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Chapter 4

Stress

Mechanical Engineering Design

Seventh Edition

Shigley • Mischke • Budynas

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Fig. 4.10 The general three dimensional stress elements.

Stress Components

The element is in equilibrium, hence

xy yx

yz zy

zx xz

Fig. 4.11 Two dimensional stress state.

2 2cos sin 2 cos sinx y xy

2 2( )sin cos (cos sin )x y xy

cos 2 sin 22 2

x y x y

xy

sin 2 cos 22

x y

xy

2

2

1,22 2

x y x y

xy

2

2 1 2maxmin 2 2

x y

xy

2tan 2

xy

p

x y

tan 22

y x

s

xy

Stress Transformation

Fig. 4.12 Mohr circle diagram.

2 2

2 2

2 2

x y x y

xy

Fig. 4.14 Mohr circles for triaxial stresses.

1 2

1/ 2

2 32 /3

1 31/3

2

2

2

1/ 22 2 2

1 2 2 3 3 1

1/ 22 2 2 2 2 2

1 2 3

1

3

16

3

1 1

3 3

oct

x y y z z x xy xz yz

oct x y z

Octahedral stresses

Fig. 4.15 Bending of a beam.

2 2

( ) ( )y z yz z yz y

x

y z yz y z yz

M zI yI M zI yI

I I I I I I

Normal Stresses in Flexure

Shear Stresses in Beams

1

c

y

VQ

Ib

Q ydA

max

3

2

V

A

max

2V

A

max

4

3

V

A

max

web

V

A

Torsion T

J

Tl

GJ

1 2

2

4

T

At

Tl

GA t

1 2

3TG c

Lc

Closed thin-walled tubes

Open thin-walled sections

9.55

HT

n

Rectangular cross sections

max 2 2

1.83

/

is the longer side

is the shorter side

T T

bc bc b c

b

c

3

Tl

bc G

b/c 1.0 1.5 2.0 2.5 4.0 8.0

0.208 0.231 0.246 0.258 0.282 0.307 0.333

0.141 0.196 0.228 0.249 0.281 0.307 0.333

Fig. 4.31 Stress distribution near a hole in

a plate loaded in tension.

Stress Concentration

max

0

tK

max

0

tsK

For an infinite plate containing an elliptical

hole

21

is the half-width

is the half-height

t

bK

a

b

a

In ductile materials the stress-concentration

factor is not usually applied to predict the

critical stresses, because plastic strain in the

region of the stress has a strengthening

effect.

In brittle materials the geometric stress-

concentration factor Kt is applied to the

nominal stress before comparing with

strength.

Fig. 4.33 A cylinder subjected

to both internal and externel

pressure

Stresses in Cylinders

2 2 2 2 2

0 0 0 0

2 2

0

( ) /i i i it

i

p r p r r r p p r

r r

2 2 2 2 2

0 0 0 0

2 2

0

( ) /i i i ir

i

p r p r r r p p r

r r

2

2 2

0

i il

i

p r

r r

Fig. 4.34 Distribution of stresses in a thick-walled

cylinder subjected to internal pressure.

Thin-walled vessels

i i

t

p r

t

2

i il

p r

t

Fig. 4.35 Notation for press and shrink fits. (a) Unassembled

parts. (b) after assembly.

Press and Shrink Fits

2 2 2 2

002 2 2 2

0 0

ii

i i

r R R rpR pR

E r R E R r

2 2

2 2( ) i

it

i

R rat R p

R r

2 2

0

2 2

0

( )ot

r Rat R p

r R

Rotating Rings

2 22 2 2 2

2

2 22 2 2 2

2

3 1 3

8 3

3

8

i ot i o

i or i o

r rr r r

r

r rr r r

r

Fig. 4.36 Thermal stresses in an infinite slab during

heating and cooling.

Temperature Effects

A thermal stress is one which arises

because of the existence of a

temperature gradient in a member.

( )T E

Fig. 4.37 Note that y is positive in the direction toward point O.

Curved Members in Flexure n

Ar

dA

r

( )n

My

Ae r y

i

i

i

Mc

Aer

o

o

o

Mc

Aer

( )

( )

n

c n

M r r

Ar r r

ln( / )n

o i

hr

r r

2c i

hr r 2

3

i oc i

i o

b bhr r

b b

[( ) / ]ln( / )n

o i i o o i o i

Ar

b b b r b r h r r

2c i

dr r

2

2 24(2 4 )n

c c

dr

r r d

Fig. 4.42 (a) Two spheres held in

contact by force F; (b) contact stress

has an elliptical distribution across

contact zone diameter 2a.

Contact Stresses Contact stress problems arise in the contact of a wheel and a rail, in automotive valve cams and

tappets, in mating gear teeth, and in the action of rolling bearings.

Typical failures are seen as cracks, pits, or flaking in the surface material.

The results presented here are due to Hertz and so are frequently known as Hertzian stresses.

The maximum stress occur on the z axis, and these

are principal stresses.

2 2

1 1 2 23

1 2

(1 ) / (1 ) /3

8 1 1

E EFa

d d

max 2

3

2

Fp

a

1

max 2

2

1 11 tan (1 )

2 1

x y

zp

za za

a

max

2

21

z

p

z

a

Fig. 4.44 (a) Two right circular

cylinders held in contact by forces F

uniformly distributed along cylinder

length l. (b) Contact stress has an

elliptical distribution across the contact

zone width 2b.

2 2

1 1 2 2

1 2

(1 ) / (1 ) /2

1 1

E EFb

l d d

max

3Fp

bl

2

max 22 1x

z zp

b b

2

max 22

2

12 1 2

1

y

z zp

b bz

b

max

2

21

z

p

z

b

Fig. 4.43 Magnitude of the stress

components below the surface as a

function of the maximum pressure of

contacting spheres.

Fig. 4.45 Magnitude of the stress

components below the surface as a

function of the maximum pressure of

contacting cylinders.