2

FUNDAMENTALS
OF MACHINE DESIGN

MIR PUBLISHERS MOSCOW P QRLOV

n. opjiob

OCHOBbI

KOHCTPyHPOBAHHH

H3,Il,ATEJIbCTBO
«MAUIHHOCTPOEHHE»
MOCKBA

FUNDAMENTALS
OF MACHINE DESIGN

P. ORLOV

TRANSLATED FROM THE RUSSIAN
by YU. TRAVNICHEV

MIR PUBLISHERS • MOSCOW

First Published 1976

The Russian Alphabet and Transliteration

A a a

Kk

K

Xx

kh

B 6 b

JI ji

1

An

ts

B B V

M M

m

Hu

ch

r r g

Hh

n

in m

sh

Ah d

Oo

0

id, m

shch

E e e

nn

P

T> t

i 1

E e e

Pp

r

LI H

y

JK ;k zh

Cc

s

Lb

*

3 3 z

T T

t

33

e

M h i

vy

u

K> io

yu

Hi y

(D$

f

Ha

ya

The Greek Alphabet

A a

Alpha

11

Iota

Pp

Rho

Bp

Beta

Rx

Kappa

2a

Sigma

r v

Gamma

Ak

Lambda

Tt

Tau

A 6

Delta

M p.

Mu

r u

Upsilon

Ee

Epsilon

Nv

Nu

<D q>

Phi

z?

Zeta

BE

Xi

XX

Chi

H T)

Eta

Oo

Omicron

Psi

©0

Theta

n n

Pi

Q (o

Omega

Ha amJiuucKOM H3UKe

<g) English Translation, Mir Publishers, 1976

Contents

Chapter 1. Tightened Connections. 7

1.1. Unloaded Connections. 7

1.2. Loaded Connections . 12

Chapter 2. Press-Fitted Connections . 37

2.1. Press (Drive) Fits . 37

2.2. Strength of Press-Fitted Connections. 38

2.3. Selection of Fits. 46

2.4. Calculation Diagrams. 47

2.5. Probability Calculations for Press-Fitted Connections . 59

2.6. Press-Fitting with Heating or Cooling of Parts. 64

2.7. Pressed Connections with Electrodeposited Coatings . . 65

2.8. Design of Pressed Connections.. 67

2.9. Conical Fits. 75

2.10. Serrated Connections ... 77

2.11. Glued Connections .. 78

Chapter 3. Centring Connections . 79

3.1. Design Rules . 80

Chapter 4. Screwed Connections . 88

4.1. Longitudinal and Transverse Location of Parts in Screwed

Connections . 90

4.2. Centring in Screwed Connections. 92

4.3. Design Rules. 94

4.4. Reinforcement of Fastening Joints . 98

Chapter 5. Flanged Connections . 102

5.1. Alignment of Flanges. 105

5.2. Machining the End Faces of Fastening Holes. 107

5.3. Diameter and Spacing of Bolts. 110

5.4. Three-Flange Joints. Ill

5.5. Conical Flange Joints . 113

Chapter 6. General Principles which Should Be Followed when Desig¬
ning Units and Parts. 96

6.1. Unification of Design Elements. 96

6.2. Unification of Parts . 119

6.3. Principle of Unitization. 121

6.4. Elimination of Adjustments. 124

6.5. Rationalization of Power Schemes. 126

6.6. Compensators. 129

6.7. Torsion Bars . 133

6.8. Floating Cross-Sliding Couplings. 136

6

Contents

6.9. Elimination and Reduction of Bending. 137

6.10. Elimination of Deformation Due to Tightening. 143

6.11. Design Compactness. 146

6.12. Combining Design Functions. 150

6.13. Equistrength. 152

6.14. Uniform Loading of Supports. 156

6.15. Self-Alignment Principle . 157

6.16. Cambering . 162

6.17. Effect of Elasticity upon Load Distribution. 164

6.18. Fitting to Several Surfaces. 170

6.19. Tightening Up on Two Surfaces. 171

6.20. Axial Fixing of Parts. 172

6.21. Control of Direction . 174

6.22. Mounting Surfaces . 175

6.23. Butt-Jointing on Intersecting Surfaces. 177

6.24. Interchangeability of Rapidly Wearing Parts. 178

6.25. Accuracy of the Alignment of Parts. 179

6.26. Relief of Precision Mechanisms. 181

6.27. Coupling Parts Made from Hard and Soft Materials . . . 183

6.28. Elimination of Local Weak Spots. 186

6.29. Strengthening of Deformable Areas. 188

6.30. Composite Constructions . 189

6.31. Shoulders. 194

6.32. Chamfers and Fillets . 196

Chapter I

Tightened Connections

Depending on their operating conditions tightened connections
can be divided into unloaded and loaded ones.

1.1. Unloaded Connections

These include joints of non-bearing covers, non-bearing housing
components, etc. In this case the necessary bolt (or stud) tightening
force is determined proceeding from the condition that the joint
should remain tight and unparted with all possible deformations
in the system and with possible slacking of the screwed connections
as a result of the threads or the supporting surfaces of the nut and
bolt being crushed in the course of time. If the loads arising in the
system during operation are not considered, then the bolt load
is only the force of preliminary tightening (preload).

Such connections in the majority of cases are not calculated. The
material, diameter and thread pitch of bolts are selected on the
basis of existing practice, and the tightening force is set so that
the stresses developing in the bolt correspond to a 3- 5-fold safety
margin (in terms of the yield strength) as usual.

In non-critical connections the tightening force is not specified,
but left to the fitter to decide upon by his experience. In mechanized
assembly shops such joints are tightened mechanically with the aid
of electric or pneumatic torque-control bolt drivers and nut runners.

The tightening torque M tight , which is equal to the product of
the force applied to the wrench end and the wrench arm, produces
axial force P ax (Fig. 1) stretching the bolt and overcomes the moment
of friction in the threads and on the nut bearing surface

M mht = 10- 3 ( Pax <p * > + n + h ) kgf-m (1.1)

where P ax = axial force arising as a result of the bolt tightening,
kgf

d 0 — pitch diameter of the thread, mm
D = mean diameter of nut bearing surface, mm

8

Chapter 1. Tightened Connections

h and / 2 = coefficient of friction in the threads and on the
nut bearing surface, respectively
<p = thread helix angle

Introducing tan (jp = (where s is the thread pitch) gives

M mh , = i(r^ (4"T+/■ T +/.4) W m <‘- 2 >

where d is the nominal diameter of the thread, mm.

For the current diameter ranges of fastening bolts it is usual to
take on the average s/d = 0.15; d 0 /d = 0.9; and D/d = 1.3.
Substituting these values into Eq. (1.2), we get

Mtight = \0~ z P ax d (0.024 + 0.45/j + ,

+ 0.65/ 2 ) kgf-m

whence

p _10 3 _ ^Jight -

d (0.024+0.45/i + 0.65/ 2 )

(1.3)

Let f 1 = 0.22 and / 2 =0.11. Then

5 M tight

<1

do

d,

■D

$

10 3 -

d

(1.4)

Fig. 1. Determination of bolt
tightening torque

Force P ax induces tensile stresses
in the bolt

rr — ^ ax

Ufens '

0.785df

where d x is the minor diameter of the thread (for light bolts
the diameter of the bolt stem), mm.

The moment of friction in the threads — fi causes torsional
stresses in the bolt

_ Pax^ofi

2W tors

where W tOTB — 0.2 d\ is the torsion resisting moment of the bolt
cross-section.

Consequently

_ P

T ~ 0.4di*

The total stress, according to the third theory of strength, is
a^Vohns-r 1.6 + 25/? (1.5)

1.1. Unloaded Connections

9

Assuming (on the average) that = 0.8d, and substituting the
values / x = 0.22 and d 0 /d = 0.9, we obtain

2.6 ^

d 2

( 1 . 6 )

Substituting into this expression the value of P ax from Eq. (1.4),
we have

10 3 - ^ gM kgf/mm 2

(1.7)

where M tight = tightening torque, kgf*m

d = nominal diameter of the thread, mm
The values of a versus M tight , shown in Fig. 2, are estimated from-
Eq. (1.7) for bolts of different diameters. The diagram can be used

Fig. 2. Tightening torque Mti g ht an d stresses a for bolts of different diameters^

for approximate determination of stresses developing in a bolt
tightened with various torques. From the permissible stress it is=
possible to find the ultimate tightening torque.

10

Chapter 1. Tightened Connections

The inverse-cube relationship between the stresses due to tighten¬
ing and bolt diameter [see Eq. (1.7)] accounts for the abrupt rise
■of the stresses, occurring when the bolt diameter is reduced. When
manually tightening small-diameter bolts, one may overstress the
holts, thus stretching and even breaking them.

Approximate values of forces and torques when tightening bolts
by hand are given in Table 1.

Table 1

Forces and Torques when Tightening Bolts Manually

Bolts

Wrench arm,
m

Tightening
force, kgf

Tightening
torque, kgf-m

.Small (M4-M8)

0.1-0.15

» 10

1-1.5

Medium (M10-M14)

0.15-0.2

«15

2-3

Large (M16-M24)

0.2-0.25

«20

4-5

Assume, for example, that the tightening torque is 1.5 kgf •m.
Drawing in Fig. 2 a horizontal line Mu g ht — 1.5 kgf-m, we read
the stress values on the abscissa axis: for bolts M8—37 kgf/mm 2 ;
for bolts M6—80 kgf/mm 2 . The latter figure is well above the yield
limit of commercial carbon steels. Consequently, bolts smaller than
ZM6 may easily be broken when tightened by hand, and with the
application of excessive forces bolts M8 may also be broken.

The magnitude of stresses due to tightening, in accordance with
ZEq. (1.3), strongly depends on the coefficient of friction in the threads
and on the nut bearing surface. Friction blocks, as it were, the tighten¬
ing force, so that a major part of the latter is spent in overcoming
friction, only an insignificant portion being taken up by the bolt
stem.

For example, when f x = 0.22 and / 2 = 0.11 the portion of the
torque utilized for the bolt tightening, according to Eq. (1.3), will be

0.024

0.024+0.1 + 0.72

• 100 %

12 %

The remaining 88% of the torque is spent in overcoming friction.

Equation (1.7) and the diagram in Fig. 2 are based on rather high
values of the friction coefficients (f x = 0.22; / 2 = 0.11) correspond¬
ing to non-lubricated surfaces. If some lubricant gets onto the friction
surfaces, then, with the same torque, the bolt stresses will rise.

Table 2 gives the values of stresses, calculated from Eq. (1.7),
-which arise in bolts tightened by means of standard wrenches with
the application of 15-kgf tightening force. From the Table it is clear
that with low friction values it is possible to break even M10 bolts
-when tightening them manually. The possibility of overstres-

1.1. Unloaded Connections

11

* Table 2

Stresses in Bolts when Tightening with Standard Wrenches

Bolt dia.

j Stresses in kgf/mm 2 when

fi = 0.22
/ 2 = 0.11

/i = 0.11

h = 0.055

M6

100

180

M8

50

90

M10

30

54

M12

17

30

M14

12

22

M16

9

16

Note. The heavy line delimits stresses exceeding the yield limit of commercial carbon
steels.

sing bolts with threads larger than M12 when tightening them with
standard wrenches is practically excluded.

If, however, small-sized bolts must be used for some design reasons,
then appropriate measures must be taken to limit the tightening

Fig. 3. Methods of eliminating bolt twist when tightening

torque or the bolts must be made from a high-grade heat-treated
steel.

The simplest way to limit the tightening torque is to shorten the
wrench arm correspondingly with the decrease of the bolt diameter.
This practice is, in effect, stipulated by current standards covering
wrenches and spanners.

12

Chapter 1. Tightened Connections

The twisting-off of a bolt can be avoided if during tightening it
is held by a special element (Fig. 3 a) or if the bolt end is locked to
the housing (Fig. 3 b). In these cases only tensile stresses occur in
the bolt.

Assume that in Eq. (1.5) r = 0, and dy = 0.8d (as before), then

a = 1.56 ijs.

a i

Comparing this expression with Eq. (1.6), we see that the stresses
1 56

amount only to 0.65 of the stresses when tightening a bolt

with its being twisted.

Torsional stresses arise only during tightening and subsequently
vanish due to the bolt elasticity. Therefore, when designing tightened
connections for long-term strength the torsional stresses are usually
neglected, the bolt calculations being limited to the axial force
Pax only [see Eq. (1.3)].

1.2. Loaded Connections

These include connections subjected to the action of a force which
stretches the joint and imposes an additional load upon the fastening
bolts. This force can be constant (e.g., the static pressure of gases
and liquids in vessels) or variable (the pressure of gases in internal

combustion engines and piston
compressors, inertia forces of
moving masses in connecting
rods and bearings of crank
mechanisms).

Here the bolt preload is
selected so that, regardless of
possible working force fluctua¬
tions, a constant tension at
the joint is maintained, avert¬
ing its opening which could
break the joint seal and, in the
case of variable forces, cause
the crushing and work hard¬
ening of the metal surfaces.

The connection may be additionally loaded by thermal forces aris¬
ing as the system gets heated.

Figure 4 shows a bolted connection subjected to the action of
internal working pressure P work . To assure normal functioning of
the joint the bolts must be pretightened by a force P tight sufficient
to keep the joint tight under the action of working force P wor k-

Now let us clarify what deformations occur in the system when force
Ptight is applied to it. For the sake of simplicity we neglect changes

12. Loaded Connections

13

in the length of the threaded bolt ends and consider that the working
length l of the bolts is equal to the thickness of the joined parts.

Under the action of force Ptight the bolts will lengthen by the
value

, Plight
** ~ EiFi

while the joined flanges will shrink by the value

, Plight
E 2 F 2

where E i, E 2 and F 2 , F 2 are the elasticity moduli of the materials

and cross-sectional areas of the bolts and
housing, respectively.

The force with which the housing members are compressed is equal
to the tightening force, i.e.,

Pcomp = P tight (1-8)

After the application of the working force P wor h the bolts will
additionally extend by the value AX, and, accordingly, the housing
compressive deformation will decrease by the same value. As a result,
the pressure exerted by the housing upon the bolts is reduced by
the amount A P.

The force tensioning the bolts becomes

P tens — Pwork "l” Ptight &P (1 -9)

and the force of the housing compression

P comp = P tight — &P (1-10)

Force A P can be deduced from the following relationships.

In accord with Hooke’s law, the value of the housing deformation
decrease is

In bolts the same amount of deformation is produced by the dif¬
ference of forces prior to and after the application of force P w0Th , i.e.,

Ptens Ptight = Pwork -f* Ptight AP Ptight = Pwork A P

Consequently, in respect to the bolts

AX

(Pworh AP) l
EiF j

Equating Eqs. (1.11) and (1.12), gives

A P

Pwork

1

E,F i
E 2 F 2

( 1 . 12 )

(1.13)

14

Chapter 1. Tightened Connections

Substituting this expression into Eqs. (1.9) and (1.10) gives the
bolt tension force

Ptens = Pwor k + Ptitht - A Pw °[ h fV = Ptitht + Pw £$ - 2 ( 1 - 14 )

1+ E 2 F 2 l+ EiFi

and the joint compression force

Pcomp ~ Ptight -( 1 - 15 )

If the working force varies from zero up to P worh , then the bolt
tension force will pulsate with an amplitude

At ens = Ptens- Ptight = Pw ° T f- (1.16)

1-1-LA

+ E\F \

while the housing compression force will pulsate with an amplitude

*comp •

= Ptight—P‘

work

comp '

EjFj

E 2 F 2

(1.17)

Length does not enter into Eqs. (1.14)-(1.17). This means that
the forces acting at the joint are theoretically equal when tightening
low flanges and high housing components by bolts.

Actually the forces under consideration are affected by the elastic
and plastic deformations of the threads and bearing surfaces of the
nuts and bolt heads, etc., which may appreciably reduce the forces
that tension the bolts and compress the connection.

The relative significance of terminal conditions is much greater
for shorter bolts than for longer ones. That is why the joints fastened
with short bolts weaken more quickly in use, particularly when sub¬
jected to pulsating loads. As a general rule, it is better to use long
bolts (high flanges) or introduce elastic elements compensating for
local plastic deformations of the system.

On the basis of Eqs. (1.14) and (1.15) it is often concluded that
smaller ratios E 1 F 1 /E 2 F 2 are more beneficial, in other words, the
best combination is elastic bolts and rigid housings.

From Eq. (1.14) it is evident that the bolt tension force is minimum
E F

(Pt ens = Ptight) when E ' ~p - = 0 (perfectly elastic bolts or perfectly

^ ^ Fa F

rigid housings) and increases with an increase in ~r -, reaching

2* 2

E F

its maximum (P ten s = Ptight + Pworh) when ^r- = °o (per¬
fectly elastic housings or perfectly rigid bolts). The amplitude of
the pulsating tension force [Eq. (1.16)] also falls down with a decrease

in tt-tt- • In the limiting case (~~ = 0) the bolt tension force

2 /

1.2. Loaded Connections

15

remains constant and is Ptens = Ptight > i.e., the load on the bolts
becomes static despite the pulsating working force.-With .an increase

in ~^r- the load on the bolts becomes cyclic. When — oo

A 2^2 £>2*2

the pulsation amplitude is A fens = P W0T h-

J? F

With a decrease in g~ l the joint compression force P CO mp

[Eq. (1.15)] is also reduced. This is beneficial for the housing strength,
but is disadvantageous for the joint sealing, since the force sealing
the split connection is equal to P comp . At the same time, with the

decrease of - ~iS the pulsation of force P CO mp [Eq. (1.17)] is increas-
■^ 2^2

ed, but since the housing strength is usually much greater than
the bolt strength and the compression force fluctuations are not so
dangerous as the bolt tension force fluctuations, then it is recommend-

ed to employ low ^ values, as it is considered that this lowers
£-2^2

the bolt load. From this stems the time-honoured design rule for
split connections: elastic bolts—rigid flanges.

To this rule certain essential corrections are necessary.

To fully reveal the essence of this phenomenon, it is necessary to
ascribe definiteness to the member Ptight entering into Eqs. (1.14)
and (1.15), i.e., to agree upon how to select the tightening force.
There are two methods of such a selection. In accord with the first
method, widely applied until recently, the tightening force was
taken in proportion to the working force P W0T k

Ptight— yP work (1.18)

where y is the tightening factor (usually y = 1-2).

According to the second method, the tightening force is found
on condition that the joint compression force P co mp is proportional
to the working force, i.e.,

P comp — QP work (1.19)

where 9 is the proportionality factor (generally 0 = 0.25-1).

Let us consider both of these cases.

The case when Ptight — yPwork■ Substituting into Eqs (1.14
and 1.15) Ptight = yPwork, gives

Ptens — Pwork (y-\ -( 1 . 20 )

V 1+ ^7 /

Pcomp —Pworh ^7 EtF\ ^ (1.21)

\ EzPz )

F P

From Eq. (1.20) it is clear that variations of the ratio even
within the widest limits affects the magnitude of the force Ptens

16

Chapter 1. Tightened Connections

comparatively little. In the limiting cases Ptens = Pwork! (when
|= 0) and P tens = P WOTh (1 + y) (when = oo) . Hence,
the entire range of force Ptens is confined within the limits .

TT P

It should be noted that in the expression ■ - 1 the number of

•& 2^2

possible values of the ratio EJE 2 is limited. Bolts are made almost

fig. 5. Ratios P tens /P work and Pcomp 1 p work as a function of F 1 /F 2 for diffe¬
rent values of tightening factor y

exclusively of steels (E = 20"lO 3 - 22-lO 3 kgf/mm 2 ) and in very
rare cases (in special constructions) — of titanium alloys (E =
= 11.5-10 3 -12.5-lO 3 kgf/mm 2 ).

Joined members are made of steels, cast irons, light and titanium
alloys (Table 3).

There are three values of the ratio E r IE 2 having practical impor¬
tance: EJEz — 1 (steel-steel; titanium alloy-titanium al\oy)',Ei/E 2 w
« 2.5 (steel-cast iron); EJE^ w 3 (steel-aluminium alloys).

With specified materials the value can be influenced only

i

by changing the ratio FJF 2 , which entails changes in the strength
of the bolts and housings.

When clamping a split aluminium housing with steel bolts
(EJE 2 w 3) the ratio Ptens/Pwork, for different values of y taken
in a wide range of FJF^ values (from zero to unity), will vary inap¬
preciably, only by 1.5-2 times on the average, as evident from Fig. 5a.

1.2. Loaded Connections

17

, Table 3

Material Combinations for Bolts and Joined Components

Bolt material

Joined components

Ei/E%

material

® 2 , kgt/min!

Steel

Steel

i

Ei =21000 kgf/mm 2

Cast iron

2.6

Aluminium alloys

2.9

Magnesium alloys

4 500

4.7

Titanium alloys

12 000

1.75

Titanium alloy

Titanium alloys

12 000

1

Ei = 12 000 kgf/mm 2

Thus, the gain from decreasing F 1 /F 2 is not very great. The picture
is similar for other EJE 2 ratios.

Decreasing the force Pt en s still does not mean an improvement

in the bolt strength which is determined by the stress a = .

Such an improvement will only be reached if the ratio FJF 2 is
reduced by increasing the housing cross-section and not by reducing
the bolt cross-sections. It can easily be proved that reducing the
bolt cross-sections decreases relatively little the force Pt enS i and,
at the same time, sharply increases the bolt stresses.

On the other hand, increasing the ratio FJF 2 by enlarging the
bolt cross-sections provides a definite gain in the bolt strength.

In relation to joints for which good sealing is imperative it is
extremely important to find how the compression force P CO mp changes
with F x !F 2 , for this force determines the tightening of the joint seal.

The values of the ratio Pcomp/Pwork (for EJE 2 = 3), estimated
from Eq. (1.21), are plotted in Fig. 5b. The compression force decreas¬
es with the reduction of the ratio F x /F 2 (rigid housings), the decrease
being the sharper the less the value of y. For example, with y = 1.25
force P C omp within the range of FJF 2 of from 0 to 1 is reduced four
times. Hence, the decrease of the ratio FJF 2 adversely affects the
reliability of the joint.

To assure reliable sealing in the case of rigid housings it is neces¬
sary to increase the tightening force, i.e., to increase the bolt tension,
which makes the gain from lowering F X !F 2 fictive.

Conversely, making the housing more pliable lowers the necessary
tightening force. With compliant housings it is possible to tighten

2-01418

18

Chapter 1. Tightened Connections

the joint with y < 1 without running the risk of the joint loosening
( Pcomp/P work > 0) and without impairing its reliability.

Let us now examine the effect of the ratio FJF^ upon the pulsation
amplitude of forces Pt ens and P CO m P (Fig. 6). It is evident from the
graph that decreasing the ratio FJF^ (rigid housings) lowers the
pulsation amplitude of force P tens , which is advatageous for the bolt
strength. However, the pulsation of force P com p increases as the
ratio Fi/.F 2 is reduced, this adversely affecting the reliability of
the joint.

Thus, we may conclude that with P tirht = yP WO rh, low ratios
FJF t (rigid housings) are advantageous for the bolt strength when

Fig. 6. Ratios A tenJPwork and &comp/ p worh as a function of Fi/F 2 for diffe¬
rent values of tightening factor y

the loads on the joint are pulsating. For static (in particular thermal)
loads, and also when the reliability of the joint is of prime importan¬
ce, higher ratios FJF 2 (compliant housings) are preferable.

The foregoing conclusions are of special significance for connect¬
ions in which the rigidity of the joined parts is commensurable with
that of the clamping bolts. Such a situation is encountered, for
instance, when clamping hydraulic and air power cylinders (Fig. la)
and cylinders of internal combustion engines and piston compressors
of half-block (Fig. lb) or full-block (Fig. 7c) designs. Here the design¬
er can vary within broad limits the ratio F-JF 2 by changing the
cross-sections of the clamped components and thus make it most
suitable for the given operating conditions.

With conventional flange connections manoeuvring possibilities
are less. As shown by experiments, actually in operation is the cylin-

1.2. Loaded Gonnicfions-

19:

drical flange volume of external diameter D « 2 d 0 and internal
diameter d « d 0 (Fig. 8a). The ratio of the bolt cross-sectional

Fig. 7. Tightened connections

area to the design area of the clamped parts is

F z ~ D 2 -d%

This ratio can be increased by placing an elastic gasket between
the parts being connected, or reduced by placing under the bolt
some massive washers of '~d H

increased diameterjand also l"~ j°~j i i °~j

by decreasing the bolt stem ( r I 7 1 (j tS

diameter (Fig. 8b). I I I ( J -*—

Assuming that D x = 3d„

and d x = 0.8 d 0 , we have ///fcjf Ikxaa FaIxx 1 1xXavv'

(0.8d 0 ) 2

= 0.08

The ratio F x /F 2 can also \\\K< | X\v vf<x> aona;

be reduced by placing ela- aw \v\jgg_._j) . ggcw

stic elements between the Aw Sw t \ \ toa

flange and bolt faces. . vv p al.,. . — Avt ori l l | .H &gj w

The case when P comp = I 111 'l I ! I \

=QP wor k- The relationships ' / ) ^fb)

become different when the ' a '

tightening force is chosen Fig. 8. Tightened flanges

so that Pcomp of the joint

is proportional to P worh . This condition is quite logical: the higher
the working pressure, the greater must be! the compression to ensure
the required reliability of the'joint. Selecting the' tightening force
in such a way establishes a direct relation between the joint com-?

to

Chapter 1. Tightened Connections

pression force and the working force, whereas in the previous case
this halation is a derivative of the preliminarily chosen tightening
factor y and of elasticity characteristics of the system.

Thejbolt tension force is always equal to the sum of the working
and compression forces. Hence,

Ptens — Pwork Pcomp — (1 “l - 6) Pwork (1.22)

Tjhus, the forces P tens and P comp (for a given force P worh ) in this
case are constant and independent of the ratio E^FJE^F^.

From Eq. (1.15) the required tightening force

Pwork
, ElFi

(1.23)

Substituting into this relation the value P comp = QP worh , we get
Ptight ~ Pwork (Q “I-\ (1-24)

The ratio PtightIPworh as a function of FJF 2 (when E X IE 2 = 3)
is plotted for various values of 0 in Fig. 9. The graph shows that
p increasing the ratio FJF 2

^tiqht

(compliant housings) redu¬
ces the required tightening
force.

The determination of the
tightening force on condi¬
tion that Pcomp — 0 Pwork
is undoubtedly more ration¬
al than on condition that

= yP,

The lat-

. 9. Ratio Ptight!Pwork as a function of
Fi/F 2 for different values of factor 0

ter method must be reject¬
ed as being wrong in prin¬
ciple. In this connection
the problem of the influen¬
ce of the E 1 F 1 /E 2 F 2 factor
on the operation of the jo¬
int should be reconsidered.

As is obvious from the fore¬
going, when Pcomp—QPwork
the ratio E 1 F 1 /E 2 F 2 has no
influence on the forces Ptens
and Pcomp which are de¬

fined exclusively by the magnitude of factor 0.

The ratio EiFi/E 2 F 2 affects the tightening force only. Higher
values of EiF 1 /E 2 F 2 (compliant housings) are more advantageous as
they enable the required tightening force to be decreased.

1.2. Loaded Connections

21

In connections loaded by a pulsating force the ratio EiFi!E 2 F 2
also affects the pulsation amplitude of forces Ptens and P com p.
According to Eqs. (1.16) and (1.17), as E X F X !E 2 F 2 decreases (rigid
housings), the pulsation amplitude of Pt ena is reduced, while that
of Pcomp, increased. 1 Conversely, increasing E 1 F 1 /E 2 F 2 (compliant
housings) augments the pulsation amplitude of Pt en s and diminishes
that of Pcomp• Consequently, we may conclude that under pulsating
loads rigid housings are more advantageous for the bolt strength
and compliant housings, for reliable sealing.

In joints loaded by a constant force the magnitude of E 1 F 1 /E i F„
seems indifferent both for the bolt strength and for the quality of
sealing. All the above considerations on the relative advantages of
rigid and compliant housings under a static load are valid only
when P tight = yPwork and lose their value when P eo mp = QPworh-

The magnitude of factor 9, which in the given case is of decisive
importance for the joint parameters, is chosen to suit the reliability
requirements for the seal: for non-critical joints .6 = 0.25 to 0.3
and for critical joints, 0.5 to 1.

The calculation on condition that P CO mp = QPworh is simpler.
There is no need for the cut-and-try procedure of finding Ptight
and checking the resulting P co mp, and the calculation comes to the
application of the simplest formulae (1.22), (1.23) and (1.24) which
at once yield all the values determining the strength and reliability
of the joint.

(a) Temperature Factors

If a joint operates at elevated temperatures, and the temperatures
of the clamping bolts and joined parts differ, or if the parts are made
of materials possessing different coefficients of linear expansion,
a thermal stress P t then develops in the joint. In accord with Eq. (7.1)
[see Fundamentals of Machine Design , Vol. 1, Chapter 7], this stress is

Pt = (a 2 * 2 — <Mi) ^ EI e\f{ • 25 )

1+ “E^2

where oq, a 2 = coefficients of linear expansion of the materials
of the bolts and housing, respectively
*i, t 2 = working temperatures of the bolts and housing
The total bolt tension force is equal to the sum of forces Ptens
and P t

Pu

tens z

tens

+ P t =P

tight'

work

{afa—ajh) EjFj

EjFt

1 -

‘EjFt

E^Fz

(1.26)

22

Chapter 1. Tightened Connections

The total joint compression force is equal to the sum of forces
Pcomp find P t

r>/ n p p Pwork i (^2*2 ~ °Ml) ,a <yn\

Pcomp — Pcomp i — “tight * E\F\ ' A EiFi

i+ ~E^Fz i + %Fz

( b) Housings with Changing Sections

Often met in practice are cases when the parts being clamped have
changing (variable) sections (Fig. 10a) or are made of materials
with different moduli of elasticity.

Fig. 10. Complex systems

(a) with elements of changing section; (b) with elastic elements

Let V, l ", 1"', ... be the lengths of the dissimilar sections (/' +
+ l" -f- Z", . . . = l) and let each have its corresponding rigidity
factor E'F\ E"F", E"F m , ....

Then the rigidity factor E 2 F 2 in the previous expressions should
be changed to a corrected factor whose magnitude is determined
from the following considerations.

Let a complex system be acted upon by a force P. The total defor¬
mation % of the system, produced by the force, will be the sum of the
deformations of its individual elements

PV , PI"

A,==r+r-i-

E'F'

E"F‘

1.2. Loaded Connections

23

The relative deformation of the system

X P / V l’ \

e l 2 V E'F' “** E’F’

The corrected rigidity factor of the system

«g”T=- y ~ > , - d- 28 )

Introducing this factor into the previous equations in place of
E 2 F 2 , the calculation can be carried out as previously.

In this case the temperature deformation is

Ae = + Va’Jl + ... — la^

where a 2 = coefficient of linear expansion of the bolt material
ti = bolt temperature
In relative units

Ae=*L a '/ 2 +I-a;t;+...- ai t l (1.29)

The sum of the deformations of the individual elements in the
system may be expressed as a function of the thermal stress in the
following way:

Ae--

tPl'

P t l"

E'F'l ' E’F’l 1 • ' • 1 E^t
Equating Eqs. (1.29) and (1.30), one will obtain

~i —b a a*5 “+'••• —F i

— p * ( eg + EiFi)

Pt=

EjFj

eg

(1.31)

(c) Elastic Elements

The elasticity of a bolt-housing system can be varied without
changing the sections of the system components by introducting
elastic elements (Fig. 106). This is widely employed in practice.

Depending on their arrangement, such elastic elements increase
the elasticity of either the bolts or housings. In order to evaluate the
effect of the elastic elements it is necessary first to find out which
elements belong to the bolt system and which to the housing one.
If the application of the working force P WO rh increases the load
on an element, the latter then belongs to the bolt system, regardless
of whether the load is tensile or compressive. If the working force
lessens the load on the element, this element then belongs to the
housing system.

24

Chapter 1. Tightened Connections

For instance, the load on element 1 (Fig. 10b) is increased when
the working force P WO rk Is applied. Therefore, this element belongs
to the bolt system; its elasticity should be included into that of the
fastening bolts. Element 2 , on the contrary, is unloaded when the
working force is applied and, hence, must be regarded as belonging
to the housing system.

The deformation of an elastic element under the effect of the
applied load is

V = Pfl.

where l e = length of the element

/ = elastic ratio of the element (relative deformation pro¬
duced by 1-kgf force)

The deformation related to the total length l of the joint is

e' = Pf l f

The total relative deformation of a bolt with an elastic element
having length l a and elasticity f a is the sum of the deformations
of the bolt and element

e = e 1 + e ' = p(-JL

The quantity

1

EiFi

P

(1.32)

is the corrected rigidity factor of the system.

The corrected rigidity factor of a housing with an elastic element
of length l b and elasticity f b is

® 6 & 6 =-T - J —IT (1 - 33)

~E^ +fb T

These quantities can be introduced into the previous equations
in place of E 1 F 1 and E 2 F 2 and the calculations carried out as pre¬
viously.

With the help of elastic elements the operating parameters of
a connection can be beneficially influenced. For example, intro¬
ducing elastic elements into the housing system can improve the
reliability of sealing of the connection and reduce the necessary
bolt tightening force. The pulsation of the bolt tension force can
also be reduced by introducing elastic elements.

Elastic elements are an efficient means for preventing the gradual
weakening of the tightening force due to relaxation.

1.2. Loaded Connections

25

( d) Relaxation

Tightened joints (particularly those operating under elevated
temperatures) in the course of time weaken due to a slowly deve¬
loping plastic deformation of the bolts (and, occasionally, of the
clamped parts as well) under the influence of loads acting for a long
time. The plastic deformation phenomenon occurring under stresses
well below those corresponding to the yield point of the material
is called relaxation. The property of metals to flow when being
acted upon by a continuous load can be ascertained only in special
tests, during which specimens are held under load for 3000 to
10 000 hours.

Relaxation is often defined as a spontaneous change with time
of stresses with unchanging deformation. This definition cannot be
accepted. The fall in stresses during relaxation is inevitably accom¬
panied by plastic deformation. Moreover, plastic deformation is
the first reason for relaxation. It is more legitimate to speak about
a room-temperature creep of materials, which is akin to high-tem-
perature creep, the difference being that the room-temperature creep
deformation develops slower and has a smaller magnitude.

The process of relaxation can be followed schematically on an
example of housing-type components clamped with bolts. For the
sake of simplicity we assume at first that the housing under consi¬
deration is perfectly rigid and the only deformable element is the
holt. At first glance it seems that the system operates under the
conditions of constant deformation. Actually this is not true. Even¬
tually the bolt elongates plastically. The original relative elastic
deformation Ae of the bolt, due to the bolt preload, now decreases;
by the amount of relative residual deformation A res so that the
resultant elastic deformation becomes Ae' = Ae — A res .

If the initial tightening force equalled Ptight — A eE x F x (E x —
— elasticity modulus of the bolt material, F 1 = bolt cross-section),
then after elongation the force becomes P'ught = (Ae — A res ) E X F X ,
i.e., it decreases in comparison to the original tightening force by

(1 — times. Upon removing the bolt and measuring its length,

it is found that the bolt has lengthened by the value lA rea (l —
= original bolt length).

With the decrease of the tightening force the stresses in the bolt
fall down. When they reach the level at which the bolt elongation
discontinues or becomes negligible, the process of relaxation ceases
and the stresses in the system become stable.

In real systems of clamped elastic parts the course of this process
is somewhat different. As the bolt elongation proceeds, the clamped
parts, while expanding elastically continue to exert pressure on
the bolt, although somewhat lower in comparison with the initial

26

Chapter 1. Tightened Connections

pressure. Under such circumstances the relaxation process stops and
the system becomes stable with a relatively greater bolt elongation
than in the previous case.

We have analysed a non-loaded connection. In systems subjected
to a constantly acting static or cyclic load relaxation progresses
continuously. The state of practical stabilization sets in only as
a result of the slowing down of the elongation process, which is

•observed in most materials in the
nection has not failed before this

Fig. ila. Total deformation e of a car¬
bon-steel specimen (0.1%C) after
being held under tensile stress o.
Test temperature 20°C (after Rogan
Alexander)

course of time, provided the con-
due to the separation of the joint

Fig. 116. Residual stresses in per cent
of original stress (o 0 = 25 kgf/mm 2 )
as a function of holding time.
Test temperature 400-450°C

faces. The higher the stress, the greater the creep of materials
■(Fig. 11a), and it sharply increases with the rise in temperature.
The creep is greater under cyclic loads than under static ones. It
varies with different metals and alloys and depends on the type of
heat treatment.

This requires the introduction of a concept of relaxation resistance
understood as the ability of a material to resist plastic deformation
mnder continuous loads.

The method of determining relaxation resistance is not as yet fully
-established. The majority of the methods applied at present are
based on inducing in specimens a certain, rather high stress; the
specimens are held at this particular stress and at an elevated tern-

1.2. Loaded Connections

27

perature for a long period of time (up to 10 000 hours). Generally,
the chosen stress amounts to 0.5-0.6 of the yield limit of the material
at the given temperature.

A split-ring method is the one most often employed. The specimen
is a ring with a wedge-shaped cut. The required stress is produced by
forcing a wedge into the cut, which causes bending stresses to develop
in the opposite ring section shaped so as to have the same bending
resistance throughout its entire length. In this state the specimen
is kept at the given temperature for the given period of time.

At definite time intervals the load is removed and the residual
deformation measured. This is used for calculating the stress remain¬
ing in the specimen at the end of each interval.

The diagram shown in Fig. 11 is a result of such tests. The re¬
maining stresses are shown in Fig. 11 b as a function of time and
characterize the relaxation resistance of the material. The higher
the stress remaining in the specimen (i.e., the lower the residual
deformation of the specimen), the greater the relaxation resistance
of the material.

The remaining stresses sharply drop during the first 1000 hours
of tests and then lower at a slower pace. This shows that as the
holding period grows longer the yielding of the material decreases
(seemingly as a result of strain hardening).

The relaxation resistances of various materials are different. Thus,
for example, a specimen made from steel grade 40X, after being
held at 400°C for 3000 hours, retains 80% of its initial stresses,
while one made from steel grade 50X<I>A retains only 40%. If we
convert the results obtained for bending loads to the case of tensile
loads, then a bolt made from steel 50X0A will, after 3000 hours’
work, extend to a length equal to 60% of its initial elastic deforma¬
tion due to tightening, in which case the tightening force will be
reduced by 60%. Naturally, account should also be taken of the
above-described effects of the elasticity of the clamped parts and
of the working forces which add to the bolt elongation.

Estimating the relaxation resistance by the magnitude of the
residual stresses is controversial. To reflect the physical essence of
the phenomenon and make calculations more convenient it is advis¬
able to characterize the resistance to relaxation by the magnitude
of the deformations remaining in the specimen after its being held
under a load corresponding to the actual loading conditions (for
fastening bolts, under a tensile load).

Methods to prevent the weakening of tightened connections as
a result of relaxation consist in making bolts from relaxation-re¬
sistant materials and subjecting them to appropriate heat treatment.
Silicon steels possess a high relaxation resistance. Normalizing
followed by high tempering is an optimum heat treatment from the
standpoint of relaxation resistance. It seems possible to obtain

28

Chapter 1. Tightened Connections

a significant effect by strengthening the bolts through their preli¬
minary elongation (ageing) which acceleratively simulates the
greatest yield stage of the material.

All measures should be taken to reduce stresses in bolts and the
pulsation amplitude of the tension force in cyclically loaded con¬
nections. It is also advisable to introduce elastic elements of relaxa¬
tion-resistant materials so as to compensate for residual deforma¬
tions as they develop.

(e) Graphical Calculations of Tightened Connections

The totality of the phenomena occurring in tightened connections
is easily amenable to graphical interpretation with the help of P-e
(force vs. relative deformation) diagrams.

Let us take the simplest case, namely, the tightening of a split
connection with force Pu g ht (Fig- 12a). In solving the problem the
forces will be plotted on the ordinate axis, and relative deforma¬
tions — on the abscissa axis, assuming tension to be a positive,
and compression, a negative deformation. Line Oa shows the bolt
elongation and its slope relative to the abscissa axis is

tan a = — EiF ±

where T] = scale of forces

fi = scale of relative deformations

The compression of the parts being tightened is presented by the
straight line Ob whose slope relative to the abscissa axis is

tan P — -jj- E 2 F 2

If we draw on the diagram a horizontal line ba at a distance cor¬
responding to the tightening force Pttght from the abscissa axis,
it will intersect the tension-compression lines at points a and b,
the abscissae of which are equal to the relative deformations e 1
of the bolts and e 2 of the housing when tightened.

It is more convenient to plot the diagram as shown in Fig. 12 b,
i.e., by drawing the compression line through point a on the tension
line, which corresponds to P = P tight-

Now assume that a force P W0T k develops in the joint and that
this force imposes an additional load on the bolts while unloading
the joint (Fig. 12c).

When determining the bolt tension force, we assume, as previously,
that the relative deformations of the bolts and clamped components
are the same.

At a distance P W0T h (along the vertical) from the bolt tension
line Oa draw a straight line (dashed) parallel to it. Through point k.

jvtyd -

1.2. Loaded Connections

29

where this line intersects the clamped parts’ compression line, draw
a vertical line until the latter intersects the tension line (point c).
This vertical line will cut off a line-segment of length Ae on the

Fig. 12. Graphical calculations of tightened connections

abscissa axis. Clearly, this scheme reproduces the condition of equal
Ae for the bolts and clamped components.

This construction gives all the necessary data. The ordinate of
point c shows to scale the bolt tension force Pt ens ', the ordinate of
point k, the joint compression force P com p\ line-segments e[ and e ,

30

Chapter 1. Tightened Connections

relative deformations of the bolts and clamped components after
application of working force P wor k-

If the working force pulsates from zero to P w0Tk , the pulsation
amplitude of force P tens will then equal Ptens — Pttght (the right-
hand wavy curve) and that of force P comp , Pttght — Pcomp (the
left-hand wavy curve).

Now let us consider how the system’s behaviour is influenced by
the variation of the rigidity of bolts and clamped parts (i.e., by
changes in the slope angles a and ft of the tension and compression
lines, respectively).

Figure 12 d illustrates the case of a rigid housing and elastic bolts,
while Fig. 12e, that of rigid bolts and an elastic housing. The tighten¬
ing force Pttght is assumed to be the same in both cases. Figures 12 d
and 12e clearly illustrate the above-described regularities. The
increase of the bolt elasticity with a simultaneous increase in the
housing rigidity (reduction of the ratio E^FJE^F^) lowers the bolt
tension force and the pulsation amplitude of this force (see Fig. 12 d).
At the same time this decreases the joint compression force and in¬
creases its pulsation amplitude.

Increasing the ratio EtF^E^F^ (elastic housing and rigid bolts)
acts in the opposite direction (see Fig. 12e): force Ptens and the
amplitude of its pulsation increase, P CO mp rises too, but the ampi-
tude of its pulsation decreases.

As stated earlier, when determining connection parameters it
is more proper to proceed from the condition of proportionality
between the compression force and working pressure (P comp =
= 6P work)-

In this case the graph is plotted in the following order (Fig. 12/).
First, draw horizontals at distances P CO mv = QPwork and Ptens =
= (1 + 0) P WO rk from the abscissa axis (in Fig. 12/ it is assumed
that 0 = 0.6). Then from any arbitrary point draw at an angle (1
an inclined line representing the housing compression. From point d
at which this line intersects the horizontal line P comp draw a ver¬
tical until it meets the horizontal line Ptens and through point e
of their intersection draw at an angle a an inclined line representing
the bolt tension. The ordinate of point /, where the two inclined lines
intersect, gives the value of the tightening force Pttght-

The difference Ptens — Pttght shows, as before, pulsation ampli¬
tude of force Ptens , and the difference Ptight — Pcomp i the pulsation
amplitude of force P comp .

Figures 12 g and 12 h show how the rigidity of the housing and
bolts influences the operation of the joint (with P worh and 0 values
being the same as above). In the case of elastic bolts and a rigid
housing (Fig. 12g) force Pttght increases, the pulsation amplitude of
force Ptens decreases and the pulsation amplitude of force P com p
increases.

1,2. Loaded Connections

31

In the case of rigid bolts and an elastic housing (Fig. 12 h) the
force Ptight decreases, the pulsation amplitude of force Ptens increas¬
es and the pulsation amplitude of force P CO mp decreases.

For an unloaded connection (P WO rk = 0) subjected to heating,
which causes a thermal stress P t to develop in the joint (Fig. 12i) r
one should draw a line parallel to and spaced from the compression
line at a distance of a 2 t 2 — a 1 t 1 (along the horizontal), equal to
the relative deformation due to the heating. The ordinate of point a',
at which this line intersects the tension line, is equal to Ptens (iden¬
tical to force Pcomp compressing the housing) after the heating; the
difference between the ordinates of points a and a’ represents the
thermal stress P t .

For connections which are simultaneously subjected to heating
and to the action of force P WOT k the plotting differs in that the inclin¬
ed tension and compression lines are set apart (horizontally) at a
distance of a 2 t 2 — a 1 t l (Fig. 12;). The rest of the construction is
as before.

The elasticity characteristic of the clamped parts does not always
follow the linear relation P = eE 2 F 2 . Such is, for instance, the case
with thin-walled intricately shaped components whose walls in
certain areas are either inclined or perpendicular to the direction
of the compressive force. Here the compressive deformations of the
walls are complemented with the elastic bending deformations of
their inclined and horizontal portions. Furthermore, the walls may
be subject to buckling. If so, the rigidity of the clamped parts sharply
drops and the characteristic curve assumes a gently sloping
shape.

Occasionally the elasticity characteristic becomes curvilinear
because the individual elements of a construction come into opera¬
tion not at the same time, but one after another as the load increases.
The elasticity characteristic of such complex constructions can
only be determined experimentally. To this end a fully finished
construction is subjected to compression on a test rig and its elasti¬
city characteristic is plotted point by point on the basis of the test
data.

In this instance the graphical method of calculation is the only
way possible. The curve obtained from the tept is plotted on the
graph in place of the inclined line P = eE 2 F 2 (Fig. 12 k, l). Further
construction is as described above.

When calculating for a known force P CO mp = QPworki the curvi¬
linear nature of the characteristic has an effect on the magnitude
of the tightening force and of the pulsation amplitude of forces
Ptens and Pcomp • With a convex characteristic (Fig. 12 k) the required
force Pttght diminishes, the pulsation amplitude of force Ptens
increases and that of force P CO mp decreases. If the characteristic
is concave (Fig. 121) the picture is reversed.

32

Chapter 1. Tightened Connections

(/) Methods of Controlling the Tightening Force {Preload)

Because the preload has a great effect on the bolt tension and joint
compression forces, it is very important during assembly to strictly
Jceep to the calculated magnitude of the tightening force. This is
•achieved by tightening nuts with torque wrenches, by turning them
through calculated angles or by measuring the elongation of bolts.

The application of torque measuring wrenches does not provide
for adequate accuracy, since the force needed to tighten nuts depends
to a large extent on the thread condition, coefficient of friction in
the threads and on the bearing surfaces, etc. For this reason, bolts
tightened with the same torque may, in fact, be loaded differently.

To lessen the friction effect critical connections are sometimes
tightened on a vibration table. The rather sharp decrease in the
friction forces because of vibration must be considered when choosing
the design tightening torque.

When tightening nuts by rotating them through a calculated
angle they are first brought into close contact with the bearing sur¬
faces, i.e., up to the point when bolts begin to extend. After this
the nuts are turned with a wrench through the required angle v.

This angle is determined from the given tightening force Ptight ,
proceeding from the following considerations.

When tightening a nut, it is necessary to choose the bolt elonga¬
tion = P g 8 p ■ and clamped parts contraction X % = P ^ S p - .

The axial displacement of the nut relative to the bolt is

X = X z = Pught 1 (-e~fT+~e^) ( 1>34 )

The tangent of the thread helix angle

tan( P = i < L35 )

where s — thread pitch

d 0 = thread pitch diameter

The displacement of the nut by value X corresponds to its rotation
over arc C of the circumference of radius dj 2

C

%

tan q>

(1.36)

On the other hand, the value of C is

C = iLv_4-.^V> (1.37)

where v is the nut rotation angle in radians, and v°, the same in
degrees.

1.2. Loaded Connections

33

Equating Eqs. (1.36) and (1.37), we have

o X 360°

t3Il (p 7t£?Q

Substituting into this equation % and tan from Eqs. (1.34) and
(1.35), we have

v° = 360 Pught — (1.38)

Clearly, the tightening angle is independent of the bolt diameter.

Nuts are practically tightened by the following method. First
all the clearances in the system are taken up (by tightening all
the nuts home in a certain succession). This done, the nuts
are slackened off and again tight¬
ened by hand or by means of
a light torque-control wrench un¬
til the nuts are in close contact
with the bearing surfaces. The¬
reafter all the nuts are tightened
in a definite sequence (staggered,
cross-like, serpentine-like) de¬
pending on the configuration of
the joint faces and assuring a 4 uni-
form tightening of the connec¬
tion by turning them first through
an angle of v/2 and then,
in the same succession, through
the full angle v.

To measure the nut rotation
angle the wrench is provid¬
ed with pointer and the tight¬
ened component, a graduated
disk.

This method is more accurate
than the first one, although it also
contains sources of errors due to the difficulty of determining the
right moment to start the tightening. It is only by fitting experience
that this error can be reduced to minimum.

In respect to slender bolts and studs the accuracy of measurement
is influenced by their twisting during tightening, which is due to
friction in the threads. To eliminate the twisting, the bolt end is
held with a spanner when tightening (see Fig. 3a). However, this
method is not convenient in actual fitting. Figure 13 depicts a wrench
design which excludes any effect of the bolt twisting upon the accu¬
racy of the tightening angle measurement.

Inserted into the wrench stem is a spring-loaded locking device 1
having a tapered end with a tapered shank of cross-like section.

Fig. 13. Wrench design enabling ac¬
curate measurement of nut turning
angle relative to bolt

3-01418

34

Chapter 1. Tightened Connections

which fits into a corresponding socket in the bolt end. The locking
device is spline-connected with rod 2 whose projecting end holds
a friction-mounted pointer 3 positioned over graduated dial 4 attach¬
ed to the wrench end face.

As the wrench is applied to the nut, the locking device enters the
bolt socket, thus providing a direct connection between the bolt
and pointer. Prior to tightening, the pointer is set to the dial zere
position. During the tightening the pointer indicates the rotation

angle of the nut relative to the bolt,
i.e., the nut tightening angle.

The most accurate method is that
whereby the bolt elongation, i.e.,

% = * is measured directly.

The elongation of short bolts is
measured by means of a micrometer cal¬
liper gauge. The technique is appli¬
cable whenever the calliper jaws can
be directly applied to the bolt ends.
In this way, for example, is measured
the elongation of connecting rod bolts,
pinch bolts in clamp connections, fastening bolts of hydraulic cylin¬
ders, etc. For greater measuring convenience the bolt head and
stem end are made spherical.

The elongation of studs can be measured with the aid of an indica¬
tor, if its foot is set up on an independent support. When installed on
the component being tightened the indicator will read the total of
the elongation of the stud and the contraction of the tightened parts.

Sometimes the tightening force is controlled by means of a system
of deformable washers (Fig. 14). Rigid washers 1 and 2 and measuring
ring 3 of a plastic metal (e.g., annealled copper) are placed under
the nut. Reference washer 4 is placed concentrically around the
measuring ring. The thickness of ring 3 is selected so that after the’
preliminary, light tightening of the nut a design clearance e (equal
to the sum of elastic deformations in the tightened system) remains
between the ring and reference washer. As the nut is finally tightened
the measuring ring is flattened out. The tightening is stopped when
the clearance e is taken up completely (this being indicated by the
reference washer losing its mobility).

The measuring ring must be changed whenever the nut has to be
re tightened.

(g) Calculation Example

An engine cylinder block having a cross-section as shown in Fig. 15 is fasten¬
ed to the crankcase by means of studs 400 mm long; the stud stem diameter
is 18 nun, thread M24 (pitch s = 1.5 mm).

Fig. 14. Control of tightening
force by means of deformable
ring

1.2. Loaded Connections

35

Assume that the ignition force P u , orh = 10 000 kgf is taken up by the four
studs nearest to the cylinder and that the studs' tightening force spreads over

0 0

Fig. 15. Cross-sectional view of cylinder block

the block portion adjacent to the cylinder (delimited by lines 0-0 in the Figure)
with a cross-sectional area of 7000 mm 2 .

The block is made from aluminium alloy grade AJI5 ( E 2 = 7500 kgf/mm 2 ,
a 2 = 22 • 10 -6 ); the studs are of steel grade 30XTC (E 1 — 21 000 kgf/mm 2 ;

= 11 - 10 _fl ). The temperature of the block and studs in the running engine
is 80°C.

Find the maximum stresses in the studs and crankcase in both the cold (dead)
and hot (running) engine.

Assume that 0 = 0.5.

J oint compression force

Pcomp = 0.5P work —5000 kgf

Total stud tension force [Eq. (1.22)J

Ptens = (1 + 0.5) P worh = 15 000 kgf

Tension force per stud

n 15 000 l.~t

Ptens. l— / —3750 kgf

Tensile stress in studs

Gtens

3750

0.785-182

= 15 kgf/mm 2

Compressive stress in cylinder block

a,

comp

5000

comp -

F 2

7000

= 0.7 kgf/mm 2

Total tightening force Pfight [Eq. (1.24)] required

Ptight = Pwork (o~\--

Factor E 1 F 1
Factor E 2 F 2
Hence,

V ■ e z f 2 )

EiFf—21 000 4-0.785-18 2 = 2.1-10 7 kgf
E 2 F 2 = 7500 ■ 7000 = 5.3 -10 7 kgf
1

Ptight —i0 000 I 0.5-f

1 +

2.1

5.3

= 12 000 kgf

3 *

36

Chapter 1. Tightened Connections

Angle through which nuts must be turned when tightening [Eq. (1.38)]
v=360°P, /gW —( 0 25E(fi +1^) =

= 360 °- 3000 ~ ( 1 -± w + lp ^ w ) _ ®-

Thermal stress in the joint when heated [Eq. (1.25)]
Pt = («2*2 - «l*l)- E %\p~

fey hypothesis, t 2 = tj = 80°C. Substituting numerical values, we have

? 1.107

P t = 80 (22 -11) • lO- 6 ■ ■ -f , ■ = 13 000 kgf
14-—.

^ 5.3

After heating:
stud tension force

p, ten» = Ptens + Pt = 15 000+13 000 = 28 000 kgf
joint compression force

P'comp = Pcomp+Pf = 5000 +13 000 == 18 000 kgf
tensile stress in studs

P’

a' ten , = _ tem ■

te 4-0.785d 2

4-0.785-182

= 28 kgf/mm 2

compressive stress in cylinder block

<^mp=^T=2.6 kgf/mm 2
tension force pulsation amplitude

A ten, = P'tens ~ ( P tight +Pt) = 28 000 - (12 000 + 13 000) = 3000 kgf
coefficient of cycle skewness

Pfenj A tens 3000 . „

r ~~ JZZ

compression force pulsation amplitude

A C omp = Plight +Pt - P’ comp = 12 000+13 000—18 000 = 7000 kgf
coefficient of cycle skewness

Pcomp 18 000 _

~~ PfigAf+Pf - 12000+13000 _U ’ 7 '

The fatigue limit of steel grade 30XTC under a pulsating cycle with a coef¬
ficient of skewness of 0.9 equals 75 kgf/mm 2 . The fatigue limit of aluminium alloy
grade AJI5 under a pulsating cycle with a coefficient of skewness of 0.7 equals

8 kgf/mm 2 . Consequently, the safety factor for the studs is — = 2.7 and that

40

g

for the cylinder blocks, ="3=3.

4.0

Chapter 2

Press-Fitted Connections

Press (drive) fits are extensively applied in general engineering
to make permanent joints or those seldom dismantled. The relative
displacement of press-fitted parts is prevented by the forces due to
elastic compressive deformation (in the male part) and tensile
deformation (in the female part), which are proportional to tlie
amount of interference in the joint.

2.1. Press (drive) Fits

The USSR State Standard TOCT 7713-62 stipulates the following
press fits for the basic hole system.

1st grade of accuracy

2nd grade of accuracy

1st heavy drive fit (DM i) .
2nd heavy drive fit (Dh2 x ).

.. n P u

.. np2 t

Light drive fit (Dl) ....
Heavy drive fit (Dh) . . .
Shrink fit ( Sh) .

. Ha
. lip

• r P

2a grade of accuracy

3rd grade of accuracy

1st heavy drive fit (Dhl 2a ) • ■
2nd heavy drive fit ( Dh2 ia )..

• n Pha
• Hp2 2a

1st heavy drive fit (Dhl 3 ) ..
2nd heavy drive fit ( Dh2 3 ) .,
3rd heavy drive fit ( Dh3 3 ) .,

■ n P i 3

,. Hp2 9
.. Up $3

Figure 16a gives mean interferences A mean pm as a function of
the shaft diameter d mm for different press fits; Fig. 166—mean

relative interferences pm/mm and .

From Fig. 166 it is seen that the relative interferences sharply
increase in the small diameter range. For this reason one must be
especially careful when designing small diameter joints since the
strength of press-fitted parts depends first of all on the amount
of their relative interference.

Light drive fits ( Dl) are obtained with the least interferences.
Fits Dhl 1 are close to the latter in terms of the mean interferences.
Fits Dh2 x , Dh, Dhl ia and Dhl z are practically the same as far as

38

Chapter 2. Press-Fitted, Connections

the mean interferences are concerned, the only difference being that
the fits to lower grades of accuracy have wider tolerances. Mean
interferences are practically the same in the case of fits Sh and

Mtn

Fig. 16. Mean interference S mean and relative interference A moan ld as a func¬
tion of connection diameter d for different fits

Dh2 3 . The greatest (and almost identical) mean interferences are
in the case of fits Dh2 2a and Dh3 :i .

The relationship between the mean interference and diameter is approxi¬
mately expressed by the formula

Ajnean pm « 1 1' mm + 60) (2.1)

The proportionality factor ip for various fits is as follows

Fits. Dl Dhli Dh Dh2i Dhl 3 Dhl 2a Sh Dh2 3 Dh,2 2a Dh3 3

f. 0.23 0.28 0.46 0.44 0.5 0.54 0.95 0.98 1.13 1.16

2.2. Strength of Press-Fitted Connections

The axial holding power of a press-fitted connection is

Pax = kFf kgf (2.2)

The torsional holding power

Afior. = 0.001*F/4 kgf.m (2.3)

where k — unit pressure on the surface of joint, kgf/mm 2

F = jt dl = joint surface area, mm 2

2.2. Strength of Press-Fitted Connections

39

/ = coefficient of friction between mating surfaces (for steel
and cast iron, on the average, / = 0.1-0.15)
d, l = diameter and length of the surface of joint, mm
The unit pressure on the surface of joint is found from Lame’s
equation

_1_

c l — Pi i C 2+P2

Ei ' El

kgf/mm 2

(2.4)

where 4- = relative diametral interference (4- = , 77 ,4 ^ ^

d \d 1000 d mm/

Ei,E 2 and p lf p 2 = elastic moduli (in kgf/mm 2 ) and Poisson’s ratios

of the materials of the male and female parts,

respectively

c x and c 2 = coefficients expressed as

Cl

(2.5)

Cl

■+(ir

'-nr

( 2 . 6 )

where d x and d 2 are the inner diameter of the male part and the
outer diameter of the female part, respec¬
tively.

Let us designate = a x and — a 2 . The quantities a x and a 2

may be called the relative wall thickness of the male and female
parts, respectively. When a x = a 2 = 1 the thicknesses of the male
and female parts equal zero. Values a x = a 2 = 0 correspond to the
case of massive male and female parts.

Coefficients c x and c 2 may be expressed in a general form as follows

(Fig. 17):

Equation (2.4) shows that unit pressure k and, consequently,
the joint strength are proportional to the relative diametral inter¬
ference A Id, increasing with the increase of Young’s moduli of the
materials and decreasing with the increase of c x and c 2 , i.e., with
the enhancement of the wall thickness factor a.

40

Chapter 2. Press-Fitted Connections

The compressive stress in the male part is maximum on its internal
surface and is expressed as

2k A 2 1

a i '

d i—a\ ci — pi c 2 + n 2
Ei "T E 2

(2.7)

The maximum permissible unit pressure determined by the bearing
strength of the material is

k = Ob e ar ( 2 - 8 )

where a bea r is the ultimate bearing strength of the material.

The reduction in the internal diameter of the male part

(2.9)

The tensile stress in the female
part is maximum on its inter¬
nal surface and is expressed as

2k A 2
a 2 — 1 _ a *~ — *1

X

d 1 — a\
1

m

Ei

( 2 . 10 )

c 2~fP2
Ez

The increase in the external
diameter of the female part

( 2 . 11 )

Fig. 17. Coefficient c as a function
of wall thickness factor a

Equations (2.4), (2.7) and
(2.10) enable some general con¬
clusions to be drawn. ,

To simplify the problem as¬
sume that both the [female and

male parts are made of the same material (E i — E 2 = E; =
= p 2 = p). Then Eqs. (2.4), (2.7) and (2.10) become

, A E
K ~ d ' Cl + c 2

kgf/mm 2

° l = ~d'
A

d ’

E

1 — a\

Oo = —

c l + c 2

E

C1+C2

kgf/mm 2

kgf/mm 2

( 2 . 12 )

(2.13)

(2.14)

Figure 18a shows as a function of a x and a 2 the relative pressure
k 0 = —-— which is the value of pressure k when AEld = 1.

c i + c 2

2.2. Strength of Press-Fitted Connections

41

The pressure (and consequently the strength of the joint) is maximum
when % = = 0 (both the male and female parts are massive)

and decreases with the increase of a x and a 2 (i.e., with the decrease
of the wall thickness of both the male and female parts), tending
to zero when fli = a 2 = 1.

The reduction of the pressure with the decrease of the wall thick¬
ness of the male and female parts can be compensated for by increas¬
ing the diameter and length of the surface of joint. If, as is usual.

Fig. 18. Effect of wall thickness factors a, and a 2 on values of k 0 , a 01 and a 02 .

the length of connection is proportional to its diameter, i.e., I =
= nd (n = coefficient of proportionality), then, according to

Eqs. (2.2) and (2.3), P ax = kf red 2 and Mtors = kfn

Hence, the resistance of a joint to an axial shift is proportional
to the square of its diameter, and its resistance to torsion, to the
cube of the diameter.

From Eqs. (2.13) and (2.14) the relative stresses o 01 and o 02 (the
stresses with A Eld = 1) are

<toi

1 —a\

and or „2

2fep
1 — a.\

These relationships are diagrammed in Fig. 186. From the graph
the following conclusions can be drawn:

stresses a 01 in the male part (heavy lines) are maximum (cr 01 = 1)
with a massive female part (a 2 — 0); they decrease as the wall thick¬
ness of the female part is reduced (a 2 -> 1) and increase as the wall
thickness of the male part is reduced (aj -> 1);

42

Chapter 2. Press-Fitted Connections

stresses cr 02 in the female part (light lines) are maximum (o 02 = 1)
with a massive male part (a x = 0); they decrease as the wall thick¬
ness of the male part is reduced (a x —1) and increase as the wall
thickness of the female part is reduced (a 2 -> 1).

Assuming that the male part is a shaft, and the female part,
a hub, the following rules can be formulated:

to increase the strength of the shaft it is necessary to increase
the thickness of its walls and decrease the wall thickness of the hub
(massive shaft and thin-walled hub). This rule is applied when the
hub strength is of no concern;

to increase the hub strength it is necessary to increase the thick¬
ness of its walls and to decrease the wall thickness of the shaft (mas¬
sive hub and thin-walled shaft). The rule is applied when the shaft
is of sufficient strength.

When the male and female components of a joint are made from
dissimilar materials, the above relationships are affected by the
rigidity of the materials. If one of the parts (female or male) is made
from a material having a smaller elastic modulus ( E ') than the other
■one ( E "), then the unit pressure and the stresses in the parts fall
approximately in proportion to the ratio E'/E” (to a greater degree
in the part made from the material of the greater elastic modulus).

The best relationship between the wall thicknesses of the hub
and shaft should be established in each particular case by individual
■calculations.

(a) Coefficient of Friction

The strength of press-fitted connections is directly proportional
to the coefficient of friction between the mating surfaces [see Eqs.
(2.2) and (2.3)].

The coefficient of friction depends on the pressure on the contacting
surfaces, the magnitude and profile of surface microirregularities,
the material and state of the mating surfaces (presence of oil) and
also on the assembly technique (connection with the aid of a press,
with heating or cooling the parts).

The coefficient of friction rises with an increase in the surface
roughness and lowers with a rise in the unit pressure. It is higher
when the assembly is done with heating or cooling as compared with
a press-fitting. The coefficient can substantially be enhanced by
-electroplating.

Depending on the above factors the coefficient of friction ranges
from 0.08 to 0.3, and sometimes may be even higher. Obviously no
accuracy in calculations can be expected with such an extensive /
value scatter. The primary value of the calculation is that it enables
one to determine the influence of the geometry and rigidity of the
•connection elements on the joint strength and outline rational ways
to improve it.

2.2. Strength of Press-Fitted Connections

43

In practice lower coefficients of friction (/ = 0.1-0.15) are gene¬
rally adopted, including possible increase in the coefficient above
these values- in the safety factor.

(i b ) Effect of Surface Finish

An adequate surface finish of the mating components is an essential
condition for the strength of pressed connections.

The measured hole and shaft diameters include the height of
microirregularities which are flattened in the course of press-fitting.

i_i_i_i_l_i_l, ...J_l_l_l

0 10 20 30 60 50 60 70 80 d,mw

Fig. 19. Mean interferences with Dh grade fit (heavy line) and roughness
heights for various classes of surface finish as a function of connection diameter d

If the height of surface microirregularities is commensurate with
the interference of mating parts, the actual interference in the pressed
connection will then be appreciably smaller.

Shown in Fig. 19 are the mean values of interferences in the case
of a heavy drive fit (Dh) for different shaft diameters, and the total
roughness height 2 Rz of the shaft and the hole when machined
to the 4th-8th class of surface finish. From the graph it is clear that
small-diameter connections (less than 30-40 mm) must not be machined
worse than to the 6th class because the total roughness height in
this case becomes close to the interference value. Hence, in such
connections the interference will either be substantially reduced or
disappear altogether after the microirregularities are flattened.

44

Chapter 2. Press-Fitted Connections

Connections with diameters of the order of 50 mm and more, as
well as connections with greater interferences, may be machined
somewhat rougher. In practice the surfaces of the mating parts in
medium-size press-fitted connections are machined to the 8th-10th
and 7th-9th classes of surface finish (for shafts and holes, respec¬
tively).

Microirregularities have a certain positive effect on the strength
of the connection as they act like spurs enhancing the bond between
the contacting surfaces. Experiments show that machining
better than to the 11th class of surface finish impairs the joint
strength because of the reduced coefficient of friction between the
contacting surfaces.

Design Eqs. (2.4), (2.7) and (2.10) include the actual interference
value. Therefore, when designing pressed connections the assigned
nominal interference A nom should be reduced by the amount of the
expected flattening of microirregularities

A* = 2<p (R z i + R z2 ) (2.15)

where R zl and R zi = roughness height of the shaft and hole,
respectively, jim
<p = flattening coefficient

The amount of the flattening of the microirregularities depends
on the interference in the joint, the height of the irregularities, their
shape and profile, the density of distribution of the irregularities,
the hardness and strength of the material of the mating parts, and
the hardness relationship between the surfaces of the female and
male parts.

The amount of flattening is largely dependent on assembly con¬
ditions. When assembling in a press, the surface irregularities are
successively sheared off in the course of longitudinal movement
and are flattened much more than in the case of assembly procedures
whereby the parts are heated or cooled (i.e., when the surface irre¬
gularities are reduced radially).

The actual amount of flattening, determining the operational
strength of the connection, depends on the magnitude and type of
the load acting upon the connection and also on the number of
assembly-dismantling procedures. It is established only after a certain
period of operation.

The height of the surface microirregularities decreases with each
assembly-dismantling procedure and settles at a definite level
after three or four such procedures.

Naturally, it is impossible to duly account for all these factors.
Therefore, as a first approximation in calculations it is assumed
that the flattening of surface microirregularities averages 0.5-0.6
of the original mean roughness height. The effect of the subsequent

2.2. Strength of Press-Fitted Connections

45

operation of the joint is accounted for by a safety factor which
for pressed connections is taken at 2-4.

Assuming <p = 0.5, we find A' = R zl + R z2 and introducing
the value A nom — A' into Eq. (2.3), obtain

7. A nom (#zl“f-#z2)

lOOOd

1

c i —Pi i C 2 + P2
£1 ^ E 2

kgf/mm 2

(2.16)

When the required nominal interference is to be found, the amount
of the microirregularity flattening should be added to the calculated
interference magnitude

A n o;n = A ca ; c -4* Rzl —RzZ (2*17)

Then this nominal interference value is used as a reference for
selecting the corresponding fit from the standard.

For the 6th-llth classes of surface finish R z has the following
values:

Class of surface finish ... 6 7 8 9 10 11

R z , pm.10 6.3 3.2 1.6 0.8 0.4

Corrections for the flattening of microirregularities are substantial
when dealing with small-diameter connections. For instance, when
d = 30 mm the average interference for a heavy drive fit (Dh) is
33 pm. If the mating surfaces are machined to the 8th class of surface
finish the correction R zl + R z 2 — 6.4 pm amounts to approximate¬
ly 20% of the nominal interference. For a joint with d = 150 mm
(interference 110 pm) the correction will amount to approximately
6 %.

(c) Effect of Thermal Deformations

For connections undergoing heating during operation, the influence
of temperature upon the fit must be considered. If the female part
is made from a material having a higher coefficient of linear expan¬
sion, or is heated in operation more intensively than the male part,
the original (cold) interference in the joint will then be reduced during
heating.

Conversely, if the male part is made from a material having
a higher coefficient of linear expansion, or is heated in operation
more intensively than the female part, the original interference
will be enhanced during heating.

Whenever heating is involved, Eqs. (2.4), (2.7) and (2.10) must
include the temperature interference (with its sign)

A t = 1000d(a 1 Afi —a 2 Af 2 )pm (2.18)

where a x and a 2 = linear expansion coefficients of the male and
female parts, respectively

46

Chapter 2. Press-Fitted Connections

and A t 2 = temperature increments in the heating
male and female parts, respectively
Equation (2.15) will then take the following form

k =

1

lOOOd

&nom ~ Rz?) ~h (ct(Ati — 0.2^2) lttXW kgf/ mm 2
Cl — Pi , c 2 + p 2 ®

Ei E 2

of the

(2.19)

According to Eq. (2.19), the original relative interference necessary
to keep the assigned pressure k during heating is

&nom

lOOOd

*(

Cl —Pi
Ei

c 2 ~r P 2 \ | Rz l4~Hz2 I
E z } ' 1 “ lOOOd '

a 2 t 2 — a^i

( 2 . 20 )

When fitting components onto high-speed rotors the hub cen¬
trifugal expansion should be taken account of and compensated
for by increasing the original interference.

2.3. Selection of Fits

As a general rule, large-interference fits should be avoided, par¬
ticularly with low-grade accuracies ( Dh3 3 , Dh2 3 , Dh2 2a ). Because
of the wide tolerance range in these classes of fits, there may occur,
in the case of unfavourable tolerance combinations, interferences
detrimental to the strength of the joint.

When large interferences are necessary it is advisable to use the
shrink fit ( Sh) to the second grade of accuracy.

Large-interference fits are used for thin-walled housings, housings
made of light metals, housings expanding during heating and hubs
of high-speed rotors.

Particular care must be exercised when selecting fits for thin-
walled bushes (e.g., plain bearing bushes). In the press-fitting
process the bush bore decreases, thus necessitating additional hole
reaming operations after pressing-in (or the hole is made initially
oversize by the contraction value).

With large interferences plastic deformation may occur, the bush
contracts and the joint strength sharply falls. In practice it is often
observed that the fit becomes loose due to the bush expansion during
heating, particularly if it is made from a material with a high linear
expansion coefficient (e.g., bronze).

Thin-walled bushes are generally assembled by a fit not heavier
than Dh. In most cases it is necessary to prevent bushes from rotation
and axial shift.

In each particular case a connection should be designed with due
regard for all the factors which influence the, operation of the con¬
nection.

The work capacity (holding power) of press-fitted connections
is calculated proceeding front the minimum interference which may

2.4. Calculation Diagrams

47

arise with an unfavourable combination of the hole and shaft sizes
(hole made to the upper tolerance limit and shaft, to the. lower tole¬
rance limit).

Stresses arising in female and male parts, as well as the force
needed to assemble and dismantle press-fitted connections, are
calculated on the basis of the maximum interference (hole made to-
the nominal size, and shaft, to the upper tolerance limit).

2.4. Calculation Diagrams

On the basis of Eqs. (2.4), (2.7) and (2.10) diagrams are drawn
which ease time consuming pressed-connection calculations, allow
the influence of pressed-connection parameters on the joint strength
to be traced and indicate rational ways of strengthening.

Shown in Fig. 20 is a calculation diagram for male and female
parts made from the same material [Eqs. (2.12)-(2.14)].

The lower portion of the diagram presents the values of relative

pressure k 0 = —:— as a function of a 1 for different magnitudes

C1+C2

of a 2 . The top part of the diagram gives relative stresses o 0 , equal
for the male and female parts

G oi

k 0 _
1—of ’

°02

kp

1 —of

Absolute values of k, a 1 and <r 2 are obtained by multiplying the
values of k 0 , a 01 and a Q2 by the factor A Eld.

Let us now consider the use of the diagram by an example.

Given: a hollow steel shaft with external diameter d = 100 mm and internal
diameter d 1 = 70 mm (a 1 = 0.7) is pressed into a steel hub with external dia¬
meter d 2 = 125 mm (a 2 = 0.8) by a shrink fit (Sh). The connection length is
100 mm. Both the shaft and hole are machined to the 8 th class of surface finish
(R z i + R z 2 = 6.4 pm). The coefficient of friction f = 0.1.

Find: pressure k on the surface of joint, torsional holding power Mt 0T & and
maximum stresses cr, in the shaft and o 2 in the hub.

First determine the A Eld value. For the Sh fit the mean diametral interfe¬
rence when d — 100 mm is 120 pm. Hence, the actual interference is 120—6.4 =
= 113.6 pm. Young’s Modulus E = 21 000 kgf/mm 2 . Then

A E
d

113.6-21000

1000-100

24 kgf/mm 2

From the point aq = 0.7 (lower portion of the diagram) draw a horisontal
line to intersect curve a 2 = 0.8 (point a) and read on the abscissa the value
k 0 = 0.135. The pressure is

AF

k = k 0 --AL =0.135-24 = 3.24 kgf/mm 2
d

The torsional holding power [see Eq. (2.3)] is

Af <ors = 0.001 • 3.24 • 0.1 ,n • 100-.100 • 50 = 500 kgf • m

Fig. 20. Diagram for computing press-fitted joints (parts made of the

material)

2.4. Calculation Diagrams

49

To determine the stresses draw from point a a vertical line until it intersects
the straight lines a = 0.7 (shaft) and a = 0.8 (hub). Read on the ordinate
<j 01 = 0.52 and ct 0 i = 0.75.

Consequently, the stresses in the shaft and hub

Oi = 0.52-24 = 12.5 kgf/mm 2
a 2 = 0.75»24 = 18 kgf/mm 2

Let the shaft be solid (a x = 0) and the external diameter of the hub increased
to 165 mm (a 2 = 0.6).

Point b in which the abscissa a x = 0 intersects curve a 2 = 0.6 will give
k 0 — 0.32. Hence

k = 0.32 • 24 = 7 • 7 kgf/mm 2

7 7

The torsional holding power will increase by 3 -^ ~ 2.5 times.

Drawing through point b a vertical line until it intersects the straight lines
a = 0 (shaft) and a = 0.6 (hub), read on the ordinate cr 01 = 0.64 and ct 02 = 1.

Consequently, the stress in the shaft rises by — 1 (approximately

U.Oa

1

by 20%), and the stress in the hub, by 77 ^— 1 (approximately by 30%) as

u. /d

compared with the previous case. The rise is rather small, if one takes into
account that the strength of the joint increases by 2.5 times.

For a press-fitted connection made up of parts of dissimilar mate¬
rials, the quantity k, in accord with Eq. (2.4), is

k

A__ 1

d, ci —m . c 2 -j-p2
Ei ^ E 2

Transform this equation in the following way

k = _ 1

(«i — Pl)-^- + < ; 2 + P2

( 2 . 21 )

Substituting this value of k into Eqs. (2.7) and (2.10) gives the
following expression:

A E 2 2

d 1 — a\

1

E

( c l~Ri) -£^ + c 2 +P 2

A £2
d 1 —a\

1

F

(Cl— Pi) -g^- + < : 2 + P2

( 2 . 22 )

(2.23)

Table 4 gives the ratio E 2 !E X for various material combinations
(assumed: for steel E = 21 000 kgf/mm 2 ; for cast iron E —

— 8000 kgf/mm 2 ; for aluminium alloys E = 7200 kgf/mm 2 and
for bronzes E = 11 000 kgf/mm 2 ).

4—01418

50

Chapter 2. Press-Fitted Connections

Table 4

E 2 /E t Ratios for Various Material Combinations in Press-Fitted

Connections

Material of male part

Material of female part

steels

cast irons

aluminium

alloys

bronzes

Steels

1

0.38

0.33

0.53

Cast irons

2.6

1

0.9

1.38

Aluminium alloys

2.9

1.1

1

1.52

Bronzes

1.9

0.73

0.65

1

The figures in bold type indicate the EJE 1 ratios having practical
application. These ratios have been employed to plot calculation
diagrams (Figs. 21 to 25) on the basis of Eqs. (2.21), (2.22) and (2.23).
For plotting these diagrams it has been assumed that p = 0.15
for cast iron and p = 0.3 for all other materials. In conjunction
with the diagram presented in Fig. 20 {EJE 1 = 1), these diagrams
cover all practical cases of pressed connections.

Here are some calculation examples. For greater simplicity the
calculations are based on the mean interferences for the given grade
of fit. In actual design procedure the extreme interference limits
should be used (see pp. 46-47).

Pressing steel components into cast iron ones (Fig. 21). A hollow steel column
with external diameter d = 100 mm and internal diameter = 70 mm
(a 1 = 0.7) is pressed into the hub of a cast iron bed. The external diameter of the
hub is d 2 = 125 mm (a 2 = 0,8). The fit is Dh2 2a (mean interference A =170 pm).
The joint surface on the column is machined to the 8th class of surface finish
(R zl —3.2 pm), and the bore, to the 7th class (i? z2 =6.3 pm); R zl + R z2 =9.5 pm

A E 2
d

(170—9.5) 8000
1000-100

= 12.7 kgf/mm 2

Using as reference values a x = 0.7 and a 2 = 0.8 (point a), we find on the
diagram that k 0 = 0.175. Pressure

k = 0.75-12.07 = 2.2 kgf/mm 2

Using as reference values k„ = 0.175; a x = 0.7 and a 2 = 0.8, we find that
a 01 = 0.69 and a 02 = 0.98.

Stresses

a 1 = 0.69-12.7 =8.8 kgf/mm 2
a 2 = 0.98-12.7 = 12.4 kgf/mm 2

5 ?

Chapter 2. Press-Fitted Connections

Reduce the internal diameter of the column to 60 mm (a x = 0.6) and
increase the external diameter of the hub to 165 mm (a 2 = 0.6). In this case
<point b) fc 0 = 0.34 and o 01 = o 02 = 1.06. Hence,

* = 0.34.12.7 = 4.3 kgf/mm 2
01 = 02 = 1.06.12.7 = 13.5 kgf/mm 2

The strength of connection is increased by —■:

2 times, the stresses in the

column and hub rise approximately by

13.5

8.8

1 (approximately by 50%) and

13 5

— 1 (approximately by

10%), respectively.

Pressing steel components into those of aluminium alloys (Fig. 22). A hollow
steel shaft with d = 100 mm and d x = 70 mm (a x = 0.7) is press-fitted into the
hub of a housing cast in an aluminium alloy. The external hub diameter
d 2 = 150 mm (a 2 = 0.65). The grade of fit is Dh2 2a (A = 170 pm). The class of
surface finish of the shaft is the 8th ( R zl = 3.2 pm) and that of the hole, 7th

(R z2 = 6.3 pm); R zl -f- R z2 = 9.5 pm.

A E z (170-9-5) 7200

d

1000-100

= 11.5 kgf/mm 2

On the diagram, for a x = 0.7 and a 2 = 0.65 (point a) we find that k = 0.275;
c 0 i = 1.09; a 02 = 0.92.

Hence

* = 0.275-11.5 = 3.15 kgf/mm 2
01 = 1.09-11.5 = 12.5 kgf/mm 2
02 = 0.92-11.5 = 10.6 kgf/mm 2

Stresses 0 2 in the hub exceed the yield limit for cast aluminium alloys in
tension (0 O2 = 10 kgf/mm 2 ). To reduce the stresses we use the fit grade Dhl 2a
obtained with a smaller interference (A = 90 pm).

Then

A Ez
d

(90-9.5) 7200
1000-100

= 5.8 kgf/mm 2

5 8

The values of *, a x and 0 2 decrease by a factor of ytt ~ 0-5. The stress o 2

11<D

5>ecomes acceptable: o 2 = 0.5-10.6 = 5.3 kgf/mm 2 .

Let us now consider the case of pressing a disk of a wrought aluminium alloy
onto a hollow steel shaft with external diameter d = 100 mm and internal
-diameter d x = 70 mm (% = 0.7). The disk can be regarded as a solid part
ia 2 = 0). The grade of fit is Dh (A = 65 pm). The class of surface finish of the
-shaft is the 9th ( R zl = 1.6 pm), and that of the disk bore, 8th ( R z2 —
3.2 pm); R n + R z2 = 4.8 pm

Af?2

d

(65—4.8)7200'

1000-100

■ 4.3 kgf/mm 2

From the diagram we find (point b) that*o = 0.465; o 01 = 1.82; 0 O2 = 0.92.
Hence

* = 0.465-4.3 = 2 kgf/mm 2
01 = 1.82:4.3=8 kgf/mm 2
02 = 0.92-4.3 = 4 kgf/mm 2

Fig. 22. Diagram for computing press-fitted joints (steel parts pressed in
parts of aluminium alloys)

54

Chapter 2. Press-Fitted Connections

Assume that during operation the disk temperature rises by 100°C (in compari¬
son to the assembly temperature) and the shaft remains cold. As the linear
expansion coefficient of an aluminium alloy a 2 = 22-10~ 6 , the disk bore dia¬
meter will increase in heating by the value

A< = a2td‘10 3

r22-100-100-10 3
106

= 220 pm

Thus, the original press-fit interference vanishes and instead a clearance
amounting to 220—(65—4.8) = 160 pm develops in the joint.

To keep the alignment of the parts a fit of greater interference should be
employed, for example, that of the grade Dh2 2a (A = 180 pm). In this case during
heating of the joint there develops a clearance amounting to 220—(180—4.8) =
= 45 pm which does not disturb the alignment.

With an interference of 180 pm

d

(180 - 4.8) 7200
1000-100

= 12.6 kgf/mm 2

The values of k, o 1 and o 2 increase by a factor of ^ /y ss 3. Stress o 2 in the

■disk hub (in cold state) becomes o 2 = 3-4 = 12 kgf/mm 2 , which is acceptable
for wrought aluminium alloys.

Pressing bronze components into steel ones (Fig. 23). A tin-bronze bush
with external diameter d = 40 mm and internal diameter d t = 35 mm
(a t = 0.87), is press-fitted into a steel hub with external diameter d 2 = 53 mm
(a 2 — 0.75). The grade of fit is Dh2 2a (A = 80 pm). The joint surface of the bush
is machined to the 9th class of surface finish (i? zl = 1.6 pm) and that of the hub,
to the 8th class (H z2 = 3.2 pm); fi zt + R z2 = 4.8 pm

A E z (80 - 4.8)21 000
d ~ 1000-40

40 kgf/mm 2

From the diagram we find that when a r = 0.87 and a 2 = 0.75 (point a),
* 0 — 0.06; o 01 = 0.5; o 02 = 0.27.

Hence

* = 0.06-40 = 2.4 kgf/mm 2
= 0.5-40 = 20 kgf/mm 2
a 2 = 0.27-40 = 10.8 kgf/mm 2

Stress a, in the bush exceeds the yield limit of tin bronze in compression
(o 02 = 15 kgf/mm 2 ).

We reduce the internal diameter of the bush to d x = 30 mm (a 2 = 0.75).
On the diagram (point b) we find that *„ = 0.1; o 01 = o 02 = 0.46.

Hence

* = 0.1-40 = 4 kgf/mm 2
Oj = o 2 = 0.46-40= 18.4 kgf/mm 2

It is clear that the increase of the bush wall thickness is of little effect; the
stresses decrease only by 8% and still exceed the yield limit of the material.

The reduction of the hub wall thickness does not help either. Let a 2 = 0.85
(d 2 = 47 mm). From the diagram o 01 = 0.35, whence o x = 0.35-40 =

- 14 kgf/mm 2 .

Let us now try to reduce the interference. We take the Dhl 2a grade fit

50_^ g

(A=50 pm). Then, the actual interference decreases by a factor of =— -2— = 0.6;

oU-—4.0

the stress in the bush (with the original a t = 0.87) then becomes acceptable:

1

56

Chapter 2. Press-Fitted Connections

Ox = 0.6-20 — 12 kgf/mm 2 , and when a 2 = 0.85 it becomes a, = 0.6-14 =
= 8.4 kgf/'mm 2 .

Now assume that during operation the joint temperature rises by 100°C.
The coefficient of linear expansion of bronze is = 18-10“ 8 ; that of steel,
a 2 = 11-10 -6 . The temperature interference will then be

A t = 1000-100-40(18 —ll)-10- 6 =28 pm

The interference in the joint A = 50—4.8 -f 28 = 73 pm.

73

The unit pressure and stresses are increased by ^—— = 1.6 times. For

ou—4.o

the bush with a 1 = 0.87 the stress becomes aj = 1.6-12 = 19 kgf/'mm 2 and still
exeeds the yield limit of the material.

Now we take the Dh grade fit (A = 40 um). In contrast to the previous case,

40 — 4 8

the actual interference decreases by a factor of —~ 0.49 and the stresses

in the bush reach an acceptable level: a 2 = 0.49-19 = 9.3 kgf/mm 2 .

Pressing bronze components into those of cast iron (Fig. 24). A bronze bush
having the same parameters as in the previous example (d = 40 mm, a x = 0.87)
is pressed into a cast-iron hub (a 2 = 0.75) by the Z)h2 2a grade fit (A = 80 pm).
The surface finish is the same as previously (R tl + R z n = 4.8 pm)

__ ---= 15 kgf/mm 2

On the diagram we find that for a 2 = 0.87 and a 2 = 0.75 (point a), k 0 =
= 0.108; Ooi — 0-91 and 002 = 0.5.

Hence

k — 0.11 -15 = 1.65 kgf/mm 2
a t = 0.91-15 = 13.6 kgf/mm 2
a 2 = 0.5-15 = 7.5 kgf/mm 2

Owing to the lower modulus of elasticity of cast iron the stresses in this case
are fairly lower than those encountered when pressing the bush into a steel com¬
ponent (the previous example). Nevertheless the stress in the bush is close to the
yield limit of bronze. Let the grade of fit be Dhl 2a (A= 50’pm). The actual inter¬
ference then decreases by a factor of •=:—= 0.6 and the stress in the bush

becomes 04 = 0.6-13.6 - 8.2 kgf/mm 2 .

Let the joint temperature rise in operation by 100°C. In the joint there
develops a temperature interference equal as previously to 28 pm (the coefficient
of linear expansion of cast iron is approximately the same as that of steel).
According to the previous example, the stress in the bush will increase by
1.6 times and become Ox = 1.6 -8.2 = 13 kgf/mm 2 (in comparison with ct x =
= 19 kgf/mm 2 when heating a steel hub). However, in this case too it is recom¬
mended that the interference be reduced further. Now we use the Dh grade of fit.
Then, according to the previous conditions, the stress will drop by a factor of
0.49 and become = 0.49-13 = 6.4 kgf/'mm 2 (in comparison with Ox =
= 9.3 kgf/mm 2 for a steel hub).

Now consider the case of a solid cast iron part (a 2 = 0). Assume that the
bush parameters remain as before (a 2 = 0.87). The grade of fit is Dh (A = 40 pm).
On the diagram (point b) we find that k 0 = 0.15; cr 01 = 1.25 and 0 O2 = 0.3.

A E 2 (40 — 4.8) 8000

d 1000-40

7 kgf/mm 2

0.05 OJ 0.15 01 015 0.3 0.35 01 015 0.5 0.55 0.6 0.65

Fig. 24. Diagram for computing press-fitted joints (bronze parts pressed in

cast-iron parts)

2.5. Probability Calculations for Press-Fitted Connections

59

Hence

k = 0.15-7= 1.05 kgf/mm 2
<Ti = l.25-7 = 8.7 kgf/mm 2
02 = 0-3-7 = 2.1 kgf/mm 2

Assume that the bush temperature rises by 60°C during the start-up and
that the housing temperature remains constant. In the joint there develops
a temperature interference

A t =1000-18-10-«-60-40 = 43 pm

The actual interference thus becomes 40—4.8 4- 43 « 78 pm, and the stresses
78

increase by a factor of ^ = 2.2. Consequently, the stresses in the bush

Oj = 2.2-8.7 « 19 kgf/mm 2 , i.e., they exceed the yield limit of the material.
Obviously, the original interference corresponding to the Dh grade fit in this
case is excessive.

Let us use the Dl grade fit (A = 25 pm). The actual interference in heating
will then become 25—4.8 4- 43 = 63 um and the stresses will decrease by a factor
63 ' ^

of -yg= 0.8. The stresses in the hot bush will be 0.8-19 * 15 kgf/mm 2 , which

is more or less acceptable. Under such circumstances the bush must be prevented
from rotation.

Pressing bronze components into those of aluminium alloys (Fig. 25). A bron¬
ze bush is press-fitted into a solid housing made from an aluminium alloy
(a 2 = 0). The bush has the same parameters as above ( d = 40 mm, a y = 0.87).
The grade of fit is Dh (A = 40 pm). The surface finish is also the same as above
<i? zl -f R z2 = 4.8 pm). From the diagram we find that when a 2 = 0.87 and
a 2 — 0 (point a), k 0 = 0.175; o 01 = 1.45 and o 02 = 0.35

Hence

AE 2

d

'(40-4.8) 7200
1000-40

kgf/mm 2

k = 0.175-6.4= 1.1 kgf/mm 2

01 =1.45-6.4 = 9.3 kgf/mm 2
02 = 0.35-6.4 = 2.2 kgf/mm 2

Assume that the joint temperature rises in operation by 100°C. The diameter
of the bush will increase by 1000-18 -10- 8 -100-40 = 72 pm. The diameter of the
Lore (with the linear expansion coefficient of the aluminium alloy being a 2 =
= 22-10~ 6 ) will expand by 1000-22-10 -8 -100-40 = 88 pm. Accordingly, the
original interference will decrease by 88—72 = 16 pm and become 40—4.8—16 »
» 19 pm. The bush must be prevented from rotation.

2.5. Probability Calculations for Press-Fitted Connections

The methods for calculating press-fitted connections by the extreme
tolerance limits of shafts and holes do not take account of the regu¬
larities of the size dispersion and frequency of size distribution. The
probability for the manufacture of shafts and holes of extreme sizes
is generally very small. And the probability of such extremes being
assembled is even less.

60 ■ Chapter 2. Press-Fitted Connections _

In many cases size distribution can be represented by a Gaussian
(normal distribution) curve plotted in size vs. size occurrence fre¬
quency system of coordinates (Fig. 26). The equation of the curve
(with the origin of coordinates on the axis of symmetry) is

y = —e 202

where cr = standard deviation of size
e — 2.718 = Napierian base

Ordinates y indicate the probability of occurrence of each given
size. The area under the curve is numerically equal to unity (100%
y of the parts). The maxi-

A mum ordinate is

1 0.4

i/max — , r — ^

a]/ 2n a

The branches of the curve
asymptotically approach
the abscissa axis. The curve
has two inflexion points
at distances + cr and — cr
from the symmetry axis.

The ordinates of these
points are
+x

-30 —26 ft ~L x 9/r ^ ?/r _ . a r*.. _,0.24

0 +6
-\r-+Xi(+Z{)-

With such a distribution
Fag. 26. Gaussian ^ (normal distribution) kw 68 o /o of all parts fall

in the interval +cr; 87%,
in the interval ±1.5cr; 95%, in the interval ±2o; and
99.73%, in the interval ±3cr. For practical purposes the curve is
confined to the ±3o limits. Then the ±3a interval can be considered
to be equal to the margin tolerance 8, i.e., 8 = 6a, and the curve
used for computing the probability of occurrence of sizes within the
limits of this margin tolerance.

The percentage of parts falling upon the extreme points of the
curve at distances from the origin is expressed by the ratio v
of the shaded areas to the total area under the curve, taken at 100%.
According to the equation of the Gaussian curve

— sc

>=1 - 7 = (

a^f 2 ji J

e 2<j2 dx

2.5. Probability Calculations for Press-Fitted Connections

61

Introducing z = , we have

v = l

_1 _

~\f 2 ji

+Z1

ii
2 dz

The integral function of this expression is given in tabular form
in manuals on the theory of chances.

The Table below gives the values of v (second column) as a func¬
tion of Z which presents the ratio of the sum of line-segments 2z x

to the basis 6a of the

curve

II

SS3

Z '

V

V 2

Risk, %

0.9

0.0069

0.000048

0.0048

0.8

0.0164

0.00027

0.027

0.7

0.0357

0.00127

0.127

0.6

0.0719

0.00517

0.517

0.5

0.1336

0.01783

1.78

The probability of occurrence of combinations of parts with
dimensions, corresponding to the given Z values is equal, in accord
with the theory of chances, to v squared. The value of v 2 expressed
in per cent shows the risk percentage, i.e., the probability of occur¬
rence of combinations within limits exceeding Z (Fig. 27).

With Z >• 0.5 the risk percentage is negligible. Thus, when
Z = 0.7, for every 1000 connections approximately one is a risk,
and when Z = 0.6, approximately five connections have parameters
exceeding the given limits.

Consequently, there is very little risk in narrowing the margin
tolerance to Z8 and introducing in place of the extreme nominal
deviations A mln and A max probable deviations

Amin — A

min"

6(1—Z)

and

Amai —— An

6(1 -Z)

where Z is a value ranging from 0.9 to 0.5, depending on the adopted
risk percentage.

Let us carry out a comparative numerical computation of a press-fitted
connection by both conventional and probability methods. Assume a connec¬
tion comprising a solid steel shaft, 80 mm in diameter, and a steel sleeve with
an external diameter of 120 mm. The connection length Z = 80 mm. The grade
of fit A/Dh. The shaft and hole are machined to the 8th class of surface finish
(R z = 3.2 pm). Hole tolerance:+30 urn; shaft tolerance: lower limit +45 pm,
upper limit +65 pm. Coefficient of friction / =0 .1.

Nominal interferences: maximum 65—0 - 65 pm, minimum 45—30 = 15 pm.
With a microirregularity flattening coefficient of 0.5, the flattening amounts

62

Chapter 2. Press-Fitted Connections

to 6.4 (xm. The actual maximum interference is, thus, 65—6.4 = 58.6 pm, and the
minimum, 15—6.4 = 8.6 pm.

For the probability computation assume that Z = 0.6 (risk percentage
\—Z

is 0.517). The value -equals 0.2. Probable deviations of dimensions: hole —

minimum 0 + 30-0.2 = 6 pm, maximum 30 — 30-0.2 = 24 pm, shaft—maxi¬
mum 65 — 20-0.2 = 61 pm, minimum 45+ 20-0.2 = 49 pm. Probable inter-
erences: maximum 61 — 6 =55 pm, minimum 49 — 24 = 25 pm. Accounting for
fthe microirregularity flattening the maximum interference is 55 — 6.4 =
= 48.6 pm, and the minimum interference, 25 — 6.4 = 18.6 pm (Table 5).

As seen from Table 5 the probability computation gives more
favourable characteristics which at the same time are closer to the
actual parameters of completed connections.

The normal distribution law is valid wherever a large number
of occurrences is involved, and, hence, for mass production and
with controlled operations at that. Substantial deviations from this
law occur in individual and small-lot production, attributable
in the first instance to the small number of occurrences and, secondly,
to certain specific features of the machining process. With individual
production the operator involuntarily keeps to the lower hole toler-

2.5. Probability Calculations for Press-Fitted Connections

63

Table 5

Calculation Results for a Pressed Connection

Characteristics

Computation by

nominal values

probable values

Maximum interference, pm

58.6

48.6

Minimum interference, pm

8.6

18.6

Press-fitting force (with maximum
interference), kgf

Holding power (with minimum
interference):

8500

7200

axial, P ax , kgf

1250

2750

Aftorsi kgf-m

Stresses (with maximum interfe¬
rence):

50

110

in shaft, a u kgf/mm 2

8.6

7.1

in hub, a 2 , kgf/mm 2

15.3

12.7

ance limit and the upper limit for the shaft, orienting to the NO-GO
side of gauges. Because of this the holes are made, on the average,
closer to the minimum (nominal) size, while the shaft dimensions
show a tendency toward the ma¬
ximum (upper) limit of toleran¬
ce. The grouping centres on the
distribution curves are thereby
shifted (Fig. 28) and the proba¬
bility of obtaining maximum
interferences is thus increased.

An asymmetry of size distri¬
bution also periodically occurs
in mass production with cont¬
rolled operations. As the cutting
tools wear down, hole dimen¬
sions approach the minimum, and
shaft dimensions, the maximum. Fig 2 8. Distribution curves with
The periodicity of this pheno- displaced, grouping centre

menon depends on operation ad¬
justment frequency and is absent only with automatic readjust¬
ment. To establish any regularities of the dispersion changes in
a general form is difficult.

However it is still necessary to note that any probability com¬
putation gives average figures of the expected size distribution over
a long time interval and for large batches of products. The possibility

64

Chapter 2. Press-Fitted Connections

of a temporary crowding of low-probability size combinations is
not excluded, as a result of which comparatively large batches of
defective connections will appear, although the mean level of risk
related to very large batches will still remain within the computed
limits.

All this restricts the practical value of the probability computa¬
tion technique.

When designing press-fitted connections it is better to keep within
narrow boundaries such an interference as will ensure good working
capacity of the connection without inducing high stresses in the
mating components. With the existing systems of press fits, which
are noted for their large deviation scatter, it is difficult to achieve.
The selective assembly method is undesirable as it complicates
production. The general rule is to employ fits to the highest grades
of accuracy, in particular, those to the first grade of accuracy.
However, this class of fits only covers a limited range of interferen¬
ces. Therefore it seems advisable to develop a single class of press
fits with lower margin tolerances that would encompass the entire
range of interferences necessary for the industry.

2.6. Press-Fitting with Heating or Cooling of Parts

The press-fitting force may reach substantial values, especially
with great interferences and large surfaces of joint. This force pro¬
gressively increases as the part being press-fitted is forced deeper
into the hole, reaching its maximum at the end of the procedure.
The maximum press-fitting force may be determined from Eq. (2.2).

Let us find the force necessary to press a solid steel shaft (a x = 0) with
diameter d = 100 mm into a cast-iron hub 150 mm long and having external
diameter d 2 = 165 mm (a 2 = 0.6) by the Dhl ia grade fit (A = 90 pm).

On the diagram shown in Fig. 21 we find that for a x = 0 and a 2 = 0.6,
k 0 = 0.39. The unit pressure

k = k 0

A E 2
d

= 0.39

90-8000

1000-100

= 2.8 kgf/mm 2

The maximum press-fitting force is

P = kfF = 2.8 • O.Iji • 100 • 150 = 13000 kgf

The press-fitting procedure can be facilitated by heating the
female part or cooling the male part, or else by practising both of
these at once. When press-fitting parts into large housing structures
only the male component is cooled.

The heating temperature for the female part to have zero inter¬
ference

A

1

2.7. Pressed Connections with Electrodeposited Coatings

65

where t and t 0 = heating temperature and workshop temperature,
respectively

^= relative interference in the joint

cc 2 = linear expansion coefficient of the female part
material

d = diameter of the joint, mm

Similarly, when cooling the male component

i '+^ = 4"a 1 1 1000 (2,25)

where t' is the cooling temperature.

It should be remembered that at subzero temperatures the linear
expansion coefficient a decreases (see Fundamentals of Machine
Design, Vol. 1, Fig. 257).

Assume that in Eq. (2.1) A mean «ij)D and that f 0 = 20°C. Then,
from Eqs. (2.24) and (2.25) one can find temperatures t (heating)
and t' (cooling) required to obtain zero interference when assembling
joints by different fits. For cast-iron and steel components (a «
« 10 -6 ) these temperatures are as follows:

Fits .... Dhli Dh and Dh2^ Dhl 3 Dhl Za Sh Dh2 3 Dh2 2a Dh3 s
t,° C . . . 43 60 65 68 105 108 122 124

t', °C . . . —8 —30 —36 —40 —86 —90 —105 —110

Generally the cooling procedure is effected by using dry ice (evap¬
orates at — 80°C); or, for more intensive cooling, liquid nitrogen
(—196 °Ci and oxygen (—183°C).

It must be borne in mind that heated components rapidly cool as they are
being delivered from the furnace to the press. Furthermore, during the press¬
fitting operation the temperature of the heated hub quickly lowers due to its
contact with the cold shaft. Therefore, due corrections must be introduced
to compensate for the decrease of temperature during handling and press-fitting.
In practice this implies that the components to be press-fitted must be overheat¬
ed by 30-50°C as compared with the calculated temperature.

The cooling temperature should also allow for the heating effect.

2.7. Pressed Connections with Electrodeposited Coatings

The strength of press-fitted connections (axial and torsional
holding power) can significantly be increased by electroplating
the contacting surfaces. Figure 29 shows the results of comparative
tests of pressed connections assembled by the A/Dh grade fit. The
electrodeposited layer on the contacting surfaces was 0.01-0.02 mm
thick. Two assembly procedures were used: one by means of a hyd¬
raulic press and the other with cooling the shaft in liquid nitrogen.
In the latter case the radial clearance between the mating surfaces
was 0.05 mm.

5—01418

66

Chapter 2. Press-Fitted Connections

A connection without coating assembled in a hydraulic press
(without shaft cooling) was used as a reference specimen, its axial
holding power P 0 being taken as the unit of comparison.

From the diagram it is seen that electroplating sharply (by 2-4.5
times) enhances the connection strength, and that the assembly with
cooling the shaft assures a much higher connection strength than

f

0.8

0.7

0.6

0.5

OA

0.3

0.2

0.1

Without Cd Cu In NL Cr
coating

Fig. 29. Relative strength of pressed connections with deposited coatings.
Blackened columns indicate assembly with shaft cooling, while nachured columns
are for assembly in a press. The strength of a pressed connection without coating
and assembled in a press is taken as unity (after G. A. Bobrovnikov)

the press-fitting assembly. For uncoated connections assembled
with cooling the joint strength is doubled, and for connections with
soft coatings (Cd, Cu, Zn), improved by 20-30% in comparison with
the assembly in a press.

The strength of connections with hard coatings (Ni, Cr) and
assembled with cooling is worse than that of the same connections
assembled in a press.

The greater strength of connections with electroplated coatings is probably
due to the occurrence of strong molecular bonds at the joint surfaces. The pro¬
longed dwell under the high pressure existing on the contacting surfaces causes
a diffusion process whereby the atoms of the base and coating metals mutually
penetrate each other, thus forming some intermediate structure. In other words,
a “cold welding” process seems to occur. As for the rather high (approaching
unity) values of the coefficient of friction (see the right-hand ordinate of the
diagram in Fig. 29), this can readily be explained. The friction coefficient
concept in its classical mechanical interpretation in these conditions is no
longer valid: the value of the friction coefficient here shows the shear resistance

2.8. Design of Pressed Connections

67

of the intermediate layer associated with the base and plated metals and not,
as is usual, the resistance to the displacement of one surface relative to another.

The lower strength of connections assembled with the aid of a press is attri¬
buted to the fact that the ridges of surface microirregularities are flattened
and/or cut away. During the assembly with cooling the ridges remain intact and
after heating penetrate the valleys in the mating surface, thus adding to the
connection strength.

Joints with soft electroplating are disassembled without damage
to the surfaces, but when disassembling connections with hard
coatings, scores, cracks and deep tears in the base metal are observed,
sometimes over considerable areas of the contracting surfaces, due to
which reassembly is difficult or even becomes impossible.

The situation is aggravated by the fact that hard electroplating
lowers the fatigue strength of the joint.

Due to all these reasons preference should be given to soft coatings.

The application of soft coatings in conjunction with the assembly
with cooling allows the connection strength to be improved 3-4 times
as compared with that of the press-fitted connections without coat¬
ings. Thus, it becomes possible, while maintaining the given joint
strength, to employ smaller interferences, reducing, in this way,
the tensile stresses in the female part and compressive stresses in
the male part. In addition, the electrodeposited coatings protect
the fitted surfaces against corrosion and eliminate the “welding”
of surfaces subjected to cyclic loads.

The strength of pressed connections can seemingly be improved by metalli¬
zation and thermodiffusion saturation (e.g., by thermodiffusion zinc plating)
of fitting surfaces.

Still further strengthening of pressed connections can probably be attained
by depositing dissimilar coatings upon the joint surfaces, for instance, a zinc
coating on one surface and a copper one on the other. From the atomic diffusion
of the two metals so.ne intermediate structures can be expected in the contact
zone, possessing higher strength than homogeneous platings (e.g., brass-type
alloys when zinc and copper deposits are combined).

2.8. Design of Pressed Connections

The specific feature of pressed connections is that their joint
surfaces are already prestressed by interference forces prior to the
application of working loads, subjecting the female part to a biaxial
tension which is unfavourable from the strength viewpoint. When
adding the prestresses to the working stresses, their resultant may
exceed the yield limit of the material and cause connection failure.

Moreover, formal calculations of press-fitted joints, based on the
assumption of constant cross-sections throughout the entire part
length where the end conditions can be ignored does not express
true stress values. In practice the strength of a joint strongly depends
on the shape of the female and male parts. Uneven rigidity of parts
(stepped shafts, hubs with disks, etc.) causes an unequal distribution

68

Chapter 2. Press-Fitted. Connections

of contact pressures and stresses over the connection’s length. Sharp
stress gradients arise at the joint edges.

Formal calculation, even with high safety margins does not always
assure a reliable connection. The more so if one takes into account
that the magnitude and distribution of working stresses throughout
the part cross-section and their relationships with prestresses remain
mostly uncertain, especially in joints subjected to cyclic stresses.
Therefore, regardless of computation results all constructive measures
to strengthen pressed connections must be taken.

In order to enhance the strength and reliability of pressed con¬
nections it is advisable to:

enlarge the diameter and increase length of connections so that
unit pressure upon contacting surfaces is reduced;

< a > lb) (c) id) ie> If)

Fig. 30. Lead-in chamfers and recesses in pressed connections

select interference values within narrow magrins, making use of
more accurate fits;

avoid sharp changes in cross-sections of the connected parts over
their contact areas (areas close to these parts), thus preventing
abrupt stress relations;

improve contact surfaces by appropriate heat treatment (e.g.,
low temper hardening, induction hardening, etc.) or strengthen
them by mechanical hardening (shot blasting, rolling of shafts,
rolling out or burnishing of holes);

use assembly technique in which the female part is heated or the
male part is cooled;

electroplate contacting surfaces, using soft metals (Zn, Cu, Cd).

The reliability of pressed connections depends greatly on correct
assembly.

To facilitate the press-fitting procedure leading chamfers should
be provided on the shaft and in the hole. The chamfers should be
made at 45° (Fig. 30a) or, better, at 20-30° (Fig. 306, c). If great
interferences are expected, the shaft must have more gradual cham¬
fers of 5-10° (Fig. 30d). The chamfer leading diameter d is made
0.1-0.2 mm less than the hole diameter d 0 .

Occasionally the shaft or the hole is designed with leading cylin¬
drical collars having an aligning fit, e.g., a slide fit (Fig. 30e, f).

2.8. Design of Pressed Connections

69

The location of the aligning collar in the hole (Fig. 30 f) necessitates
the employment of the basic shaft system.

It is very important to prevent seizure anti distortion of the joined
parts, which impede the press-fitting process or may even spoil
the connection altogether.

Bush-type thin-walled components are guided when being press-
fitted by means of a centring mandrel (Fig. 31a). Figure 31b shows

Fig. 31. Methods for press-fitting
thin-walled bushes

Fig. 32. Methods for axial fixing of
parts when press-fitting

a method of press-fitting used for through holes. A bush is mounted
upon a screwed-on mandrel with a pilot shank 1 , introduced into
the hole with a slide fit. After the press-fitting is over, the mandrel
is screwed off.

Axial position of the part is fixed by pressing the part home to
a stop shoulder (Fig. 32a, b), a hole step (Fig. 32c), flush with the
hole edge (Fig. 32c?), or until the hole edge is flush with a shoulder
on the part (Fig. 32e). Plain parts can be fixed in any position by
means of distance rings 1 placed under the press plunger (Fig. 32/).

A typical error made when designing pressed connections is the
insufficient length of the joint collar. The connection with a short
collar quickly fails due to the crushing of the contact surfaces under
the effect of working loads. Examples of wrong and correctly designed
pressed connections are illustrated in Fig. 33.

For general purposes and rough determination of the minimum
length of pressed connections the following formula may be used:

^i„ = 4d 2/3 (2.26)

70

Chapter 2. Press-Fitted, Connections

where Z m in = length of the press-fitted portion (less chamfers), mm
d = connection diameter, mm
A graph based on Eq. (2.26) is plotted in Fig. 34.

If the connection is subjected to high bending or shearing loads,
especially alternating ones, or if it is necessary to assure a precise

Fig. 33. Pressed connections:

(a) connecting rod endl (b) press-fitting cup-shaped component on shaft; 1 — wrong; 2 —

correct

positioning and sound anchoring of the press-fitted component
(e.g., bed columns), the length of the press-fitted portion is con¬
siderably increased (l — 1.5-2 d).

Avoid, whenever possible, press-fitting in blind holes, because
the latter are more difficult to machine and hamper dismantling.

Should the fitting in blind holes be unavoidable, it is necessary
to assure free exit of air during the press-fitting process. Compression
of air when pressing, accompanied by increase of its specific volume,
can break the female part, particularly when it is thin-walled or
made of a less strong material (e.g., light alloys). Air is vented via
holes (Fig. 35a, c) or grooves (Fig. 355).

Never press-fit components through two sections with holes of
the same diameter (Fig. 36a): while passing the component through
the first section (the upper in the Figure) a misalignment frequently
occurs, that impedes the introduction of the male part into the
second section. In addition, scores may form on the surfaces of the
mating parts.

Under such circumstances each section should have holes of dif¬
ferent diameters (Fig. 365). The axial dimensions of the connection
must be so designed that the part enters the second section first to
a depth m = 2-3 mm (Fig. 36c), thus obtaining a definite direction,
and only after that enters the first section.

It would be wrong to force a bush into a casing as illustrated in
Fig. 36 d. Here, in order to reduce the amount of precision machining,

2.8. Design of Pressed Connections

71

the bore has two short joint collars. The error is that both collars
have the same diameter. Besides, the distortion of the bush on the
fitting areas is inevitable.

Imin, mm

Fig. 34. Minimum length l m i n of pressed connections as a function of joint

diameter d

If strict linearity of the hole walls is important, the bush then
must be reamed after the assembly, or fitted over its complete length
Fig. 36e) or, at least, over a major portion of its length (Fig. 36/).

Fig. 35. Air evacuation when press-fitting parts in blind holes

Female components must be given ample rigidity to avoid their
deformation during the forcing procedure.

In the case shown in Fig. 36 g the upper eye will deflect, thereby
making impossible the press-fitting into the lower eye. If for some

72

Chapter 2. Press-Fitted. Connections

s cannot be made th
employ some stiffe:
The simplest device
etween the eyes.

(d) (e) (f) (g)

Fig. 36. Press-fitted connections

The employment of such a spacer must be thought of beforehand
by designing the distance between the eyes to suit one common spacer
which can be used for a series of parts.

The female and male parts should be of uniform rigidity in the
radial direction. Local weakenings, cut-outs, etc., should be avoided.

Fig. 37. Press-fitting a bush

In the case depicted in Fig. 37 a press-fitting is impeded by the una¬
voidable deflection of the bush toward the cut-out. Furthermore, at
the cut-out area the bush will be given additional deflection due to
the unilateral radial interference. The situation can be improved

2.8. Design of Pressed Connections

73

slightly if the bush is press-fitted into two sections located in the
non-cut-out areas of the hub (Fig. 376). In this case it is better to
assemble the bush by a location fit and fasten it in position with
bolts (Fig. 37c).

Press-fitting is inapplicable when the female or male part has
through-cuts extending to the end face (Fig. 37 d). If cut-outs cannot
be eliminated the only reason¬
able way is to employ a location
fit.

Occasionally it is necessary to
position press-fitted parts at a
certain angular attitude. Such,
for example, is the case when
press-fitting a keyed shaft into
its hub. The alignment of the
key with its slot is assured if the
leading end of the shaft (Fig. 38a)
has a location or free fit over
length l which exceeds the dis¬
tance h of the key from the shaft
end face. The key is first intro¬
duced into the keyslot and after that the shaft is press-fitted.

Another method: the key protrudes out from the shaft to a distance
h which is sufficient to assure the shaft location in the keyslot before
the press-fitting procedure begins (Fig. 386).

The best practice is to assemble the connection with previously
heated hub or cooled shaft in order to obtain a clearance in the

m

Fig. 38. Fitting a keyed shaft in
a hub

Fig. 39. Press-fitting cams in a disk

connection. The angular location of the shaft in the hole will then
present no difficulties.

Lobed cams with given edge setting angles (Fig. 39) should be
pressed in place by means of a guiding adapter having radial cut¬
outs for the cam lobes, which is located from the central hole
in the disk. The possibility of applying such an adapter must be
considered in designing.

74

Chapter 2. Press-Fitted Connections

Figure 39a shows an incorrect design: large shank at the cam root
prevents the cam passing through cut-outs in the adapter. In alter¬
native, presented in Fig. 396, the shank size is reduced but the dis¬
tance m between the cam edges is less than the fitting diameter d,
therefore, the cut-outs in the adapter must have formed contours.
In the correctly designed version, Fig. 39c {m> d) a simple adapter
can ensure a correct cam orientation.

(a) Pressing-Out

jwai

Fig. 40. Pressing-out conditions

When designing a pressed connections the possibility of its disas¬
sembly must be considered. Parts to be pressed out must have sur¬
faces (preferably flat) which
can rest during the pressing-
out upon solid plates or bush¬
es.

A poor design is illustrated
in Fig. 40a. Here, the pulley
press-fitted upon the hollow
shaft has to rest during
the pressing-out procedure on
its conically-shaped portion,
which complicates the shape
of the bearing plate intended
to receive the pulley during
the pressing-out. The sharp edges of the shaft are unsuitable for
resting against the end face of the press ram.

In the design shown in Fig. 406 the pulley has a bearing cylindrical
web; the shaft end face is made flat. However, in the course of pres¬
sing-out excessive stresses may arise in the pulley disk, particularly
if it has a large diameter.

Therefore it is better to position bearing surfaces close to the hub
(Fig. 40c).

The pressing-out force reaches its peak value (rather high) at
the initial moment, when the friction of rest is being overcome.
After this the pressing-out force decreases because the friction of rest
concedes to that of motion, while the length of the press-fitted section
continuously decreases as the female part moves off the shaft.

Over the last few years hydraulic systems are used in which pres¬
surized oil, whose pressure exceeds the contact pressure (of the
order of up to several hundred atmospheres) is admitted into an annular
recess in the joint surface through a hole in the shaft (Fig. 41a) or
in the hub (Fig. 416). The pressure of oil produces elastic radial
deformation of the parts to be pressed out. At the same time the

2.9. Conical Fits

75

presence of oil minimizes friction during pressing-out. Moreover,
the layer of oil, penetrating into a circular slot in-between the parts

Fig. 41. Pressing-out hydraulically Fig. 42. Pressing-out a

bush hydraulically

due to capillary forces, acts as a wedge, which greatly facilitates
the pressing-out.

When pressing-out conical joints hydraulically, no mechanical
force is required to force the female part off the shaft.

Figure 42 shows how a bush can be pressed-out hydraulically from
a blind hole. The bush space is filled up with oil and the bush forced
out by a sharp blow on plunger 1.

2.9. Conical Fits

Alongside cylindrical press fits extensive use is made of conical
press fits (Fig. 43). Tapers K usually range from 1 : 50 to 1 : 100.

Fig. 43. Conical pressed connections

The required connection interference is obtained by applying a defi¬
nite force when press-fitting the shaft in place. The force is rigorously
controlled since the slow taper easily causes excessive radial inter¬
ferences.

76

Chapter 2. Press-Fitted Connections

The press-fitting force may be determined from the relation
P = Fk (sin a + /)

where F = joint surface area, mm 2

k = unit pressure on the contacting surfaces, kgf/mm 2
a — cone generatrix angle (Fig. 43a)

/ = coefficient of friction between the mating surfaces

As the magnitude of sin a is very small in comparison with /,
it can be omitted. Then

P = Fkf

As the values oi D and d are rather close, the joint surface area
(i.e., the curved surface of the truncated cone) will be

F » nDl

Finally

P « nDlkf (2.27)

Another method is forcing the male component into the female
one to a definite depth h (Fig. 43 b), counting from the moment the
mating surfaces come into close contact. The amount of h (axial
interference) can be found from the relation

fe = 10-3 A + 2( P (foi + r z2) = 1Q -3 A+2<P (Rzi + Rzz) mm (2 28)
2tana K v '

where K = 2tan a = taper

A = required diametral interference in the joint, \im
R zl and R z2 — mean roughness height of the shaft and bore,
respectively

(p = flattening coefficient of microirregularities

(ip 0.5)

Let us find the magnitude of h for a pressed connection when D= 50 mm
and K = 1 : 50.

Assume that Ai 2l =6.3 pun (7th class of surface finish) and R z2 = 10 pun (6th
class of surface finish). Let the interference be 37 pim, which corresponds to
the mean interference in the case of the Dh grade fit for a diameter of 50 mm.

From Eq. (2.28)

h = 10 3 —— = 2.6 mm

The h value is held either to the difference between the positions
of marks on the shaft prior to and after the press-fitting or by pres¬
sing the shaft until it rests against a stop shoulder positioned at
distance h from the hole edge (see Fig. 436). The latter method requir¬
es a selective assembly of parts, for inevitable inaccuracies in the
joint diameters cause substantial deviations in the dimension h
due to the slow taper.

2.10. Serrated Connections

77

Often press-fitting is effected by a calibrated blow, i.e., by drop¬
ping a weight 7 (Fig. 43c) from a certain height. The mass of the weight
and the dropping height are found experimentally or calculated by
a design subsidence value h obtained when press-fitting a reference
shaft into a reference hole by a blow.

A still better fitting method is that with a preliminary heating of
the female part or cooling of the male part, because in this case the
male part is introduced into the female one either vithout any or
with a very slight effort. After the female part has cooled (or the
male part has warmed) a tight interference fit occurs in the joint.
The magnitude of interference (and, for a known interference, the
required heating temperature) may be found from Eqs. (2.18) to
( 2 . 20 ).

2.10. Serrated Connections

In some instances, to improve the torsional holding power of pres¬
sed connections, the shafts are serrated, i.e., given longitudinal
grooves having a triangular cross-section (Fig. 44). The external
diameter of serrations is made
0.05-0.2 mm greater than the
hole diameter. During press-fit¬
ting the sharp ridges of the ser¬
rations cut into the material of the
female part, thus ensuring a
strong connection between the
shaft and the part.

Generally serrations are pro¬
duced by cold rolling. The shaft
surface must have a hardness of
not below 35-40 Rc and the hole
surface 10-15 units lower than
that.

Serrations are formed either
throughout the entire length of
the joint surface (Fig. 44a) or
(which is more preferable) over
a limited section on the side fur- Fig- 44. Serrated connections
thest from the press-fitting direc¬
tion (Fig. 44ft). The latter technique ensures more accurate alignm¬
ent. Serrated connections allow lower interferences to be used (we
mean the interferences on plain sections) than in conventional
pressed joints. Small-diameter rods (d -< 15 mm) are often intro¬
duced into drilled holes without any interference, relying on the
strength of connection between serrations and the bore.

A-A

78

Chapter 2. Press-Fitted Connections

Serration is only practicable for permanent connections, since
the repeated installation of serrated shafts weakens the connection
because the first assembly destroys the hole surface.

2.11. Glued Connections

In many cases press-fitted connections can successfully be replaced
by glued connections whose strength is comparable with that of
the former.

The axial holding power of pressed and glued connections P = kFf and
F%, respectively, and their torsional holding power, M = kFf and

rgi

M g i= F% 7 p , where k is the pressure on the joint surfaces and t, the shear strength
of the glue layer.

Equating P = P gi and M = M g i , gives a general equistrength relationship

*-f

which defines the unit pressure k in pressed connections equivalent (in terms
of strength) to glued joints. For epoxy base glues x = 2-3 kgf/mm 2 . Calculating

2

by the lower limit, we have k = With f— 0.15, k = 13.5 kgf/mm 2 .

This value of k corresponds to moderate interference fits of grades Dhl lt Dh2 x
and Dh.

The advantage of glued connections is that they do not cause
stresses in the connected parts. They exclude the need for pressing,
or heating, or cooling and considerably facilitate the assembly pro¬
cedure. For thermosetting glues, however, they must be held at a
temperature of 150°C for approximately 2 hours.

Glued connections are assembled by slide or transition fits. In
pressing-out the glue film is destroyed. For the repeated assembly
it is necessary to wash off the traces of the old film with a solvent
and then apply a fresh layer of glue.

Glued connections remain stable at temperatures up to 200°C,
which limits their field of application. Under the effect of cyclic
loads, even in cold connections there may develop local seats of
increased heat liberation which destroy the glue film.

Chapter 3

Centring Connections

Cylindrical surfaces are generally centred by slide ( S) of transi¬
tion [push ( P ), wringing (W) and tight (7 1 )] fits. Occasionally easy
side ( Se) fits are also employed.

Figure 45 shows the mean values of clearances and interferences
for various grades of fits as a function of the centring surface dia¬
meter. .

With slide fits the clearan- n rn/an 1 ^tn
ce in the joint is zero only un- /
der extreme conditions when
the female and male surfaces j
are made to the nominal size.

In the majority of cases the
connections have some clearan¬
ce reaching significant val¬
ues for the low accuracy clas¬
ses. Consequently, slide fits
cannot ensure accurate cen¬
tring.

Clearances can also occur '
with the push fits ( P ). The
wringing fits ( W) are without
clearance and should be used
when accurate alignment is

necessary. Tight fits (T) are m 2 00 300 WO D,mm

obtained with insignificant in¬
terferences in the connection. Fig- 45. Mean clearances and interferen-

m. , . , . . ... ces Aior diflerent lits as a lunction

Tight and wringing fits en- 0 f centring surface diameter D
sure accurate alignment with¬
out complicating assembly and

disassembly if the centring surfaces have short lengths (e.g.,
collar flanges). Parts with long centring surfaces (hubs) are bet¬
ter assembled by slide and push fits, if the requirements
as to high alignment accuracy are not imperative and there is no
danger of the joint surfacs being damaged under the action of loads.

80

Chapter 3. Centring Connections

Average clearance values depend not only upon the grade of fit,
but on the grade of accuracy as well. The easy slide fit to the 1st
grade of accuracy (Se^ is practically equivalent to the slide fit (S)
to the second grade of accuracy (disregarding the narrower margin
tolerance); the S i fit is equivalent to the P fit, and the Se fit, to
the S ia fit.

When nominating fits due consideration must be given to the
temperature conditions in which the joint is expected to operate.
The original (cold) fit may sharply change during heating, particu¬
larly when the female and male components are made of materials
with different linear expansion coefficients. In such conditions heat
calculations are obligatory.

If during heating the female component expands more than the
male part, tighter fits should be nominated ( P , W and T), and if
the male part expands more, free fits ( S , Se and even R) should be
preferred.

Let the diameter of the centring surface be D = 200 mm. The female part
is made of a light alloy (c^ = 24 -10~ 8 ), and the male part, of steel (a 2 =
= 11 -10 - ®). The working temperature of the connection is 100°C. The connec¬
tion is assembled by the S 2a grade of fit (diametral clearance A = 0 to 0.12 mm).

After heating the clearance becomes

Aj = A + 200(ai — cc 2 ) 100 = (0 to 0.12) + 20(M3-10- 6 = (0.26 to 0.38) mm

Obviously, the centring accuracy is disturbed. A tight fit will somewhat
impove the situation. The maximum clearance with this grade of fit amounts to
0.056 mm, and the maximum interference, 0.064 mm. Consequently, after heat¬
ing the resultant clearance in the joint will range from 0.26 — 0.064 = 0.196 mm
to 0.26 4- 0.056 = 0.316 mm.

With larger radial dimensions of the connection and higher working tempera¬
tures the original fit often alters to such a degree that it is impossible to centre
the cylindrical surfaces, and a temperature-independent centring must be used
(see Fundamentals of Machine Design, Yol. 1, Section 7.3, p. 472).

3.1. Design Rules

To enhance the alignment accuracy and reduce the effects of tem¬
perature-induced deformations, it is undoubtedly better to centre
parts from the smallest possible diameter. Ways of reducing the
centring diameters are illustrated in Figs. 46 and 47.

The screwed bracket (Fig. 46 d-f) is an example. When the centring
is accomplished from a large diameter (Fig. 46d) equal to, say,
200 mm, the maximum clearance with the S 2a grade of fit will be
0.12 mm, but when centring from the minimum diameter (Fig. 46/)
this clearance decreases to 0.037 mm, i.e., approximately thrice.
The centring is sharply improved and becomes practically tempera¬
ture-independent.

3.1. Design Rales

81

When] centring franged parts a sufficiently long centring shoulder
should be provided, allowing for the leading chamfers in the hole

Fig. 46. Reduction of centring diameter

(a-c) gear rim; (d-f) bolted bracket

and on the male part, as well as gaskets, that considerably decrease
the actual length of the centring surfaces.

Shoulder height H is chosen so as to ensure positive centring from
section h (Fig. 48)

H = h -f 2c + Aff + 2Ac 4- s

where c = length of the leading chamfers with the most unfa¬
vourable combination of manufacturing deviations
s = gasket thickness (in compressed state)

Ac = plus deviation of the leading chamfer dimensions from
the nominal

Aff = minus deviation of the centring shoulder height from
the nominal

The dimensions of the centring section h and chamfers (in the
hole and on the male part) for mass produced connections may be
taken as follows:

Diameter of cen-

mm. up to 100 100-200 200-300 300-500 over 500

h, mm. 3-4 4-5 5-6 6-7 7-8

Chamfers. . . . 0.5-45° 0.8-45° 1-45° 1.5-45° 2-45°

82

Chapter 3. Centring Connections

The shoulder height H is practically determined from the rela¬
tion H « 0.5 YI), where D is the centring surface diameter'(provid¬
ed that ordinary 0.1-0.2 mm thick gaskets are used).

Fig. 47. Reduction of centring diameter in connections of housing-type com¬
ponents

In housing-type components (Fig. 49) the centring surfaces should
be made in the form of holes which can readily be machined on
boring machines (Fig. 49<2, e ). The observance of this rule is particu¬
larly important when the centring is ac¬
complished from co-axial surfaces position¬
ed on different sides of the housing (Fig.
49/, g). The design illustrated in Fig. 49/
is extremely poor as the centring shoulders
on such a housing have to be machined in
different tool settings and proper alignment
can be assured only with the aid of spe¬
cial attachments. In the correctly designed
version presented in Fig. 49g the centring
surfaces of the housing can be through-
machined in a single tool setting, thus
ensuring good alignment.

Simultaneous centring from two surfaces
(Fig. 50a) must be avoided whenever possib¬
le: the centring must be accomplished from
one surface only, leaving a positive cearance s on the other
surface (Fig. 50b, c).

In the flange unit shown in Fig. 50d the centring of splined flange 1
from the shaft is not only superfluous (the centring being effected

Fig. 50. Elimination of double centring

84

Chapter 3. Centring Connections

by the splines themselves) but harmful since it hinders the correct
clearance-free fitting of the splines. In the correctly designed ver¬
sion pictured in Fig. 50e clearance s is provided between the compo¬
nents.

The next Fig. 50/ shows an incorrect valve design with a guide
shank. The small clearance between the shank and the bore prevents
the accurate seating of the valve cone. In a better alternative
(Fig. 50g) ajgreater clearance is given.

Figure 50 h presents a wrongly designed assembly of a ball bearing
inside a sleeve with two centring surfaces serving simultaneously

Fig. 51. Reducing the number of centring surfaces

as a sealing bush with split spring rings. Correct alternatives are
illustrated in Fig. 50i, j.

In units consisting of several concentrically arranged components
the number of centring surfaces should be reduced to the minimum,
since the accuracy of the unit as a whole is considerably impaired
by the cumulative error introduced by each centring surface, which
is the greater, the larger the number of the surfaces.

In the design shown in Fig. 51a, an antifriction bearing is mounted
in two intermediate bushes. Apart from the clearances between
the rolling bodies and races, there are four centring surfaces. Reduc¬
ing the number of the centring surfaces down to two (Fig. 516) impro¬
ves centring accuracy approximately twice.

When centring from a cylindrical pin (Fig. 51c) pressed in member 1
and fitted by a slide fit in the hole of member 2, the centring errors
from the two surfaces are summed. In units requiring accurate centr¬
ing it is necessary either to machine the centring surface of the
pin concentrically to the precision surfaces on the part to be centred,
after the pin is pressed in, or the centring pin must be made directly
on the part in question (Fig. 51d).

Figure 51e depicts the case of an irrational centring from two sur¬
faces: surface m on the shaft, and surface n on a detachable disk.
The centring from the surface n is either fictitious (if the disk is
fitted on the centring part with a clearance) or disturbs the centring
from the surface m (if the disk is fitted on the shaft with an interfe¬
rence). It is better to centre the part from the shaft, the disk being
freely mounted (Fig. 51/).

3.1. Design Rules

85

centring from teeth (Fig. 52b), splines (Fig. 52c) or even from indi¬
vidual lugs, when the latter are at least three in number and are
spaced symmetrically over a circumference.

A common mistake of young designers is to introduce centring
where for operation it is superfluous and can be dispensed with.
For instance, for the intermediate mounting of an antifriction bear¬
ing the centring of the lateral cheeks (Fig. 53a-d) is actually not
necessary. Here it is sufficient to fix the cheeks in radial direction
by means of fastening bolts (Fig. 53e), this considerably simplifying
the manufacture.

Chapter 4

Screwed Connections

The principal requirement for correct functioning of screwed
connections is that the threads must be free from bending and shear¬
ing stresses.

A bolt loosely mounted in the holes of the clamped parts and subjected
to the action of transverse forces (Fig. 54 a) deforms. With a full elimination of
clearance at bolt sections near the joint more shear stresses arise. Furthermore
the bolt is subjected to tension due to which it elongates under the displace¬
ment of the pulled part. All'these stresses are added to the tensile stresses induc¬
ed in the bolt during the preliminary tightening. As a result, in the bolt body

(b)

(o)

Fig. 54. Flexure of threaded parts

a complex state of stress occurs, produced by the simultaneous action of bend¬
ing, shearing and tension forces, and the bolt’s strength abruptly drops.

A screwed-in bolt has still worse working conditions (Fig. 546) when in its
critical section (close to the joint plane) a thread is cut, appearing as a very
strong stress concentrator. Most unfavourable is the case of the bending of
a stud screwed home in a hole (Fig. 54c). Here high tensile stresses arise in the
critical section, comprising the prestresses and stresses produced as the stud
is tightened against the hole bottom. As in the previous case, here the stress
concentration occurs in the section where the thread comes to the critical sec¬
tion plane.

Also unfavourable are the working conditions for the material of the thread¬
ed holes of tightened components. The transverse forces, acting at the joint,
wedge out the threads of the screwed hole, producing higher local bearing stres¬
ses which in time weaken the thread and loosen the fit of the screwed stem,
particularly under alternating loads.

The problem of strengthening tightened connections consists in
the elimination of the complex stressed state in the fastening ele¬
ments and assure conditions in which these elements would operate
only in tension when under the influence of preliminary tightening

Chapter 4. Screwed Connections

87

and working forces. Transverse acting forces must be taken up by
additional shear load-carrying elements.

Let us consider the case of a cantilevered rod, screwed in a hous¬
ing (Fig. 55a). Design 1 is obviously unsatisfactory, because the
maximum bending moment from the transverse force P is in the
weakened threaded portion. High stresses at the imbedment point
inherent in the cantilever make the rod bend and crushes the threads
in the hole and on the rod.

Provision of a shoulder in the tightening area (version 2) helps
little, since the shoulder thrust face is nearly parallel to the direc¬
tion of bending displacement and the deformation is resisted only
by the friction forces arising at the thrust face when tightening.

In more correct designs 3 and 4 the rod has a cylindrical or tapered
section which closely fits the housing hole and efficiently impedes
transverse strains and rod displacement. Since it is rather difficult to
assure strict alignment of the threaded and plain sections, the screwed
end must have clearance.

The best are designs 5 and 6 which have a cylindrical or tapered
fitting section in the housing. In this case the threads are completely
relieved from bending and take up only the tightening tension.

Figure 556 illustrates various methods of fixing a cast stand loaded
with a transverse force to a housing. Version 7 is absolutely incorrect:
the fastening stud is subjected to bending from the transverse force.
Version 8 is hardly better: here the stand is aligned by the plain
stud portion. In the improved version 9 the stud has a centring cy¬
linder which closely fits into the holes of the housing and stand.
In design 10 shearing forces are taken up by check pins and in design
11 by the stand centring shoulder.

Illustrated in Fig. 55c are various methods of taking up shearing
stresses in a counterbalance unit joint loaded with a centrifugal
force, and in Fig. 55 d, in a flanged connection transmitting a torque.
Designs 12 and 17 are wrong, and the rest to a more or less degree
provide correct operating conditions for bolts.

Bending of bolts is often the result of their incorrect positioning
relative to the acting loads (Fig. 55e). The bracket in design 22 has
two errors: lack of an element taking up the shear stresses and the
bending of the bolt stems due to the off-centre application of the
axial load. Under the action of force P the bracket tends to turn
about point A. With the relationships presented in the Figure, the

PI

force acting upon each bolt is TV = ^ = 1.7P. The bolt is subjected

Nd

to bending by a moment M = -j-, where d is the diameter of the
bolt head.

With the flange turned through 90° (version 23) the load upon the
bolt becomes practically central (owing to the extension of arm b).

Chapter 4. Screwed Connections

89

Since in the given case only one bolt is active (the right-hand bolt
is relieved), the force applied to the bolt still remains rather great
(N — 1.7 P).

The introduction of a triangular flange (design 24) in no way en¬
hances the joint strength, as the added bolts do not take part in the
work. In the rational alternative 25 two bolts are active. The force
acting upon each of the bolts is thus reduced to N = P. The bolt

shearing is forestalled by means of the bracket centring shoulder.
In design 26 (the square-shaped flange) the base area is enlarged
and, as a result, the bolt load is reduced to N = 0.7 P.

Fastening elements are often bent due to the distortion of the
support surfaces, which causes an off-centre load application (Fig. 56).
Should the supporting surface be angled, then tapered washers (de¬
sign a) or, better, spherical washers (design b ) are obligatory. To
prevent off-centre loads the flat end faces of pressing, lifting and
other screws (design c) must be changed to spherical ones (design d).

Off-centre flexure may also arise with an unsymmetrical shape
of bolt heads, for example, a bolt head having a flat preventing its

90

Chapter 4. Screwed Connections

rotation when tightening (version e ). The flexure here can be eli¬
minated by machining flats on two sides (alternative /) or by decreas¬
ing the head rigidity in the area opposite to the flat (version g).

Self-alignment of fastening elements is a radical means to avoid
flexure. In design h (clamping down adjacent angles of hydraulic
communication system with the aid of a crossbar) distortion of the
crossbar is inevitable, also inevitable are bending of the bolt and
non-uniform tightening of the angles. These defects are eliminated
in version i by providing a self-aligning crossbar.

Wrong j and correct k alternatives of clamping the flanges
of adjacent cylinders to the crankcase are shown in Fig. 56.

Flexure may also arise as a result of elastic strains emerging in
the tightened parts. When tightening clamp-type connection l
the ends of the clamps distort and the load becomes eccentric. In
design m the flexure of the clamping bolt is prevented by the intro¬
duction of spherically shaped washers.

In all connections where the bolts are displaced from the plane of
acting forces flexure is inevitable (e.g., in flanged connections n
loaded by the internal pressure). Self-aligning bolts o are recommend¬
ed for heavily loaded critical connections.

4.1. Longitudinal and Transverse Location
of Parts in Screwed Connections

Screwed connections do not ensure accurate mutual location of
the clamped parts.

Generally, fastening bolts and studs are introduced into their holes with
some clearance whose amount is dependent on the number and spacing, as well
as accuracy of centre-to-centre distances, of holes, and on the average varies
from 0.5 to 1 mm. If such a clearance is absent, the assembly becomes impossible
due to certain mismatching of mated holes in the parts to be coupled and the
displacement of holes one relative to another in each part, a fact inevitable in
practice.

To ensure accurate mutual location of parts it is necessary to
introduce certain locating elements. The methods of such a location
are illustrated by an assembly example of a bearing cover to a hous¬
ing (Fig. 57). Often the location is accomplished from longitudinal
shoulders (Fig. 57 a-c) machined in the cover or the housing, which
are tightly fitted in corresponding slots.

The design alternative with a locating lug (Fig. 57d) is not suit¬
able because it require 0 a close contact over four surfaces at once.
The location with fitted-in prismatic keys (Fig. 57e) is also not recom¬
mended.

The cylindrical shape of the centring shoulder (Fig. 57/) facilita¬
tes accurate manufacture of the shoulder and its housing seat and
ensures longitudinal and transverse cover location. The cover,
however, is not prevented from turning about the line of joint.

4.1. Location of Parts in Screwed Connections

91

The design seems to be more advantageous for single bearings; for
series-arranged bearings it is better to employ longitudinal shoul¬
ders and slots machined in a single tool setting.

Fig. 57. Methods of fixing bearing caps to housings

Occasionally the cover is located by means of endwise splines
having a triangular section (Fig. 57 g) anti; produced by outside broach¬
ing. This method is technically possible (provided the necessary
equipment is available) and enables a sturdy joint to be obtained.

92

Chapter 4. Screwed Connections

Higher robustness is assured by mounting the covers in lengthwise
housing recesses (Fig. 57 h, i).

The methods illustrated in Fig. 57 a-i ensure a transverse location
of the cover. If the bearing has to take up axial loads, a lengthwise
location is necessary to assure the matching of the cap and housing
end faces. Such a location is accomplished by means of cylindrical
collars on studs (Fig. 57;), insert-type cups (Fig. 57 k) or dowel pins
(Fig. 571).

Detachable taper locating pins (Fig. 57 m) ensure a more accurate
location, but are much more difficult to produce and assemble.
Moreover, they have to be prevented from falling out. Location by
means of lengthwise dowel pins (Fig. 57 n) is practicable only when
the bearing end face can be directly drilled and reamed. This method
cannot provide for the taking up of longitudinal loads.

The design in Fig. 57o shows a unique method of fastening the
cover with the aid of taper pins press-fitted into a row of lugs milled
in the housing and cover. Lengthwise location is effected by fitting
closely the lugs of the cover (or a pair of lugs) inside the receiving
slots of the housing.

Still another method is the location by means of lengthwise pins
(Fig. 51p) introduced into the holes machined in the vertical joint
surfaces between the cover and housing. The system is tightened
transversely by bolts. If it is impossible to drill or ream directly
in the bearing end face, broached triangular-section splines must
replace the pins.

4.2. Centring in Screwed Connections

Screwed connections of conventional accuracy do not ensure pre¬
cise centring due to the practically unavoidable run-out of the thread
pitch diameter, as well as thread clearances. Exceptions from this
rule, however, are encountered occasionally when dealing with
precision centring threads—mostly large trapezoidal threads pro¬
duced by milling and grinding.

With the usual manufacturing accuracy the centring from the
threads is not permitted (Fig. 58 a-g). If threads must be used, then
additional centring surfaces should be introduced. Most often the
problem is solved by using plain cylindrical collars. Threads in
this case should be made loose so that the centring is unobstructed.
The position of such centring collars relative to the threads depends
on the actual operating conditions of the joint. From technological
viewpoint it is wise to make the diameter of such a centring collar
slightly smaller than the thread minor diameter, locating the collar
behind the thread (Fig. 58/j). This allows precision through-machin¬
ing of the centring bore. The loading scheme of the connection must
also be considered. Should the force act as shown in Fig. 58i then

O. Centring in Screwed Connections

93

the correct position of the centring collar is before the thread, even
though it incumbers slightly the machining of the bore.

Screwed joints of conventional accuracy also fail to assure true
perpendicularity of the screwed component’s end face relative to the

Fig. 58. Centring in screwed connections
1 — wrong; 2 — correct

thread effective diameter. For this reason the nut end face cannot
be used as a bearing surface taking up axial forces in frictional units
(Fig. 58/). In this case the inevitable misalignment of the nut end
face relative to the shaft axis causes a unilateral force application
and increases the wear of the bearing shoulder. The correct design
is illustrated in Fig. 58A;.

94

Chapter 4. Screwed Connections

A plunger assembly with a rod sliding in the bore of a screwed-on
detachable cover (Fig. 581) does not ensure axial alignment of the
holes in the cylinder and cover. An aligning collar tailored beyond
the thread boundaries (Fig. 58m) rectifies but partially the displa¬
cement of the cover.

An aligning collar located nearby the cylinder end face (Fig. 58n)
ensures alignment only if the cylinder outside centring surface
is made exactly concentric with the hole. Still better is the design
with a female thread (Fig. 58o); here the collar is centred directly
from the cylinder walls. The most reliable way of effecting the align¬
ment is by means of a cylindrical shoulder with the cover clamped
down by a captive nut (Fig. 58p).

Figure 58 q-u shows units enabling a continuous adjustment of the
axial position of the shaft mounted in a bearing and loaded with
a force of constant direction. The alternative in which a thrust plate
is screwed on (Fig. 58 q) is unsatisfactory due to the plate distortion
which causes wear on one side of the plate bearing face. An increase
of the threaded portion (Fig. 58r) only worsens the situation. A satis¬
factory solution is the introduction of a cylindrical centring collar
(Fig. 58s) which eliminates the distortion of the plate end face, if
it is machined strictly perpendicular to the centring cylindrical
surface and the thread clearance is large enough not to interfere
with the plate placement on that surface. A still more reliable mount¬
ing of the plate is illustrated in Fig. 58* where the plate is fitted
upon a plain centring collar and its axial position is adjusted by
means of a nut and a locknut. A still better design is presented
in Fig. 58 u in which the plate can align itself on the spherical sur¬
face of the nut.

4.3. Design Rules

Screw threads in heavily loaded connections must always be tight¬
ened, otherwise they will quickly fail, particularly under cyclic and
dynamic loads, owing to the crippling, work hardening and, some¬
times, welding of the threads.

The design of a valve in which the valve plate is screwed into the
valve stem (Fig. 59a) is poor, as under the action of forces and bend¬
ing moments, when the drive cam runs against the plate, the threaded
connection works loose. Moreover, the free thread does not accura¬
tely locate the plate relative to the stem. Lenghtening the thread¬
ed portion (Fig. 595) only partially eliminates this disadvantage.
It is better to tighten the screwed connection by a locknut (Fig. 59c).
The turnbuckles shown in Fig. 59d, e are similar examples.

Screwed connections of the usual accuracies are not seal-proof.
When applying screw threads to vessels containing gases or liquids
under pressure, preliminary measures must be taken against leakage
through the threads. Fitting gasket washers under nuts (Fig. 60a)

4.3. Design Rules

95

is insufficient because liquid penetrates along the threads. In such
cases one should use cap nuts (Fig. 606) or install below the nuts
bushes having inserts made of some elastic material (rubber, plas¬
tic), which will assure the sealing of the joint over the plain cylin¬
drical part on the bolt (Fig. 60c).

Studs the ends of which penetrate into the internal cavity contain¬
ing the liquid must not be used (Fig. 60d). However, interference-

(a) (b) (c) ^(d) ^(e) *7f)

Fig. 60. Sealing of screwed connections

correctly. Tightening studs onto sealing gaskets (Fig. 60e) complicat¬
es the design and is not fully reliable. The best way to prevent leak¬
age is to fit the studs in blind bosses (Fig. 60/).

Through-holes for studs (ref. No. 1, Fig. 61) and bolts are permitted
in spaces where the liquid is present in the form of droplets, splashes
or films covering the walls. In oil baths the fastening studs must be
blind-mounted (2).

Plugs in oil-containing cavities must have sealing gaskets
(Fig. 62a). Tapered threads (Fig. 626) have self-sealing properties,
particularly when screwed into housings made of plastic metals.

Chapter 4. Screwed Connections

Parts, for which an accurate angular position is essential, should
not be screwed in. When positioning a threaded angular pipe con¬
nection (Fig. 63a), its accurate location can be ensured either by
scraping the end face surface of the bearing shoulder or by selecting
a gasket of a suitable thickness. These measures complicate assembly.

In later reassemblies the correct posi¬
tion of the connection will inevitably
be disturbed. The most advisable so¬
lution in this case is to mount the
connection on a flange (Fig. 63b).

Figure 63c-/ shows various bra¬
cket designs for a handrail. When

Fig. 61. Installation of
fasteners in oil-contain¬
ing cavities

m

(b)

Fig. 62. Plugs

using a threaded connection it is practically impossible to put
the bracket at its required angular attitude (Fig. 63c). It is better
to fasten the bracket with a cylindrical stem (Fig. 63d), tightening
the nut after the handrail is introduced into the bracket head. The

Fig. 63. Angular location of

threaded parts

angular position of the bracket can also be set with a key (Fig. 63c),
or by means of a flange (Fig. 63/). Readjustment in the first case is
accomplished at the expense of the clearances between the key and
its seat, and in the latter case, at the expense of the clearances be¬
tween the bolts and their holes in the flange.

4.3. Design Rules

97

Power tightening should always be made possible for threaded
connections. In the poor designs shown in Fig. 64a, c friction forces
produced during tightening on the bearing flange surfaces, being
applied over a large radius, sharply increase the tightening torque,
thus making power-tightening impossible. In the correctly designed

Fig. 64. Reduction of tightening torque

versions (Fig. 64 b, d) the friction forces act at the minimum distance
from the bolt axis, this distance being equal to the average radius
of the bearing surface of the bolt head.

In joints where tightening is effected over gently tapered surfaces
(Fig. 64e) power-tightening is impossible because of the impeding

Fig. 65. Tightening over soft gaskets

effect of frictional forces on the tapered surfaces and also due to
the compression of the stem threads by those of the nut over the tape-
red area. Cone angles not exceeding 90° (Fig. 64/) are recommended.

Tightening on soft materials (packings, gaskets, etc.) should be
avoided.

For instance, tightening of a packing gland nut (Fig. 65a) causes
twisting and collects the packing on the nut end face. This can be

7—01418

98

Chapter 4. Screwed Connections

avoided by installing an intermediate metallic ring 1 (Fig. 65b)
or by replacing the nut with a flanged bush (Fig. 65c).

Figure 65 further illustrates an incorrect (Fig. 65d) and correct
(Fig. 65e, /) examples of a cover with a sealing gasket.

In general, screwed joints with large diameter threads should
be avoided (Fig. 66a) and bolted and studded connections used instead

(Fig. 66b). Large diameter threads
are difficult to produce, par¬
ticularly in big housings. Thread¬
ed holes for fastening elements
are more economic, even with
a large number of holes. Fur¬
thermore, small-sized fasteners
are more convenient in assemb¬
ly and allow commercially pro¬
duced fasteners to be employed.

It is strongly recommended
that large diameter threads in
components made of ductile and
plastic materials (light and zinc
alloys, stainless steels) be avoid¬
ed. Ductility and low antifriction properties of these materials
cause tearing of threads thus hampering tightening.

5*..

WA

( a )

Fig. 66. Replacing threaded connec¬
tion by bolted joint

Fig. 67. Threaded hole designs
(a) and (c) wrong; (i>) and (d) correct

Threaded holes with skew (Fig. 67a) or stepped (Fig. 67c) edges
are unacceptable under any circumstances, since it is extremely dif¬
ficult to screw a fastener into such a hole. They can tap such holes
only having left beforehand a flat area on the hole end face which
will be removed after tapping.

4.4. Reinforcement of Fastening Joints

Fastening bolts and studs should be positioned at the rigidity
nodes of the assembled parts so that the fastening force is spread
over as large an area as possible.

Fig. 69. Methods of reinforcing fastening units

7*

100

Chapter 4. Screwed Connections

The spacing of the lid fastening bolts shown in Fig. 68a is wrong
as the bolts are arranged in bosses positioned in areas weakly con¬
nected with the part body. Somewhat better is the design in Fig. 68 b
where the lid has a stiffening edge improving the force distribution
over the tightening area. In the correctly designed version (Fig. 68c)
the bolts are spaced at the rigidity nodes at the corners. The rigidity

Fig. 70. Fastening a bearing cap to crankcase

of the lid as a whole is further enhanced by diagonal stiffening ribs,
tying up the fastening points with the part body.

Figure 68 d shows badly positioned bolts in an intricately shaped
flange. The correctly designed version is pictured in Fig. 68e where
the bolts are positioned at the rigidity nodal points within the part’s
external contour, which enhances clamping rigidity and improves
the outer appearance.

The bosses in cast parts intended for receiving screwed ends of
bolts and studs (Fig. 69) must be reinforced with ribs oriented pre¬
ferably in line with the bolt axis (Fig. 69 b-d). Load-carrying studs
and bolts (Fig. 69c), particularly when the clamped parts are made
from low-tensile alloys, should be screwed deeper into the parts

4.4. Reinforcement of Fastening Joints

101

so that the maximum cross-sectional area of the wall is engaged in
the work. Furthermore, longer studs under cyclic loads are stronger
and ensure a more reliable fastening.

Figure 70 shows the subsequent stages of strengthening a crank¬
case and bearing cap assembly loaded in tension. The weakest
design is that with the short studs (Fig. 70a). Here is a further defect,
namely, the crankcase strengthening ribs are offset relative to the
stud axes and do not share the load. In improved versions (Fig. 70b, c)
the studs are positioned deep into the case body and the bosses
reinforced with ribs. In the design illustrated in Fig. 70 d the studs
ends are tightened with nuts, relieving the case thread from load;
the crankcase cross-section, which works in tension, is reinforced
with an arch-shaped strengthening collar.

Studs can be relieved from bending (in one plane) by installing
their screwed ends in cylindrical swivelling inserts (Fig. 70e).

The most advisable design (Fig. 70/) has the stud ends outside.
The tensioning of the material inherent in the previously illustrated
versions is practically eliminated and the load is taken up by the
entire crankcase cross-section.

Chapter 5

Flanged Connections

When designing flanged connections it is necessary to ensure
strength and rigidity with the minimum weight, as well as the
rigidity of the sections connecting the flanges to the walls of the part.

Illustrated in Fig. 71 are typical flange designs for turned steel
cylindrical parts (a sleeve held to a housing), approximately in the
order of increasing rigidity.

The design in Fig. 71a is unsatisfactory: the flange connection to
the walls is too thin and inadequately rigid. The principal flange
strengthening methods are: introducing fillets (Fig. 71 6, c) and tapers
(Fig. 71 d-f) in the areas where the flanges are connected to the walls.
When large fillets and tapers are used, nut bearing surfaces are
machine faced (Fig. 71c, e, f) so that the fastening studs may be
brought closer to the walls.

The weight of flanges is reduced by holes machined between the
fastening studs (Fig. 72a), by removing superfluous material off the
periphery (Fig. 726, c) and surface (Fig. 72 d,e), and by face recesses
(Fig. 72/, g) and radial recesses (Fig. 72 h).

Typical designs of cast flanges are pictured in Fig. 73. The rigidity
of flanges is improved by ribbing (Fig. 736), by making local bosses
in the fastening hole areas (Fig. 73c), by making the flanges higher
(Fig. 73 d-f). To eliminate excessive solidness high flanges are under¬
cut. Flanges with through undercuts (Fig. 73 d) have the disadvantage
that the flange is subjected to bending when tightening fastening
studs. This shortcoming is eliminated in the design having bosses
around the fastening holes (Fig. 73e, /).

Flanges can be reinforced with a continuous peripheral rib
(Fig. 73 g) linked by transverse ribs to the walls of the part. Occasion¬
ally a flange is made in the form of two flanges interconnected by
bosses of the fastening holes (Fig. 73 h).

Rigidity is significantly enhanced by locating fastening studs
in recesses, which have semi-circular cross-sections (Fig. 73 i-k).
Still further improvement can be attained by increasing the height
of fastening stud bosses (Fig. 73 l, m).

In the designs depicted in Fig. 73 n-p the stud-receiving bosses are
made in the form of columns; the nut-bearing surfaces are slightly

104

Chapter 5. Flanged Connections

raised above the top horizontal wall of the part. This enables column
end faces to be through-machined and facilitates nuts tightening.
The highest strength and rigidity are possessed by the design pictured
in Fig. 73 p in which the vertical wall of the part is matched with
the column extreme points.

Figure 74 shows flange designs of conical and spherical castings.
With small (in respect to flange diameter) cone sizes the flange is
connected to the part walls by means of a tulip-shaped mouth

Fig. 74. Flanges on conical and spherical

parts

(Fig. 74a), thus ensuring a streamlined transition of the force flow
from the part walls to the flange. In cones of large diameter sizes
the joint between the flange and the part walls is reinforced with
ribs (Fig. 74 b-d) or by positioning fastening holes in recesses
(Fig. 74e, /).

With small slope angles the recess walls are extended excessively.
In these cases they are given a semi-closed arched form (Fig. 74 g).

To obtain the greatest rigidity and strength the walls are position¬
ed at the flange periphery and the recesses ceiling linked with the
walls by internal ribs (Fig. 74k).

The sizes of the vaulted recesses in height and cross-section must permit
easy assembly of fastening parts. With insufficient height (i.e., when the

Fig. 75. Mounting fasteners in recesses

clearance h between the recess ceiling and stud end face is less than nut height h 0 ,
Fig. 75a), the unit can be assembled only by raising the part (Fig. 756) and
placing all the nuts on the studs ends, a procedure seriously hampering the
assembly. The correct version (h > h 0 ) is pictured in Fig. 75c. When using bolts
(Fig. 75 d) the recess height must be greater than the bolt length.

5.1. Alignment of Flanges

105

To reduce the weight of low flat flanges the latter are often shaped
(in plan), making them narrower in width at the areas between fasten¬
ing bosses (Fig. 76a-c, g-i). In the extreme limit the flange actually
vanishes so that only bosses remain which are cast directly onto the
part walls (Fig. 76 d, j). These methods must be used very carefully-

4

(a) W (c)

Fig. 76. Reducing the weight of cast flanges

since the flange rigidity and strength is lowered and the connection
between the bosses and part walls weakened. When reducing the
flange width to trespass beyond the fastening holes centre line is
not recommended (Fig. 76 b, h). It is useful to strengthen the connection
between bosses and part walls with local ribs. More preferable are
solid flanges (Fig. 76/, l) ensuring higher rigidity and better fasten¬
ing stability.

5.1. Alignment of Flanges

Cylindrical flanges are usually aligned by a shoulder and a recess
which are machined on the flanges and fit to one another (Fig. 77a, b).
In the joints fastened with screwed-in bolts or studs the above-said

Fig. 77. Centring of cylindrical flanges

recess is substituted by the minimum internal flange diameter of
one of the flanges (Fig. 77c). Alignment can also be accomplished
on an external flange rim (Fig. lid).

The aligning shoulder must be at least 3-4 mm distant from the
extreme points of fastening holes (Fig. 78a), otherwise thin rags (m)
or sharp-pointed burrs ( n) will form which easily break in service
and impair the shape of sealing gaskets.

106

Chapter 5. Flanged Connections

In exceptional cases, in order to obtain smaller sizes, the aligning step can
be provided at the area where fastening holes are located (Fig. 78 b-d). This
method is practicable only for bolted connections; cutting threads in stepped
holes and screwing-in fastening elements into such holes offer formidable diffi¬
culties. Stepped holes are machined on assembly which complicates manufactur¬
ing. The designer should not position the centre-aligning step behind the axial

Fig. 78. Centring of flanges

line of fastening holes (Fig. 786, c) as this would inevitably lead to the formation
of sharp rags and burrs. Furthermore, such connections bend as the fastening
bolts are being tightened. As obvious from Fig. 78 c a gap formed in the joint
accumulates dirt. An acceptable interval at which the centre-aligning step may
be positioned is 0.25 hole diameter from the holes axial line (Fig. 78d).

To attain close contact between the joint surfaces it is necessary
to preclude the corners of aligning surfaces from touching each
other. This can be achieved by providing a chamfer that clears the
fillet radius where the aligning shoulder meets the flange (Fig. 79a),
or it can also be accomplished by radial, face or diagonal undercuts
(Fig. 79 b-d).

The height H (see Fig. 79a) of centre-aligning steps can be taken
for conventional joints at 0.5 Y D {D — centring diameter).

5.2. Machining the End Faces of Fastening Holes

107

A usual mistake made when designing flanged connections is
that the aligning recess weakens the flange (Fig. 80a, region 1).

R)l

Fig. 79. Centring shoulder designs

The defect can be cured by making the flange thicker (Fig. 806) or
(if sizes and casting technology permit) by reducing the diameter
of the centring surface (Fig. 80c).

Flanges other than round are made flat, and positioned by fitted
pins. Non-critical detachable parts (covers, casings, etc.) are held
in place by fastening elements alone.

5.2. Machining the End Faces of Fastening Holes

When designing flanged connections the designer must anticipate
and specify in the drawing the methods by which the bearing surfa¬
ces under nuts and fastening bolt heads will be machined.

Cylindrical flanges may most simply be lathe-turned (Fig. 81a).
However, with cast components the lathe-turning adversely affects
the strength as it removes the hard surface crust and undercuts
the flange in the black surface transition region. It is also inadvisable
to lathe-turn flanges with jutting bosses (Fig. 816, c) as the tool
receives multiple repeated impacts from the bosses and quickly
becomes blunt making it difficult to obtain high standards of accu¬
racy and surface finish.

It is more reasonable to machine bosses by milling cutters
(Fig. 81d) or core drills, centre-guided by previously made pilot holes
(Fig. 81e). The most productive process is machining with a combi¬
nation tool that is a core drill combined with a twist drill (Fig. 81/).
In view of the inevitable dimensional inaccuracies encountered in

108

Chapter 5. Flanged Connections

moulding practice, the core drill diameter D is made larger than the
nominal boss diameter D 0 (on average D = 1.2Z> 0 ). This must be
allowed for when specifying joint dimensions.

When spot facing sunk bearing surfaces (Fig. 81g) the flange con¬
tour should be 3-4 mm distant from the extreme points of the machin-

Fig. 81. Machining methods for nut and bolt head bearing surfaces

ed surfaces. Otherwise the possibility of forming the easily broken
rags and burrs occurs.

If a core drill cannot be used directly on the work surface then
inversed core drilling is practised, where the core drill is mounted
upon an arbor passed through a previously drilled hole (Fig. 81 h).
Machining efficiency is sharply reduced with this method, so that
the latter is applicable in the event of holes not less than 10-12 mm
in diameter.

The end faces of holes in half-closed recesses are machined by
milling (Fig. 82a) or inversed core drilling (Fig. 82b). The height
and radius R of recesses, taken in cross-section, must be consistent
with the dimensions of cutting tools.

The modem highly accurate casting techniques (shell mould casting, chill
casting, pressure die casting, investment casting, etc.) obviate, in certain cases,
machining of support surfaces. However, for critical, heavy-loaded structures

5.2. Machining the End Faces of Fastening Holes

109

machining is obligatory with the assurance that the bearing surfaces are strictly
perpendicular to the hole axis, thus avoiding fastening bolt distortion and
flexure.

Table 6 gives the moulded flange design relationships for the
conventional range of bolt diameters (d — 8-20 mm).

The machining allowance t depends on the size and grade of accu¬
racy of the casting. On drawings which depict cast components

Fig. 82. Machining bearing surfaces in recesses

the value of t is generally omitted, which, nevertheless, in no way
implies that the designer may neglect this value when calculating
the dimensions of parts.

The minimum distance s between the machined and the nearest
black surfaces is established proceeding from the accuracy standards
of casting, dimensions of parts and distances of surfaces from the

110

Chapter 5. Flanged Connections

Relationships of Cast Flange Elements

Table &

Flange

elements

Mini¬
mum flange
height h
Mini¬
mum di¬
mensions of
machined
surfaces
(a, R, D/2)
Mini¬
mum bolt
axis distan¬
ce b from
machined
wall

Minimum
bolt axis
distance A
from flange
end face

1.5 d 1.2 d

1 . 2 d 1 . 2 d

1.3d 1.2 d

1.7 d 1.5 d

black and machined datum points. With respect to the small- and
medium-sized components (200-500 mm) obtained by conventional
sand casting s = 3-5 mm; these values can be reduced by 30-50%
by applying higher-accuracy casting methods.

5.3. Diameter and Spacing of Bolts

The proper choice of the diameter and spacing of bolts depends
on many factors, the most important of which are the operational
conditions, material of parts and rigidity of structures. The require¬
ments are quite different for connections subjected to small static
loads and connections undergoing high cyclic and dynamic loads,
working under pressure and where complete sealing is necessary.

For the simplest cases (flange-coupled connections, loaded with
moderate forces, free of internal pressure and elevated tempera¬
tures), the following approximations can be recommended.

5.4. Three-Flange Joints

111

The diameter of bolts fastening cylindrical flanges

d = 6 + (0.015 to 0.018) D

where D is the mean flange diameter.

The thickness of the flanges:

for components made of grey cast iron and light alloys
h = 6 + (0.022 to 0.025) D

i

h l

i— h

p!-^

j| —

u

The pitch of bolts

(a)

Fig. 83. Bolt spacing for flanges of different rigidity
for components made of steel or high-strength cast iron
h = 4 + (0.022 to 0.025) D j

l = ad

For small-sized non-rigid flanges (Fig. 83 a) a = 6-8; for mode¬
rately rigid flanges ^(Fig. 83b) a = 8-10; for flanges of higher
rigidity fastened with large bolts (Fig. 83c) a = 10-12.

The parameters of joints subjected to cyclic loads and operating
at elevated temperatures are obtained by calculation (see Chapter 1).

5.4. Three-Flange Joints

When designing housing-type components it is often necessary
to connect three flanges in one unit. Let us take, as an example,
a unit comprising an intermediate partition (membrane) installed
between the joint faces of two housings (Fig. 84).

The simplest method is tightening the membrane between the
flanges of the housings (Fig. 84a-e), the alignment being accomplished
from internal or external shoulders. The installation accuracy is
the highest in Fig. 84b, c and e (the alignment is effected from one
cylindrical surface).

The design presented in Fig. 84e is less strong than the others. To

112

Chapter 5. Flanged Connections

avoid interference the flange is usually assembled with an axial
clearance of 0.1-0.2 mm, owing to which proper membrane tighten¬
ing cannot be assured.

Should it be necessary to preserve integrity of mechanisms during
dismantling, then it is better to secure the membrane on one of the
housings (Fig. 84/-t). After the housings are separated from each

(a) (3) (C) (d) ie)

(t> (7”) (7J J 1°) (p)

Fig. 84. Three-flange connections

other the membrane remains secured on one of the housings, carrying
all the members mounted on it. Designs differ in alignment methods.
In terms of accuracy the most preferable versions are those shown
in Fig. 84 h, i.

Independent mounting of the membrane can also be implemented
by holding (pinching) the flange with various forms of pinch bolts
(Fig. 84 j-p), which may be shouldered (Fig. 84;'), with stop rings
(Fig. 84fc), tapers (Fig. 84 1) and nuts (Fig. 84m).

Occasionally the membrane is secured with the aid of tapped
bushes 1 (Fig. 84 n), into which the fastening bolts of the other hous¬
ing are screwed, or the membrane can be held to the housing with
independent bolts 2 , 3 (Fig. 84o, p) spaced between bolts 4, 5, which
hold the housings together. The heads of type 2 bolts are lodged in
the housing flange apertures, while those of type 3 are sunk in the
membrane.

5.5. Conical Flange Joints

113

5.5. Conical Flange Joints

For butt-jointing pipelines, cylindrical sections and loaded con¬
nections use is made of quick-acting split clamps which tighten
upon outer conical flange surfaces (Fig. 85). Parts may be connected
at any angle in the joint plane. When angles must be fixed or when
the connection transmits torque fitted pins are used.

Joined parts are aligned with the aid of cylindrical shoulders
(Fig. 85a). Sometimes the joint is made plain (Fig. 85 b) relying

upon the alignment given by the clamp’s conical surfaces. The latter
method is preferred when the parts to be joined cannot be brought
together axially and must be manoeuvred in the joint plane.

Conical flange joints ensure a good strong joint with a relatively moderate
tightening force being applied to the joint. Assuming the simplest transfer
of the tightening force P to two points (as in Fig. 85c, a rigid clamp), then the
axial tightening force will be

p - 2P

ax tan a/2

where a is the cone angle. For the usual a values varying from 20 to 30°,
Par = (8 to 10) P.

The cone angle a on the clamp collars is made 1-2° less than that
of the clamping flanges (Fig. 86a) so that the tightening force is
applied closer to the collar base with the aim of increasing joint
rigidity and sealing reliability. Similar results are obtained if
the clamp walls are made flat (Fig. 86 b).

Pictured in Fig. 86c-e are conical flange connections with sealing
gaskets.

8—01418

114

Chapter 5. Flanged Connections

Figure 86 f-h illustrates various designs of thin-walled tube conic¬
al flange connections.

In certain cases inverted conical flange connections are employed. The invert¬
ed conical flanges of the connected parts have through slots (Fig. 87a). During
assembly the projecting lug of one flange enters the slot in the other, thereby
forming between the flanges a conical cavity into which is placed the central
tightening collar (Fig. 876).

The binding clamp of a taper flange joint must open fully so that
it may be installed on the flanges from the side and axially and

Fig. 86. Conical flange connections

assure uniform circumferential flange tightening, i.e., be compliant
radially. The clamp must be quick-acting.

The clamp comprising two halves fastened together by bolts
(Fig. 88a) is not quick-acting. A more practicable design has its

Fig. 87. Inverted conical flange connection

halves linked with a pin and tightened by one bolt (Fig. 88b, c).

In the quick-action lock with a swing bolt (Fig. 88 d) radial slits
are provided in the hoop walls to enhance the compliance of the
system. A circlip-type lock is applied to fasten light joints (Fig. 88e).

In a load-carrying structure furnished with a quick-action lock
(Fig. 88/) the coupling bolt extends through a cylindrical nut 1

5.5. Conical Flange Joints

115

mounted in a half-open cup; the bolt end terminates in articulated
joint 2. As the bolt is being unscrewed, the nut leaves the cup and the
bolt can be freely swung away by rotating it about the joint axis.

In the design pictured in Fig. 88g the two halves are tightened
together by means of a swing lever with pressure screw 3. In the

Fig. 88. Collar clamps

version presented in Fig. 88k use is made of a swing-type three-link
mechanism (“frog”). In the latter case an elastic packing must be
inserted between the flanges, which would compensate for the rigi¬
dity inherent in such a mechanism in the locked position.

Figure 88i illustrates an elastic clamp made of a steel ribbon with
welded-on channel-sectioned segments 6. The coupling bolt extends
through pin 4 of the articulated joint and is screwed into the cylin¬
drical nut 5.

8 *

Chapter 6

General Principles
which Should Be Followed
when Designing Units
and Parts

6.1. Unification of Design Elements

When designing one should repeatedly use elements for all parts
of the design which in the assembly process reveal average rated
parameters and allow the parts’ list to be reduced to its mini¬
mum.

Subject to unification in the first instance are fitted connections
(their nominal sizes, type of fits and classes of accuracy), threads
(diameter, pitch and classes of accuracy), splined and keyed con¬
nections, fasteners, specifications, etc. It is also advisable to reduce
the number of material grades, unify surface finish classes, finishing
operations and electroplating techniques, welding methods, forms
of welded seams, etc.

Figure 89a-c illustrates the arrangement of a typical engineering
unit (a shaft carrying some elements and mounted in a bronze bush).
In the design presented in Fig. 89a the choice of fitting diameters
is badly thought over. The basic fitting size (diameter of the shaft
bearing journal), taken from standard specifications (<j> 50), is cor¬
rectly chosen, but from this point mistakes occur. Wishing to eco¬
nomize on the consumption of scarce bronze, the designer used a bush
wall thickness of 3.5 mm, thus obtaining a non-standard size for
the external diameter of the bush(<£ 57) and, moreover, while striving
to enhance the shaft strength at tbe fitted connection, he reduced the
shaft diameter relative to the journal diameter by 3.5 mm on each
side, thus obtaining again a non-standard diameter (<j> 43), which,
in turn, meant choosing an M42 screw thread size for the clamping
nut.

In the arrangement in Fig. 89 b, which is based on standard dimen¬
sions, the external diameter of the bush is 60 mm and the diameter
of the fitted connections 40 mm. Hence, the clamping nut thread
size is M37. However, standardization of the dimensions in this
case has lowered somewhat the shaft strength and also increased
the weight of the bronze bush. The most rational version depicted
in Fig. 89c has its shaft journal diameter equal to 55 mm, bush
O.D. 60 mm and the fitted connection diameter 45 mm.

M42

Comparative Table of the Back Gearing Dimensions

Table 7

Design element

Diameters
and fits

Design para¬
meter according
to Fig. 89d

jj|

Design para¬
meter according
to Fig. 89e

Qnty

Design element

Design para¬
meter according
to Fig. 89d

Qnty

Design para¬
meter according
to Fig. 89e

Qnty

SOP

1

Threads

M25

■

M30

2

35,4/.R

l

M37

H

40 A/S

1

35P

2

Keys

6

i

40A/P

1

b, mm

8

3

8

40,4//?

2

42,4/S

1

10

1

45 A/S

2

45 A/S

1

Tooth mo-

4

1

5QA/Dh

2

dule, m

5

1

5

1

45A/Dh

1

5QA/R

6

1

1

55A/Dh

1

Total

18

7

6.2. Unification of Parts

119

Naturally, other solutions are possible, but under any circumstan¬
ces it is important for the diameters to be standard. Another example
is given in- Fig. 89 d, e (back
gear).

The design in Fig. 89<2 has
significant irregularities in
the sizes of fitting diameters,
threads, keys and tooth mod¬
ules. In the rational version
shown in Fig. 89e the number
of fitting dimensions has been
reduced and the keys and to¬
oth modules unified. The ne¬
cessary tooth strength for the
small gears is obtained by the
increased face width. The uni¬
fication results are presented Fig. 90. Unification of wrench sizes
in Table 7. All-in-all the num¬
ber of machine elements is cut down from 18 to 7 items.

To illustrate the unification of wrench sizes, let us consider a
unit for adjusting a reduction valve (Fig. 90). In the design shown
in Fig. 90a three wrench sizes (1-3) are used, whereas the unified
design in Fig. 90b uses only a single wrench size (4).

6.2. Unification of Parts

Individually designed parts must be unified as much as possible.
This is particularly important for labour-consuming parts and
those in general use (gears, clutches, chain links, etc.).

Figure 91a shows a conveyer chain composed of links of two types.
The rational design (Fig. 91b) has unified links. The tightening
clamp illustrated in Fig. 91c consists of two labour-consuming com¬
ponents. Connection by an intermediate link (Fig. 91 d) enables the
clamp to be made of two similar halves. Figure 91e, / shows the uni¬
fication of pressed steel components in the unit of a built-up pulley,
while Fig. 91g, h shows unification embodied in the design of a
cylindrical pressed reservoir.

Often unification is only attained as a result of purposeful, ori¬
ginal, constructional working out of the design.

In an angle drive presented in Fig. 92a the required gear ratio
is ensured with the aid of two different shafts having different bevel
gears. To position the lower gears within preset spaces, one of the
gears is displaced relative to the other and the tooth face width of
the driving gear 1 made longer. As a result, the design contains five
gears (numbered 1 to 5) and two shafts (numbered 6 and 7).

6.3. Principle of Unitization

121 -

In the alternative, Fig. 926, the diameter of lower gears 2 is de¬
creased so that they can be driven by the one gear 1. To keep the
gear ratio the same the diameter of upper gears 3 is increased. As
a result of the conversion the required number of gears is reduced
to three (1 to 3) and of the shafts, to one (4).

Now, let us examine a reduction gear unit with two concentric
shafts running at the same speed in opposite directions (Fig. 93a).
The unit driving shaft carries two gears, one of which ( 1) engages
with gear 2 of the reduction unit, while the other (3) meshes via
idler 4 with gear 5. The total number of gear units is four {1-3, 2,
4, 5). The large number of components and complexity of the design
are caused by the necessity to avoid fouling between gears 3 and 5,
which means reducing the diameter of gear 3 and, accordingly, that
of gear 5 (in order to retain the same gear ratio).

In the unique solution (Fig. 936) the design is simplified as fol¬
lows: gear 1 of the drive shaft meshes on one side with the reduction
unit right-hand pinion and on the other side, with gear 2 of the
drive. The number of gears is thus cut down to two and the pinions
and gear wheels of the unit are identical pairs. To realize this design
it is sufficient to separate the gear wheels by a distance s sufficient
to engage the pinions.

6.3. Principle of Unitization

It is advisable to design units as self-contained assemblies, which
are individually assembled, adjusted, run-in, tested and finally
installed as completely ready-to-operate units upon a machine.
Such successive unitization allows parallel and independent assembly
of machine units, facilitates mounting, prompts prototype testing
and eases the employment on new machines of finished units which
were checked during manufacture. Furthermore, unitization simpli¬
fies repair as worn out units can be directly replaced by new ones.
Sometimes unitization may complicate the design, but in the long
run, it always gives large savings in the total machine manufactur¬
ing costs, reliability and convenience in use.

Some examples of the unitization of small assemblies are presented
in Fig. 94. In the design illustrated in Fig. 94a the reduction valve
is mounted directly in the housing body. When the valve is assembled
in a separate bush the design becomes unitized (Fig. 946).

The end face seal depicted in Fig. 94c is unsatisfactory because
during dismantling the sealing disk 1 is forced by the spring to
leave the ways and slots which prevent its rotation, and the unit
disintegrates. Equally unsuitable is the seal assembly. The intro¬
duction of locking ring 2 unitizes the construction (Fig. 94 d).

Another example of unitization is offered in Fig. 94 e-g where the
unit of a distributing slide valve is shown progressively improved^

122

Chapter 6. General Design Principles

The construction given in Fig. 94e is grossly wrong: an accurate
hole intended to receive the valve was made directly in the frame
-casting. At the spot where the valve is located (i.e., in the concentra¬
tion of material) there may occur pits and voids which make valve
sealing impossible. Moreover, as the hole wears out during use it
•can only be repaired by introducing bushes.

The first step to an improved design is to mount the valve in an
intermediate bush (Fig. 94/) made of a high-grade material with a

Fig. 94. Examples of unitization

good wear resistance. In the most rationally designed version
(Fig. 94 g) the component has been fully unitized. Here the unit is
made separately; it is assembled, ground-in, verified for geometry
and attached to the frame on a milled surface.

Shown in Fig. 95a is a worm reducer which is coupled directly
with the machine driving shaft. The worm wheel shaft is mounted
in supports installed in different housings. To maintain support
alignment when machining is difficult. The assembly procedure is

6.3. Principle of Unitization

123

very ambiguous as it is necessary to fit the worm wheel first onto
the main shaft, install the reducer housing and then assemble the
worm by screwing it into the worm wheel teeth. To verify the cor-

Fig. 95. Unitization of gear transmissions

rectness of engagement and align the worm wheel under these cir¬
cumstances is very difficult.

In the unitized construction (Fig. 95b) the worm wheel shaft is
mounted in two supports, one of which is accommodated inside the
housing and the other in the housing cover. Both supports can be
machined together, thus obtaining good alignment. The shaft end

124

Chapter 6. General Design Principles

is coupled with the driving shaft by the splined adapter 1. Hence,
the reducer assembly is considerably simplified.

The gearing mounted in a frame (Fig. 95c), in addition to the draw¬
backs inherent in the non-unitized constructions, presents diffi¬
culty in introducing the idle gear shaft into the support housing.
As soon as the housing is removed from frame, it loses both supports
and falls to pieces. To verify the correctness of meshing and align¬
ment of the shaft and supports is impossible.

In unitized constructions the gear supports are mounted in diaph¬
ragm 2 (Fig. 95 d), or in bracket 3 (Fig. 95e) screwed by its feet to
the gearing casing. With the latter design it is easy to inspect the
mechanism during assembly.

Figure 95/, g shows the unitization of a gear drive mounted on a
frame; Fig. 95 h, i, a reduction unit driven from an engine crankshaft.
In the unitized version (Fig. 95i) the reduction unit is mounted in a
separate housing; the shaft torque is transmitted by means of adap¬
ter 4.

6.4. E limination of Adjustments

Fitting and adjustment of units and parts when in position must
be eliminated. Adjustments, particularly when accompanied by
machining or fitting operations, lower assembly productivity and
deprives the construction of interchangeability.

An example of fitting in situ is shown in Fig. 96a, b. The gear is
placed in position on the shaft so that it meshes with its mating
gear. This done, the position is fixed by a self-piercing screw (Fig. 96a)
or by a taper pin (Fig. 966), which requires drilling and reaming
in situ. Chips inevitably fall into the unit. After the machining it
is necessary to dismantle the unit, flush it and then reassemble.
Marking during assembly with a subsequent transfer to machine
tools further complicates assembly.

A better engineering method is to lock the gears in position by
ring-shaped stop washers which are placed into grooves previously
machined on the shaft (Fig. 96c).

If a bearing has been installed in a housing by the “in situ” tech¬
nique (Fig. 96 d), its originally correct position will be disturbed
every time the bearing is dismantled, thus necessitating readjust¬
ment. Fixing the bearing by dowel pins (Fig. 96e) entails machining
on assembly. The correct solution is displayed in Fig. 96/, where the
bearing is aligned against the bore in the housing, this bore being
machined to the required accuracy, which ensures correct functioning
of the unit.

If a straight machine way is mounted on a bed as illustrated in
Fig. 96g, then its position must be verified on assembly and holes
drilled to suit fastening bolts. In this case the way will not be gua¬
ranteed against offset within the clearance values existing between

126

Chapter 6. General Design Principles

the bolts and their holes. Fixing by dowel pins (Fig. 96 h) means drill¬
ing and reaming holes in the way and bed. In the rational design
(Fig. 96i) the way is mounted in groove machined in the bed.

The gear drive illustrated in Fig. 97a is unsatisfactory. The gear
supports are bolted to the housing. The fitter is compelled to adjust
the support position so as to assure correct gear meshing. Dismantl¬
ing will inevitably disturb the arrangement, thereby necessitating
readjustment on assembly. The supports can be fixed in position
by dowel pins (Fig. 976), but this will involve additional job opera¬
tions in assembly.

In the correct design (Fig. 97c) the gear studs have been aligned
against holes whose spacing relative to each other has been maintai¬
ned within close tolerances during machining of the housing. In
the best version (Fig. 97 d) all the gears are enclosed inside a common
housing, which brings about full unitization and advantageous ope¬
rating conditions.

Figure 97e gives a wrong, and Fig. 97/, a correctly designed ver¬
sion of a back gearing with a V-belt drive.

6.5. Rationalization of Power Schemes

The perfection of a structure, its weight, size and, to a significant
degree, its capability of work are dependent on the rationality of
its power scheme. A power scheme is rational if all acting forces are
mutually equalized over as short a section as possible with the help
of elements loaded mostly in tension, compression or torsion (but
not flexure).

As an example let us examine a screw-type conveyer (Fig. 98 a)
driven from an electric motor via a worm-type reduction unit 1
and chain transmission 2. The conveyer housing, which is several
metres long, is made of steel sheet and rests upon four tubular legs.
The main error consists in the fact that the housing is loaded with
the drive force (the direction of force is shown by an arrow), which
bends and deforms the non-rigid housing, mounted on unsteady sup¬
ports. Owing to a small internal clearance between the conveyer
screw and housing the screw edges scrape the inner walls. The inten¬
sified friction increases the driving torque which, in turn, increases
the bending force still more and, consequently, the friction and
finally the screw jams in the housing.

The deffect can partially be eliminated by changing screw rota¬
tion with a corresponding change in screw helix. The driving side
of the chain drive is now the lower side of the chain drive and the
torque moment acting upon the housing is substantially reduced.
Alternatively, it is possible to move the reducer into the symmetrical
plane of the entire assembly, redistribute the legs, strengthen the
housing and mount it upon a stiff foundation. Nevertheless, all

6.5. Rationalization of Power Schemes

127

these measures fail to eliminate the principal defect, namely, the
presence of external forces in the system.

A radical solution is presented in Fig. 98ft. Here the screw is driven
from a flange-mounted electric motor via a reduction unit installed
coaxially on the housing end face. The drive torque and reactance?

Fig. 98. Improvement of power scheme of screw-type conveyer

torque on the housing quench each other at the connection joint.
The housing is absolutely free from the action of external forces and
deformation is eliminated.

In the drive of a suspended conveyer (Fig. 99a) comprising a reduc¬
tion unit 1, bevel gearing 2 and spur gears 3, which impart motion
to the driving sprocket 4 of a chain drive, the load-carrying elements
are irrationally arranged. Indeed, the support units of the drive,
the fastening bolts and foundations are all loaded with drive forces
and a large number of elements are in flexure. The drive units are

128

Chapter 6. General Design Principles

separated; they are installed on individual bases and are not fixed
one relative to the other. Such a design only ensures satisfactory
functioning after the mechanisms have been carefully aligned.

Furthermore, the cast iron gears are not protected from fouling
■and can only be periodically lubricated by packed grease. The

Pig. 99. Improvement of power scheme of suspension-type conveyer drive

supports of the horizontal and vertical shafts are also lubricated
periodically.

The entire unit is excessively large, explained by the isolation of
units and also by the employment of a low-strength material (cast
iron) for manufacture of the most critical components (gears).

In fact, the design is typical of old design practice.

In the modern unitized enclosed construction with a rational power
arrangement (Fig. 99 b) the drive is from a flange-mounted electric
motor, installed vertically, which obviates the necessity for the
angular transmission. The required gear ratio is obtained in one co¬
axially mounted reduction unit. The employment of gears made of
a high-grade and appropriately heat-treated steel in conjunction
with the application of a centralized oiling system, sharply reduces
overall sizes. The drive forces are absorbed in one casing. The casing
and base are loaded only by the force of the final driving sprocket.
These changes result in tremendous savings in the weight and size of
the installation, simplicity of manufacture, convenience in mounting
and attendance, reliability and a long service life.

6.6. Compensators

129

6.6. Compensators

For multistation machine systems having mechanical drives the
design of couplings which transmit torque is of extreme importance.
Such a coupling must compensate for lengthwise displacements,

(a) (b) (c)

Fig. 100. Misalignments in coaxial connections

parallel misalignment e and conical misalignment a of the units
being connected (Fig. 100).

For this purpose involute splined connections are most commonly
employed (Fig. 101). The advantages of involute teeth, are as fol¬
lows:

due to the thickening of
the tooth towards the root
(particularly with positive
correction) the tooth posses¬
ses increased strength and
stress concentration at the
root is not great;

involute teeth, both
external and internal (with
a sufficiently large rim dia¬
meter), can be machined to
close tolerances on conven¬
tional gear cutting machi¬
nes.

External involute teeth Fig. 101. Compensators with involute teeth
can be given high surface

hardness by heat and chemical-heat treatments, followed by fi¬
nish machining on conventional gear-grinders.

The operating conditions for teeth in compensating connections
are much more harder than for the centring splined connections. The
enhance their compensation ability such connections are made with
an increased circular clearance s = (0.05 to 0.07) m, where m is
the tooth module. In the case of a misalignment the forces are con-

9-01418

130

Chapter 6. General Design Principles

centrated at the tooth extreme edges lying in a plane perpendicular
to the misalignment. As a result the linear contact over the tooth
face changes to point contact, thus sharply increasing local bear¬
ing stresses. Since each tooth passes through the zone of loading twice
per revolution, the load upon the teeth will be cyclic, regardless
of torque behaviour.

The working capacity of the connection can materially be inten¬
sified by enhancing surface hardness of teeth. To eliminate work har¬
dening and promote removal of heat, produced during tooth impacts
and warpage, ample lubrication is obligatory. The most effective

Fig. 102. Determination of maximum conical misalignment in a compensator

method of obtaining an efficient connection is to increase the gear
rim diameter since this offers certain advantages in manufacture,
namely, the internal teeth can be gear-cut in place of using expensive
broaches.

The amount of misalignment admissible in a coupling is limited,
first of all, by the tooth edge contact in a plane perpendicular to
the misalignment direction (Fig. 102a). The teeth positioned in the
misalignment plane have much displacement' freedom as the clear¬
ance in the radial direction with a standard pressure angle (20°) is
approximately three times greater than the circular clearance.

The maximum possible misalignment angle a may be determined
from the relation

tan a = ~f

where s = circular clearance in teeth
l — tooth face width

The maximum displacement of the compensator’s extreme points

S = L tana^- (6-1)

where L is the compensator length.

6.6. Compensators

131

Compensation can be improved by decreasing the tooth face width,
obtained without weakening the tooth by increasing the gear rim
diameter.

The circumferential force acting upon the splined rim

n_

D

( 6 . 2 )

where Mtg e = torque transmitted

D = mean diameter of the splined rim
For small displacements the tooth strength is determined from the bearing
stress on the spline faces

Gbear
where l

P

Izh

P

Izam

(6.3)

= spline width
2 = number of splines
h = am = working spline

height, proportional
to the spline
tooth module
a = constant
Since z = Dim, then

P 2M t „

Dla

Gbear

whence
l =

2 M

tge

D2la

const

D 2 Obea

D2

(6.4)

The spline length l [Eq. (6.4)1
and maximum possible displa¬
cement S of the compensator’s
extreme points [Eq. (6.1)] as a
function of diameter D are
shown graphically in Fig. 103
(s and l when D = D t are as¬
sumed to be equal to unity).

Fig. 103. Spline length l and maximum
displacement S vs. spline diameter

To decrease the load upon the tooth edges and increase the permis¬
sible misalignment angle, it is good practice to give a barrel-like
shape to the teeth. The tooth end face edges should be rounded along
the entire tooth contour. For large conical misalignments the tooth
faces and roots should be made spherical (see Fig. 1026).

Illustrated in Fig. 104 is an intermediate compensating bush
which permits large free displacements and in its general scheme
resembles a cardan joint. The bush has incomplete internal tooth
rims; the toothed portions of both rims are at right angles. The disks
with external teeth have complete gear rims, which ensures correct
assembly at any angular attitude of the flange relative to the inter¬
mediate part.

9 *

132

Chapter 6. General Design Principles

Some of the design alternatives of compensating connections are
presented in Fig. 105. Connection by means of splines cut directly
on drive shafts (Fig. 105a) is unreasonable, because the compensa¬
tion ability of such a joint is
rather low and it is determi¬
ned only by the amount of
spline displacement in the
gap between spline edges. In¬
creasing the drive shaft shank
length (Fig. 1056) only worsens
the position since the spline d
shank owing to inevitable
manufacturing inaccuracies
and assembly is apt to runout
proportional to the distance
at which the end is spaced
from the drive shaft supports.

If a splined reducing
bush is freely mounted upon
the splines of shafts and acts as an adapter (Fig. 105c), then the
compensating ability, determined by the total clearance in the
splines will be twice that of the design given in Fig. 105a.

Fig. 104. Cardan-type compensator

The best construction has the driving ends of the splined bush
spaced apart longitudinally (Fig. 105 d). Compensation in this instan¬
ce is improved due to the displacement of the bush itself. Still better
is the case when a longer bush is used (Fig. 105e). However, this
alternative is more difficult to produce because of the different
splined rims on the bush ends.

The best design has a long splined shaft, a torsion bar (Fig. 105/),
having high compensation characteristics.

6.7. Torsion Bars

133

6.7. Torsion Bars

Torsion bars (torsion bar springs) not only compensate for mis¬
alignments, they also damp torque-induced vibrations, thus contri¬
buting to quieter and smoother operation. This advantage is par¬
ticularly important in machines operating under pulsating torsional
loads (piston-operated machines).

Thanks to their small radial dimensions, torsion bars can readily

Fig. 106. Torque transmission by means of torsion bars

be included in shaft interiors, thereby adding to the compactness
of design (Fig. 106a).

To avoid the breakage of torsion bars by overloading, provision
is occasionally made of a torque limiter. The latter is generally made
in the form of a splined bush mounted concentrically with the tor¬
sion bar (Fig. 106b). The side clearance in the torque limiter splines
exceeds that of the torsion bar splines so that the limiter takes up
the load as soon as a certain angle of torsion is reached.

General-purpose torsion bars are made of silicon spring steels, grades 60C2A,
70C3A or 60C2XA, which, when subjected to appropriate heat treatment (harden¬
ing plus moderate tempering), possess a fatigue limit t 0 = 65-70 kgf/mm*
under pulsating torsional loads and x_ x = 30-35 kgf/mm 2 under symmetrical
alternating torsional loads.

134

Chapter 6. General Design Principles

For stressed structures, as well as for structures operating under elevated
temperatures, use is made of silicon-nickel, silicon-tungsten and silicon-vana¬
dium steel grades 60C2H2A, 65C2BA and 60C2XOA, for which t 0 = 80-
90 kgf/mm 2 and t-j = 40-50 kgf/mm 2 .

The elastic torsion of torsion bars can be enhanced by increasing the design
stress level. For pulsating cycles t = 30-40 kgf/mm 2 is generally adopted,
which corresponds (in terms of fatigue limits) to a safety margin of the order
of 1.5-2. In structures designed for a limited service life the admissible stresses
can be increased to 80-100 kgf/mm 2 .

The fatigue strength of torsion bars can materially be improved by means of
plastic strain hardening. For instance, torsion bars intended for operation under
alternate cyclic loading, are hardened by shot blasting, while torsion bars,
designed for pulsating loads are hardened through prestraining (i.e., applying
a static moment of the same direction as the operating one, the stress level
being 20-40% in excess of the material yield point). The shot blast technique
and prestraining procedures enable torsion bar service life to be approximately
doubled. The best results are obtained by means of strain hardening (shot blast¬
ing with prestrain) which further increases service life by 20-30%.

Torsion bar splines are strengthened by rolling in the axial direction with
rolls profiled to suit the spline tooth space. Involute splines are strengthened
by rolling against a hardened sizing gear.

The torsional stress in a torsion bar

i = kgf/mm 2 (6.5)

vv tor8

where M tqe = transmitted torque

1 Ytors — 0.2 d 3 (1—a 4 ) = torsion resisting moment of the tor¬
sion bar section

a = d 0 /d = ratio between the I.D. and O.D. of
the bar section (for solid torsion bars
6 a = 0)

The torsion bar twist angle is

<P

Mfq e l

GItOT8

[rad]

360°

2jt

Mfq e l

Glfors

Mt ae l

[deg] = 57.3 -Qp— [deg]

tors

( 6 . 6 )

where G — modulus of torsional shear (for steels

G = 8000 kgf/mm 2 )
l = acting torsion bar length, mm
I tort = 0.1 d* (1 — a 4 ) = polar moment of inertia of the torsion

bar section

The twist angle expressed as a function of stress t

9-44* [rad] = 57.3- J4 Z [deg] (6.7)

Let a solid torsion bar transmit power N = 100 hp at a speed n = 2000 rpm.
The acting torsion bar length l = 200 mm.

The torque

M tq ,

75 N
(0

75-100 75-100

0.104re 210

= 36 kgf-m = 36-10 3 kgf-mm

6.7. Torsion Bars

135

Assume that the design stress t = 40 kgf/mm 2 . According to Eq. (6.5),
the diameter of the torsion bar will be

The twist angle

= 1 = 57 ^--^=-200 = 0.118 rad = 57.3-0.118 deg = 6°45'

Cr d oOUU 17

Assume that the torsion bar drives a gear with pitch circle diameter D 0 =
= 200 mm. The elastic displacement of the gear rim under load at the pitch
circle radius i? 0 = 100 mm is / = i? 0 q) [rad] = 100-0.118 * 12 mm. Such
a result can hardly be obtained with other absorbing elements (e.g., an elastic
clutch with coiled springs placed between the drive shaft and gear rim).

Some typical torsion bar design alternatives are shown in Fig. 107.
For transmitting low torques the simplified design version 1 with

Fig. 107. Torsion bars

squared shanks is used. Much more sturdy are torsion bars which
have involute splines (design examples 2 to 10). Version 2 is unac¬
ceptable because of the different strength properties offered by the

136

Chapter 6. General Design Principles

stem and splines. Versions 3 to 6 are better designs due to the increa¬
sed spline diameter. The larger the spline diameter (designs 7, 8
and 9), the smaller the spline load and the better the torsion bar
compensator. For a given compensation value the increase of dia¬
meter enables spline clearance to be decreased, thus improving
spline working conditions and prolonging service life.

To reduce weight, torsion bars are generally made tubular (designs
10 and 11). Cross-shaped torsion bars (12) have better elasticity.

Lengthwise fixing of torsion bars is most frequently accomplished
with ring-shaped stop washers (Fig. 107b).

6.8. Floating Cross-Sliding Couplings

If considerable misalignment is involved it is good practice to
employ compensating floating cross-sliding couplings (Oldham’s
couplings). A typical example of the kind is illustrated in Fig. 108a.
Each of the two connecting flanges has two jaws milled on its face,
which enter into slots in the intermediate plate 1. The flanges can

Fig. 108. Floating cross-sliding couplings

freely move in the plate slots one relative to the other in any radial
direction. The plate continuously moves in the plane, normal to
the shafts axes and slides on the side faces of the jaws. For this rea¬
son the plate is made of some antifriction material (cast iron, bronze
or, for light loads, from plastics) or, at least, is provided with anti¬
friction shims.

The joint loadability can be increased if the drive elements are
made of hardened steel and if an abundant flow of lubricant is
brought to the rubbing surfaces. The floating element can be made
in the form of a star-shaped steel plate with hardened working sur¬
faces (Fig. 108b), or manufactured as a set of selfaligning cylindrical

6.9. Elimination and Reduction of Bending

137

insert pieces (Fig. 108c), which ensure uniform load distribution on
the working surfaces.

The compensator, designed to transmit huge torques (Fig. 108d)
is a combination of a floating cross-sliding coupling and a cardan
joint. The principal component is the crosspiece 2 mounted on needle
bearings in the yokes of the driving and driven shafts. The misalign¬
ment of the shafts is compensated for by the displacement of the
yokes along the trunnions, and angular displacements, by the yokes
turning about the trunnion axes; the axial inaccuracies of the shafts
are rectified by shifting the shafts along the yoke splines during
assembly.

6.9. Elimination and Reduction of Bending

Whenever the design permits, bending must be changed to other
more preferable kinds of loading—tension, compression or shear.
Rod, bar or other similar structures, whose elements work mainly
in tension or compression, are preferred.

If bending stress is unavoidable, then the moment arm of the
bending force must be decreased and the resisting moment of criti¬
cal sections increased. This is especially important for cantilevered
loads which are most unfavourable from the standpoint of strength
and rigidity.

Some practical suggestions aimed at eliminating or, at least,
reducing bending stresses are presented in Fig. 109. The arms of
an angular lever (Fig. 109a) are subjected to bending from the forces
applied at the extreme points. As obvious from Fig. 109&, the rib
introduced between the lever ends eliminates bending completely.

If a roller unit is secured to the frame in a way illustrated in
Fig. 109c, the fastening lug will be subjected to bending. The design
with stiffening ribs (Fig. 109<2) is somewhat better. However, the
best way is to mount the roller under the frame wall, which in this
instance is subjected to compression (Fig. 109e).

In the unit of a roller tappet (Fig. 109/) the stem acts like a can¬
tilever; the forces acting upon the roller bend the tappet, causing
also increased guideway support reactions. Finally, the slot in the
tappet stem is loaded by a moment which tends to rotate the tappet
about its axis.

The operating conditions are improved by mounting the roller
centrally (Fig. 109g).

The design of the roller tappet drive unit (Fig. 109fe) is bad, since
the cantilevered tappet guide sleeve is bent excessively by the drive
cam. Fastening the sleeve end in the frame eliminates bending

(Fig. 109i)-

The successive improvements to the welded joint of a lug and pipe
is illustrated in Fig. 109/-Z. The design in Fig. 109/ is irrational: the
lug overhang is excessive and the whole joint is subjected to bending*

-138

Chapter 6. General Design Principles

made longer and is now mostly in shear. In the still sturdier design
(Fig. 109Z) the lug is cut into the pipe.

Pictured in Fig. 109m, n is a cross-sliding coupling comprising
drive plate 1 and driven plate 2 interlinked with a plastic floating
plate 3.

6.9. Elimination and Reduction of Bending

139

In the version given in Fig. 109 to the radial lugs of the intermediate
plate are located in parts inside the slots between the drive jaws
and inside the slots of the driven plate. The driving forces (indicated
by black arrows) and reactive forces on the driven plate (indicated
by clear arrrows) tend to bend the lugs of the intermediate plate.

In the inversed scheme (Fig. 109re) the lugs are changed to slots
into which the jaws of the driving and driven plates enter in pairs.
The design has a more favourable force distribution. Furthermore,
the sections between slots are subjected mainly to compression and
the cylindrical portion, to tension, from the resulting drive and
reactive forces.

In the step ball bearing (Fig. 109o) the circular flange undergoes
severe bending from the working load. In the improved version
(Fig. 109p) the flange is ribbed for rigidity. In the best alternative
(Fig. 109<?) the force is transferred directly onto the housing walls
which in this case work in compression.

In the connecting rod yoke joint (Fig. 109r) the pin and yokes are
subjected to bending.

In the design shown in Fig. 109s the pin is passed through a comb
of spurs, thus eliminating bending for shear across several planes.

Figure 109 t-y depicts some methods by which a flanged connection
loaded in tension can successively be improved. The design in
Fig. 1092 is poor due to the excessive fastening bolt overhang from
the walls of the mating parts. Bending at the joint face can be redu¬
ced by decreasing the bolt overhang to its minimum permitted by
the bolt head sizes and their facing requirements (Fig. 109u). Still
greater sturdiness can be achieved by introducing stiffening ribs
(Fig. 109y, w) and bringing the wall surfaces closer to the bolts
axes (Fig. 109a:, y).

A source of bending stresses, which often escapes the designer’s
attention in practice, is also the curvilinear shape of parts subjected
to tension or compression.

For example, let us consider curvilinearly profiled ribs (Fig. 110a).
Elongation bends the ribs which is accompanied by high tensile
stresses along the rib edges. Straightening the profile (Fig. 1106)
and particularly positioning the ribs along the acting force line
(Fig. 110c) lessen these internal stresses.

Tensile or compressive forces applied to parts with eyes (Fig. llOd)
bend and deform the walls. The defect can be rectified by designing
streamlined transitions (Fig. llOe). The eyes can be strengthened
with stiffening ribs (Fig. 110/) or webs (Fig. llOg).

A lifting hook (Fig. 1106) undergoes bending from the moment
PI (l = arm of force P due to the lifted load relative to the neutral
axis of the critical section A). In a two-horn hook (Fig. llOi) the
bending moments on both sides of the hook will equalize each other;
hence, the hook stem is in tension. The bending moment acting

140

Chapter 6. General Design Principles

in the critical section B decreases to PI'12 (V = arm of force). With
the relations like those in the Figure, the bending moment will
drop by 211V » 5 times as compared with the original design.

In the case of tensile-compressive loading the bending is often
caused by an off-centre load application.

Fig. 110. Elimination of bending

In a high-speed rotor with a disk displaced from the axis of sym¬
metry of the rim (Fig. 110/) the centrifugal forces tensioning the rim,
cause spatial bending of the disk. In the design with a conical disk
(Fig. 1101c) which connects on the symmetry axis of the rim the
bending will be notably less. The best design has a central disk
(Fig. 1101) which works in tension.

6.9. Elimination and Reduction of Bending

141

If a counterweight is secured to a crank web as shown in Fig. 110m,
the web will be subjected to bending due to the off-centre position
of the counterweight and the rivets will also work partially in bend¬
ing. In the designs shown in Fig. llOn the bending has been elimi¬
nated by securing the counterweight in a fork. The rivets now work
primarily in shear and the number of shear planes double that of
the previous designs.

If conveyer chain links are joined as in Fig. 110o, they will bend
when the chain is tensed because of the asymmetrical arrangement
of the eyes. Positioning the eyes symmetrically eliminates the
defect (Fig. llOp).

Bending may also be caused by local form irregularities in parts
at the spots to which forces are applied. Figure 111 illustrates bolt

Fig. 111. Elimination of off-centre force application

heads. The asymmetry of the bearing surfaces of the bolt heads
(Fig. 111a, b) and their non-uniform rigidity (Fig. 111c, /) cause
the bolt stems to bend when in tension. Eliminating the asymmetry
assures central force application (Fig. lllg, h).

The successive improvement in a connecting rod and piston unit
is illustrated in Fig. 112. In the version shown in Fig. 112a the pis¬
ton crown, bosses and pin are subjected to bending under the gas
pressure. The bending is sharply reduced if the bosses are
fixed to the crown with ribs (Fig. 1126) or solid webs (Fig. 112c).
The rigidity and strength of the crown are enhanced by shaping
it as a concave sphere (Fig. 112c).

Tapering the connecting rod small end (Fig. 112 d) minimizes
the area of non-supported surface of the crown and will simultaneous¬
ly decrease the bending of the wrist pin. Furthermore, this lowers
the gas pressure specific load on the operating surfaces of wrist pin.

Finally, the piston crown and wrist pin can be completely relieved
from bending if the underneath side of the crown rests directly against
the connecting rod small end (Fig. 112e) or upon the pin through a
cut-out in the connecting rod end (Fig. 112/).

Figure 113a, b shows irrational designs of a suspended frame bra¬
cket loaded by a tensile force. Due to the curvilinear shape bending
stresses occur in the frame rods. Bending can partially be reduced

(d) (e) (f)

Fig. 112. Elimination of bending in connecting-rod-to-piston joint

Fig. 113. Elimination of bending in suspended frame bracket

6.10. Elimination, of Deformation Due to Tightening

145

by introducing a stiffening web (Fig. 113c) or it may be completely
eliminated by introducing a solid web between the rods (Fig. 113d).
The last design, however, loses the weight advantage.

If for some reasons the frame-like structure is preserved, it is
better to employ straight rods (Fig. 113e). In this case the system
will be similar to a truss frame. Bending (of a secondary order) will
occur only as a result of the positive imbedment of the rods in the
joint (in ideal girder structures the bending of rods is prevented by
articulated joints). In the most preferable design (Fig. 113/) the load
is taken up by the reinforcing central rod subjected to tension.
The side rods enhance the stability of the system in the transverse-
direction.

6.10. Elimination of Deformation Due to Tightening

When structural parts are tightened, appropriate measures must
be taken to avoid their deformation. Fastening with studs and bolts
in a “hanged-up” state (Fig. 114a) entails deflection of the walls and
overstresses the material. Studs and bolts passing through hollow

Fig. 114. Elimination of deformation during tightening

components must be enclosed in rigid columns (Fig. 1146). Occasion¬
ally reinforcing the assembled walls with ribs will suffice (Fig. 114c),
provided they are close to the fastening element.

The bearing cap design shown in Fig. 114c? is wrong as the bolts
during tightening will deform the cap and badly affect the true
cylindrical form of the bearing. Moreover, bending stresses will

144

Chapter 6. General Design Principles

occur in the bolts. In the design presented in Fig. 114e the cap is
relieved from the tightening forces.

Figure 114/ shows an incorrect and Fig. 114g, a correct fastening
of a connecting rod cap.

If a pin is fastened as in Fig. 114ft, the end eye lugs will be subjec¬
ted to bending. Fastening the pin to one lug with a dowel (Fig. 114i)
alleviates the system from internal stresses, but deprives it of rigi¬
dity. Reinforcing the eye lugs with a spacer bush (Fig. 114/) makes
it necessary to accurately keep to the spacer length and lug span
width, thus complicating the manufacture. It is more correct to
tighten the pin in one lug (Fig. 114ft), leaving the other end of the
pin free for self-centring.

Mounting a roller in a yoke (Fig. 114 l) brings the fork lugs tow¬
ards each other as the bolt is being tightened and the roller loses its
mobility. The introduction of a spacer bush (Fig. 114m) improves
the situation but complicates manufacture. For the most correct alter¬
native (Fig. 114 m) the pin is fixed in one lug only. The necessary
end play enabling the roller to rotate freely is ensured by longitudinal
•clearance between the roller and lugs.

Figure 114o illustrates a wrong and Fig. 114p, a correct tightening
of flanges.

Fastening of a connecting rod in a yoke (Fig. 114 q) requires accura¬
te machining or exact fitting of the mating surfaces on both parts.
Otherwise, either the yoke will be spread out by the rod, or the yoke
will be bent as the bolts are tightened. In an improved design (Fig.
114r) the lower bolt is changed for a dowel pin working in shear. The
eyes are bent (if clearance is present in the joint) only as a result of
tightening the upper bolt, which, however, in this case will induce
less stresses than in the version shown in Fig. 114g. In the design
pictured in Fig. 114s the parts are fastened by means of dowel pins
and are free of bending stresses. Yet, the joint lacks the capability
of being tightened.

In terms of averaged indices of strength and rigidity the best alter¬
native seems to be that in Fig. 114r.

Figure 114* depicts a badly fixed bearing bush. The locking screw
is driven to stop against the bush, thus deforming it (shown by
dashed line). In a correct design version (Fig. 114u) the screw head
rests against the bearing body, with a clearance ft being left between
the thread and the bush.

An example of an erroneous tightening of a thrust washer juxtapo¬
sed onto a plain bearing end face is given in Fig. 115a. The shaft
shoulder is insufficiently high; in the course of tihgtening the was¬
her is deformed by force P. To preserve the washer flatness the shoul¬
der height must be increased and the diameter of the nut, decreased
(Fig. 115ft), or the washer must be given a stiff collar (Fig. 115c).

The defect of the gear fitted on aligning tapers (Fig. 115 d) is that

6.10. Elimination of Deformation Due to Tightening _ 145

the tapers are positioned under the teeth which deform as the assem¬
bly is tightened (shown by dashed line). In the correct design (Fig.
115e) the tapers are brought beyond the gear rim boundaries.

The design of a sealing unit with split spring rings (Fig. 115/)
is bad. During tightening the housing crests are deformed and the

Fig. 115. Elimination of deformation during tightening

rings lose their mobility. Correct designs, free of rings seizure, are
illustrated in Fig. 115g-t.

Occasionally a limited deformation is introduced in order to enhance the
rigidity and stability of assembly. For example, when securing a column in
a pedestal (Fig. 115;') a clearance h is left between the column flange and bearing
surface. This clearance is taken up in the process of tightening. The'amount of
such a clearance is chosen either through calculations or experimentally so as
to keep the stresses in the flange within safe limits.

Similar results can be obtained by making the flange bearing surface slightly
tapered (Fig. 115fc).

Sketched in Fig. 115Z is an assembly of a guide-vane unit and an impeller
, of a centrifugal pump. The vane ends are machined to a taper and they intima¬
tely close the impeller blades in tightening (Fig. 115m), thereby precluding
blade vibration when running.

10-01418

146

Chapter 6. General Design Principles

6.11. Design Compactness

Compactness is a feature of a rationally designed component.
Efficient utilization of space enables the overall size and weight
to be decreased and the strength-to-weight ratio, increased.

Axial dimensions may often be reduced by spreading the structure
in the radial direction. In the unit of a mechanical face seal as shown

Fig. 116. Reduction of overall dimensions

in Fig. 116a bush 1 is forced by the spring to the sealing disk 2\
now, if the spring is installed over the bush (Fig. 1166) the entire
system becomes more compact without any adverse effect to the para¬
meters determining the efficiency of the unit.

An example of dimensional reduction through structural incorpo¬
ration of elements into parts is shown in Fig. 116c, d which pictures
a step ball bearing unit.

147

_ 6.11. Design Compactness

In the spherical connection of a pipe unit (Fig. 116e) considerable
reduction in dimensions is achieved by changing one of the external
spherical surfaces to an internal spherical surface (Fig. 116/).

In the unit of a shaft end mounting loaded with alternating radial
and axial forces (Fig. 116g) the axial load is taken up by two single¬
row thrust bearings. The design is cumbersome. The shaft fixing
in the lengthwise direction is not accurate: the thrust bearings spa¬
ced at a fair distance from each other must be installed with an
axial clearance which compensates for temperature deformations;
also axial backlash is unavoidable in this system.

In the design presented in Fig. 116ft the axial load is taken up by
a double-row thrust bearing installed between radial supports.
The overall dimensions are decreased and the axial play minimized.
Preserving the same dimensions as in Fig. 116ft, one may increase
the radial spacing of the supports 1.5 times, thus beneficially influenc¬
ing the shaft stability.

Figure 116i-ft illustrates the design and operational improvement
of a unit intended to lock an internal nut mounted on a shaft end.
In the original design (Fig. 116i) the nut is locked with a spring-
loaded stopper provided with two hexagons, the larger one of which
slides in the shaft hexagonal hole and the other smaller one enters
the hexagonal bore of the nut. Unscrewing the nut should be preceded
by sinking the stopper as this will bring the smaller hexagon out of
engagement with the nut. The axial size of the unit is unreasonably
large. The nut internal hexagon is brought to the nut end face so that
a poorly experienced fitter may try to unscrew the nut without having
released the stopper beforehand. As the stopper is not fixed length¬
wise it will drop out of the shaft hole during unscrewing.

Now, if the spring is mounted inside the stopper hexagon (Fig.
116;) the nut internal hexagon becomes shorter, thus preventing the
nut unscrewing without preliminary release of the stopper.

In the most rational alternative (Fig. 116ft) the stopper is made of
a hexagonal bar. Hexagonal holes in the shaft and nut are machined
by the same broach, whereas in the previous examples two different
broaches were needed. Mounting the spring outside the stopper
allows plan dimensions of the unit to be reduced 1.5 times as compar¬
ed with the original design. Here the stopper is fixed axially by
means of a circlip and does not drop out of the hole after unscrewing
the nut. The nut cannot be unscrewed unless the stopper is released.

Figure 117a illustrates a bevel gear unit in which the gear is in¬
stalled conventionally in a cantilevered mode. An improved design
shown in Fig. 117ft has two supports: one end of the driving gear shaft
is installed in the casing wall and the other, in detachable cover 1
with a port provided in the area where the gears mesh. In this alter¬
native the size of the gears is decreased and their stability improved.

The axial size can be reduced to half that of the original construc-

10 *

148 _ Chapter 6. General Design Principles _

tion if the gear is moved to the other side of the driving shaft
(Fig. 117c).

Apart from the excessive overall dimensions, the bevel gear redu¬
cer sketched in Fig. 117d is inefficient in that its gears are mounted
in various housing components and accurate engagement is therefore
difficult to obtain. The accuracy of engagement can only be verified
by blacking in.

In the improved design (Fig. 117e) the housing volume is utilized
in the most efficient way as the bearings of the major bevel gear are
incorporated into the housing contour. One of the bearings of the
minor gear is also located inside the housing. Further decrease of the
overall dimensions has been reached through changing the gear key-
type shanks to internal splines. All the bearings are housed inside
a single component, this being very beneficial from the viewpoint of
the accuracy of the bevel gears positioning. The design is unitized:
after the cover is removed, the gear unit remains in place and is
easily accessible for inspection and adjustment.

Figure 118 shows how the axial dimensions of a gear reduction
unit can be reduced. In the original design (Fig. 118a) end gear 1
is cantilever-mounted in diaphragm 2. The drive gear is mounted
in two bearings, one of which is installed in cover 3 and the other,
in the end gear recess. This latter bearing is arranged in a cantileve¬
red way in relation to the main bearings. The intermediate gears shaft
is supported at one end in diaphragm 2 and at the other, in cover 3.
The axial dimensions of the system are unreasonably large. The sup¬
ports of the driving and intermediate shafts are located in different
components. The bearing surfaces of the drive pinion cannot be
machined together, this imposing higher demands for the alignment
of the end gear shaft bearing surfaces. The assembly procedure of the
unit is extremely complicated. As the diaphragm is fitted together
with the cover the ends of the driving and intermediate gears shafts,
which have previously been fixed each in its support, hang; the
fitters are thus compelled to blindly place them into the second
supports. In the improved version (Fig. 1186) the shafts supports are
positioned within the shaped diaphragm 4 and tightened against
cover 5. Owing to this feature the entire system becomes unitized.
Now all the fitting surfaces can be machined in one setting with the
diaphragm being screwed to the cover. The reduction gear unit can
be assembled and checked as an individual unit. Any part of the
unit is readily accessible for inspection during the assembly proce¬
dure. Axial dimensions are reduced by half as compared with the
original design. This is achieved by the following measures (apart
from the introduction of the detachable diaphragm):

the rim of the end gear drive splines (which in the previous design
is machined on a separate fitted part) is made integral with the end
gear as extension of the teeth;

150

Chapter 6. General Design Principles

the end gear is mounted on a needle bearing installed upon a cy¬
lindrical flange of the diaphragm;

all the external splines of the input shaft are replaced by internal
ones.

In the alternative illustrated in Fig. 118c the gearing is mounted
on a single component, namely, on diaphram 6. Cover 7 is non-bear¬
ing and is connected to the reduction gear mechanism only by the
seal which encompasses the tip of the driving shaft. In this case
manufacture and assembly are still more simplified.

6.12. Combining Design Functions

The overall size and weight of a structure can sometimes be decreas¬
ed materially by combining several functions in one component.

In a paired radial-thrust ball bearing intended to take up axial
loads in both directions (Fig. 119a), the actual load is taken up by
one of the bearings (at a given moment), while the other is idle.

In the single-row double-thrust ball bearing (Fig. 1196) the balls
are encased in races which have deep grooves; for assembly convenien¬
ce the outer ring is split. When subjected to load the balls are forced
against one side of the groove, leaving, accordingly, the opposite
side. With a changed load direction the opposite groove is loaded.
Such bearings of the same load capacity have half the axial length
of the paired bearings. Furthermore, by encasing outer split races
inside a common ring (Fig. 119c) the assembly becomes a unitized
component.

Another example, similar to the above, is a helical ball drive
(the system is very commonly applied in power drives). In the alter¬
native depicted in Fig. 119 d two balls are arranged in each helix,
consequently, the axial load is taken up by one half of the total num¬
ber of balls only. If only one ball is installed is each helix (Fig. 119e)
the axial load will then be shared by all the balls (regardless of the
load direction), thus the load capacity is doubled.

In the rod journal joint of a two-part crank the connection is clam¬
ped with a bolt (Fig. 119/). In the improved version (Fig. 119g) the
connection is accomplished by using the shank of the left-hand part
of the crank.

In the flexible mounting of a gear rim on its hub (Fig. 119A) the
alternating torque is transmitted from the gear rim via a number of
springs mounted (in order to avoid distortion) on cylindrical inserts 1
resting in specially provided nests in projections 2, 3 of the rim and
hub. Here one half of the springs damp the torque-induced vibrations
in one direction, and the other half, in the opposite direction.

In the improved alternative (Fig. 119i) the projections of the rim
and hub are superimposed (in plan view); hub projections 4 are
introduced into recess in the rim projections 5. Owing to such an

152

Chapter 6. General Design Principles

arrangement all the springs take up torque-induced vihrations in
both directions simultaneously. The unit is capable of transmitting
doubled torque or damping vibrations whose amplitude is twice that
of the previous system.

When mounting gears as shown in Fig. 119/, each gear is secured
in place on its own splines. The small spline lengths encountered
in the system forces the introduction of aligning elements, the
cylindrical collars 6, 7 and 8. In the design shown in Fig. 1191c the
gears are slipped onto one common spline. The result: special tools*
list (broaches, hobs) is cut down by half; the assembly is simplified;
the shaft strength, enhanced; the shaft right-hand journal diameter
and the length of splines, increased, which enables the additional
alignment of gears to be obviated.

In the most perfect alternative (Fig. 119Z) the gears are held on an
extension of the pinion gear’s teeth. The only special tool required
is a broach for machining the splined holes.

In the unit of a valve drive mechanism (Fig. 119m) rockers 9
are mounted each on a stud of their own. In the better version
(Fig. 119n) both rockers are cantilever-mounted on a common pin.

Figure 119o gives an example of gears mounted with intermediate
spring coupling 10. In the alternative shown in Fig. 119 p damping
springs are arranged in the gear wheel of the drive. As a result
the weight and axial dimensions are reduced and rigidity im¬
proved.

In the planetary drive unit of a cam plate 11 (Fig. 110 q) the satel¬
lites are mounted on an individual, in fact, superfluous part 12.
In the improved design (Fig. 119r) the satellites are mounted directly
on the plate. As a result, the unit becomes lighter and more compact
and manufacture is eased.

6.13. Equistrength

Some ways in which machine components can be imparted uni¬
form strength are illustrated in Fig. 120.

When a simply supported shaft is loaded with a transverse bending
force (Fig. 120a), a body of equal bending resistance with the same
maximum stresses in all sections will have the profile of a cubic
parabola (shown in the Figure by a thin line). The design in this case
is not equistrong: the equal-resistance parabola is twice (on the shaft
tapered section and beyond the cylindrical lug) outside the actual
part contour. These sections are weak in comparison with the remain¬
ing ones.

In a correctly designed version (Fig. 120&) the parabola is contain¬
ed within the part contour. The through hole provided in the part
hardly affects the strength, but considerably reduces the weight of
the part.

6.13. Equistrength

153

Under complex loading conditions (Fig. 120c-/) the shaft undergoes
bending from the force of the tooth drive and in the section between
the teeth and splines which transmit the driving force the shaft is
subjected to the action of operating torque M.

In the design shown in Fig. 120c the shaft section is designed to
meet maximum flexural and torsional stresses across AA, without

y//Av/M.

mkU\i

|.»ii

131

IliiS! 1 .!

|l;Oll|

Fig. 120. Imparting equal strength properties to parts

considering the drop of the bending moment towards the supports-
Also neglected is the fact that the shaft right-hand end is subjected
to bending alone, whereas the left-hand end undergoes bending and
torsion. Hence, the latter is loaded more severely.

In the equal-strength design (Fig. 120d) the shaft is made tapered,
thus meeting accordingly the bending moment variation along the

154

Chapter 6. General Design Principles

axis. The equal strength of the left-hand (heavily loaded) and right-
hand (lightly loaded) sides of the shaft is achieved by stepping the
internal bore.

Equal strength can also be achieved by increasing the outer
dimensions of the tapered portion m (Fig. 120e), or by decreasing
the diameter of the right-hand ball bearing (Fig. 120/). Here the
internal hole is again stepped but on the opposite side to the previous
examples, accompanied also by an increase in the diameter of the
left-hand support bearing.

The designs illustrated in Fig. 120 d-f are equivalent, therefore,
the design choice is dictated by technological and operational con¬
siderations.

In a cluster gear (Fig. 120g) the gear teeth offer uneven resistance
to bending and crushing. The circumferential force on the minor gear
rim will under any conditions (i.e., with multiplying and reducing
gear ratios) be less than that on the major gear rim by a factor equal
to their diametral ratio. The design can be made more balanced in
strength if the width of the major gear teeth is decreased (Fig. 120 h)
■even though not strictly in proportion to the acting forces (with due
regard for the higher circumferential speed).

In a bar bracket loaded with a tensile force (Fig. 120i) the middle
bar carrying the greater portion of the load is overloaded in compari¬
son with the lateral, slightly loaded bars. The system becomes
equally strong if the middle bar cross-section is increased (Fig. 120/).

For units to have equal strength means that all their parts must
have equal stresses (for similar materials) or possess equal strength
(for materials of different strength). The design version of a splined
joint illustrated in Fig. 121a has a non-uniform strength on two co¬
unts: the strength of shafts in plain sections (diameter d) is much
more better than that across the splined part (diameter d 0 ); the splin¬
ed bush has a considerably higher resistance to torsion than the
shafts.

The equal-strength condition requires that both the bush and the shafts
have equal resisting moments in torsion: 0.2 D 3 (1 — a 4 ) = 0.2a 3 , where
-a = d/D. For the relationships in the Figure (a = 0.68) the strength
1 —

of the bush is—=—= 2.5 times greater than that of the shafts.
a 3

In a rational design (Fig. 121b) the diameter of the shafts is reduced
in comparison with the internal diameter of the splined joint and
the external diameter of the bush (D 1 ) is reduced to the technolo¬
gically acceptable minimum value.

In the connection of a screw propeller blade (aluminium alloy)
with a steel hub (Fig. 121c) the predominant stress is that of tension
produced from the centrifugal force as the propeller rotates. The
■design has unequal strength. The thread profile both at the blade

6.13. Equlstrength

155

root and in the hub is the same although the permissible stresses
in bending and shear for aluminium alloy are lower than those for
steel.

In the design shown in Fig. 121d the height (in the axial direction)
of the blade threads is made higher than that of the bush so as to
render them equally strong. The bush is given a shape offering equal
resistance to tension. The bush cross-section progressively increases

Fig. 121. Imparting equal strength properties to units

downward coping accordingly with the rising tension forces transmit¬
ted to the bush by the blade threads. A relieving recess is made in the
blade.

The square-profiled toothed-slot fastening of turbine blades in the
rotor (Fig. 121e) is irrational for two reasons. First, the blade root
is uniform in thickness and therefore is not equally strong. The
full centrifugal force is taken up by the cross-section lying nearest
to the blade root platform and this section is overloaded as com¬
pared with the other sections in which the acting force progressively
decreases towards the root tip. The picture is opposite in the rotor
webs, where the most severely loaded section lies close to the root
tip, while the above-lying sections are loaded to a lesser degree.

Secondly, because of the square-shaped teeth the material of the
root is not utilized to its best. The total section working in shear
and flexure is approximately equal to one half of the root section
(over height h).

156

Chapter 6. General Design Principles

The entire root height can share the load if the teeth are made
triangular (Fig. 121/) or trapezoidal (Fig. 121^). In this case the tooth
shear strength improves approximately twice and the flexural
strength, 4 times. Trapezoidal teeth have an additional advantageous
feature, namely, lower bearing stress on the teeth working faces.

In the most rational version (Fig. 121 h) the root is wedge-shaped,
thus coping with progressively decreasing forces acting upon the
root. Such a shape increases the critical sections of the blade root
and rotor webs about 1.5 times, with a corresponding increase in
the joint strength.

6.14. Uniform Loading of Supports

When designing units with antifriction bearings, the latter should
advisably be made of equal durability.

Let us consider an example of a gearing (Fig. 122a). The drive load
Px on the pinion exceeds load P 2 on the gear by a factor of

Fig. 122. Imparting equal durability to supports

« 4. With the supports spaced as in the Figure, the left-hand bear¬
ing is loaded with force N x which is 2.5 times greater than force
N 2 acting upon the right-hand bearing. The supports can be made
of equal durability if on the shaft right-hand end is mounted a smal¬
ler bearing (Fig. 1226) whose load capacity is 2.5 times less than that

157

_____ 6.15. Self-Alignment Principle

of the left-hand bearing. If in the interests of unification it is still
preferable for both bearings to have the same size, then the position
of the supports relative to the acting forces should be altered so that
the load on both bearings is the same (Fig, 122c).

In a cantilevered impeller unit of a centrifugal compressor (Fig.
122d) the shaft is acted upon by radial force P x (resulting from impeller
imbalance) and axial force P 2 (from the operating fluid). The design
is inefficient. The front (nearest to the impeller) bearing is loaded
by a large radial force Ni and axial force P 2 , while the rear bearing
is loaded by a very small radial force N 2 . In the version shown in
Fig. 122e the axial force is taken up by the rear bearing, this result¬
ing in more uniformly distributed bearing loads. The alternative
illustrated in Fig. 122/ has its shaft mounted in different bearings
whose capacities are consistent with the acting loads.

6.15. Self-Alignment Principle

In sliding mechanical connections, where distortions and deflec¬
tions are possible, preliminary thought should be given to free self¬
alignment so that the connection works correctly with all assembly
and manufacturing inaccuracies.

The most simple example is a toe bearing. If the bearing plate is
rigidly mounted in the housing (Fig. 123a) the toe edges will work
over the plate due to which distortions are inevitable in the system.
In the design shown in Fig. 1236 the plate is placed on a spherical
support, which assures intimate contact along the entire friction
surface, thus increasing loading capacity and durability.

The self-alignment principle is extensively used when designing
shaft supports subjected to bending and distortion. Self-alignment
is particularly important for plain bearings with great length-to-dia-
meter ratios. If the mounting is rigid (Fig. 123c) the shaft’s bending
and distortion causes higher edge pressures thus sharply worsening
bearing operating conditions. To enable self-alignment the bearings
are installed on spherical supports (Fig. 123d).

In radial ball bearings (Fig. 123e) shaft flexure distorts the bear¬
ing and unilaterally loacls the balls by a value which may exceed the
designed load. The defect can be eliminated by placing the bearing
into a spherical ring (Fig. 123/) or by applying double-row spherical
bearings (Fig. 123g).

It should be noted that double-row spherical bearings have lower load capaci¬
ties than the single-row radial bearings (due to the outer race shape which is
unfavourable to contact strength) and cannot withstand large axial forces.
For this reason when designing units intended to receive higher axial loads
it is better to employ single-row hearings on spherical supports or double-row
self-aligning bearings with barrel-shaped rollers.

158

Chapter 6. General Design Principles

Another example — a two-stage air compressor piston (Fig. 123fe).
Piston 1 slides in a low-pressure cylinder, while the rod slides in
a high-pressure cylinder (air passages are not shown in the Figure).
The defect of the system is that the piston and rod are made as one.
This means that accurate alignment of working surfaces is necessary,

Fig. 123. Imparting self-alignment properties

i.e., those of the piston and rod on one hand and bores of the high-
and low-pressure cylinders on the other. Since the clearance between
the rod and high-pressure cylinder walls is much less than that bet¬
ween the piston and low-pressure cylinder walls, transverse drive
forces are taken up mostly by the rod, which causes its excessive
wear.

In the advisable design (Fig. 123i) the rod can deflect and move
about the piston axis as it is separated from the piston. The drive
forces are taken up by the piston, and the rod is relieved of transverse
loads. Thus, rigorous alignment of the bores of the low- and high-
pressure cylinders is now not necessary.

The plate valve design (Fig. 123/) in which the plate is positively
fixed to the stem end fails to assure leak-tight valve-to-seat contact

6.15. Self-Alignment Principle

15 »

because of the inevitable non-perpendicularity of the seating sur¬
face in respect to the stem axis. Another mistake is that during clo¬
sure the plate rotates together with the stem relative to the seat. The
latter error is eliminated in the design in Fig. 123 k where the plate is
fixed to the stem end by means of two transverse pins 1. As the
valve is being closed, the stem rotates relative to the plate, however,
a tight valve seating is still not achieved.

In the correctly designed version (Fig. 1231) the stem end is made
spherical, owing to which the plate can freely self-align and seat
tightly regardless of the manufacturing inaccuracies. The free self¬
alignment is due to clearance s between the transverse pins and the
stem end shoulder.

A flap which alternately closes two pipes, square to each other
(Fig. 123m) cannot closely seat, especially as the pipes have soft
gaskets which can appreciably alter pipe positions during reassem¬
blies. In the correct design (Fig. 123 n) the flap is mounted on a sphe¬
rical pivot 2. Moreover, it is locked endwise with transverse pins 3
and prevented form rotation about the stem axis by placing the pins
in shallow holes in the stem.

In the clamp illustrated in Fig. 123o the clamping effort is taken,
in fact, by one point of the rifled surface. Hence, the fastening bolt
thread is subjected to bending.

The design version in Fig. 123p is free from distortions in all its
links. The articulated bolt also adds to the convenience in operation.

Sketched in Fig. 124 is a wedge-type gate shutting off coaxial
pipes. When the gate is rigidly fastened to the driving rod (Fig. 124a)
it cannot be simultaneously pressed tightly against both seats;
the gate self-alignment is only possible with clearances in the system
and flexible deformation of the rod.

A somewhat better design is shown in Fig. 124 b, in which the gate
can freely move relative to the rod thanks to a cylindrical hinge. Here
any inaccuracy in the rod position relative to the seat will in no way
impair the functioning of the system. However, manufacturing inac¬
curacies between the gate and seat tapered surfaces, axial misalign¬
ments and pipe distortions, pipe angular displacements in the plane
normal to the axis, etc., are not considered.

Figure 124c shows a gate composed of two halves coupled to each
other and to the driving rod by a cylindrical pin. To preclude spon¬
taneous opening of the halves with the gate up, distance springs are
incorporated between the halves. The problem has been solved but
partially, because the gate halves can self-align in the plane of rota¬
tion about the axis only and not in the plane running normally to
the pipe axis. This drawback is eliminated by making the pin-receiv¬
ing holes in the rod and eyes oval-shaped. Then (Fig. 124 d) self¬
alignment is ensured in any direction (within the clearance between
the pin and hole walls).

Fig. 125. Levelling out loads on gear teeth
(a-c) planetary gearings; (d-f) multitrain gearings

11-01418

162

Chapter 6. General Design Principles

Most correct designs are those in Fig. 124e, /. In the first case the
eye halves are spherical and the pin which links the mechanism parts
in the free state is mounted in the rod yoke. In the second case the
gate halves are fixed by semi-spherical hinges at the rod head.

The self-alignment principle is utilized to equalize gear tooth
loads in planetary and multiple gearings.

In the planetary gear drive (Fig. 125a) crown gear 1 is freely moun¬
ted on satellites 2 and prevented from rotation by a splined connec¬
tion with the gearing casing. Gear 3 is also freely mounted on the
splines of the driving shaft. Both these gears are free to shift (within
splined connection clearances) in radial directions, thus the loads
upon the satellites are balanced.

In the design shown in Fig. 1256 self-alignment is accomplished
by fitting satellite holder 4 freely on the splines of the final shaft,
while in the version in Fig. 125c the object is achieved by imparting
elasticity to the rims of gears 5 and 6.

Self-alignment in multiple gearings can be ensured by installing
intermediate gear 7 (Fig. 125 d) in cage 8 locked against rotation by
casing splines; by fitting the driving gear 9 (Fig. 125e) and driven
gear 10 on free splines; by coupling driving gear 11 with the driving
shaft by means of a flexible bush 12 made of elastomer, and driven
gear 13 with the final shaft, by means of splines (Fig. 125/).

6.16. Cambering

Any linear or flat contacting surface operating under load should
be cambered outwards to assure central load application and elimi¬
nate excessive edge pressures caused from assembly and manufactur-

(a) (b) (c) (a)

Fig. 126. Ensuring uniform contact over roll length

ing inaccuracies. The method known as cambering is extensively
applied to parts working under heavy loads, rolling and sliding
friction.

Figure 126 shows the application of this principle to an antifric¬
tion bearing roller. If the roller has sharp edges (Fig. 126a) excessive
edge pressures are inevitable. Chamfering the edges (Fig. 1266)
does not rectify the situation. The only difference is that the edge
pressure is applied to dulled edges. Things are slightly better in
Fig. 126c where the sharp edges are radiused.

6.16. Cambering

163

In the cambered design (Fig. 126<2) the roller has a slight barrel
shape. The roller profile is either found by calculation or determined
experimentally so that the load is uniformly distributed throughout
the entire contact length. The difference between the maximum

Fig. 127. Ensuring self-alignment of tappet drive unit

and minimum diameters of the roller is generally some hundredths
of a millimetre.

Figure 127a shows an example of a cam-actuated drive of a cylin¬
drical tappet. Sharp edges on the contact surfaces (Fig. 127a) are
inadmissible. It is necessary, at least, to
round off the end faces (Fig. 1276). In
Fig. 127c the cam is cambered. It is techno¬
logically simpler to camber the working sur¬
face of the tappet (Fig. 127d).

In Fig. 127e it is the guide surface of the
tappet that is cambered. With off-centre lo¬
ads the tappet will self-align within certain
limits, maintaining a more or less uniform 128 . Barrel-shaped

contact over the working surfaces. Another tooth

way to ensure self-alignment is to make the
tappet guide surface slightly tapered (Fig. 127/).

Recently cambering has been rather extensively applied to gear
teeth, so that excessive edge pressures in the event distortions are
avoided. The teeth of a pair of gears (or one gear only) are given
a barrel-like shape (Fig. 128). Such a shape is produced (provided
the tooth hardness is not in excess of 40-45 Rc) by shaving on a ma¬
chine tool with an oscillating table. When hard teeth are ground by
dished grinding wheels (i.e., by the basic rack method) such a surface
can be obtained by periodically moving the work to the grinding
wheel or by oscillating the table in the feed plane.

To avoid intricate profile processing methods when cambering flat
and cylindrical surfaces a method of preliminary elastic deformation is used.
The technique is illustrated in Fig. 129a-d, where the plate of a tappet is given
a slightly spherical shape. During finishing the tappet stem is clamped in

11 *

164

Chapter 6. General Design Principles

a cylindrical arbor, which causes the plate to camber inwards and assume the
shape exaggeratedly illustrated in Fig. 1296. The working surface is then ground
flat (Fig. 129c) and after the tappet is detached from the arbor, the plate

Fig. 129. Cambering procedures

straightens and assumes a slightly outward-cambered shape (Fig. 129d). The
amount of convexity is controlled by the arbor tightening force.

A similar cambering technique as applied to a roller is pictured in Fig. 129 e-h.

Another method is based on the application of a strictly controlled load to the
finished part, which induces therein residual deformations of the required level.

6.17. Effect of Elasticity upon Load Distribution

Elastic deformations substantially influence load distribution
and the magnitude and distribution of stresses in the body of the
part. The directions of elastic deformations in the part must be preci¬
sely known and used to lower the stresses and balance the loads.

As an example Fig. 130a presents a splined connection unit
of a gear and shaft. The gear disk is displaced relative to the splines.
The torque transmitted by the gear is taken up mostly by the part
of splined connection positioned at the rigidity node, i.e., in the
disk plane. The qualitative picture of the bearing stress distribution
over the splines working face is presented by the curve. Obviously,
the stresses reach their maximum in the disk plane and are insignifi¬
cant over most of the spline length. If the splines rim is inverted
(Fig. 1306), the torque from the shaft end twists the shaft. As a
result, the splines on the left of the gear come into close contact
with the hub splines over their entire length and also twist the hub.
Thus, the torque is transmitted more uniformly throughout the con¬
nection. In some measure the system exhibits self-levelling properties:

6.17. Effect of Elasticity upon Load Distribution

165

the greater the torque and, hence, the shaft twist, the more uniform
is the load distribution along the splines.

In the press-fitted connection (Fig. 130e) the pressure on the con¬
tacting surface is concentrated [predominantly at the'rigidity node,
i.e., in the plane of the fitted disk. The pressure can be distributed
more uniformly by placing the disk centrally and by reinforcing
the hub with ribs (Fig. 130 d).

Another example of using elasticity to level out force distribution
is given by the seating of a column into its pedestal. With the usual
design (Fig. 131a) the bulk of the load is applied at the rigidity node,
namely, at the transition section where the flange meets the column
body.

If the flange support surface is made slightly conical (Fig. 131b),
the flange rim will first touch the bearing surface and then, during
tightening, will elastically deform and distribute the tightening
force more uniformly over the bearing surface. The joint, as a whole,
obtains greater rigidity and steadiness.

Figure 131c, d shows a method of strengthening the fastening of
a turbine blade in a fir-tree-shaped rotor slot. In the design shown in
Fig. 131c the working surfaces of the blade’s trapezoidal teeth, taking
up centrifugal force P, in the initial position contact the rotor slots
thrust faces. With the centrifugal force applied the blade root exte¬
nds, while the rotor body deforms to a lesser degree, thanks to its
greater rigidity. Due to this the load is taken mainly by ^the first
teeth (see the curve).

In Fig. 131d the blade teeth fit the slots with clearances h u h z
and h 3 which increase successively from the root to the blade plat¬
form. As the blade extends, the working surfaces of the teeth succes¬
sively contact the rotor slots thrust faces, thus a more even load
distribution among the teeth results and, consequently, the connec¬
tion becomes stronger.

In practice, when dealing with fir-tree-shaped connections,
allowances should be made for thermal deformations due to the
non-uniform blade heating in the interblade rotor sections and for
the creep of the blade root.

In conventional threaded connections (Fig. 132a) the thread load
distribution is uneven. This condition is aggravated by the fact that
the elastic deformations of the bolt and nut differ in direction; the
bolt elongates by a value f x under the action of tightening forces and
operating loads and the nut contracts by a value / 2 . As a result, the
first threads of the bolt (from the nut bearing surface) under load
tighten onto the first threads of the nut (Fig. 1326) and take up most
of the load. Both theoretical and experimental data proves that the
first thread carries approximately 30% of the total load.

Recently efficient methods have been devised which enable the
load to be equalized throughout the thread by correcting the thread

6.17. Effect of Elasticity upon Load Distribution

167

geometry. The simplest of these methods consists in making the nut
thread pitch larger than that of the bolt by a factor that depends on
the magnitude of the acting force established either by calculations
or experimentally. On the average this factor varies from 1.002 to
1.004. In screwed connections with usual pitches (1-2 mm) the dif¬
ference between the nut and bolt pitches is 3-5 p,m. Furthermore,
increased radial clearances are provided in such connections in order
to ensure self-alignment of the nut relative to the bolt and the bear¬
ing surface. In the free state the bolt topmost thread contacts the
topmost thread of the nut (Fig. 132c) and progressively increasing
axial clearances h 1 , h % and h 3 are formed between the adjacent
threads. As soon as a load is applied, the bolt elongates and the nut

contracts, thus the bolt threads tighten successively onto the
threads of the nut (Fig. 132d). With a correctly chosen pitch difference
the load will be taken up by all the threads. Uniform loading can
be assured only at a certain design magnitude of the force acting
on the joint. But even with forces close to the above the load is dist¬
ributed more evenly along the threads than with threads of the same
pitch.

Another method is based on reversing the direction of the nut
deformation. If the threaded section is subjected not to compression
(Fig. 133a) but to tension (Fig. 1336), then the first nut threads acted
upon by the load shift in the same direction as those of the bolt.
As a result, the load is distributed more uniformly over the threads.
Such nuts are called tension nuts.

In a semi-tension nut (Fig. 133c) the bearing surface is positioned
approximately in the middle of the screwed section whose lower
part takes up tensile stresses. To involve into efficient use the upper
threads, it is necessary to utilize the effect of elastic deformation
of the nut upper portion. The forces acting upon the bearing surface
cause a radial deformation of the nut body (in the direction shown by
light arrows) and, as a result, squeezing of the upper bolt threads.
Today both tension and semi-tension nuts are widely used in criti¬
cal screwed connections.

The profile of a tension nut with a uniformly distributed load is determined
proceeding from the following considerations. Let the length of the threaded nut
section be h (Fig. 134a). For uniform distribution of thread load the tensile
force acting upon the nut in any section at a distance x from the thread start

168

Chapter 6. General Design Principles

must he

Pnut = P T

where P is the force applied to the bolt.

The tension force acting upon the bolt in the same section

Pbolt = P-Pnut = P (l—f-)

For deformation compatibility the relative elongation of the nut in any
section must be equal to that of the bolt

Pnut Pbolt ,a

PnuiPnut EboltFbolt

where F nu t, Fb 0 it and E nu t, E bo i t are the cross-sectional areas and moduli of
elasticity for the materials of the nut and bolt, respectively.

Fig. 134. Determination of tension nut profile

Substituting in Eq. (6.10) the values of P nu t and P bo it fr° m Eqs. (6.8) and
(6.9), we obtain

r> 17 ^ Eboit

Fnut-Ftottj^-j—

If the nut and bolt are made of same material (E nUt = E bo n)

IFnut^Fboitj^ ( 6 . 11 )

Substituting in Eq. (6.11) the values F bo it= 0.785d 2 and Fnut — 0.785 X
X(D 2 — (P), where d is the bolt diameter and D running external diameter of the
nut, we get _

“-tVth i6 - ,2 >

When x = h, diameter D = oo.

Obviously, this condition can only be approximated. For any finite nut
diameter (where thejthread plane finishes) the upper threads will be loaded more
than thej lower ones.

Uniform distribution of thread loads can be achieved by increasing the bolt
end compliance. Let a conical recess be made in the bolt [stem (Fig. 1346) so that
its peak begins in the same plane as the thread. The running diameter 0 of the
recess is

h

where d. is the initial recess diameter (assumed to be equal to that of the bolt).

6.17. Effect of Elasticity upon Load Distribution

169

The running cross-sectional area of the bolt is

^bolt = 0.785(d2-02) = 0.785d2 [V— (-f-) 2 ]

Substituting this relation in Eq. (6.11) gives

= (6.13)

When x = h, diameter D = d\/~3 = 1.73 d.

This nut shape can be realized in practice.

The cross-section of the nut above the threaded portion is determined from
the equal strength condition for the nut and bolt in tension

Fnut = Fbolt (6.14)

Let the diameter of the recess in the nut above the threaded portion be 6,
Then, according to Eq. (6.14)

0.785 [(1.73d)2—02J = O.785d2

whence

0 = dty r 2 = 1.41d

Figure 134c shows a design approximation of the theoretical nut and bolt
shapes, where the load is uniformly distributed along’the threads.

Figure 135 illustrates how a heavily loaded screwed connection
(namely, a propeller blade fastening unit) can be rationally redesign¬
ed.

The factor of elasticity must
necessarily be taken into conside¬
ration when designing bearing
units. Figure 136a, b shows a
paired installation of antifriction
bearings. In Fig. 136a the greater
part of the load is taken up by
the bearing positioned in the ri¬
gidity node (housing wall plane).

The other bearing mounted at
the hub end is lightly loaded
because of the hub being compli¬
ant. The bearing loads can be
evenly distributed (thus contri¬
buting to a better load-carrying
capacity of the unit as a whole)
if the hub end is stiffened with another web (Fig. 136b).

Another example of elasticity utilization is given in Fig. 136c
in which the bearings are installed in a thin-walled steel bush.
Thanks to the elasticity of the bush the system can adapt itself to
shaft distortions, and resembles a system which has bearings mounted
in a spherical support.

Thus, by utilizing elasticity one can obtain optimum distribution
of load between bearings. In the bearing unit loaded with radial

blade fastening

170

Chapter 6, General Design Principles

force P and unilateral axial load Q (Fig. 136d) it is better to distri¬
bute the load according to the bearings function: one of the bearings
should be loaded with the radial force and the other, with the axial

Fig. 136. Effect of housing elasticity on load distribution in paired bearing

installation

force. This can be realized by mounting the bearings in an elastic
cantilever-type bush. Bearing 1 located in the rigidity node (i.e.,
in the fastening flange plane), takes the radial load. Bearing 2,
mounted at the cantilever end receives axial load only.

6.18. Fitting to Several Surfaces

It is good practice to avoid, whenever possible, fitting parts to
several surfaces (Fig. 137a, b). As a rule, parts should be fitted to one

6.19. Tightening up on Two Surfaces

171

tion, elastic deformations, thermal expansion of the system or under
compression of sealing gaskets.

Rough errors, like those sketched in Fig. 137a, c, are only charac¬
teristic of inexperienced designers. Still more typical errors include
superfluous fitting or alignment, etc. For example, fitting a prisma¬
tic loose key throughout the entire contour of the keyway (Fig. 137/)
will heavily, and, which is still more important, quite unnecessarily
complicate the production. It is more reasonable to fit the key to the
working edges only, providing clearances on the key end faces, as
well as between the key top surface and the keyway bottom (Fig.
137g, h).

6.19. Tightening Up on Two Surfaces

'"Occasionally it may be necessary, due to some design features,
to^tighten a unit of two surfaces. Figure 138 (tightening three flan¬
ges) reviews techniques applied under these conditions.

Simultaneous and uniform tightening of all surfaces (Fig. 138a)
presumes] that the end flange faces should be machined together

Fig. 138. Tightening against two surfaces. Fastening of intermediate flange

and fitted or have undergone extremely accurate manufacture. If
a flange protrudes from its recess (Fig. 1386), an interference may occur
and the tightened part will bend. On the other hand, if the flange
is'sunk in its recess (Fig. 138c), the flange axial fixing will
be lost.

The introduction of elastic gaskets (Fig. 138d-/) improves the design
and seals the joint if the gasket is sufficiently thick and flexible
and overlaps non-flush sealing surfaces.

Whenever the good sealing and accurate axial fixing of a flange
are necessary, it is recommended to employ gaskets of some soft
metal (red copper, lead, aluminium, etc.) whose thickness exceeds
the depth of the recesses where the gaskets are to be fitted. During

172

Chapter 6. General Design Principles

tightening the metal gasket plastically deforms (Fig. 138g), seals
the joint and fixes the flange. Some space must be provided to enable
excessive metal to freely flow out. The bearing stresses arising in

the gasket under the action of working axi¬
al forces must be below the yield point of
the gasket material. Otherwise, accurate
axial fixing will be lost.

If harder metals (brass, bronze, low-car¬
bon annealed steel) are used for the gask¬
ets, these must be made corrugated or
comb-like (Fig. 138ft) to assure plastic defor¬
mation. Spring gaskets (Fig. 138i, /) are also
applied.

Figure 139 shows two-piece crank conne¬
cted in a main journal by means of trian¬
gular-shaped end-face splines, an antifri¬
ction bearing being simultaneously clamped
between the two parts of the crank. jThe
tightening of the splines and inner bearing
race in this case is ensured on the account of the deformation of
metallic rings 1 placed on either side of the inner race.

6.20. Axial Fixing of Parts

Parts must be fixed axially at One point so that they can freely
self-align throughout their length. If, for instance, a pin is fixed by
self-tapping screws at both supports (Fig. 140a) then additional
stresses are likely to occur as a result of thermal dimensional
changes. In the correct design shown in Fig. 1406 only one end of
the pin is fixed, while the other end is free to move in its support.

In the badly designed herringbone gearing (Fig. 140c) the lower
gears are fixed in position twice: by their teeth and stops m. To
obtain the complete coincidence of fixing positions is impossible.
The error can be eliminated by providing clearances s, which assures
gear self-alignment by their teeth (Fig. 140<2).

The gear shaft mounted in plain bearings (Fig. 140e) is fixed in
two points spaced far away from each other.

Accurate fixing in this case is impossible because a large clearance
must be provided between fixing surfaces in order to avoid seizure
of bearing surfaces due to thermal expansion of the housing. Further¬
more, due account must be taken of the inevitable inaccuracies of
manufacture and assembly due to the required clearances between
mounting surfaces.

The design is somewhat improved by bringing the fixing surfaces
closer together (Fig. 140/). In the correct designs (Fig. 140g, ft) the
shaft is fixed over a short distance (in the design presented in Fig.

Fig. 139. Tightening
against several surfaces.
Two-part crank journal

174

Chapter 6. General Design Principles

140 h it is fixed practically in one plane); the other end of the shaft
is allowed to self-align in its support.

Free areas of parts must be given certain margins for self-align¬
ment and compensation for manufacturing dimensional inaccura¬
cies. Let us consider the case of a shaft installed inside a housing on
antifriction bearings. In the design illustrated in Fig. 141a the axial
dimensions, predetermining the mutual location of the shaft, bear¬
ings and housing, are held to the nominal size. In the alternative

Fig. 141. Introducing margins on fitting surfaces

pictured in Fig. 1416 the following clearances are provided: m — on
the fitting surface of the housing to suit the floating bearing; h —
on the fitting surface of the housing relative to the fixing stop rings;
k —in the thread for the fastening nut; n —on the fitting surface of
the shaft to suit the floating bearing.

The clearance values are found by calculating the dimensional
chains and thermal expansions in the system. The maximum clear¬
ances should be provided in the areas of transitions to black cast
surfaces where dimensional variations are particularly great (for
medium-sized castings of moderate accuracy such clearances are
taken at 3-4 mm).

6.21. Control of Direction

Parts reciprocating linearly along two guideways should be guided
along one way only, the other guide only supporting the part (Fig. 1426,
d). A simultaneous guiding along two ways (Fig. 142a,c) means
much more stringent machining requirements to the guides and ways.
Any change in temperature conditions may impair orientation and,
hence, the part may seize in its guideways.

6.22. Mounting Surfaces

175

If the use of two guideways is inevitable, their manufacture must
be simplified as much as possible. In a design with two guiding rods
(Fig. 143a) the necessity to accurately keep the centre distance bet¬
ween the rod sockets in the driven part and the guiding holes in the

mmmmm

(a)

J ^\\\\\\\W

WMMM/M
( 6 )

(c)

Fig. 142. Guiding of parts
(o) and (e) wrong; (6) and ( d) correct

(d)

housing can be eliminated by increasing the diameter of the sockets
(Fig. 1436) and fixing the rods in their sockets after aligning them
with the guiding holes. Another method consists in machining the

Fig. 143. Ensuring accurate guidance over two surfaces

rod sockets and guiding holes together. In this case the diameters
of the guiding holes and the rod sockets must be the same (Fig.
143c). However, different fits for the rods are required, namely
a sliding fit in the housing and an interference fit in the rod sockets.

6.22. Mounting Surfaces

Generally speaking, any mounting surface for detachable parts
should be flat. Fastening on cylindrical surfaces should be avoided
(Fig. 144a). The manufacture of such connections is labour consuming.
The fitting surface of the detachable component must be machined
in a fixture to assure the equality of the diameters of the mounting
surfaces on the part and housing and it is rather difficult to uni¬
formly tighten angularly-spaced bolts. The correct design with a
flat mounting surface is pictured in Fig. 1446.

Sometimes it may be necessary to fasten parts to surfaces position¬
ed at an angle to one another (Fig. 144c) .J The connection seems
strong and rigid. Its manufacture, however, is difficult as it is

176

Chapter 6. General Design Principles

absolutely necessary to keep the angular equality between the fitted
part and housing to preclude the deformation of the part during
fastening. Fastening bolts must be alternately tightened and each
time by small amounts in order to assure the close fitting of the part
to both mounting surfaces. Obviously, a design with a plane mount¬
ing surface is much more preferable (Fig. 144 d).

Flat attachment is of particular importance when air-tight con¬
nections have to be obtained. Sealing surfaces must be free of any

Fig. 144. Clamping on curved surfaces

steps, as well as of outer or inner corners. Fitting to curvilinear sur¬
faces is inadmissible. An example is presented in Fig. 144e, which
shows a cover enclosing an angular space in a housing. The design
has two defects. First, it is practically impossible to seal off the space
end walls forming reentrant angle a. Secondly, the proper tightening
of the cover is made impossible by the fact that screwing up one row
of bolts will interfere with screwing up the other row
of bolts. The latter of these two defects is eliminated in the design
pictured in Fig. 144/ where the cover is tightened up by a row of
diagonally arranged bolts. This technique is rather common in
fastening shields over spaces which do not require sealing, and
also when enclosing through tunnels. Should the air-tightness be
necessary, the only correct solution will be that illustrated in Fig.
144 g where the cover is fitted on a flat surface.

Also erroneous is the design of a cover intended to enclose a hatch
in the corner of a welded sheet-steel housing (Fig. 144A), since it
is practically impossible to provide close tightening over a curvi¬
linear surface even when using a thick gasket. A correct design
sketched in Fig. 144i incorporates a flat frame welded to the hatch
and forming a flat fitting surface.

6.23. Butt-Jointing on Intersecting Surfaces

177

6.23. Butt-Jointing on Intersecting Surfaces

Butt-jointing on intersecting surfaces complicates the manu¬
facture and sealing of joints.

An incorrect joint design is pictured in Fig. 145a. Here the side
cover 1 is mounted upon the joint of the housing and the upper

Fig. 145. Fitting on intersecting surfaces

cover, which means that its fitting surface must be machined after
the assembly of the housing and the upper cover, and to assure
the tightness of the joint, a thick elastic gasket must be used. In

Fig. 146. Methods of fastening split housings

the correct design shown in Fig. 1456 the housing-to-upper-cover
joint is moved beyond the side cover.

A non-technological design is that of a housing made up of two
halves joined in vertical plane AA (Fig. 145c). The upper cover
is mounted on the joint of the halves. Still worse is the design in
Fig. 145c? where the cover is joined with the housing halves in two
mutually perpendicular planes. In the correct design (Fig. 145e)
the joint surfaces are separated.

Figure 146a shows a rotor machine housing split in the horizon¬
tal plane with two end covers also split horizontally, which makes
the joint sealing difficult. The slightly improved version in Fig. 1465
has solid end covers. The best designs have either a housing and
covers (Fig. 146c) joined in vertical planes (axial assembly), or
a housing (Fig. 14 6d) split in the horizontal plane (radial assembly).

12-01418

178

Chapter 6. General Design Principles

6.24. Interchangeability of Rapidly Wearing Parts

It is good practice to manufacture any rubbing or rapidly wearing
part of a machine component as an easily interchangeable piece.
The latter can be made of a material with special properties which
are absent in the basic material of the component.

Figure 147a illustrates a T-slot made in the body of a cast-iron
bed. To provide for machining convenience, longer service life

Fig. 147. Introduction of replaceable parts

and changeability it is advantageous to make the T-slot as a sepa¬
rate piece from some durable material and to fasten it to the bed,
e.g., in a slot (Fig. 147b).

To ensure reliable operation of the threaded pair in an inter¬
nally threaded steel rod for a lead screw (Fig. 147c), it is advisable
to use a bronze bush (Fig. 147d) which possesses antifriction pro¬
perties and can easily be changed.

Figure 147c illustrates an erroneously designed seal with split
spring rings fitted in shaft grooves, the external cylindrical sur¬
faces of the rings sealing against the housing. When the grooves
in the shaft and the hole in the housing get worn, both these expen¬
sive components have to be rejected. In the correct design (Fig. 147/)
the rings are assembled in a changeable bush and work in a sleeve
made from a harder material.

When installing an antifriction bearing in a light-alloy housing
(Fig. 147g) the seating surface rapidly crushes during operation.
Machining the hole even slightly oversize causes the rejection of

6.25. Accuracy of the Alignment of Parts

179

the entire housing unit. It the correct design (Fig. 147 h) the bear¬
ing is installed in an intermediate bush made from a harder mate¬
rial, which reduces the seat wear and enables defective housings
to be rectified.

To install the valve of an internal combustion engine in the
manner depicted in Fig. 147i, i.e., directly in a cast iron head,
is bad practice. It is better to install the valve in a guide bush made
of a harder material (Fig. 147;) and introduce an insert-type seat
of a heat-resistant material.

Figure 147 k-m shows a water-cooled engine block. If the cylinder
working surfaces are machined directly in the cast block (Fig. 147/c)
the design is bad. To obtain high-quality mirror-finish bores in
a large casting is difficult. Furthermore, the cylinder walls may
have defects (blisters, voids, pits, etc.) which sometimes/are
exposed only during finish machining. Clearly, a single defective
cylinder means rejection of the whole block. Finally, excessive
wear in one of the cylinders during operation means failure of the
entire expensive unit.

The correct solution is to use insert liners (Fig. 1471). The best
design has wet liners which are directly surrounded by water
(Fig. 147m). This system has significant additional advantages: simp¬
ler casting techniques, reduced weight and better cooling of the
cylinders.

6.25. Accuracy of the Alignment of Parts

Parts requiring accurate mutual location are preferably installed
in one housing with a minimum number of transitions and fits.

Fig. 148. Reduction valve design

For example, let us take a reduction valve unit (Fig. 148). The
most important factor predetermining the reliability of the unit
is the fitting of the valve tapered chamfer on the seat. In our exam-

12 *

180

Chapter 6. General Design Principles

pie it is accomplished with a number of transitional connections,
each being the source of inaccuracies, namely:

running fit between valve stem 1 and guide bush 2;

press fit between bush 2 and cover 3\

slide fit between cover 3 and casing 4\

press fit between valve seat 5 and casing 4.

The design requires accurate alignment of the following elements:
in the valve—guide surface and bevelled edge of the plate;
in the bush—bore and seating surface;
in the cover—hole and centring shoulder;

in the casing—centring bore for the cover and hole for the seat;
in the seat—chamfer and seating surface.

Fig. 149. Angle gear drive

When being ground-in to its seat, the valve is centred in guide
bush 2. The seal obtained by the grinding-in procedure is spoilt
during reassembly because of the displacement of cover 3 in relation
to casing 4.

In the rational design (Fig. 148 b) the valve is centred directly
on the seat. The accuracy of the valve motion is now dependent
on a single joint only, i.e., on the slide fit of guide shank 6 in seat 7
of the valve.

In this case strict alignment of the following elements will assure
good functioning of the unit:

in the valve—guide surface of the shank and chamfer;

6.26. Relief of Precision Mechanisms

181

in the seat—chamfer and seating surface.

The remaining elements can safely be manufactured to lower stan¬
dards of accuracy.

In the grinding-in process the valve is centred in its seat and
reassembly does not destroy the seal obtained by the grinding-in
process. Another example is an angle gear drive. In the design
shown in Fig. 149a the gears are installed in different housings 1
and 2, which hampers the manufacture. The joint surface of hous¬
ing 2 must be machined strictly parallel to the pinion axis. Never¬
theless, the mounting accuracy is disturbed when the sealing gasket
in the joint is tightened. Another defect is the obstructed access
to the gears. The axial position of the gears can be adjusted only
by blacking-in in several repetitive tests, the gear being detached
each time. The adjustment accuracy may be impaired during reas¬
semblies due to a non-uniform tightening of the gasket.

With the gears installed in one housing, their positional accuracy
is not disturbed during installation and subsequent reassemblies
(Fig. 1496). The gears can now be easily inspected during assembly*
The gears’ engagement adjustment is greatly simplified.

6.26. Relief of Precision Mechanisms

Precise moving connections and mechanisms should be relieved
from the action of external forces which may cause excessive wear
or disturb the correct working of the mechanism. The working
surfaces when in use must be protected against extraneous forces
and negligent treatment.

Figure 150a illustrates a conical plug-type cock with a handle
provided directly on the plug shank, which is wrong for the follow¬
ing reasons: the effort exerted in turning the handle is taken up by
the cock ground-in surfaces; occasional impacts against the handle
may spoil the sealing surfaces. Furthermore, an inexperienced
operator may pull the cock handle axially, thus injuring the tight¬
ness of the cock. The self-alignment of the plug in the conical seat
is hampered by the fact that the shank should simultaneously be
centred in the cock cover.

In the design presented in Fig. 1505 the driving shaft with the
handle are mounted in a separate body and spline-connected with
the plug. Here the plug is relieved from external forces and can
self-align in its seat.

This design has an additional improvement: the axial position of the plug
is adjusted by pressure screw 1. At the beginning the screw is loosened, allowing
the plug (actuated by a spring) to tightly fit its seat. Then the screw is slightly
screwed in, thus slightly lifting the plug. This hardly impairs the tightness,
but enables the cock to be turned much easier. As the plug wears down, the
adjustment is repeated.

182

Chapter 6. General Design Principles

In the design of a flat face slide valve illustrated in Fig. 150c
the tightness may easily be spoilt by depressing inadvertently the
handle and thus displacing the slide valve away from the surface
being sealed. The defect is eliminated if the driving shaft and the
valve are separated (Fig. 150d).

In the unit of a distributing cam-operated slide valve (Fig. 150e)
the actuating forces from the cam are taken up by accurately
ground-in surfaces of the slide valve. During operation these susfaces

Fig. 150. Relieving mechanisms from extraneous forces

wear down. In the better alternative (Fig. 150/) the drive forces are
taken up by an individual tappet, hence the slide valve is relieved
from the action of the transverse forces and the wear on the sealing
surfaces is reduced to the minimum.

Another example is the drive of an internal combustion engine
valve. In the version presented in Fig. 150g' the cam acts directly
on the plate which is screwed into the hollow valve stem. As the
valve opens, the cam runs against the plate and the valve distorts
(within the guide clearance); the sealing chamfer of the head leaves
its seat and forms a crescent-shaped slit. This is particularly dan¬
gerous with exhaust valves because a jet of hot gases gushes through

6.27. Coupling Parts Made from Hard and Soft Materials

183

the slit, causing a one-sided erosion and burning of the valve. When
the cam runs off the plate, the valve rests sideways in its seat.
Consequently, the wear of the valve sealing chamfer and seat occurs
on one side.

In the design shown in Fig. 150/i the transverse force components
are taken up by intermediate sleeve 2 , so that the valve is acted
upon by an axial, centrally-applied force. The increase in the mass
of the reciprocating parts means some limitation to the engine
speed. This defect is eliminated in the design depicted in Fig. 150t
where the valve is driven through intermediate lever 3. Now the
valve is practically relieved from the action of transverse forces
(not as fully as in Fig. 150 h).

6.27. Coupling Parts Made from Hard and Soft Materials

Frictional units incorporating parts made from hard and soft
materials must be designed so that the rubbing surface of the harder
and more wear-resistant material overlaps completely the rubbing
surface of its mating member made of a softer and less wear-resistant
material. Observation of this rule assures a uniform wear of the

Fig. 151. Combination of parts made from materials of different hardness

softer part. If the soft surface overlaps the harder one, a stepped
worn spot appears which impairs the operation of the unit.

Let us now examine this fact through examples. In the end trun¬
nion bearing installed in a bronze bush (Fig. 151a) the trunnion end
is shorter than the bush. After wear, part l of the bush will be step¬
ped and hinder the axial self-alignment of the trunnion. It is also
incorrect to hold the axial dimensions to the nominal sizes, for
manufacturing errors, assembly inaccuracies, as well as thermal
deformations in the system may produce a displacement of the
trunnion end inward of the bearing with the same result as previ-

184

Chapter 6. General Design Principles

ously. In the correctly designed version (Fig. 1516) the trunnion
projects beyond the bush for a distance, which allows all possible
variations of the longitudinal dimensions to be catered for.

In the self-aligning spherical footstep bearing (Fig. 151c) the
rubbing surface diameter of the steel disk is less than that of the
bronze support. Eventually the disk will inevitably wear steps
in the bronze support, thus obstructing the shaft self-alignment.
The correct design is shown in Fig. 151d.

A wrongly designed face-type mechanical seal is pictured in
Fig. 151e, in which an immovable textolite! bush 1 is pressed by
springs 2 against a hardened steel disk 3 rotating together with

Fig. 152. Design versions of conical stop cocks

the shaft. Here the frictional surface of the steel disk is smaller
than that of the textolite bush, thus causing uneven wear. In the
correct design (Fig. 151/) the steel disk face overlaps the textolite
bush.

Figure 151 g, h shows respectively wrong and correct designs
of a cylinder-piston unit.

In the tappet unit in which the tappet slides in a bush (Fig. 151i)
the oil-distributing recess is provided in the tappet stem. The more
reasonable design has the recess machined in the bush (Fig. 151/),
thus ensuring equal wear of the stem and bush.

Figure 152a-c illustrates a stop-cock whose conical plug, made
of a hard material, is mounted in the body of a soft material. The
design in which the plug is shorter than the conical socket in the
body (Fig. 152a) is wrong as during the grinding-in process a step
is formed at the part h of the socket. This step hinders the deepening
of the plug into the socket. The same occurs as the socket wears
in operation.

In the improved version (Fig. 1526) the plug end extends out
of the socket, thus ensuring uniform wear. An insignificant step,
however, may appear at part m of the plug. The best design is given
in Fig. 152c, where the plug top is slightly below the socket surface.
Now the wear of the socket and plug in no way hampers the deepening
of the plug. The design in operation has a self-grinding-in property.

For the inversed case, i.e., a soft plug and hard body, the afore¬
said remains valid. The construction in Fig. 152d is incorrect.

6.27. Coupling Parts Made from. Hard and Soft Materials

185 -

because from the grinding-in processes and wear a step is formed
at part n. This step prevents the plug from going deeper. The defect
can be eliminated by lowering the plug below the socket face (Fig.
152e). A still better design is shown in Fig. 152/, where any possi¬
bility of step formation is utterly excluded, both on the plug and
in the socket.

Thus, we may generalize all cases, including those when the plug^
and body are of equally hard material, and say that the plug top
end must be below the surface and the lower smaller end must
protrude beyond the socket.

This statement remains true for fixed connections made of mate¬
rials which differ in hardness. It would be wrong to fit a soft huh

Fig. 153. Tightening of tapered joints

onto a conical portion of a steel shaft (Fig. 153a). Here the hub front
face, i.e., the one nearest to the nut, extends beyond the shaft taper.
As the hub is ground-in to the taper, and also while the hole loses
its true roundness in running, a step is formed at part h. This step
obstructs the hub assembly on the shaft in the course of repeated
tightenings. In a correct version (Fig. 153b) the shaft taper extends
beyond the hub, so that changes in the hole size when fitting and
because of bearing deformations in no way interfere with the proper
tightening. Should the material of the hub be harder than that of
the shaft (a fact rare in practice), then the most dangerous case is the
one when the hub rear face does not reach the start of the taper
(Fig. 153c), because now, in the process of grinding-in and tightening,
a step is formed at part m of the shaft. In the correct version (Fig. 153d).
the hub end overlaps the taper.

Conical fits fail to assure accurate longitudinal fixation. The mutual posi¬
tion of the fitted parts depends to a great extent upon the manufacturing accu¬
racy of the tapers both on the shaft and in the hub, upon the tightening force,
number of reassemblies, bearing deformations and wear of the surfaces. For

186

Chapter 6. General Design Principles

these reasons conical fits must not be used when accurate axial positions are
required.

For example, let us consider the pinion carrier of a planetary drive,
whose disk is attached to the casing by means of the satellite shafts. In the
design presented in Fig. 153e it is practically impossible to maintain precisely
distance l in respect to all fixture points. Because of the inevitable inaccuracies
in the taper diameters and axial distances between them, the lengthwise disk
displacements in the course of tightening will differ from shaft to shaft. This
displaces, buckles and overstresses the disk. Furthermore, it is difficult to
maintain strict centre distances between the tapers. It is also impossible to ensure
registration of holes in the connected parts by machining them together (as is
often done with cylindrical holes). In fact, the parts cannot be accurately as¬
sembled.

The design showing shafts with only one tapered end (Fig. 153/) is better
only because the errors are halved.

Such kinds of assemblies should have cylindrical fitting surfaces with the
disk tightened up on stops (Fig. 153g). The distances between the fixing steps
on the shafts can be held to close tolerances. Registration of the holes’ centres
in the cover and casing is attained either by machining the holes in a jig or by
through machining them together.

6.28. Elimination of Local Weak Spots

Local weak spots due to decreased cross-sectional areas and stress
■concentrators can sharply reduce the strength of components.

Often such a weakening results from a miscalculation of the cross-
sectional areas of the part. This mistake often occurs when designing
small parts deemed not worth calculations, which allegedly take
on loads (except tightening forces). A characteristic example is

illustrated in Fig. 154a. A nipple has a hole 0 20A. To enable free
retraction of cutting tools the hole bottom has an undercut 020.5.
The threaded nipple portion (M24-1.5) also has an undercut 022
to clear the thread-cutting tool. For the given dimensions the wall
thickness at part h (over the internal groove) is 1.25 mm, and at
part m (under the external groove) the wall thickness is 1 mm. Clear¬
ly, the nipple wall will break even in the case of a weak tightening
against the hexagon.

In the improved design (Fig. 1546) the diameter of the part across
the internal groove section is increased to <f> 26; hence, the minimum

6.28. Elimination of Local Weak Spots

187

wall thickness is increased to 2.75 mm. A larger thread M27 with
a finer pitch (s — 1 mm) is used, which increases the wall thickness
at the external groove to 2.75 mm.

Very often a local weak spot can be eliminated by shifting the
weakening element into a zone of larger cross-sections. The
example given in Fig. 155a illustrates this. The bush is sharply
weakened by two grooves in the same plane (position h), which

Fig. 155. Elimination of local weak spots

enable a free withdrawal of cutting tools. Moving the internal
groove into the flange plane (Fig. 1556) strengthens the part.

In the nozzle cap nut (Fig. 155c) the undercut should be provid¬
ed in the hexagon plane (Fig. 155<2).

The internal nut (Fig. 155e) is seriously weakened at the section m
where a tool retraction groove is provided and at the sections n
where spanner-receiving holes are drilled. In the improved design
(Fig. 155/) the holes are replaced by splines and at the weak section
the wall thickness is enhanced.

If a lever is secured to a shaft in the way shown in Fig. 155g,
the hub will be seriously weakened by the keyway. In the better
design (Fig. 1551i) the keyway is moved to the area of increased
cross-section, namely, to the transition where the hub joins a longi¬
tudinal rib.

An eye bolt loaded with a tension force P (Fig. 155i) is weakened
by hole d located in the most severely stressed area and intended
to receive an oiler. In the strengthened construction (Fig. 155/)
the hole is arranged in the thickened portion of the eye.

188

Chapter 6. General Design Principles

The bolt head with an internal hexagon produced by broaching
(Fig. 155fe) is heavily weakened by a groove at the section /, which
is intended to clear the broach where the bolt stem turns into the
head (this section undergoes bending). Changing the broaching to
upsetting (Fig. 155Z) eliminates the weakness and the bolt acqui¬
res additional strength thanks to a more favourable grain orien¬
tation.

The strength of the leaf spring fastening unit (Fig. 155m) is lowered
by the screw-fastening hole at the critical section. The strengthening
methods in Fig. 155w-p complicate the manufacturing procedure
and cause greater waste since the springs are produced by blanking
from steel sheet or ribbon.

The strongest and most technological design of the spring made
from steel strip and secured by strap v is shown in Fig. 155g. Now
the spring takes up stresses with its unweakened section and is
safely protected by the strap at the most critical section.

6.29. Strengthening of Deformable Areas

The strength of a part can be substantially enhanced by reinforc¬
ing its non-rigid areas which are readily deformed under the action
of operating forces.

Let us take, for example, the slot-tang joint unit of shafts illu¬
strated in Fig. 156a, b. In the irrational design (Fig. 156a) the tang
of the driving shaft deforms the forks of the slotted shaft when trans¬
mitting torque. This loosens the fit. A press-fitted binding hoop over
the fork end (Fig. 1565) sharply increases the joint strength and
rigidity.

In the universal joint (Fig. 156c) the torque is transmitted by
a pin press-fitted into the driving shaft knuckle and passing into
slots machined into the driven shaft end. In the reinforced construc¬
tion internal slots are used instead of the forked end in the solid
shaft (Fig. 156d).

The fork-end connection in Fig. 156e is irrational, because the fork
opens under the action of tension forces (shown by light arrows).
The unit strength will considerably improve, if the forks are addition¬
ally tightened on to an auxiliary bush (Fig. 156/).

Figure 156g shows the fastening section of turbine blades mounted
in a T-shaped tenon on the rotor rim. Under the action of centrifugal
forces the blades roots move apart (shown by dashes). In the better
design (Fig. 156/i) the roots are provided with spurs m entering the
annular recesses in the rotor rim, thus preventing the roots from mov¬
ing apart.

The two-part crank pictured in Fig. 156i is coupled in such a way
that its crankpin tightens against the flat web face n. The elastic

6.30. Composite Constructions

189

deformations of the unit from the working loads cause conical defor¬
mation of the crankpin end, and work hardening and welding of the
fitting surfaces.

Fig. 156. Reinforcement of units

In the strengthened design (Fig. 156/) the crankpin end is coned
and fits into the conical recess l in the web. The tightness of the fit¬
ted collar stops the conical deformation and work hardening.

6.30. Composite Constructions

In many cases it is advisable to separate parts, connecting them
correctly later by press-fitting, welding, brazing, riveting, etc., or
by means of fastening bolts. Such composite designs simplify machin¬
ing and geometry, save weight and economize on rare or expensive
materials.

The separating into difJerent components in many instances saves
considerable labour when manufacturing large bed castings. A bed
casting with the lower halves of longitudinal shaft bearing housings
cast integrally (Fig. 157a) is not correct technologically, as it is
necessary to bore the cylindrical holes in the bearing caps and bed
simultaneously and maintain strict parallelism of the hole centres,
etc. The machining is particularly difficult when the bearings
arranged in a line are spaced at substantial distances.

In the design with separate housings (Fig. 1576) machining is
much easier, as it only means milling or planing the surfaces to which

190

Chapter 6. General Design Principles

the bearing housings are attached. The housings are positioned on the
bed by dowel pins.

Another example is the mounting of rollers on the corners of a mo¬
vable bed (Fig. 158a). To obtain axial alignment of the rollers and

Fig. 157. Bed with bearings

position them in the same horizontal plane is very difficult, especial¬
ly when the rollers are spaced rather far from each other. The use
of separate eye fastenings simplifies the processing (Fig. 1585).

Fig. 158. Simplification of machining of housing-type components through
introduction of detachable parts

The best version is given in Fig. 158c. Here the seating surfaces for
the roller lugs can be through-milled, which ensures that the rollers
are positioned in the same plane. Furthermore, the lugs are more
rigid than the eye fastenings and are reliably fixed against rotation
relative to the bed.

Easing the machining of slots in beds through the application of
detachable parts is illustrated by the examples in Fig. 158 d-g.

Figure 159 illustrates how fabrication by welding can simplify
machining. The cluster gear (Fig. 159a) is very difficult to produce
because of the shaped cavity between the gears. The composite design
(Fig. 1596) in which two separate gears are joined by resistance

Fig. 160. Composite constructions as substitutes for forgings

192

Chapter 6. General Design Principles

welding enables easy machining of the gears of required configuration.
The angle lever which is rather difficult to blank (Fig. 159c) can
advantageously be changed to a welded design composed of two simi¬
lar parts of simple shape (Fig. 159d).

Other examples of composite constructions are given in Fig. 159e-;\

Composite constructions are extremely helpful for intricately
shaped components. The cross-piece (Fig. 160a) which serves to
transmit motion from a cam to two valve tappets can be produced
only by stamping in enclosed dies, the latter being economical only
in the event of batch production. In piece production it is reasonable
to use a built-up component made from two lathe-turned parts press-
fitted together (Fig. 160b).

Composite constructions are popular substitutes for forgings when
dealing with shafts having large-diameter flanges, etc.

Figure 160c-e shows an example of how a forged flanged shaft is
transformed into a welded construction. In Fig. 160 d the problem
is solved but partially: the shaft here is produced from round rolled
stock and involves difficult drilling and boring operations.

In the improved design presented in Fig. 160e the shaft is produced
of a seamless pipe; the shaft end is formed by welding a length of
a smaller-diameter seamless pipe. Such a construction reduces ma¬
chining operations to the minimum.

It is very essential to note that the employment of composite
constructions as substitutes for forgings is justifiable only under
small-batch production conditions. Generally it is more advantageous
to produce parts from forgings that approximate as close as possible
in configuration, as this results in substantial strength gains,
lesser machining, greater productivity and, in the final analysis,
less cost.

However, even in batch production it is sometimes more reasonab¬
le to resort to composite designs as a means of simplifying forgings.
Let us take, for instance, the shaft of a planetary reduction unit
with a pinion carrier (Fig. 160/). Clearly, an attempt to forge these
units as one will cause quite a lot of difficulties. In the composite
design (Fig. 160g) the pinion carrier is made as a separate part and
fitted to the shaft on splines. The forging of both parts is very much
simplified.

In a number of cases components are made of light alloys, which
enables the weight of the unit to be radically decreased.

The cam plate (Fig. 161a) is better made from aluminium alloy,
to which a steel cammed rim is attached (Fig. 161b). Other examples
are: the axial compressor rotor with a steel drive splined rim (Fig.
161c, d); the brake drum with a steel liner on its frictional surface
(Fig. 161e, /).

Composite constructions find wide application as a means of eco¬
nomizing on expensive and scarce materials.

Fig. 161. Composite constructions made of light alloys

Fig. 1G2. Composite constructions as a means of saving scarce materials

13-01418

194

Chapter 6. General Design Principles

Worm gear teeth (Fig. 162a) working in sliding friction conditions
are normally made of antifriction bronze which is rather expensive.
To economize on bronze, bronze hoops are used (Fig. 162a), which
are then press-fitted upon carriers made of some cheaper material
(e.g., carbon steel, cast iron). Compared to Fig. 162a the amount of
bronze in Fig. 162fc is reduced to one third.

Clearly, with the gears shown in Fig. 162c only the gear rim need
be made of high-grade steel (Fig. 162d).

Figure 162e illustrates a rod made completely of alloy steel, with
its spherical end being hardened. The more economical construction
in Fig. 162/ has only the spherical end made of alloy steel.

Lead bronzes and babbits surely belong to short-supply materials.
Whenever conditions permit, low-lead or even leadless bronzes and
babbits, as well as adequately qualitative substitutes, e.g., antifric¬
tion aluminium alloys should be used.

Should the application of difficult-to-obtain non-ferrous alloys
be avoided, their consumption must be reduced to the minimum.
Let us take, for instance, a housing with many friction surfaces
(central bore and lug holes). In the alternative presented in Fig. 162g
the housing is made completely of expensive antifriction bronze,
but the rationalized version, illustrated in Fig. 162 h uses cast iron
(or some other abundant material), and bronze bushes for the
friction surfaces.

An example of how a material in short supply can be spared in the
production of bearing bushes is given in Fig. 162t and /. Here the
thick bronze bush (Fig. 162i) is changed to a thin-walled one rolled
from strap brass (grade JIC-59-1) and strengthened by rolling and
expanding in the seating bore (Fig. 162/).

An effective way of economizing on lead babbits is by decreasing the thick¬
ness of babbitting. These days the thickness is reduced to 0.2-0.3 mm (instead
of 1-3 mm practised quite recently). The most economic bearings are those with
a double-layer babbitting and consisting, in effect, of a babbit layer whose
thickness varies within several hundredths of a millimetre. The layer is deposited
by an electrodeposition technique upon a sublayer of porous bronze. The deposi¬
tion of babbit in the pores of the bronze sublayer assures good cohesion between
babbit and bronze and creates in the bronze sublayer a structural interface which
is rather similar to lead bronze in terms of antifriction properties.

6.31. Shoulders

Shoulders (Fig. 163a-d) serve as positive rests for components in
fixed joints or as stops which restrict the axial displacement of parts
in movable connections. The most rationally designed shoulders are
those that have (owing to their shape) equal resistance to bending
(Fig. 163d). Such shoulders possess the least weight and are simple
to produce. The non-working surfaces of shoulders are sloped at 45°

6.31. Shoulders

195

so that they can easily be turned by means of a bull-nose lathe tool
whose nose angle is 45°.

Shaped shoulders are uneconomical because they are difficult to
produce (Fig. 163ft).

Shoulders must be kept as low in height as possible: the higher
the shoulder, the greater the metal wastes in the form of chips and

Fig. 163. Shoulder designs

the more difficult the production process. Figure 164 shows how a
high shoulder in a gear unit can be eliminated. Shoulder m is for
holding the gear shaft in the housing and fixing the gear axially
(Fig. 164a). Itis the latterfunc-
tion which needs such a high
shoulder. Now, replacing the
shoulder by the washer n
which works upon a low step
of the shaft (Fig. 164ft) makes
the shaft manufacture easier.

Illustrated in Fig. 165a-o are
various techniques enabling
one either to make lower sho¬
ulders or to eliminate them
at all (for fixed joints only).

In Fig. 165 b-d the fitted part is tightened against an intermediate
washer resting against a step or shoulder of reduced height.

Shoulders are often replaced with ring stops having rectangular
cross-sections (Fig. 165e). The strength of the unit can further be
enhanced by fitting the ring into a cylindrical recess made in the
part or in some intermediate member, thus preventing the ring from
loosening and leaving the recess (Fig. 165/, g).

A very strong stop is ensured by rings enclosed in a conical recess
either in the part or in an intermediate washer (Fig. 165ft-/).

Fig. 164. Reduction of shoulder height

13 *

196

Chapter 6. General Design Principles

Occasionally use is made of half-rings placed into grooves on the
shaft and locked either by means of a recess in the part to be tighten¬
ed up (Fig. 165fc, l) or by means of an embracing ring (Fig. 165m).
In the design alternative given in Fig. 165rc embracing ring 1 is

Fig. 165. Reducing the shoulder height and replacing shoulders

fixed on half-rings by locking spring ring 2. The joint is easily
disassembled manually by merely shifting the ring axially.

In the version pictured in Fig. 165o the shoulder is formed by swag¬
ing a plastic metal ring into a groove on the shaft. The process
is done on rotary swaging machines; after swaging, the ring is
machined together with the shaft.

It should be noted that grooves weaken shafts, and fitting rings
into such grooves is not recommended for joints subjected to high
cyclic loads. In some cases weakening can be avoided by making the
shaft thicker across the section where a groove is needed (Fig. 165/).

6.32. Chamfers and Fillets

Generally all external corners must be chamfered (Table 8) and
all internal corners, filleted (Table 9).

Conventionally the chamfers are made at 45°. The leg c of a cham¬
fer for usual cylindrical components is obtained from the relation¬
ship c — 0.1 D, where D is the diameter of the cylinder. The va¬
lues obtained with the aid of this relationship are rounded off to
standard values:

c = 0.2; 0.5; 0.8; f; 1.2; 1.5; 1.8; 2; 2.5; 3; 3.5; 4; 5

6.32. Chamfers and Fillets

197

Free, non-contacting surfaces are given chamfers within 0.1-0.2 mm. Such
chamfers are not indicated in drawings (in contrast to the design chamfers);
a note merely says “Remove sharp comers”. Often the necessity for removing

Fig. 166. Overlapping of fillets

sharp corners is indicated in the Technical Specifications pertinent to the part,
the chamfer sizes and tolerances being specified.

Figure 166 shows the methods of overlapping fillets. The overlap¬
ping is effected by a fillet whose radius is larger than that of the part
being encompassed (Fig. 166a), or by a recess (Fig. 166fc) or else by
a chamfer (Fig. 166c). The latter method is the simplest one.

Chamfers

Table 8

Design

Design

wrong

correct

wrong

correct

Table 8 ( continued)

Application

Purpose

To facilitate screwing of nuts and
threaded rods

Threaded

connections

To facilitate use of spanners

To form an annular bearing surface
(with diameter £>). To prevent point
contact on bearing surface

Fillets

Table 9

Design

wrong

correct

Application

Purpose

Design

wrong

correct

Application

Purpose

Cb

vT

R1 m

B

■

p

■I

m

Heat-

treated

parts

Parts sub¬
jected to
chemical
and thermal
treatments
and thermal
diffusion
saturation

To prevent
edges from
overheating
and decarboni¬
zation. To re¬
duce quenching
stresses at tran¬
sitions

To ensure
uniform satu¬
ration of sur¬
face layer with
introduced ele¬
ments

Ml

I

mm,■

Electro¬
plated parts

To prevent
local variations
in current den¬
sity. To ensure
uniform depo¬
sition of metal

Painted,
varnished
and poly¬
merized
parts

To ensure uni¬
form coating.
To prolong ser¬
vice life of
coatings

Cast parts

To ensure
uniform solidi¬
fication of me¬
tal in cooling.
To reduce shrin¬
kage stresses

Application

Purpose

Stamped

parts

To improve
metal flow and
fill die reentry
angles

Parts
stamped
from sheets

To improve
metal flow. To
prevent disrup¬
tions at transi¬
tion areas

Reser¬

voirs

To eliminate
corrosion at
corners. To fa¬
cilitate flushing

Parts
subject to
heat erosion

To prevent
edges from
overheating
and burning

Table 9 ( continued)

Application

Purpose

Heat-dis¬

sipating

fins

To improve
heat transfer
from part body
to fins

Decorati¬
ve parts

To improve
outer appearan¬
ce. To facilita¬
te polishing

Hand

controls

To protect
hands from in¬
jury and facili¬
tate manipula¬
tion. To facili¬
tate polishing

Design

Design

Table 9 (continued)

Application

Purpose

Any loaded
part

To increase
static and cyc¬
lic strength at
transitions

Parts subject- To reduce

ed [to contact edge pressures
loads

Machined

parts

To enhance
durability of
cutting edges

Index

Accuracy of parts alignment, 179
Alignment of flanges, 105
Axial fixing of parts, 172

Bending, elimination and reduction
of, 137

Butt-jointing on intersecting surfaces,
177

Cambering, 161
Chamfers, 196

Combining design functions, 150
Compactness of design, 146
Compensators, 129
Composite constructions, 189
Connection(s),
centring, 79
conical flange, 113
design rules for, 80
glued, 78
loaded, 12
pressed, design of, 67
pressed, with electro-deposited
coatings, 65
press-fitted, 37
screwed, centring in, 92
serrated, 77
strength of, 38
three-flange, 111
tightened, 7
unloaded, 7

Control of direction, 174
Coupling of parts of different hardness,
183

Coupling(s), floating cross-sliding, 136
Design,

compactness of, 146
of equally strong parts, 152
general principles of, 116
of pressed connections, 67
of screwed connections, 94

Elasticity, effect on load distribution,
164

Elimination,

of adjustments, 124
of deformations due to tightening,
143

of local weak spots, 186
Equistrength, 152 ,

Fillets, 196

Fitting to several surfaces, 170
Fit(s),

conical, 75
drive, 37
press, 37
selection of, 46

Interchangeability of rapidly wearing
parts, 178

Methods of controlling preloads, 32
Mounting surfaces, 175

Press-fitting,

with cooling of parts, 64
with heating of parts, 64

Rationalization of power schemes,
126

Relaxation, 25

Relief of precision mechanisms, 181

Self-alignment, 157
Shoulders, 194

Strengthening of deformable areas, 188

Tightening up on two surfaces, 171
Torsion bars, 133

Unification,

of design elements, 116
of parts, 119

Uniform loading of supports, 156
Unitization, principles of, 121

TO THE READER

Mir Publishers welcome your comments on the content, trans¬
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We would also be pleased to receive any suggestions you care
to make about our future publications.

Our address is:

USSR, 129820, Moscow, 1-110, GSP, Pervy Rizhsky Pereulok, 2,
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