) a freely
supported beam (Fig. 1206) and a constrained beam (Fig. 120c).

From the comparison of the maximum bending moment and flex¬
ures the advantages in rigidity and strength clearly lie with the
freely-supported beam, than with the cantilevered one. With iden¬
tical loads, beam lengths and sections, the maximum bending
moment (and, hence, the maximum flexural stresses) with the simp¬
ly-supported beam are four times, and with the constrained beam

4.3. Enhancing Rigidity at the Design Stage

281

eight times less than that of the cantilever. Even greater is the advan¬
tage in rigidity. The maximum flexure of the simply supported beam
is sixteen times and that of the constrained beam sixty four times
less than that of the cantilever.

In practice the difference between the simply-supported and can¬
tilevered beams is not as sharp as illustrated. In comparable con¬
structional variants of the cantilevered and simply-supported beam
the length of a cantilever rarely equals the beam span distance,
generally the cantilever length is very much less.

Let us consider a cantilevered (Fig. 120 d) and simply-supported
(Fig. 120e, /) arrangement of a pinion shaft. In this particular case,
because of overall sizes, the span distance in the simply-supported
arrangement cannot be less than 21, owing to which we have to com¬
pare these two alternatives on the basis of different and not equal
lengths as in the preceding comparison.

The maximum bending moment for the simply-supported case is,
in accordance with Fig. 120e, half that of the cantilevered arrange¬
ment. The gain in maximum flexural stresses is much greater since
the resisting moment at the dangerous section (i.e., in the plane
of the acting force P) of a simply-supported shaft is significantly
greater than the resisting moment at the same section (i.e., in
the plane of the leading bearing closest to the load) of the cantileve¬
red shaft. For the relationships in Fig. 120, stresses in the critical
section of the simply-supported shaft are 5 times less than those of
the cantilevered shaft.

The maximum simply-supported shaft flexure is formally half
that of the cantilevered shaft, but if the large moment of inertia /'
of a simply-supported shaft, as. compared to the moment of inertia I
in a cantilevered shaft is considered, flexure will be still less (for
the relationships in Fig. 120, the decrease will be 6.5 times).

The load on the bearings of the simply-supported shaft is less
than the load on the front bearing of the cantilevered shaft
P (1 + l!L) by 2 (1 + l/L) times.

The situation is better with the constrained shaft. The actual im¬
provement in this case is obtained by increasing the rigidity of
supports, e.g., employing roller bearings and strengthening the
casing walls (Fig. 120/). Here the maximum bending moment is
one quarter that of the cantilevered shaft, and half, when compared
with the simply-supported shaft, mounted in ball bearings. The
maximum flexure of the shaft, installed in rigid supports, is corres¬
pondingly less by 8 and 4 times (without considering the difference
in I and /' values).

However, it is necessary to note that short and rigid shafts increas¬
ing support rigidity does not give practical advantage since the
rigidity washes out the difference between the freely supported and
constrained shafts.

282

Chapter 4. Rigidity of Structures

(i d ) Increasing Rigidity and Strength
of Cantilevered Constructions

If the use of a cantilevered component cannot be avoided, then
its inherent disadvantages must be eradicated. First of all measures
must be taken to shorten cantilever overhang and increase rigidity
and strength of the construction’s cantilevered portion.

Fig. 121. Rational design of a canti- Fig. 122. Shapes of shoulders
levered shaft

The example in Fig. 121 shows an improved design of the cantile¬
vered shaft presented in Fig. 120J. The cantilever length has been
reduced to the limit permitted by the construction and the moments

Fig. 123. Constructional methods of reducing overhang in cantilevered bevel

gear assemblies

of inertia and resistance at the most loaded areas increased. The
front bearing, accepting the highest load, is strengthened.

One of the commonest types of cantilevers, applied in mechanical
engineering, is the shoulder of a supporting bush for a cylindrical
component. In the incorrect design (Fig. 122a) the collar is excessive¬
ly large. If the collar is reduced, for example, three times (Fig. 122&),
the stresses across the critical section are decreased in the same pro¬
portion and the maximum deformations 27 times.

4.3. Enhancing Rigidity at the Design Stage

283

In a number of cases it is possible to shorten the cantilever length
by altering the shape of the part.

Example: The overhang of the bevel gear (Fig. 123a) can be de¬
creased by altering the position of the gear hub relative to the rim
(Fig. 1236) or by using a stem gear (Fig. 123c).

Fig. 124. Determining forces acting upon supports
(a) cantilevered shaft; ib) inverted cantilever

The loads on the cantilevered shaft bearings depend upon the
distance L between the supports and cantilever length l (Fig. 124a).
Thus, the load on the front bearing

A '-- p ( 1 +-r)

and that on the rear bearing

n ^ pJ l

where P is the force acting on the cantilever.

From the graph, showing the dimensionless ratio NJP and NJP
as a function of Lll (Fig. 125a), it is seen that the bearing loads sharp¬
ly rise as the support span distance decreases. With the increase
L

in the -y ratio the bearing loads fall; then asymptotically tends

to the P value and N 2 , to zero. When LU > 2-2.5 the bearing load
becomes practically constant, but when LU < 1 it sharply rises.
Thus, the best recommended range for the Lll ratios is 1.5-2.5 (shown
hatched in Fig. 125a).

For a general rule it may be assumed that the distance between
the supports must be twice the cantilever length. Naturally, higher
Lll ratios have the advantage of fixing the shaft more accurately.

284

Chapter 4. Rigidity of Structures

Figure 125a, shows as a function of Lll the load ratios of the front
and rear bearings NJM s = 1 + Lll, which can be used as a guide
when selecting bearings for those cases when their life expectancy
has to be the same. Thus, the recommended values are LU = 2,
NJN % = 3.

Permissible loads on antifriction bearings are determined from
the formula

<9 =—£—

^ (nh) 0 - 3

where C — bearing work capacity coefficient;
n — speed, rpm;
h = bearing service life, h

Since n — const, and h — const, the work capacity coefficients
of the front and rear bearings, when Lll — 2, will be within the
ratio CJC% — 3.

Fig. 125. Loads on supports
(a) cantilevered shaft; (6) inversed cantilever

Often an inverted cantilever is employed. In this case (Fig. 1246),
the gear, so that load is applied in the span between the supports,
is given a bell-shaped form; it thus works as an inverted cantilever.

The values of the bearing loads are shown in the graph, Fig. 1256,
as dimensionless ratios NJP and NJP against AIL (A —distance
from the rear bearing to the plane of force P). The position of the
inverted cantilever must be within AIL = 0 to 1; if AIL > 1 the
system becomes a direct cantilever.

It is evident from the graph that the maximum values of N 1 and
N % , in the region of the inverted cantilever, are equal to the acting
force P {NJP and iV a /P ratios are equal to unity). Loads N t and
iV 2 have identical and equal values — 0.5P when AIL — 0.5, when
the force plane lies midway between the supports.

4.3. Enhancing Rigidity at the Design Stage

285

A second constructional example of an inverted cantilever is
illustrated in Fig. 126a, b.

To fully eliminate the cantilever, the part is mounted on a sta¬
tionary support 1 (Fig. 126c). Through this passes the flexure-relie¬
ved driving shaft, which imparts to the part through a splined rim

a- pure torque moment. The bearings are loaded as in the simply-
supported shaft. However, their working conditions are less favou¬
rable since the outer race rotates (and not the inner one as in the
simply-supported shaft), due to which their durability period is
shortened.

The above-described disadvantages of cantilevered systems by no
means deprive the designer of their use. Cantilever systems are
fully lawful elements of the construction and they are widely applied
in practice. It is only necessary to know their peculiarities and eli¬
minate the disadvantages by suitable design measures.

The use of cantilevers often assures more simple, compact, tech¬
nological convenient construction for assembly than the double-
supported shaft assemblies. Shown as an example in Fig. 127 are
the constructions of two centrifugal pumps: the first (Fig. 127a)
has the simply-supported shaft and the second (Fig. 1276) the can¬
tilevered mounting.

4 The cantilevered version simplifies assembly, provides easier
access to the vane and hydraulic cavity, betters the entry of operat-

286

Chapter 4. Rigidity of Structures

ing liquids to the impeller, eliminates one seal and improves shaft
alignment. Furthermore, shaft supports are accommodated inside
one housing and the hearing fitting holes can he machined at one
setting.

In the simply-supported arrangement the supports are aligned
to each other through the housing joint, the two halves of which

Pig. 127. Centrifugal pump with (a) simply supported and (6) cantilevered shaft

are, owing to the unit’s design, fixed only by fitted dowels; in other
words, shaft fitting holes cannot be machined at a time.

Taken as a whole, the cantilevered design strongly gains in sim¬
plicity, accuracy of manufacture, reliability and operation conve¬
nience.

(e) Rational Arrangement of Supports

The flexure of a simply-supported beam is proportional to the
third power of the span, therefore reducing span distance—an effec¬
tive means of enhancing rigidity.

Figure 128 shows a simply-supported gear assembly. If the span
between the supports is reduced three times, then the maximum bend¬
ing moment and stresses in the shaft are also reduced three times and
the maximum flexure by 27 times. When the shaft diameter d==
— 40 mm, length L — 200 mm and load P = 1000 kgf, the shaft
flexure, Fig. 128a, will attain a comparatively large value (of the

4.3, Enhancing Rigidity at the Design Stage

287

order of 0.1 mm), disadvantageous for gear operation. After decreas¬
ing the span three times (Fig. 1286), the flexure reduces to a negli¬
gible value (of the order of 0.004 mm).

Very often the rigidity of a system is improved by the introduction
of additional supports (Fig. 129).

Fig. 128. Reducing span distance between supports

In the construction, shown in Fig. 129a, the crankshaft is mounted
in two bearings. The system has low rigidity; to increase its value
it is necessary to enlarge the web and neck sections of the shaft.

The rigidity characteristic can also be improved by means of an
additional central support (Fig. 1296) or, the more so, by several
supports (Fig. 129c). The last version is now almost always used.

Fig. 129. Arrangement of crankshaft supports

Figure 130 shows schematically methods of increasing rigidity and
strength of the fastening assembly of a connecting rod and fork.
Since the connecting rod generally oscillates about the fork end at
small swings it is possible to introduce additional supports which
practically eliminate flexure, j

The original, rather popular design, presented in Fig. 130a, pos¬
sesses poor rigidity, since the pin here is subjected to flexure.

In the design, illustrated in Fig. 1306, the pin is relieved of flexure
owing to a thrust-pad, provided in the fork.

Flexure can be sharply decreased by increasing the length of the
connecting rod upper bearing surface (Fig. 130c, d). Over section A
the pin works in compression. Since compressive deformations are

238

Chapter 4. Rigidity of Structures

infinitesimal in comparison to those in flexure, practically all the
load taken up by the pin is a compressive one.

The designs, depicted in Fig. 130 b-d, are suited for mostly unila¬
teral loads, acting in the direction shown by an arrow. Moreover,
in these designs the amplitude of the connecting rod’s oscillations
about the fork is limited.

In the designs, intended for loads in both directions with large
oscillation amplitude, strengthening is obtained by increasing the
number of supports and lessening span distance over which the

Fig. 130. increasing rigidity of a connecting rod and fork assembly

flexural loads are taken (Fig. 130e). In this design in view of the
doubled shortening shoulders l the force of the flexural stresses are
also decreased by half, and the deformations, by eight times as
compared with the original design (Fig. 130a).

With the increased number of supports (Fig. 130/), the loading
scheme approximates pure shear. Changing to shear and increasing
the number of thus loaded sections significantly adds to the strength
and rigidity of the assembly.

In some instances, when dealing with unilateral loads, acting
forces can he transmitted directly to the supports, fully relieving
the pin of load (Fig. 130g, h). The manufacture of such construc¬
tions is more complex than the previous ones, since here it is neces¬
sary to precision-machine the cylindrical bearing surfaces k coaxial¬
ly with the pin bearing surfaces. Otherwise, the scheme of applied
forces becomes indeterminate.

(/) Rational Sections

It is important that the increase in rigidity is not accomplished
by increasing weight of the detail. In the general form, the solution
to the problem means strengthening sections, which under the given
loads are subjected to the highest stresses, and removing weight

Table 17

19—01395

290

Chapter 4. Rigidity of Structures

from the unloaded or slightly loaded areas. In flexure the stressed
sections are those farthest from the neutral axis. In torsion the exter¬
nal fibres are mostly stressed; moving radially and inwards, the
stresses become weaker and at the centre are zero. Consequently,
for these cases it is more rational to concentrate material at the
peripheries and remove it from the centre.

Generally the greatest rigidity and strength characteristics with
smallest weight are possessed by components' with thin-walled sec¬
tions, i.e., parts such as box sections, tubes and shells.

Fig. 131. Effect of increase of shaft diameter upon the rigidity, strength and
weight of construction and durability of antifriction bearings

Table 17 gives rigidity and strength comparisons for differently-
shaped sections. The base of the comparison depends upon similar
weight conditions of parts, expressed as similar cross-sectional areas.
The strength and rigidity improvements are obtained by successful
application of the material distribution principle in the regions of
the highest acting stresses.

For cylindrical sections the moment of inertia J 0 and the moment
of resistance W 0 of a solid round section are taken as the units of
comparison; with respect to the other parts, a solid square-shaped
section.

The dependence between weight, strength and rigidity of cylindri¬
cal shafts with different d/D ratios is described in a general form in
Figs. 36, 37 and 39.

A constructive example of a gear and integral shaft running in
antifriction bearings is given in Fig. 131. The parameters of rigidity /,
strength W , weight G and durability h of bearings are given for
successive increases of shaft diameters (and sizes of bearing sup¬
ports).

Unit characteristics are those of the solid shaft (Fig. 13la).

(g) Improving Transverse Rigidity

Together with the increase in external sizes and thinner wall
sections it is necessary to increase rigidity in the transverse direction
of acting bending forces in order to avoid constructional stability
losses.

4 ^ 4-644

^s ssss ss S

(d)

Fig. 132. Methods of increasing the radial rigidity of hollow components

Fig. 133. Increasing the rigidity of beams
(a) by stiffening partitions; ( b) by stiffening boxes; (e) by semi-circular stiffeningSelements

Fig. 134. Beams reinforced with diagonal stringers

Fig. 135. Increasing the rigidity of shapes by local section stiffening

19 *

292

Chapter 4. Rigidity of Structures

For cylindrical shafts the problem is solved by using stiffening
collars and webs (Fig. 132a, b). Stiffening collars are advisably posi¬
tioned in the plane of acting loads, on bearing, fixing areas and also
at the free ends of a part (Fig. 132c, d).

Figure 133 shows reinforcement of beams by transversal ribs and
stiffening boxes.

Snake-like diagonal stringers in the form of. webs strongly improve
rigidity (Fig. 134a, b), and also local section stiffening (Fig. 135).
Thus, constructions with longitudinally formed stiffening rib angles
at the transition points where vertical walls change to horizontal
ones (Fig. 1355) have greater rigidity than the original construction
(Fig. 135a) in spite of the formal lessening of inertia moment. The
rigidity parameter increases also when the longitudinal rib has
transverse stiffening ribs spaced over the part length (Fig. 135c, d).

Table 18 illustrates how longitudinally arranged webs affect
the rigidity of profiles during flexure and torsion. Diagonal webs
have the strongest effect. One diagonal stringer will suffice,* another
stringer will enhance the rigidity but to a small degree.

Table 18

Increasing Rigidity of Sections by Longitudinal Webs

Profile

< 5 =

3 $

9K

B

■

*

Factors

*/fea

*fors

G

I flea

G

*tors

G

1

1

1

1

1

1.17

2.16

1.38

0.85

1.56

1.55

3

1.26

1.23

2.4

1.78

3.7

1.5

1.2

2.45

293

_ 4.3. Enhancing Rigidity at the Design Stage

(h) Ribbing

Ribbing finds wide application in improving rigidity, particular¬
ly of cast bousing-type components. Due care, however, should be
exercised Jwhen employing this technique since wrongly related
ribbed sections and ribbed details may weaken the part instead of
strengthening it. ^ ^ " „

If a part has externai ribs and is subjected to iiexure acting in the
plane of ribs (Fig. 136a), then considerable tensile stresses may occur

(a) (b) (f) (g)

Fig. 136. Rib forms (in the order of increasing strength)

at the rib ridge, attributable to the comparatively small rib width
and cross sectional area. The situation is aggravated when thin
ribs, tapering to their tip, are used (Fig. 1366, c); the failure of
parts always begins at the rib tip. Strength is significantly increa¬
sed when thickening the rib, particularly at the critical region, i.e.,
the rib tip (Fig. 136 d-g). ,

Fig. 137. Effect of ribs on the rigidity and strength of shapes

Weakening of details by ribs is formally expressed by the reduc¬
tion of the details moment of resistance. Table 19 shows how stiffen¬
ing ribs affect the moments of resistance and inertia of a rectangular
section. The moment of resistance of a non-ribbed rectangular sec¬
tion is taken as unity.

The influence of relative height and width of ribs upon the rigidity
and strength of a part is easily expressed in a generalized form. Com¬
pare the strength and rigidity of a rectangular section profile
(Fig. 137a) with the similar profile provided with a rib (Fig. 1376).

Calculations show that correlation between the moment of iner¬
tia / of a ribbed profile and the moment 7 0 of the original profile

Effect of Ribs Upon Strength and Rigidity of Parts

Table 19

4.3. Enhancing Rigidity at the Design Stage

295

is expressed by the relation

I/I 0 -1 + 6 if + 38 ti (1 + 611 ) gL ) 2 (4.29)

where t] = h/h 0 = ratio of rib height h to original profile height h 0
8 = &/ 6 0 = ratio of rib width b to original profile width b 0
When related to sections with a series of parallel ribs (Fig. 138c),
the reciprocal of 8 will be the relative rib pitch i 0 , i.e., the ratio
of rib pitch to rib thickness

In this case the value 8 can be termed as the density of rib spacing.

Fig. 138. Rigidity and strength of ribbed sections as a function of relative rib
height h/hf, for different values of relative rib width b/b 0
(a) rigidity; (b> strength

The correlation of moments of the compared profiles is equal to

W I 1 + 6-n

Wq 7 0 1+2t] + St] 2

(4.30)

On the basis of formulae (4.29 and 4.30) are constructed the graphs
(Fig. 138), which show the relative influence of rib dimensions on
strength and rigidity.

As evident from the graph (Fig. 138a), the introduction of ribs
in all cases enhances the moment of inertia of a section and, conse¬
quently, the flexural rigidity of a part. The higher the rih and the
greater its relative thickness the better the rigidity.

The picture is quite different when dealing with the moments
of resistance (Fig. 1386). Introduction of ribs, whose section is small
in comparison with that of the part to be ribbed (small values of

___ Chapter 4. Rigidity of Structures

h/h 0 and b/b Q , large pitch), worsen the moment of resistance, i.e.
weakens the part. Under unfavourable conditions (h/h 0 = 2; b/b 0 —
— 0-01) the moment of resistance will drop to a third of that of the
original profile.

The picture becomes more expressive if we plot the relative pitch
values t 0 on the abscissa and variations of the moment of resistance,
for different values of relative rib height h!h 0 , on the ordinate
(Fig. 139). The portions of curves, located below the line W/W 0 =' 1,

2 3 4 5 6 7 8 3 10

5 S 7 8 310

a 2 ai

30 4 0 50 SO 70 80 100
0.033 0.02 0.01

Fig. 139. Strength of ribbed sections as a function of relative rib pitch i 0 —
= b/b 0 for different values of relative rib height h/h 0

describe relationships at which strength begins to fall (large relative
pitches t 0 , small relative rib thicknesses b/b 0 ). The less the rib'height,
the sharper the weakening is expressed.

Weaking may be prevented by increasing rib height. Ribs, whose
relative height h/h 0 > 7, will not worsen strength of parts up to the
largest values of relative pitch encountered in practice (t {) —
= 100 ), • •

However, for cast components increases in rib height are limited
by the casting technique. In practice the relative height rarely
exceeds h/h Q = 5. Casting technique also restrict rib^thicknesses.
Generally thickness is kept within (0.6-0.8) h 0 .

4.3. Enhancing Rigidity at the Design Stage _ 297

Actually tlie other way is to decrease relative pitch. When t 0 > 6
weakening does not begin even with the smallest ribs (h/h 0 ~
= 1 ).

From toe graph, plotted in Fig. 439, it is possible to find the pitch
values of different height ribs which do not weaken the part. These
values correspond to abscissas of points, in which curves k/h Q inter¬
sect the ordinate W/W 0 — 1-

If relative pitches t 0 are expressed as a function of relative rib
height h/h 0 , then the curve W/W 0 = 1 (Fig. 140) will correspond to

to

Fig. 140. Relative rib pitch t 0 as a function of relative rib height h/h 0 for diffe¬
rent ratios W/ W 0

the case when strength of a part is not affected by the increased
number of ribs and the curves W/W 0 = 4.5 and W'/W 0 — 2 will
correspond to the cases, when strength of a part rises in proportion
to the increased number of ribs.

For practical determination the maximum allowable pitch is
found from the relation

'.=t < 2 (vf (4 ; 31)

expressed as the averaged <! 0 values when

W/W Q = 1.5-2

The maximum adm?"ible rib pitch

« = »(£)’ (4.32)

where b is the rib thickness, mm.

Based on Eq. (4.32) a graph (Fig. 141) is constructed which allows
limiting t values to be found for different rib parameters.

298

Chapter 4. Rigidity of Structures

It should be noted that if stresses in a part are insignificant, usual¬
ly the case in housing-type components, then the decrease in strength
due to the introduction of unfavourably shaped ribs is not dangerous.

*-1-1-1-1-1-1-1-1_i-1_i_> » »

1 t 3 4 5 6 7 8 9 10 11 1 Z 13 14 b mm

Fig. 141. Graph for determining the maximum admissible rib pitch

In such cases the designer may freely apply ribs (including those,
working in tension), as a means of enhancing rigidity, without con¬
sidering the lessening strength.

For a heavily loaded part all the above recommendations remain
valid and are important for correct design practice.

Triangular ribs. Very often use is made of ribs, whose height dimi¬
nishes in the plane of the bending moment (triangular-shaped ribs).

4.3. Enhancing Rigidity at the Design Stage

299

With such rib forms whatever their initial height sections are ine¬
vitable where weakening of the detail begins.

Figure 142 pictures typical shapes of triangular ribs on a cylindrical
cantilevered part being bent by a force applied at the cantilever
end. Under each figure is shown a qualitative picture of changes
in the moment of resistance W and flexural stresses a along the part

Fig. 142, Influence of triangular ribs on the strength of a cantilevered Part

axis. For the unit moment of resistance the moment of resistance
W 0 of a non-ribbed portion of a part is taken; for stresses, the value
of the basic cantilever stress a 0 , i.e., at the point where the cylinder
joins the flange. Stress values for the non-ribbed part are indicated
by dashed lines.

For the rib form, shown in Fig. 142a, weakening begins at the
part m where the rib joins the cylindrical wall. Such a rib form is
particularly unfavourable as weakening occurs at the region of large
bending moment values and at the weakened part a sharp rise in
stresses takes place.

Somewhat better are long ribs (Fig. 142 b). The weakened section m
is displaced to the region of smaller bending moment. Stresses at
the weakened section slightly exceed maximum stresses in the part.

The most advantageous case is when the rib extends to the canti¬
lever end (Fig. 142c). Here the weakened area is at the region of
minimum bending moment and almost has no effect upon stress
values.

Ribs in compression. Internal ribbing. The relationships for ribs
undergoing tension during flexure (Fig. 137) formally hold for ribs
working in compression during bending (Fig. 143). Ribs, subjected

300

Chapter 4. Rigidity of. Structures

to compression, work in more favourable conditions since the majori¬
ty of cast materials resist compression rather better than tension.
With the introduction of ribs the phenomenon of the fall in strength
working in compression has less significance than when working
in tension.

The better strength iniierent in ribs working in compression as
well as the wide free choice of their parameters compels, in all cases,
the preference to return to ribs in compression than those intension.

Pig. 143. {Flexure of a cantilevered Fig. 144. Housing component

member with internal ribs working in (a) external ribs; (b) internal ribs

compression

This means that in housing-type components preference is given to
internal ribbing and not external ribbing.

For a housing, having external ribbing loaded with bending force P
(Fig. 144a) the main part of the load is taken by ribs m, positioned
away from the side of acting load, and also by the housing walls
(especially those portions parallel to the bending moment plane).
The opposite ribs, which in accordance with the force scheme should
work in compression, hardly take part, because the load reaching
them is strongly reduced. Hence, in this loading mode the ribs work
largely in tension, i.e., nnfavourably.

With internal ribbing (Fig. 1446) the ribs m positioned on the
side of the acting load work in compression. The opposite ribs, which
according to the loading scheme should work in tension, are practi¬
cally free of load.

Apart from the improved rib strength, the internal ribbing allows
the strength and rigidity of the housing as a whole to be sharply
increased, as the radial dimensions of the housing walls can be
increased. Within the same overall sizes (being determined by the
rib outline for an externally ribbed housing) one can obtain sub¬
stantial improvement in the values of the moment of resistance
and the moment of inertia.

The moulding of internal ribs is simpler (particularly when the
part] interior is formed hy cores). Furthermore, internal rihbing

4.3. Enhancing Rigidity at the Design Stage

301

betters the outside appearance of the components. In general, inter¬
nal ribbing is preferred in all cases, except in special ones (when outer
fins, for example, are necessary for cooling purposes).

Design Rules

When designing ribs it is necessary to observe the following basic
rules:

avoid loading ribs in tension, apply in all cases, when permitted
by the design, ribs working in compression;

for strength reasons be sure that the height of ribs, intended for
housing-type component, is not below (8-10) h 0 (h 0 —wall thickness),

if the ratio between ribs total
thickness and wall width is
small (i.e., of the order of
5/5 0 = 0.01). Should such a
height be unobtainable, owing
to size or casting conditions,
then rib pitch must be incre¬
ased according to Eq. (4.32);

ribs of triangular form must
extend to the plane of : acting
bending forces;

ribs must extend to rigidity
points, around fastening bolts
in particular (Fig. 145).

It is advisable to thicken rib
ridges, because ribs with sharp
tips invite higher stresses on

Pig. 145. Arrangement of u ribs on a part
used as a cover
( a > poor design; (b) good design

their edges.

Occasionally it is most advantageous to obviate ribs at all: this

will even add to the part strength.

Ribbing of parts subjected to torsion. When loading cylindrical
parts, or other similar shapes, by a torque moment longitudinal
straight ribs insignificantly increase^rigidity (Fig. 146a). Such ribs
quickly become harmful since they are subjected to flexure (in the
plane normal to the rib edge), which causes higher stresses in the

ribs.

During torsion it is more favourable to use diagonal ribbing
(Fig. 1465), which under the influence of the torsional moment work
in compression and substantially increase rigidity. The described
design is a particular case of applying the principle of diagonal
stringers.

The design, depicted in Fig. 1465, is calculated for a torque of one
constant direction. For an alternating torque moment or a snake¬
like pattern (Fig. 146 d) crossed ribs (Fig. 146c) are advisable.

302 _ Chapter 4. Rigidity of Structures _

Diagonal and helical ribs are less prone to internal stresses, which
occur during casting shrinkage due to its non-uniform setting.

However, the moulding of diagonal ribs] on outer cylindrical,
conical and the like surfaces is difficult.

Pig. 146. Rib forms on a cylindrical component working in torsion

Therefore, when dealing with cylindrical and similar-shaped
components, working in torsion, it is advisable, as in the case of
flexure, to use internal ribbing.

Circular ribs. Circular ribs, along with the usual straight ribs, are
used to enhance the rigidity of round components, such as disks,
cylinder ends, etc.

The mode of action of these ribs is peculiar. Let us assume for
example, that a round plate with a circular rib is bent by some axial
force, applied in the centre (Fig. 147a). The deformations of the

4.3. Enhancing Rigidity at the Design Stage

303

plate are imparted to the circular rib; its walls tend to extend towards
the periphery (Fig. 1476). The tensile stresses, arising in the ring,
check the plate flexure.

The circular rib, facing the load (Fig. 147c) will act in a similar
way, the difference being that it is subjected to inward radial com¬
pression.

To impove rigidity it is recommended that the circular ribs be
made higher and positioned at the radius, where the sag angle of

VMM

m

I

■

■

1

i

■

1

MHMFNB m

B

(a)

Fig. 148. Rib forms

the plate has its greatest value; for plates, resting on their rim, it
will be closer to the periphery and for plates with built-in rims, clo¬
ser to their mean radius. Positioning ribs at relatively small distan¬
ces from the plate centre is almost useless.

The most effective combination of circular and radial ribs are
shown in Fig. 147 d-h.

Structural varieties of ribs. To impart especially high rigidity
the following types of ribs are employed: wafer (Fig. 148a), honey¬
comb (Fig. 1486) and diamond (Fig. 148c).

Often use is made of hollow ribs (Fig. 149), open relief (1-9, 13)
or closed profile (10-12-14). In contrast to conventional ribs hollow
ribs under all circumstances enhance the rigidity and strength of
constructions.

Closed ribs are more rigid than the open ones, but their moulding
is more difficult. Practically the same results are obtained when
applying open ribs, strengthened by cross partitions (3, 6, 9 and 13).

Internal hollow ribs (13, 14) are more preferable than external
ones. In the range, where rectangular-shaped internal closed ribs
join each other, the most rigid and strongest double-walled box-
section construction is obtained (15).

4.4. Improving Rigidity of Machine Constructions

305

Structural Examples

Figure 150 shows examples of correct and incorrect rib design.

A housing-type component with a rib working in tension at the
transition point of two sections (Fig. 150a) is not at all advantageous
to strength. Removal of the rib increases the housing strength
(Fig. 1506). When introducing a rib, it is better to make it T-shaped
(Fig. 150c) or position it so that the rib works in compression
(Fig. 150c?).

Figure 150c-; illustrates compartments of cylindrical housing
with a partition (diaphragm), loaded with a transversal force P
or bending moment M. The short ribs (Fig. 150c, /) weaken the
partition.

The best constructions have ribs of constant height (Fig. 150g)
or ribs which widen to the place of fixing (Fig. 150k).

The highest strength is possessed not by ribbed components but
by ones which have fluted partitions (Fig. 150?), or box sections
(Fig. 150;), especially when reinforced with inner transversal
ribs.

Depicted in Fig. 150k-p are spherical cantilevered housings. Some¬
times such components are ribbed on the outside (Fig. 150k). If the
rib height is not great in relation to the wall thickness, the ribbing
weakens the component. Removal of the ribs (Fig. 150?) enhances
component strength. Still stronger components are obtained when
their walls are enlarged within the limits of the overall sizes
(Fig. 150m). Even farther increase in strength may be obtained by
providing the component with internal longitudinal (Fig. 150n)
or wafer-like (Fig. 150 o) ribs. High rigidity and strength are posses¬
sed by components with fluted walls (Fig. 150p).

4.4. Improving the Rigidity of Machine Constructions

Ways of improving rigidity and strength of typical engineering
structures are given in Table 20.

(a) Fixing of Cantilevers

The rigidity of cantilevered systems is strongly affected by the
method of fixing. Constructional measures can give a cantilever of
any rigidity, but these measures may be brought to nought if the
cantilever fixing is weak or if the cantilever is set into a non-rigid
component. Ways of improving the rigidity of cantilevered systems
are given in Table 21.

20—01395

306

Chapter 4. Rigidity of Structures

Table 20

Rigidity Improvements of Engineering Structures

Original design

Improvement

Essence of
improvement

Cantilever fastening of
a roller lever

Lever weak, roller
shaft fastening non-rigid

Flat bearing washer
loaded in flexure by axial
force

fist

Cantilever span
is reduced. Lever,
shaft and assembly
strengthened

Cantilever elimi¬
nated; roller shaft
mounted in two
supports in. lever
fork

Reinforced by
circular collar

Wasner body
given an equal re¬
sistant form

Washer given
conical form; fle¬
xural stresses shar¬
ply decreased.

4.4. Improving Rigidity of Machine Constructions

307

Table 20 ( continued )

Original design

Improvement

Essence of
improvement

Internal combustion
engine valve

Non-rigid valve plate;
plate-to-stem link poor

Valve plate given
tulip-shaped form.
Plate rim rigidity
still poor

Stem and plate
made more solid;
rigidity of plate
rim improved

Rim-to-cylinder
joint strengthened

Shoulder-to-sho-
ulder web is pro¬
vided (most rigid
design)

Skirt of
piston en¬
gine cylin¬
der

Skirt de¬
forms under
transverse .
piston load

Annular stiffe¬
ning ribs at skirt
bottom end

20 *

308

Chapter 4.'Rigidity of Structures

Table 20 ( continued )

Original design

Sleeve loaded by
transverse force

Improvement

Essence of
improvement

Sleeve edges
strengthened by
bead

Under loading, sleeve
edges loose their cylind¬
rical shape

Terminal-connection

Journal defor¬
mation eliminated
by introducing wed

Tightening of termi¬
nal deforms shaft journal

Terminal connection

Terminal streng¬
thened. Coupling
bolt positioned
closer to shaft

Terminal lugs bend
when tightened. Power
tightening not possible

4.4. Improving Rigidity of Machine Constructions

309

Table 20 ( continued )

Original design

Improvement

Essence of
improvement

Arch or base
reinforced by in¬
ternal ribs, -wor¬
king in tension

Arch of base
reinforced by ex¬
ternal ribs, wor¬
king in compres¬
sion

Cast lug

Arch of base is sub¬
jected to heavy flexure

Arch of base
given rigid pyra¬
mid form

Arch of base
provided with pe¬
ripheral frame;
frame rests on
seating surface

Lug provided
in acting load pla¬
ne which takes the
load (most light
weight construc¬
tion)

310

Chapter 4 . Rigidity of Structures

Table 20 ( continued )

Original design

Improvement

Essence o!
Improvement

4.4. Improving Rigidity of Machine Constructions

311

Table 20 ( continued )

312

Chapter 4. Rigidity of Structures

Table 20 ( continued)

Original design

Improvement

Essence of
improvement

Cast multigroove
V-belt pulley

Spoke construction.
Rigidity poor

Rim joined to
hub by solid rib¬
bed disk. Hub len¬
gthened

Section given
conical box form
(best design for
rigidity)

Spur gear

Disk of gear
made conical

Ribs added to
spur gear disk (for
cast gears)

Rigidity poor

d.4. Improving Rigidity of Machine Constructions

313

Table 20 ( continued)

Rigidity poor

314

Chapter 4. Rigidity of Structures

Table 20 ( continued)

Original design

Bevel gear

Rigidity poor

Improvement

Essence of
Improvement

Disk made co¬
nical

Disk made
spherical

Ribs added (lor
cast gears)

Welded box-
shaped prestressed
construction.

Clearance, bet¬
ween added body a
and shoulder b, is
eliminated before
welding. Teeth and
splines machined
after welding

4.4. Improving Rigidity of Machine Constructions

315

Table 20 ( continued )

316

Chapter 4. Rigidity of Structures

Table 20 ( continued )

Original design

Improvement

Essence of improvement

Trunnion lengthened
and force-fitted into hou¬
sing body

Disk attached to hou¬
sing body by additional
central bolt

Disk attached to hou¬
sing body by two rows
of radially spaced bolts

4.4. Improving Rigidity of Machine Constructions

317

Table 20 ( continued )

Original design

Improvement

Essence of improvement

Connected by j box-pat¬
tern members (box-pat¬
tern members are expen¬
sive to manufacture)

Composite beam, built
from two thin-walled U-sec-
tions (direction, of work loads
shown by arrows)]

Rigidity poor

Connected by bent sec¬
tions' (transverse rigidity
not assured)

Connected) by bent sec¬
tions (longitudinal rigi¬
dity not assured)

Connected $ by
tions ^(longit udinal
dity^not assured)

U-sec-

rigi-

318 _ Chapter 4. Rigidity of Structures ____

Table 20 (continued)

Original design Improvement Essence of improvement

Diagonal connec¬
tions (rigidity assured
in all directions)

Trapezoidal connect¬
ions (rigidity assured
in all directions)

Trapezoidal connect¬
ions arranged alterna¬
tely alond the beam.

Best constriection in
terms of rigidity,
weight and manufac¬
ture

4.4. Improving Rigidity of Machine Constructions

319

Table 20 (continued)

Original design

Improvement

Essence ol Improvement

Frame bracket loaded by force P

Rods work mainly in flexure.
Stresses in the system very high.
Rigidity poor

Construction given
a truss form, rods ar¬
ranged in triangular
diagonal pattern. Rods
work mostly in ten¬
sion-compression
When loaded by a
transverse force, the
side rods, positioned
parallel to the plane
of acting bending mo¬
ment, accept the load;
the rods, positioned
perpendicularly to
that plane also parti¬
cipate in the work as
space trusses, whose
side rods serve as bra¬
cing struts. The truss
form is closed by the
front stiffening ring
To give the system
full determinancy, the
ring-to-rod joints are
articulated

Shell-type constru¬
ction. A highly rigid
construction. Weight
can be reduced by cut
outs

Shell lightened by
box-shaped cut outs
Non-rational design,
since parts between
cut outs work in fle¬
xure

Shell construction
given more rationally
shaped cut outs

320

Chapter 4. Rigidity of Structures

Table 20 (continued)

Original design

Improveme t

Essence of
improve¬
ment

Weight-

relieved

construc¬

tion

4.4. Improving Rigidity of Machine Constructions _ 321

Table 20 ( continued )

Original design Improvement Essence of improvement

Internal ribs added

Internal partitions ad¬
ded

Cover made vault-sha¬
ped

Deformation restricted
(degree of deformation
determined by initial
clearance a between the
cover and limiting stop
shoulder of stud)

External circular and
radial stiffening ribs pro¬
vided :

Circular ribs provided
and cover made vault¬
shaped

Cover given arched
form

21-01395

322

Chapter 4. Rigidity of Structures

4.4. Improving Rigidity of Machine Constructions

Table 20 ( continued )

Original design

Improvement

Essence of improvement

Cast bracket, loaded
in flexure

Ribbedijbraeket column

Radial size ofjbracket'column
increased

tTColumn Jmade conical. Con¬
nection to fastening!; flange
strengthened

Rigidity poor

Radial size of bracket column
increased. Column-to-flange
joint made conical

Chapter 4. Rigidity, of Structures

Table 20 ( continued )

Original design

Improvement

Cast revolving drum.
Reciprocating along cy¬
lindrical guides are ope¬
rating rods with rol¬
lers, which roll on a
stationary templet as
the drum revolves

The through slot
strongly weakens the
lugs. Under action of
operating forces > the
lugs walls move apart
(see arrows), thus spoi¬
ling rod directional
movement

Essence of improvement

Radial size of bracket increa¬
sed to maximum overall size.
Internal ribbing introduced
(construction very rigid and
strong)

Non-through slot. Design not
technological. Very difficult to
assemble rod and roller unit

Lugs reinforced by external
ribs 1 ‘

Lugs reinforced by continuous
circular ribs

4.4. Improving Rigidity of ■ Machine Constructions

325

Table 20 (continued)

Original design

Improvement

Cast turntable of rotary
machine; loaded by ben¬
ding forces acting at the
operative sections

Essence o t improvement

Radial drum size inc¬
reased. Lugs strengthened
by ribs

Radial drum size in¬
creased to maximum size
(best rigid and strong
construction)

Ribs linking the cent¬
ral, hub to the peripheral
bosses are introduced

Peripheral rigidity is
enhanced further by cir¬
cular rib

326

Chapter 4. Rigidity of Structures

Table 20 ( continued)

Original design

Improvement

Essence of improvement

Peripheral bosses hoop
still further strengthened
by second circular rib

Turntable rigidity im¬
proved by radial and cir¬
cular interconnected sys¬
tem of ribbing

Turntable given box
shape (most rigid con¬
struction)

Improving Rigidity of Cantilevered Systems

Table 21

Original design

Improvement

Essence of improvement

Installation of cy¬
lindrical column into
a cast housing

Rigidityjjj poor

Axial fixing. Extended end of
column fixed in rigid housing
boss

Radial fixing. Column given
a flange and attached to streng¬
thened housing surface

4.4. Improving Rigidity of Machine Constructions

327

Table 21 { continued )

Original design

Improvement

Essence of improvement

Radial and axial fixing

Attachment by welding
rod to U-shaped channel

/pjcj-jLKj

Rod unstable in tran¬
sverse direction

'bzd

E222

3 E Z3

Rod fixed to both channel
flanges

Channel flanges strengthened
by webs where the rod is wel¬
ded to channel

Fixing a column to a
cast housing

Column is unstable
owing to compliance of
housing top

Fixing strengthened radially
(housing top still remains
compliant)

Fixing strengthened
(housing top still
compliant)

axially

remains

328

Chapter i. Rigidity of Structures 1

Table 21 { continued )

Original design

Improvement

Essence of improvement

Local internal ribbing
which only strengthens
central part of top

Extended internal rib¬
bing which strengthens
entire top

Ribbing extended fur¬
ther, strengthens comers,
top and. part of vertical
walls

Ribbing extended still
further, walls strengthe¬
ned down to base

4.4. Improving Rigidity of Machine Constructions

329

(b) Column Bases

Cast steel footings joined with the column body by welding
(Fig. 151a) impart high rigidity and strength to the entire support.
However, such constructions are very complex to manufacture and
heavy.

In the design shown in Fig. 151 b the foot is a plate welded to the
column end. The rigidity of the joint is insufficient. Such construe-

Fig. 151. Column bases

tions are suitable for light props subjected to moderate loads. The
rigidity of column fixing can be increased further by the addition
of a welded formed collar (Fig. 151c) or gussets (Fig. 151d). The latter
design is widely used as it is simple to manufacture and sufficiently
rigid.

The column end (Fig. 151e) is flared when it is necessary to make
the external appearance more attractive. With large column sizes
flaring can be difficult.

More practical from the viewpoint of manufacture are welded-on
cone constructions (Fig. 151/, g). The stiffening element is often
made in the form of a streamlined torus (Fig. lblk).

330

Chapter 4. Rigidity of Structures

A tulip-shaped base butt-welded to the column walls has the hig¬
hest rigidity (and best outward appearance) as can he seen in
Fig. 151i.

(c) Rigidity of Housing Components

The main ways to improve the rigidity of housing components
without increasing their weight (sometimes even with lowering it)
are: rounding of transition points, making walls vault shaped, effi¬
cient (internal) ribbing, introduction of interwall struts (preferably
diagonal).

The rigidity of housings can also be improved constructionally
uniting housing elements (monoblock constructions).

Figure 152 shows (approximately in the order of historical succession)
constructional strengthening of in-line internal combustion engi¬
nes. In an engine with separate cylinders (Fig. 152a) the rigidity is
determined only by the rigidity of the crankcase. With the flexure
forces occurring during ignition, the crankcase is deformed and so
is the entire engine.

More rigid is the half-block construction (Fig. 1526) where the
cylinder heads are united into one general block. The summed
moment of inertia of the system, strengthened by a cylinder head
block and overall camshaft cover, sharply increases.

The systems having wide use and preferred in the motor industry
are the block systems (Fig. 152c, d). Here rigidity is enhanced by
making the cylinder jackets in one common block which is either
attached to the crankcase (Fig. 152c) or cast integrally with it
(Fig. 152 d). The latter alternative gives the most rigid and strongest
design with the smallest number of joints between elements.

Of no less importance is the rigidity of the crankcase incorporating
crankshaft supports, since the former is subjected to flexure in the
plane of acting ignition forces, i.e., in the longitudinal plane of
crankcase symmetry. To increase rigidity, it is advisable to improve
the moments of inertia across the transverse sections of the crankcase
and prevent the side walls from “opening” by providing stiffening
transverse webs between the walls.

Figure 153 shows simplified constructional examples of crankcase
designs used with detachable cylinder jacket units (ref. to Fig. 152c).

The system, presented in Fig. 153a, comprises the main crankcase
(top half, depicted in heavy lines) and sump, and has rather poor
rigidity, though it is very convenient for installation and assembly
of the crankshaft.

The split plane of the main crankcase and sump lies above the
shaft axis. The shaft is held in suspension bearing 1,

The main crankcase can be made very robust by shifting the split
plane to the crankshaft axis and diminishing accordingly the sump
height (Fig. 1536). In the design illustrated in Fig. 153c, the crank-

(c)

W)

Fig. 152. Constructional evolution of internal combustion engines

332

Chapter 4. Rigidity of Structures

case transverse partitions are stiffened with arched ribs. Bearing
suspensions 2 have been developed in the transverse direction and
secured to the case by two rows of bolts, as a result of which rigidity
points are formed around the crankshaft supports.

The rigidity of the main crankcase can be further improved by
shifting the split plane below the shaft axis (Fig. 153d). To strengthen
the bond between the crankcase side walls, the arched ribs are exten¬
ded to the walls and the bearing suspensions. are mounted in the
crankcase cut-outs.

Wall-to-wall strength is enhanced with the aid of bolts which
join bearing suspensions and crankcase partitions together
(Fig. 153e). In the design pictured in Fig. 153/ the crankcase walls
are joined by tie-bolts; to avoid overtightening, the value of the
internal free span is adjusted by nuts.

A more rational construction is the one in which the tie-bolts
are tightened on to the suspension wall support (Fig. 153g).

Further improvements in the rigidity of the system can be obtained
by making the main crankcase in two parts with the split line plane
along the shaft axis (Fig. 153 h, i). .The suspended bearings in this
case are integral with the lower crankcase half.

The most rigid design is presented in Fig. 153/': the crankcase is
comprised of two parts (halves) connected together at the shaft axis
plane by two rows of clamping pins. Both halves are load-carrying
elements and equally take part in flexure.

( d) Plates

Figure 154 illustrates methods of improving rigidity and strength
of moulded plates. It is assumed that the plate is loaded in the centre
and rests upon four side supports (at the corners).

The original design (Fig. 154a) has poor rigidity and strength.
When provided with longitudinal ribs which offer uniform resistance
to flexure (Fig. 1545), a higher longitudinal rigidity is obtained,
but ' transverse rigidity remains inadequate.

Equal rigidity in the longitudinal and transverse directions is
obtained when the plate is furnished with radial ribs (Fig. 154c).

Another design principle has been embodied into the alternative
presented in Fig. 154d, the rigidity of which is improved by the
addition of vertical walls around the plate perimeter. Flexure defor¬
mations are restrained by the walls which work in tension. The rigi¬
dity can be enhanced still further by increasing the wall height,
by increasing edge sections and connecting the plate body to the
walls by ribs (Fig. 154e) which will transfer the flexure deformations
to the edge walls.

Drawing the side walls together by tie-bolts (Fig. 154 f) create
stresses in the plane of opposite sign to those of the working stresses

334

Chapter 4. Rigidity of Structures

High rigidity and strength are possessed by constructions having
a sheet steel cover which works in tension (Fig. 154g). By heating
the cover prior to mounting prestressed conditions can be created
if it is positively secured to the plate (e.g., by means of locating
pins).

Another way of imprqving rigidity is by making the plate vault¬
shaped (Fig. 154 h). Side walls with diagonal (Fig. 154i), wafer-like
(Fig. 154/), checkered (Fig. 154 k), diamond (Fig. 154 1) and honey¬
comb (Fig. 154m) ribbed constructions have high rigidity.

When fastening points on the plate are present, the ribs must be
arranged so as to connect the centres of rigidity (Fig. 154n).

The highest rigidity is given by double-walled plates when inter¬
nal slanted ribs (Fig. 154o, p) formed during casting by means of
through cores secured on prints in the plate side walls.

Close to these are h'alf-enclosed plates with internal cells moulded
during casting by mould core assemblies secured on prints through
holes in the plate bottom (Fig. 154g) and double-walled plates with
concave bottoms (Fig. 154r).

Figure 154s illustrates a light easily producible design of lattice
type. For a smooth external surface such plates are usually faced
with thin-sheet metal skins.

(i e) Rigidity of Thin-Walled Constructions

In constructions made from sheet materials (shells, thin-walled
sections, reservoirs, skins, panels, covers, etc.) it is necessary to
consider not only the deformations caused by working forces but
also the strains arising during welding, machining, joining and
tightening of assembled elements. The possibility of chance damage
to walls during transit, installation and carelessness in use must also
be considered.

In loaded shell constructions prevention of stability losses is of
primary importance.

The chief methods of increasing rigidity remain the same: maxi¬
mum relief of flexure, changing flexure stresses to tensile-compres¬
sive ones, introducing ties between planes of maximum deformation,
enlarging sections and moments of inertia at critical points, employ¬
ing reinforcing elements at places where loads are concentrated
and at parts of discontinuous force flows, using conical and arched
forms.

Compartments. The radial rigidity of large size thin-walled cylin¬
drical components (compartment-like units) is enhanced by means
of annular stiffening collars (Fig. 155a-j).

Compartment constructions with double walls are more rigid
and strong (Fig. 156a). To improve radial rigidity, it is advisable

4.4. Improving Rigidity of Machine Constructions

335

for compartment walls to be joined with each other. Sometimes one
restricts himself to introducing local ties, i.e., welding strengthened

to compartment walls
(Fig. 1565) or by welding
tubes into them (Fig. 156c).

The best results are ob¬
tained when annular stif¬
fening collars are used
(Fig. 156 d~g). No less effi¬
cient is subdividing the
entire compartment into
several shorter subcompart¬
ments (Fig. 156A, j). In
the latter case the role of
stiffening collars is fulfil¬
led by subcompartment
butt-joints.

Introduction of cones
(Fig. 156/) or arched ele¬
ments (Fig. 156 k, l) increa¬
ses not only radial but also
longitudinal rigidity.

Figure 157a-c presents
satisfactory compartment
designs, reinforced by coni¬
cal elements.

Fig. 155. Stiffened zones in cylindrical shell
components

Longitudinal rigidity of compartments is attained by ribs posi¬
tioned along the cylinder generatrices (Fig. 1586-g). Components

Fig. 156, Enhancing the radial rigidity of double-walled compartments

of the highest rigidity are obtained when combining longitudinal
ribs and stiffening rings (Fig. 158a).

336

Chapter 4. Rietdttu of Structures

Helical and zigzag ribs (Fig. 159) better torsional ^gidity. thei 1
manufacture, however, is more difficult than that of straight longi¬
tudinal ribs.

Double compartments are joined together by means of externa
(Fig. 160a-c) and internal (Fig. 1603) flanges. The latter contribute
to still higher rigidity and substantially decrease the radial dimen¬
sions of components.

iron

338

Chapter 4. Rigidity of Structures

When planning the introduction of bolts from inside it is necessary
to provide holes in the internal walls large enough to introduce and
tighten the bolts.

Figure 161a-/ illustrates enhancement of radial rigidity in conical
compartments; Fig. 162 shows the construction of a double-walled
spherical bracket-type component.

Shell constructions with spatial lattices. The highest rigidity of
a shell-system can be obtained by filling the shell interior space
with uniformly distributed stiffening elements connecting all their
sections and turning the system into a spatial lattice working as
one unit. The appearance of strong synthetic resins and adhesives
allows this problem to be solved rather closely.

There are two basic kinds of spatial reinforced shells in use: foam
plastic and honeycomb constructions.

In the first case the space between the metallic shells is filled
with foamable plastics based on thermo-setting or curable resins.
The plastics are introduced in a liquid state, with the addition of
gassing agents and emulsifiers. When heated to 150-200 °C the plas¬
tic composition foams and hardens, forming a spongy mass up to
80-90% in volume and with specific weight y = 0.1-0.2 kgf/drn 8 .

As a result, the strength, rigidity and stability of the reinforced
system considerably increase though not so much as when metallic
spatial ties are introduced. The system is generally applied in con¬
junction with metallic stiffening elements, either transverse (ribs,,
frames) or longitudinal (spars, stringers).

Honeycomb constructions are prepared by joining together honey¬
comb-pattern impressed fabric or glass-fibre cloths impregnated
with thermo-setting dr curable resins. Mantles are made of sheets of
the same material or of metal sheets. The size of honeycomb cells
is usually 8 to 15 mm.

Still better in strength and rigidity are metallic honeycombs made
by gluing together embossed metal sheets covered with a film of phe¬
nol-neoprene adhesives or modified epoxides. The same adhesives
serve simultaneously for joining metallic mantles to honeycomb
structures whose strength is dependent on the adhesion (the resistance
of strongest synthetic adhesives to shear comprises 2-5 kgf/mm 2 ,
and bond strength 5-10 kgf/mm 2 ).

Steel sheets can be joined more strongly by furnace brazing with
bronze alloys in vacuum or in a reducing atmosphere.

New possibilities for making strong honeycomb structures are
opened by fine-focus electron-beam welding technique. The welding
temperature is produced only at the focus point, the remaining
zones do not undergo significant heating. This enables butt-welding
to be done at any depth with one and the same position or attitude
of the welding unit. Depth adjustment in welding is effected by
refocusing the electron beam with the aid of converging magnetic

4.4. Improving Rigidity of Machine Constructions

339

coils and in the transverse and longitudinal directions hy means
of deflecting coils. Thus, all inner joints in the component can be
successively welded.

- Stability of shell constructions. The increase in overall dimensions
and reduced wall thicknesses in the first instance mean enhancing
radial rigidity and elimination of component stability losses from
loads.

Fig. 163. Rectangular thin-walled beams with transverse ties

The rigidity of welded thin-wall square-section beams is improved
by embossing reliefs (preferably oblique) on the walls (Fig. 163a),
introducing transverse partitions (Fig. 1636), providing tubular or
box-section connection elements (Fig. 163c), and diagonal stringers
(Fig. 163d).

r i i

f i \

-4-

j

VL/

(c)

(d)

Fig. 164. Thin-walled beams with
transverse struts

Fig. 165. Cross-sections of thin-walled
high-rigidity beams

High rigidity is ensured by serpentine-like diagonal partitions
(Fig. 164).

Figure 165 illustrates cross sections of beams, possessing high
rigidity and stability.

Reinforcement of areas subject to concentrated loads. In the design
of thin-walled components particular attention should he paid to
areas subjected to concentrated loads. Insufficient rigidity at these
parts causes local wall deformations making the construction unfit
for use.

22 *

340

Chapter 4. Rigidity of Structures .

For cylindrical shell parts the simplest procedure is to introduce
straps which distribute the force over a larger area (Fig. 166a, b).
Still more efficient are stiffening collars and partitions (Fig. 166c-e)
distributing the force over the entire component cross section.

Fig. 166. Reinforcement of shell structures at load concentration points

The flexure of thin-walled components in holt fastening areas
(Fig. 167a) is prevented by using large-diameter washers (Fig. 1675),
flanging the walls (Fig. 167c, d) or introducing some stiffening
elements (Fig. 167e, /). The most efficient technique is the taking

Fig. 167. Reinforcement of fastenings

up of the lightening forces by a strut-like element (e.g., a tubular
column) working in compression (Fig. 167g, h ).

Figure 168 shows how a thin-walled cover is attached to the main
component by means of a non-loosable bolt. In the original design
(Fig. 168a) the cover deforms even under light tightening pressure.

To avoid sag, the tightening is (limited to the preset gap m
(Fig. 168&-d).

In the design presented in Fig. 168’d the limiting element is made
in the form of a conical catcher easing the introduction of the threaded
bolt end when mounting the cover. The spring keeps the bolt straight
when removing the cover and facilitates reassembly.

4.4. Improving Rigidity of Machine Constructions

341

Butt joints of sheet structures. The rigidity of butt joints in
thin-walled components is particularly important when the joints
must be hermetic.

Fig. 168. Fastening of a thin-walled cover to a housing

When connecting flanges of two thin-walled cylindrical components
of large diameter (Fig. 169a) no hermetic sealing can be effected
in sections between bolts because of the insufficient rigidity of the

Fig. 169. Joints of thin-walled cylindrical parts

flanges. Washers placed under the bolt heads and nuts hardly impro¬
ve rigidity (Fig. 169b). It is possible to make the joint tight by intro¬
ducing additional solid rings, clamped or welded in position
(Fig. 169c, d).

Fig. 170. Fastening of a formed sheet steel sump to a housing

If a stamped sheet steel sump is to be secured to a housing
(Fig. 170a), tight sealing is accomplished by bending the flange
(Fig. 1706) or by reinforcing with a solid frame tack-welded to the
sump (Fig. 170c, d).

342

Chapter 4. Rigidity of Structures _

Stiffening reliefs. Rigidity can. be enhanced by embossing reliefs
in walls (Fig. 171), generally in the form of beads. To simplify
the cold stamping process, the height of reliefs should not be greater

Fig. 171. Design forms of stiffening reliefs

than (3-5)s, where s — thickness of material. Reliefs of larger heights
are stamped in several passes with intermediate annealing, this
making the production more expensive.

Fig. 172. Various relief forms on a rectangular cover

In hot-stamped designs higher and more extensive reliefs can be
made.

Apart from better strength and rigidity due to purely geometrical
relationships (increased moments of resistance and inertia of sec¬
tions), cold embossed reliefs add to the toughness as a result of
cold working.

Fig. 173. Increasing the bottom rigidity of thin-walled cylindrical components

Relief beads should be positioned along the acting plane of the
bending moment (Fig. 172a). The reverse position (Fig. 1726) does
not improve rigidity, it only makes the component more compliant.

Reliefs must be directed toward the rigidity points of the system.
For this reason, the best arrangement of stiffening beads for a square¬
shaped plate is a diagonal one (Fig. 172c, d).

4.4. Improving Rigidity of Machine Constructions

343

Embossing the reliefs on the bottoms of cylindrical thin-walled
vessels (Fig. 173) not only increases rigidity, but also improves
stability and imparts definiteness to the vessel installation on a plane.

An effective way of increasing rigidity at angle transition points
from edges to bottoms is by forming local triangular-shaped beads
(Fig. 173/).

Figure 174 illustrates edge reinforcement of cylindrical reservoirs.

Fig. 174. Flanged edges on thin-walled cylindrical components

Lightening holes. Thin-walled constructions are often made with
holes in order to reduce their weight. To increase local rigidity,
to reduce stress concentrations and to enhance fatigue strength impai¬
red by punching tools, the edges of the holes are reinforced by flang¬
ing (Fig. 175a-c), flanging with half-and full-curled edges (Fig. 175 d-f),
flanging with swaged edges (Fig. 175g, h ) and also by reinforcing
straps (Fig. 175 i-k).

R*(2±3)S h~

— D - ,

, n-m-uw

"PI

_i_

j ««/

^ ....J

la)

_j

(W

— s

(C)

1

2 —

^ WM p7yyJ ..

LI J

(f) 55 1 . W 1 . 1 (j)

Fig. 175. Strengthening the edges of lightening holes

_ The flange height h (see Fig. 175a) should never be made too great
since this entails greater manufacturing difficulties. The flange
height attainable in a single cold flanging operation may be as great
as (0.15-0.25)7). Higher flanges and also flanges with curled edges
require several successive operations.

Another way to improve fatigue strength of material in the vici¬
nity of holes is bilateral crimping of hole edges, using for the purpose
dies and caulkers of rounded profile (Fig. 176).

Reservoirs. When designing reservoirs subjected to internal pres¬
sures it is necessary to avert wall bulging. Rectangular-section reser¬
voirs (Fig. 177a) are not advantageous, because their walls bulge from
the internal pressure (shown exaggerated by dash-lines). With such

344

Chapter 4. Rigidity of Structures

shapes the addition of transverse stiffening webs is obligatory
(Fig. 177 b).

Better rigidity have oval, elliptic {Fig. 177 c-e) and particularly
cylindrical (Fig. 177/) reservoirs.

When reinforcing cylindrical reservoirs with external ribs, the
direction of wall deformations must be considered.

Tensile stresses in sections along generatrices

where p — internal pressure

D — diameter of reservoir
s = thickness of wall (Fig. 178a)
Stresses in cross-sections

nD* P D
° 2 ~ P 4nDs 4s

0.5oi

i.e., half those along generatrices.

This is the reason why reservoirs fail along generatrices (Fig. 178&).

Longitudinal ribs (Fig. 178c) contribute very little to the strength
and rigidity of reservoirs, only to the extent of their resistance to
bending in the longitudinal plane.

More advantageous are circular ribs (Fig. 178d) which work in
tension.

The shape of the cylinder ends is of great importance. Flat ends
(Fig. 179a-c) are unacceptable for high internal pressures. Concave
ends are stronger and more rigid (Fig. 179d-/). However, the pres¬
sure-oriented deformations will cause additional tensile stresses in
the cylinder course. Moreover, the concave end appreciably dimi¬
nishes the reservoir usable capacity.

Conversely, convex ends (Fig. 179g', h) and similar to them conical
ends (Fig. 179i-/c) restrain radial deformations in the course.

Shields and screens. The rigidity of covers, shields, screens, panels,
dashes, etc., is enhanced by flanges (Fig. 180a~g), embossing reliefs
(Fig. l&Oh-k) and by making them convex (Fig. 180f, m).

Fig. 178. Arrangement of ribs on the walls of reservoirs subjected to internal

pressure

(b) (c) (d) (e) (f) (g) fh) (i) (J) (ft) tt)

Fig. 180. Designs of panels and screens

Chapter 4.4. Improving Rigidity of Machine i Constructions

347

Illustrated in Fig. 181 are typical shapes of covers (in plan) pro¬
vided with rectangular (Fig. 181a-e) and diagonal (Fig. 181/-/) pat¬
terns of relief. Also shown are covers with diamond convex patterns
(Fig. 181/c-o).

The pattern of relief is often dictated by aesthetic requirements,
particularly when faces are in view. Pleasant to look at and suffi¬
ciently rigid are rustic forms of surfaces.

Very large shields are usually made of several compartments each
of which is strengthened by one of the described methods (Fig. 181p).
For strengthening longitudinal rigidity compartments are inter¬
connected by a frame or by longitudinal ribs (Fig. 181g).

Chapter 5

Any part subjected to a sustained repeated-alternating load will
fail under stresses which are well below the ultimate strength the
material displays when exposed to static load. This feature can
hardly be overestimated when dealing with modern multirotary
machines, whose parts run under continuous cyclic loads with a

total number of cycles through¬
out the entire period of machine
service reaching many millions.

According to statistical data
at least 80% of failures and acci¬
dents encountered when using
modern machines are from fati¬
gue phenomena. That is why the
problem of fatigue strength is the
key enhancing dependability and
service life of. machines.

Cyclic loads manifest them¬
selves most notably in machines
and mechanisms whose parts
perform reciprocatory movements
(piston-operated machines, cam-
actuated mechanisms, etc.).
However, cyclic loads are inevitable even in machines with smooth
running parts (e.g., rotary machines of turbine type) due to disba¬
lance of rotors, radial and end play of rotors, etc.

With rare exceptions all modern machines incorporate gear drives
whose teeth are always subjected to cyclic loading. Also subject
to cyclic loads are shafts working under constant directional loads
(such are, for example, the shafts of gear-, belt- and chain-gearings).

Let ns consider, for instance, a simply-supported gear shaft
(Fig. 182) subjected to a driving force P causing flexure in a con¬
stant plane. As the shaft makes one revolution, the four points
a, b, c and d on the shaft successively move through the plane.
With each revolution the cycle repeats. Thus, despite the constant
force typical cyclic loads occur.

Pig. 182. Diagram of the development
of cyclic loads in a gear shaft

Chapter 5. Cyclic Strength

349

In short, static loads are exceptions in modern machines. Gene¬
rally loads change cyclically with a large or small frequency and
amplitude.

The number of load cycles which the material can sustain without
failure depends on the maximum stress and also on the interval
between the extreme values of cyclic stresses. With a stress decrease

Fig. 183. Fatigue diagrams (er D =
fatigue limit)

the number of cycles necessary to cause failure will increase and
at some sufficiently small stress value the material acquires the
ability to withstand an infinite number of cycles without failure.
This stress value is called the fatigue limit, assumed as the basis
for strength calculations of parts subjected to cyclic loading.

The fatigue limit is ascertained by plotting fatigue curves. The
number of cycles N is plotted on the abscissa, and on the ordinate
found from testing standard specimens the maximum stresses o
■of the cycle which cause failure at a given number of cycles. In the
■domain of small values the rupture stress approximates the static
strength. With the increase in the number of cycles this value decre¬
ases. At a certain number of cycles the rupture stress becomes cons¬
tant .

The ordinate of the horizontal part of the curve is the fatigue
limit.

350

Chapter ,5. Cyclic Strength

Fatigue graphs are plotted in 0 , N coordinates (Fig. 183a),
in 0 , log N semilog coordinates (Fig. 1836) and in log 0 , log N
log coordinates (Fig. 183c). The first method is now almost out of
use since it does not allow the shape of the fatigue curve to be ascer¬
tained in the region of small or large numbers of cycles. Most often
used is the semilog coordinate method.

For most structural steels the fatigue limit becomes evident at
1-10 megacycles. These figures are taken as the basis for determining
the fatigue limit of steels (the so-called basic number of cycles).
For non-ferrous alloys (e.g., aluminium alloys) the number of load
alternation cycles required to determine the fatigue limit is substan¬
tially higher (50-100 megacycles). Even after this number of cycles
a further drop of fatigue limit is often observed, which shows that
in reference to some metals the fatigue limit, as formulated above,
does not exist. In this case the nominal fatigue limit is determined
as the stress not causing specimen failure within a certain number
of cycles (usually 50 megacycles).

There are also no clearly expressed fatigue limits for contact stres¬
ses, for cyclic loads in conditions of increased temperatures, and
for components working in a corrosive medium. Under such condi¬
tions the rupture stress value will continuously fall with the increase
in number of cycles. The absence of a clearly defined fatigue limit
for large-sized parts is also noticed.

(a) Stress Cycles

Four basic types of stress cycles are distinguished:
symmetrical alternating — the highest and the lowest stresses
are opposite in sign and identical in value (Fig. 184a);

asymmetrical alternating — the highest and the lowest stresses
are opposite in sign and differ in value (Fig. 1846);

pulsating— the highest and the lowest stresses have the same sign
but differ in value (Fig. 184c, d );

complex —various combinations of the above-described cycles
(Fig. 184e~g).

The cycles have the following fundamental characteristics:

Umax —the greatest algebraic stress value in a cycle (tensile stres¬
ses are considered positive, compressive stresses—negative);

Omin—the smallest algebraic stress value in a cycle;

o m =£saS5i2s!fl — mean stress of a cycle;

amplitude of stresses in a cycle (the value 2 <s a is ■ termed
cycle stress fluctuation swing);

cycle asymmetry ratio (cycle stresses are taken with alge¬
braic sign).

„ _Uinax-“Oinin

o a --

g min

Chapter 5. Cyclic Strength

351

For a symmetric cycle r — —4; for asymmetric —1 < r < 0;
for pulsating (non-symmetric, of fixed sign) 1 > r > 0; for pulsating
in which the maximum or minimum value equals zero, r — 0.

The fatigue limit for symmetric cycles is indicated by: in flexure
(bending) a_i; tension-compression torsion for pulsating

cycles respectively cr 0 , a op and to-

The most used method for determining the fatigue under sym¬
metric cyclic flexure is that of Weller. A cantilevered or simply-sup¬
ported sample revolving at a constant speed about its own axis is
loaded by a constant directional force. For each revolution all points
on the sample surface in the critical section pass once through the
zone of maximum tensile stress and once through the zone of
maximum compressive stress, making complete cycle of sym-

.352

Chapter 5. Cyclic Strength

metrical alternating flexure. The cycle frequency is equal to
the number of revolutions the specimen makes in unit time; the
number of revolutions prior to failure is the number of cycles causing
failure.

Such a kind of flexural loading (termed circular bending), is natural
for many engineering parts (e.g., gear shafts, belt and chain drives,
etc.).

It should be emphasized that under such type of loading the mate¬
rial behaviour will radically differ from other types of repetitive
bending (loading a stationary part with a symmetric cyclically
alterating load invariant in direction). In the latter case the fatigue
loading will be exerted to only two diametrically opposite zones in
the plane of the acting bending moment, whereas with circular bend¬
ing all peripheral zones of the section will be successively loaded.
This cannot but affect the sample life. The compressive-tensile stres¬
ses, while travelling over the sample periphery in a crescent-encom¬
passing like motion, will inevitably involve the entire periphery of
the sample; under these circumstances each point on the sample sur¬
face, taken across the critical section, will be subjected (apart from
maximum stresses emerging when the point passes through the bend¬
ing moment plane) additionally to the action of stresses, which are
successively arriving and leaving during sample rotation.

Moreover, the stresses occurring under circular bending, while
overlapping fully the periphery of the sample section, will “feel”
the weakest points, which may give origin to fatigue cracks; in
a stationary sample the weak points are not necessarily found in the
bending moment plane.

On the other hand, with circular bending, those portions of material
which are leaving the loaded zones are subjected to periodic thermal
relief. Under plane bending the loaded portions are constantly under
load.

( b ) Limited Durability

The left-hand'descending branch of fatigue curves corresponds to
the area of limited durability. It enables the durability to be deter¬
mined (in cycles), which will be typical of components loaded with
stresses surpassing the fatigue level, or with such stresses which will
he ultimate for a given durability.

Fatigue curves in the zone of limited durability can he, within
-certain limits, expressed by the equation

o m N = C (5 1)

>or

N m

where N = number of cycles
m = exponent
C— constants

The values of m and C can be derived from q d and N 0 (i.e., fatigue
limit and number of cycles corresponding to such limit), as well as
from the original values of 0 * and JVj (i.e., original stress, close to
the yield limit cr y and preset number of cycles, Fig. 185).

Fig. 185. Determining exponent m of a fatigue curve

For these two points

oTN^C

Having equated Eq. (5.2) and Eq. (5.3) we obtain

oTN^o^No

or

/ Pi N 0

\ <y D ) N i

Having taken the logarithm, we find that

log

m--

N 0

N,

log

o D

(5.2)

(5.3)

23-01395

354

Chapter 5. Cyclic Strength

Substitute the value of tn in Eq. (5.3), then

log

Wo.

Ni

log

<H

C = N 0 Cj)

In log coordinates factor m is equal to cotangent of angle a, the
inclination of fatigue curve descending branch relative to the abscissa

m-.

log

log

No.

N t _ log JY 0 — log Ni
gl ~ lOgOi — log 0D

— cot cc

Generally the log scale for stresses is larger than the scale for
the number of cycles. In this case

m= log iy 0 —log Ni
a (log log a D )

where a is the coefficient accounting for differences in scales.

The values of m depend on material properties and shapes of
components. On the average, for plain components m — 8-15, while
for components with stress concentrators m = 3-8. The m-parameter
may, to a certain degree, be regarded as a resistance criterion of
material fatigue. The smaller the m value (i.e., the steeper the fatigue
curve slope), the shorter the service life of parts at stresses exceeding
the fatigue limit and, as a general rule, the lower the fatigue limit.

Generally, components designed for a limited service period are
those made of materials lacking a definite fatigue limit or having
a steeply dropping fatigue curve (concentration-sensitive materials),
as well as the parts which, because of size or weight considerations,
cannot be given dimensions determined by the fatigue limit. Conside¬
red in the same way are machines and mechanisms operating under
low-cycle conditions and mechanisms in which operative periods
alternate with prolonged downtimes or operation under low loads
(e.g., hoisting machines of intermittent action, etc.), i.e., mechanisms
in which the total number of cycles throughout the entire service
period is less than the number of cycles corresponding to the fatigue
limit.

Parts subjected to high-frequency loads of continuous action are
considered against the fatigue strength limit with the necessary
safety factor. The service life of such parts is sharply reduced if
the fatigue limit is exceeded.

Assume that a mechanism works at 2000 rpm, i.e., its parts are
subjected to 2000 cycles per minute. Its service life limit is determined
according to Eq. (5.1) from the relation

Chapter 5. Cyclic Strength

355

Let N 0 = 10® cycles, m — 5. From Eq, (5.5) we find that at
stresses reaching 1.5, 1.2 and 1.1 times that of the fatigue strength

limit ^-^- = 0.666; 0.833; 0.91 j , the service period will be equal

to 1 h, 3 h 20 min and 5 h, respectively. At stresses equal to the
fatigue limit the service life becomes infinite. Thus, even the most
insignificant increase of designed stresses over the fatigue strength
limit severely lowers service life and does not save weight or size.

(c) Fatigue Limits,

Fatigue strength limit cannot be regarded as a constant characte¬
ristic inherent in a given material. It is actually subject to much
greater fluctuations as compared with mechanical properties of
material under static loading. The magnitude of this limit is depen¬
dent on the conditions of loading, type of cycle (particularly on the
degree of its asymmetry), test methods, shape and absolute dimen¬
sions of a part, manufacturing conditions, state of a surface, etc.

Thus, it is more correct to say that when testing standard samples
for fatigue it is not the fatigue strength of a material but the fatigue
limit of a sample made of this material which is being tested. When
passing from a test sample to a real part it is necessary to introduce
corrections which consider the shape and size of parts, their surface
condition, etc. Therefore, we may speak about fatigue strength
of parts.

In this connotation the fatigue strength limit implies a rather
different notion in contrast to its original meaning of a material
characteristic, although the fatigue limit determined by testing
standard samples is still quoted as one of the basic strength properties
of materials.

Now we speak also of fatigue strength of units (threaded, press-
fitted and other assemblies). Thus, the fatigue strength notion inclu¬
des not only the factors of material properties and geometry of parts,
but also the interaction factors of adjacent components.

Full-scale tests become more popular in which service periods
and fatigue strength limits of parts and assemblies are determined.

Flexure fatigue limits have their minimum value during a sym¬
metrical alternating cycle, increase with the growth of cycle asym¬
metry, rise in the range of pulsating loads and approach to the values
of material static strength with the decrease of pulse ampli¬
tudes.

Fatigue strength to tension-compression is approximately 1.1-1.5
times more, and in torsion 1.5-2 times less than for the case of a sym¬
metrical alternating flexure.

No definite relationship exists between the characteristics of
fatigue and static strength. The most stable relationships exist

23 *

356

Chapter 5. Cyclic Strength

between. cr_i (fatigue limit in flexure with a symmetric cycle) and
Gb (ultimate strength) and also 0 O . 2 (yield limit under static tension).
Experiments give the following correlations:
for steel

o+— (0.2 — 0.3) o 6 (l + -^-)

for steel castings, high-strength cast iron • and copper alloys

o_j = (0.3 — 0.4) 0&

for aluminium and magnesium alloys

0_i = (0.25 — 0.5) o &

for grey cast iron

G-i — (0.3 — 0.6) Gb

Schimek obtained the following graphical relationships between
fatigue strength and tensile strength G b (Fig. 186) found by analyzing

the results of fatigue tests
kg f/mm carried out on refined struc-

g 0 _ZZ tural steels:

/ for tension-compression un-

80 —- y/ - der a symmetrical cycle

1 k/1 I 0_i = O.330 6 + 1.25

mmmumfz

go 60 80 WO no ~c b , kgf/mm z

for tension-compression un¬
der a pulsating’cycle

jo 0 =0.5806 +2.3

for flexure under a symme¬
tric cycle

0„ 1 6 = O.40 b + 5.7

for torsion under a symmet¬
rical cycle

t_ 1= = 0.206+ 4.8

Fig. 186. Fatigue limits for various eye- , * or torsion under a pulsa¬
tes of loading as a function of tensile ting cycle

strength cr 6 {after Schimek)

= 0.2506 + 24.2

Under a symmetric cycle the fatigue limits are related together
by the following tentative relationships

0_i = (1 — 1.5) 0_ lp

t_i = (0,5—0.7) 0_ 4

Chapter 5. Cyclic Strength

357

Under a pulsating and alternating symmetrical cycle the fatigue
limits are connected together by the following approximate rela¬
tionships:
in flexure

o 0 = (1.4 —1.6) o_j

in tension

a 0p = (1.5 —1.8) c_i p

under torsion

— 2) t:_]

The fatigue limits for asymmetric cycles can be approximately
determined by an empirical relationship between maximum stress
a m ax of cycle, the mean cycle stress a m and ultimate cycle ampli¬
tude G a .

One of such relationships

Omax = 0„jp j^l ) ]H" a ra

and

°*= <r -*d 1 -(w) 2 ]

where cr 6 is the static ultimate tensile strength.

These correlations give some idea about general relationships.
For calculations it is necessary to refer to technical literature on
fatigue strength.

(d) Generalized Fatigue Diagrams

The interrelationships between the fatigue strength, mean stress
of cycle and coefficient of cyclic asymmetry are shown in the form of
generalized diagrams. The most used are Smith’s diagrams (Fig. 187).

The line of the mean stress cycle g m = t which is at the

same time the zero line of amplitudes, is drawn at 45° to the abscissa;
the stress scale is plotted on the ordinate axis. Plotted on the zero
line are the stress amplitudes found from experiment and safe for
each given value of a m and characterized by the stress values a max
and cr min . The ABC envelope of the o max points represents the
fatigue limit in tension and the DEF envelope of the (—o m ax)
points, the fatigue limit in compression.

The higher limit for cr niax is assumed to be the tensile yield limit
cr y tens (line BC) and for (—a max ), the compressive yield limit
Gy compr (line DE ).

The coefficient of the asymmetry cycle r — —— f or arbitrary

Umax

point a is determined as a relation of lengths ~ (the first correspond-

Chapter 5. Cyclic Strength

359

ing to a ml) , and the second, to a mas )■ Each length is taken with
its own sign.

The magnitude of r may also be determined from

2 A

r — -- -1

tan a

where a is the incidence angle of a beam connecting point a with
the origin of the coordinate axes.

By arranging the Smith’s diagrams for definite materials and
load forms it is possible to perform fatigue calculations for any value
of the asymmetry cycle coefficient.

Fig. 188. Smith’s diagram plotted for Fig. 189. Fatigue limits in eircu-
circular bending, cyclic tension-compres- lar bending, cyclic tension-comp-
sion and cyclic torsion ression and cyclic torsion as a func¬

tion of cycle asymmetry coeffi¬
cient r

Shown schematically in Fig. 188 is a Smith’s diagram for a struc¬
tural steel subjected to three kinds of loads: circular bending, cyclic
tension-compression and cyclic torsion. Diagrams for bending and
torsion are plotted only on one side of the ordinate axis as they encom¬
pass in that region all possible kinds of stresses.

In practice it is more convenient to employ diagrams depicting
fatigue limits for different kinds of loading as a function of the
cycle asymmetry coefficient r (Fig. 189) since these diagrams con¬
tain, in concise form, the same data as Smith’s diagrams.

(a) Damage ability Curves

The value of fatigue strength is affected by overloads which the
part is subjected to before loading. One method accounting for such
overloads, suggested by French, consists in plotting damageability

360

Chapter 5. Cyclic Strength

(French’s) curves. The method consists in preliminarily loading
the samples with stresses exceeding the fatigue limit at different
number of cycles and consequently testing the samples at stresses
corresponding to the fatigue limit level.

Let samples be subjected to a test stress which is 1.5 times the
fatigue limit at 10 4 ; 5*10 4 ; 10 5 ; 5-10 s , etc. cycles. In the course
of the following fatigue test some samples when subjected to an overs¬
tress duration, say, higher than 10 3 cycles,fail; while other samples,
subjected to overstressing for a smaller number of cycles remain
sound. This means that at a number of cycles greater than 10 5 some

irreversible damages occur in the metal, which make the part incap¬
able of further cyclic loading, even if the stresses are maintained at
the fatigue strength limit. On the other hand, loading duration less
than 10 5 cycles is safe. The point corresponding to the stress equal
to 1.5 times the fatigue limit and duration of 10 s cycles is plotted
onto the fatigue diagram. The locus of such points taken for diffe¬
rent overstress levels and corresponding safe cyclic durations clearly
outlines the zone of safe overloads on the diagram.

A typical damageability curve (in log c vs. log N coordinates)
is shown in Fig. 190a. Overloads positioned below damageability
curve 2 are safe, while those between curves 1 and 2 are inadmissible.

The closer curve 2 to curve 1, the better the ability of material
to withstand overloads. For some high-strength materials subjected
to efficient heat-treatment, the second curve 2 of French practically
coincides with the slopes of the first curve 1 of Weller. In some other

Chapter 5. Cyclic Strength

361

materials (e.g., annealed carbon steels) the third curve 3 of French
is the continuation of the horizontal portion of Weller’s first curve 1.
This means that such materials are quite incapable of sustaining

Fig. 190&. Relationship between admissible shear stress i and tension stress cr
for biaxial state of stress (symmetric bending and torsion)

overloads. For this reason, parts made from these materials must
he calculated by the fatigue limit, even in the region of limited
durability.

(/) Fatigue Strength in the Case of
Complex States of Stress

The problem of fatigue strength under complex states of stress
is still not fully studied. Investigated better than others is the biaxial
state of stress in which simultaneously act symmetrically varying
cyclic stresses, both normal and tangential (cyclic tension-compres¬
sion and torsion, cyclic flexure and torsion).} Found experimentally
for the case the ultimate values of normal stresses and ultimate
values of shear stresses x uU can be formulated through an ellipse-
type relation:

where g- 2 and t_ x are fatigue limits under pure tension-compres¬
sion and pure symmetrical torsion, respectively.

At a given value of a u it the admissible magnitude of concurrently
acting shear stress

362

Chapter S. Cyclic Strength

and, conversely, at a given value of x uli the admissible magnitude
of simultaneously acting tensile-compressive stress

This relation is shown graphically in cuiwe

Any combination of stresses positioned between the limiting curvp
x ,.—a u and axes of coordinates (e.g., point a) is safe, t he mar B
of safety for any"combination can be determined by plotting a net¬
work of curves of equal safety with the magnitudes of _T-i and <j_i
rcre^rproportionally to the safety coefficient n(ttan^
Diagrams'Jike those presented m Fig. 1906 have been plotted tor
syxnmetrical>lternating cycles with cophasally varying values o
x and a. The regularities typical of these diagrams appb’ a so
asymmetrical cycles, as well as to the cases of acophasal variations

Th/fatigue strength behaviour under non-stationary conditions
of t and /variations, as well as in triaxial stressed conditions has
so far been studied insufficiently.

(. g ) Effect of Load Character on the Fatigue Limit

The effect of cycle frequency and speed of stress Ranges within
a cvcle on the fatigue limit has not been completely studied.

It has been proved that increasing the number of cycles per unit
time increases fatigue strength, particularly noticeable at frequencies
greater than 1000 cycles per minute. For some materials the follow
ing relation is established:

l

N = -V

where N — number of cycles prior to failure
p, = cyclic frequency

ImpwvTm C e°nt i/the fatigue limit with increasing cyclic frequency
mav be attributed to the fact that plastic strains occur at a low rate
(hundreds of times less than the speed of elastic strains equal, as is
fen to the Tonic speed in a given medium).
of cycles suppresses plastic strains in microvolumes of metal preced

‘“C mTSons If&Trt We include: fatigue under

cyclic impact load (impact fatigue), fatigue Y/^'/Ho/c/Jyvarying
(contact fatieue), fatigue under increasing and periodically varying
temperatures (thermal fatigue) The mecbmism oSJa ,gMjgwgf
under the above-described conditions is still not fully meg

Chapter 5. Cyclic Strength

363

(h) Nature of Fatigue Failure

Fatigue failure is the result of repetitive and quickly alternating
elastic and elastic-plastic deformations distributed, due to non-
homogeneity of the material, non-uniformly through the volume
of the part. Initial failures arise in the microvolumes unfavourably
orientated relative to the acting loads, prestressed by residual
stresses and weakened by local defects. Such local damages, as they
gradually accumulate and pile up, may initiate the process of com¬
ponent general failure.

In the fatigue failure processes a great role is played by the sources
of liberated heat in the micro volumes undergoing deformations.
As a result of increased temperature the strength of material in the
microvolumes lowers. This facilitates the inception of additional
plastic shears which, in their turn, cause further rises in temperature.

The greater the heat liberated in the microvolumes, the larger
the stress amplitude and the less the cycle asymmetry factor. On
the other hand, the magnitude of local rise of temperature is depen¬
dent on the properties of material and on its structural constituents.
The greater the increase in temperature in a microvolume, the less
the heat-conductivity and heat-capacity of the material and the
higher its cyclic toughness which (at the stage of elastic deformations)
determines the amount of irreversible conversion of vibrational
energy into thermal energy.

This explains why the fatigue strength has its minimum value
in the case of symmetrical cyclic stresses causing the greatest oppos¬
itely directed shears. This also explains why high but short-time
overloads do not lower fatigue strength: the heat evolved in the
overstressed micro volumes quickly dissipates into the surrounding
masses of material, thus enabling the strength of overstressed volumes
to be recovered.

The initiation process of a fatigue crack has several stages. At
the initial stages of loading cracks originate at the boundaries
of crystallites (grains) as a result of displacements (shears) of packs
of crystalline planes oriented in parallel to the action of maximum
tangential stresses, i.e., directed approximately at 45° to the ten¬
sile stresses (octahedral shearing stresses). Depending on the orien¬
tation of a grain the shears can occur in one plane, or simultaneously
in two (Fig. 191 Ilia, b) or three (Fig. 191 1 lie) planes.

At a certain loading stage the mass of metal presents a mosaic
of grains subjected to plastic strains (Fig. 192), and grains free of
plastic strains thanks to more favourable orientation of crystalline
planes relative to the action of tangential stresses.

The inception of initial cracks within the boundaries of a grain
is essentially the result of a directed propagation (diffusion) of
dislocations (such as vacancies) toward the grain boundaries. The

364

Chapter 5. Cyclic Strength

rate of diffusion is proportional to the magnitude of stresses and
temperature. The diffusion process is intensified by microheating
of metal.

Fig. 191. Orientation of crystallites relative to active forces
1-11 — favourable; HI — unfavourable

The accumulation of vacancies leads to loosening of the structure,
formation of submicropores and, finally, the appearance of initial
cracks.

At the initial stages the process is reversible. As soon as stresses
discontinue (i.e., during intervals of repose), vacancies migrate

Fig. 192. State of stress in a surface layer subjected to a tensile load (heavy
lines discriminate the grains with crystal planes parallel to tangential stresses

\

in the reverse direction. As a result, cluster vacancies are gradually
reabsorbed and distributed uniformly throughout the grain microvo¬
lumes. Thereby, the material returns to its original state. The process
can be stimulated by raising the temperature. Experience proves
that initial disruptions can be cured by short-time heating.

Chapter 5. Cyclic Strength

365

If the.stresses continue to act, the disruption accumulation process
develops. Gradually propagating, initial cracks emerge to the grain
surface. Here their development stops mainly due to obstacles being
created by other crystalline orientation of adjacent grains. This
disorientation of crystalline planes leads to plastic shear displace¬
ment.

Grain interlayers due to their inherent impurities possessing
a strongly distorted crystalline lattice, sometimes different in type
from the grain crystalline lattice, serve as another obstacle.

Thus, self-formed intergrain barrier is produced which effectively
brakes crack propagation. To overcome this barrier, stress is required
significantly in excess of the stress causing intergrain shear. The
magnitude of the penetrating (break-through) stress depends on the
interlayer strength and on the degree of disorientation of the grain
crystalline planes separated by the interlayer. Obviously, interlayers
between grains with identically oriented (directed) crystalline planes
are the easiest to overcome. However, these cases are statistically
rare.

The mean value of stress necessary to overcome intergrain barriers
determines the fatigue strength of the material. The fatigue limit
can be regarded as the average stress level at which the crack nuclei
still remain within grain boundaries and are partially or totally
restored during rest intervals.

The resistance of material to intergrain shear depends on its
physical and mechanical properties and the fine crystalline structure
of a grain. Of great significance is the size of minute (some hundredths
of a micron) crystalline clusters (subgrains) comprising the grain.
Diminution of crystalline clusters and their still greater disorienta¬
tion, as well as distortions of the atomic-crystalline lattice caused
by impurities, strain hardening, precipitation of secondary phases
and inception of non-equilibrium (hardenable) structures — all these
factors enhance the resistance to intragrain shear and improve the
fatigue strength of material. Actually, it is this that the strengthen¬
ing effect of alloying, heat treatment and plastic deformation is
aimed.

Should the level of stresses related to the entire thickness of the
material, be below the fatigue limit, then the initial cracks may
practically remain for an unlimited time within individual grains
without causing significant loss in part strength. If the stresses
throughout the entire part or in part of its volumes exceed the fatigue
limit (e.g., due to local stress concentrations), then the cracks will
overcome intercrystallite barriers and extend into the mass of metal.

As soon as it extends beyond the grain boundaries, the crack
intermittently widens developing into a macrocrack which changes
the direction of its propagation extending now normally to the action
of maximum tensile stresses. The development of the crack is now

366

Chapter 5. Cyclic Strength

expedited by the onset of abrupt stress concentrations which occur
at the crack leading edge. Local failure entails heating which softens
the metal, thus, contributing to the crack propagation.

A macrocrack can extend under the action of stresses well below
those necessary to overcome the interbarrier resistance, in addition
the stresses necessary for crack propagation lessen as the crack grows.

Having reached the part surface, the crack then starts to penetrate
into the depth of material progressing along the material weakest
regions.

Simultaneously, a large number of cracks develops. However, at
a definite'stage,^the process is localized: mainly one or a group of
local cracks expands, having outstripped the rest in their develop¬
ment by virtue of material defects concentrated at the given place,
localized prestressed rupture, or on the strength of unfavourable orien¬
tation of crystals relative to the acting stresses.

The adjacent cracks blend together forming a dee], branched system.
No new plastic shears or cracks come into existence, while those
having had time to occur either discontinue or slow down their deve¬
lopment as all strains are taken up by the main crack. Propagation
of the main crack results in the part failing owing to decrease of its
net-section.

Contrary to the first stages in the appearance of intergrain and
intragrain cracks developing over a long period final failure begins
abruptly and has the form of brittle fracture.

Fatigue fractures generally display two sharply visible zones.
The zone of fatigue crack propagation has a dull porcelain-like sur¬
face typical of fractures in which transcrystallite failures are predo¬
minant (the so-called state-pattern fracture). Crack edges often show
smooth, shiny work-hardened regions—the result of impacts, crushing
and abrasion of crack walls during the periodic deformations of the
material.

The zone of final failure has a crystalline surface typical of brittle
fractures in which intercrystallite failures are predominant (e.g., im¬
pact fractures and fracture of brittle materials). A streak-like pattern
is usually seen in the zone of failure, i.e., a series of parallel lines
which are, in fact, traces of intermittent advance cracks in proportion
to the cycle accumulation of alternating loads.

As a rule, initial cracks originate, for all forms of loading, in the
surface layer whose thickness, on the average, does not exceed three
grain diameters (which amounts to 0.05-0.2 mm for steels). Usually
cracks occur in grain fragments located on the surface and cross¬
cut during preceding machining.

Thus, for fatigue strength the surface layer plays the decisive
part. It is particularly important because in the majority of loading
cases (flexure, torsion, complex stressed states) it is the surface
layer that is subjected to the maximum stresses.

Chapter 5. Cyclic Strength

367

A number of reasons explain the particular role of the surface
layer.

Firstly, the pure physical factors should be noted. As known from
the physical regularities, the packing of atoms in the surface layer
is closer than those lying below.

As a result of the interaction with the underlying less densely
packed layers, in the surface layer there develop tensile stresses
giving rise to loosened spots which are potential sources of the crack
formation.

Secondly, the metal particles which emerge to the surface, posses¬
sing only one-way metallic bonds with the underlying metal, have
a greater activity and easily combine with particles of the surround¬
ing medium. On the bare surface of the metal there are formed stable
adsorbed films of vapours, gas, moisture, oxides, etc. which cannot
be removed by mechanical or chemical methods. Penetrating deeper
into the depth of the material through microcracks, the adsorbed
films disturb the solidness of the metal, and weaken the surface
layer. Of much significance is the cleaving action of thinnest films
of surface-active agents infiltrating into submicroscopic slits on
the metal surface. With slits widths of the order of hundredths of
a micron the films may build up extremely high pressure (occasion¬
ally reaching hundreds and thousands of atmospheres), and contri¬
bute to metal failure.

Thirdly, some processing factors must be mentioned. The surface
layer inevitably, to a greater or less degree, is impaired by the prec¬
eding processing. Machining, even the finest, inevitably causes
radical changes in the surface layer. Being actually a combination
of plastic deformation and destruction of metal, the machining pro¬
cedure is accompanied by grain cutting, breaking and tearing of
individual grains, appearance of microcracks and onset (in the sur¬
face layer and adjacent regions) of high residual rupture stresses,
close to the material’s yield limit. The heat evolved in the machining
process induces partial recrystallization of the surface layer and some¬
times is accompanied with phase and structural changes. 1

The heat treatment process is often followed by surface layer
decarburization, decomposition of pearlite and cementite with the
formation of unstable ferrite scale.

Fourthly, the metal surface is attacked by all kinds of corrosion
encountered in practice and causing deep damages to the surface
layer. Usually corrosion spreads along grain interlayei’s and micro¬
cracks.

Any surface working in friction is subjected to one more form
of weakening-—wear. Wear lowers fatigue strength significantly and
is accompanied by changes in microgeometry and disturbances in
the surface layer structural pattern.

Thus, concentrated in the surface layer are numerous and diverse

368

Chapter 5. Cyclic Strength

submicro-, micro- and macrodefects, which are caused by mechanical,
physical and chemical factors unavoidable due to the technical con¬
ditions of producing the surface layer as well as the particular role
of the surface layer as the surface separating the metal from the
surrounding medium. One may rightfully state that the surface
layer of each part is a stress concentrator (or stress raiser), whose
effect can be lessened by a number of measures but which can never
be eliminated completely.

All those factors, that disturb the continuity and homogeneity
of the surface layer causing the appearance of higher tensile stresses,
promote the onset and development of initial cracks and sharply
lower the fatigue strength of a material. Conversely, compaction of
originally loose structure of the surface layer and creating prelimi¬
nary compressive stresses in it even at small depths (shot-blasting,
rolling) materially enhance the resistance of material to cyclic loads.

Strength under cyclic loadings can also be improved by removing
the defective surface layer by some techniques which do not introduce
further damage (microfinishing, polishing). It has been found that
deep repolishing of specimens in the course of fatigue tests sharply
improves their service life. This is attributed to the partial removal
of the surface layer together with the initial cracks formed in it
at the previous testing stage and packing of the surface layer by
repolishing and partial curing of the existing microcracks.

Therefore, the improvement of fatigue strength requires first
of all work-hardening of the surface layer. This is obtained by chem¬
ical heat treatment processes, surface thermo-diffusion alloying,
surface strain hardening, etc. Of much importance is the elimination
of macro- and micro-defects in the surface layer, particularly defects
brought about as a result of machining.

In hollows (e.g., tubular components) being subjected to tensile
or complex stresses in which tensile stresses are predominant, the
conditions of the inside surfaces are no less important than those
of the outside surface. In such components the inside surfaces should
be subjected to strain hardening and carefully checked for defects.

Experience proves that fatigue strength (in contrast to static
strength) only slightly depends upon grain size, which seems at
first glance paradoxical: fine-granular metals with deep strengthening
lattice of cleavage surfaces would appear to resist cyclic loads better
than metals possessing coarse grains and loose lattice. In actual
fact this phenomenon is quite natural.

Fatigue strength is determined by the stress necessary to overcome
the initial intercrystallite barriers. As soon as these barriers have
been broken through, the propagation of a crack is materially facili¬
tated. Widening, the primary crack propagates in a way typical of
macrocracks, easily crossing all the following barriers (at moderate
temperatures, usually in a transcrystallite mode).

Chapter 5. Cyclic Strength

369

(i) Stress Concentration

The fatigue strength of parts is seriously aggravated by the pre¬
sence of weakened spots, sharp transition areas, entering angles,
etc., leading to local stress concentrations. The abrupt changes of
stresses occurring at weakened spots may exceed 2-3 and more
times those average values which are commonly encountered in the
same section of a part (Fig. 193a).

Inasmuch as severity of initial fatigue damage is dependent on
the diffusion rate of vacancies, while the latter is proportional to

HHHm

(a)

iff

,, i l

ll

’ ■ f 1

<h'

(h)

(d)

Fig. 193. Force flow in a part undergoing tension

the magnitude of active stresses, it is not surprising that at the spots
of stress concentrations the metal becomes loose, this generally prec¬
eding the inception of fatigue cracks. Due to this the fatigue damage
in the stress concentration zones outstrips damages in the remaining
parts of the piece.

The intensity of stress increase depends primarily on the kind
and form of weakening. The greater the section gradient at the part
of transition and the sharper the transition, the higher the local
maximum stress.

The stress concentration phenomenon has been carefully studied —
both theoretically and experimentally.

Given below is a simplified description of how stress concentra¬
tions occur. The concept is based on the phenomenon of a distorted
force flow which occurs in a weakened zone. Though not reflecting
all the complexity of the picture, the suggested scheme seems to be
rather illustrative and true, thus enabling certain practical conclu¬
sions to be drawn.

Let us assume that a beam is subjected to a tensile load P
(Fig. 1936) and that the latter is uniformly distributed throughout

24-01395

370

Chapter 5. Cyclic Strength

the entire sectional area. Through the agency of internal bonds the
load applied to each sectional point is transmitted to adjacent points.

The paths of load transmission from one point to another along
the beam body are called force lines (shown by fine lines in the figure)
and the aggregate of these lines is referred to as force flow.

The force lines are continuous and cannot be broken since this
would signify local breakage of bonds between adjacent points,
i.e., the beginning of material failure. Hence, the number of force
lines must be the same at any cross section.

The density of force flux (the number of force lines per cross section
unit area) defines the stress value. If the part sectional area reduces
(e.g., due to a central bo.re in the part body—Fig. 193c), then in

IIIHHIIIII

(a)

tmnmiM timtiMm

Hiimuui minimi!

(b) (c)

ttmiHH ii

(d)

Fig. 194. Force flow in a part with changing sections

constricted sections the force lines get thickened causing a rise
in the stresses. This is always accounted for when carrying, out strength
calculations at the design stage.

At the same time the force lines bend while passing the obstacle.
This means that under the influence of tensile loads the material
fibres positioned close to the obstacle are subjected not only to
tensile stresses but to bending as well.

In the event of an internal hole the fibres being bent by tensile
forces will tend to converge to the centre, thus producing transverse
material compression at the hole, as if contracting the latter.

These stresses when added to the tensile stresses in a weakened
region close to the hole cause a sharp stress increase (Fig. 193d).

Analogous picture is seen in the case of recesses positioned along
the sides of a part (Fig. 194a). Here the force lines are bent (curved)
in the recess area. Here the bent fibers tend to spread the sides,
causing transverse tension of the material near the recesses as if
opening them.

In the event of abrupt stepped-like changes of sections (Fig. 194b)
the force lines are also bent. At areas above the transition (shoulder)
the materia) undergoes transverse compression, while in areas below.

Chapter 5 .' Cyclic Strength

. 371

(a)

Fig. 195:

the transition the material is subjected to transverse tension. The:
fibers under tension tend to open the transition entering angles,'
thus producing local tensile stresses of higher values. The intensity:
of transverse tensile-compressive stresses is proportional to the slope
angle of the force lines.

Tensile stresses can be considerably reduced by'making transitions
streamlined (Fig. 194c, d): this decreases the force line curvature and
increases the strength of the ' - I : ■ ■ ’ • 1 ' ■ •

weakened portion.

Pictures similar to those
illustrated in Figs 193-194 are
also observed in cases of fle¬
xure and torsion in parts with
local weakenings.

From the above it is clear
that one active way of lowe¬
ring stresses in weakened- sec¬
tions is to give the transition
portion smooth outlines.

A well-known positive effect
is produced by stress deconce¬
ntrators, i.e., additional local
weak spots made close to the
chief source of stress concen¬
tration.

In a part subjected to ten¬
sion and having a stress raiser
in the form of a central hole
(Fig. 195a), stress deconcent¬
rators can be made in the
form of additional small-dia¬
meter holes located in line'
with the main hole along the

force flux direction (Fig. 1955). • ; '■■■ . . ■ •

Passing the obstacle the force lines'curve less than in the. case Of
one central hole. Consequently, the stress. concentration lowers.

Naturally, stress peaks occur around deconcentrators, however,
it is evident that the height of peaks both near the chief stress raiser
and deconcentrators is less than in the case ,of lone stress raiser;

The position of the deconcentrators is of decisive! importance. 1
They are useful only when they make for .the Straightening of the
force lines; otherwise they increase the force flux-curvature and.make
the stress concentration still more intensive. ■* i; . . ■ . PhV'

This can be easily seen in Fig. 196. With correctly positioned
notches (Fig. 1965) the deconcentrators make the force lines' more
smooth, as if taking the material of the steps’outside.<the force.flux

24 *

TMIT

tytriMir

mi

■

TTTTTiTTi

d".

t$mm

Force flow in parts

(a) with a stress' concentrator (hole); ( b ; with
stress deconcehtrators

372

Chapter 5. Cyclic Strength

area. In essence they-have the same effect (but only weaker) as would
have an improved smoothness of the transition section between the
steps and specimen body, as is shown in Fig. 1965 by dashed lines.

Incorrect position of additional notches (Fig. 196c) produces
further force lines distortion and, hence, larger stress concentrations.

Another technique usedito minimize the harmful effects of con¬
centrators consists in cold reduction of the material on the surfaces
of the concentrators and in the sections close to them. Thus, hole-
walls are made more dense by furnishing, while the surfaces around
the hole are embossed; round fillets and annular recesses are rolled
over. These procedures are aimed at producing in the material
residual compressive stresses. Adding to the local tensile stresses

WHWW mm

(a) (b)

Fig. 196. Effect of additional notches on the uniformity of force flow

occurring in concentrators after, the working load has been applied,
such residual compressive stresses materially decrease the intensity
of tensile stresses, thereby improving the strength at the areas of
concentrators.

The above-described squeezings are misinterpreted as deconcen¬
trators; their action is quite different. The purpose of deconcentrators
is to straighten the force flow lines and that of squeezings—to streng¬
then the material. This difference is very important in practice since
the arrangement of deconcentrators and squeezings is not the same:
the former are positioned in the stream of force lines before the con¬
centrator and after it, and the latter—at the focus of stress concentra¬
tion (Fig. 196d, e).

The phenomenon of stress concentration caused by the factor of
shape is aggravated in practice by the fact that the areas in which
stress concentrators are located are almost always weakened by
manufacturing conditions.

Accordingly, there are two types of concentrators: geometric con¬
centrators (form concentrators) and manufacture concentrators.

In machined parts the weakening of transition sections is caused
by cutting through the fibers which have been formed in the course
of: the hot working of blanks. In cast parts the transition sections

Chapter 5. Cyclic Strength

373

are usually weakened by casting defects which have been originated
by structural distortions during crystallization of metal and solidif¬
ication of ingot. The affected areas concentrate, as a rule, loose spots,
voids, microcracks ansi internal stresses. In forged and stamped
parts the transition sections have poor strength as a result of metal
drawing in the areas.

(/) Stress Concentrators

Figure 197 shows schematically stress concentrators typical of
such parts as plates, bars, etc. when the latter are subjected to ten¬
sion-compression or flexure.

Stress concentrators typical of cylindrical parts (e.g., shafts)
are cited in Table 22.

Stress concentrators on insides of hollow shafts are depicted in
Fig. 198.

Fig. 198. Stress concentrators in hollow shafts (indicated by arrows)

Stress concentrations can also be originated by the internal defects
of material: microvoids, microcracks, flakes, fine flaws, non-metallic
inclusions (oxides, silicides), etc.

Concentration of stresses can be caused not only by the form of
a part but also by the action of its mating part. An example is given
in Fig. 199, which shows the stress distribution in the body of a cou¬
pling bolt obtained by experiment. Owing to the bolt shape, the

374

Chapter 5. Cyclic Strength

Chapter 5. Cyclic Strength

375

highest stress is found at the transition position where the bolt
stem meets the head and is three times greater than the mean

stress 0 0 in the stem. Still
much greater stress differen¬
tial occurs in the plane of the
nut end face ( 0 max = 50 o ).

(k) Coefficient of Stress
Concentration

The enhancement of stresses
at the areas of local weaken¬
ings is expressed in terms of
the stress concentration coeff¬
icient.

There are two kinds of the
coefficients: theoretical (found
by methods of the mathema¬
tical theory of elasticity pro¬
ceeding from an assumption

of material homogeneity and Fig. 199. Concentration of stresses in
ideal elasticity) and effective a bolted connection

(found experimentally). The

latter illustrates the properties of actual materials and some other

factors which define actual increase of stresses.

(Q'tLs

6MB

®nom s(8 3 ~d 3 )

Fig. 200. Determination of nominal stresses
(a) and (6) in tension and in flexure for a stepped beam; (c) and (d) in tension and in
flexure for a beam with a bole (s=beam thickness)

The theoretical coefficient of stress concentration

k ms gmax
®nom

where 0 max = maximum theoretical stress in the weakened area
Onom — nominal stress in the weakened area, calculated for
the smallest section of the weakened area by the
general formulae of the strength of materials
(Fig. 200)

376

Chapter 5. Cyclic Strength

The theoretical coefficient of stress concentration has been defined
for most forms of weakened sections found in practice (see, for exam¬
ple, Fig. 201).

The amount of increased stress i:
deviate from theoretical values. It

Fig. 201. Theoretical coefficient of stress
concentration as a function of relative
radius Bib at conjugated step sections
{(for a stepped beam in tension)

i actual conditions can strongly
depends on the kind of loading
(static or cyclic), type of
material and its mechanical
properties.

The actual amount of inc¬
reased stress is evaluated
through the stress concentra¬
tion effective coefficient

k e = 5™L

Vnom

where 0 max —maximum actual
stress at concentration area.

With cyclic loads the stress
concentration effective coeffi¬
cient is deduced as a ratio
between the fatigue strength 0
of a plain specimen and the
fatigue limit 0 ' of a specimen
with a stress concentrator:

Under static loading the amount of stress concentration will pri¬
marily depend on material plasticity. In plastic materials the pheno¬
menon of stress concentration is rather vague. With the growth
of stresses in the weakened zone the material begins to yield and
a plastic hinge is formed there, which helps transfer forces to adjacent,
less stressed areas.

For highly plastic materials the k e parameter approaches to unity,
which means that stress concentration does not occur.

In brittle materials the favourable yield effect is absent, so that
higher stresses occur in weakened areas. As soon as local stresses
pass beyond the ultimate strength of material, the latter undergoes
brittle failure. For such materials the stress concentration effective
coefficient approaches the theoretical coefficient (k e » k t ).

There are, however, some exceptions. Thus, for instance, in the
case of grey cast irons k e — 1. This is attributed to the structural
features of the material. Grey cast irons contain flaky inclusions of
graphite which, owing to infinitesimal strength of the latter, act
as equivalent internal notches and cause numerous local stress con-

Chapter 5. Cyclic Strength

377

centrations surpassing built-in stress concentrators (boles, recesses,
etc.).

The phenomenon of stress concentration becomes more apparent
under cyclic loading, this being attributable to the specific features
typical of this type of loading. Plastic strains occurring as a result
of stress concentrations follow one another at a high frequency and
change their direction (under alternating loads), thereby gradually
upsetting the material structure and
leading to a fatigue failure.

The intensity of stress concentrati¬
on will depend on material of a part,
its chemical composition and homoge¬
neity, heat treatment, mechanical
strength of material, nature of a cycle,
kind of concentrators, state of surface
and absolute dimensions of a part.

Given below are approximate data
which show how the material influences
the effective coefficient of stress con¬
centration:

Material \ ratio

Alloy steels. 0.85-0.9

Carbon and low-alloy steels . . 0.75-0.8

Wrought aluminium alloys . . . 0.7-0.75

Castable aluminium alloys . . . 0.6-0.65

Titanium alloys.0.55-0.6

As a rule, the lower the plasticity
of the material and the higher its
strength, the more extensive the stress
concentration (Fig. 202).

However, the relationship between
the stress concentration effective coef¬
ficient and material properties is very
complex. Thus, for example, steels
with martensite and troostite structure (hardened respectively with
low and medium tempering) are less sensitive to stress concentra¬
tion than refined and normalized steels with sorbite and sorbite-
pearlite structures.

When taken within certain restricted limits, the coi'relation bet¬
ween the effective and theoretical coefficients of stress concentration
can be expressed in the following form:

k e = i + q(k t ~ 1) (5.6)

where q — coefficient of material sensitivity to stress concentration.

Pig. [202. Theoretical ( k t ) and
effective ( k e ) stress concentra¬
tion coefficients for bending of
a cylindrical shaft having an
undercut and made from mate¬
rials of different strength(after
Buchner)

1 — low-carbon annealed steel;

2 — low-carbon Shot-rolled steel;
S — medium-carbon normalized steel;
4 — medium-carbon structurally-
improved steel; S — low-carbon work-
hardened steel; 6 — structurally-

improved alloy steel

378

Chapter 5. Cyclic Strength

In respect to high-strength alloy steels the q value approaches
unity, while in relation to carbon and low-alloy steels this parameter
varies within 0.6-0.8 (higher values corresponding to steels better
in strength). For grey cast irons q — 0 (k e — 1).

Figure 203 shows graphically the ratio calculated from Eq. (5.6)

and taken in regard to different values of q and k t .

k s /k t

Fig. 203. Ratio of the effective stress Fig. 204. Increase of strength margin
concentration coefficient k e to the when, changing over to steels of higher
theoretical stress concentration coeffi- fatigue strength with different values
cient k t as a function of the coeffi- of sensitivity coefficient q. The value

cient q of the sensitivity to stress of q for the original steel q is taken

concentration (for different values at 0.6

of k t )

Highs sensitivity of high-strength steels to stress concentration
conceals their strength advantages. In many cases it is more advanta¬
geous to apply steels of moderate strength with lower values of the
sensitivity coefficient.

Assume that a part is made of steel with moderate fatigue strength
and with q — 0.6. Now let us determine the gain obtained by using
steels with greater fatigue strength and correspondingly with higher q.

With geometric sizes of the part remaining the same, the nominal
stresses a nom will also be the same. The relationship of the safety
factor (determined in terms of maximum stresses cr max ) when changing
to high-strength steels

n __ °-i g max „ ^Uk-H a nom G-i^e f5 71

n *•*-? °max k e O nom O-A

where cr.i and ol, — fatigue limits taken respectively for original

and higher-strength steels

k e and k e = effective coefficients of stress concentration,
taken respectively for original (q = 0.6) and
higher-strength steels

Chapter 5. Cyclic Strength

379

Let kt — 2.5. The relation between' k' e (for different values of q )
and initial k e (? = 0.6) is presented graphically in Fig. 203.

*The graph shown in Fig. 204 has been plotted with the aid of
calculations based on Eq. (5.7). Here the variation of n'in is given
as a function of ol 1 /o_ 1 for different values of q.

As evident from the graph, the safety factor can appreciably be
increased ( n'/n — 1.2) when applying steel whose fatigue strength
is at least 1.35 that of original steel when q = 0.8, 1.5 times when
q = 0.9, and 1.7 when q — 1. With oh/o^ values below the above
magnitudes, one may advantageously, as to the cost of manufacture,
employ steels with a moderate strength.

The effect of the kind of loading on the magnitude of the stress
concentration effective coefficient depends on the type of stress
concentrator:

for notches and transverse holes

ke tens-compr ■ k e flex • kg tors === 1 * 0.85 : 0.6o

for fillets

ke flex ' k e tens-compr ■ k e tors ~ 1.2:1:0.8

Stress concentrations depend also on the kind of cycle. The
higher the cycle asymmetry coefficient r and the greater the mean
stresses cr m , the sensitivity of q to notches and k e is reduced.

Fig. 205. Decreasing the maximum stress by lowering the nominal stress

The main influence on the stress concentration effective coefficient
is given by the form of the weakened areas. Coefficient k t , similarly
to^coefficient k e , abruptly drops with the improvement in transition
smoothness and lessening sectional gradient between non-weakened
and weakened areas of the part (see Fig. 202).

Another way of increasing strength of weakened areas is reduction
of nominal\ stresses at their concentration area.

380

Chapter 5. Cyclic Strength

Since <? max = k e a nom < the maximum stresses at the area of con¬
centration (for a given k e ) depend on the value of a no m• Hence,
all measures, contributing to the reduction of nominal stresses
(larger sectional areas and higher moments of resistance where
concentrators are positioned) lower the maximum stresses.

For a simple example we take a beam with a central hole (Fig. 205a).
With k s remaining constant it is possible to halve the maximum
stresses "by doubling the beam thickness s (Fig. 205b). Approximately
the same result is obtained when increasing the beam strength
locally with the aid of bosses positioned each side of the hole
(Fig. 205c).

Strength of weakened areas can substantially be improved by
strengthening processes. Some forms of processing (nitriding, shot¬
blasting, etc.) practically paralyse stress concentrations even in
concentration-sensitive steels.

(1) Scale Factor

Fatigue strength becomes lower with increase in part dimensions. ’
It is usual to characterize the influence of details’ sizes by the
scale factor coefficient which is the ratio between the fatigue limit a

of a given specimen to the fatigue
limit Oq of a laboratory specimen
of small sizes (sectional diameter
5-10 mm)

a

Figure 206 presents averaged va¬
lues of the scale factor coefficient
for structural steels.

The fatigue strength drops parti¬
cularly sharply in the range of sec¬
tional diameters from 5 to 100 mm.
With further increase of dimensions
the scale effect is smoothed out.

The influence of dimensional cha¬
racteristics on the fatigue strength
of a rectangular section specimen from normalized carbon steel
in the conditions of unilateral bending is illustrated graphically
in Fig. 207. A specimen of the cross-sectional area F — 60 mm-*
and width b — 11 mm was assumed as a 100% strength reference
piece. With the increase of the sectional area up to 10 4 mm 2 (b =
= 140 mm) the fatigue strength fell to 55% of the original value.
The presence of weakening holes magnify the scale effect to a greater
extent the less the hole diameter. With a hole diameter d = 0.1 b

Fig. 206. Scale factor as a fun¬
ction of part diameter (average
values (after Lehr, Faulgaber and
Peterson)

Fig. 207. Effect of size on fatigue strength (after Kermes and Biizek)

Fig. 208. Force flow lines in a stepped specimen
(a) small-size specimen; (b) geometrically similar larger specimen

382

Chapter 5. Cyclic Strength

the fatigue strength, when F = 10 4 mm 2 , comprises only 30%
of the original value.

Several assumptions have been put forward to explain the influ¬
ence of dimensional sizes upon the fatigue strength of parts. Accord¬
ing to statistical theory the enlargement in part size enhances prob¬
ability of heterogeneities and internal defects.

From the technological point of view advanced in the first instance
are the difficulties in obtaining uniform strength throughout the
entire section of large parts, as, for example, during hot forming
and heat treatment.

The increase of part size definitely makes stronger the influence
of stress concentrations. This can be understood from the distribu¬
tion of force flow lines in a stepped part subjected^ tension (Fig. 208).
Should the dimensions of a part be increased, while preserving fully
the geometric similitude and with equal stresses (Fig. 208b), then
the flow of force lines will change (with the same force line spacing):
in the step zone the force lines curve more sharply than in the smaller
part, thus indicating the enhancement of stress concentrations.

(m) Surface Condition

The fatigue strength strongly depends on the condition of surface,
particularly in those cases of loading, when the severest stresses

occur in subsurface layers
(flexure, torsion and comp¬
lex stressed states).

Rough machining, cau¬
sing plastic shear, tears
and microcracks in the sur¬
face layer sharply lowers the
fatigue limit. Finish machi¬
ning (polishing, superfinis¬
hing, etc.) improves fatigue
strength. This feature is
particularly expressed in
2 small-sized parts.

Fig. 209. Fatigue strength of steel specimens
depending on the surface condition and ten¬
sile strength of the steel

1 — polished specimen; 2 — ground specimen;

5 — rough-turned specimen; 4 — specimen with
a circular notch; s — specimen with f scaly surface;

6 — specimen subjected to corrosion in fresh water;

7 — specimen subjected to corrosion in sea water

Fatigue strength is adve¬
rsely affected by chance
scratches and damages to
the surface layer and also
surface wear. The fatigue
strength is also badly affec¬
ted by corrosion.

Figure 209 illustrates the fatigue strength of steel specimens
subjected to various kinds of machining and surface damages as
a function of tensile strength o&. For unity the fatigue strength

Chapter 5. Cyclic Strength

383

of a polished specimen of steel with tensile strength a b = 30 kgf/mm 2
was taken. From the graph it is seen that the influence of surfaces
damages grows with the increase in material strength, which indi¬
cates the higher sensitivity of these materials to stress concentrations.

( n) Other Factors

Pressed, tapered and clamped connections with high compressive
stresses on the hearing surfaces lower fatigue strength. Especially
sharp is the fatigue strength drop in the compressive stress interval
up to 4 kgf/mm 2 (Fig. 210).

With further increase of
pressure the effects of cold
working decrease.

Galvanized metallic coa¬
tings negatively influence
fatigue strength, particu¬
larly those of hard and
durable metals (chrome,
nickel). Coatings of plastic
metals (copper, zinc, cad¬
mium, tin, lead) have
almost no effect upon the
fatigue strength.

Reduction of fatigue
strength, caused by elect¬
rolytic coatings, is mainly
due to hydrogen embrittle¬
ment of metal — both of
the part and coating.

In the course of electrolytic deposition the metal coating is satu¬
rated with hydrogen, acquiring a closely-packed hexagonal lattice
typical of ; metal-hydrogen compounds. This induces considerable
tensile stresses in the surface layer (6-15 kgf/mm 2 for chrome and
nickel platings).

Furthermore, the fatigue strength of metal coatings is generally
less than that of the metal part.

For all these reasons the initial cracks occur first of all, in the
plating and then they propagate into the depth of the part.

Due to the yield in plastic metal coatings significant stresses
do not occur.

The fatigue strength of chrome- and nickel-plated parts can
substantially be increased by annealling at 400-500 °C: this lowers
residual tensile stresses in the surface layer. The most efficient
way to - improve fatigue strength is to strain-harden the surface
layer before coating and also to strain-harden the coating by shot-

0 1 2 3 4 5P c ,kj}f/mm'<

Fig. 210. Fatigue strength of press-fitted
connections as a function of unit pressure p c
on contacting surfaces

384

Chapter 5. Cyclic Strength

blasting or rolling, which results in the inception of compressive
stresses, Combined employment of the aforelisted technique will
practically eliminate the weakening effects of electrolytic plating
and even improve the fatigue strength as compared to the original
value inherent in the material in the non-strained state.

(o) Fatigue under Non-Stationary Loading Conditions

Tests on fatigue according to Weller and damage tests to French
are carried out under stable time conditions and continuously
acting cyclic loads. This kind of load is typical of several machines,
working continuously in a constant regime (stationary power motors,
electric generators, machines built into automatic continuous pro¬
duction lines, etc.). The majority of machines operate in changing
conditions with regular or irregular alternating cycles and different
stress levels (transporting, construction, road-building and hoisting
machines, machine tools, presses, hammers, etc.).

Depending on working conditions, the level of stresses can vary
within broad limits (idle running, normal loading, overloads).
In the course of operation some machines may be subjected to rather
high overloads, which cannot only exceed the fatigue limits, but
also often go beyond the material ultimate yield limit, thereby
causing plastic deformations in the part.

Obviously, the fatigue limits, determined in tests under stationary
conditions of cyclic loading are inapplicable to machines running
under non-stationary conditions. The problem of fatigue strength
under non-stationary working conditions is at present studied
most attentively.

Non-stationary loading conditions may occasionally be schemat¬
ically presented as operating in an alternating definite continuous
group of cycles (block stresses) with rest periods (work —rest).

In the general case the conditions of non-stationary load are
defined by the following parameters:

kind of load (tension-compression, bending, torsion); type of
cycle (alternating symmetrical or asymmetrical loads, pulsating
cycles);

number of cycles in a block (block size);

relative duration of work and rest periods;

regularity of stress changes within the block;

the value of maximum stress in a block and its relationship with
the original fatigue limit under stationary conditions (degree of
overload);

frequency of block (degree of load periodicity);

frequency of cyclic stresses within the block.

The combination of these parameters are named the load spect¬
rum.

Chapter 5. Cyclic Strength

385

Some of several possible load spectra are presented schematically
in Fig. 211. A special kind of non-stationary loading is presented
by the block alternation of cyclic loads with periods of static loads
(Fig. 211s, /).

Fig. 211. Schematic stress spectra under non-stationary loading conditions
(a) and (i) pulsating sign-constant cycles; (e) asymmetric alternating cycle; <c0, (e), (/)
and ( g ) symmetrical alternating cycles; Qi) complex loading with random alternation of
cycles; (i) and 0) combination of cyclic loading with periods of static loading

When testing -under non-stationary conditions, the stress spectrum
is given on the basis of possible or actual working conditions. The tests
are undertaken at a changing value of some dominating factor (mostly
the one, showing the degree of overloading). In this way a network of
secondary fatigue curves is obtained, biased relative to the primary
fatigue curve (i.e., a curve, typical of the stationary load).

35-01395

386

Chapter 5. Cyclic Strength

Depending on the value of overloads, type of cycle, periodicity
size of specimen and other factors, alternating loads may act either
as strain-hardening or strain-softening ones, and have an effect on
the secondary fatigue curves (Fig. 212). When secondary fatigue

Fig. 212. Secondary fatigue curves in circular bending for various overloads
(specimens made from steel grade 40)

Curves 1 — plain cylindrical specimens; curves 2—specimens with annulas recesses. Heavy
lines distinguish initial fatigue curves

curves are biased upwards and to the right this will signify strain¬
hardening of the material and enhancement of service life period
within the limits of restricted durability. Opposite biasing (down
to the left) indicates strain-softening of material and a reduction
in restricted durability.

The fatigue strength limit is slightly increased when operating
under short-time overloads about 1.5 those of the original fatigue

Chapter 5. Cyclic Strength

387

limit which alternate either with rest periods or with less strained
loads.

In the majority of cases the increase of fatigue strength is the
result of a reduced mean amplitude of stresses. Thermal rest of
metal favourably effects the latter when operating under alternat¬
ing work—rest periods. Heat, accumulated in microvolumes during
work periods, dissipates during following pauses into adjacent cooler
volumes so that an overstressed volume will meet the next cycle
in a cooled, i.e., strain-hardened state.

The improvement of fatigue strength under the action of short-
time overloads can be attributed to the strain-hardening, which
occurs during plastic deformation of material microvolumes and
is similar to cold working.

It has been established that irrecoverable strains bring about strain¬
hardening effects: randomization of crystalline lattices; thicker
density of dislocations; refinement of crystalline clusters and still
more extensive disorientation of the latter; displacement of grain
boundaries; deformation of cleavage surfaces as a result of plastic
shear reaching the grain surface, thereby reinforcing intergranular
bonds. Elements C, O and N show poorer dissolubility in a-iron;
these elements precipitate from solid solution, forming highly-dis¬
persed carbides, oxides and nitrides, which occur as clouds arresting
the extension of dislocations. In hardened steels the residual austenite
decomposes and turns into finely acicular martensite typical of strain
phase.

Growth of strength, observed when raising the overload to a known
limit, can be attributed to the increased number of microvolumes,
undergoing plastic strains, and also by the enhancement of strain-
hardening intensity. After a certain stage the strain-hardening
process discontinues. In fact, this situation arrives at such level
and rate of stress alternation, when irreversible intra- and intercrys¬
tallite damages occur in the material, affecting the continuity of
the latter.

The studies of fatigue strength under non-stationary conditions
are of theoretical and practical importance as they allow a still
deeper penetration into the nature of the fatigue phenomenon, more
rational use of the material and a more accurate determination
of a structure’s service life under working conditions. However
the calculations are very complicated. A large quantity of experim¬
ental data is necessary to explain the regularities of change in fatigue
limits with different load spectra. Also to be duly accounted for
are stress concentration factors, surface conditions, etc., since their
influence upon the fatigue curves under non-stationary conditions
may differ much from those under stationary loads and often signif¬
icantly (Fig. 212).

25 *

388

Chapter 5. Cyclic Strength

Efforts have been made to determine the durability under non¬
stationary conditions using the Palmgren hypothesis on the cumul¬
ative summation of damages. A stress curve is subdivided into
separate segments (steps) having approximately the same stress
amplitude of stresses. Since the load character at separate steps
may be different, the mean stresses at each step lead to stresses
equivalent in terms of damaging action in the symmetrical cycle.

According to the Palmgren hypothesis the amount of fatigue damage
is linearly dependent upon the number of cycles at a given stress
level, thus comprisinga proportional quota of the full damage (failure),
ensueing at the durability, ultimate for the given stress level.

Let N-i be the number of cycles to failure at a given level of stresses
Oi (Fig. 213). Then, one cycle of stress c ;) will cause damage equal
to 1/iVj., and % of cycles— n x lN x of the full damage. If cyclic durab¬
ility under stress o' 2 is equal to Ah of cycles, while the number of
cycles for the given stress is %, then the fatigue damage will com¬
prise njN 2 of the full damage, etc.

Further it is assumed that accumulation of damages is not depen¬
dent on the order of the alternating steps.

The terms of damage summation are written as follows

N 2 r

»3

N 3

where the full damage (failure) is taken as unity.

Dividing both halves of the equation by durability N . correspond -
ing to failure gives

i ”2 | »3 ( _.JL

NNi ^ NN Z " r NN 3 t ** N

Chapter 5. Cyclic Strength

389

Here == ~ — t 2 ; — t 3 . . . —duration of stress action

at cycle steps in relation to the expected limit cycle duration.
Substituting these values into the previous expressions, one will
obtain

•ei

Ni

+

N

In the generalized form (Minor’s equation)

2

1

where = relative durations of acting stresses actions on the
cycle steps

Ni — durabilities, corresponding to the levels of these stresses
r = total number of steps

Experiments do not prove this relationship. According to exper¬
imental data

i=l

The variations are so great that the possibility of using Minor’s
equations in calculations is questionable. The Palmgren theory is
based upon rather primitive assumptions far from true physical
nature of the phenomena. In particular, it does not consider the
actual kinetic development of damage with the increase of a number
of cycles, influence of the stress level upon the magnitude and kinetics
of damage, influence of the alternating steps on the damage, rest
periods between steps and blocks of steps.

At present the philosophy of fatigue strength (particularly strength
under noinstationary conditions) is most strongly influenced by
the science of physics of metals. If not accompanied by the creation
of an orderly physical theory of metal fatigue strength, the empirical
study of this phenomenon will boil down to a mere accumulation
of statistical data, suitable for use in particular engineering calcul¬
ations.

A problem also emerges how to use most rationally the observed
regularities of strain-hardening in order to improve endurance and
durability of structures. The problem consists in the development
of rational training of parts through higher cyclic loads, alternating
with intervals of rest. In addition it is necessary to develop techniques
of strain-hardening of parts through dosed plastic deformation by
applying static and cyclic loads.

390

Chapter 5. Cyclic Strength

It has been established that fatigue limit of specimens, loaded
with cyclic tensile stresses, is substantially increased by preliminary

deformation as this work
hardens the material (Fig. 214).
The effect of plastic defor¬
mation is particularly promi¬
nent.when the deforming load
and the working load have-
the same sign.

Deformation also causes
residual stresses in the sur¬
face layers of opposite sign
to those of working stresses.
This feature is now utilized
in the process of spring pre¬
setting and can be applied
for strengthening other parts,
e.g., shafts subjected to a
circular of repeated unilate¬
ral bending.

An even greater effect is
obtained by combining zonal
plastic strain with outside
cold working of. zones being
deformed (strain-hardening in
a stressed state).

Fig. 214. Effect of preliminary deforma¬
tion of the fatigue strength of specimens
14 mm in diameter and made from steel
grade C-r. 7 (upper curves are for plain
specimens; lower curves, for specimens
with a stress concentrator — an annular
recess)

1 — without deformation; 2, s and 4 — with
torsional deformation of 25, 50 and 75%, res¬
pectively (after N. V. Kudryavtsev and
V. V. Rrunynova)

(p) Fatigue Limits of Parts

When generalized, the fat¬
igue limits can be expressed
in the following form

c jo part — k i kjt i kje i k z k’ikz

where u j = fatigue limit of a plain polished specimen of a given
material at a given kind of strain-hardening process
and type of loading
k x — coefficient of surface finish
/c 2 = coefficient of corrosion attack

k 3 — coefficient, accounting for surface damage due to wear
in the process of service

*4 — coefficient, accounting for cycle frequency factor
k 5 = coefficient, accounting for the degree of load
impact

5.1. Improvement of Fatigue Strength

391

k 6 = coefficient, accounting for thermal conditions under
■which the part operates

k 7 — coefficient, accounting for heterogeneity of material
and scatter of strength characteristics
k$ — coefficient of load spectrum
e ft = coefficient of scale factor
k e = stress concentration effective coefficient
The k u as and k e coefficients can be inferred from available
experimental data, /c 8 — from test data at a given load spectrum.
In usual circumstances the influence of the & 4 -& 6 coefficients are
insignificant. The other uncertainties are generally accounted for
through the factor of safety, taken as equal to 1.5-2.

Much more reliable is the full-scale test of a part under the condi¬
tions, which most closely approach the actual working conditions
and load spectrum. In this the specific features of the part design
are directly considered. The factor of safety must include the scatter
factors of material characteristics, damages inflicted in the process
of work and also deviations of actual loading conditions from those
during the test.

5.1. Improvement of Fatigue Strength

Up to now the physical phenomena of fatigue strength, have still
not been studied to the degree which allows orderly fatigue strength
calculations of parts to be undertaken. The lack of fundamental
physical principles forces one to accumulate experimental data which
does not always give true answers. The situation is aggravated by
the fact that the data, being supplied by various experimenters,
have extensive scatter and frequently due to the differences in the
testing techniques are incompatible and even contradictory. Calculat¬
ing formulae become more complicated because of the different levels
of fresh data, introduction of correction coefficients and also the
varied subjects of registered factors.

In these conditions comprehension of general regularities which
determine the fatigue strength of parts is of very great importance.
The designer has at his disposal enormous possibilities of improving
fatigue strength. Sensible design, based on knowledge of fatigue
strength regularities at times gives rather more than formal calcula¬
tions and avoids mistakes that would otherwise have to be rectified,
for instance, through strengthening processing.

The designer must know and confidently apply his own technol¬
ogical and constructional recommendations on methods of improving
fatigue strength.

In many cases one can eliminate the source and achieve if not
full exclusion of cyclic loads then at least their partial elimination.
Even in machines having definite cyclic action it is possible to lower

392

Chapter 5. Cyclic Strength

significantly the maximum values of cyclic stresses and their amplitu¬
des and also mitigate impact loads.

One of the principal ways is to enhance elasticity of parts in the
direction of acting load and introduce elastic links between parts
transmitting and receiving loads.

For example, increasing the elasticity of bolts in joints subjected
to cyclic loads reduces both the value of acting loads on the bolts
and the interval between the maximum and minimum loads, i.e.*
favours the improvement of fatigue strength.

During cyclic torsion peak stresses may be cushioned, consequently
decreasing cyclic amplitude, by using flexible couplings between
members accepting the torque. For example, the provision of spring
dampers between shafts and gears will lower peak stresses in teeth
and make gears run more smoothly and quietly.

The change-over from rolling contact bearings to plain bearings
in conrod-crank mechanisms lowers peak loads as the oil film has
a damping action. The work being spent for displacing oil from the
bearing clearance absorbs the impulse of the acting forces,, which
helps toj reduce the loads imposed on the mechanism.

Cyclic loads, arising in shafts, may in some cases (non-driving
gear wheels, idlers) be eliminated by mounting rotating parts
on axles.

In many cases the onset of high alternating loads is associated
with the appearance of resonance vibrations in parts of the mechanism.
This dangerous kind of cyclic loading can be removed by employing
spring, pendulum, hydraulic or friction type dampers.

Vibrations of machines and units, presenting sources of alternat¬
ing loads, are eliminated or cushioned by suspending them on anti-
vibration) or vibration-damped mountings.

Sometimes cyclic loads can be completely or almost completely
removed by means of more accurate manufacture of parts and their
supports. The elimination of static and dynamic disbalance in
high-speed rotors, causing alternating loads in supports and housings,
is an example of this. Better teeth manufacture, smaller errors in
pitch and tooth-thickness, more precise profiles, reduced pitch
circle runout, accurate gear helices, etc., will remove cyclic loads
being caused by such errors.

It is very important to note that any measure, aimed at decreasing
nominal value of stress simultaneously improves fatigue strength.
Belonging to such measures are rational spacing of supports, elimina¬
tion of irrational loads, increasing of cross-sections at parts subjected
to cyclic loading, enlarging of surface contact areas (with cyclic
contact stresses).

All rational design rules being applied to statically loaded construc¬
tions not only remain valid but are also of significance for cyclically
loaded structures.

5.1. Improvement of Fatigue Strength

393 :-'

In the cases when liquidation of cyclic loads or at least a reduction
in cyclic stresses is not successful, special methods of improving
fatigue strength should be used. These techniques can be divided
into process and constructional. In the first case strengthening
is attained through special processing, while in the second case
by imparting shapes which are most rational to fatigue
strength.

(a) Technological Methods of Improving Fatigue Strength

Extremely heavy influence on the fatigue strength value is exerted 1
by the steel melting process. Vacuumized steels have improved fatigue
strength qualities. The same is true for steels, produced by electric
remelting in a vacuum or under a layer of synthetic slag. A substan¬
tial improvement in fatigue strength is also assured by thermal-
mechanical processes (particularly LTTP).

Thermal and chemical-thermal treatment of steels have a favour¬
able effect* especially when stress raisers are present. According
to their influence on the fatigue strength these processes can be arran¬
ged approximately in the following order: structural refinement and
normalizing, low-temper hardening, case-hardening, induction hard¬
ening, cyaniding and nitriding.

The strain-hardening in the latter four cases is dictated in the main,
by the appearance of compressive stresses (of the order of 40-
80 kgf/mm 2 ) in the surface layer due to a larger unit volume forming-
in the structure (martensite—in case-hardening and induction harden¬
ing, nitrides and carbonitrides—in cyaniding and nitriding) which
is bigger than the basic one.

Creation of prior compressive stresses is equivalent to a reduction
of mean stresses in tensile load cycles and increased asymmetry
cycle coefficient which as easily seen from the Smith’s diagram leads;
to a higher fatigue limit.

Moreover, increased surface hardness, obtained by thermal and
chemical-thermal treatment, averts loss of strength due to wear,
accidental scratches or some other defects.

Optimum surface layer thicknesses: in case-hardening, 0.4-0.6 mm;
cyaniding 0.15-0.2 mm; nitriding, 0.3-0.5 mm; induction hardening,
2-4 mm. It is essential that strain-hardening processes involve all
parts on the surfaces with stress concentrators (Fig. 215a, b) since
substantial tensile stresses may appear at the boundaries of treated
and non-treated zones, these stresses being extremely dangerous
if located in the stress concentration zone.

Nitriding is the most effective since it practically eliminates the
influence of stress concentrations. For nitrided parts the coefficient
q, showing sensitivity to stress concentration, is close to zero (i.e., the
effective coefficient of stress concentration k e « 1).

-394

Chapter S. Cyclic Strength

Since no appreciable changes in dimensions and shape of a part
are caused by nitriding, it can be employed as a final stage manufactur¬
ing process. This eliminates the final grinding process, which affects
the strength due to the defects inherent to that operation (burnings,,
-cracks, etc.). Combination of a high hardness,, good fatigue strength
and corrosion-resistance makes the nitriding treatment extremely
valuable in processing the parts expected to be used under high
cyclic loads and wear (crankshafts, heavily loaded gears, etc.).

The surfaces of parts working under high cyclie loads must be
machined to the maximum economically acceptable surface finish
standards. All kinds of finishing operations (polishing, lapping,
superfinishing, etc.) smoothing down micro-roughness which remains
on the surface after previous coarse machining operations contribute
to fatigue strength, especially for details made from high-strength
and hard materials.

Very good results are achieved by polishing under pressure, which
■affects deeply the structure of surface layers. Crystallites, separated
'during preceding machining, close together and are united by the
action of pressure and heat evolved in friction. The surface layer is
thus “compacted”. Sharp edges of mieroirregularities are smoothed
down, while micro-valleys and microcracks are levelled up. The
smoothed surfaces of parts offer better corrosion-resistance.

For this reason all important components, subjected to high cyclic
loads, are polished all-round not only on fitting and friction sur¬
faces.

Moreover,, it is easier to detect defects such as flakes, flaws, heat-
treatment cracks, etc., on polished surfaces.

Press polishing is accomplished under 1-2 kgf/mrn 2 pressure,
■using laps in the form of bronze or cast iron shoes, whose working
•surface is charged with micropowders (flours) of abrasive matter
(carborund, boron carbide, diamond dust). Superfine polishing (buf¬
fing) is carried out with the aid of soft laps (babbit, wood, leather,
suede, felt) employed in combination with lapping pastes of TOIi
type. Final finishing is done without any abrasives, using only
kerosene or solvent naphta lubricants.

Excellent results have been obtained when polishing rubbing
surfaces under pressure with colloidal graphite or (which is still
better)molybdenum disulphide: this process improves wear-resistance
and imparts better corrosion-resistance.

The principal methods of strengthening by plastic surface layer
deformation are shot-blasting, rolling and caulking (cold calibration
stamping); holes by expansion, ball sizing, piercing with smoothing
broaches (driftpeening). Applicable also are diamond smoothing tools
(i.e., hardening the surface with a rounded diamond tool), hardening
by turning with carbide-tipped tools having large negative rake
.angle; caulking by explosion or electrohydraulic shock; abrasive-jet

5.1. Improvement of Fatigue Strength

395

polishing; ultrasonic hardening; hydraulic-pulse hardening with
a high-pressure jet (10-20 thousands of atmospheres).

Exceptionally good results have been reached on surfaces, which
were case or induction hardened prior to the cold working procedure.
Hardening effects of plastic strains in this case can be attributed not
■only to the inception of compressive stresses, but also due to structu¬
ral changes occurring in the material surface layer as a result of work
hardening (the residual austenite decomposes and highly dispersible
martensite , emerges). Regardless of the kind of strain-hardening
applied, it is necessary to induce in the surface layer stresses, which
exceed the material’s yield limit under volumetric compression.

Hardening through plastic deformation is practised mostly for
steels. Brittle materials, for example, grey cast irons, are less prone
to this type of structural improvement. Cold work hardening can
also be applied to ferrite and ferrite-pearlite cast irons, even though
the effect of strain-hardening is weaker in this case when compared
with plastic materials. Other kinds of cast irons hardenable in the
above-said way, are malleable cast irons, as well as modified and
high-strength irons with globular graphite.

Shot blasting is one of the simplest and most universal of modern
-strain-hardening techniques. When chosen correctly, this procedure
enables fatigue strength' to be enhanced 1.5-1.6 times. However,
it slightly impairs surface quality therefore precision surfaces should
be carefully finished after being shot-blasted.

As the optimum schemes of hardening by plastic deformation are
insufficiently developed, their use is determined experimentally for
each individual case.

The shot peening procedure is selected to suit the properties of the
material, its hardness and strength. Overhardening will readily
cause brittleness and hairline imperfections in the surface layer.
Approximate shot blasting recommendations for steel products:
velocity of shot—50-60 m/s, flow rate 50-80 kg/min, angle of attack
'(jet incidence relative to component surface) 60-90°. Duration of
process 2-5 min. Shot diameter (hardened steel balls)—0.5-1 mm.

When rolling the surface,'’the pressure force P, exerted onto a roll
is’chosen so as to produce "in the|surface layer stresses (to Hertz)
amounting to^350~600 kgf/mm 2 . An empirical formula for the force

P = ZAQr 3 al'zDB~~ kgf (5.8)

where D — component diameter, mm
o 0 , 2 — material yield limit, kgf/mm 2
B = usable width of a roll, mm

a = P r °‘L = ratio between diameters of roll and component

396

Chapter 5. Cyclic Strength

In the range of most widely applied values a = 0.7-1.0, the
value & 0.5. Substituting this value into Eq. (5.8)
jP = 1.5-10 _3 (jg 2 Z)S kgf

Generally roll width B — 10-20 mm. The component surface
speed 10-20 m/min. Longitudinal feed s — (0.05-0.1) B, on average
0.5-1 mm/rev. Number of passes 1-3.

In order to relieve pressure, exerted upon a roll, vibrorolling has
become very popular, in which a roll is imparted intermittently radial
displacements from a mechanical or air-operated actuator.

Fig. 215. Enhancement of fatigue strength
(a) and (b) surface hardening: I — incorrect; II — correct; <c) ali-rouna swaging o hole
edges; (d) rolling of annular grooves; (e) embossing around a keyway; l — Original part;

2 and 3 — strengthened parts

When the rolling conditions have been correctly chosen,, the res¬
idual compressive stresses in the surface layer will comprise 70-
80 kgf/mm 2 . The depth of the strain-hardened layer will reach
0.1-0.5 mm.

In contrast to shot peening, which slightly worsens the origi¬
nal surface finish, rolling improves it.

The finer the preceding machining the more favourable the rolling
effect. To obtain the best results, it is better to finish machine (even
polish) the component surface prior to rolling.

As component dimensions are hardly changed by rolling, the
latter may be regarded as a final manufacturing operation. The
possibilities of size change or final lapping after work hardening
is not excluded: experience proves that removal of a work-hardened
layer to the depth of 0.05-0.1 mm (the total thickness of the layer
being 0.3-0.5 mm) brings about no appreciable decline of hardening
effects.

It is useful to subject strain-hardened parts to stabilizing treat¬
ment, by heating to 100-150°C. At higher temperatures the residual
compressive stresses in the hardened layer reduce and at 400-500°G
vanish completely.

5.2. Design of Cyclically Loaded Components

397

Rolling of threads improves their strength 1.5-2 times, and pract¬
ically eliminates stress concentrations at thread roots, this being
of particular merit when dealing with stress concentration sensitive
steels.

Plastic deformation is widely used for strengthening hole edges
punched in sheet metals. Hole edges are stress concentrators, due to
plastic displacements, tears and microcracks which occur under the
action of a punch. Circular reduction of hole edges (Fig. 215c) to a
significant measure liquidates their weakening effects. Shafts, wea¬
kened by undercuts, ai’e work hardened by rolling adjacent areas
{Fig. 215 d).

To improve strength of shafts weakened by keyways, the keyway
is caulked around the key introduced into the way (Fig. 215e). Apart
from reduced stress concentration, this measure also adds to the key
f'tting strength.

A markedly improved fatigue strength is displayed by elements
which have been obtained by direct squeezing (rolling of threads,
gear teeth and splines). In these cases the metal fibres are not cut as
when machining large solid blanks, but are forced Into the required
configuration by the pressing process.

Normally threads are cold rolled, but gear teeth and splines are
first hot rolled and then dimensionally sized cold.

5.2. Design of Cyclically Loaded Components

(a) Reduction of Stress Concentrations

The reduction of stress concentrations—is one of the major means
<of increasing fatigue strength.

Sometimes stress concentrators cannot be removed entirely. Under
such circumstances the designer should do his best to substitute sharp
•concentrators with moderate ones. Example: screwed holes, belon¬
ging to most intensive stress concentrators, must, whenever feasible,
be replaced by plain holes, whose negative effect is less and can be
brought still lower by some additional measures.

Stress concentrators should be removed from most stressed zones
of a part and disposed (if permitted by the design) in the zones of least
stresses.

In order to decrease nominal stresses it is better to enlarge those
cross-sections where stress concentrators are located.

Some examples of eliminating and decreasing stress concentrations
are illustrated in Table 23.

Fillets. Concentration of stresses in entering angles of stepped parts
{e.g., stepped shafts) can substantially be reduced by imparting
rational forms to the conjugated steps.

398

Chapter 5. Cyclic Strength

Removal of Stress Concentrations

Table 23'

Original design
Bolt head

The locking lug is arran¬
ged in the most stressed
section, thus causing sharp
stress concentration. It also
reduces the area of the cri¬
tical section

Mounting of a slip-on
rotor upon a shaft

Fastening bolts and loca¬
ting pins enter the hub bo¬
dy, the most stressed part
of the rotor

Securing a plug in a
shaft

Thread causes stress con¬
centration

Improvement

Essence of Improvement

The lug is fitted in the'
bolt head. Nominal stresses-
reduced, but stress concent¬
ration'still remains

The lug is now an in¬
tegral part of the bolt head-
Stress concentration elimi¬
nated

The rotor is clamped-
with a circular nut. Torque-
is transmitted by teeth in
the hub face which fit into-
slots on the shaft flange

Threads removed, press-
fit substituted

Threads removed, expen¬
ded fitting substituted

Direct thread connect¬
ions to shaft eliminated.
Nut and bolt form of fit¬
ting substituted

Screwed joint eliminated.
And fastening with threaded
hole substituted

5.2. Design of Cyclically Loaded Components

399 >

Table 23 ( continued )■■

Original design Improvement Essence of improvement

Oil hole in crankshaft
rod journal

Hole positioned at the
part of maximum bending
stresses, occurring due to
ignition

Hole transferred to neut¬
ral zone (hole position'musfe'
be correlated with load vec¬
tor diagramme)

FFixing a solid turbine
rotor to a divided shaft

Fastening bolts, passing
through the rotor, sharply
weaken it

Bolt receiving holes po¬
sitioned in thickened annu¬
lar bosses on the rotor ar¬
ranged beyond stressed (sec¬
tions

Bolt receiving holes po¬
sitioned in flanges brought
from the rotor body

■400

Chapter 5. Cyclic Strength

Table 23 ( continued )

Original design Improvement Essence o£ improvement

Two concentrators (an
• external step and an inter¬
nal recess) positioned in the
-same plane. The stress fi¬
elds, being produced by the
•concentrators, accumulate.
Reduced cross-sectional area
increases nominal stresses

Securing a bevel gear rim
do a disk

The stresses, caused by
bolt holes, add to the stres¬
ses near tooth root

Stress concentrators posi¬
tioned in different planes

Holes are repositioned
away from gear teeth on an
extension disk

Bevel gear rim

Two stress concentrators
• combine (i. e. tooth gashes
.and sharp face edges)

5.2. Design of Cyclically Loaded Components

401

Table 23 (continued)

Turbine rotor

Rotor disk •weakened by
relieving holes

Hollow shaft

Two concentrators com¬
bine (external and internal
entering angles)

Essence of improvement

Screwed portion enlarged

Sectional areas of wea¬
kened portions enlarged

1. Holes strengthened
with bosses

2. Holes positioned in
■a strengthening ring

Internal stress concentra¬
tor moved

Internal angles given
smooth streamlined forms

26-01395

402

Chapter 5. Cyclic Strength

Table 23 ( continued )

Original design Improvement Essence of improvement

Thread portion made
thicker

Shaft cross-sectional are¬
as with stress concentrators
enlarged; internal surfaces
streamlined

Bearing installation on
shaft

Shaft weakened by cir¬
clip grooves

Stress concentration at
spline roots

Shaft strengthened
weakened portions

Shaft strengthened
stress concentration areas

at

5.2. Design of Cyclically Loaded Components

401

Table 23 ( continued)

26 4

404

Chapter 5. Cyclic Strength

Table 23 ( continued )

Original design

Improvement

Crankshaft neck with
an oil delivery system
to crankpin bearing

. The end-cap, plug
and an oil inlet pipe are
threaded: thread causes
stress concentrations

Essence of improvement

The end-cap is sli¬
de-fitted in the shaft
and locked by a bolt
used instead of the
threaded end-cap; the
oil inlet pipe is exten¬
ded through the end-
cap into the shaft
bore

Acute entering angles at a transition section (Fig. 216a, b) cause
sharp stress concentrations. Tapered conjugations (Fig. 21.6c) add
to the strength of transitional areas, hut shorten the cylindrical

Fig. 216. Decreasing stress concentrations in the entry angles of stepped shafts

surface length of the smaller diameter. That is why such conjugations
are applicable only at free transitions when this part of the detail
does not mate with adjacent parts.

Most often stress concentrations at transitional areas are reduced
by fillets (Fig. 216 d-g).

The effectiveness of a fillet depends on its radius (Fig. 217).

Stress concentrations fall with reduction in diameter gradient
and with an increase in ratio p. Significant gains in strength are

5.2. Design of Cyclically Loaded Components

405

obtained with p value approximately equal to 0.1 for large diame¬
ter gradients and when p = 0.05-0.08 for small diameter gra¬
dients.

The maximum relative radius of a fillet, equal to p ma jc —
= 0.5 — l\ is limited by the diameter gradient. Usually in.

0.4, the stress concentration

practice Did & 1.2, so that the relative fillet radius cannot exceed
0.1. If a component fitted on the
shaft has a rectangular supporting
face at its step ( h , see Fig. 216e),
the radius must be even smaller.

Elliptic fillets (see Fig. 216/)
at the same diameter gradient
assure a comparatively greater
(approximately by 2 0 %) strength.

The effectiveness of such fillets
depends on the ratio of the ellipse
major semiaxis b to the shaft
diameter d.

When b = (0.4-0.45) d and

Fig, 217. Effective stress concentra¬
tion coefficient of a stepped shaft in
flexure as a function of relative fillet
radius p — Rid and ratio Did of adja¬
cent (diameters (after Serensen)

a

T

coefficient will not exceed 1.5.

The disadvantage of elliptic
fillets is the shortening of the
cylindrical portion of a shaft,
which is undesirable when mo¬
unting fitted details or when
fitting shaft journals in sliding
bearings.

The reduction in the length of a shaft’s cylindrical portion can be
avoided by using face undercut fillets (Fig. 216g).

Face undercut fillets in terms of their effectiveness approximate
round fillets, having similar Rid values.

Face undercutting is recommended when cylindrical shafts are
mated with prismatic components, when there is place for a fillet
of sufficiently large radius.

Figure 218 shows how higher strength fillets can be overlapped when
mounting fitting components, for example, antifriction bearings,
having small lead radii or small chamfers. For large-radius and ellip¬
tic fillets (Fig. 2185,'c), intermediate washers which have appropriate
recesses are used.

Fillets must be provided at all transitional entering angles of parts
subjected to high cyclic loads (Figs 219 and 220).

Holes. Stress concentrations caused by holes can be reduced by
enlarging the sections of a part where holes are located by rounding

408

Chapter 5. Cyclic Strength

hole rims, crimping of edges, strain-hardening hole walls and caul¬
king the material around the periphery.

Figure 221 illustrates the sequence of operations when machining
holes in highly loaded components (e.g., discharge holes of turbine

Fig. 218. Installation of ball bearings Fig. 219. Shapes of some typical
i on shafts with strengthened fillets machine elements

(a) irrational; (6) rational

disks): a—drilling, b— chamfering, c—countersinking, d —reaming,
e —rounding of hole rims, /—compacting the fillet, g —broaching the
hole with a ball.

Fig. 220. Conjugation of gear teeth with rim
(a) irrational; (b) improved; (c) rational

Hollow shafts. The internal cavities of critical hollow shafts
nndergoing high cyclic loads, should be machined to the highest;
economical surface finish: grinding, polishing, rolling, sizing, com¬
pacting broaching, etc. Recesses, threads and other stress concentra¬
tors should be avoided on inner surfaces.

In stepped holes smooth transitions between steps should be in¬
troduced. Incorrect designs are presented in Fig. 222a, b. The acute

Fig. 221. Successive stages in machining holes in cyclically loaded parts

408

Chapter 5. Cyclic Strength

entering angles close to steps cause stress concentrations and impair
shaft strength. The versions presented in Fig. 222c, d show how fillets
improve strength. Fig. 222e-j, illustrate shafts with bottle-neck-sha¬
ped holes.

Fillets in bottle-neck-like holes are turned to a templet, which
monitors the cross-slide traverse. Finish turning is effected by a
form tool, secured in a boring bar, centered in the smaller diameter
hole (Fig. 223a).

More difficult to machine are hollows with limiting fillets at both
ends (barrel-like holes). One machining technique is illustrated in
Fig. 2236. The form tool is clamped in rod 1, secured eccentrically
in boring bar 2. By turning the rod, the tool is retracted, the boring
bar is introduced into the hole and the cutting tool then moved into
position.

The most productive method of machining bottle-necked holes is
with a swing tool mounted in a boring bar and monitored with a
connecting rod (Fig. 223c), rack (Fig. 223d) or worm gearing
(Fig. 223c).

The fillet radius is determined by the position of tool-holder.

The most advantageous swing mechanism design is when the tool-
holder axis is positioned at the boring bar centre (Fig. 223c, d), i.e.,
when the fillet forms a sphere with its centre on the shaft axis (see
Fig. 222/, g). Such a shape ensures a smooth transition from one hole
diameter to another.

An even smoother transition is obtained by shifting the tool-holder
pivot point off the shaft axis (Fig. 223/).

To determine the maximum radius of an internal fillet the following
approximate formula can be used

•^max = 0-5 (£>+ OJd)

where D and d are the m axirnum and minimum hole diameters, res¬
pectively (see Fig. 222a).

When machining inner steps in preformed large tube blanks the
material fibres are cut through at the most stressed areas of transition
from one step to another.

To attain better strength, shafts with barrel-shaped interiors have
their tube ends hot pressed (Fig. 224). Used as blanks are thick-wall
drawn tubes whose outside diameter is turned with extra metal for
reduction of ends (Fig. 224a). Surface m is the datum for subsequent
operations.

The ends are reduced by pressing (Fig, 2246) until the hole is fully
closed (Fig. 224c). Then, from surface m holes n are bored for shaft
journals and the barrel-shaped bore h machined employing one of the
above-listed techniques (Fig. 224d).

Then using holes n as a datum, the outside diameter is finish
machined.

5.2. Design of Cyclically Loaded Components

409

Crankshafts. An example of successive fatigue strength enhance¬
ment of crankshaft throws is pictured in Fig. 225.

The initial design (Fig. 225a) has low strength. In the version,
illustrated in Fig. 225 b, the strength is increased due to the larger

(c) (d)

Fig. 224. Process stages im manufacturing one-piece forged shafts with barrel-

shaped interior

diameters of the main and rod necks and increased web sections. The
increased neck diameters shorten lengths m between necks parts which
are most critical in terms of strength.

Fig. 225. Enhancing the fatigue strength of a crankshaft

Still better strength is obtainable by offsetting the conrod neck
internal recess from the neck geometric axis by value k (Fig. 225c):
this will strengthen the connection between rod necks and webs and
increase neck strength resistance against ignition flexure.

410

Chapter 5. Cyclic Strength

In the most rational design (Fig. 225 d) the neck diameters are
increased -until the main and conrod necks overlap, thereby ensuring
direct connection of necks (part n). A barrel-shaped bore between the
main and conrod necks lowers stress concentration from oil holes in
crankshaft webs and betters the neck-to-web connection strength.
A combination of all of the above-described measures substantially
improves crankshaft strength.

Forms with deep internal cavities are practicable in cast crank¬
shaft designs (Fig. 225e, /).

(i b ) Elimination of Load Concentrations

A most important design rule for cyclically loaded parts is the
elimination of local stress surges arising where the concentrated
loads are applied.

Let us take, for example, the design of gear teeth (Fig. 226a).
Out-of-straightness of spur gear teeth, helix errors of helical gear

Fig. 226. Elimination of load concentration on tooth faces

teeth, misalignment of gear axes as a result of improper mounting or
inaccurately spaced supports—these and/or other defects may cause
load concentrations on tooth end faces, the consequence being higher
bending and crushing stresses.

It is advisable to increase tooth compliance at the end faces by
the introduction of relieving recesses in the gear rims (Fig. 2265) or
by lessening rim rigidity near the end faces (Fig. 226c).

An effective means of preventing high-edge pressures is by giving
teeth a barrel-shaped form (camber) simultaneously rounding-off
face edges (Fig. 226d). This method is beneficial because even in the
event of distortions and inaccuracies the gear contact spot will remain
approximately in the tooth centre, thereby assuring most favourable
loading of teeth.

Pressed connections. Presented in Fig. 227 are some ways of
strengthening pressed connections. The simplest of these consists in
increasing the diameter of fitting surface (at least by 5-10%) in
relation to the shaft main diameter d (Fig. 227 a).

Advisably lowered are the stresses at the edges of pressed fitted
connections by relieving recesses in the hub (Fig. 2276), shaft

5.2. Design of Cyclically Loaded Components

411

(Fig. 227 c), by tapering the hub down towards its end faces (Fig. 227d)
or by cambering the shafts (Fig. 227e).

Shafts strength can be improved by rolling load relieving grooves
near the joint ends (Fig. 227/).

Fig. 227. Improvement of fatigue strength

The most effective method of improving fatigue strength of pressed
connections is by strengthening the contacting surfaces through ther¬
mal-chemical treatment and plastic deformations (rolling).

Fasteners. Constructional ways of increasing the fatigue strength
of fasteners (bolts, studs, etc.) largely means increasing their elasti¬
city, thus assuring a lowering of maximum stresses and stress ampli¬
tudes. If bolt lengths are the same higher elasticity can be achieved
by lessening the bolt stem diameter d to 0.7-0.8 of its thread nominal
diameter d 0 (Fig. 227 h); in this case the sections of the bolt stem and
thread become approximately equistrong.

The head-to-stem bolt transition must have the maximum possible
radius or face undercuts (Fig. 2270- Equally necessary are smooth
fillets (i? g ) at the stem-to-thread transit section.

412

Chapter 5. Cyclic Strength

The strength of thread itself is strongly dependent on fillet radii
near the bolt thread root (this effect is less in a nut because it is
usually stronger than a bolt).

The current standard (GOST 9150-59) does not stipulate the root-
form between bolt threads: they may he flat (Fig. 227;) or rounded to
a radius (Fig. 227 k). In this case the maximum radius for bolts is
equal to 0.144s (s — thread pitch).

For critical screwed connections subjected to high cyclic loads it
is advisable to employ threads with more stream line-rounded root
radii: for nuts R = 0.1s,, for bolts R = 0.2 s (Fig. 227 1).

Particularly high fatigue strength is obtained by rolling rough-cut
and heat-treated threads and also when rolling threads outright on
the solid metal.

As a general rule parts subjected to high cyclic loads should have
smooth forms to assure evenness of force flow. To avoid stress jumps
in the sections of parts it is necessary to determine from conditions of
approximate similarity the stresses by considering all active loads.

Well designed parts undergoing high cyclic loads have a characte¬
ristic smooth form (Fig. 227 ri), often conveniently called “stream¬
line”.

5.3, Cylindrical Joints Operating under
Alternating Loads

Any joint transmitting a pulsating torque or subjected to alterna¬
ting radial loads undergoes work hardening, welding and fretting
corrosion.

In the main these defects are caused by numerous repetitive strains
and microshears in the mated surfaces, in the circumferential and
lengthwise directions, which heat the material. Joints, operating
in the severest conditions, are sometimes heated to 500-600°C. Momen¬
tary peak temperatures at parts of micro-irregularity contiguities
reach 1000-1500°C.

Under such circumstances work hardening easily occurs, crushing
the surface, forming irregularities and partial cohesion of mated
metallic surfaces. During the next stage such surfaces may weld
together so that their disengagement is possible through destruction
only.

This kind of welding can occur at a temperature well below the
welding temperature of the material. Under usual conditions the
metal surface becomes coated with strong adsorbed films of lubri¬
cants, oxides, moisture and vapours, which prevent direct metal-
to-metal contact. Heat and excessive pressures, particularly when at
points of conjugated microirregularities destroy the films, and the
metal particles close to a distance, at which the molecular and
crystalline interaction forces cause metal fusion.

5.3. Cylindrical Joints Operating under Alternating Loads

413

Greater propensity to welding is shown by identical metals and
metals which have similar crystalline'structures. Structural hetero¬
geneity presence of several phase changes and non-metallic inclusions
(carbides, oxides) in metals stop the welding phenomenon. Ample
.resistance to welding is offered by hardened steels with a martensite
structure (provided, however, no tempering has occurred due to
overheating).

Frictional (fretting) corrosion is, in fact, oxidation of surfaces of
metals as a result of microshear caused by local elevations of tem¬
perature. On steel and cast iron surfaces iron oxides form (mostly
Fe a 0 3 ) either as rusty spots or (if the process has developed far enough)
as clusters of brown powder. On bronze surfaces appear green films
of copper oxides.

Work-hardening and welding defects can.be averted by:

reducing strains and microshear in mated surfaces (making the
design more rigid, employing power-clamped joints, clearance-free
torque transmissions);

removing heat, evolved' during micro-displacements (applying
gaskets made from heat-conducting materials, use of cooling pil in
joints having clearances);

application of separating coatings (phosphating, copper-cladding,
tinplating, cadmium-plating, galvanizing, coating with polymeric
films, introduction of solid lubricants having a molybdenum disul¬
phide base, colloidal graphite, etc.); .

increasing hardness and thermal stability of the surface layer; creating
structures which show good resistance to welding (aluminizing, sul¬
phiding, nitriding, diffusion chrome-plating, boronizing (see Table 24).

The main design method, enabling work-hardening and the welding
to be avoided, is to provide interference fits to the mating surfaces,
radial (on cylindrical surfaces) or axial (on end faces). An interference
fit materially enhances the rigidity of a unit as a whole, lessens elas¬
tic deformations in a system and efficiently constrains relative dis¬
placements of mated surfaces.

Figure 228 shows some methods of fitting parts to shafts.

Mounting without tightening or with a weak clamping (Fig. 228a, b)
is unacceptable for power-transmitting connections.

Often used now is the axial tightening of a hub resting against a
shaft shoulder (Fig. 228c). Here the amount of radial interference
depends upon the grade of the hub fit. The heavier the operating
conditions, the tighter should be the fit. In shaft-end connections a
central bolt is also used for clamping (Fig, 228d), or an internal nut
(Fig. 228e) which; provides for a greater tightening.

A purely radial interference is provided by a press'fit (Fig. 228/).
By introducing taper pins (Fig. 228g) into the joint a practically
clearance-free torque transmission may be obtained, thus eliminating
angular micro-displacements of .the mating surfaces. It should be

414

Chapter 5. Cyclic Strength

Table 24

Measures Which Prevent Welding

Process

Essence of process

Procedure

Purpose

Phosphating

Deposition of a
crystalline phosp¬
hide film on the
surface

Treatment in a
solution of phosp¬
hates Fe, Mn and
Zn

Creation of a
separate wear-resis^
tant film

Aluminizing

Deposition of
crystalline film
Al 20 3 on the trea¬
ted surface. Forma¬
tion in the surface
layer of A1 solid
solutions in a-iron

Soaking in a
blend of ferroalu-
minum powder and
A1 2 0 3 at 900-1000°C
(5-8 h)

Increases ther¬
mal and corrosion
resistance of the
surface layer

Sulphiding

Formation of
iron sulphides in
the surface layer

Soaking in a
blend of sulphates
(NaS-9H 2 0) and
cyanates (cataly¬
zers) at 550-580°C
(2-4 h)

Imparts anti-sei¬
zure properties;

increases welding
resistance

Nitriding

Formation of Fe,
Al, Mo hard nitri¬
des etc. in the sur¬
face layer

Soaking in an
ammonia atmos¬
phere at 500-550°C
(20-30 h)

Enhances ther¬
mal and corrosion
resistance and also
hardness (800-
1200 VPH)

Diffusion

chrome-plating

Formation in
the surface layer
of chrome carbides
and Cr solid solu¬
tions in the a-iron

Soaking in a
medium of vola¬
tile chrome chlori¬
des CrCl 2 , CrCl 3
(.gaseous chrome¬
plating) at 800-
1200“C (5-6 h)

Betters thermal
resistance and
hardness , ( 1200 -

MOO VPH)

Boronizing

Formation in
the surface layer
of Fe borides and

B solid solutions
in the a-iron

Soaking in a
blend of powders
of boron carbide
(B 4 C) and borax
(Na 2 B 4 0 7 ) at 900-
1100°C (5-6 h)

Increases ther¬
mal resistance and
surface hardness
(1500-1800 VPH)

Fig. 228. Tightening of cylindrical connections

416

Chapter 5. Cyclic Strength

emphasized, however, that in this case a permanent joint is obta¬
ined.

A rather strong connection is attained when tightening on a taper
(Fig. 228ft). The value of radial interference will depend on the force,
with which the nut is tightened. In particular instances it is obliga¬
tory to use torque wrenches although this method does not fully
guarantee the degree of tightening since it depends to a great extent
upon the state of the mating surfaces.

Slide-fit splined connections (C to the aligning diameter, S and C
to the splines working faces) must be tightened by a nut (Fig. 228i)-

Fig. 229. Clamp connections

For permanent or rarely disconnected connections a press- or force-fit
on the aligning diameter is usually practised or a drive-fit to the
working faces (Fig. 2287).

Figure 228ft, illustrates how radial interference can be obtained by
press-fitting a plug into the shaft bore. A permanent connection is
thus accomplished. If dismountable connections are required, the
above tightening is implemented either by a taper threaded plug
(Fig. 228Z) or with a cone tightened by a central bolt (Fig. 228 m).
In the latter case the plug must be internally threaded to allow use
of an extractor.

In critical heavy-duty splined connections to hubs are tightened
on tapers (Fig. 228«, 0 ).

When connecting cylindrical components with prismatic elements
(e.g., journals to webs in split-type crankshafts) use is made, apart
from the above-listed techniques (press- and taper-fits, Fig. 228p, r)
of clamps (Fig. 228s, t), as well as internal conical plugs (Fig. 228u).
To avoid work-hardening the mating surfaces, a thin-walled liner 1
made from bronze or sheet brass is used.

When dealing with clamp connections it is necessary to assure
uniform tightening throughout the entire circumference of the clamp.
Fig. 229a, illustrates an incorrect design. The torque from the journal

5.3. Cylindrical Joints Operating under Alternating Loads

417

to the web is transmitted by two tenons. During tightening the upper
edges of the tenons thrust against slot walls and the AA section re¬
mains loose hence work-hardening here is unavailable.

In a correct design (Fig. 229fr) the tenon is positioned at the axis
of clip symmetry. Uniform tightening is also obtained when the
torque moment is transmitted by a bolt fitted into a groove machined
in the journal.

Owing to the above clamps cannot be used for heavily loaded spli-
ned connections.

27-01395

Chapter 6

Contact strength

With contact loading the force acts upon a rather small area of
surface, thereby causing higher stresses in the metal surface layer.
This kind of loading is encountered most frequently when spherical
and cylindrical bodies come into contact with flat, spherical or cylin¬
drical surfaces. Such are, for instance, antifriction hearings, gears,

rollers of overrunning clutches,
friction speed variators, etc.

In the theoretical solutions
on the stressed state of elastic
bodies (Hertz, Belyaev, Fap-
ple) in the zone of contact it is
assumed that the applied load
is static, that the materials
of the contacting bodies are
homogeneous and isotropic,
that the contact area is small
in comparison with the total
surface area and that the
acting force is directed nor¬
mally to the area of contact.

In the zone of contact a
flat area is formed whose
dimensions depend upon the
elasticity of the materials and the shapes of the compressed bodies.
During compression of spheres (Fig. 230a) this area will have the
shape of a circle with diameter

Fig. 230. Contact loading schemes
(o) spheres; (6) cylinders

6 — 1,4 ~ mm

where P — load, kgf

c 1 Tv*

E —- = reduced modulus of elasticity for the material of the

spheres, kgf/mm 2
d*D

O' = -jyzTd ~ rec ^ lce< l diameter of spheres, mm (the minus sign rela¬
tes to the case when a convex surface contacts a con¬
cave surface with diameter D)

Chapter 6. Contact Strength

419

The maximum pressui'e in the centre area is 1.5 times greater than
the average pressure

P

ipmax = l-5 o,7856 2

When cylinders are compressed (Fig. 2305), the area will have a
reactangular form whose width

5= 1.5 j/" q-^jr mm

where •& = reduced diameter of cylinders

q — load upon cylinder unit length, kgf/mm 2

The maximum pressure along the centre line of the area is 1.27
times higher than the average pressure

p m ax = 1.27|

The material fibres in the zone of acting maximum pressures are
in a state of omnilateral compression; also in the fibres form mutually
perpendicular compressive stresses o x , a y , g z and directed to them

at 45° octahedral shearing stresses . The

distribution of these stresses (in fractions of maximum pressure
Pm ax per contact area) in terms of surface layer depth (in fractions of
contact .area width b) is shown in Fig. 231. Normal stresses (Fig. 231a)
have their greater value (a z = o y = p max ; — 0-5pmax) on the-
surface, while shearing stresses (Fig. 231 b) are maximum at a distance-
of (0.25-0.4) b from the surface.

Under omnilateral compression the yield point of hardened high-
strength steels reaches 300-500 kgf/mm 2 , which is approximately
4-5 times that under unilateral compressive stresses.

In mechanical engineering structures the load, as a rule, is cyclic,
due to the periodically varying active force, as well as the usual
relative movements of contacting bodies.

Pictured in Fig. 232 are the main operating schemes of coupled
bodies with contact load (in the descriptions design analogies are
given in brackets):

a—static load (lifting screw with spherical end); b— impact load
(tappet with spherical end); c —sphere rotating about axis normal to
contact area (ball toe); d— sphere moving parallel to support surface
(lever-mechanism with spherical striker); e —sphere rolling over sup¬
port surface (ball-mounted straight-line guide); /—sphere rolling
and simultaneously moving parallel to support surface (rolling ac¬
companied by skidding); g —transmission of rotation from, one cylin¬
der to another without resistance on driven cylinder (rolling of a
cylindrical surface by a roll); h —same but with resistance on driven

Chapter 6. Contact Strength

421

cylinder (disks of friction speed variator); i —same but accompanied
by slipping; 7 —rolling of one cylinder over another (roller bearing);
k —same but accompanied by slipping in-between cylinders (gear
teeth).

In the schemes Fig. 232 e-k, the load bears cyclic character even
if the acting force is static; the load is exerted successively over
various points on the surface.

Relative movements of conta¬
cting bodies disturbs Hertz’s
stress distributions in the zone
of contact, where the surface
layer is subjected to compression
and tension in a tangential
direction. Disposition of comp¬
ressive and tensile zones depends
on the kinematics of movement.

In sliding (Fig. 232d) and pure
rolling (Fig. 232e, g, 7 ) the com¬
pression zones on both conju¬
gated surfaces are found on one
side of the contact centre (i.e.,
opposite to the movement), while
on the other side the material
is subjected to tension (Fig. 233a).

In rolling accompanied by
slipping (Fig. 232t, k) the comp¬
ression zone on the leading
(anticipating) surface is in front
of the contact centre (opposite
to the movement), whereas on the
lagging (delaying) surface it is
behind; on the opposite lying
areas the material is subjected
to tension (Fig. 233 b).

In the leading surface compression zone (Fig. 233c) the fibres of
material converge and shift in the direction, shown by the arrows.
In the zone of tension the fibres are elastically stretched and shift
in the same direction. On the lagging surface the fibres move in the
opposite direction. As a result, friction forces appear on the con¬
tacting surface deflecting the acting forces from the normal to the
area of contact. The tangential compression and tension change the
type of the state of stress in the contact zone.

Periodic compression and tension of fibres causes, even in the case
of pure rolling with resistance on the driven body (see Fig. 232 h),
systematic lagging of the driven body. The length of surface of the
leading body at angle a (see Fig. 233c) is equal to b/2 — A b, where

Fig. 233. Compression and tension in
the zone of contact

422

Chapter 6. Contact Strength

Ab —elastic shortening of the surface. The length of surface of a driven
body over the same area will be b !2 + Ab', where Ab'—elastic
lengthening of the surface. Hence, the rotary speed of the driven
body is less than that of the driving one by the ratio

6/ 2-A& 1—2A&/6

l ~ b/2+&b' ~ 1 -f- 2Ab'/b

In practice, i — 0.99-0.995.

Clearly from the above actual conditions in the zone of contact
are more complex than those obtained under a static load, thus for¬
mulae deduced for a static load can only be applied as a first appro¬
ximation.

With contact loading durability of cyclically loaded connections
is determined from the fatigue strength of material. The fatigue
strength curves are dependent on the kind of loading. In general
they are similar to the fatigue curves, taken for the case of a uniaxial
stressed state (tension, compression, bending), the difference being
that the numerical values of breaking stresses are much higher and
the curves do not have a clearly expressed plateau (fatigue limit).

The hardness of the surface layer is the fatigue strength decisive
factor under contact load conditions (Fig. 234).

The process of fatigue failure in rolling, as well as with contact
surfaces sliding at a low speed, develops rather peculiarly. Initial
cracks emerge in the zone, where maximal tangential stresses are
active—i.e., at a depth, equalling approximately to 0.3-0.4 of the
contact area size. While gradually developing, the cracks propagate
onto the surface, producing a typical pit-like eruption pattern.
Further on these pits grow in size and blend into chains, in-between
which large metal particles flake and spall off. This phenomenon is
called pitting, which, as a rule, leads to failure as the pits unite
together.

Increasing speeds of the relative movements (rolling accompanied
by slipping) have up to a limited degree a beneficial effect. In the
process of wear the damaged layer is gradually removed and, con¬
sequently, no spalling off occurs. The durability of a joint will now
be dependent upon the intensity of the abrasive wear, changing in
the course of time the original shape of the contacting surfaces.

A typical example of a contact fatigue failure is the pitting of gear
teeth flanks. Here the pitting is generally concentrated on the areas
of the teeth lying most closely to the gear pitch circle. This can be
attributed to the fact that at conventional values of the overlapping
coefficient e — 1.2-1.8 the loading upon these areas is taken up by
one tooth only, whereas the loading, exerted onto the areas, close to
the heel and toe, is shaped by two teeth. Moreover, while middle
area of tooth profile roll without slip, the portions close to the
heel and toe roll with slip. Thus, the latter areas are, in effect,

423

Chapter 6. Contact Strength

being ground by conjugated surfaces, eliminating surface damage,
but in time leading to distortion of tbe involute profile.

A lubricant acts in two ways. Under moderate pressures an oil
film in the zone of contact contributes to a more uniform pressure
distribution and increases the actual contacting surfaces. The sur-

Fig. 234. Fatigue contact strength as a function of Rockwell hardness C

l —.steel grade 45XH (induction hardened); 2 — steel grade 20X2H4A (case hardened);
-3 — steel grade MX 15 (hardened and low tempered). Heavy lines — ultimate stresses at
JV — 10’ cycles; tine lines — at (V = 10' cycles

face rolling produces a certain hydrodynamic effect: higher pressures
occur in the film, forced from the gap which helps separation of the
metallic surfaces.

The hydrodynamic effect is still more expressed when slip is pre¬
sent: the oil, entrained by the moving surface, is continuously fed
into the narrowing part of the gap, thus separating the metallic
surfaces. Under favourable conditions (high slipping speeds, small
unit pressures, higher viscosity of oil) fluid-type friction begins.

424

Chapter 6. Contact Strength

Under high pressures the oil in the zone of contact has a negative
influence. Under the influence of the running surface and capillary
attraction, oil will penetrate into loose spots and microcracks expan¬
ding them, causing accelerated metal spalling. This phenomenon is
particularly sharp when one of the surfaces in the high pressure zone
is subjected to tension (Fig. 2335) this helping microcracks to open.

The problem of increasing the strength of contacting joints con¬
sists, first of all, in reducing contact area pressures by giving the
conjugated surfaces as rational a form as possible.

6.1. Spherical Joints

The maximum stress a max in a surface layer, during compression
of two spheres according to Hertz

<w-o. 6 yig- VF*W < 61 >

Kgf/mm 2

where P = load, acting upon the joint, kgf

E — Young’s Modulus for the material of the spheres, kgf/mm a
d = diameter of the smaller sphere, mm

a = — = ratio of diameters of the large and small spheres

The minus sign is for the case when the sphere acts on

a concave spherical surface.

With given d, P and E values
the maximum stress will be pro¬
portional to the dimensionless
value

The value can be defined as
the maximum effective contact
stress. The maximum actual
stress is equal to the product of
2 /~PE%

o Q and the factor 0.6 y -p-.

The values of a 0 as a function
of a are shown in Fig. 235 for
three kinds of loads: sphere-on-
1 ZZhS W W SO i33 ZOO a sphere, sphere on a spherical

Fig. 235. Effective maximum stress seating and sphere on a flat sur-
o 0 as a function of a = Did (comp- face (D = oo, o 0 = 1). From the
ression of spheres) graph the following conclusions

can be drawn.

The stress has its maximum value (cr 0 = 1.59) when two spheres
of the same diameter (a = 1) are compressed. As the diameter of one

MMim

'■■mill

(MHHini

6.1. Spherical Joints

425

of the spheres increases, the stress falls becoming equal to c 0 = 1
when a = oo (i.e., when a sphere rests upon a flatjjsurface).

If a sphere is in contact with a concave spherical surface, the stres¬
ses will be significantly less than in the preceding case and will

sharply drop with the decrease in a, i.e., when the diameter of the
concave spherical surface approaches closer to the sphere diameter;
tending to zero at a = 1, when the concave spherical diameter is
equal to the sphere diameter.

426

Chapter 6. Contact Strength

This does not imply that the stresses vanish completely; but only
means that Hertz’s formula is inapplicable when a « 1, because
this renders invalid one of the assumptions underlying the basic
theory, namely, the assumption that the size of the compression area
is infinitesimal in contrast to the dimensions of spheres. When a — 1
(and even with values very close to unity) the stresses should be
defined as crushing stresses.

Now, let us transform Eq. (6.1) as follows. Substitute the radicand^
^ by Ccompr — Q , where o compr is the compressive stress kgf/mm 2 , ;

produced by the action of force P in the central section of a d-diameter
sphere (actual stress for solid spheres and conditional for truncated
spheres and bodies having a limited spherical surface).

In the overwhelming majority of cases contact joints are made of
steel (E = 21 000 kgf/mm 2 ). Substituting this value into Eq. (6.1)

gives _

“ 430 yfficompr ^0 (6.3)

where a 0 —effective stress Eq. (6.2).

The diagram, presented in Fig. 236, embraces the three kinds of
loadings and shows a mas as a function of <y compr for various magnitu¬
des of a and enables any problem connected with calculations of
spherical joints to be easily solved.

A few examples for illustration.

Example 1. Find the maximum stress in a ball, 10 mm in diameter, res¬
ting upon a flat surface and loaded with force jP == 15 kgf.

P- 16

The compressive stress a compr = q ' — 1-27 ^«0.2. Starting from

the point ccompr = 0.2 on the abscissa moving vertically until meeting line
a = co, we find on the ordinate axis that cr m3X = 250 kgf/mm 2 .

Example 2. The diameter of a hall and its load are the same. The hall rests
in a spherical socket with a = 1.02. Clearly, the compressive stress in the ball
is equal to that in the preceding case (cr compr = 0.2). Now, moving upward from
this point until meeting line a = 1.02, we find that o mas = 18 kgf/mm 2 .

Example 3. The given load is 100 kgf; permissible stress a ma x — SO kgf/mm 2 .
Find the diameter of a ball suitable for these conditions when the ball is to be
mounted in a socket with ratio a = 1.02.

Follow from the point cr max = 50 kgf/mm a along the horizontal, until mee¬
ting line a = 1.02, we find on the abscissa that a com p r = 4 kgf/mm 3 . Hence,
the ball diameter

Y

1.27 P

ompr

■■V

f 1.27-100

: 5.7 mm

As the material in the contact area is subjected to omnilateral
compression, high stresses (100-200 kgf/mm 2 ) are generally permissible
when designing contact joints. With shock loads the permitted stres¬
ses are lowered 2-3 times.

Contact joint members are made of thermally hardened steels,
mostly of bearing grades IHX15 and HIX15Cr (hardness after har¬
dening and low tempering 62-65 Rc).

Fig. 237. Maximum stresses a max (with P = 1 kg!) as a function of sphere diame

ter d

Fig. 238. Spherical connections
l and 2 — point contact: s and 4 surface contact

428

Chapter 6. Contact Strength

We now explain in a generalized form the influence of sphere dia¬
meter on strength. From Eq. (6.3)

<W = 430 ya co mpr a 0 = 430 ]/" 1.27 -J- o 0 (6.4)

Consequently, the maximum stress is inversely proportional to d 2/s .

The graph, plotted in Fig, 237, is based on Eq. (6.4). It is evident
from the graph, that increase in sphere diameter sharply lowers
stresses in the area of small diameters but as diameter increases the
stress lowering proceeds more slowly. Beginning from a certain value
of d, the lowering becomes hardly perceptible. In the given case when
a — 1.02-1.1, this begins from sphere diameters of 20-30 mm. The
range of advisable diameter increase is rather larger for spheres
working on spheres.

In conclusion, let us compare the strength under point and surface
contact (Fig. 238). Assume o compr = 0.1 kgf/mm 2 for all cases. For
cases S and 4 having surface contact this stress, obviously, is equal
to the bearing stress cr& on the bearing surfaces (cr& = G C0Jnp! . =
— 0.1 kgf/mm 2 ).

Listed below are the values of 0 max , for cases 1 and 2, as well as
the cj raa .Jo b ratios, characterizing the comparative strength of joints
having point and surface contact. Since the permissible stresses under
contact loading are, on average, 5 times higher than the admissible
bearing stresses a b , the comparison is given in terms of the ratio
0max/5o&

Case i] Case 2

G max* kgf/mm 2 • . 200 15

G tix3ixI G b . 2000 150

« - • # 400 30

Obviously, the strength of a joint, when a sphere rests upon a flat
surface, is 400 times, and when seating in a spherical socket at
a = 1.02, is 30 times less than the strength obtained under surface
contact.

6.2. Cylindrical Connections

According to Hertz, the maximal stress in cylindrical connections

o/max = 0.6 1 / / 'j/" 1 ± ~ kgf/mm 2 (6.5)

where P — acting force, on the connection, kgf

E — Young’s modulus for the material of the cylinders, kgf/mm 2
d — diameter of small cylinder, mm
l = length of cylinders, mm
a = D/d — ratio of large and small cylinders diameters

6.2. Cylindrical Connections

429

The minus sign is for the case when a cylinder acts on a concave
cylindrical surface.

Denoting

1 \ _ . P

( 1 ± -J ) — O'o; 7 J- — tfompr
— 21 000 kgf/mm 2 , we find that

and assuming E — 21 000 kgf/mm 2 , we find that

Cfmax = 87 VGcompr <*0 kgf/mm 2 (6.6)

where o 0 = maximum effective contact stress
O'compr — compressive stress, taken across the meridional section
of a cylinder having diameter d

The values of <j 0 as a function of a are plotted in Fig. 239, for the
three kinds of loads: cylinder on a cylinder, cylinder in a cylindrical
seat and cylinder on a flat
surface (D — oo ; cr ? = 1). f°

In their general appea- 16

ranee the o 0 curves are sim--

ilar to the curves of the u r-

spherical connections (see A— -

Fig. 235). The stresses have 12 —A

their maximum value during_

compression of identical fg _

cylinders (o 0 = 1.41), are ' ”

lowered when a cylinder n J

contacts a flat surface °- 8 ]

(o 0 = 1) and. sharply fall T

when a cylinder seats in a 0.6—h-

cylindrical socket, approach- -4— -

ing zero if the diameter of 0.4 -

the socket becomes equal to I_

that of the cylinder (a — 1). q 2 \ _

A diagram, used when ' i
designing cylindrical conne-

ctions and based on Eq. (6.6), f z 345 10 20 SO too ZOO a

is presented in Fig. 240. nnn ..

W hile comparing this B raph function of a — Did (compression of cylin-

with the graph, shown m ders)

Fig. 236, one can clearly

see, that the cr max stresses, for the same a compr , values are much
less in the case of cylindrical joints than for the spheres.

The ratio between stresses o max sph in spherical connections and
stresses o max cyi in cylindrical, connections agreeing to equations
(6.3) and (6.6) is equal to

Umax sph 430 Y Ocompr Po sph

Umax eyl 87 Y O'compr cyl

values are much

430

Chapter 6. Contact Strength

0.0001 0.001 0.0! O.i 1 W 6 C om P >!<Of/m0 s

Fig. 240. Maximum stresses o max as a function of compressive stresses o COmp
for different values of a — D/d

If Ocompr is the same then

q max sph ^ 1 q o sph

VaxcsJ a Umpr Coc y l

Calculated in accordance with this equation and plotted in Fig. 241
is the ratio — a * sph as a a compr function, for various values of a.

q max cyl

From the graph it is clear that the G max sph stresses exceed the

6.2. Cylindrical Connections

m

CTjnax ey i stresses 7-15 times for low <j C ompr values and 1.1-2.5 times
for high. ones.

Let us compare the strength of cylindrical connections having
linear contact (cases 1 and 2, Fig. 242) with the strength of connec-

a max.sph
c max.cyL
15

W
8

5
5

3

2.5
2

1.5
1

0.01 0.1 1 a c ompikgf/mmz

Fig. 241. Ratio of stresses cr m ax in spherical connections to stresses c maS c!jl
in cylindrical connections as a function of o compr

tions with surface contact (cases 3 and 4). As in the case of spherical
connections, assume that the compressive stress is equal to a com p T ~

Fig. 242. Cylindrical connections
l and s — point contact; 3 and 4 — surface contact

= 0.1 kgf/mm 2 . For surface contact connections the bearing stresses
are equal to the compressive stresses (<j 6ear — Oaompr — 0.1 kgf/mm 2 ).
The results of the comparison are given below

432

Chapter 6. Contact Strength

<W> kgf/mm2
ff aax/ 0 6eor • • •
Omax/So&ear • -

Case l

30

300

60

Case 2
4
40
8

Clearly, the strength of connections, in which a cylinder rests on
a flat surface is 60 times, and when a cylinder rests in a cylind¬
rical socket with a = 1.02, is 8 times less than that with surface
contact.

We now explain the influence of cylinder diameter on the strength
of a connection.

Present Eq. (6.6) in the following way

_

'max '

• 87 "$/"Ocompr Oo — 87 j/^'

6d2

(6.7)

where b = -j — ratio of cylinder length to its diameter.

For geometrically similar cylinders b — const. Thus, it is evident
from Eq. (6.7) that, other conditions being equal, the tr majt stresses

are inversely proportional to

Fig. 243. Maximum 'stresses c; mas as
a function of cylinder diameter d {with
P = 1 kgf and lid = 1)

the cylinder diameter.

The cr mas values given in
Fig. 243, for cylinders of dif¬
ferent diameters are calculated
from Eq. (6.7). The general
view of the c max curves is
analogous to those of spheri¬
cal connections (see Fig. 237),
the only difference being that
in the former case the stresses
are smaller and the influence
of diameter upon the strength
is greater, than in the case of
the spheres.

With assigned loads and
cylinder diameters the load¬
carrying capacity of cylind¬
rical connections can further
be enhanced by making eyli-
the compressive stress a<

* compr •

nders longer, i.e., by reducing
This is impossible with spherical connections

In practice, however, imperfections of manufacture cause irregular
load distribution over the cylinder length as soon as the l/d ratio
exceeds 1.5-2. As a result, loads are concentrated over cylinder rims.

Now, let us compare the strength of spherical (point contact) and
cylindrical (linear contact) connections with that of surface contact
connections (Fig. 244). Assume as unity- the bearing stresses of a

6.2. Cylindrical Connections _ 433

Table 25

Stresses in Various Types of Connections

Stresses

Type of connection

a max’

kgf/mms

^bear*

kgf/mm*

a maxl°bear

a maz' 5a bear

Sphere in seat (a =1.02)

15

150

30

Cylinder in seat (a = 1.02)

4

—

40

8

Surface contact (a— 1)
Surface contact (cylinder

0.1

1

1

with l/d = 1)

r . — . * . ....•*.—. .

_

0.0785

0.785

0.785

sphere seated in a socket with a — 1 (a bear = 0.1 kgf/mm 2 ). The
result of the comparison is given in Table 25.

Obviously,, when a = 1.02, the strength of point-contact connec¬
tions is 30 times, and linear-contact connections is 8 times less than

Fig. 244. Diagrams of contact in connections
l — point contact; 2 — line contact; 3 — surface contact

the strength of connections having surfaces of a sphere and a plane
in contact. In the case of a cylinder and plane surfaces in contact the
respective figures are 40 and 10.

Hence, linear surface contact in terms of strength is more advan¬
tageous than point-contact (in the case considered it is approximately
4 times better), while surface contact is as many times better than
the compared linear contact.

(a) Design Rules

The design rules for spherical and cylindrical connections carrying
high loads are the following:

The contacting parts must be heat-treated to a hardness not lower
than 60-62 Kc and processed to a surface finish not below VlO value.

28—01395

434

Chapter 6. Contact Strength

With the aim of reducing contact stresses in the case when the
working conditions for the connection allow the body accepting the

• di lb) 1C) id) IS) {ft

Fig. 245. Strengthening of a ball step bearing

load should be supported in a socket which has a diameter close to
the body diameter (Did, = 1.01-1.02). An example of the progressive

Fig. 246. Size reduction of a spherical working surface

improvement in strength of a spherical connection is illustrated in
Fig. 245 (a spherical shaped contact unit). The most advantageous

shape is a large diameter sphere-
seating in a spherical socket,,
Fig. 245/.

In view of the fact that even
in the case of the sphere and
socket of large diameters having-
diameters close to one another
the surfaces in contact are very
small, thus, in the interest of
reducing precision machining it
is better to give the working
surfaces the largest sizes which
can be achieved by the manufa¬
cturing process (Fig. 246).

In all instances when this is
possible in the design linear con¬
tact should be preferred to point contact and surface contact to linear.

As an example, let us consider a pin and two lever connection. The
pin is fixed to one lever and slides in a fork of the other. The design^

Fig. 247. Changing line contact to
surface contact in a lever joint

6.2. Cylindrical Connections

435

shown in Fig. 247a, is irrational because contact on the rubbing sur¬
faces is linear and the pin rapidly wears the fork. In, the rational
design (Fig. 247b) the pin is held in a block the sides of which slide
in the fork. Thus, we have obtained surface contact between the pin
and the block and also between the block and the fork. As a result
the durability of the connection has been greatly increased.

Another example deals with a rotary pump blade (Fig. 248a).
One end of the blade slides in a slot of rotor 1 , while the other end,

Pig. 248. Changing, Hue contact to surface, contact for a rotary pump, vane

loaded with a strong spring and centrifugal force, is displaced relative
to the eccentric stationary housing 2. The contact between the blade
and housing wall is linear; while moving the blade abrades the housing
cylindrical surface. Operation conditions can be improved by intro¬
ducing cylindrical insert 3 (Fig. 248b): thus, the contact between the
insert and cylinder is a surface one and will materially decrease wear.

(b) Connections Operating under Impact Loads

The operating conditions in cyclically loaded, connections are
seriously impaired if the joint has clearance: coupling, surfaces
periodically move apart and close and shock loading occurs. When
designed improperly, such a connection quickly becomes unservice¬
able due to overheating, work hardening and wear of surfaces.

To improve the work capacity of connections undergoing shock
loads, it is advisable to:

raise the elasticity of the system by introducing shock-absorbers
which mitigate the shock;

lessen stresses on working surfaces by changing point and linear
contacts to surface contact and by increasing surface areas; , „

28 *

486

Chapter 6. Contact. Strength

impart greater strength, hardness and thermal stability {e.g., by
stelliting) to the working surfaces;

lessen or fully eliminate connection clearances;
provide ample lubricant to connections with the aim of creating
shock-absorbing oil film, remove heat, emanating during impacts
and (in the case of steel surfaces) eliminate tempering;

decrease weight of mechanism links to lower inertia loads.

For loads under control (e.g., in cam-actuated mechanisms) all
measures must be taken to reduce the load and its degree of impact by
decreasing the accelerations in the system (applying rationally-de¬
signed cams, e.g., parabolic and polynomial).

An example of progressive strengthening «©f a reciprocating rod
drive unit is displayed in Fig. 249. Since the clearance h is inescapable
between contacting surfaces (Fig. 249a), the load has an impact
character. The operating conditions are aggravated still more by
the fact that owing to the kinematics' of the system the movement
of the rocker is accompanied by displacement of the striker over the
rod’ end face. '

The work capacity of the connection can be improved by the appli¬
cation of a screwed-in striker and rod end cap, hardened to a high
degree (Fig. 2496). Disadvantage of the system is the point
contact.

The connection operation can also be heightened by applying a
striker with a large radiused cylindrical surface which ensures
linear contact with lower contact stresses (Fig. 249c).

Still better is the design, shown in Fig. 249d, in which sliding fric¬
tion is changed to rolling friction.

In the most rational design (Fig. 249e, /) the striker is made as a
•spherical insert with a flat working surface. Here the linear contact
is changed to surface contact due to which pressures on the working
•surfaces are sharply lowered. Owing to its spherical shape, the connec¬
tion acquires self-alignment which means uniform load distribution
over the working surface regardless of possible distortions.

6.2. Cylindrical Connections

437

In the designs Fig. 249 d, /, the rod end face is stellited. Stellite
has high hardness (50-55 Re) and keeps its hardness at elevated tem¬
peratures in contrast to hardened steels.

Fig. 250a, illustrates an example of how elasticity of a tappet can be
enhanced. During the overloads the preset force of the spring is
overcome mitigating the shock. The system is applicable only in the

Fig. 250. Tappets
(o) spring-loaded; (6) hydraulic

cases with excess values of the driving force when some deviation!
of the motion may be permitted in the final link of the mechanism)
from the designed value given by the cam profile.

In addition to the increased hardness and decreased unit pressures
on the working surfaces all measures should be taken to reduce con¬
nection clearances. Provision of adjustment (see Fig. 2496) allows
minimum clearances, consistent with the mechanism correct functio¬
ning to be set and also compensation for wear. However, adjustments
complicate use since periodic control of the mechanism state is
required.

The best solution is automatic elimination of clearance. One of
the ways is the application of hydraulic compensators.

Fig. 2506, illustrates a hydraulic tappet, used in internal com¬
bustion engines t in the valve drive mechanism.

438

Chapter 6. Contact Strength

The tappet consists of sleeve 1, which reciprocates in a gnide bush
when actuated by a cam, linked by a rod to the valve drive mechanism.
The tappet reverse stroke is actuated by a valve spring. In cylinder 2
there slides plunger 8 which has a spherical seat which receives the
valve mechanism drive rod. A ball valve loaded by a light spring
is arranged in cavity A . Oil from the pressure pipe of the engine is
continuously fed into cavity A through a system of channels. Slightly
raising the ball from its seat the oil enters the cavity and forces the'
plunger out of the cylinder, until the clearance h in all mechanism’s
links is eliminated. The plunger area is so calculated that the oil
pressure cannot open the engine valve nor materially reduce the
force, developed by the valve spring.

As the cam runs against the tappet, the pressure of oil in the space
under the plunger increases, causing the ball valve to close. The
driving force is transmitted through the column of oil, locked up in
cavity A . Since oil is practically an incompressible medium, the
mechanism acts as a solid system.

As the cam leaves the tappet the oil pressure under the plunger
decreases and oil from the pressure pipe flows under the plunger
compensating the loss which occurred during the tappet working
stroke as oil leaked away through the clearances between the plunger
and cylinder.

The system automatically assures mechanism operation without
clearance regardless of thermal expansion and wear of the linking
mechanisms working surfaces.

Clearly, oil leakage from under the plunger in no way affects the
operation of the mechanism. Moreover, it is indispensible for correct
functioning. If the system were sealed then a decreasing clearance in
the joint, as a result of temperature drop (due to lower pressure,
idle running, etc.), would probably cause incomplete valve closures.
The plungers, moved out from the cylinders by an amount, corres¬
ponding to the preceding increased clearance, would be prevented
from seating, thus keeping the valves a little open, which disturbs
correct gas distribution. The oil leakage enables the mechanism to
adapt itself to reduced clearances.

At the starting stages, when the pressure in the oil pressure pipe is
absent, the system will operate for a short period with excessive
clearances, because the driving force at this time is transmitted by
the plunger thrusting against the tappet end face. Only after the
oil pump has developed pressure, does the system come into action.
To reduce the excessive clearance working period, it is reasonable to
reduce the volume of the oil piping and also the volumes of oil spaces
in the tappets, e.g., by using plungers made of light materials (plun¬
ger 4, shown in Fig. 250b). To increase oil pump capacity is also
feasible. ■

Chapter 7

Thermal stresses and strains

Elevated temperatures are observed not only in internal combus¬
tion engines, turbines, higb-pressure compressors, etc., in wbicb
beating is a consequence of working processes. In “cold” machines
tbe mechanisms operating at high speeds and under heavy loads
(gears, bearings, cam-actuated mechanisms, etc.) are also heated.
Parts undergoing cyclic loads under multiple “loading-unloading”
cycles are also heated due to the elastic hysteresis.

Elevation of temperature is accompanied by changes in parts
dimensions and may cause high stresses.

7.1. Thermal Stresses

Thermal stresses occur in the material if the latter cannot freely
expand and contract under fluctuations of temperature. Such a be¬
haviour is either the result of constraint (impediment) of thermal
strains in a part by mated parts (contiguity constraint) or constraint
of material fibres strain by contiguous fibres (configuration con¬
straint).

(a) Contiguity Constraint

This kind of constraint can be illustrated by an example of rigid
connection of several parts, which are either subjected to different
temperatures in the course of operation or are made of materials
possessing different coefficients of linear expansion.

Let stud 1 (Fig. 251) and bush 2 be made of materials whose coef¬
ficients of linear expansion are respectively a x and cc 2 , and <x 2 >• a v
The temperatures of parts are respectively t x and t 2 . When heated
from some initial temperature (say, from zero) the stud and bush
in the free state would lengthen by the amounts la x t x and la ? t 2
(l —length of connection). In the restrained system the extension dif¬
ference equals l (a 2 t 2 —- a :! t L ) (in relative units a 2 f 2 — a ih)> causes
temperature interference. The thermal force P t in the connection will
cause (according to Hooke’s law) lengthening of the stud, equal in
relative units to i > f /i? 1 .F 1 , and contraction of the bush equal to

440

Chapter 7. Thermal Stresses and Strains

P t /E 2 F 2 (E t and E %—are respectively Young’s moduli of the mate¬
rials; F 1 and F 2 — cross-sections of the stud and bush).

Proceeding from the strain compatability law the sum of the above
values must be equal to the amount of relative temperature

interference

Fig. 251. Contiguity constraint dia¬
gram

Pt

EiFi 1

hence
P t - (a z t 2

■E Z F 2

■«A)*

: « 2 h — Uiti

EiPi

EiFi

(7.1)

! E 2 F 2

According to Eq. (7.1) the
tensile stress in the stud

I '-^Ei (a*—<Mi) Eft,

+ e 2 f 2
(7.2)

<^1

the compressive stress in the
bush

°2—7T- = E Z ( a zk—^ih)-

(7.3)

With the given value <x 2 t 2 — the stresses will be proportional
to the factors

bush, respectively.
The ratio

gpi

a 0 2

E i

(7.4)

—

.EiFi

h e 2 f 2

e 2

(7.5)

°02 —

1-J

E 2 F 2
h EiFi

relative thermal stresses in the

stud and

1-

E i •

, e 2 f 2

h EiFi Fz

(7.6)

~ e 2 ,

, E^i t\

E 2 F 2

i.e., is independent of the elasticity moduli of the stud and bush
materials, but is completely dependent on the cross-section ratio oi
the parts.

7.1. Thermal Stresses

441 '

Table 26

jPi/J ? 2 Ratio. Studs and Casings Have Equal Strength

Casing material

a compr> kgt/mmS!

Fi/&2 ^COTfipT^b

Steel 20JI

80

0.8

Grey cast iron CT32-52

100

1

High-strength cast iron BH60-2

150

1.5

Aluminium alloy AJI4

25

0.25

When F x /F 2 = 1 the tensile stresses in the stud are equal to the
compressive stresses in the bush (<y 01 /(j 02 = 1).

As a rule, the material of the stud is stronger than that of the bush,
(e.g., studs used to tighten together cast aluminium parts). Moreover,;,
cast materials have generally
higher resistance to compression
than tension.

Let studs be made of quality
steel with tensile strength o b —

— 100 kgf/mm 3 . and casings of
cast steel 20JI, grey cast iron
G t 132-52, high-strength cast iron
B *160-2 or aluminium alloy AJI4
(Table 26).

If the studs and casings have
equal strength (equal safety fac¬
tor), their sections, according to
Eq. (7.6), must be consistent
with the ratio F x /F z = o c0mpr /<j b .

In practice the cross sectional
areas of castings are usually
determined by their function and
by the casting technology. Generally the strength of casings is many,
times that of fastening studs. Therefore when designing tightening
connections it is necessary to begin from stud strength.

From Eq. (7.2) and Eq. (7.3) with given values of E x and E 2 thermal,
stresses in studs become higher with the decrease in the./’ 1 /E 2 ratio,
(Fig. 252), approaching the value E x (a 2 f a — ct^) when F 2 = oo
(absolutely rigid casing). In the casing thermal stresses grow with the-

Fig. 252. Relative thermal stresses cr 01: -
in the outer member 'and cr 02 in the
inner member jas a function of their;
sections ratio F x !F i

442

Chapter 7. Thermal Stresses and Strains

increase of the F t /F 2 ratio approaching the value E % (a 2 t 2 — )

when F 2 = 0 (absolutely elastic casing).

On the graph, presented in Fig. 252, the values of cr 01 and cr 02 are
■.given as a function of ratio FJF % . Assumed as unity for <r 0 i is the

(a) (b) -(c)

Fig. 253. Ways of reducing thermal stresses in tightened connections

value Ex (a a f 2 — a,^), and for u 03 the value E„ (a 2 t 2 — aih). From
the graph and Eq. (7.4), (7.5) it is possible to conclude that:

to reduce thermal stresses in studs it is necessary to make them
rigid and the casing elastic-,

to reduce thermal stresses in the casing studs have to be elastic
and the casing rigid.

Since the strength of a casing is not decisive for the strength of the
tightened connections therefore all thermally loaded connections
must keep the following principle: elastic casing—rigid studs.

Equations (7.4-7.5) do not contain the length of the connection.
This means that thermal forces and stresses are independent of the
stud and bush lengths provided they are the same. Under similar
conditions the theoretical values are identical, for example, in the
• case^when a stud fastens a flange 10 mm thick or a casing 500 mm
high.

7.1. Thermal Stresses

443

In practice,, the thermal stress value is affected by the elastic de¬
formation of threads, washers, etc.,, which may decrease thermal
stresses. This influence is relatively greater the shorter the studs.
Elastic strains are connected with residual strains (disto¬

rtion of threads and bea- 6
ring surfaces). Short studs a '®
weaken in use quicker
than longer studs which
maintain their elasticity
better.

Apart from thermal stre¬
sses, the strength of tigh¬
tened connections is depen¬
dent on the preliminary
tightening force as well as
the working forces, acting
upon the connection. The
combination of these factors
and practical recommenda¬
tions on the design of faste¬
ning connections are con¬
sidered further in Vol. 2,

Section 1.

From the structure of Eq.

‘(7.1), it is evident that
thermal forces can be redu¬
ced by:

decreasing the tempera¬
tures difference of parts to
be joined together (e.g., cooling the part being fastened or incre
using the temperature of the fastening part);

decreasing the differences in the linear expansion coeffi
-cients (by appropriate selection of materials for the parts to be
mated).

If the materials of a fastened and fastening parts cannot be varied,;
then, thermal stresses can be lowered by introducing some additional
■elements (Fig. 253a) between the fastened and fastening parts made
of a material with a lower linear expansion coefficient (e.g. s
invar).

Invar (H36) is an iron-nickel alloy (36% Ni). The coefficient of its
linear expansion in the temperatures interval from minus 100°C
to plus 100°G equals a — (1.1-2) -10“® "C -1 , and sharply rises with
■t > 100°C (Fig. 254).

In this case the temperature interference

4oa*a +th

444

Chapter 7. Thermal Stresses and Strains

where a 2 , <4 — coefficient of linear expansion of respectively fas-
and a x tened, intermediate and fastening parts
f 2 , t' % — their corresponding temperatures and lengths
and t x
and Z a , Z 2
and l x

Conditions for absence of temperature interference
h a zh + — lt a it t — 0

Introducing l' z — h — h and assuming t 2 — i' = t x , we find that

« 2 ~ «1
cti—-<%

(7.7)

Substituting <x 2 — 22-lO -6 (aluminium alloy), a 2 = 2 40~ 8 (invar
at t <t 150°C) and a x — 11 •10~ 6 (steel), we find that

Z;«1.2Z 2 (7.8)

i.e., to fully eliminate temperature interference, the length of an
invar insert must be 20% greater than the length of parts being
iastened. Such conditions are difficult to fulfil in practice.

There are some materials, whose coefficient of linear expansion is
zero or even negative. In the latter case parts shrink when heated.
Belonging to such materials are some si tails (a = —2-10~ 6 ). Cal¬
culations, based on Eq. (7.7) will give in this case V 2 = 0.8Z 2 . Clearly*
even under such conditions the intermediate bush must have sub¬
stantial length.

In order to reduce thermal stresses, fastening bolts must occasional¬
ly be made of materials possessing high coefficients of linear expan¬
sion, for instance, of Cr-Ni austenitic steels (Fig. 2535), for which
a = (14-18)-10" 3 . Let us compare fastening of an aluminium casing
(a 2 = 22-10 -8 ) by means of bolts made of conventional structural
steel (a x = 11 -10 ~ 6 ) and by bolts made of austenitic steel (a —
= 16*10~ 8 ). When changing to the austenitic steel the thermal stres¬
ses decrease in the ratio

a 2 — a t 22—It _
a 2 ~ai ' 22 —16 “

(7.9)

i.e., approximately by half.

It should be remembered that the strength of austenitic steels is
significantly lower than high-grade structural steels. For example,
austenitic steel, grade 91169, with a = (16-18) -10 ~ 6 has the tensile
strength level o b = 70 kgf/mrn 2 and yield limit o 0 .a — 40 kgf/mm 2 .
Structural steel, grade 30XFC (a = 11 -10“ 6 ) extensively used for
producing load-carrying bolts has: o & == 110 kgf/mm a and o 0 . 2 =

446

Chapter 7. Thermal Stresses and Strains

— 85 kgf/mm 2 . From the strength point of view the latter steel, for
the given case, is definitely more advantageous in spite of its small
coefficient of linear expansion.

In addition, austenitic steels are much more expensive than low-
and medium-alloy steels.

An efficient means of reducing thermal stresses in bolts is to en¬
hance elasticity of the part being fastened (see Eqs. 7.2 and 7.3).
If, however, this is impossible, then some spring elements (Fig. 253c)
must be placed under the bolts, which is equivalent, in effect, to
increasing the casing elasticity. If the spring element has sufficient
elasticity, the thermal forces can practically be fully cancelled, the
spring element considered as a preliminary tightening force and as
a working force, acting on the connection.

This method is often applied to absorb thermal deformations when
several details are assembled on one shaft if the details are made from
alloys having high linear coefficient of expansion (e.g., rotors of
multi-stage axial compressors). To fix and mount such details
requires considerable axial forces. That is why in a given case spring-
elements are made as a stack of numerous, strong and very rigid
elements (Fig. 255), which when summed give the required elasticity.
Elastic element calculations are described in Vol. 2, Section 1.

(b) Form. Constraint

Thermal stresses, produced by form constraint, occur during non-
uniform heating of the part, when individual fibres of material cannot,
(due to the shape of the part) expand in accordance with the thermal
expansion law. In contrast to the contiguity constraint, stresses here
are due only to the existence of a temperature gradient in the part
(with stationary heat flow, when heat flows from hot areas to cooler
ones, or during unstable heat flow, e.g., thermal shock, when a wave
of heat propagates throughout the part).

As a rule, hot areas of a part, having temperatures exceeding the
average, undergo compressive stresses and the more cool areas—
tensile stresses.

Thermal stresses do not occur in a uniformly heated part, where
the temperature at any portion is the same.

(c) Flat Walls

Imagine a flat wall of thickness s (Fig. 256a), through which passes,
normally to its plane, a uniform thermal flow. Assume also that the
wall surface facing the heat source has temperature t x , and that the
opposite surface temperature is t 2 and that t 2 . The temperature
across the wall according to the heat transfer theory in this case

7.1. Thermal Stresses

447

changes to a linear law. The average wall temperature

= ^ (7.10)

Now imagine that the plate consists of a number of thin parallel
layers. If each layer could expand freely when heated, then the layers
with temperature above t av would expand relative to the average

Fig. 256. Determination of thermal stresses in a flat wall

layer, and the layers with temperatures below t av would shrink.,
Hence, the plate would take the shape, pictured in Fig. 2565.

The relative extension of the extreme, most heated layer

ei = a(ti—tav) = a (7.11)”

The relative shrinkage (shortening) of the coolest layer

( ,

e z =a(t av — t z ) = a — # 2 ) =«^- (7.12).

i.e.,

e 2 = e 1 = e max = aiiZ±. (7.13)-

If the plate maintains its flat shape during heating, then all its
layers by virtue of deformation compatability, must have sizes equal
to the sizes of the average layer. In such a plate the most heated layers
are compressed by the constraint of adjacent less heated layers
(Fig. 256c), while the coolest layers are stretched by the action of the
hot layers (Fig. 256.<f), each along two, mutually perpendicular
directions. The highest stresses arise in the extreme surface layers.

As known from the theory of elasticity, relative elongation under-
the conditions of a biaxial stressed state
along axis x

(7.14)

448

Chapter 7. Thermal Stresses and Strains

along axis y

(7.15)

where g x — stresses along x and y axes respectively

and Gy

E — Young’s Modulus

m — Poisson’s ratio (transverse strain coefficient), which is,
in fact, the ratio between the value of transverse com¬
pression and longitudinal elongation within the limits
of elastic strains, taken for the case of simple tension in
one direction

With symmetrical tension-compression (as in the case under
•consideration)

tfjc — c y = cr, e x — e y = e

Consequently

G^Ee-j-^ —

1 — m

Substituting the e-value from Eq. (7.13) into the above expression,
.gives the maximum stress value in the extreme layers

(7.16)

where the plus sign refers to tension, and the minus sign to compres¬
sion.

The stresses across the wall are distributed according to a linear
law similar to the temperature change law.

The temperature gradients may be expressed in terms of the quan¬
tity of heat Q, passing through a wall in unit time and unit surface.
According to Fourier

cal/m 2 -h (7.17)

where X = heat conductivity coefficient of the material, cal/'m -h - C C
s == wall thickness, m

Substituting t x — t 2 from Eq. (7.17) into Eq. (7.16), gives

Q Ea 1 yrr A

O m ax^Y S "‘T^m ( 7-18 )

( d ) Thermal Strength of Materials

As obvious from Eq. (7.18) the maximum thermal tension-com¬
pression stresses for a given heat flow intensity are proportional to

the wall thickness and factor , which is specific for any mate¬

rial (Table 27).

7.1. Thermal Stresses

449

Table 27

Thermal Strength Characteristics of Materials

Materials

a

£

<n i?
i

o bn
««h.M

JL

oO

o

sB

o

o

4

g

Si

S

1

6

a S

►o bn
to ■«

Grey cast irons

8

ii

35

0.15

3

30

10

Garbon steels

21

ii

40

0.3

8.3

60

m

Alloy steels

21

12

3S

0.3 j

'

10.4

120

11.5

Stainless austenitic steels

21

16

15

0.3

32

70

| 2.2

Aluminium J

alloys j

f castable

1

j

7.2

22

150

0.33

1.6

20

12.5

wrought

50

31

Magnesium J

alloys

|" castable

4.2

28

70

0.33

2.5

15

6

wrought

<

25

10

Bronzes

| 11

18

70

0.33

4.2

60

14

Titanium alloys

12

8.5

B

0.3

21

120

5.7

Materials, with a zero linear expansion coefficient are free from
thermal stresses due to form constraint.

Fig. 257 gives the values of linear expansion coefficient for various
materials, plotted against temperatures.

For all materials the j--~- factor is close to 1.5 (except cast irons,
for which the factor equals 1.18). When roughly comparing the
materials in terms of thermal stresses, a simplified expression
may be employed.

29 — 01395

450

Chapter 7. Thermal Stresses and Strains

A comparative study of Table 27 clearly shows that according to
the thermal stress values (low valued factor a ^ oys

are the most advantageous and titanium alloys and stainless steels
the worst.

Fig. 257. Linear expansion coefficient of metals as a. function of temperature

l — titanium alloys; 2 — martensitic steels; s — pearlitic steels and cast irons; 4 — auste¬
nitic steels; s — copper-base alloys; « — aluminium alloys; 7 — magnesium alloys

Thermal strength, i.e., material resistance to the action of thermal
stresses, can be expressed by the ratio between the tensile strength

Ob of the material and the factor '{—m ( aiia ^ 0 8' ous to safety strength)

Ea

The values of this factor are presented in the extreme right-hand
column of Table 27.

By their resistance to thermal stresses (high value of the factor)
wrought aluminium alloys occupy the first place. Much less profit¬
able are titanium alloys and austenitic stainless steels.

7.1. Thermal Stresses

451

The above cited correlations are valid only for temperatures up
to 200°C when the characteristics of strength, elasticity, linear
expansion and heat conductivity for usual structural materials
undergo comparatively small changes. These correlations become
invalid at higher temperatures, since now it is the heat-resis¬
tance, that is of the most important significance Belonging
to heat-resistant materials are steels, alloyed Ni, W, Mo, Ta, nickel-
based alloys, titanium alloys, etc. In the high-temperature domain
the qualitative correlations between materials alter. With the in¬
crease of temperature most of the above-examined materials (e.g\,
steels of conventional composition) lose their strength; some of them
are unable to endure high temperatures at all (light alloys). Conver¬
sely, titanium alloys, which under moderate temperatures are hardly
not the worst, according to their thermal stress values, now, due to
their high heat-resistance take one of the first places.

(i e) Curvilinear Walls

In our previous argument it was assumed that a flat plate sub¬
jected to thermal strains preserved its flatness, that it was either
arranged in rigid guide-ways or had sufficient rigidity against ben¬
ding. Now, if the plate is freely deformed under the influence of the
temperature gradient, then the thermal strains will decrease, comple¬
tely vanishing under certain conditions, for example, if the plate is
thin enough, made of a material with low elasticity and can bend
completely so that its external fibres elongate and internal ones
shorten by the value a (4 — 1 2 ). In this case the plate will bend to
a sphere (Fig. 258a), whose mean radius

^ ==s [a(j 1 -i 2 r + *]

Should free bedning be possible in only one direction, then the
plate will bend to a cylinder (Fig. 258b), whose mean radius

Rmean= a^t z )

Stresses along y-axis in this case will weaken or vanish completely,;
while stresses along the ar-axis will maintain. The amount of these
stresses can be found by putting a y = 0 in Eq. (7.14).

Then

Since according to Eq. (7.13)

29 *

452

Chapter 7. Thermal Stresses and Strains

then

F*h=h.-&8 —
; £a ___—-s %

(7.19)

Hence, Eq. (7.18) gives the maximum value of thermal stresses m
a wall, which is unable to change its shape, while Eq. (7.19) the

Fig. 258. Flexure of a thin plate under the effect of thermal stresses

value of stresses, arising when the wall shape can change m °^e direc¬
tion only. In the intermediate cases the thermal stress values w

vary from 1 to (i.e., from 1 to 1.5 on average).

(/) Cylindrical Tubes

In practice there are cases when the part due to its configuration
maintains its shape fully or nearly in the presence of a temperature

Fig. 259. Deformation of an unrestrained cylinder end

gradient. A typical example is a cylindrical tube of a sufficientiy large
length. In the event of a unilateral heating, for example, from inside,
ipi~ s 259a the tube will expand both radially and axially, keeping,
omthewh'olefits cylindrical shape. The internal, warmer wall layers,

7.1. Thermal Stresses

453

will be subjected to compressive stresses, and those on the outside,
the cooler layers, to tensile stresses.

The stresses will fall only at the free end of the tube, where the
constraining influence of the circular sections is weaker as a result
of which the tube expands in a flare-like way.

The picture is quite the opposite when the tube is heated from
outside (Fig. 259b): the outside, warmer layers are subjected to com¬
pression, and the inside cooler
ones, to tension. The tube free
ends “flare” inwards.

When it is necessary to preserve
the true cylindrical shape circular
rigidity ribs should be introduced
on the tube ends (Fig. 259c).

The correlations, formulated
for a flat wall, remain valid for
a cylindrical tube as well, pro¬
vided due corrections are made
for eurvilinearity of walls and
different distribution of tempe¬
rature across the wall.

If temperature distribution
across the wall is arbitrary
(Fig. 260), the average wall tem¬
perature will be

ta

t =4{ tdz

ti

Fig. 260, Determination of the average
temperature of a wall, with the tempe¬
rature across the wall changing nonu-
niformly

being, in fact, the height of a rec¬
tangle with base s, area abed ,

having a value equal to the area of the temperature diagram.

When a stationary heat flux is flowing outwardly, the tempera¬
ture across a cylindrical wall changes according to a logari¬
thmic law

f + Qr In R

t—ti — - -

where Q — heat quantity, flowing through the wall in unit time
i = heat conductivity coefficient
r and R = respectively internal and external radii of the cylinder
For the case under consideration the average temperature of wall

R

tav== s "(R + r)" }

where p is the present radius.

454

Chapter 7. Thermal Stresses and Strains

The tau value is determined either analytically or gjaphical y.
The maximum thermal stresses in the extreme layers of the cylin¬
drical wall are similar to Eq. (7.16)

(7ma X « ± Ea - —y— c

where c is the correction factor for the wall cylindncity.

For tensile stresses (wall cold side)

2y 2 _1_

c ~~ y 2 —1 lay

for compressive stresses (wall hot side)

2 1

c = -

iny

where y == ~.

For small wall thicknesses the effect of curvilineanty can be
neglected, and the thermal stresses determined from Eq.

Thermal stresses may reach substantial values, and m certain
cases can limit the strength of a part. , nAmm

We now consider a steel tube heated from outside (l.D. - iw mm,
O.D. — 120 mm) and having a wall-to-wall temperature dilieren

of 30°. £ „ . n

According to Eq. (7.16) thermal stresses for a flat wall

1 . = 22•10 3 • 11• 10' 6 . 1 .44 • 15 = 5.5 kgf/mm 2

a max — Ea i __ m - 2

The correction factor c :
for tension

2-1.2 2 1

c — 1.22—1 "

for compression
2

1.2 2 -1

2.88

In 1.2 0.44 0.182

:6.55 — 5.5=1.05

Css-

0.182

:4.55 —5.5;

-0.95

In 1.2 0.44

Hence, the tensile stresses

<f (w -5.5-1.05:
compressive stresses

Ocompc — 5.5-0.95 — 5.2 kgf/mm 2

: 5.8 kgf/mm 2

The thermal tensile stresses obtained are as if the tube were su. -
jected to a burst of internal pressure, equal to (when calculating e
tube wall in tension to the formula of Boyle-Mariotte)

p = 10 2 -5.8

2s

1 10 2

5 . 8-20

100

: 116 atm

7.1. Thermal Stresses

455

(g) Addition of Thermal and Working Stresses

Generally thermal stresses are associated with stresses, produced
hy outer loads. Such a combination can be beneficial, if the thermal
■and working stress association results in lower stresses and can also
be unfavourable when the resultant stresses are higher. This depends
•on the relationship between the thermal and working stresses and the
law of their change through the wall.

Figure 261a, shows a thin-walled tube, carrying a hot fluid or a
gas under high pressure and cooled from outside (continuous arrows
show the direction of pressure, while dashed arrows indicate the
direction of heat flow). Distribution of working stresses across the wall
is shown according to the Boyle-Mariotte formula, with a straight
line. The summation of working o w and thermal a th stresses produces
the tensile stress peak a on the outside (Fig. 261a, III).

For a tube, carrying an operating fluid or a gas under high pressure
and heated from outside (Fig. 2166), the summation of working a w
and thermal o ih stresses produces the tensile stress peak cr on the
wall inside.

If the tube is subjected to an external pressure, then heating both
from inside (Fig. 261c) and outside (Fig. 261d) causes only the com¬
pressive stress peaks, which are less harmful than tensile stresses.

In thick-walled tubes the distribution of working and thermal
stresses across the wall is different. Here with favourable relation¬
ship thermal stresses may decrease the resultant stresses and lead to
a more uniform distribution of stresses across the wall (Fig. 261c).

It should be emphasized, however, that an increase in wall thick¬
ness does not always imply an improvement of tube strength in terms
of its resistance against the joint action of external and thermal loads.

According to the Boyle-Mariotte formula, the tensile stresses, pro¬
duced by internal pressure, for thin-walled tubes will equal

decreasing with the increase of wall thickness. The thermal stresses,
as seen from Eq. (7.19), at a given heat flow intensity Q, will rise
with the increase of wall thickness. Consequently, there exists some
optimal wall thickness at which the summed stresses across the wall
have a minimum value.

Determining thermal stresses from Eq. (7.18), one obtains the
following relation for total stress

v — G w¥ G th —

pd . Qs Ea,
__

1

1 — m

Differentiating this relation with respect to $ and equating the
derivative to zero, the maximum value of s, for which a has its

7.1. Thermal Stresses

457

minimum value will be found:

■-/r/iS < 7 - 20 >

It can be seen that this value depends on the operational fa tors
( Q,p) and also upon the characteristics of the material (X, E, a and m).

Presented in Fig. 262 as a function of wall thickness are the stresses
in a steel tube, having the following parameters: d = 100 mm, p =
= 100 kgf/cm 2 and Q = 100 000
cal/m 2 -h. The total stresses will
have a sharply defined minimum
at a wall thickness of s 10 mm.

Increasing wall thickness beyond
this optimal value leads to a
stress increase.

(h) Disk-Like Parts. Rotors

Thermal stresses play a very
important role in the strength
of high-speed rotors of thermal
machines (e.g., turbines, centri¬
fugal and axial compressors,
etc.). The rotors, being subjected
to bursting loads produce cent¬
rifugal forces and thermal stres¬
ses caused by uneven tempera¬
tures in the rotor body. Gene¬
rally the temperature is higher
at the rotor periphery. Here
thermal compressive stresses
occur. At the hub, i.e., where the tensile stresses from the centrifugal
force have their greatest value, arise thermal tensile stresses. Addi¬
tional tensile stresses are added for interference fitted rotors.

The distribution of thermal, centrifugal and total stresses in the
symmetrical cross-sectional plane of the rotor is shown diagramma-
tically in Fig. 263a.

Adding the thermal stresses o th and centrifugal stresses o ct j
shows a tensile stress peak at the rotor hub.

To determine the thermal stresses in the rotor is difficult since the
regulating of the changing temperature in the rotor body depends on
the operating conditions. Moreover, rotors in the majority of cases
have intricate profiles which affect the magnitude of thermal stresses
in the axial and circumferential directions.

Rather critical is the starting period when the blades and peri¬
phery of rotor quickly heat under the influence of operating gases.

0 10 20 30 s, mm

Fig. 262. Optimum wall thickness of
a tube subjected to tensile stresses a op
and thermal stresses

•458

Chapter 7. Thermal Stresses and Strains

while the hub still remains cold. In this case the tensile stresses near
the hub are maximum. Under working conditions the rotor tempera¬
ture equalizes due to which thermal stresses decrease.

When the blade temperatures fall, during idle running, the picture
is the opposite: the rotor periphery becomes cooler than the hub

Fig. 263. Stresses in a rotating disk
— temperature; a t — thermal stresses; a centr — stresses due to centrifugal forces; cr —

summed stresses

(Fig. 263&), and at the periphery occur thermal tensile stresses and
■at the hub, compressive ones. The peak of the total stresses then shifts
to the periphery. Since rotor speed is less during idle running these
working conditions are less harmful to rotor strength.

7.1. Thermal Stresses

459

(i) Decreasing Thermal Stresses

Methods for decreasing thermal stresses being caused by configura¬
tion constraint, are aimed primarily at eliminating or, at least, dimi¬
nishing the source of trouble-irregularities of temperature field over
the part section. Sometimes this is obtained by cooling the part. Thus,
for turbine rotors it is advisable ^

to cool their peripheral parts. W& Jp|f 11

Cooling of the central portion | ^ M Cl J 0

of rotor is irrational as, when 11 11 II 111

working, the drop of tempera- | ^ M i P \ t

ture can cause even greater bur- | ^ MCI 5 I

sting stresses in the hub. Mil \\

If the temperature diffe- ^ W M M

rential cannot be eliminated ^ ^ ^

owing to some functional rea- ^ \ \ 5

son (e.g., heat-exchanger tu- ^-4-- 0 0 -ph “

bes), then it is advantageous W

to use materials which have Fig. 264. Reduction o! thermal stresses
beneficial combinations of through introduction of a thermal buffer
strength, heat-conductivity and

thermal expansion characteristics (see Table 27). For instance,
sitall tubes with a zero coefficient of linear expansion are perfectly
immune to thermal stresses.

Thermal stresses can be broken down by using thermal buffers,
i.e., by imparting compliance to areas where the temperature differs
from the temperature in adjacent regions.

Figure 264 shows ways of improving the design of an engine cylin¬
der cooling jacket. In the rigid design (Fig, 264a) significant thermal
stresses are possible due to temperature differences between the
jacket and cylinder walls. In the cylinder walls which have the

460

Chapter 7. Thermal Stresses and Strains

higher temperature axially directed compressive stresses arise and
in the jacket walls tensile stresses. The corrugated jacket walls,
materially increasing the elasticity of the system, Fig. 264a, sharply
reduce the thermal stresses, Fig. 264b.

In box-shaped components stiffening collars, flat partitions and
abrupt transitions (Fig. 265a), which enhance the configuration con¬
straint factor, should be avoided. More beneficial are conical, sphe¬
rical and other similar shapes, which assure smooth transitions from
o.ne part of the detail to another part (Fig. 2656). These measures help
to level the temperature gradient and increase the compliance of
parts in the direction of acting thermal forces.

(j) Expansion Joints

In several cases without detriment to the functional purpose of the
part is complete elimination, or almost complete elimination of

Fig. 266. Expansion joints in circular radiator fins

form restraint as the original source of thermal stresses. Expansion
joints, which serve as an example are, in fact, radial grooves made

in the circular fins of sleeves in
air-cooled engines (Fig. 266a).
To avoid any distortion of cor¬
rect cylindrical shape such gro¬
oves are arranged either in a
staggered (Fig. 2666) or helical
(Fig. 266c) pattern.

Expansion joints do not sig¬
nificantly worsen heat-dissipa¬
tion of ribs.

Fig. 267. Cooling surface formed by The ribbing can be totally de¬
coded spirals prived of thermal stresses if the

grooves are arranged so thickly
that the circular ribs turn into separate pillars (needle-like surface),
like those pictured in Fig. 266d. Loss of the cooling surfaces at the

7.2. Thermal Strains

461

areas of grooves is compensated by the new cooling surfaces formed
at the groove faces. Compensation can be complete if the groove
width equals rib thickness. Moreover, thermal dissipation is still
better owing to the enhanced turbulenc of air flow amidst the ribs.
The design is considerably lighter, approximately by half, when the
groove width equals that of the cooling needles.

Still further development of this cooling principle is in the creation
of ,a brush-like surface, for example, by soldering spiral wire coils to
the walls (Fig. 267).

7.2. Thermal Strains

Thermal strains sometimes change completely the size of parts
and their relative position in a unit. This feature must be accounted
for when designing units,
comprising parts having
different working tempera¬
tures or made of materials
possessing different linear
expansion coefficients.

(a) Axial Clearances

Thermal strains may to
a large degree change axial
clearances in mechanical
joints.

Let us take for an example
the design of a fixed plain
bearing (Fig. 268). Let the
shaft be made of a steel
whose linear expansion coe¬
fficient is a,, and the be- Fig-268. Determination of thermal end face

aring body be made of an clearance m a locating plain bearing

alloy with a 2 . The working

temperatures are respectively and t 2 .

When assembled the end face clearance (cold) is

A ~L sh — Lb

where L sh and L b are respectively lengths of shaft journal and
bearing.

When heated to the working temperature f x , the length of the shaft
journal will be

L’sh = L s h [1 + — 4 )]

462

Chapter 7. Thermal Stresses and Strains

length of bearing

L{, = Lb [1 + a 2 (4—4)1

The end face clearance in the working state
A — L s h Lb — L$h Lb -\-L 3 )jttj (4 — 4) — 4&oc 2 (4 — 4) —

= A -{-Lsh £04(4 -—4)—x^" a 2 (4 — 4) J

where t 0 is the assembly temperature.

As the relation is very close to unity, one may say

A' — A -j- A;

where A ( is the thermal change of the clearance.

Aj = L s h (oq (4—4) — ®2(£24)3 (7.21)

Depending on the relationship between 01, a 2 and t t , 4 the original
(cold) clearance may increase or decrease. The latter case is dangerous
as the shaft end faces may seize.

Let the bearing body be made of aluminium alloy with a linear
expansion coefficient a a = 23 ■10~ 6 , and the shaft be made of a steel
with «i = 11 •10"®. If we assume 100°C for the working temperature
of the bearing body and 50°C for the shaft, the length of the shaft
journal 100 mm, the assembly temperature 20°C and the amount of
the original cold clearance equal to 0.05 mm, then the thermal change
of the clearance according to Eq. (7.21)

A« = 100 [11 • 10 -6 (50 — 20) — 23.10" 6 (100 - 20)) =

= 100 ( — 0.0015)——0.15 mm

Hot clearance

A' = AAi = 0.05 — 0.15=—0.1 mm

Thus, in the connection interference of 0.1 mm arises; consequent¬
ly, the shaft will seize in the bearing. If a minimum clearance is
necessary in the working state, e.g., 0.05 mm, then the original cold
clearance must equal 0.05 + 0.15 — 0.2 mm.

The selection of correct face clearances is of utmost importance
for multi-supported shafts with bearings spaced at fairly large
distances one from another (Fig. 269). Let the front bearing A be
the fixed one. To avoid shaft seizure due to the crankcase heating
it is necessary to provide adequate clearances between the webs of
the crankshaft and the faces of the corresponding bearings. These
clearances, designated as A x , A a and A 3 , must be proportional to the
distances L x , L 2 and L 3 , of the bearings from the fixed one. Using
the numerical values of 04, 04, 4, 4 and 4 from the previous example

7.2. Thermal Strains

463

and assuming L x = 300, L % = 500 and L s = 700 mm, we obtain the
following values for the thermal changes in the clearances

Aj = 300 (-0.0015) = -0.45 mm
A 2 = 500 (-0.0015)= -0.75 mm
A 3 = 700 (-0.0015)= -1.05 mm

Fig. 269. Determining thermal end face clearances in a multi-support crankshaft

design

When fixing the clearances one should add to them the magnitudes
of the original cold clearances, which were established with due
account to the corresponding shaft and crankcase dimensions..

( b) Position of Locating Bases

The position of locating bases should be chosen with such consi¬
deration that with all possible temperatures, the accuracy of the
sizes of the details positioned in the system is not impaired or,
if so, to the very smallest degree.

In the bevel gearing unit, accommodated inside a light alloy'hou¬
sing (Fig. 270a). the fixing bearing 1 is spaced at distance L far from
the centre of engaged gears. Elongation of the housing because
of heating will make the minor gear shift in the direction shown by
the arrow. The major gear will also shift in the same direction but
over a smaller distance (this being due to a lesser coefficient of linear
expansion of the steel shaft). As a result, the engagement clearance
decreases. With certain relationship the gears will start operating
under thrust.

464

Chapter 7. Thermal Stresses and Strains

ilV.I

u&iii

ia

mi

mil

Fig. 270. Fixing a horizontal shaft in a bevel gear unit

of engagement. The displacement of the bevel gears one relative to
the other, during heating is much less; in this instance the engage¬
ment clearance increases, not decreases as in the previous case.

(c) Assurance of Free Thermal Deformations

Axial fixing of parts at two points should be avoided as thermal
deformations in such cases due to form constraint can cause thermal
stresses.

mi

ISil

V////////y//////77/777/7/Z///7/7/ 7//Y///77777Z/ZFZ/Z/ZW/////////

(a) (b)

Fig. 271. Fixing a shaft in antifriction bearings

An example of incorrect mounting is the fixing of a shaft in two
antifriction bearings simultaneously (Fig. 271a). If the bearing
body is made of a material whose linear expansion coefficient dif¬
fers from that of the shaft, and if the shaft and the casing have
different working temperatures, then either clearance or interference
occur in the unit, causing bearing seizure. Inevitable errors in the
axial dimensions of connections during manufacture may, in turn,
cause clearance or interference. ■

7.2. Thermal Strains

46 5

Hence/ the shaft must be fixed at one bearing (Fig. 2716). The
second bearing must float, i.e., have free movement in the axial
direction.

Another example is presented in Fig. 272, which illustrates a cylin¬
der liner of, an internal combustion engine directly being cooled
by water. The version, shown in Fig. 272a, in which'^the liner is

Fig. 273. A turbine housing mounting foot

fixed in two points, by the top shoulder and the sealing shoulder
is a mistake. Thermal forces arise in the unit, the liner is heated,
compressing the liner and stretching the jacket. In the correct design,
pictured in Fig. 2726, the liner is fixed by its top shoulder only
and the seal can float, therefore the liner freely moves relative
to the jacket.

30—01395

466

Chap ter 7. Thermal Stresses and Strains

Fig. 274- Thermal expansion compensators

Figure 273 depicts a typical fastening foot design, which fastens
•a' turbine housing to a foundation (the direction of thermal expan¬
sion of the housing is shown by an arrow). The foot is secured in place
bv an anchor holt, extending through an oblong hole. A clearance
%0.05^0tl ; mm is'provided betweerrthe bolt washer and the.fpot
end face.

7.2. Thermal Strains -■

In connections of pipes, conveying hot fluids hr gas'es, 'it is neces¬
sary to preliminary supply thermal expansion compensators pre¬
venting the onset of thermal forces and pipe deformation.

“Lyre"—shaped compensators (Pig.-,274a) have, large.sizes.:More
compact are the. cap-type (Fig. 2745) or, even better, hellows-type
compensators (Fig. 274c).

( d) Positional Changes During Heating

When designing connections working at high temperatures, it is
absolutely necessary to calculate the probable changes in dimensions
and relative position of the heated parts. '. ■

Fig. 275. Position of & valve in its seat:

As an example consider the;, fit in a seat of a discharge valve in an
internal combustion engine (Fig. 275a).

After being heated, the diameter of the valve head increases by the
value | - /

■ (7.22)

and the valve seat diameter by

A ' ttd 0 a s (t 3 — t 0 ) . (7.23)

where d 0 = diameter of valve head 1 ;

a„- == corresponding coefficients .of linear; expansion of mate-
and a s rials of valve and seat ;. !,. ,■

t v = corresponding.working;temperatures pf.valve' head and
and t s seat • • . ■ n'•

t 0 — initial temperature .(temperature of assembly)

Since the working temperature'of the valve head is well above
that of the seat, the valve during heating will move out ; of the seat
(Fig. 2755) by an amount

a = 0.5(A-A').texia/2 !U ’* "

• ■ ,v> -

where a .is the included -valve face, angle.

When a v = 90°" a'^O.S (A — ''A *)'•'' "

30 *

468

Chapter 7. Thermal Stresses and Strains

Taking into account Eq. (7.22) and Eq. (7.23)
a — 0.5tig [ct c (f„ — t G ) cc a (i t 3 tg)]

(7.24)

In high-energy engines the discharge valves and seats are made of
Cr-Ni austenitic steels (e.g., X13H7G2), whose coefficient of linear

expansion at temperatures up
to 800°C is (equal to a —
(18-20)10- 6 , Assuming the
working temperature of the
valve head equal to t 0 = 700°G,
and that of the seat t s — 300°C,
and the temperature of asse¬
mbly t 0 = 20°C, we obtain

a = d 0 0.5-20.10' 3 X
X (680° — 280°) = 0.004d 0
With a valve head dia¬
meter d 0 = 60 mm

a = 0.004 ■ 60 = 0.24 mm

To ensure a correct fit of
the valve in the seat it is
necessary to reduce the minor
diameter d of the head
(Fig. 275c) by the value
2 a & 0.5 mm.

Now let us analyse how
thermal strains affect the geo¬
metry of the valve drive unit.
In the simplest version, pre¬
sented in Fig. 276, the valve is actuated by a cam shaft, moun¬
ted in bearings in the engine head (overhead valve drive) and acts
directly upon the valve plate.

The clearance between the cam rear surface and the valve plate
in cold state

e—H — R—l ( 7 * 25 )

In the hot state

e'=*H[ 1 + ct A {t h -f 0 )l — R [1 + ( t sh — *«)]-

— I [1 -f- a„ (t„ — t Q )]+a = e + Ha h {t h -~t 0 ) —

- Ra ah (t Bh -t 0 ) - (t v -t a ) + a (7.26)

7.2. Thermal Strains

469

where a h , a sh , a v — coefficients of linear expansion of the engine

head, cam shaft and valve, respectively
t sh , t h and t„ — respective average temperatures

a = displacement of valve in seat as a result of
valve head expansion [see Eq. (7.24)]
Given: a h = 11-10" 6 (cast iron); a sk — 11 ‘lO -6 (structural steel);
a„ = 20-lO” 6 (austenitic steel); t h — 100°C; t sh = 50°C; t v = 450?C;
H = 150 mm; R — 20 mm; l — 130 mm and a — 0.24 mm. Clea¬
rance variation according to Eq. (7.25) and Eq. (7.26) will be

e l — e — 150 * 11 - lO-e (ioo°- 20°) - 20 • 11.10~« (50°- 20°) —

—130 • 20 • 1O- 0 (450° — 20°) + 0.24 » - 0.7 mm

To avoid any disturbance of gas distribution phases during star¬
ting period the cold clearance in the above case must be made equal to

e' — 0.7 -)- Sq

where e 0 is the warranted clearance.

In other valve drive designs, for instance, with a bottom distribu¬
tion or with the use of tappets, conrods, levers or rockers, the magni¬
tudes of clearance variation maiy be still greater and can be calcula¬
ted in the similar way.

In the latest engines designs, use is^made of automatic compensators,
which enable clearance in the valve distribution to be maintained
nearly constant, regardless of the engine thermal conditions.

(e) Shape Corrections of Parts

In a number of cases uneven heating distorts the shape of parts.
In such instances the initial form of the part is corrected to caldula-
tions which assure that the part takes the required shape as it is
heated to operational temperatures.

Similar methods are applied to the pistons of internal combustion
engines. The piston has its maximum temperature at the head
(Fig. 277a). Towards the skirt the temperature falls owing to the
removal of heat by piston rings, which transfer heat to the cylinder
walls, and also due to the cooling effect of oil splashed from the
case onto the internal walls of the piston. When heated, the piston
will acquire approximately a conical shape (shown by dashes in
Fig. 277a). To prevent piston head seizure the piston is machined
to an inverted conical form, converging to the head (Fig. 2775).

The magnitude of the “cold” clearance between the piston and
cylinder walls and the amount of required convergence of the piston
head can be found from the following relationship.

.470 Chapter 7. Thermal Stresses and Strains

Diametral clearance between the piston and cylinder walls in
coid state

A = D-d

where D and d

are the nominal diameters of cylinder and piston,
respectively.

In the working state the clearance is
A — A - D otj, (tp — fa) ■—-jj ct c {t c —i?p)J ^

, A — Djap ltp— 1 0 ) — a c (t c — * 0 )] (7.27)

where a.p

■■ and a c — coefficients of linear expansion of materials of
piston and cylinder, respectively
tp

'■>! and t c = average temperatures of piston and cylinder,
respectively

Let the diameter of cylinder 'D'— 100 mm, a p — 23-10 -6 (alu¬
minium alloy), a c — 11 ” 10 “® (steel), temperature of cylinder walls

■ ' . ; 300 0 ■

Fig. 277. Piston shape correction to allow for thermal deformation in heating

100°C (liquid-cooled engine), temperature of piston top collar 300°C
and bottom collar 200°G.

To assure that the piston will acquire a cylindrical form when hea¬
ted, it is necessary to assure that in the cold state the diameter of the
piston head is less than that below lower piston ring groove by the
value

Ad —100-23- 10“ fi (300°-200°) = 0.23 mm

; The alteration in clearance between the piston and cylinder in the
hot state according to Eq. (7.27) is equal to

A — A' = 1001 [23 -10 ' 6 (200° — 20°) —11 - 10 -s x
; X (100°—20°)1 — 0.32 mm

. 7.2. thermal Strains ■ ......._ 471

Now, if the minimum clearance between the piston and cylinder
in the hot state should be equal to, say, 0.8 mm, then the cold state
clearance must be equal at the head to 0.3 -j- 0.32 -f 0.23, = 0.85 mm
and below the bottom ring groove 0.3 + 0.32 = 0.62 mm..

Now determine the design clearance between the rear surface of
piston rings and the bottom of piston grooves (Fig. 278).

Fig. 278. Determining the
Clearance between the rear
surface of a piston ring and
piston groove bottom^

After the piston has been heated to the working temperature, the
diameter d 0 of the piston groove bottom increases by

Ado = (tp if 0 )

and the diameter of the cylinder

AD — Da c (t c — 1 0 )

If we neglect the change in ring thickness during heating, then the
change in diametral clearance between the ring rear surface and
inside of the piston groove

A6 = Ad 0 — A£> = d 0 a p (t p —1 0 ) — Dct c ( t a — t 0 ) =

= £>[-$- a p (t p —1 0 ) — a c (t c — «o)]

Assuming d 0 /D — 0.85 and substituting numerical values, we
obtain

A6 = 100 [0.85 • 23• 10" 6 (300°- 20°) -11-10~ 6 (100°- 20°)] = 0.46 mm

Let the clearance necessary to meet the requirements of ring
normal functioning in working state be equal to 1 mm. Hence, the
design (cold) clearance must .be equal to 1.46 mm.

472

Chapter 7. Thermal Stresses and Strains

Another example of correcting shape is the tapering of exhaust
valve stems of internal combustion engines (Fig. 279a). Since the ope¬
rating temperature of the rod upper end is lower than that at the
neck (i.e., where the rod joins the head), the diameter of the rod upper
end must be (to maintain a constant clearance throughout the guide
bush) larger than the diameter of rod near the neck by the value

6 — da v A t

where d = nominal diameter of rod

a„ = coefficient of linear expansion of valve material
At — difference in temperature between the neck and upper
end of the rod

For a valve, made from austenitic steel («„ — 20*10 -6 ), rod dia¬
meter d — 12 mm and At — 200°C

S = 12.20-10- 6 .200^0.05 mm

Correction can also be accomplished by tapering the valve guide
bore making it larger at the valve head end (Fig. 2796).

7.3. Temperature-Independent Centering

The usual methods of centering cylindrical surfaces are unsuitable
when substantial thermal strains occur in the connecting parts.

If a female part has a higher temperature or is made of a material
with a higher thermal expansion coefficient than the male part, then
clearance will occur in connection, thereby disturbing the accuracy
of centering. If, however, the situation is opposite to that described
above, then interference occurs which overloads the connection and
deforms the mated parts, which finally also disturbs centering.

This should especially be considered when designing heat engines
(e.g., gas turbines) with large-diameter housings, which are often
made of different materials.

Assume that the circular housing members of an axial compressor
and turbine are connected and centred relative to each other by
a shoulder to the class A 2a IC 2a fit; one of the members is made of
a light alloy with a coefficient of linear expansion a x = 23 -10 ~ 6 ;
the other of steel with a 2 — 11-10 -6 . The diameter of the centering
shoulder D 0 = 1000 mm.

A hole, made to 2a class accuracy may deviate within the limits
of 0 to 0.13 mm, and the shoulder diameter from 0 to -—0.09 mm.
Consequently, the connection, assembled cold may have clearance
from 0 to 0.22 mm.

Let the operating temperature of the housings be 150°C. Then the
diametral expansion difference of the centering surfaces

AT) = D 0 t op (cC|.— c&g)

7.3. Temperature-Independent Centering

473

Substituting numerical values we shall obtain
AD =1000-150(23 —11). 10-s = 1.8 mm

Adding this value to the cold clearance value (0-0.22 mm), we
obtain the hot-state clearance: minimum 1.8 mm and maximum
2.02 mm. Naturally, the centering accuracy is completely lost.

Fig. 280 presents the designs of flanges, made of metals with dif¬
ferent coefficients of thermal expansion. Shown also are some techni¬
ques, by which centering can be accomplished in the presence of
thermal strains.

In the designs shown in Fig. 280a steel flange 1 is centred by the
shoulder of the housing component 2 made from aluminium alloy.

Fig. 280.Methods of centering flanges made from materials with different linear

expansion coefficients

After the system is heated clearance appears in the joint. Centering,
in effect, is implemented by an indefinite^action of fastening bolts.

More positive centring is effected when the joint is tightened by
fitting bolts (Fig. 280&). However, with heating interference occurs
at the joint deforming the unit.

Interference also occurs when centeringbyan external shoulder on
the steel flange (Fig. 280c).

The above-described methods of centering are applicable when one
of the connecting parts can yield in the radialdirection, for instance,
if the steel flange is part of a thin-walled cylindrical housing 3
(Fig. 280d), which may somewhat expand radially. In this case the
stresses in the unit decrease.

Occasionally a double centering system is applied (Fig. 280e).
In the cold state the connection centres against the internal shoulder
of the steel flange. The external shoulder is made with clearance m
equal to the difference of the thermally increased diameters of the
aluminium and steel flanges. When heated, the centering function is
implemented by the external shoulder and clearance occurs at the
internal shoulder. In the heating period the centering becomes uncer¬
tain between the extreme limits of temperature.

An alternative of this technique is the centering by means of
a shoulder which enters with an internal clearance h into a circular
recess of the mating part (Fig. 280c).

474

Chapter 7. Thermal Stresses and Strains

(a) Star-Radial Centering

In cylindrical parts subjected to uniform thermal expansion all
elements will displace along radii, converging on the axis of sym¬
metry. If the centering elements are arranged along radial lines, the
centering will be maintained under any thermal deformations in the
system. The number of centering elements must not be less than three.

Such a type of centre alignment is termed star-radial.

Examples of star-radial centering are pictured in Fig. 281 (the
female part is made from aluminium alloy, and the male from steel).

Fig. 281. Star-radial centering

In the version, Fig. 281a, the centering elements are bolt necks 1
with flats. The necks tightly lit in radial slots in the flange. The
flange is tightened to the flange with such a force that the friction
force in the joint is less than the thermal forces arising during heating
or cooling. Occasionally the system can be tightened until the nut
thrusts against the bolt neck in order to obtain a minimum axial
clearance (some hundredths of a millimetre) in the connection.
Centring by a shoulder is not necessary in this case (the shoulder
shown in Fig. 281a serves only to accommodate packing).

A version of the star-radial method is pin (stud) centering. The
centering pins are tightly fitted into holes which are machined on
assembly in the mating part (Fig. 281b). The shoulder in this case
is only a preliminary centering flange for the machining.

This technique does not implement axial tightening of parts; the
pins only fix the parts in the axial direction. The sealing of the
joint can be attained by introducing elastic packing elements into
the joint (Fig. 231c .

Figure 282 a-i shows a number of star-radial centering techniques
applied to parts transmitting torque. Centering is accomplished by
means of keys—prismatic (Fig. 282a, b) or circular (Fig. 282c),
bolts with flats (Fig. 282d), face cams (Fig. 282c), splines (Fig. 282/)
and radial pins (Fig. 282g, h).

Preliminary centering of connecting flanges by a cylindrical sur¬
face (Fig. 282gj is used whenever a female part expands under working
temperatures more than its male counterpart. In the opposite case
radial clearance is provided between the male and female parts

Fig. 285. Star-type suspension of combustion chambers

7.3. Temperature-Independent Centering

477

(Fig. 282 h, i). Machining of holes intended to receive centering pins
is done separately (to a jig) or in assembly using dummy centring
rings.

Figures 283 and 284 illustrate star-radial centering of plain bearings
and antifriction bearings whose housings are made of light
alloys.

A star suspension is often applied when parts operate under high
temperatures and temperature gradients, for example, in furnaces.
Figure 285 illustrates a suspension
design of aircraft turbine engine
combustion chambers 1 inside casings
2 by means of radial centering sle¬
eves 5.

Figure 286 suggests an alternative
of star-radial suspension of fire-box
1 on longitudinal ribs 2 which
assure freedom for radial and axial
strains.

( b ) Alignment of Fitted-On Parts

The problem of temperature-
independent alignment of parts is
encountered when fitting turbine
rotors on shafts of centrifugal and
axial compressors, etc. If the rotor temperature is too high (turbine
runners) or if it is made from some light alloy (centrifugal and axial
compressors), then clearance will appear in the fitting zone leading
to disbalance and runout. In high-speed rotors the clearance increa¬
ses still more under the action of tensile centrifugal forces which
reach their maximum near the rotor bore. Under such circumstances
it is necessary to paralyze the effect of thermal strains and hub ten¬
sion.

An effective means is the cooling of rotors. This technique is
widely used in gas turbines. Cooling air, taken from compressor first
stages, flows around the runners and then enters the turbine duct.
Cooling of steam turbine rotors is more difficult.

Examples of temperature-independent centering of fitted parts
are illustrated in Fig. 287.

With double centering (Fig. 287a) the cold rotor is centered by its
bore upon the shaft. In the working state, when the diameter of the
hub increases, centering is accomplished by means of ring shoulders,
which embrace the hub from both sides. In the interval between the
extreme positions the rotor is decentralized and this can cause harm¬
ful vibrations.

7.3. Temperature-Independent Centering

479

Multi-stepped centering is accomplished by comb-shaped disks
with clearances successively increasing towards the periphery
(Fig. 2876). As the hub is heated its dimensions increase and new
ridges come into action, thus the alignment is kept at all stages of
heating.

Sometimes use is made of spring C-shaped rings (Fig. 287 c) fitted
between the shaft and hub. In this case the possible rotor displace¬
ments relative to the shaft are confined within the elastic strain
limits of the rings. As a result radial and axial runout of the rotor
may occur.

Star centring is effected by side faces of splines 1, whose planes
converge at the shaft axis (Fig. 287d). With uniform heating and
axisymmetrical huh tension by the centrifugal forces the system
retains its geometric similarity, and the centering is kept under any
operating conditions.

Practically the same result is obtained with splines 2 centering
on their working faces. Deviation from correct centring is less with
thinner splines, i.e., the larger their number.

Figure 287a, shows a star method where centering is(on the faces
of teeth entering radial slots of the drive disks.

When parts with long hubs are fitted upon shafts changes"in hub
axial dimensions should be considered. When heating symmetrically
in the equatorial and meridional planes each heated point of the
part will move along rays outgoing from the geometrical centre.

The simplest solution is to arrange the centering elements within
the part’s meridional plane of symmetry along radii, which con¬
verge on the axis. This principle underlies the system of pin bushes
(Fig. 287/) often applied in turbine design practice. The pins are
mounted in intermediate bushes since other ways of assembly are
impossible. The bush is introduced into the rotor hub, the pins are
fitted from inside and the rotor then assembled onto the shaft. -

The bush is interference fitted upon the shaft (sometimes by
tightening on a conical surface). Since the bush is loaded with inap¬
preciable centrifugal forces, its actual dimensions remain constant
and the interference fit is preserved. The resultant system provides
freedom of thermal deformation in the radial and axial directions
(on each side of the pin location plane).

The manufacture of joints presents considerable difficulty. First,
the. holes in the hub are drilled and reamed with the aid of special
heads in which, the cutting tools are arranged at the right angle to the
axis. Secondly, the holes in the bush and the rotor must he fully
concentric and coincident with each other.

’ Correct star-like alignment can he accomplished in another, still
more practical way-—by introducing pins from outside' into the
holes, which.have been "machined in assembly,. To preserve .the centr¬
ing and avoid any changes in the position of the rotor meridional

480

Chapter 7. Thermal Stresses and Strains

symmetrical plane the hole axes must converge onto the shaft axis
in this plane (Fig. 287 g). The same effect is attained when pins are
arranged in a row (to the left or to the right from the rotor plane
of symmetry).

Nevertheless, the system of oblique pins fails to assure true align¬
ment if the hub changes its dimensions when affected by tensile
forces. The centrifugal forces acting normally to the shaft axis will
bend the pins. Hence, the’system described above is applicable only
in those cases when thermal deformations predominate, and tensile
ones are small. The closer the pins to the axis of symmetry of the
part, the better the centering accuracy in the presence of centrifugal
forces.

Correct alignment can be ensured also under tensile stresses, if the
pins are arranged radially and displaced relative to the axis of sym¬
metry (position A, Fig. 287 h). However, in this case the axial ther¬
mal deformations will be directed away from the plane of the pins
and the meridional symmetrical plane of the rotor will, when affec¬
ted by thermal deformations, shift along the shaft.

The rotor plane, which does not change its position relative to the
shaft, is determined by the position of points, where the axes of the
pins intersect the shaft axis (positions A, B and C).

If a part to 4 be aligned has a butting face, which fixes the direction
of axial deformations and if axial thermal deformations predominate
(the case of long hubs), then the pin axes must converge in the thrust
shoulder plane (Fig. 287i). This provides free thermal expansion
of the hub.

Another form of star centering is mounting the rotor on cones whose
generatrices converge in the meridional symmetry plane of the rotor
(Fig. 287/). In this case the requirements for correct centering during
thermal movements and for a constant position of the meridional
symmetry plane are fully met. Torque can be imparted to the rotor
through the agency of a key, splines or bevel teeth (Fig. 287k).The
system will not assure centering^!, under the action of tensile forces,
the dimensions of the bore increase. An exception is the case when
the cones are tightened by a spring, which continuously eliminates
clearance on the fitting surfaces. The cone angle must be less than
friction angle (to return the hub to its initial state after cooling).

Should several rotors be set in series (Fig. 287 1), the cones ensure
correct radial alignment and keep the position of the meridional
symmetry planes of each rotor constant on the shaft, and also prevent
axial compressive stresses in the hub and tensile stresses in the shaft
during temperature variations.

Occasionally spring tightening (Fig. 287m). is applied, which sof¬
tens axially directed stresses in the system, but does not assure
radial alignment of the rotor or its constant axial position on the
shaft. In this instance the symmetry plane of the rotors during

7.4. Heat Removal

481

thermal deformations shifts by an amount proportional to their
distance from the fixed shoulder.

Correct alignment can also be accomplished by removing the
centering collars from the zone of active tensile stresses. With this
aim the centering surfaces D are separated from the rotor body by
annular recesses (Fig. 281 n). Thus, the centering collars are practically
relieved from tensile stresses and keep their original dimensions
and fit on the shaft. If the transition from the rotor body to the
centring collars has a certain form, the fit can become even tighter
as a result of rotor body tension which closes and compresses the
fitting collars.

The system will assure good alignment during thermal deforma¬
tions of the rotor if the heat transfer from the rotor body to the fitting
collars is reduced by decreasing the cross sections of the transient
areas and if the fitting collars are simultaneously cooled by fins
(Fig. 287o).

A unique design is presented in Fig. 287p. Here the rotor hub is
nearly cut into two parts by deep annular grooves: a solid part cal¬
culated to take up centrifugal and thermal forces, and a thin-walled
subhub centering bush. The dimensions of the bush isolated from
tensile stresses and from heat transfer from the rotor remain practica¬
lly unchanged, thus providing correct rotor alignment under any
conditions. The design is used in stationary installations. In aircraft
turbines and compressors the gyration forces arising in the course
of aircraft manoeuvres may cause overstresses in the connec¬
ting web.

7.4. Heat Removal

An active means of lowering thermal stresses and deformations, of
avoiding buckling and of keeping material strength is the reduction
of temperature and temperature gradient. This can be realized by
isolating the component from the source of heat or by increasing
thermal transfer into the ambient medium. With particularly high
temperatures it is good practice to introduce cooling systems with
forced delivery of cooling agent (air, oil, water, etc.).

The design of a disk friction clutch, having one friction lining
attached to the clutch body and the other, to the pressure disk
(Fig. 288 a) is nonrational because the heat, emanating during engage¬
ment, flows into the thin disk and overheats it. Much more advanta¬
geous is the design where the friction linings are attached to the clutch
disk (Fig. 288&). Due to their heat-insulating properties the linings
reliably protect the thin disk against overheating; the heat, evolved
during engagement, passes to the solid clutch housing and pressure
disk, which due to their heat capacity are heated only slightly;

Heat transfer can be intensified by eliminating thermal resistan¬
ces. In a bank-type water-cooled engine with dry liners (Fig. 289a)

31-01395

(b) ., (c)

Fig. 290. Exhaust pipe designs in air-cooled engines

7.4. Heat Removal

483

heat transfer from the liners into the cooling water is impeded by
the superfluous wall, the inevitable oil film and pollutions on the
cast surfaces. The temperature of liners directly in contact with
circulating water (Fig. 289&)

is much lower.

Obsolete designs of an air¬
cooled engine exhaust pipe
are presented in Fig. 290a,&,
with one modern design of
the same element, having
strongly developed ribbing
and better heat transfer,
pictured in Fig. 290c.

Uniform cooling of the
seat and guideway areas of
the valve is essential, oth¬
erwise the seat may lose

Fig. 291. Cooling of the guide bush and seat
of an exhaust valve

its cylindrical form, which

consequently disturbs valve function. An incorrectly designed exhaust
pipe for a water-cooled engine is shown in Fig. 291a. The error con-

Fig. 292. Increasing heat removal from the piston head in an internal combustion

engine

sists in the single-sided intake of cooling water, which results in poor
cooling of solid sections m and n. In the correct design, shown in
Fig. 291 b, cooling water is circulated around the seat and guideway.

31 *

484

Chapter 7. Thermal Stresses and Strains

Figure 292, shows methods of intensifying heat flow from internal
combustion engine pistons. The piston crown is cooled mostly with
oil thrown up from the engine crankcase. To improve heat removal
the underside of the crown is ribbed with cruciform (Fig. 292 b),
longitudinal (Fig. 292c) or wafer-like (Fig. 292 d) ribs, which, in
addition, increase the strength and rigidity of the crown. The grea¬
test cooling surface in conjunction with the smallest weight is given
tfy acicular coolers (Fig. 292e), but they do not add to the rigidity
of the crown.

, In high thermally stressed pistons forced oil cooling is used
(Fig. 292/). Cooling oil is delivered from the crankshaft journal
through bores in the connecting rod. Through a bore in the connecting
rod small end oil flows to a cavity under the piston crown and thence
back into the crankcase.

(a) Increasing Internal Heat Transfer

The use of materials with high thermal conductivity facilitates
heat transfer from hot areas to cooler ones and lowers temperature
gradients.

In components made from materials of low heat conductivity the
internal transfer pf heat can be assisted by introducing inserts made

Fig. 293. Successive steps oi extruding hollow valve

from metals of high heat conductivity (aluminium, copper) or by
filling internal spaces with a liquid heat carrier (e.g., some low-
melting metal). The latter method is widely applied in the design
of exhaust valves with sodium-cooling. Here the use of a liquid heat-
carrier is particularly advantageous because, owing to valve recipro¬
cations, the heat-carrier continuously moves, thus energetically
transferring heat from the valve hot head into relatively cooler
valve stem.

Metallic sodium possesses a number of valuable properties, which
make it a good heat-carrier: low melting temperature,, (97°G), high
heat capacity (0.27 cal/kg, °C), low density (0.97 kgf/dm s in solid

7.4. Heat Removal

485

state and 0.74 kgf/drn 3 in liquid state). Boiling point is 880°C»
An extremely high latent heat of evaporation (1100 cal/kgf) assures
good heat absorption potentialities in the event of short, abrupt
temperature elevations above 880°C.

Production of hollow valves is difficult.

Nevertheless, the higher expenses are well
repaid due to greater reliability and longer
service terms.

The production of hollow valves by ext¬
rusion technique begins with drawing
a blank in the form of a hollow sleeve
(Fig. 293a), which is then upset in several
passes until the cylindrical portion of the
cavity is completely confined by forging
(Fig. 293 b-d). This (is followed by drilling
and reaming the hole and dressing of exte¬
rnal surfaces (Fig. 291e). For forging the
rod end an allowance s is left. After forging
the end (Fig. 293/), a taper hole is drilled
and reamed to suit a sealing plug (Fig. 293g).

The external surfaces of the valve are
preliminarily machined and it is then filled in' an inert atmosphere
with sodium at 200-300 e C to approximately 60% of its volume. The
hole is then sealed with a taper plug, and the rod end coated with
stellite. Then the valve is finish machined.

Hollow valves can be made simpler by welding the heads on sepa¬
rately (Fig. 294). After welding the valve head spherical surface,
the chamfered surfaces and end of the rod are coated with stellite.
Then the valve is ground and polished all over.

However, welding can only be applied to some valve steels. The
highest heat-resistant steels of martensitic-austenitic grades cannot
be welded. Furthermore, welded valves are not as strong as extruded
valves.

Chapter 8

Strengthening of structures

This section deals with strengthening techniques, which allow
certain stresses to be induced in engineering structures that are
opposite in sign to the working stresses. Two main methods are
applied, elastic and plastic.

8.1. Elastic Strengthening

With elastic’strengthenirig the system is preliminary given defor¬
mations which* are opposite to those of working loads.

A classical example^of this type of strengthening is in truss beams
{Fig. 295). The system incorporates tensors 1 which are tie-rods made

8.1. Elastic Strengthening

487

(a)

of a high-strength material. As these rods are tautened, preliminary
stresses are induced in the beam: compressive stresses on the side
closer to the rods, and tensile stresses on the opposite side (Fig. 295a).
Application of working load P WO rk causes stresses of the opposite
sign (Fig. 2905). Summation of the preliminary and working stresses
significantly lowers the final stresses exis¬
ting in the beam (Fig. 295c). Obviously, the
tensile stresses in the rods will increase.

Recently the manufacture of prestressed
beams has been mastered. Into the flange
opposite to the loaded side of the beam
are rolled rods, made from high-strength
wire, which are prestressed mechanically or
thermally (Fig. 296a). Such beams can
safely be cut into any lengths without
impairing the prestressed properties of resu¬
ltant pieces.

In the other design (Fig. 2965) a prestres¬
sed strap made from high-strength sheet steel
is attached to the lower flange, it is wel¬
ded to steel beams or riveted to light-
alloy ones.

Another example of elastic strengthening
is the hooping of containers, made from
light alloys, with steel wire (or ribbon), in
one or several rows (Fig. 297a-c). During
coiling compressive stresses are induced in
the vessel walls (Fig. 297 d). Being ded¬
ucted from the tensile stresses due to inter¬
nal pressure (Fig. 297e), these stresses
significantly lessen the resultant stresses in
the vessel walls (Fig. 297/). Tensile stres¬
ses in the wire increase due to the inte¬
rnal pressure.

Such systems are practical, however, only if the material of fasten¬
ing elements is stronger than that of fastened parts. Introduction of
preliminary tightening relieves the weaker material and makes the
structure, as a whole, stronger.

A variant in elastic strengthening is the hooping of hollow thick-
walled cylindrical components subjected to high internal pressures
(strengthening of high-pressure vessels, frettage of gun barrels, etc.).
In this case it is not obligatory for the fastening elements to be
stronger than the fastened ones; the strengthening effect is achieved
here owing to the singular distribution of stresses throughout the
cross-sections.

Fig.

296. Prestressed
beams

488

Chapter 8. Strengthening of Structures

According to Lame, the maximum stresses in a thick-walled vessel
subjected to internal pressures, occur at the inside wall and decrease
towards the outside (Fig. 298a). To attain greater strength the com-

Fig. 297. Strengthening of cylindrical containers

ponent is made of two tubes; the inner tube is press-fitted into the
outer one so that high interference is assured. Thus, the outer tube
is subjected to tensile stresses, and the inner tube to compressive

Compression t

- Tension

Fig, 298. Hooping of gun barrels

stresses (Fig. 2986). As a result of the addition of the preliminary and
working stresses (Fig. 298c) the tensile-stress peak at the inside wall
is lowered (Fig. 298d), the stresses throughout the entire cross-section
are levelled out and the strength of the system is enhanced.

8.2. Plastic Strengthening

489

8.2. Plastic Strengthening

In applying this technique the parts undergoing the greatest
loads during operation are subjected to preliminary plastic defor¬
mations, thus creating residual stresses which are opposite in sign
to the working stresses.

(a) Strengthening by Overloading

Strengthening by overloading means that the part is subjected
to a higher than working load in the same direction which causes
plastic deformation in the most heavily stressed areas.

Fig. 299. Strengthening by overloading

When bending a beam by a transverse force P^otk in the upper
fibres of the material arise compressive stresses, and in the lower
ones tensile stresses (Fig. 299a). Let us now exert upon the beam
larger force P, that produces plastic deformation in the extreme
fibres (Fig. 299b). As a result, the upper fibres are shortened, and the
lower ones lengthened. The middle fibres remain in the elastic state.
After the strengthening load is removed, the elastic middle while
returning into original state stretches the previously compressed
upper fibres and compresses the previously stretched lower fibres,

490

Chapter 8. Strengthening of Structures

producing in them stresses of opposite sign to the working stresses;
and reactive stresses will be induced in the middle (Fig. 299c).

Should the beam so stressed be subjected to the action of the work
load P wor k (Fig. 299 d), then the residual and working stresses add
algebraically. The resultant stresses in the extreme fibres are much
less (Fig. 299c) than the stresses in a beam not subjected to strengthen¬
ing. Consequently the beam can be loaded, with a heavier force,
provided the critical limit is not exceeded.

Similarly, thick-walled cylindrical vessels are strengthened by
preloading them with higher internal pressure (e.g., autofrettage of
artillery gun barrels).^

In the vessels pressures are created which cause plastic tensile
stresses in the internal wall layers (Fig. 299/). After releasing the
pressure the elastically stressed basic material of the wall regains
its original state compressing the plastically deformed internal
layers producing in them residual compressive stresses (Fig. 299g).
The tensile stresses, which occur in the wall under the action of
working pressure (Fig. 299 h), are partially equalized by the compres¬
sive prestresses. The peak of stresses at the internal surface is lowered
and the.stress distribution throughout the wall becomes more uniform
(Fig. 299i), making the vessel stronger.

The overloading method is applied also for strengthening torsion
bars (e.g., presetting of helical springs). In this case the bar is sub¬
jected to a higher torque moment M tT q, which induces plastic defor¬
mations of shear in the extreme fibres (Fig. 299/). After the streng¬
thening load is removed the elastic core is stretched entraining with
it the plastically deformed fibres causing in them stresses opposite
in sign to the shearing stresses given by the working load (Fig. 299A).

Now if some working torque moment M wor h trq is applied to the
bar (Fig. 299 1), the residual stresses add to the working stresses,
thus lowering the resultant stresses (Fig. 299m).

Strengthening by overloading is applied only to materials possess¬
ing sufficient plasticity. In brittle materials overstresses may cause
microcracks and flaking in the stretched layers leading to material
failure. Similar picture may be observed in plastic materials subjec¬
ted to high deformation. For these reasons the level of plastic defor¬
mation is confined within certain limits, admitting only overstres¬
ses which do not exceed 1.1-1.2 of the cr 0 , 2 yield point.

Furthermore, one should bear in mind that any kind of overstress
strengthens the material only against loads applied in one direction,
but weakens with a load, acting in the opposite direction. Hence,
this technique is applicable only for loads acting in one direction,
pulsating and alternate, provided the load of one sign is prevailing
(asymmetric cycles).

Clearly, any system subjected to the loads of permanent direction
and made from sufficiently plastic material, possesses the self-

8,2. Plastic Strengthening

491

strengthening property. Temporary increase of the working load
np to a value, inducing moderate plastic deformations, strengthens
the material. If, however, a part is subjected to alternate loads
surpassing the yield point under the unidirectional load action, the
material’s ability to withstand the load action of opposite sign falls.

The advantage of the overloading method is that it allows the
most heavily stressed areas to be relatively strengthened. It seems
as if the overload finds the weakest spots in the structure and auto¬
matically strengthens them.

(5) Strengthening by Work Hardening

Another form of plastic strengthening is surface work hardening.
It consists in compacting the surface layer to the depth of 0.2-0.8 mm
and the induction in this layer of compressive stresses favourable
to strength.

Mechanism of the surface hardening is illustrated in Fig. 300a.
During hardening process the surface layer spreads. If it were able

Fig. 300. Effect of work hardening

to extend freely it would separate from the basic metal (Fig. 3005),
but the extension is prevented by cohesive forces in the metal. As
a result, biaxial (longitudinal and transverse) compressive stresses
occur in the hardened layer while in the thick basic metal negli¬
gible reactive compressive stresses arise (Fig. 300c).

In addition the hardening process strengthens the surface layer
owing to structural and phase changes occurring in.the material.
In a beam, bent in a constant direction by a transverse force
(Fig. 301), it is advantageous to harden the surface, opposing the
active force. The extension of surface layer caused by the hardening
is accompanied by bending of the beam in the same direction as the
working load. The elastic counteraction of the basic metal tending
to unbend the beam will compress the plastically stretched out
layers, thus inducing compressive stresses in them (Fig. 301a).
When applying the working load (Fig. 3015), the compressive stresses
are subtracted from and lower the tensile stresses (Fig. 301c).

With two-sided hardening the picture is somewhat changed. In
this case the residual compressive stresses originate from both sides

492

Chapter 8. Strengthening of Structures

of the beam (Fig. 301(2). When adding the residual and working stres¬
ses (Fig. 301 e) the final tensile stresses are decreased and those of
compression increased (Fig. 301/). But since it is the tensile stress
which defines the strength, the resultant load capacity of the part
is enhanced. In addition the part acquires the ability to carry higher
loads in both directions.

The surface hardening techniques were ■ described in Section 5.

Hardening in a stressed state is extremely effective. This is actually
a combination of overloading and hardening. With this method the
part is loaded with a force that acts in the same direction as the
working load thus producing in the material elastic or plastic defor¬
mation. The surface of the part in this condition is subjected to work

Fig. 301. Strengthening by work hardening

hardening (e.g., shot blast treatment). After removing the load,
residual compressive stresses occur in the surface layer, which are
much higher than those when overloading or work hardening is per¬
formed separately.

Applied recently is an explosive hardening technique. This techni¬
que is much superior to others in capacity and versatilities. Explosive
hardening enables parts having intricate configurations to be streng¬
thened with simultaneous consolidation of all external and internal
surfaces. The consolidation intensity and depth are controlled by
the explosion power.

Surface hardening (induction hardening, also hardening of steels
which have limited hardenability) and chemical-thermal treatment
(case hardening, nitriding) not only strengthen materials but also
induce (as work hardening) residual compressive stresses in the
surface layer owing to the formation of higher specific volume struc¬
tures. Extension of the surface layer is impeded by the core which
preserves the original pearlite structure. As a result, biaxial com¬
pressive stresses occur in the surface layer (in cylindrical parts—
triaxial) and in the core only insignificant tensile reactive stresses
develop.

(c) Volumetric Consolidation

Volumetric consolidation is the deep reduction of parts of com¬
ponents undergoing tensile stresses during operation. Generally
the reduction takes place when the part is a blank in its cold or semi¬
plastic state (thermal deformation).

8.2. Plastic Strengthening

493

;Let us now consider a beam being bent by a transverse force P W0T h
(Fig. 302). The reduced areas are those opposite to the acting load
(hatched section in Fig. 302a). Plastic deformation of the material
causes the beam to sag. After reduction the elasticity of the material

Comprssslon

Fig. 302. Strengthening by spatial deformation

straightens the beam. In the reduced areas biaxial compressive
stresses arise and in the non-reduced areas tensile stresses occur
(Fig. 302b). During action of the working load (Fig. 302c), the summed
residual and working stresses lessen the resultant stresses (Fig. 302c?).

tt) (/) Ik) (l) (m)

Fig. 303. Examples of strengthening by spatial deformation

. The magnitude and distribution of the resultant stresses will
depend on the cross-sectional relationship of the reduced and non-
reduced zones, on the degree of reduction and the changes along the
cross-section of the part. With a rational choice of these parameters
it is possible to significantly (sometimes completely),'decrease the
resultant stresses. • r ... .

Examples of volumetric consolidation by reduction are illustrated
in Fig. 303 (reduced areas are shown black). The beams (pig. 303a, 6)

494

Chapter 8. Strengthening of Structures

are strengthened by rolling the flanges and holes (Fig. 303c)—by
broaehing; flat parts (Fig. 308<?) by pressing from end faces and rings
(Fig. 303d, /, g)—by eccentric burnishing.

Figure 303 k-m depicts some methods of strengthening of disk-shaped
components by metered deformation between plates with a simul¬
taneous enlargement of the hole. S?:

(d) Thermal Strengthening. iff

The thermal strengthening technique is based on the fact that in
the course of uneven heating the hot areas of the part have compres¬
sive stresses and the cold areas tensile stresses. The magnitudes
of the stresses depend on the temperature gradient, coefficientjof

Fig. 304. Thermal strengthening

linear expansion and the elasticity modulus of the material. With
sufficiently high temperature gradients local plastic strains arise,
which can be utilized for the strengthening.

Let a beam be bent by a working force P work (Fig. 304a-e). With
thermal strengthening the beam must be heated from the side of
acting forces. The heated layers are lengthened and then compressed
by adjacent more cold layers, in which tensile reactive'stresses appear.
The magnitudes of tensile and compressive stressed, as well as their

8.2. Plastic Strengthening

495

distribution across the section depend on the temperature gradient
in the section. In the case being considered it is advantageous to heat
the beam uniformly to a substantial depth (Fig, 304a) so that slight
compressive stresses appear on the heated side and high, exceeding
the yield limit, tensile stresses in the thin cold layer on the oppo¬
site side (Fig. 304 b).

After cooling to the initial temperature, the stresses, which appea¬
red as a result of the temperature difference, disappear and the plasti¬
cally stretched layers will be compressed by the elasticity of the
basic material. As a result, compressive stresses arise in these layers
and on the opposite equalizing tensile stresses (Fig. 304c). The
beam becomes advantageously prestressed. Under the influence
of working loads (Fig. 304c?), the residual stresses are subtracted
from the working ones thus reducing the resultant stresses (Fig. 304e).

Intensive heating of a thin upper layer (Fig. 304/) induces residual
compressive stresses in it (Fig. 304g); after cooling tensile stresses
occur in this layer and in the lower lying layers insignificantly equa¬
lizing compressive stresses (Fig. 304ft). When applying the working
force P W0Tk the residual stresses are subtracted from the working
ones (Fig. 304i), thus the resultant stresses (Fig. 304/) reduce. It
should be noted, however, that the gain in the value of the tensile
stresses is much lower than in the previous case.

Thermal strengthening is most often applied to engineering parts
made from light alloys, which possess a number of the necessary
properties for' the case in question, i.e., high coefficient of linear
expansion, low yield point and low transition temperature to the
plastic state. Thermal strengthening is applied to light alloy high¬
speed rotors. The problem in this case is to balance centrifugal ten¬
sile stresses which are maximum at the rotor hub. Still higher tensile
stresses arise in the hub if rotor during work heats from the periphery
and still more if the hub is press-fitted upon the shaft.

Figure 304 k-n, shows thermal strengthening of a disk, subjected
to tensile working stresses, whose curves are presented in the form
of epure in Fig. 304m. The heating of the disk begins from its peri¬
phery (Fig. 304ft). The heating temperature and the temperature
gradient along the radius are selected so that tensile residual stresses
are induced in the internal cold layer. After cooling, the stretched
layers are compressed by the elasticity of the external layers. In the
internal layers compressive prestresses are produced and tensile
prestresses in the external layers (Fig. 304?). When the working load
is applied (Fig. 304m), the residual and working stresses add algeb¬
raically; the resulting stresses (Fig. 304?z) are less and more favou¬
rably distributed than in a disk not thermally strengthened.

The actual thermal strengthening procedure may vary. Tempera¬
ture gradient can be increased by simultaneously cooling the hub and
heating the periphery. Sometimes deep cooling of the hub will suf-

496

Chapter 8. Strengthening of Structures

fice (e.g., cooling the hub with liquid air). The procedure sequence
may also be different. It is possible to uniformly heat the disk and
then quickly cool the hub. The same result is obtained when the
disk is first cooled to a minus temperature and then heated from its.
periphery.

Thermal strengthening must agree with the magnitude and sign
of working stresses. If the core of the part is subjected to compressive
stresses (e.g., to compressive thermal stresses, arising when the
working temperature of internal layers is higher than that of the
external ones), then the aim of the thermal strengthening procedure
is to obtain tensile prestresses in the internal layers by prematurely
providing residual compressive strains in these layers. In this case
the process of thermal strengthening must be opposite to the above,,
i.e., the part has to be heated from inside and cooled on the outside.

To generalise the situation: compressive prestresses require cooling
of the part area; tensile prestresses—heating.

The drawback of the thermal hardening method is the delivery
of factors which define the magnitude, sign and distribution of
prestresses, and also the difficulty of strictly maintaining thermal
conditions which predetermine the stability and reproducibility of
the results.

The thermal strengthening temperature should be lower than that
of the previous heat treatment, otherwise the thermal strengthening
effect is lost.

If the part, when working, heats to a temperature close to the
strengthening temperature, especially in the presence of high stresses,
the strengthening effect will again be lost.

(e) Improving Strength of Trussed Systems

Similar methods are used to improve the strength of trussed and
analogous structures.

As an example let us consider the case of a trussed bracket, loaded
with tensile force P wor k (Fig. 305a). The bracket middle bar carries
more load than the side bars. The elastic deformation of the middle
bar under the load (and, hence, according to Hooke’s law—the
tensile stresses in the bar) exceeds the deformations of the side bars
in the proportion s/s' « 1/cos a (when a — 60-70°, sis' == 2-3).

The system may be strengthened by overloading, causing tensile
plastic deformations in the middle bar. After removing the strength¬
ening load, the middle bar will be compressed due to the elasticity
of the side bars (Fig. 305 b) in which arise tensile stresses (light
arrows). If such a prestressed system is subjected to the working load
then the addition of the residual and working stresses, i.e., the load
on the bars becomes more uniform (Fig. 305c) and the load capacity
of the system is enhanced.

8.2. Plastic Strengthening

497

Load capacity can also be increased by thermal strengthening,;
i.e., by heating the side bars until residual tensile strains are pro¬
duced in the middle bar. After cooling, in the middle bar there
occur compressive stresses and the system becomes prestressed.

Fig. 305. Strengthening of a truss system

It is possible to employ an elastic strengthening method. For
this the side bars should be pulled tight or the length of the middle
bar increased above the nominal so that compressive stresses are
induced in it during assembly.

In castings beneficial prestresses can be assured by earlier crystal¬
lization of material of the middle bar and by cooling it quicker in
comparison to the side bars (e.g., by providing cooling condensers
in the mold).

l/ 2 32-01 395

Fig. 306. Microphotographs ol surfaces machined by different methods (X30)

1— rough turning; 2 — reamings 3 — diamond Boring; i— honing; S —tine honing;

6 — superfinishing

ficial irregularities appreciably affect tbe surface behaviour and
service properties of the part.

Rough machining (e.g., rough turning) leaves on the surface
notches up to 0.3 mm deep and also ridges with rugged edges. Plastic
shear and tearing of metal particles by the cutting tools causes high

Chapter 9. Surface Finish

499

residual tensile stresses in the surface layer. The flakes and micro¬
cracks which deteriorate the strength of the surface layer become
sources of local corrosion. Furthermore, as the metal heats during
machining structural changes occur in the surface layer (phase
thansformations, recrystallization, etc.), which change the mecha¬
nical properties of the metal.

Finishing operations are aimed at the complete or partial elimi¬
nation of the damaged and overstressed layer, removing or smoothing
out the irregularities.

Fig. 307. Microprofiles

As a rule, improvements in smoothness of surface better fatigue
strength, improve wear- and corrosion-resistance of parts, help to
maintain dimensional stability of parts during operation, increase
contact rigidity, assure uniform load distribution and reduce the
friction factor in movable joints.

Generally the quality of a surface (or surface finish) is evaluated
through the averaged values of microirregularities, existing on it, :
in other words, the degree of roughness, though traditionally the
term “surface finish” (opposite in meaning to the above) is
used.

Evaluating the surface finish by the averaged height of microirre¬
gularities is incomplete as it does not account for the nature of micro-
relief, i.e., the distribution density of the irregularities, their profile,,
sharpness of undercuts at valley bottoms and other factors which
determine wear-resistance, fatigue strength and contact rigidity
of the surface.

The microrelief may consist of an alternating series of ridges and
valleys (Fig, 307a), which may be rarely spaced (Fig. 3076), or
ridges separated by plateaus (Fig. 307c), or valleys separated by
plateaus (Fig. 307<i). The parameters of surfaces in the above
mentioned cases would be different although in the case of equal
mean height of the microirregularities the quality of surface is
accepted as homogenious.

Being only a geometrical index, the averaged height of microirre¬
gularities does not reflect the physical and mechanical changes, which
occur in the surface layer in the machining process. Nevertheless, it
provides, in general, true picture of the quality of surface, because
damages, as a rule, are the lesser, the finer the machining.

32 *

500

Chapter 9. Surface Finish

9.1. Classes of Surface Finish

The Soviet State Standard GOST 2789-59 stipulates the following
rules for determining the roughness of a surface. _

The base for determining magnitudes of the microirregnlarities
is the mean profile line m—m (Fig. 308a), which runs parallel to the
geometrical surface of the profile, dividing it such that within the

limits of the measured part (base length l) the sum of the squares
of the heights y } , y 2 , y 3 , .... y n , taken at the closest distances one
from another, is minimum.

The mean line of the profile may be approximately considered as
the line which divides within limits of the base length so that the

9.1. Classes of Surface Finish

501

areas on both sides of this line are equal (Fig. 3086)

■F1 + F3+ ... -j-F n — F 2 -\-Fk-{- - - • + F n -i

Surface roughness is determined from the following parameters.

Mean arithmetic deviation S a is the mean value of the heights
(|/i, y 2 , y 3 , . . ., y n ) of points on the measured profile (Fig. 308a)
from the mean line (regardless of algebraic sign)

1

= ~ j [if] da;

a

Approximately

n

1

Height of irregularities ~R Z is the mean distance between five highest
crests and five lowest valleys within base length l (Fig. 3086)
measured from a line, running parallel to the mean line

r> _ (fti + ^3 • • • + A9I — (^2 4- *4+ • • • + hio)

n z -g

The value of R a can be represented as the height of a rectangle
whose area is equivalent to the area of profile on each side of the
mean line, while the R z value can be expressed as the mean height
of irregularities at the extreme points of the profile (Fig. 308c).
R z is considerably (4-5 times) higher than R a . In practice the R z value,
expresses the roughness level more vividly than the R a value.

GOST 2789-59 establishes 14 classes of surface roughness
(Table 28). The classes of surface roughness from 6 to 14 are subdivi-

Table 28

Classes of Surface Roughness

502

Chapter 9. Surface Finish

ded into subclasses a , b and c, which allow finer classification within
each of the classes. The R a scale is regarded as basic for classes from 6
to 12 and the R z scale—for classes from 1 to 5 and from 13 to 14
(thick black lines in Table 28). The R a and R z values are correlated
to each other as folows: R z = 4 R a for classes 1-6; R z — 5 R a for clas¬
ses 7-14.

In drawings the surface roughness classes are denoted by an equi¬
lateral triangle V with a number, indicating the class (and accom¬
panied by a letter of the subclass whenever necessary). The numerical
values of the microirregularities, which underlie the classification
into classes, only limit their maximum heights. Should the maximum
and minimum values of microirregularities be necessary, then two
class numbers are quoted. For instance, writing V9-V10 indicates
that the roughness must be within the R a and R z values, for classes
9 and 10.

Surfaces rougher than the 1st class are denoted by the symbol V,
above which the ultimate height of irregularities R z is given in

500

microns. For example, the V symbol indicates that the acceptable
height of surface irregularities shall not exceed 500 urn.

Surfaces, whose roughness does not require any classification,
are shown by the symbol co.

One should identify certain contradiction, which exists between
what is implied by the surface finish designation and what is imp¬
lied by the symbols depicted on drawings. In accordance with
GOST 2789-59 the class of surface finish indicates merely the degree
of surface roughness and ignores the method by which this surface
has been produced—by machining or in as-received state (e.g., moul¬
ding, casting, etc.). Clearly, this is insufficient because shop produc¬
tion requirements reject such ambiguity and suggest that all the
surfaces to be machined are strictly specified.

For this reason the appropriate symbols of surface finish are
depicted on the drawings only against those surfaces that require
definite machining. The rest of the surfaces are denoted by the
symbol oo, regardless of actual surface finish of these surfaces which,
in fact, remains in the as-received state.

Thus, the symbol v with its accompanying figure shows that the
surface so designated must be machined to the denoted standard of
surface finish, while the co symbol indicates the black surface in
as-received state.

The values of R a and R z parameters are presented on a log scale
in Fig. 309, for the different classes and subclasses of surface
roughness.

Table 29 cites the classes of surface finish obtainable by various
machining operations. Surface roughness classes mostly used in
mechanical engineering, are listed in Table 30 for reference.

VI V2 Vi V» VJ Vi V7 VS V5 VW V!! V/2 V/3 V/4
Fig. 309 R a and R x values for various classes of surface finish

504

Chapter 0. Surface Finish

Surface Roughness Values Obtainable

Ciass of surface roughness

V 1

V 2

V 3

V 4

'

nm, not more than

*

320 1

160

’ 80

40

B a , (tm, not more than

80

40

20

10 ■

Gas cutting (machine)

Filing

Drilling

Planing

finish

finest

)

I

Face milling

finish

finest

Circular milling

finish

finest

Turning

finish

finest

!

Boring

finish

finest

j

i

Core drilling

Face undercutting

finish

finest

Cutting of male thread

tool, chaser, circular die,

• screw die, milling
rolling
grinding

Cutting of female thread

tap, tool

milling

grinding

Machining of gears

shaping, milling

hobbing

shaving

9.1. Classes of Surface Finish

505

Table 29

by Various Machining Methods

33—01395

506

Chapter 9. Surface Finish

Class of surface roughness

v 1

V 2

V 3

V 4

R z , y,m, not more than

320

100

80

40

R a , nm, not more than

j 80

20

10

Machining of gears

grinding

finishing, generating

Anode-mechanical machi¬
ning

ordinary

fine

Electro-chemical machi¬
ning to size

ordinary

fine

Electric discharge machi¬
ning

usual

fine

Ultrasonic machining (holes, recesses)

Scraping

usual

fine

Reaming

usual

fine

|

i

i

Broaching

usual

fine

Plane grinding

1 usual

1 fine

Circular grinding

usual

fine

Regrinding

usual

fine

Polishing

usual

fine

9.1. Classes of Surface Finish

V 6

V 7

V 8

V 9

V 10

V 11

7 12

10

6.3

3.2

1.6

0.8

0.4

[ 0.2

2.5

1.25 ;

0.63

0.32

0.16

0.08

| I

__ 507

Table 29 (continued)

0.1 0. u5

0.02 0 .01

508

Chapter 9. Surfa.ce Finish

Rolling

1-2 classes higher than

Shot-blasting

1-2 classes lower than

Liquid^polishing (hydrohoning)

2-3 classes higher than

Electropolishing

2-3 classes higher than

9,1. Classes of Surface Finish 509

Table 29 ( continued)

the original surface finish (upje class 14)

the original surface finish

the original surface finish (up to class i2)

the original surface finish (up to class 14)

510

Chapter 9. Surface Finish

Table 30

Surface Finish of Finished Blanks

Class of surface finish

V I

V 2

V 3

V 4

V 5

V 6

V 7

V 8

V 9

V iO

R z , hm, not more than

320

160

30

40

20

10

6.3

3.2

1.6

0.8

Rolled metal sections

! I

1

Casting

floor mould
core mould
shell mould
metal mould
investment
pressure
(non-fer¬
rous allo¬
ys)

1

1

i

1 I

! 1

i

!

Hot

stamping

ordinary

precision

Precision forging
(coining)

Drawing

Plastic products (pres¬
sing and press¬
moulding)

9.2. Selection, of Surface Finish Classes

The class of surface finish must be agreed with the class of manu¬
facturing accuracy. The higher the accuracy class, the higher must
be the surface finish. Otherwise, the value of the surface microirre¬
gularities becomes commensurable with tolerance margins. Measu¬
rements taken over the extreme points of the profile give false
dimensional values. In use the part quickly loses its accurate dimen¬
sions due to wear and crushing of ridges (in movable joints) or the
action of working loads (in fixed joints).

For classes of rougher manufacturing accuracy with wider tole¬
rances the class of surface finish can be lowered, thus reducing pro¬
duction costs.

9.2. Selection of Surface Finish Classes

511

To obtain different classes of accuracy the minimum surface
finish class is as follows:

Class of accuracy ... 1 2 2a 3 3a 4 5 7

Class of surface finish . V7 V7 V6 V5 V5 V4 V4 V4

When selecting the surface finish class properties of the material
and hardness of the part surface must be considered. For steels
a high degree of surface finish can be obtained with hardness values
not below 30-35 Rc. Steel products, undergoing finishing treatment,
must be at least structurally improved or normalized. Fine finishes
on raw low-carbon steels are difficult to obtain.

Due to machining conditions holes are more difficult to finish
than shafts. For this reason the surface finish standards assigned
to holes should be somewhat lower than those for shafts.

In general lower classes of surface finish should always be accepted
if commensurate with reliable component operation, since higher
surface finish requirements mean additional finishing operations
and, hence, increased production costs.

Furthermore, higher surface finish does not always better the
functioning of joints. Press-fitted joints, for example, have certain
optimum standards of surface finish and deviations from these
standards to any side worsen joint strength.

In the interest of economy free surfaces (i.e., those, which are
either out of engagement or are separated by a gap from neighbou¬
ring surfaces) should be machined to the lowest possible surface
finish standard. Exceptions are heavily-loaded parts subjected to
cyclic loads. To attain higher fatigue strength such parts are machi¬
ned, polished or burnished all over to high surface finish standards.

Listed below and taken on the basis of general engineering practice
are approximate values of surface finish for typical engineering parts.

Surface Finish of Typical Engineering Parts
Plain bearings:

low-loaded, running at moderate surface speeds:

hole . V7— V9

shaft ...V8— V10

high-loaded, running at high surface speeds:

hole . V8— V10

shaft .V10— V12

Plain thrust bearings (working surfaces):

low-loaded .V6— V8

high-loaded, running at high surface speeds . V8— V12

Spherical surfaces (of self-aligning bearings, etc.) . V9— V12

Fixed connections accomplished with a sliding fit:

hole . V8— V9

shaft . V9— VI1

Connections with transition fits:

hole ..V7— V9

shaft .. V8— V10

512

Chapter 9. Surface Finish

Pressed connections:

hole . V7— V10

shaft . . ■.V8-V11

Thrust shoulders of fixed cylindrical connections (working surfaces) V6—V8
Fits of antifriction bearings:

hole in housing, for bearings of class:

standard (H) .. V8— V9

higher (II) .V9— ViO

high (B) ' V10-V11

precision (A) VI1—VI2

shaft, for bearings of class:

standard (II) . V8 — VIO

higher (H) .VIO—Vll

high (B) V11—V12

precision (A) .. V12-V13

Rolling contact bearings, contact load connections.VIO— V13

Cylinders (highly polished):

lor pistons with soft packing (cups). V7— VIO

for pistons with metallic rings . V9— V12

(lapped in)

Pistons (working surfaces):

cast iron and steel .. V9— ViO

light alloy . VIO— V12

Piston gudgeon pins:

hole . V8— Vll

pin . .....' .V9— V12

High-pressure plunger pumps:

cylinders ...VIO—V12

{lapped in)

plungers ... Vl2— V14

Cylindrical slide valves:
oil distributing:

hole . . . ..

valve .

gas distributing:
hole .

valve .

Flat slide valves:

body .

gate ..

Taper plug cocks (working surfaces):
hole ...

plug .

(lapped in)

V7— V9

(lapped in)

V9— Vll
(lapped in)
V9— V12
(lapped in)

V7—VIO
(lapped in)
V8— Vll
(lapped in)

V8— VIO
(lapped in)
VIO— V12
(lapped in)

Valves with conical sealing surfaces:
guide surfaces:

guide ... V8—V9

stem . V9— ViO

9.2. Selection of Surface Finish Classes 513

sealing surfaces:

seat working surface .. V9—Vll

(lapped in)

valve working chamfer ...V10— V12.

(lapped in)

Cam mechanisms (working surfaces):

cam . V9— Vll

drive roller .. V9—VJ2

tappet . V8— Vll

Templets (working surfaces):

templet .. v8—VIO

roller . V9— Vll

Splined connections (centering surfaces):
centering on outer diameter:

hole . V7—VIO

shaft . V8— Via

centering on inner diameter:

hole . V9— V12

shaft . V7— V9

centering' on spline faces:

female surfaces . V7—VIO

male surfaces. VS—Vll

splined connections with clearance:

spline (working faces) . .. 77— VIO-

hole ...78— Vll

shaft . V7—V8

Keyed connections (working faces):

key ways . .. V5—V7

keys . V6— V8

Vee ways:

female surfaces . VS— VIO

male surfaces. V9— V12

Male threads:

commercial grade .V5— V6

higher accuracy .. V6—V7

precision . V7—V9

Female threads:

commercial grade . V4— V5

higher accuracy . .. V5—V6

precision . V6— V8

Leadscrews (working surfaces):

nut .V8— VIO

screw . V8— VI2

Spur gears (working faces teeth):

non-critical . V6—V7

operating under moderate loads and

surface speeds . V7—V8

operating under medium loads and

surface speeds . V9— VIO

heavily-loaded, subjected to impact loads

and running at high surface speeds. VlO— V12

(lapped in
or run-in)

Helical and herringbone gears (working faces of teeth):

operating under moderate loads and moderate surface speeds . . V6—V8

heavily-loaded and running at high surface speeds. V8— VIO

514

Chapter 9. Surface Finish

Bevel gears (working faces of teeth):

operating under moderate loads and moderate surface speeds V6— V8
heavily-loaded and running at high surface speeds ...... V8— V10

'Worm wheels (working faces of teeth):

operating under moderate loads . V7— V8

heavily-loaded ...’ V8— V10

Worms (working laces of threads):

operating under moderate loads .. . .. V8— V9

heavily-loaded ....V10—Vll

Ratchet wheels (working faces of teeth) ..’. V8— V9

Roller free-running wheels (working surfaces):

female cage . V8-~ V10

male cage.V10—V12

roller . V12— V13

Frictions, brakes (working surfaces):

cylindrical surfaces ... V9— V12

flat surfaces. V8— V10

Contact cylindrical packings (working surfaces of shafts):

with soft packing elements (cups) . V8— V10

with metallic packing elements . V9— Vll

Packing faces (working face of disks):

with soft packing elements . V9— V10

with metal packing elements ..V10—V12

(lapped in)

Sealing surfaces of nipples, pipe unions, etc. V7— V9

Belt pulley drives (working surfaces):

tor flat belts. V9— V12

for V-belts .. V8—V10

Hermetic joints with gaskets:

with soft gaskets.. V6— V8

with hard gaskets . V8— V9

with gaskets of soft metals .. V9—V10

Hermetic metal-to-metal joints ..V10— V12

(lapped in)

Attaching surfaces (without gaskets):

commercial .. V5— V7

precision . V8— V10

Free surfaces (end faces and non-bearing surfaces of shafts, chamfers,
non-working surfaces of gears, pulleys, flywheels, levers, eonrods,
crankshaft webs, etc.):

low-loaded parts... V4— V6

cyclically high-loaded parts . V6— V9

(and higher
up to poli-

■ shing)

Fillets:

non-critical . V5— V6

for cyclically high-loaded parts . V8— V10

(and higher
up to poli¬
shing)

Hexahedrons, tetrahedrons flat keyways, slots for keys, etc ... . V4— V5

Drilling (to hold loosely inserted parts) . . V4—• V5

Supporting surfaces for nuts and bolt heads. V4™ V5

9.2. Selection of Surface Finish Classes _ 515

Aligning shoulders (o£ flanges, covers, housings, etc.):

hole . V5— V6

shoulder . V6— V7

Control members, levers, knobs, handwheels, etc.V8— V10

(polished)

Compression springs (dressing of end face) . V4— V5

Measuring tools (working surfaces) . V12— V14

(fine fini¬
shing)

Alloy(s),

aluminium, 227
light, 226
magnesium, 230
titanium, 234
Alloying, 212
Anisotropy, ol metals, 198

Centring, temperature-independent, 472
Coefficient,
ol elasticity, 254
of hysteresis, 215
of operational expenditures, 12
of rigidity, 253
of stress concentration, 375
Compactness, of constructions, 172
Compounding, 63
Connection^), cylindrical, 428
Constraint, form, 446
Construction, from pressed sheet me¬
tal, 159

Content, metal, 131
Cost, of machine, 57

Derivative(s), machine, building of,

61

Design,
choice of, 92

of cyclically loaded parts, 397
economic factors of, 10
economy-oriented, 10
enhancing rigidity of, 272
equal strength, 141
general rules of, 84
methods of, 88
plate, 332

rational schemes of, 170
safety margins of, 207
Development,

consecutive, of machines, 76
of design versions, 93
Dimension(s), standard linear, 81
Dislocation(s), 216
Downtime, 33
Durability,
criteria of, 28
design, 33

influence on output, 23
influence on size of machine
fleet, 19

fimits to increase, 46

means of enchancing, 37.
theory of, 35

Durability and obsolescence, 50

Effect,

economic, 11

of fillets, chamfers and tapers, 156
of loading schemes, 176
of strength of mating parts, 194
ol system resilience, 188
of type of loading, 164
Elimination,

of load concentrations, 410
of superfluous links, 170
Expenditures, 12

Extrusion, reducing weight by, 162

Factoris),

accidental breakdown, 32
economic, 10
load, 31

machining time, 31
off-day, 30

operational, influence upon econo¬
mic effect, 13
repair downtime, 31
scale, 380
seasonal work, 30
shift, 30
use, 11, 30
Fatigue,

diagrams of, ? “357

under non-stationary loading condi¬
tions, 384

Field, machine application, study
of, 91

Holes, as stress concentrators, 405

Indices, specific rigidity, ol materials
260

Joint(s),
expansion, 460
spherical, 424

Life, service, 29
Limit(s), fatigue, 355

I ndex

517

Materials),

of improved strength, 211
non-metaliic, 235
glassceramics, 238
plastics, 235
reinforced concrete, 239
reinforced wood, 237

Number(s),
preferred, 78
basic series of, 79
derived series of, 80
used in designing, 82

Parameter(s), machine, rational selec¬
tion of, 182

Plate(s), design of, 332
Principles, of machine design, 9
Procedures, composition, 108
Profile(s),

round hollow, strength and rigidity
of, 138

strength and rigidity indices of, 135
Profitability, of machine, 10

Recoupment, of equipment, 12
Reduction,

of machine cost, 15
of product range, 70
of stress concentration, 397
Reliability,

means for improving, 54
operational, 52
Ribbing, 293
Rigidity,
criteria of, 253

transverse, improvement of, 290
of structures, 252
of thin-walled constructions, 334

Section(s), rational, 133, 288
Sectionalization, 61
Series,

parametric, 71
size-similar, 73
unified, of machines, 68
Standardization, 60
integrated, 65
Strain, thermal, 439, 461
Strength,
contact, 418
cyclic, 348

equal, of units and connections, 147
fatigue,

for complex stresses, 361
improvement v of, 391, 393
of materials,
improvement, of, 211
specific indices of, ' 246
thermal, 448, 450
Strengthening
elastic, 486,
plastic, 489

of pressed connections, 410
of structures, 486
Stress(es),
design, 207
correction of, 184
thermal, 439
reduction of, 459
Succession, design, 89
Support(s), rational arrangement of, 286
Surface finish, 498
classes of, 500
& yslem(s) >

cantilevered, 2S0
■'double-support, 215
multi-station machine, 65

Unification, 58

Universalization, of machines, 75

Versions, design, development of, 93

Weight, as design parameter, 131
Whiskers, 218

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