1
Problem 3-4-1
A B
y
z
x
500N
300N
150mm
100mm
500N
300N
Rz=500N
Ry=300N
My=500(0.15) =75N·m
Mz=300(0.15) =45N·m
Ry
Rz
My
Mz
A B A
B
( )( ) 494
232
m)10(4002.04
m)10(4.002.0
ππ
ππ
−
−
===
==
yz II
A
0.02m
Problem 3-4-1
B
B Ry
Mz
Rz
My
( )( ) MPa16.7)10(40020459 −=−==− π
σ.
IcM z
x
( )
( )MPa531.0
04.0)02.0(3
)02.0(42
)02.0(5004
2
===π
ππ
τItVQ
B
Side View
Top View
A
A Rz
My
Ry
Mz
Side View
Top View
A
( )( ) MPa8.29)10(40020759 ==
=
− π
σ
.IcM y
x
( )
( )MPa318.0
04.0)10(403
)02.0(42
)02.0(3009
2
===− π
ππ
τItVQ
x
z
y
x
z
y
Problem 3-4-2
c
10in
14in 500lb
800lb
500lb
800lb
T
Mx
Mz
Rz
y
x
z Ry
T
Mx
Mz Rz Ry
T
lb800;0800;0
lb500
;0500;00
=
∑ =−=
=
=+∑ −=
=∑
z
zz
y
yy
x
RRF
R
RFF
inlb000,7;0)14(500;0
inlb200,11
;0)14(800;0inlb000,8
;0)10(800;0
⋅=
=−=∑
⋅=
=−=∑
⋅=
=−=∑
z
zz
y
x
xx
MMM
T
TMMMM
kjikjiFrM
jirkjF
7000112008000
1014800500
++−=
++=×=
+=
−=
zx MTM
r y
Maximum Stress?
Problem 3-4-2 Axial Load: Ry; Normal (tensile) stress
Shear Force: Rz; Shear stress
Bending Moment: Mz; Normal Stress
A
A
A
A
A A
psi283=
=c
yy AR
σ
( )[ ]
4
4
3
in2813.0'' wherepsi604
)5.1(4/75.0)in2813.0(800
==
=
==
AyQ
ItVQ
πτ
( )( )psi
IcM z
126,214/75.0)75.0(7000
4
=
==π
σ
A
Bending Moment: Mx; Bending Stress
A
No stress on A
N.A. y
( )psi901,16
2/)75.0()75.0(200,11
4
=
==π
τJTc
A
A
Torsion: T; Shear Stress
Summing up
A
τyz=16.9+0.6 =17.5ksi
σy
σz
τyz
σz = 0 psi σy = 21.1+0.28 =21.4 ksi
2
Problem 3-4-2 ksi4.241.243.0 =+=yσ
Axial Load: Ry; Normal (tensile) stress
Shear Force: Rz; No Shear stress
Bending Moment: Mz; No Normal Stress
B B B psi283=yσ
Bending Moment: Mx; Bending Stress B
( )ksi1.24
4/)75.0()75.0(8000
4
=
==π
σIcM x
y
B Torsion: T; Shear Stress
ksi9.16=τ
B B B
B B
B ksi9.16=τ Problem 8.2.4
3/2
)3/4)(2/(3
2max
rrryAQ
=
== ππ
( )ksi21.1630.1609.04/)5.0()5.0(1600
5.075
42
=+−=
+−=
+−=
ππ
σIcM
AR
x
c
yy
Mz=125(8)-75(3) =775 lb-in
Mx=200(8) =1600 lb-in
A
B
Ty=200(3) =600 lb-in
A
( )( ) ( ) ksi84.2
2/5.0)5.0(600
)1(4/5.03/)5.0(2125
44
3
max
−=−
+=
+=
ππ
τJcT
ItQR yx
xy
B ( )ksi78.7
4/)5.0()5.0(775
5.075
42
=
+−=
+−=
ππ
σIcM
AR
z
c
yy
€
τyz =RzQmax
It+MycJ
=200 2(0.5)3 /3( )π0.54 /4( )(1)
+600(0.5)π0.54 /2( )
= 3.4ksi
A
B
y
x
z
3’
8’
Fy=75lb Fx=125lb
Fz=200lb
d = 1”
Mz=125(8)-75(3) =775 lb-in
Mx=200(8) =1600 lb-in
Ty=200(3) =600 lb-in
75lb 125lb
200lb
775 lb-in
1600 lb-in 125lb
200lb
inlb82.17771600775 22
−=
+=RM
lb85.235200125 22 =+=RV
32o
25.84o
€
V =VR cos 32° − 25.84°( )= 235.85cos6.14 = 25.31lb
€
σ = −MRcI
−Ry
A= −1777.82⋅ 0.5π 0.5( )4 4
−75
π 0.5( )2
= −18.1− 0.095 = −18.195ksi
€
τ =VQIb
+TcJ
=25.31⋅ 2c 3 3πc 4 4( ) 2c( )
+600cπc 4 2
= 0.085 + 3.055 = 3.14ksi
€
D = 2R = 2 9.12 + 3.1442 =19.25ksi25.84°
Problem 8.2.4 Curved Beam in Bending
( )
nc
c
n
n
n
rrerr
yrAeMyrdAAr
−=
=
=
−=
=
∫
axiscentroidalofradiusaxisneutralofradiuswhere
σ
See Table 3-4 o
oo
i
ii
AerMcAerMc
=
=
σ
σ
( )ion
ic
rrhr
hrr
ln
2
=
+=
h
b rn rc
ri
ro
3
Contact Stresses • Hertzian Stress
– Spherical Contact ( ) ( )
( )
2
2max
3
2
21
max
21
2max
3
21
2221
21
1
12
111tan1
23
1111
83
azp
azaza
zp
aFp
ddEEFa
z
yx
+
−==
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−+⎟⎟⎠
⎞⎜⎜⎝
⎛−−
====
=
+
−+−=
−
σσ
ν
σσσσπ
νν
Contact Stresses • Hertzian Stress
– Spherical Contact ( ) ( )
( )
2
2max
3
2
21
max
21
2max
3
21
2221
21
1
12
111tan1
23
1111
83
azp
azaza
zp
aFp
ddEEFa
z
yx
+
−==
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−+⎟⎟⎠
⎞⎜⎜⎝
⎛−−
====
=
+
−+−=
−
σσ
ν
σσσσπ
νν
Cylindrical Contact ( ) ( )
2
2max
3
2
2
2
2
max
2
2
max
max
21
2221
21
1
21
21
12
2311112
azp
bz
bzbz
p
bz
bzp
lFp
ddEE
lFb
z
y
x
+
−==
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−
+
+−=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−+−=
=
+
−+−=
σσ
σ
νσ
π
ννπ
Figure 10.1 Internally pressurized thin-walled cylinder. (a) Stress element on cylinder; (b) stresses acting on element.
Internally Pressurized Thin-Walled Cylinder
Stresses in Thin- Walled Cylinders:
CylinderwalledThick40
CylinderwalledThin40
−<
−>
h
i
h
i
tdtd
4
Stresses in Pressurized Cylinders ( )
( )
22
2
22
22222
22
22222
io
iil
io
iooiooiir
io
iooiooiit
rrrp
rrrpprrrprp
rrrpprrrprp
−=
−
−+−=
−
−−−=
σ
σ
σpo
pi
Thick-Walled
( )
tpd
ttdp
tpd
il
ir
it
4
2
2
=
+=
=
σ
σ
σThin-walled
p
Problem 3-1
MPatrp
MPatrp
MPap
iiz
ii
ir
50)05.0(2)5.0(10
2
10005.0)5.0(10
10
===
===
−=−=
σ
σ
σ
θ
( )
( ) MParrpr
MParrrrpMPap
io
iiz
io
ioi
ir
62.475.055.0)10(5.0
24.1055.055.0)5.055.0(10
10
22
2
22
2
22
22
22
22
max.
max.
=−
=−
=
=−
+=
−
+=
−=−=
ππ
σ
σ
σ
θ
MPatrp
MPatrp
MPap
iiz
ii
ir
100)025.0(2)5.0(10
2
200025.0)5.0(10
10
===
===
−=−=
σ
σ
σ
θ
High Pressure Cylinder (pi=10MPa, ri=0.5m, t=5cm)
Thin
-wal
led
Cylin
der
Thick-walled Cylinder
High Pressure Cylinder (pi=10MPa, ri=0.5m, t=2.5cm)
( )
( ) MParrpr
MParrrrpMPap
io
iiz
io
ioi
ir
56.975.0525.0)10(5.0
12.2055.0525.0)5.0525.0(10
10
22
2
22
2
22
22
22
22
max.
max.
=−
=−
=
=−
+=
−
+=
−=−=
ππ
σ
σ
σ
θ
40>h
i
td
Pressurized Cylinders
( )( )
( )( )22
22
22
22
1
1
io
oii
io
oiir
rrrrrp
rrrrrp
−
+=
−
−=
θσ
σ
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
1
1
2
2
22
2
2
2
22
2
rr
rrrp
rr
rrrp
i
io
oo
i
io
oor
θσ
σ
( )
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−
+==
−
+=
−==
ν
νε
σ
σ
θ
θ
22
22
22
22
22
22
max.
max.,At
io
ioii
io
ioi
i
io
ioi
iri
rrrr
Erpu
rrrr
Ep
ru
rrrrp
prr
22
2
max.
max.
2
,At
io
io
oro
rrpr
prr
−−=
−==
θσ
σ
3-16 Press and Shrink Fit
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
−+
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
−+
=
ii
o
io
i
o
o rRrR
ERrRr
ER
pνν
δ
22
22
22
22 11
( )( )( )[ ]222
2222
2 io
io
rrRrRRr
REp
−
−−=
δ
oi δδδ +=
For same materials
At the interface,
( ) ( ) ( ) ( )
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−
+=
−==−+
=
oo
o
oo
o
oro
o
otooot
RrRr
EpR
Ev
ERRRR
νδ
σσδπ
πδπε
22
22
222
pr −=σ( )
( )22
22
22
22
RrRrp
rRrRp
o
oRrot
i
iRrit
−
+=
−
+−=
=
=
σ
σ
The total radial interference
( ) ( ) ( ) ( )
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−
+−=
−=−=−−
=
ii
i
ii
i
iri
i
itiiit
rRrR
EpR
Ev
ERRRR
νδ
σσδπ
πδπε
22
22
222
5
Press Fit - Front View
Interference Fit
( ) ( )( )
( )( )
( )22
2
2222
223
22
22
22
22
2),0(shafthollowFor
2,materialssameFor
fo
offri
iffo
ioffr
s
s
ifs
if
h
h
foh
foff
rsrhr
rrErpr
r
rrrrErrpr
ErrErr
ErrErr
pr
−==
−−
−=
⎥⎥⎦
⎤
⎢⎢⎣
⎡−
−
+++
−
+=
−=
δ
δ
νν
δδδ 3-15 Stresses in Rotating Rings
⎟⎟⎠
⎞⎜⎜⎝
⎛−−+⎟
⎠
⎞⎜⎝
⎛ +=
⎟⎟⎠
⎞⎜⎜⎝
⎛
+
+−++⎟
⎠
⎞⎜⎝
⎛ +=
22
22222
22
22222
83
331
83
rrrrrr
rrrrrr
oioir
oioit
νρωσ
ννν
ρωσ
tro 10≥For
Rotating disk ( )222
222
83
331
83
rr
rr
or
ot
−⎟⎠
⎞⎜⎝
⎛ +=
⎟⎠
⎞⎜⎝
⎛+
+−⎟
⎠
⎞⎜⎝
⎛ +=
νρωσ
ννν
ρωσ
Rotating Cylinders
ρωσσσθ rdrd
rrr 2=−
−
⎟⎟⎠
⎞⎜⎜⎝
⎛−−+
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛
+
+−++
+=
22
22222
22
22222
83
331
83
rrrrrrv
rrrrrrv
oioir
oioi
ρωσ
νν
ρωσθ
( )222
222
83
331
83
rrv
rrv
or
i
−+
=
⎟⎠
⎞⎜⎝
⎛+
+−
+=
ρωσ
νν
ρωσθ
( ) ( )
( ) ( ) oioir
ii
o
rrrrrv
rrrrv
=−+
=
=⎟⎟⎠
⎞⎜⎜⎝
⎛
+
−+
+=
at83
at31
83
22max
222
max
ρωσ
νν
ρωσθ
( ) ( ) ( ) 0at83 2
maxmax =+
== rrvor ωρσσθ
Press Fit
( )rhhf
rhrh Er
σνσδ
ε θ −==1 ( )rs
sf
rsrs Er
σνσδ
ε θ −==1
( )
( ) ( )22
22
22
222
22
222
1
1
fo
fof
fo
foff
ffo
foffr
rrrrp
rrrrrp
prrrrrp
−
+=
−
+=
−=−
−=
θσ
σ
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−
+−== s
if
if
s
ffsrs rr
rrEpr
u νδ 22
22
For Hub For Shaft
( )22
22
2
2
22
2
2
2
22
2
1
1
if
iff
f
i
if
ff
ff
i
if
ffr
rrrrp
rr
rrrp
prr
rrrp
−
+−=⎟
⎟⎠
⎞⎜⎜⎝
⎛+
−−=
−=⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=
θσ
σ
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−
+== h
fo
fo
h
ffhrh rr
rrEpr
u νδ 22
22
6
THERMAL STRESS Increase in Temperature, ΔT, in both bars
F
where α = Coefficient of Thermal Expansion (CTE)
ΔT = Temperature change
Reaction Force F = 0 F = E A α ΔT Thermal Stress σT = 0 σT = E α ΔT Thermal Strain εT = α ΔT εT = 0 Thermal Displacement δΤ = α ΔT Lo δΤ = 0
Shrink Fit
Tru
Tru
r
fr
ff
rr
Δ==
Δ===
αδ
αδ
ε
( )[ ]
( )[ ]
( )[ ] TvE
TvE
TvE
yxzz
zxyy
zyxx
Δ+−−=
Δ+−−=
Δ+−−=
ασσσε
ασσσε
ασσσε
1
1
1
Achieve Interference by cool down the ‘hot’ hub after assembly