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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.