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