Chapter02.pdf

CAPITULO 2 MECANICA DE MATERIALES BEER

1. CCHHAAPPTTEERR 22

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PROBLEM 2.1
An 80-m-long wire of 5-mm diameter is made of a steel with 200 GPaE= and an ultimate tensile strength of
400 MPa. If a factor of safety of 3.2 is desired, determine (a) the largest allowable tension in the wire, (b) the
corresponding elongation of the wire.

SOLUTION
(a)
6222 62
400 10 Pa (5) 19.635 mm 19.635 10 m
44
U
Ad
ππ
σ
−
=× = = = = ×

66
all
(400 10 )(19.635 10 ) 7854 N
7854
2454 N
.3.2
UU
U
PA
P
P
FSσ
−
==× × =
== =

all
2.45 kN=P 
(b)
3
69(2454)(80)
50.0 10 m
(19.635 10 )(200 10 )
PL
AE
δ
−
−
== =×
×× 50.0 mmδ= 

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PROBLEM 2.2
A steel control rod is 5.5 ft long and must not stretch more than 0.04 in. when a 2-kip tensile load is applied to it.
Knowing that
6
29 10 psi,E=× determine (a) the smallest diameter rod that should be used, (b) the
corresponding normal stress caused by the load.


SOLUTION
(a)
6
(2000 lb) (5.5 12 in.)
0.04 in.
(29 10 psi)
:
PL
AE A
δ
×
==
×


221
0.11379 in
4
π==Ad

0.38063 in.d= 0.381 in.d= 
(b)
2
2000 lb
17580 psi
0.11379 in
σ== =
P
A 17.58 ksiσ= 

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PROBLEM 2.3
Two gage marks are placed exactly 10 in. apart on a
1
2
-in.-diameter aluminum rod with
6
10.1 10E=× psi and
an ultimate strength of 16 ksi. Knowing that the distance between the gage marks is 10.009 in. after a load is
applied, determine (a) the stress in the rod, (b) the factor of safety.

SOLUTION
(a) 10.009 10.000 0.009 in.δ=−=

6
3
(10.1 10 )(0.009)
9.09 10 psi
10
E
LE Lδσ δ
εσ ×
== = = = ×
9.09 ksiσ= 
(b)
16
..
9.09σ
σ
==
U
FS . . 1.760=FS 

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PROBLEM 2.4
An 18-m-long steel wire of 5-mm diameter is to be used in the manufacture of a prestressed concrete beam.
It is observed that the wire stretches 45 mm when a tensile force P is applied. Knowing that
200 GPa,E=
determine (a) the magnitude of the force P, (b) the corresponding normal stress in the wire.


SOLUTION
(a) ,or
PLAE
P
AE L δ
δ
==
with
22 6211
(0.005) 19.6350 10 m
44
Ad
ππ
−
== = ×

62 9 2
(0.045 m)(19.6350 10 m )(200 10 N/m )
9817.5 N
18 m
P
−
××
==


9.82 kNP= 
(b)
6
629817.5 N
500 10 Pa
19.6350 10 m
P
A
σ
−
== = ×
× 500 MPaσ= 

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PROBLEM 2.5
A polystyrene rod of length 12 in. and diameter 0.5 in. is subjected to an 800-lb tensile load. Knowing that
E
6
0.45 10 psi,=× determine (a) the elongation of the rod, (b) the normal stress in the rod.

SOLUTION

22 2
(0.5) 0.19635 in
44
ππ
== =Ad
(a)
6
(800) (12)
0.1086
(0.19635) (0.45 10 )
PL
AE
δ== =
× 0.1086 in.δ= 
(b)
800
4074 psi
0.19635
P
A
σ== = 4.07 ksiσ= 

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PROBLEM 2.6
A nylon thread is subjected to a 8.5-N tension force. Knowing that 3.3 GPa=E and that the length of the
thread increases by 1.1%, determine (a) the diameter of the thread, (b) the stress in the thread.


SOLUTION
(a) Strain:
1.1
0.011
100Lδ
ε
== =
Stress:
96
(3.3 10 )(0.011) 36.3 10 PaEσε== × = ×

P
A
σ=
Area:
92
68.5
234.16 10 m
36.3 10
P
A
σ
−
== = ×
×
Diameter:
9
6
4(4)(234.1610)
546 10 m
A
d
ππ
−
−
×
== =×
0.546 mmd= 
(b) Stress:
36.3 MPaσ= 

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PROBLEM 2.7
Two gage marks are placed exactly 250 mm apart on a 12-mm-diameter aluminum rod. Knowing that, with an
axial load of 6000 N acting on the rod, the distance between the gage marks is 250.18 mm, determine the
modulus of elasticity of the aluminum used in the rod.

SOLUTION

0
0
22 2 62
6
6
250.18 250.00 0.18 mm
0.18 mm
0.00072
250 mm
(12) 113.097 mm 113.097 10 m
44
6000
53.052 10 Pa
113.097 10δ
δ
ε
ππ
σ
−
−
=Δ = − = − =
== =
== = = ×
== = ×
×
LLL
L
Ad
P
A


6
9
53.052 10
73.683 10 Pa
0.00072
Eσ
ε ×
== = ×
73.7 GPaE= 

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PROBLEM 2.8
An aluminum pipe must not stretch more than 0.05 in. when it is subjected to a tensile load. Knowing that
6
10.1 10E=× psi and that the maximum allowable normal stress is 14 ksi, determine (a) the maximum
allowable length of the pipe, (b) the required area of the pipe if the tensile load is 127.5 kips.


SOLUTION
(a) ;
PL
AE
δ=
Thus,
6
3
(10.1 10 )(0.05)
14 10
EA E
L
Pδδ
σ ×
===
×


36.1 in.=L 
(b)
;
P
A
σ=
Thus,
3
3
127.5 10
14 10
P
A
σ
×
==
×

2
9.11 in=A 

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PROBLEM 2.9
An aluminum control rod must stretch 0.08 in. when a 500-lb tensile load is applied to it. Knowing that
all
22σ=ksi and
6
10.1 10E=× psi, determine the smallest diameter and shortest length that can be selected for
the rod.

SOLUTION

3
all
2
all 3
all
2
500 lb, 0.08 in. 22 10 psi
500
0.022727 in
22 10
4 (4)(0.022727)
4
P
PP
A
A
A
Ad dδσ
σσ
σ
π
ππ== =×
=< > = =
×
===

min
0.1701 in.=d 

all
6
3
all
(10.1 10 )(0.08)
36.7 in.
22 10
E
E
L
E
L
δ
σε σ
δ
σ
== <
×
>= =
×

min
36.7 in.=L 

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PROBLEM 2.10
A square yellow-brass bar must not stretch more than 2.5 mm when it is subjected to a tensile load. Knowing
that
105E= GPa and that the allowable tensile strength is 180 MPa, determine (a) the maximum allowable
length of the bar, (b) the required dimensions of the cross section if the tensile load is 40 kN.


SOLUTION
63
93
180 10 Pa 40 10 N
105 10 Pa 2.5 10 m
P
Eσ
δ
−
=× =×
=× =×

(a)
93
6
(105 10 )(2.5 10 )
1.45833 m
180 10
PL L
AE E
E
L
σ
δ
δ
σ
−
==
××
== =
×

1.458 mL= 
(b)
3
62 2
6
40 10
222.22 10 m 222.22 mm
180 10
P
A
P
A
σ
σ
−
=
×
== = × =
×

2
222.22Aa a A=== 14.91 mma= 

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PROBLEM 2.11
A 4-m-long steel rod must not stretch more than 3 mm and the normal stress must not exceed 150 MPa when
the rod is subjected to a 10-kN axial load. Knowing that
200 GPa,E= determine the required diameter of
the rod.


SOLUTION
4mL=
36
93
310m, 15010Pa
200 10 Pa, 10 10 NEPδσ
−
=× = ×
=× =×

Stress
:
3
62 2
6
10 10
66.667 10 m 66.667 mm
150 10
P
A
P
A
σ
σ
−
=
×
== = × =
×
Deformation
:
3
62 2
93
(10 10 ) (4)
66.667 10 m 66.667 mm
(200 10 )(3 10 )
PL
AE
PL
A
E
δ
δ
−
−
=
×
== = × =
××
The larger value of A governs:
2
66.667 mmA=

2 4 4(66.667)
4
A
Addπ
ππ
=== 9.21 mmd= 

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PROBLEM 2.12
A nylon thread is to be subjected to a 10-N tension. Knowing that 3.2E= GPa, that the maximum allowable
normal stress is 40 MPa, and that the length of the thread must not increase by more than 1%, determine the
required diameter of the thread.


SOLUTION
Stress criterion:

6
92
6
9
26
40 MPa 40 10 Pa 10 N
10 N
:25010m
40 10 Pa
250 10
: 2 2 564.19 10 m
4
P
PP
A
A
A
Addσ
σ
σ
π
ππ
−
−
−
==× =
=== =×
×
×
=== =×


0.564 mmd=
Elongation criterion
:

1% 0.01
:
L
PL
AE
δ
δ
==
=


9
92
9
62
/10N/3.210Pa
312.5 10 m
/0.01
312.5 10
2 2 630.78 10 m
PE
A
L
A
d
δ
ππ
−
−
−×
== = ×
×
== = ×


0.631 mmd=
The required diameter is the larger value:
0.631 mmd= 

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PROBLEM 2.13
The 4-mm-diameter cable BC is made of a steel with 200 GPa.=E Knowing
that the maximum stress in the cable must not exceed 190 MPa and that the
elongation of the cable must not exceed 6 mm, find the maximum load P that
can be applied as shown.

SOLUTION

22
6 4 7.2111 m=+=
BC
L
Use bar AB as a free body.

4
0: 3.5 (6) 0
7.2111
0.9509
Σ= − =


=
ABC
BC
MP F
PF
Considering allowable stress:
6
190 10 Paσ=×

22 62
663
(0.004) 12.566 10 m
44
(190 10 )(12.566 10 ) 2.388 10 N
ππ
σσ
−
−
== = ×
=∴==× ×=×
BC
BC
Ad
F
FA
A

Considering allowable elongation:
3
610 mδ
−
=×

693
3
(12.566 10 )(200 10 )(6 10 )
2.091 10 N
7.2111
BC BC
BC
BC
FL AE
F
AE L δ
δ
−−
×××
=∴== =×

Smaller value governs.
3
2.091 10 N
BC
F=×

33
0.9509 (0.9509)(2.091 10 ) 1.988 10 N
BC
PF== ×=× 1.988 kNP= 

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PROBLEM 2.14
The aluminum rod ABC (
6
10.1 10 psi),E=× which consists of two
cylindrical portions AB and BC, is to be replaced with a cylindrical steel
rod DE (
6
29 10 psi)E=× of the same overall length. Determine the
minimum required diameter d of the steel rod if its vertical deformation is
not to exceed the deformation of the aluminum rod under the same load
and if the allowable stress in the steel rod is not to exceed 24 ksi.

SOLUTION
Deformation of aluminum rod.

3
62 2
44
28 10 12 18
10.1 10 (1.5) (2.25)
0.031376 in.
BCAB
A
AB BC
BCAB
AB BC
PLPL
AE AE
LLP
EA A
ππ
δ=+

=+


×
=+


×

=

Steel rod
. 0.031376 in.δ=

3
2
6
3
2
3
(28 10 )(30)
0.92317 in
(29 10 )(0.031376)
28 10
1.1667 in
24 10
δ
δ
σ
σ
×
=∴== =
×
×
=∴== =
×
PL PL
A
EA E
PP
A
A

Required area is the larger value.
2
1.1667 in=A
Diameter
:
4(4)(1.6667)A
d
ππ
== 1.219 in.=d 

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PROBLEM 2.15
A 4-ft section of aluminum pipe of cross-sectional area 1.75 in
2
rests on a
fixed support at A. The
5
8
-in.-diameter steel rod BC hangs from a rigid
bar that rests on the top of the pipe at B. Knowing that the modulus of
elasticity is
6
29 10× psi for steel, and
6
10.4 10× psi for aluminum,
determine the deflection of point C when a 15-kip force is applied at C.

SOLUTION
Rod BC:
6
7 ft 84 in. 29 10 psi
BC BC
LE== =×

22 2
3
/ 6
(0.625) 0.30680 in
44
(15 10 )(84)
0.141618 in.
(29 10 )(0.30680)
ππ
δ
== =
×
== =
×
BC
BC
CB
BC BC
Ad
PL
EA


Pipe AB
:
6
4 ft 48 in. 10.4 10 psi
AB AB
LE== =×

2
3
3
/ 6
1.75 in
(15 10 ) (48)
39.560 10 in.
(10.4 10 ) (1.75)
δ
−
=
×
== =×
×
AB
AB
BA
AB AB
A
PL
EA

Total
:
3
//
39.560 10 0.141618 0.181178 in.δδ δ
−
=+= ×+ =
CBACB


0.1812 in.
C
δ= 

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PROBLEM 2.16
The brass tube (105GPa)=AB E has a cross-sectional area of
140 mm
2
and is fitted with a plug at A. The tube is attached at B to a
rigid plate that is itself attached at C to the bottom of an aluminum
cylinder
(72GPa)E= with a cross-sectional area of 250 mm
2
. The
cylinder is then hung from a support at D. In order to close the
cylinder, the plug must move down through 1 mm. Determine the force
P that must be applied to the cylinder.

SOLUTION
Shortening of brass tube AB:

262
9
9
96
375 1 376 mm 0.376 m 140 mm 140 10 m
105 10 Pa
(0.376)
25.578 10
(105 10 )(140 10 )
δ
−
−
−
=+= = = =×
=×
== =×
××
AB AB
AB
AB
AB
AB AB
LA
E
PL P
P
EA

Lengthening of aluminum cylinder CD:

262 9
9
96
0.375 m 250 mm 250 10 m 72 10 Pa
(0.375)
20.833 10
(72 10 )(250 10 )
δ
−
−
−
===×=×
== =×
××
CD CD CD
CD
CD
CD CD
LA E
PL P
P
EA

Total deflection:
AABCD
δδ δ=+ where 0.001 m
A
δ=

99
0.001 (25.578 10 20.833 10 )
−−
=×+×
P

3
21.547 10 NP=× 21.5 kNP= 

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PROBLEM 2.17
A 250-mm-long aluminum tube (70 GPa)E= of 36-mm outer
diameter and 28-mm inner diameter can be closed at both ends by
means of single-threaded screw-on covers of 1.5-mm pitch. With
one cover screwed on tight, a solid brass rod
( 105 GPa)E= of
25-mm diameter is placed inside the tube and the second cover is
screwed on. Since the rod is slightly longer than the tube, it is
observed that the cover must be forced against the rod by rotating
it one-quarter of a turn before it can be tightly closed. Determine
(a) the average normal stress in the tube and in the rod, (b) the
deformations of the tube and of the rod.

SOLUTION

22 2 2 2 62
tube
22 2 62
rod
9
tube 96
tube tube
rod 6
rod rod
( ) (36 28 ) 402.12 mm 402.12 10 m
44
(25) 490.87 mm 490.87 10 m
44
(0.250)
8.8815 10
(70 10 )(402.12 10 )
(0.250)
(105 10 )(490.87 1
oi
Add
Ad
PL P
P
EA
PL P
EA
ππ
ππ
δ
δ
−
−
−
−
=−= −= = ×
== = = ×
== =×
××
=− =
××
9
6
*6
**
tube rod tube rod
996
3
3
9
4.8505 10
0)
1
turn 1.5 mm 0.375 mm 375 10 m
4
or
8.8815 10 4.8505 10 375 10
0.375 10
27.308 10 N
(8.8815 4.8505)(10 )
P
PP
P
δ
δδδ δδδ
−
−
−
−−−
−
−
=− ×

=×= =×


=+ − =
×+ ×=×
×
==×
+

(a)
3
6
tube 6
tube
27.308 10
67.9 10 Pa
402.12 10
σ
−
×
== =×
×
P
A

tube
67.9 MPaσ= 

3
6
rod 6
rod
27.308 10
55.6 10 Pa
490.87 10
σ
−
×
=− =− =− ×
×
P
A

rod
55.6 MPaσ=− 
(b)
93 6
tube
(8.8815 10 )(27.308 10 ) 242.5 10 mδ
−−
=× ×=×
tube
0.2425 mmδ= 

93 6
rod
(4.8505 10 )(27.308 10 ) 132.5 10 mδ
−−
=− × × =− ×
rod
0.1325 mmδ=− 

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reproduced, or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited
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PROBLEM 2.18
The specimen shown is made from a 1-in.-diameter cylindrical steel rod
with two 1.5-in.-outer-diameter sleeves bonded to the rod as shown.
Knowing that
6
29 10 psi,=×E determine (a) the load P so that the
total deformation is 0.002 in., (b) the corresponding deformation of the
central portion BC.

SOLUTION
(a)
ii i
ii i
PLLP
AE E A
δ=Σ = Σ

1
2
4
i
ii
i
L
PEAd
A
π
δ
−

=Σ =

L, in. d, in. A, in
2
L/A, in
–1

AB 2 1.5 1.7671 1.1318
BC 3 1.0 0.7854 3.8197
CD 2 1.5 1.7671 1.1318
6.083 ← sum

613
(29 10 )(0.002)(6.083) 9.353 10 lbP
−
=× = × 9.53 kipsP= 
(b)
3
6
9.535 10
(3.8197)
29 10
BC BC
BC
BC BC
PL LP
AE EA
δ
×
== =
×

3
1.254 10 in.δ
−
=× 

21. PROPRIETARY MATERIAL. © 2012 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,
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PROBLEM 2.19
Both portions of the rod ABC are made of an aluminum for which 70 GPa.E=
Knowing that the magnitude of P is 4 kN, determine (a) the value of Q so that
the deflection at A is zero, (b) the corresponding deflection of B.

SOLUTION
(a)
22 62
(0.020) 314.16 10 m
44
AB AB
Ad
ππ
−
== =×

22 32
(0.060) 2.8274 10 m
44
BC BC
Ad
ππ
−
== =×
Force in member AB is P tension.
Elongation
:

3
6
96
(4 10 )(0.4)
72.756 10 m
(70 10 )(314.16 10 )
δ
−
−×
== = ×
××
AB
AB
AB
PL
EA

Force in member BC is Q − P compression.
Shortening
:

9
93() ()(0.5)
2.5263 10 ( )
(70 10 )(2.8274 10 )
δ
−
−− −
== =×−
××
BC
BC
BC
QPL QP
QP
EA

For zero deflection at A,
BC AB
δδ=

96 3
2.5263 10 ( ) 72.756 10 28.8 10 N
−−
×−=×∴−=×QP QP

33 3
28.3 10 4 10 32.8 10 NQ=×+×=× 32.8 kN=Q 
(b)
6
72.756 10 m
AB BC B
δδδ
−
=== × 0.0728 mmδ=↓
AB


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PROBLEM 2.20
The rod ABC is made of an aluminum for which 70 GPa.E= Knowing that
6 kN=P and 42 kN,=Q determine the deflection of (a) point A, (b) point B.

SOLUTION



22 62
22 32
(0.020) 314.16 10 m
44
(0.060) 2.8274 10 m
44
AB AB
BC BC
Ad
Ad
ππ
ππ
−
−
== =×
== =×


3
33 3
610N
610 4210 3610N
0.4 m 0.5 m
==×
=−=× − × =−×
==
AB
BC
AB BC
PP
PPQ
LL


3
6
69
3
6
39
(6 10 )(0.4)
109.135 10 m
(314.1610)(7010)
(36 10)(0.5)
90.947 10 m
(2.8274 10 )(70 10 )
AB AB
AB
AB A
BC BC
BC
BC
PL
AE
PL
AE
δ
δ
−
−
−
−×
== =×
××
−×
== =−×
××


(a)
66 6
109.135 10 90.947 10 m 18.19 10 m
AABBC
δδ δ
−− −
=+= × − × = × 0.01819 mm
A
δ=↑ 
(b)
6
90.9 10 m 0.0909 mm
BBC
δδ
−
==−× =− or 0.0919 mm
B
δ=↓ 

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PROBLEM 2.21
Members AB and BC are made of steel
6
(2910psi)E=× with cross-
sectional areas of 0.80 in
2
and 0.64 in
2
, respectively. For the loading
shown, determine the elongation of (a) member AB, (b) member BC.

SOLUTION
(a)
22
6 5 7.810 ft 93.72 in.=+= =
AB
L
Use joint A as a free body.
3
5
0: 28 0
7.810
43.74 kip 43.74 10 lb
Σ= − =
==×
yAB
AB
FF
F


3
6
(43.74 10 )(93.72)
(29 10 )(0.80)
δ
×
==
×
AB AB
AB
AB
FL
EA
0.1767 in.δ=
AB

(b) Use joint B as a free body.

3
6
0: 0
7.810
(6)(43.74)
33.60 kip 33.60 10 lb.
7.810
Σ= − =
===×
xBC AB
BC
FF F
F

3
6
(33.60 10 )(72)
(29 10 )(0.64)
δ
×
==
×
BC BC
BC
BC
FL
EA
0.1304 in.
BC
δ= 

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PROBLEM 2.22
The steel frame (200GPa)E= shown has a diagonal brace BD with
an area of 1920 mm
2
. Determine the largest allowable load P if the
change in length of member BD is not to exceed 1.6 mm.

SOLUTION

3262
22 9
963
3
1.6 10 m, 1920 mm 1920 10 m
5 6 7.810 m, 200 10 Pa
(200 10 )(1920 10 )(1.6 10 )
7.81
78.67 10 Nδ
δ
δ
−−
−−
=× = = ×
=+= =×
=
×××
==
=×
BD BD
BD BD
BD BD
BD
BD BD
BD BD BD
BD
BD
A
LE
FL
EA
EA
F
L

Use joint B as a free body.
0:
x
FΣ=

5
0
7.810
−=
BD
FP

3
3
5 (5)(78.67 10 )
7.810 7.810
50.4 10 N
×
==
=×
BD
PF
50.4 kN=P 

25. PROPRIETARY MATERIAL. © 2012 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,
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PROBLEM 2.23
For the steel truss ( 200 GPa)E= and loading shown, determine
the deformations of the members AB and AD, knowing that their
cross-sectional areas are 2400 mm
2
and 1800 mm
2
, respectively.

SOLUTION
Statics: Reactions are 114 kN upward at A and C.
Member BD is a zero force member.

22
4.0 2.5 4.717 m=+=
AB
L
Use joint A as a free body.

2.5
0: 114 0
4.717
yAB
FFΣ= + =

215.10 kN
AB
F=−

4
0: 0
4.717
xAD AB
FF FΣ= + =

(4)( 215.10)
182.4 kN
4.717
AD
F
−
=− =
Member AB
:

3
96
( 215.10 10 )(4.717)
(200 10 )(2400 10 )
AB AB
AB
AB
FL
EA
δ
−
−×
==
××


3
2.11 10 m
−
=− × 2.11 mmδ−
AB

Member AD
:
3
96
(182.4 10 )(4.0)
(200 10 )(1800 10 )
AD AD
AD
AD
FL
EA
δ
−
×
==
××


3
2.03 10 m
−
=× 2.03 mmδ=
AD


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PROBLEM 2.24
For the steel truss
6
(2910psi)E=× and loading shown, determine the
deformations of the members BD and DE, knowing that their cross-
sectional areas are 2 in
2
and 3 in
2
, respectively.

SOLUTION
Free body: Portion ABC of truss




0 : (15 ft) (30 kips)(8 ft) (30 kips)(16 ft) 0
48.0 kips
EBD
BD
MF
F
Σ= − − =
=+



Free body: Portion ABEC of truss




0 : 30 kips 30 kips 0
60.0 kips
Σ= + − =
=+
xDE
DE
FF
F 

3
26
(48.0 10lb)(8 12 in.)
(2 in )(29 10 psi)
δ
+× ×
==
×
BD
PL
AE

3
79.4 10 in.δ
−
=+ ×
BD


3
26
(60.0 10lb)(15 12 in.)
(3 in )(29 10 psi)
δ
+× ×
==
×
DE
PL
AE

3
124.1 10 in.δ
−
+×
DE


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PROBLEM 2.25
Each of the links AB and CD is made of aluminum
6
(10.910psi)E=×
and has a cross-sectional area of 0.2 in.
2
. Knowing that they support the
rigid member BC, determine the deflection of point E.

SOLUTION
Free body BC:

3
0 : (32) (22)(1 10 ) 0
687.5 lb
CAB
AB
MF
F
Σ=− + × =
=

3
0 : 687.5 1 10 0
312.5 lb
yCD
CD
FF
F
Σ= −× + =
=
3
6
3
6(687.5)(18)
5.6766 10 in
(10.9 10 ) (0.2)
(312.5)(18)
2.5803 10 in
(10.9 10 ) (0.2)
δδ
δδ
−
−
== =×=
×
== =×=
×
AB AB
AB B
CD CD
CD C
FL
EA
FL
EA


Deformation diagram:

3
6
3.0963 10
Slope
32
96.759 10 radδδ
θ
−
−
− ×
==
=×
BC
BC
L



36
3
2.5803 10 (22)(96.759 10 )
4.7090 10 in
δδ θ
−−
−
=+
=×+ ×
=×
ECEC
L


3
4.71 10 inδ
−
=× ↓
E


28. PROPRIETARY MATERIAL. © 2012 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,
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PROBLEM 2.26
The length of the
3
32
-in.-diameter steel wire CD has been adjusted so that
with no load applied, a gap of
1
16
in. exists between the end B of the rigid
beam ACB and a contact point E. Knowing that
6
29 10 psi,E=×
determine where a 50-lb block should be placed on the beam in order to
cause contact between B and E.

SOLUTION
Rigid beam ACB rotates through angle θ to close gap.

31/16
3.125 10 rad
20
θ
−
== ×
Point C moves downward.

33
3
2
232
4 4(3.125 10 ) 12.5 10 in.
12.5 10 in.
3
6.9029 10 in
432
C
CD C
CD
CD CD
CD
CD
Ad
d
FL
EA
δθ
δδ
ππ
δ
−−
−
−
== × = ×
== ×

== = ×


=


633
(29 10 )(6.9029 10 )(12.5 10 )
12.5
200.18 lb
CD CD
CD
CD
EA
F
Lδ
−−
×××
==
=

Free body ACB:
0: 4 (50)(20 ) 0
ACD
MF xΣ= − −=

(4)(200.18)
20 16.0144
50
3.9856 in.
x
x
−= =
=

For contact,
3.99 in.x< 

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PROBLEM 2.27
Link BD is made of brass (105GPa)E= and has a cross-sectional area of
240 mm
2
. Link CE is made of aluminum (72GPa)E= and has a cross-
sectional area of 300 mm
2
. Knowing that they support rigid member ABC
determine the maximum force P that can be applied vertically at point A if
the deflection of A is not to exceed 0.35 mm.

SOLUTION
Free body member AC:

0 : 0.350 0.225 0
1.55556
Σ= − =
=
CBD
BD
MPF
F P


0 : 0.125 0.225 0
0.55556
Σ= − =
=
BCE
CE
MPF
F P

9
96
9
96(1.55556 )(0.225)
13.8889 10
(105 10 )(240 10 )
(0.55556 )(0.150)
3.8581 10
(72 10 )(300 10 )
δδ
δδ
−
−
−
−
== = = ×
××
== = = ×
××
BD BD
BBD
BD BD
CE CE
CCE
CE CE
FL P
P
EA
FL P
P
EA


Deformation Diagram:
From the deformation diagram,
Slope,
9
9
17.7470 10
78.876 10
0.225
BC
BC P
P
L
δδ
θ
−
−
+ ×
== =×



99
9
13.8889 10 (0.125)(78.876 10 )
23.748 10
δδ θ
−−
−
=+
=×+ ×
=×
ABAB
L
P P
P
Apply displacement limit.
39
0.35 10 m 23.748 10δ
−−
=× = ×
A
P

3
14.7381 10 NP=× 14.74 kNP= 

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PROBLEM 2.28
Each of the four vertical links connecting the two rigid
horizontal members is made of aluminum
(70 GPa)=E
and has a uniform rectangular cross section of 10 × 40 mm.
For the loading shown, determine the deflection of
(a) point E, (b) point F, (c) point G.

SOLUTION
Statics. Free body EFG.




3
0 : (400)(2 ) (250)(24) 0
7.5 kN 7.5 10 N
FBE
BE
MF
F
Σ=− − =
=− =− ×

3
0 : (400)(2 ) (650)(24) 0
19.5 kN 19.5 10 N
ECF
CF
MF
F
Σ= − =
==×
Area of one link:

2
62
(10)(40) 400 mm
400 10 m
−
==
=×
A

Length:
300 mm 0.300 m==L
Deformations
.

3
6
96
3
6
96
( 7.5 10 )(0.300)
80.357 10 m
(70 10 )(400 10 )
(19.5 10 )(0.300)
208.93 10 m
(70 10 )(400 10 )
BE
BE
CF
CF
FL
EA
FL
EA
δ
δ
−
−
−
−−×
== =− ×
××
×
== = ×
××

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PROBLEM 2.28 (Continued)

(a) Deflection of Point E
. ||δδ=
EBF
80.4 mδμ=↑
E

(b) Deflection of Point F.
FCF
δδ= 209 m
F
δμ=↓ 
Geometry change.



Let
θ be the small change in slope angle.

66
6
80.357 10 208.93 10
723.22 10 radians
0.400δδ
θ
−−
−
+ ×+ ×
== =×
EF
EF
L

(c) Deflection of Point G
.
GFFG
Lδδ θ=+

66
6
208.93 10 (0.250)(723.22 10 )
389.73 10 mδδ θ
−−
−
=+ = × + ×
=×
GFFG
L
390 mδμ=↓
G


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PROBLEM 2.29
A vertical load P is applied at the center A of the upper section of a
homogeneous frustum of a circular cone of height h, minimum radius a,
and maximum radius b. Denoting by E the modulus of elasticity of the
material and neglecting the effect of its weight, determine the deflection of
point A.

SOLUTION
Extend the slant sides of the cone to meet at a point O and place the origin of the coordinate system there.



From geometry, tan
ba
hα
−
=


11
,,tan
tan tan α
αα===
ab
ab ry
At coordinate point y,
2
Arπ=
Deformation of element of height dy:
δ=
Pdy
d
AE


222
tan
δ
π πα==
Pdy P dy
d
ErE y

Total deformation.

1
1
1
1
22 2 2
11
111
tan tan tan
b
b
A
a
a
PdyP P
y abEyE E
δ
πα πα πα

==−=− 
 



11 11
2
11
()
tan
ba PbaP
ab EabE
ππα
−−
==

A
Ph
Eab
δ
π=↓ 

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PROBLEM 2.30
A homogeneous cable of length L and uniform cross section is suspended from one end. (a) Denoting by ρ the
density (mass per unit volume) of the cable and by E its modulus of elasticity, determine the elongation of the
cable due to its own weight. (b) Show that the same elongation would be obtained if the cable were horizontal
and if a force equal to half of its weight were applied at each end.

SOLUTION
(a) For element at point identified by coordinate y,




2
0
0
weight of portion below the point
()
() ()
() 1
2
L
L
P
gAL y
PdygALydygLy
ddy
EA EA E
gL y g
dy Ly y
EE
ρ
ρρ
δ
ρρ
δ
=
=−
−−
== =
− 
==−





2
2
2
ρ
=−


g L
L
E

2
1
2ρ
δ
=
gL
E


(b) Total weight: ρ=WgAL

2
11
22δρ
ρ
==⋅ =
EA EA gL
F gAL
LL E
1
2
FW= 

34. PROPRIETARY MATERIAL. © 2012 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,
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PROBLEM 2.31
The volume of a tensile specimen is essentially constant while plastic deformation occurs. If the initial
diameter of the specimen is d
1, show that when the diameter is d, the true strain is
1
2ln( / ).ε=
t
dd

SOLUTION
If the volume is constant,
22
10
44
dL dL
ππ
=

22
11
2
0
2
1
0
ln lnε

==



==


t
ddL
Ldd
dL
L d

1
2lnε=
t
d
d


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PROBLEM 2.32
Denoting by ε the “engineering strain” in a tensile specimen, show that the true strain is ln (1 ).εε=+
t


SOLUTION

0
00 0
ln ln ln 1 ln (1 )
δ δ
εε+
== =+=+ 

t
LL
LL L

Thus
ln (1 )εε=+
t


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PROBLEM 2.33
An axial force of 200 kN is applied to the assembly shown by means of
rigid end plates. Determine (a) the normal stress in the aluminum shell,
(b) the corresponding deformation of the assembly.

SOLUTION
Let P a = Portion of axial force carried by shell
P
b = Portion of axial force carried by core.

,or
aaa
a
aa
PL E A
P
EA L
δδ==

,or
bbb
b
bb
PLEA
P
EA L
δδ==
Thus,
()
ab aa bb
PP P EA EA
L
δ
=+= +
with
22 32
232
[(0.060) (0.025) ] 2.3366 10 m
4
(0.025) 0.49087 10 m
4
a
b
A
A
π
π
−
−
=−=×
==×

939 3
6
[(70 10 )(2.3366 10 ) (105 10 )(0.49087 10 )]
215.10 10
P
L
P
L
δ
δ
−−
=× × +× ×
=×

Strain:
3
3
66
200 10
0.92980 10
215.10 10 215.10 10δ
ε
−×
== = = ×
××
P
L

(a)
936
(70 10 )(0.92980 10 ) 65.1 10 Paσε
−
==× × =×
aa
E 65.1 MPa
a
σ= 
(b)
3
(0.92980 10 )(300 mm)Lδε
−
== × 0.279 mmδ= 

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PROBLEM 2.34
The length of the assembly shown decreases by 0.40 mm when an axial
force is applied by means of rigid end plates. Determine (a) the magnitude
of the applied force, (b) the corresponding stress in the brass core.

SOLUTION
Let P a = Portion of axial force carried by shell and P b = Portion of axial force carried by core.
,or
aaa
a
aa
PL E A
P
EA L
δδ==

,or
bbb
b
bb
PLEA
P
EA L
δδ==
Thus,
()
ab aa bb
PP P EA EA
L
δ
=+= +
with
22 32
232
[(0.060) (0.025) ] 2.3366 10 m
4
(0.025) 0.49087 10 m
4
a
b
A
A
π
π
−
−
=−=×
==×

939 3 6
[(70 10 )(2.3366 10 ) (105 10 )(0.49087 10 )] 215.10 10
δδ
−−
=× × +× × = ×P
L L

with
0.40 mm, 300 mmLδ==
(a)
630.40
(215.10 10 ) 286.8 10 N
300
P=× =×
287 kNP= 
(b)
93
6
3
(105 10 )(0.40 10 )
140 10 Pa
300 10δ
σ
−
−
××
== = =×
×
bb
b
b
PE
AL
140.0MPaσ=
b


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PROBLEM 2.35
A 4-ft concrete post is reinforced with four steel bars, each with a
3
4
-in. diameter.
Knowing that
6
29 10
s
E=× psi and
6
3.6 10
c
E=× psi, determine the normal
stresses in the steel and in the concrete when a 150-kip axial centric force P is
applied to the post.

SOLUTION

2
2
3
4 1.76715 in
44π


==



s
A

22
8 62.233 in=−=
cs
AA

6
s6(48)
0.93663 10
(1.76715)(29 10 )
ss
s
ss
PL P
P
AE
δ
−
== = ×
×

6
c6(48)
0.21425 10
(62.233)(3.6 10 )
cc
c
cc
PL P
P
AE
δ
−
== = ×
×
But
:
s c
δδ=
66
0.93663 10 0.21425 10
s c
P P
−−
×= ×

0.22875
s c
P P= (1)
Also:
150 kips
sc
PPP+== (2)
Substituting (1) into (2):
1.22875 150 kips
c
P=

122.075 kips
c
P=
From (1):
0.22875(122.075) 27.925 kips
s
P==

27.925
1.76715
s
s
s
P
A
σ=− =− 15.80 ksi
s
σ=− 

122.075
62.233
c
c
c
P
A
σ=− =− 1.962 ksi
c
σ=− 

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PROBLEM 2.36
A 250-mm bar of 15 30-mm× rectangular cross
section consists of two aluminum layers, 5-mm thick,
brazed to a center brass layer of the same thickness.
If it is subjected to centric forces of magnitude
P = 30 kN, and knowing that E
a = 70 GPa and
E
b = 105 GPa, determine the normal stress (a) in the
aluminum layers, (b) in the brass layer.

SOLUTION
For each layer,

262
(30)(5) 150 mm 150 10 mA
−
== =×
Let
load on each aluminum layer
load on brass layer
=
=
a
b
P
P
Deformation
.
ab
ab
PLPL
EAEA
δ==
105
1.5
70
== =
b
ba aa
a
E
P PPP
E
Total force
. 23.5 =+=
ab a
PPP P
Solving for P
a and P b,
23
77
ab
P PP P==
(a)
3
6
6
2 2 30 10
57.1 10 Pa
77 150 10
a
a
P P
AA
σ
−
×
=− =− =− =− ×
×
57.1 MPa
a
σ=− 
(b)
3
6
6
333010
85.7 10 Pa
77 150 10
b
b
P P
AA
σ
−
×
=− =− =− =− ×
×
85.7 MPa
b
σ=− 

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PROBLEM 2.37
Determine the deformation of the composite bar of
Prob. 2.36 if it is subjected to centric forces of
magnitude
45 kN.P=
PROBLEM 2.36 A 250-mm bar of
15×30-mm
rectangular cross section consists of two aluminum
layers, 5-mm thick, brazed to a center brass layer of
the same thickness. If it is subjected to centric forces
of magnitude
30P=kN, and knowing that 70
a
E=
GPa and
105
b
E= GPa, determine the normal stress
(a) in the aluminum layers, (b) in the brass layer.

SOLUTION
For each layer,

262
(30)(5) 150 mm 150 10 mA
−
== =×
Let
load on each aluminum layer
load on brass layer
=
=
a
b
P
P
Deformation
.
ab
ab
PLPL
EAEA
δ=− =−
105
1.5
70
== =
b
ba aa
a
E
P PPP
E
Total force
. 2 3.5 Pa
ab
PPP=+=
2
7
a
P P=

33
96
6
2
7
2 (45 10 )(250 10 )
7(70 10 )(150 10 )
306 10 m
a
aa
PL PL
EA EA
δ
−
−
−
=− =−
××
=−
××
=− ×


0.306 mmδ=− 

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PROBLEM 2.38
Compressive centric forces of 40 kips are applied at both ends
of the assembly shown by means of rigid plates. Knowing
that E
s
629 10 psi=× and
6
10.1 10 psi,
a
E=× determine (a) the
normal stresses in the steel core and the aluminum shell,
(b) the deformation of the assembly.

SOLUTION
Let portion of axial force carried by shell
portion of axial force carried by core
=
=
a
s
P
P

aaa
a
aa
sss
s
ss
PL E A
P
EA L
PL E A
P
EA L
δδ
δδ==
==

Total force
. ()
as aa ss
PP P EA EA
L
δ
=+= +

aa ss
P
L EA EA
δ
ε
==
+
Data
:
3
22 2 2 2
0
22 2
40 10 lb
( ) (2.5 1.0) 4.1233 in
44
(1) 0.7854 in
44
ai
s
P
Add
Ad
ππ
ππ
=×
=−= −=
== =

3
6
66
40 10
620.91 10
(10.1 10 )(4.1233) (29 10 )(0.7854)
ε
−−×
==−×
×+×

(a)
663
(29 10 )( 620.91 10 ) 18.01 10 psi
ss
Eσε
−
==× − × =− × 18.01 ksi− 

663
(10.1 10 )(620.91 10 ) 6.27 10 psi
aa
Eσε
−
== × × =−×  6.27 ksi− 
(b)
63
(10)(620.91 10 ) 6.21 10Lδε
−−
== × =− ×
3
6.21 10 in.δ
−
=− × 

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PROBLEM 2.39
Three wires are used to suspend the plate shown. Aluminum wires of
1
8
-in.
diameter are used at A and B while a steel wire of
1
12
-in.diameter is used at
C. Knowing that the allowable stress for aluminum
6
( 10.4 10 psi)
a
E=× is
14 ksi and that the allowable stress for steel
6
(2910psi)
s
E=× is 18 ksi,
determine the maximum load P that can be applied.

SOLUTION
By symmetry, ,
AB
PP= and
AB
δδ=
Also,
CAB
δδδδ===
Strain in each wire:

,2
2
AB C A
LL
δδ
εε ε ε
== ==
Determine allowable strain
.
Wires A&B:
3
3
6
4
14 10
1.3462 10
10.4 10
2 2.6924 10
A
A
A
CA
E
σ
ε
εε
−
−×
== = ×
×
== ×

Wire C
:
3
3
6
6
18 10
0.6207 10
29 10
1
0.3103 10
2
C
C
C
AB C
E
σ
ε
εε ε
−
−×
== = ×
×
== = ×

Allowable strain for wire C governs, ∴
3
18 10 psi
C
σ=×

2
66
2
3
1
(10.4 10 )(0.3103 10 ) 39.61 lb
48
39.61 lb
1
(18 10 ) 98.17 lb
412
σε ε
π
π
σε σ
−
==

=××=


=

=== ×=


AAAAAAA
B
CCCCCC
EPAE
P
EPA

For equilibrium of the plate,

177.4 lb=++=
ABC
PP P P 177.4 lb=P 

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PROBLEM 2.40
A polystyrene rod consisting of two cylindrical portions AB and BC is
restrained at both ends and supports two 6-kip loads as shown. Knowing
that
6
0.45 10 psi,E=× determine (a) the reactions at A and C, (b) the
normal stress in each portion of the rod.

SOLUTION
(a) We express that the elongation of the rod is zero:



22
44
0
BC BCAB AB
AB BC
PLPL
dE dE
ππ
δ=+=
But
AB A BC C
P RPR=+ =−
Substituting and simplifying:
22
0
CBCAAB
AB BC
RLRL
dd
−=


2 2
25 2
15 1.25 
== 

BCAB
CAA
BC AB
dL
R RR
Ld

4.2667
CA
R R= (1)
From the free body diagram: 12 kips
AC
RR+= (2)
Substituting (1) into (2):
5.2667 12
A
R=

2.2785 kips
A
R= 2.28 kips
A
R=↑ 
From (1):
4.2667(2.2785) 9.7217 kips
C
R==

9.72 kips
C
R=↑ 
(b)
2
4
2.2785
(1.25)
π
σ
+
== =
AB A
AB
AB AB
PR
AA
1.857 ksi
AB
σ=+ 

2
4
9.7217
(2)
BC C
BC
BC BC
PR
AA
π
σ
− −
== =
3.09 ksi
BC
σ=− 

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PROBLEM 2.41
Two cylindrical rods, one of steel and the other of brass, are joined at
C and restrained by rigid supports at A and E. For the loading shown
and knowing that
200 GPa
s
E= and 105 GPa,
b
E= determine
(a) the reactions at A and E, (b) the deflection of point C.

SOLUTION
A to C:
9
232 32
6
200 10 Pa
(40) 1.25664 10 mm 1.25664 10 m
4
251.327 10 NE
A
EA
π
−
=×
==× =×
=×
C to E
:
9
22 62
6
105 10 Pa
(30) 706.86 mm 706.86 10 m
4
74.220 10 NE
A
EA
π
−
=×
== =×
=×



A to B:
6
12
180 mm 0.180 m
(0.180)
251.327 10
716.20 10
A
A
AB
A
PR
L
RPL
EA
R
δ
−
=
==
==
×
=×
B to C
:
3
3
6
12 6
60 10
120 mm 0.120 m
( 60 10 )(0.120)
251.327 10
447.47 10 26.848 10
A
A
BC
A
PR
L
RPL
EA
R
δ
−−
=−×
==
−×
==
×
=× −×

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PROBLEM 2.41 (Continued)

C to D
:
3
3
6
96
60 10
100 mm 0.100 m
( 60 10 )(0.100)
74.220 10
1.34735 10 80.841 10
A
A
BC
A
PR
L
RPL
EA
R
δ
−−
=−×
==
−×
==
×
=×−×
D to E
:
3
3
6
96
100 10
100 mm 0.100 m
( 100 10 )(0.100)
74.220 10
1.34735 10 134.735 10
A
A
DE
A
PR
L
RPL
EA
R
δ
−−
=−×
==
−×
==
×
=×−×
A to E
:
96
3.85837 10 242.424 10
AE AB BC CD DE
A
R
δδδδδ
−−
=+++
=×−×
Since point E cannot move relative to A,
0
AE
δ=
(a)
96 3
3.85837 10 242.424 10 0 62.831 10 N
AA
RR
−−
×− ×==× 62.8 kN=←
A
R 

333 3
100 10 62.8 10 100 10 37.2 10 N
EA
RR=−×= ×−×=− × 37.2 kN=←
E
R 
(b)
96
1.16367 10 26.848 10
CABBC A
Rδδ δ
−−
=+= × − ×

93 6
6
(1.16369 10 )(62.831 10 ) 26.848 10
46.3 10 m
−−
−
=× ×−×
=×
46.3 m
C
δμ=→ 

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PROBLEM 2.42
Solve Prob. 2.41, assuming that rod AC is made of brass and
rod CE is made of steel.
PROBLEM 2.41 Two cylindrical rods, one of steel and the
other of brass, are joined at C and restrained by rigid supports at A
and E. For the loading shown and knowing that
200
s
E= GPa
and
105 GPa,
b
E= determine (a) the reactions at A and E, (b) the
deflection of point C.

SOLUTION
A to C:
9
232 32
6
105 10 Pa
(40) 1.25664 10 mm 1.25664 10 m
4
131.947 10 N
E
A
EA
π
−
=×
==× =×
=×
C to E
:
9
22 62
6
200 10 Pa
(30) 706.86 mm 706.86 10 m
4
141.372 10 N
E
A
EA
π
−
=×
== =×
=×
A to B
:
6
9
180 mm 0.180 m
(0.180)
131.947 10
1.36418 10
A
A
AB
A
PR
L
RPL
EA
R
δ
−
=
==
==
×
=×
B to C
:
3
3
6
12 6
60 10
120 mm 0.120 m
( 60 10 )(0.120)
131.947 10
909.456 10 54.567 10
A
A
BC
A
PR
L
RPL
EA
R
δ
−−
=−×
==
−×
==
×
=×−×
C to D
:
3
3
6
12 6
60 10
100 mm 0.100 m
(6010)(0.100)
141.372 10
707.354 10 42.441 10
A
A
CD
A
PR
L
RPL
EA
R
δ
−−
=−×
==
−×
==
×
=×−×

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PROBLEM 2.42 (Continued)

D to E
:
3
3
6
12 6
100 10
100 mm 0.100 m
( 100 10 )(0.100)
141.372 10
707.354 10 70.735 10
A
A
DE
A
PR
L
RPL
EA
R
δ
−−
=−×
==
−×
==
×
=×−×
A to E
:
96
3.68834 10 167.743 10
AE AB BC CD DE
A
R
δδδδδ
−−
=+++
=×−×
Since point E cannot move relative to A,
0
AE
δ=
(a)
96 3
3.68834 10 167.743 10 0 45.479 10 N
AA
RR
−−
×− ×==× 45.5 kN=←
A
R 

333 3
100 10 45.479 10 100 10 54.521 10
EA
RR=−×= ×−×=− × 54.5 kN=←
E
R 
(b)
96
2.27364 10 54.567 10
CABBC A
Rδδ δ
−−
=+= × − ×

93 6
6
(2.27364 10 )(45.479 10 ) 54.567 10
48.8 10 m
−−
−
=× ×−×
=×
48.8 m
C
δμ=→ 

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PROBLEM 2.43
The rigid bar ABCD is suspended from four identical wires. Determine the
tension in each wire caused by the load
P shown.

SOLUTION
Deformations Let θ be the rotation of bar ABCD and ,,
ABC
δδδ and
D
δ be the deformations of wires A, B,
C, and D.
From geometry,
BA
L
δδ
θ−
=


BA
Lδδ θ=+

22
CA BA
Lδδ θδδ=+ = − (1)

332
DA B A
Lδδ θδ δ=+ = − (2)



Since all wires are identical, the forces in the wires are proportional to the deformations.

2
CBA
TTT=− (1 ′)

32
DBA
TTT=− (2 ′)

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PROBLEM 2.43 (Continued)

Use bar ABCD as a free body.

0: 2 0
CABD
MLTLTLTΣ= − − + = (3)
0: 0
y ABC D
F TTTT PΣ= + + + −= (4)
Substituting (2′) into (3) and dividing by L,

420 2
AB B A
TT TT−+ = = (3 ′)
Substituting (1′), (2′), and (3′) into (4),

234 0
AAAA
TTTTP+++−= 10
A
TP=

1
10
A
TP= 

1
2(2)
10
BA
TT P

==



1
5
B
TP= 

11
(2)
510
C
TPP
 
=−
 
 

3
10
C
TP= 

11
(3) (2)
510
D
TP P
  
=−
  
  

2
5
D
TP= 

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PROBLEM 2.44
The rigid bar AD is supported by two steel wires of
1
16
-in. diameter
6
(2910 psi)=×E and a pin and bracket at D. Knowing that the
wires were initially taut, determine (a) the additional tension in each
wire when a 120-lb load
P is applied at B, (b) the corresponding
deflection of point B.

SOLUTION
Let θ be the rotation of bar ABCD.
Then
24 8
AC
δθδθ== 





AE AE
A
PL
AE
δ=
62 1
416
3
(29 10 ) ( ) (24 )
15
142.353 10
A
AE
AE
EA
P
L
π
θδ
θ×
==
=×


CF CF
C
PL
AE
δ=
()
2
6 1
416
3
(29 10 ) (8 )
8
88.971 10
C
CF
CF
EA
P
L
π
θδ
θ×
==
=×

Using free body ABCD,
0:
D
MΣ=

33
3
24 16 8 0
24(142.353 10 ) 16(120) 8(88.971 10 ) 0
0.46510 10 rad
θθ
θ
−
−+−=
−×+−×=
=×
AE CF
PPP
π

(a)
33
(142.353 10 ) (0.46510 10 )
AE
P
−
=× × 66.2 lb
AE
P= 

33
(88.971 10 ) (0.46510 10 )
CF
P
−
=× × 41.4 lb
CF
P= 
(b)
3
16 16(0.46510 10 )
B
δθ
−
== ×
3
7.44 10 in.
B
δ
−
=× ↓ 

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PROBLEM 2.45
The steel rods BE and CD each have a 16-mm diameter
(200 GPa);E= the ends of the rods are single-threaded with a
pitch of 2.5 mm. Knowing that after being snugly fitted, the nut at
C is tightened one full turn, determine (a) the tension in rod CD,
(b) the deflection of point C of the rigid member ABC.

SOLUTION
Let θ be the rotation of bar ABC as shown. 

Then
0.15 0.25
BC
δθδθ==
But
turn
CD CD
C
CD CD
PL
EA
δδ=−
turn
92
4
36
()
(200 10 Pa) (0.016 m)
(0.0025m 0.25 )
2m
50.265 10 5.0265 10
CD CD
CD C
CD
EA
P
L
π
δδ
θ
θ=−
×
=−
=×−×
or
BE BE BE BE
BBEB
BE BE BE
PL E A
P
EA L
δδ==
92
4
6
(200 10 Pa) (0.016 m)
(0.15 )
3m
2.0106 10
BE
P
π
θ
θ
×
=
=×

From free body of member ABC:

0 : 0.15 0.25 0
ABECD
MPPΣ= − =

636
0.15(2.0106 10 ) 0.25(50.265 10 5.0265 10 ) 0θθ×− ×− ×=

3
8.0645 10 radθ
−
=×
(a)
363
50.265 10 5.0265 10 (8.0645 10 )
CD
P
−
=×−× ×

3
9.7288 10 N=× 9.73 kN
CD
P= 
(b)
3
0.25 0.25(8.0645 10 )
C
δθ
−
== ×

3
2.0161 10 m
−
=× 2.02 mm
C
δ=← 

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PROBLEM 2.46
Links BC and DE are both made of steel
6
(2910 psi)E=× and are
1
2
in. wide and
1
4
in. thick. Determine (a) the force in each link
when a 600-lb force
P is applied to the rigid member AF shown,
(b) the corresponding deflection of point A.

SOLUTION
Let the rigid member ACDF rotate through small angle θ clockwise about point F.
Then
4 in.
2 in.δδ θ
δδ θ== →
=− = →
CBC
DDE


or
FLEA
F
EAL
δ
δ
==
For links:
2
6
6
6
611
0.125 in
24
4 in.
5 in.
(29 10 )(0.125)(4 )
3.625 10
4
(29 10 )(0.125)( 2 )
1.45 10
5
δ θ
θ
δ θ
θ

==


=
=
×
== =×
×−
== =−×
BC
DE
BC
BC
BC
DE
DE
DE
A
L
L
EA
F
L
EA
F
L
Use member ACDF as a free body.

666
3
0: 8 4 2 0
11
24
11
600 (3.625 10 ) ( 1.45 10 ) 2.175 10
24
+0.27586 10 rad
FBCDE
BC DE
MPFF
PF F
θθθ
θ
−
Σ= − + =
=−
=×−−×=×
=×
σ

(a)
63
(3.625 10 )(0.27586 10 )
BC
F
−
=× × 1000 lb
BC
F= 

63
(1.45 10 )(0.27586 10 )
DE
F
−
=− × × 400 lb=−
DE
F 
(b) Deflection at Point A
.

3
8 (8)(0.27586 10 )
A
δθ
−
== ×
3
2.21 10 inδ
−
=× →
A


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PROBLEM 2.47
The concrete post
6
(3.610
c
E=× psi and
6
5.5 10 / F)
c
α
−
=× ° is reinforced
with six steel bars, each of
7
8
-in. diameter
6
(2910
s
E=× psi and
6
6.5 10 / F).
s
α
−
=× ° Determine the normal stresses induced in the steel and
in the concrete by a temperature rise of 65°F.

SOLUTION

2
22
7
6 6 3.6079 in
448ππ 
== =

s
Ad

22 2
10 10 3.6079 96.392 in=−=− =
cs
AA
Let
c
P= tensile force developed in the concrete.
For equilibrium with zero total force, the compressive force in the six steel rods equals
.
c
P
Strains: () ()
cc
sscc
ss cc
PP
TT
EA EA
εαεα=− + Δ = + Δ
Matching:
cs
εε=
() ()
cc
cs
cc ss
PP
TT
EA EA
αα+Δ=− +Δ

6
66
3
11
()()
11
(1.0 10 )(65)
(3.6 10 )(96.392) (29 10 )(3.6079)
5.2254 10 lb
αα
−

+=−Δ


+=×
××
=×
csc
cc ss
c
c
PT
EA EA
P
P


3
5.2254 10
54.210 psi
96.392
c
c
c
P
A
σ
×
== =
54.2 psi
c
σ=

3
5.2254 10
1448.32 psi
3.6079
c
s
s
P
A
σ
×
=− =− =−
 1.448 ksi
s
σ=− 

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PROBLEM 2.48
The assembly shown consists of an aluminum shell
6
( 10.6 10 psi,=×
a
E
αa = 12.9 × 10
–6
/°F) fully bonded to a steel core
6
(2910 psi,=×
s
E
αs = 6.5 × 10
–6
/°F) and is unstressed. Determine (a) the largest allowable
change in temperature if the stress in the aluminum shell is not to exceed
6 ksi, (b) the corresponding change in length of the assembly.

SOLUTION
Since ,
as
αα> the shell is in compression for a positive temperature rise.
Let
3
6 ksi 6 10 psi
a
σ=− =− ×

()
22 2 2 2
(1.25 0.75 ) 0.78540 in
44
ππ
=−= −=
aoi
Add

22 2
(0.75) 0.44179 in
44
ππ
== =
s
Ad

aa ss
P AAσσ=− =
where P is the tensile force in the steel core.

3
3
(6 10 )(0.78540)
10.667 10 psi
0.44179
aa
s
s
A
Aσ
σ ×
=− = = ×


33
63
66
() ()
()()
10.667 10 6 10
(6.4 10 )( ) 0.93385 10
29 10 10.6 10
sa
sa
sa
sa
as
sa
TT
EE
T
EE
T
σσ
εα α
σσ
αα
−−
=+Δ=+Δ
−Δ=−
××
×Δ= + = ×
××

(a)
145.91 FTΔ= ° 145.9 FΔ= °T 
(b)
3
63
6
10.667 10
(6.5 10 )(145.91) 1.3163 10
29 10
ε
−−×
=+× =×
×

or

3
63
6
610
(12.9 10 )(145.91) 1.3163 10
10.6 10
ε
−−−×
=+× =×
×


3
(8.0)(1.3163 10 ) 0.01053 in.δε
−
== × =L  0.01053 in.δ= 

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PROBLEM 2.49
The aluminum shell is fully bonded to the brass core and the assembly is
unstressed at a temperature of
15 C.° Considering only axial deformations,
determine the stress in the aluminum when the temperature reaches
195 C.°

SOLUTION
Brass core:

6
105 GPa
20.910/C
E
α
−
=
=×°

Aluminum shell:

6
70 GPa
23.6 10 / C
E
α
−
=
=× °

Let L be the length of the assembly.
Free thermal expansion
:

195 15 180 CTΔ= − = °
Brass core:
() ( )
Tb b
L Tδα=Δ
Aluminum shell:
() ( )
Ta
L Tδα=Δ
Net expansion of shell with respect to the core: ()()
ab
L Tδαα=−Δ
Let P be the tensile force in the core and the compressive force in the shell.
Brass core:
9
22
62
105 10 Pa
(25) 490.87 mm
4
490.87 10 m
()
b
b
Pb
bb
E
A
PL
EA
π
δ
−
=×
==
=×
=

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PROBLEM 2.49 (Continued)

Aluminum shell
:
9
22
32
32
70 10 Pa
(60 25 )
4
2.3366 10 mm
2.3366 10 m
() ()
()()
a
a
Pb Pa
ba
bb aa
E
A
PL PL
L TKPL
EA EA
π
δδ δ
αα
−
=×
=−
=×
=×
=+
−Δ= + =
where

9693
91
11
11
(105 10 )(490.87 10 ) (70 10 )(2.3366 10 )
25.516 10 N
bb aa
K
EA EA
−−
−−
=+
=+
××××
=×

Then

66
9
3
()()
(23.6 10 20.9 10 )(180)
25.516 10
19.047 10 N
ba
T
P
Kαα
−−
−
−Δ
=
×−×
=
×
=×

Stress in aluminum
:
3
6
3
19.047 10
8.15 10 Pa
2.3366 10
a
a
P
A
σ
−
×
=− =− =− ×
×
8.15 MPa
a
σ=− 

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PROBLEM 2.50
Solve Prob. 2.49, assuming that the core is made of steel (200
s
E= GPa,
6
11.7 10 / C)
s
α
−
=× ° instead of brass.
PROBLEM 2.49 The aluminum shell is fully bonded to the brass core
and the assembly is unstressed at a temperature of
15 C.° Considering
only axial deformations, determine the stress in the aluminum when the
temperature reaches
195 C.°

SOLUTION
Aluminum shell:
6
70 GPa 23.6 10 / Cα
−
==×°E
Let L be the length of the assembly.
Free thermal expansion: 195 15 180 CTΔ= − = °
Steel core:
() ( )
Ts s
L Tδα=Δ
Aluminum shell:
() ( )
Ta a
L Tδα=Δ
Net expansion of shell with respect to the core: ()()
as
L Tδαα=−Δ
Let P be the tensile force in the core and the compressive force in the shell.
Steel core:
92262
200 10 Pa, (25) 490.87 mm 490.87 10 m
4
()
π
δ
−
=× = = = ×
=
ss
Ps
ss
EA
PL
EA
Aluminum shell
:
9
22 32 32
70 10 Pa
()
(60 25) 2.3366 10 mm 2.3366 10 m
4
() ()
δ
π
δδ δ
−
=×
=
=−=× =×
=+
a
Pa
aa
a
Ps Pa
E
PL
EA
A

()()αα−Δ= + =
as
ss aa
PL PLL TKPL
EA EA
where

9693
91
11
11
(200 10 )(490.87 10 ) (70 10 )(2.3366 10 )
16.2999 10 N
−−
−−
=+
=+
××××
=×
ss aa
K
EA EA

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PROBLEM 2.50 (Continued)

Then

66
3
9
()() (23.6 10 11.7 10 )(180)
131.41 10 N
16.2999 10αα
−−
−
−Δ ×−×
== =×
×
as
T
P
K

Stress in aluminum
:
3
6
3
131.19 10
56.2 10 Pa
2.3366 10
a
a
P
A
σ
−
×
=− =− =− ×
×
56.2 MPa
a
σ=− 

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PROBLEM 2.51
A rod consisting of two cylindrical portions AB and BC is restrained at both
ends. Portion AB is made of steel
(200 GPa,
s
E=
6
11.7 10 / C)
s
α
−
=× ° and
portion BC is made of brass
(105 GPa,
b
E=
6
20.9 10 / C).
b
α
−
=×° Knowing
that the rod is initially unstressed, determine the compressive force induced in
ABC when there is a temperature rise of
50 C.°

SOLUTION

22 2 62
22 32 32
(30) 706.86 mm 706.86 10 m
44
(50) 1.9635 10 mm 1.9635 10 m
44
AB AB
BC BC
Ad
Ad
ππ
ππ
−
−
== = = ×
== =× =×

Free thermal expansion
:

66
6
() ()
(0.250)(11.7 10 )(50) (0.300)(20.9 10 )(50)
459.75 10 m
TABs BCb
LTLTδα α
−−
−
=Δ+Δ
=×+×
=×

Shortening due to induced compressive force P
:

9693
9
0.250 0.300
(200 10 )(706.86 10 ) (105 10 )(1.9635 10 )
3.2235 10
P
sAB bBC
PL PL
EA EA
PP
P
δ
−−
−
=+
=+
×× ××
=×

For zero net deflection,
PT
δδ=

96
3
3.2235 10 459.75 10
142.62 10 N
P
P
−−
×= ×
=×
142.6 kNP= 

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without permission.

PROBLEM 2.52
A steel railroad track
6
s
( 200 GPa, 11.7 10 /°C)
s
E α
−
==× was laid out at a temperature of 6°C. Determine
the normal stress in the rails when the temperature reaches 48°C, assuming that the rails (a) are welded to
form a continuous track, (b) are 10 m long with 3-mm gaps between them.

SOLUTION
(a)
63
( ) (11.7 10 )(48 6)(10) 4.914 10 m
T
TLδα
−−
=Δ = × − = ×

12
9(10)
50 10
200 10
P
PL L
AE Eσσ
δσ
−
=== =×
×

312
4.914 10 50 10 0
TP
δδ δ σ
−−
=+= × +× =

6
98.3 10 Paσ=− × 98.3 MPaσ=−
(b)
312 3
4.914 10 50 10 3 10
TP
δδ δ σ
−− −
=+= × +× =×

33
12
6
3 10 4.914 10
50 10
38.3 10 Pa
σ
−−
−
×− ×
=
×
=− ×
38.3 MPaσ=− 

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PROBLEM 2.53
A rod consisting of two cylindrical portions AB and BC is restrained at
both ends. Portion AB is made of steel
6
(2910 psi,=×
s
E
6
6.5 10 / F)α
−
=× °
s
and portion BC is made of aluminum
6
( 10.4 10 psi,=×
a
E
6
13.3 10 /°F).α
−
=×
a
Knowing that the rod is
initially unstressed, determine (a) the normal stresses induced in portions
AB and BC by a temperature rise of 70°F, (b) the corresponding
deflection of point B.

SOLUTION

22 2 2
(2.25) 3.9761 in (1.5) 1.76715 in
44
ππ
== ==
AB BC
AA
Free thermal expansion.
70 FTΔ= °
Total:
63
63
3
( ) ( ) (24)(6.5 10 )(70) 10.92 10 in
( ) ( ) (32)(13.3 10 (70) 29.792 10 in.
( ) ( ) 40.712 10 in.
TAB AB s
TBC BC a
TTABTBC
LT
LTδα
δα
δδ δ
−−
−−
−
=Δ+ × =×
=Δ= × =×
=+=×

Shortening due to induced compressive force P
.

9
6
9
624
( ) 208.14 10
(29 10 )(3.9761)
32
( ) 1741.18 10
(10.4 10 )(1.76715)
δ
δ
−
−
== =×
×
== = ×
×
AB
PAB
sAB
BC
PBC
aBC
PL P
P
EA
PL P
P
EA

Total:
9
( ) ( ) 1949.32 10δδ δ
−
=+= ×
PPABPBC
P
For zero net deflection,
PT
δδ=
93
1949.32 10 40.712 10P
−−
×=×
3
20.885 10 lbP=×
(a)
3
3
20.885 10
5.25 10 psi
3.9761
AB
AB
P
A
σ
×
=− =− =− ×
5.25 ksi
AB
σ=− 

3
3
20.885 10
11.82 10 psi
1.76715
BC
BC
P
A
σ
×
=− =− =− ×
11.82 ksi
BC
σ=− 
(b)
93 3
( ) (208.14 10 )(20.885 10 ) 4.3470 10 in.
PAB
δ
−−
=× ×=×

33
( ) ( ) 10.92 10 4.3470 10
BTAB PAB
δδ δ
−−
=→+←=×→+×←
3
6.57 10 in.
B
δ
−
=× → 
or

93 3
( ) (1741.18 10 )(20.885 10 ) 36.365 10 in.
PBC
δ
−−
=× ×=×

333
( ) ( ) 29.792 10 36.365 10 6.57 10 in.
BTBC PBC
δδ δ
−−−
=←+→=×←+×→=×→ (checks)

62. PROPRIETARY MATERIAL. © 2012 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,
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without permission.


PROBLEM 2.54
Solve Prob. 2.53, assuming that portion AB of the composite rod is made
of aluminum and portion BC is made of steel.
PROBLEM 2.53 A rod consisting of two cylindrical portions AB and BC
is restrained at both ends. Portion AB is made of steel
6
(2910 psi,=×
s
E
6
6.5 10 / F)α
−
=× °
s
and portion BC is made of aluminum
6
( 10.4 10 psi,=×
a
E
6
13.3 10 /°F).α
−
=×
a
Knowing that the rod is
initially unstressed, determine (a) the normal stresses induced in portions
AB and BC by a temperature rise of 70°F, (b) the corresponding
deflection of point B.
SOLUTION


22
(2.25) 3.9761 in
4
π
==
AB
A
22
(1.5) 1.76715 in
4
π
==
BC
A
Free thermal expansion.
70 FTΔ= °

63
63
( ) ( ) (24)(13.3 10 )(70) 22.344 10 in.
( ) ( ) (32)(6.5 10 )(70) 14.56 10 in.
TAB AB a
TBC BC s
LT
LTδα
δα
−−
−−
=Δ= × =×
=Δ= × =×

Total:
3
( ) ( ) 36.904 10 in.
TTABTBC
δδ δ
−
=+=×
Shortening due to induced compressive force P.


9
6
9
624
( ) 580.39 10
(10.4 10 )(3.9761)
32
( ) 624.42 10
(29 10 )(1.76715)
δ
δ
−
−
== =×
×
== =×
×
AB
PAB
aAB
BC
PBC
sBC
PL P
P
EA
PL P
P
EA

Total:
9
( ) ( ) 1204.81 10δδ δ
−
=+= ×
PPABPBC
P
For zero net deflection,
PT
δδ=
93
1204.81 10 36.904 10
−−
×= ×P
3
30.631 10 lbP=×
(a)
3
3
30.631 10
7.70 10 psi
3.9761
AB
AB
P
A
σ
×
=− =− =− ×
7.70 ksi
AB
σ=− 

3
3
30.631 10
17.33 10 psi
1.76715
BC
BC
P
A
σ
×
=− =− =− ×
17.33 ksi
BC
σ=− 

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PROBLEM 2.54 (Continued)

(b)
93 3
( ) (580.39 10 )(30.631 10 ) 17.7779 10 in.
PAB
δ
−−
=× ×=×

33
( ) ( ) 22.344 10 17.7779 10
BTAB PAB
δδ δ
−−
=→+←=×→+ ×←
3
4.57 10 in.
B
δ
−
=× → 
 or
93 3
( ) (624.42 10 )(30.631 10 ) 19.1266 10 in.
PBC
δ
−−
=× ×=×

333
( ) ( ) 14.56 10 19.1266 10 4.57 10 in. (checks)
BTBC PBC
δδ δ
−−−
=←+→=×←+ ×→=×→

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PROBLEM 2.55
A brass link (105GPa,
b
E=
6
20.9 10 / C)
b
α
−
=×° and a steel rod
(E
s200 GPa,=
6
11.7 10 / C)
s
α
−
=× ° have the dimensions shown
at a temperature of
20 C.° The steel rod is cooled until it fits
freely into the link. The temperature of the whole assembly is
then raised to
45 C.° Determine (a) the final stress in the steel
rod, (b) the final length of the steel rod.
SOLUTION
Initial dimensions at 20 C.T=°
Final dimensions at
45 C.T=°

45 20 25 CTΔ=−=°
Free thermal expansion of each part
:
Brass link:
66
( ) ( ) (20.9 10 )(25)(0.250) 130.625 10 m
Tb b
TLδα
−−
=Δ = × = ×
Steel rod:
66
( ) ( ) (11.7 10 )(25)(0.250) 73.125 10 m
Ts s
TLδα
−−
=Δ = × = ×
At the final temperature, the difference between the free length of the steel rod and the brass link is

66 66
120 10 73.125 10 130.625 10 62.5 10 mδ
−− −−
=× + × − × = ×
Add equal but opposite forces P to elongate the brass link and contract the steel rod.
Brass link
:
9
105 10 PaE=×

232
12
93
(2)(50)(37.5) 3750 mm 3.750 10 m
(0.250)
( ) 634.92 10
(105 10 )(3.750 10 )
b
P
A
PL P
P
EA
δ
−
−
−
===×
== = ×
××

Steel rod
:
9
200 10 PaE=×
22 62
(30) 706.86 mm 706.86 10 m
4
s
A
π
−
== =×

9
96(0.250)
( ) 1.76838 10
(200 10 )(706.86 10 )
Ps
ss
PL P
P
EA
δ
−
−
== = ×
××

96 3
( ) ( ) : 2.4033 10 62.5 10 26.006 10 Nδδδ
−−
+= × =× = ×
Pb Ps
PP
(a) Stress in steel rod
:
3
6
6
(26.006 10 )
36.8 10 Pa
706.86 10
s
s
P
A
σ
−
×
=− =− =− ×
×
36.8 MPa
s
σ=− 
(b) Final length of steel rod
:
0
() ()
f Ts Ps
LL δδ=+ −

66 93
0.250 120 10 73.125 10 (1.76838 10 )(26.003 10 )
f
L
−− −
= +×+ ×− × ×

0.250147 m= 250.147 mm
f
L= 

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PROBLEM 2.56
Two steel bars
6
( 200 GPa and 11.7 10 / C)
ss
E α
−
==×° are used to
reinforce a brass bar
6
( 105 GPa, 20.9 10 / C)
bb
E α
−
==×° that is
subjected to a load
25 kN.P= When the steel bars were fabricated, the
distance between the centers of the holes that were to fit on the pins
was made 0.5 mm smaller than the 2 m needed. The steel bars
were then placed in an oven to increase their length so that they
would just fit on the pins. Following fabrication, the temperature
in the steel bars dropped back to room temperature. Determine
(a) the increase in temperature that was required to fit the steel
bars on the pins, (b) the stress in the brass bar after the load is
applied to it.

SOLUTION
(a) Required temperature change for fabrication:

3
0.5 mm 0.5 10 m
T
δ
−
==×
Temperature change required to expand steel bar by this amount:

36
36
, 0.5 10 (2.00)(11.7 10 )( ),
0.5 10 (2)(11.7 10 )( )
21.368 C
Ts
LTT
TT
Tδα
−−
−−
=Δ × = × Δ
Δ= × = × Δ
Δ= °
21.4 C° 
(b) Once assembled, a tensile force P
*
develops in the steel, and a compressive force P
*
develops in the
brass, in order to elongate the steel and contract the brass.
Elongation of steel:
262
(2)(5)(40) 400 mm 400 10 m
s
A
−
===×

**
9*
69
(2.00)
() 2510
(400 10 )(200 10 )
Ps
ss
FL P
P
AE
δ
−
−
== =×
××
Contraction of brass:
262
(40)(15) 600 mm 600 10 m
b
A
−
== =×

**
9*
69
(2.00)
( ) 31.746 10
(600 10 )(105 10 )
Pb
bb
PL P
P
AE
δ
−
−
== = ×
××
But
() ()
Ps Pb
δδ+ is equal to the initial amount of misfit:

39*3
*3
( ) ( ) 0.5 10 , 56.746 10 0.5 10
8.811 10 N
Ps Pb
P
Pδδ
−−−
+=× × =×
=×

Stresses due to fabrication
:
Steel:
*3
*6
6
8.811 10
22.03 10 Pa 22.03 MPa
400 10
s
s
P
A
σ
−
×
== = × =
×

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PROBLEM 2.56 (Continued)

Brass:
*3
*6
6
8.811 10
14.68 10 Pa 14.68 MPa
600 10
b
b
P
A
σ
−
×
=− =− =− × =−
×

To these stresses must be added the stresses due to the 25 kN load.
For the added load, the additional deformation is the same for both the steel and the brass. Let
δ′ be the
additional displacement. Also, let P
s and P b be the additional forces developed in the steel and brass,
respectively.

69
6
69
6
(400 10 )(200 10 )
40 10
2.00
(600 10 )(105 10 )
31.5 10
2.00
sb
ss bb
ss
s
bb
b
PL PL
AE AE
AE
P
L
AE
P
L
δ
δδδ
δδδ
−
−
′==
××
′′′== =×
××
′′′== =×

Total
3
25 10 N
sb
PP P=+=×

663 6
40 10 31.5 10 25 10 349.65 10 mδδ δ
−
′′ ′×+×=× = ×

66 3
66 3
(40 10 )(349.65 10 ) 13.986 10 N
(31.5 10 )(349.65 10 ) 11.140 10 N
s
b
P
P
−
−
=× × = ×
=× ×= ×


3
6
6
3
6
6
13.986 10
34.97 10 Pa
400 10
11.140 10
18.36 10 Pa
600 10
s
s
s
b
b
b
P
A
P
A
σ
σ
−
−
×
== = ×
×
×
== = ×
×

Add stress due to fabrication.
Total stresses
:

666
34.97 10 22.03 10 57.0 10 Pa
s
σ=×+×=× 57.0 MPaσ=
s


666
18.36 10 14.68 10 3.68 10 Pa
b
σ=×−×=× 3.68 MPaσ=
b


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PROBLEM 2.57
Determine the maximum load P that may be applied to the brass bar of
Prob. 2.56 if the allowable stress in the steel bars is 30 MPa and the
allowable stress in the brass bar is 25 MPa.
PROBLEM 2.56 Two steel bars ( 200 GPa and 11.7
ss
E α==× 10
–6
/°C)
are used to reinforce a brass bar
( 105 GPa, 20.9
bb
E α==× 10
–6
/°C)
that is subjected to a load
25 kN.P= When the steel bars were fabricated,
the distance between the centers of the holes that were to fit on the pins was
made 0.5 mm smaller than the 2 m needed. The steel bars were then placed
in an oven to increase their length so that they would just fit on the pins.
Following fabrication, the temperature in the steel bars dropped back to
room temperature. Determine (a) the increase in temperature that was
required to fit the steel bars on the pins, (b) the stress in the brass bar after
the load is applied to it.

SOLUTION
See solution to Problem 2.56 to obtain the fabrication stresses.

*
*
22.03 MPa
14.68 MPa
s
b
σ
σ=
=

Allowable stresses:
,all ,all
30 MPa, 25 MPa
sb
σσ==
Available stress increase from load
.

30 22.03 7.97 MPa
25 14.68 39.68 MPa
s
b
σ
σ=− =
=+ =

Corresponding available strains
.

6
6
9
6
6
9
7.97 10
39.85 10
200 10
39.68 10
377.9 10
105 10
s
s
s
b
b
b
E
E
σ
ε
σ
ε
−
−×
== = ×
×
×
== = ×
×

6
Smaller value governs 39.85 10ε
−
∴= ×
Areas:
262
(2)(5)(40) 400 mm 400 10 m
s
A
−
===×

262
(15)(40) 600 mm 600 10 m
b
A
−
== =×
Forces
96 6 3
96 6 3
(200 10 )(400 10 )(39.85 10 ) 3.188 10 N
(105 10 )(600 10 )(39.85 10 ) 2.511 10 Nε
ε
−−
−− −
==× × ×=×
==× × ×=×
sss
bbb
PEA
PEA
Total allowable additional force:


333
3.188 10 2.511 10 5.70 10 N
sb
PP P=+= × + × = × 5.70 kN=P 

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PROBLEM 2.58
Knowing that a 0.02-in. gap exists when the temperature is 75 F,°
determine (a) the temperature at which the normal stress in the
aluminum bar will be equal to −11 ksi, (b) the corresponding exact
length of the aluminum bar.

SOLUTION

3
33
11 ksi 11 10 psi
(11 10)(2.8) 30.8 10lbσ
σ=− =− ×
=− = × = ×
a
aa
PA

Shortening due to P
:




33
66
3
(30.8 10 )(14) (30.8 10 )(18)
(15 10)(2.4) (10.6 10)(2.8)
30.657 10 in.
ba
P
bb aa
PL PL
EA EA
δ
−
=+
××
=+
××
=×

Available elongation for thermal expansion
:

33
0.02 30.657 10 50.657 10 in.
T
δ
−−
=+ ×= ×
But
() ()
Tbb aa
L TL Tδα α=Δ+Δ

666
(14)(12 10 )( ) (18)(12.9 10 )( ) 400.2 10 )TTT
−−−
=×Δ+ ×Δ=×Δ
Equating,
63
(400.2 10 ) 50.657 10 126.6 FTT
−−
×Δ= × Δ= °
(a)
hot cold
75 126.6 201.6 FTT T=+Δ=+ = °
hot
201.6 F=°T 
(b)
()
a
aaa
aa
PL
LT
EA
δα=Δ−

3
63
6
(30.8 10 )(18)
(18)(12.9 10 )(26.6) 10.712 10 in.
(10.6 10 )(2.8)
−− ×
=× − =×
×


3
exact
18 10.712 10 18.0107in.L
−
=+ × = 18.0107 in.=L 

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PROBLEM 2.59
Determine (a) the compressive force in the bars shown after a
temperature rise of
180 F,° (b) the corresponding change in length of
the bronze bar.

SOLUTION
Thermal expansion if free of constraint:

66
3
() ()
(14)(12 10 )(180) (18)(12.9 10 )(180)
72.036 10 in.
Tbb aa
LTLTδα α
−−
−
=Δ+Δ
=× + ×
=×

Constrained expansion
: 0.02in.δ=
Shortening due to induced compressive force P:




33
72.036 10 0.02 52.036 10 in.
P
δ
−−
=×−=×
But
ba ba
P
bb aa bb aa
PL PL L L
P
EA EA EA EA
δ

=+= + 


9
6614 18
995.36 10
(15 10 )(2.4) (10.6 10 )(2.8)
P P
−

=+ =×
××

Equating,
93
3
995.36 10 52.036 10
52.279 10 lb
P
P
−−
×= ×
=×

(a)
52.3 kipsP= 
(b)
()
b
bbb
bb
PL
LT
EA
δα=Δ−

3
63
6
(52.279 10 )(14)
(14)(12 10 )(180) 9.91 10 in.
(15 10 )(2.4)
−− ×
=× − =×
×

3
9.91 10 in.δ
−
=×
b


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PROBLEM 2.60
At room temperature (20 C)° a 0.5-mm gap exists between the ends
of the rods shown. At a later time when the temperature has reached
140°C, determine (a) the normal stress in the aluminum rod, (b) the
change in length of the aluminum rod.

SOLUTION
140 20 120 CTΔ= − = °
Free thermal expansion:

66
3
() ()
(0.300)(23 10 )(120) (0.250)(17.3 10 )(120)
1.347 10 m
Taa ss
LTLTδα α
−−
−
=Δ+Δ
=×+ ×
=×

Shortening due to P to meet constraint
:

33 3
1.347 10 0.5 10 0.847 10 m
P
δ
−− −
=×−×=×

96 96
9
0.300 0.250
(75 10 )(2000 10 ) (190 10 )(800 10 )
3.6447 10
as as
P
aa ss aa ss
PL PL L L
P
EA EA EA EA
P
P
δ
−−
−

=+= + 


=+
×× ××
=×

Equating,
93
3
3.6447 10 0.847 10
232.39 10 N
P
P
−−
×=×
=×
(a)
3
6
6
232.39 10
116.2 10 Pa
2000 10
a
a
P
A
σ
−
×
=− =− =− ×
×
116.2 MPaσ=−
a

(b)
()
a
aaa
aa
PL
LT
EA
δα=Δ−

3
66
96
(232.39 10 )(0.300)
(0.300)(23 10 )(120) 363 10 m
(75 10 )(2000 10 )
−−
− ×
=×− =×
××
0.363 mmδ=
a


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PROBLEM 2.61
A 600-lb tensile load is applied to a test
coupon made from
1
16
-in. flat steel plate
(E
6
29 10 psi=× and 0.30).v= Determine
the resulting change (a) in the 2-in. gage
length, (b) in the width of portion AB of the
test coupon, (c) in the thickness of
portion AB, (d) in the cross-sectional
area of portion AB.

SOLUTION

2
3
3
6
611
0.03125 in
216
600
19.2 10 psi
0.03125
19.2 10
662.07 10
29 10
σ
σ
ε
−
 
==
 
 
== = ×
×
== = ×
×
x
A
P
A
E

(a)
6
0
(2.0)(662.07 10 )
xx
Lδε
−
== ×
3
1.324 10 in.
l
δ
−
=× 

66
(0.30)(662.07 10 ) 198.62 10
yz x
vεε ε
−−
==− =− × =− ×
(b)
6
width 01
( 198.62 10 )
2
y
wδε
−
==− ×



6
99.3 10 in.
w
δ
−
=− × 
(c)
6
thickness 01
( 198.62 10 )
16
z
tδε
−
== − ×



6
12.41 10 in.
t
δ
−
=− × 
(d)
00
(1 ) (1 )
y z
Awtw t εε== + +

00
(1 )
y zyz
wtεεεε=+++

000
66
62
()
11
( 198.62 10 198.62 10 negligible term)
216
12.41 10 inεεεε
−−
−
Δ= − = + +
 
= − ×− ×+
 
 
=− ×
yzyz
AAA wt

62
12.41 10 in
−
Δ=− ×A 

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PROBLEM 2.62
In a standard tensile test, a steel rod of 22-mm diameter is subjected
to a tension force of 75 kN. Knowing that
0.3v= and
200 GPa,E= determine (a) the elongation of the rod in a 200-mm
gage length, (b) the change in diameter of the rod.

SOLUTION

32262
75 kN 75 10 N (0.022) 380.13 10 m
44
PAd
ππ
−
==× == =×

3
6
6
6
6
9
6
75 10
197.301 10 Pa
380.13 10
197.301 10
986.51 10
200 10
(200 mm)(986.51 10 )
x
xx
P
A
E
L
σ
σ
ε
δε
−
−
−
×
== = ×
×
×
== = ×
×
== ×

(a)
0.1973 mmδ=
x


66
(0.3)(986.51 10 ) 295.95 10
yx
vεε
−−
=− =− × =− × 

6
(22 mm)( 295.95 10 )
yy
dδε
−
== − × 
 
(b) 0.00651 mmδ=−
y 

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PROBLEM 2.63
A 20-mm-diameter rod made of an experimental plastic is subjected to a tensile force of magnitude P = 6 kN.
Knowing that an elongation of 14 mm and a decrease in diameter of 0.85 mm are observed in a 150-mm length,
determine the modulus of elasticity, the modulus of rigidity, and Poisson’s ratio for the material.

SOLUTION
Let the y-axis be along the length of the rod and the x-axis be transverse.

22 62 3
3
6
6
(20) 314.16 mm 314.16 10 m 6 10 N
4
610
19.0985 10 Pa
314.16 10
14 mm
0.093333
150 mm
y
y
y
AP
P
A
L
π
σ
δ
ε
−
−
== =× =×
×
== = ×
×
== =

Modulus of elasticity
:
6
6
19.0985 10
204.63 10 Pa
0.093333y
y
E
σ
ε ×
== = ×
205 MPaE= 

0.85
0.0425
20
x
x
d
δ
ε
==− =−
Poisson’s ratio
:
0.0425
0.093333
x
y
v
ε
ε −
=− =−
0.455v= 
Modulus of rigidity:

6
6
204.63 10
70.31 10 Pa
2(1 ) (2)(1.455)
E
G
v
×
== =×
+
70.3 MPaG= 

74. PROPRIETARY MATERIAL. © 2012 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,
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PROBLEM 2.64
The change in diameter of a large steel bolt is carefully measured as the
nut is tightened. Knowing that
6
29 10 psiE=× and 0.30,v=
determine the internal force in the bolt, if the diameter is observed to
decrease by
3
0.5 10 in.
−
×

SOLUTION

3
3
3
0.5 10 in. 2.5 in.
0.5 10
0.2 10
2.5
y
y
y
d
dδ
ε
ε
−
−
−
=− × =
×
==− =−×


3
3
0.2 10
: 0.66667 10
0.3yy
x
x
v
v
εε
ε
ε
−
−
− ×
=− = = = ×


63 3
(29 10 )(0.66667 10 ) 19.3334 10 psi
xx
Eσε
−
==× × = ×

22 2
(2.5) 4.9087 in
44
ππ
== =Ad

33
(19.3334 10 )(4.9087) 94.902 10 lb
x
FAσ== × = ×

94.9 kipsF= 

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PROBLEM 2.65
A 2.5-m length of a steel pipe of 300-mm outer diameter and 15-mm
wall thickness is used as a column to carry a 700-kN centric axial load.
Knowing that
200 GPaE= and 0.30,v= determine (a) the change in
length of the pipe, (b) the change in its outer diameter, (c) the change in
its wall thickness.

SOLUTION

()
3
22 2 2 32
0.3 m 0.015 m 2.5 m
2 0.3 2(0.015) 0.27 m 700 10 N
(0.3 0.27 ) 13.4303 10 m
44
ππ
−
== =
=−= − = = ×
=−= − = ×
o
io
oidt L
dd t P
Add

(a)
3
93
(700 10 )(2.5)
(200 10 )(13.4303 10 )
PL
EA
δ
−
×
=− =−
××


6
651.51 10 m
−
=− × 0.652 mmδ=− 

6
6
651.51 10
260.60 10
2.5Lδ
ε
−
−
−×
== =− ×


6
LAT
6
(0.30)( 260.60 10 )
78.180 10
vεε
−
−
=− =− − ×
=×
(b)
6
LAT
(300 mm)(78.180 10 )ε
−
Δ= = ×
oodd

0.0235 mmΔ=
od 
(c)
6
LAT
(15 mm)(78.180 10 )ttε
−
Δ= = ×

0.001173 mmtΔ= 

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PROBLEM 2.66
An aluminum plate (74E= GPa and 0.33)v= is subjected to a
centric axial load that causes a normal stress
σ. Knowing that, before
loading, a line of slope 2:1 is scribed on the plate, determine the slope
of the line when
125σ= MPa.

SOLUTION
The slope after deformation is
2(1 )
tan
1
y
x
ε
θ
ε+
=
+




6
3
9
33
125 10
1.6892 10
74 10
(0.33)(1.6892 10 ) 0.5574 10
2(1 0.0005574)
tan 1.99551
1 0.0016892
x
x
yx
E
v
σ
ε
εε
θ
−
−−×
== = ×
×
=− =− × =− ×
−
==
+
tan 1.99551θ= 

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PROBLEM 2.67
The block shown is made of a magnesium alloy, for which
45E= GPa and 0.35.v= Knowing that 180
x
σ=− MPa,
determine (a) the magnitude of
y
σ for which the change in the
height of the block will be zero, (b) the corresponding change in
the area of the face ABCD, (c) the corresponding change in the
volume of the block.

SOLUTION
(a)
6
000
1
()
(0.35)( 180 10 )
yy z
yxyz
yx
vv
E
v
δεσ
εσσσ
σσ===
=−−
== −×

6
63 10 Pa=− × 63 MPa
y
σ=− 

6
3
9
6
3
9
1 (0.35)( 243 10 )
( ) ( ) 1.89 10
45 10
1 157.95 10
( ) 3.51 10
45 10
zzxy xy
xy
xxyZ
v
vv
EE
v
vv
EE
εσσσ σσ
σσ
εσσσ
−
−−×
=−−=−+= =−×
×
− ×
= − − = =− =− ×
×

(b)
0
0
(1 ) (1 ) (1 )
()()
xz
x xz z xz x z xz
xzx z xz xzx z
ALL
AL L LL
AAA LL LL
εε εεεε
εεεε εε
=
=+ += +++
Δ=− = + + ≈ +

33
(100 mm)(25 mm)( 3.51 10 1.89 10 )A
−−
Δ= − × − ×
2
13.50 mmAΔ=− 
(c)
0
0
(1 ) (1 ) (1 )
(1 )
( small terms)εεε
εεεεεεεεεεεε
εεε
=
=+ + +
=+++++++
Δ=− = + + +
xyz
xxyyzz
xyz xyzxyyzzxxyz
xyz x y z
VLLL
VL L L
LLL
VVV LLL

33
(100)(40)(25)( 3.51 10 0 1.89 10 )
−−
Δ= − × +− ×V
3
540 mmΔ=−V 

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PROBLEM 2.68
A 30-mm square was scribed on the side of a large steel pressure
vessel. After pressurization, the biaxial stress condition at the square
is as shown. For
200 GPaE= and 0.30,v= determine the change
in length of (a) side AB, (b) side BC, (c) diagnonal AC.

SOLUTION
Given: 80 MPa 40 MPa
xyσσ==
Using Eq’s (2.28):
6
3
1800.3(40)
() 34010
200 10
xxy v
Eεσσ
−
−
=−= =×
×


6
3
1400.3(80)
() 8010
200 10
yyx v
Eεσσ
−
−
=−= =×
×

(a) Change in length of AB
.

63
( ) (30 mm)(340 10 ) 10.20 10 mm
AB xABδε
−−
== ×=×

10.20 m
ABδ
μ= 
(b) Change in length of BC.

63
( ) (30 mm)(80 10 ) 2.40 10 mm
BC yBCδε
−−
== ×=×

2.40 m
BCδ
μ= 
(c) Change in length of diagonal AC.
From geometry,
222
()()()AC AB BC=+
Differentiate:
2( ) ( ) 2( ) ( ) 2( ) ( )AC AC AB AB BC BCΔ= Δ+ Δ
But
() () ()
AC AB BCAC AB BCδδδΔ= Δ= Δ=
Thus,
2( ) 2( ) 2( )
AC AB BCAC AB BCδδδ=+

11
(10.20 m)+ (2.40 m)
22
AC AB BC
AB BC
AC AC
δδδ
μμ=+=
8.91 m
ACδ μ= 

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PROBLEM 2.69
The aluminum rod AD is fitted with a jacket that is used to apply a
hydrostatic pressure of 6000 psi to the 12-in. portion BC of the rod.
Knowing that
6
10.1 10E=× psi and 0.36,v= determine (a) the change in
the total length AD, (b) the change in diameter at the middle of the rod.

SOLUTION

6
6
6
6
6000 psi 0
1
()
1
[ 6000 (0.36)(0) (0.36)( 6000)]
10.1 10
380.198 10
1
()
1
[ (0.36)( 6000) 0 (0.36)( 6000)]
10.1 10
427.72 10
xz y
xxyz
yxyz
P
vv
E
vv
Eσσ σ
εσσσ
εσσσ
−
−
==−=− =
=−−
=−−−−
×
=− ×
=− +−
=−−+−−
×
=×

Length subjected to strain
:12 in.
x
Lε=
(a)
6
(12)(427.72 10 )
yy
Lδε
−
== ×
3
5.13 10 in.
l
δ
−
=× 
(b)
6
(1.5)( 380.198 10 )
xx
dδε
−
== − ×
3
0.570 10 in.
d
δ
−
=− × 

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PROBLEM 2.70
For the rod of Prob. 2.69, determine the forces that should be applied to
the ends A and D of the rod (a) if the axial strain in portion BC of the rod
is to remain zero as the hydrostatic pressure is applied, (b) if the total
length AD of the rod is to remain unchanged.
PROBLEM 2.69 The aluminum rod AD is fitted with a jacket that is
used to apply a hydrostatic pressure of 6000 psi to the 12-in. portion BC
of the rod. Knowing that
6
10.1 10E=× psi and 0.36,v= determine
(a) the change in the total length AD, (b) the change in diameter at the
middle of the rod.

SOLUTION
Over the pressurized portion BC,

1
() ( )
1
(2 )
x zyy
yBC x y z
y
p
vv
E
vp
Eσσ σσ
εσσσ
σ==− =
=− +−
=+

(a)
22 2
() 0 2 0
2 (2)(0.36)(6000)
4320 psi
(1.5) 1.76715 in
44
(1.76715)( 4320) 7630 lbεσ
σ
ππ
σ=+=
=− =−
=−
== =
== − =−
yBC y
y
y
vp
vp
Ad
FA
i.e., 7630 lb compression 
(b) Over unpressurized portions AB and CD,
0
xz
σσ==

() ()
y
yAB yCD
E
σ
εε
==
For no change in length,

() () () 0
( )() () 0
12
(20 12) (2 ) 0
24 (24)(0.36)(6000)
2592 psi
20 20
(1.76715)( 2592) 4580 lbδε ε ε
εε
σ
σ
σ
σ=++=
++=
−+ +=
=− =− =−
== − =−
AB y AB BC y BC CD y CD
AB CD y AB BC y BC
y
y
y
y
LLL
LL L
vp
EE
vp
PA
4580 lb compressionP= 

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PROBLEM 2.71
In many situations, physical constraints prevent
strain from occurring in a given direction.
For example,
0
z
ε= in the case shown, where
longitudinal movement of the long prism is
prevented at every point. Plane sections
perpendicular to the longitudinal axis remain
plane and the same distance apart. Show that for
this situation, which is known as plane strain,
we can express
,
z
σ ,ε
x
and
y
ε as follows:
2
2
()
1
[(1 ) (1 ) ]
1
[(1 ) (1 ) ]
zxy
x xy
y yx
v
vvv
E
vvv
Eσσσ
εσσ
εσσ=+
=− −+
=− −+


SOLUTION

1
0( )or ( )
z xyz z xy
vv v
Eεσσσσσσ== − − + = + 

2
2
1
()
1
[()]
1
[(1 ) (1 ) ]
xxyz
x yxy
xy
vv
E
vv
E
vvv
Eεσσσ
σσ σσ
σσ=−−
=−−+
=− −+


2
2
1
()
1
[()]
1
[(1 ) (1 ) ]
yxyz
x yxy
yx
vv
E
vv
E
vvv
Eεσσσ
σσ σσ
σσ=− +−
=− +− +
=− −+


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PROBLEM 2.72
In many situations, it is known that the normal stress in a given direction is zero,
for example,
0
z
σ= in the case of the thin plate shown. For this case, which is
known as plane stress, show that if the strains
εx and εy have been determined
experimentally, we can express
,,σσ
x y and
z
ε as follows:
22
()
111
xy yx x yzxy
vv v
EE
vvvεε εε
σσεεε++
===−+
−−−


SOLUTION

0
1
()
zx xy
v
E
σ
εσσ=
=−
(1)

1
()
y xy
v
Eεσσ=− + (2)
Multiplying (2) by v and adding to (1),

2
2
1
or ( )
1
x yx x xy
vE
vv
E v
εε σ σ εε
−
+= = +
−

Multiplying (1) by v and adding to (2),

2
2
1
or ( )
1
y xy y yx
vE
vv
E v
εε σ σ εε
−
+= = +
−


1
()
zxy
v
vv
E E
εσσ=− − =−
E
⋅
2
2
()
1
(1 )
() ()
11
x yy x
xy xy
vv
v
vv v
vvεεεε
εε εε+++
−
+
=− + =− +
−−


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PROBLEM 2.73
For a member under axial loading, express the normal strain ε′ in a
direction forming an angle of 45° with the axis of the load in terms of the
axial strain
εx by (a) comparing the hypotenuses of the triangles shown in
Fig. 2.49, which represent, respectively, an element before and after
deformation, (b) using the values of the corresponding stresses of
σ ′ and
σx shown in Fig. 1.38, and the generalized Hooke’s law.

SOLUTION

Figure 2.49
(a)
22 2
2222
22 22
[ 2 (1 )] (1 ) (1 )
2(1 2 ) 1 2 1 2
42 2 2
xx
x xxx
xx x x
v
vv
vvεε ε
εε ε ε ε ε
εε εε ε ε′+=++−
′′++ =+ ++− +
′′+=+−+
Neglect squares as small
422
x x
vεε ε′=−
1
2
x
v
εε
−
′=



(A) (B)

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PROBLEM 2.73 (Continued)

(b)
1
2
1
2
x
v
EE
vP
E A
v
E
σσ
ε
σ′′
′=−
−
=⋅
−
=


1
2
x
v
ε
−
=


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PROBLEM 2.74
The homogeneous plate ABCD is subjected to a biaxial loading as
shown. It is known that
0z
σσ= and that the change in length of
the plate in the x direction must be zero, that is,
0.
x
ε= Denoting
by E the modulus of elasticity and by v Poisson’s ratio, determine
(a) the required magnitude of
,
x
σ (b) the ratio
0
/.
z
σε

SOLUTION

0
0
,0,0
11
()()
zyx
xxyz x
vv v
E E
σσσ ε
εσσσ σσ===
=−−=−

(a)
0x
vσσ= 
(b)
2
2
00 0
111
()(0)
zxyz
v
vv v
EEE
εσσσ σσ σ
−
=− − + =− −+ =

0
2
1
z
E
v
σ
ε
=
− 

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PROBLEM 2.75
A vibration isolation unit consists of two blocks of hard rubber
bonded to a plate AB and to rigid supports as shown. Knowing
that a force of magnitude
25 kNP= causes a deflection
1.5 mmδ= of plate AB, determine the modulus of rigidity of the
rubber used.

SOLUTION



33
3
311
(25 10 N) 12.5 10 N
22
12.5 10 N)
833.33 10 Pa
(0.15 m)(0.1 m)
FP
F
A
τ
== × =×
×
== = ×

3
3
1.5 10 m 0.03 m
1.5 10
0.05
0.03δ
δ
γ
−
−
=× =
×
== =
h
h

3
6
833.33 10
16.67 10 Pa
0.05
Gτ
γ ×
== = ×


16.67 MPaG= 

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PROBLEM 2.76
A vibration isolation unit consists of two blocks of hard rubber with a
modulus of rigidity
19 MPaG= bonded to a plate AB and to rigid
supports as shown. Denoting by P the magnitude of the force applied
to the plate and by
δ the corresponding deflection, determine the
effective spring constant,
/,kPδ= of the system.

SOLUTION


Shearing strain:
h
δ
γ
=
Shearing stress:
G
G
hδ
τγ
==
Force:
12
or
2
GA GA
PA P
hh δδ
τ
== =
Effective spring constant:
2P GA
k
h
δ
==
with
2
(0.15)(0.1) 0.015 m 0.03 mAh== =


6 2
6
2(19 10 Pa)(0.015 m )
19.00 10 N/m
0.03 m
k
×
==×

3
19.00 10 kN/mk=× 

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PROBLEM 2.77
The plastic block shown is bonded to a fixed base and to a horizontal rigid
plate to which a force P is applied. Knowing that for the plastic used
55G= ksi, determine the deflection of the plate when 9P= kips.

SOLUTION
Consider the plastic block. The shearing force carried is
3
910lb=×P
The area is
2
(3.5)(5.5) 19.25 in==A
Shearing stress
:
3
910
467.52 psi
19.25
P
A
τ
×
== =

Shearing strain
:
3
467.52
0.0085006
55 10τ
γ
== =
×G
But
(2.2)(0.0085006)
δ
γδγ
=∴== h
h 0.0187 in.δ= 

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PROBLEM 2.78
A vibration isolation unit consists of two blocks of hard rubber bonded to
plate AB and to rigid supports as shown. For the type and grade of rubber
used
all
220τ= psi and 1800G= psi. Knowing that a centric vertical force
of magnitude
3.2P= kips must cause a 0.1-in. vertical deflection of
the plate AB, determine the smallest allowable dimensions a and b of the
block.

SOLUTION
Consider the rubber block on the right. It carries a shearing force equal to
1
2
.
P
The shearing stress is
1
2
P
A
τ=
or required area
3
2
3.2 10
7.2727 in
2 (2)(220)
τ
×
== =
P
A

But
(3.0)Ab=
Hence,
2.42 in.
3.0
==
A
b
min
2.42 in.=b 
Use
2.42 in=b and 220 psiτ=
Shearing strain
.
220
0.12222
1800Gτ
γ
== =

But
a
δ
γ
=
Hence,
0.1
0.818 in.
0.12222δ
γ
== =a
min
0.818 in.=a 

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PROBLEM 2.79
The plastic block shown is bonded to a rigid support and to a vertical plate to
which a 55-kip load P is applied. Knowing that for the plastic used
150G= ksi,
determine the deflection of the plate.

SOLUTION



2
3
3
3
3
3
(3.2)(4.8) 15.36 in
55 10 lb
55 10
3580.7 psi
15.36
150 10 psi
3580.7
23.871 10
150 10
2 in.
τ
τ
γ
−
==
=×
×
== =
=×
== = ×
×
=
A
P
P
A
G
G
h

3
3
(2)(23.871 10 )
47.7 10 in.
hδγ
−
−
== ×
=×
0.0477 in.δ=↓ 

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PROBLEM 2.80
What load P should be applied to the plate of Prob. 2.79 to produce a
1
16
-in. deflection?
PROBLEM 2.79 The plastic block shown is bonded to a rigid support and to a vertical plate to which a
55-kip load P is applied. Knowing that for the plastic used
150G= ksi, determine the deflection of the plate.

SOLUTION

3
3
1
in. 0.0625 in.
16
2 in.
0.0625
0.03125
2
150 10 psi
(150 10 )(0.03125)
4687.5 psi
h
h
G
G
δ
δ
γ
τγ==
=
== =
=×
== ×
=


2
3
(3.2)(4.8) 15.36 in
(4687.5)(15.36)
72 10 lb
τ
==
==
=×
A
PA

72 kips 

92. PROPRIETARY MATERIAL. © 2012 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,
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PROBLEM 2.81
Two blocks of rubber with a modulus of rigidity 12 MPaG= are
bonded to rigid supports and to a plate AB. Knowing that
100 mmc=
and
45 kN,P= determine the smallest allowable dimensions a and b of
the blocks if the shearing stress in the rubber is not to exceed 1.4 MPa
and the deflection of the plate is to be at least 5 mm.

SOLUTION

Shearing strain:
aG
δτ
γ
==

6
6
(12 10 Pa)(0.005 m)
0.0429 m
1.4 10 Pa
G
aδ
τ ×
== =
×
42.9 mma= 
Shearing stress
:
1
2
2
P
P
Abc
τ==

3
6
45 10 N
0.1607 m
2 2(0.1 m) (1.4 10 Pa)
P
b
c
τ
×
== =
×
160.7 mmb= 

93. PROPRIETARY MATERIAL. © 2012 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,
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PROBLEM 2.82
Two blocks of rubber with a modulus of rigidity 10G= MPa are bonded
to rigid supports and to a plate AB. Knowing that
200b= mm and
125c= mm, determine the largest allowable load P and the smallest
allowable thickness a of the blocks if the shearing stress in the rubber
is not to exceed 1.5 MPa and the deflection of the plate is to be at least
6 mm.

SOLUTION

Shearing stress:
1
2
2
P
P
Abc
τ==

3
2 2(0.2 m)(0.125 m)(1.5 10 kPa)Pbcτ== × P 75.0 kN= 
Shearing strain
:
aG
δτ
γ
==

6
6
(10 10 Pa)(0.006 m)
0.04 m
1.5 10 Pa
G
aδ
τ ×
== =
×
40.0 mma= 

94. PROPRIETARY MATERIAL. © 2012 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,
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PROBLEM 2.83
∗

Determine the dilatation e and the change in volume of the 200-mm
length of the rod shown if (a) the rod is made of steel with
200E= GPa
and
0.30,v= (b) the rod is made of aluminum with 70E= GPa and
0.35.v=
SOLUTION

22 2 62
3
6
(22) 380.13 mm 380.13 10 m
44
46 10 N
121.01 10 Pa
0
1
()
(1 2 )1
()
x
yz
x
xxyz
x
yz x
x
xyz xxx
Ad
P
P
A
vv
EE
vv
E
v
evv
E E
ππ
σ
σσ
σ
εσσσ
σ
εε ε
σ
εεε σ σ σ
−
== = = ×
=×
== ×
==
=−−=
==− =−
−
=++= − − =


Volume:
233
(380.13 mm )(200 mm) 76.026 10 mmVAL
VVe
== = ×
Δ=
(a) Steel
:
6
6
9
(1 0.60)(121.01 10 )
242 10
200 10
−−×
==×
×
e

6
242 10
−
=×e 

36 3
(76.026 10 )(242 10 ) 18.40 mm
−
Δ= × × =V
3
18.40 mmΔ=V 
(b) Aluminum
:
6
6
9
(1 0.70)(121.01 10 )
519 10
70 10
−−×
==×
×
e

6
519 10
−
=×e 

36 3
(76.026 10 )(519 10 ) 39.4 mm
−
Δ= × × =V
3
39.4 mmΔ=V 

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PROBLEM 2.84
∗

Determine the change in volume of
the 2-in. gage length segment AB in
Prob. 2.61 (a) by computing the
dilatation of the material, (b) by
subtracting the original volume of
portion AB from its final volume.

SOLUTION
From Problem 2.61, thickness =
1
16
in., E = 29 × 10
6
psi, v = 0.30.
(a)
211
0.03125 in
216 
==
 
 
A

3
00
Volume: (0.03125)(2.00) 0.0625 in== =VAL

3
3
6
6
66
6600
19.2 10 psi 0
0.03125
119.210
( ) 662.07 10
29 10
(0.30)(662.07 10 ) 198.62 10
264.83 10
xyz
x
xxyz
yz x
xyz
P
A
vv
EE
v
e
σσσ
σ
εσσσ
εε ε
εεε
−
−−
−
== = × = =
×
=−−== = ×
×
==− =− × =− ×
=++= ×


663
0
(0.0625)(264.83 10 ) 16.55 10 in
−−
Δ= = × = ×VVe 
(b) From the solution to Problem 2.61,

36 6
1.324 10 in., 99.3 10 in., 12.41 10 in.δδ δ
−− −
=× =−× =−×
xy z
The dimensions when under a 600-lb tensile load are:

3
0
6
0
6
0
3
63
0
Length: 2 1.324 10 2.001324 in.
1
Width: 99.3 10 0.4999007 in.
2
1
Thickness: 12.41 10 0.06248759 in.
16
Volume: 0.062516539 in
0.062516539 0.0625 16.54 10 inδ
δ
δ
−
−
−
−
=+=+ × =
=+=− × =
=+ = − × =
==
Δ=− = − = ×
x
y
z
LL
ww
tt
VLwt
VVV


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without permission.

PROBLEM 2.85
∗

A 6-in.-diameter solid steel sphere is lowered into the ocean to a point where the pressure is 7.1 ksi (about
3 miles below the surface). Knowing that
6
29 10E=× psi and 0.30,v= determine (a) the decrease in
diameter of the sphere, (b) the decrease in volume of the sphere, (c) the percent increase in the density of the
sphere.

SOLUTION
For a solid sphere,
3
00
3
3
6
(6.00)
6
113.097 in.
p
π
π
σσσ
=
=
=
===−
xyz
Vd

3
3
6
6
7.1 10 psi
1
()
(1 2 ) (0.4)(7.1 10 )
29 10
97.93 10
xxyz
vv
E
vp
Eεσσσ
−
=− ×
=−−
−×
=− =−
×
=− ×
Likewise,
6
97.93 10
yz
εε
−
==− ×

6
293.79 10
xyz
eεεε
−
=++=− ×
(a)
66
0
(6.00)( 97.93 10 ) 588 10 in.ε
−−
−Δ =− =− − × = ×
x
dd
6
588 10 in.
−
−Δ = ×d 
(b)
633
0
(113.097)( 293.79 10 ) 33.2 10 in
−−
−Δ =− =− − × = ×VVe
33
33.2 10 in
−
−Δ = ×V 
(c) Let
mass of sphere. constant.mm==

00 0
(1 )mVVVeρρρ=== +

00
00 0
23 23
6
60
0 1
111
(1 ) 1
(1 ) 1
293.79 10
100% (293.79 10 )(100%)
ρρ ρ
ρρ
ρρ
ρ
−
−
−
=−= ×−= −
++
=−+−+ −=−+−+
≈− = ×
−
×= ×

Vm
Vem e
ee e ee e
e
0.0294% 

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PROBLEM 2.86
∗

(a) For the axial loading shown, determine the change in height and the
change in volume of the brass cylinder shown. (b) Solve part a,
assuming that the loading is hydrostatic with
70
xyz
σσσ===− MPa.

SOLUTION

0
22 32 32
00
33 63
000
135 mm 0.135 m
(85) 5.6745 10 mm 5.6745 10 m
44
766.06 10 mm 766.06 10 m
h
Ad
VAh
ππ
−
−
==
== = × = ×
== × = ×

(a)
6
6
6
9
0, 58 10 Pa, 0
15810
( ) 552.38 10
105 10σσ σ
σ
εσσσ
−
==−× =
×
=− +− = =− =− ×
×
xy z
y
yxyz
vv
EE


6
0
(135 mm)( 552.38 10 )ε
−
Δ= = − ×
y
hh 0.0746 mmΔ=−h 

6
6
9
(1 2 )1 2 (0.34)( 58 10 )
( ) 187.81 10
105 10σ
σσσ
−
−−−×
=++= = =−×
×
y
xyz
vv
e
EE


33 6
0
(766.06 10 mm )( 187.81 10 )
−
Δ= = × − ×VVe
3
143.9 mmΔ=−V 
(b)
66
6
6
9
70 10 Pa 210 10 Pa
1 1 2 (0.34)( 70 10 )
( ) 226.67 10
105 10σσσ σσσ
εσσσ σ
−
===−× ++=−×
−−×
=− +− = = =− ×
×
xyz xyz
yxyz y
v
vv
EE


6
0
(135 mm)( 226.67 10 )ε
−
Δ= = − ×
y
hh 0.0306 mmΔ=−h 

6
6
9
1 2 (0.34)( 210 10 )
( ) 680 10
105 10
σσσ
−−−×
=++= =−×
×
xyz
v
e
E


33 6
0
(766.06 10 mm )( 680 10 )
−
Δ= = × − ×VVe
3
521 mmΔ=−V 

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PROBLEM 2.87
∗

A vibration isolation support consists of a rod A of radius
1
10R= mm and
a tube B of inner radius
2
25R= mm bonded to an 80-mm-long hollow
rubber cylinder with a modulus of rigidity G = 12 MPa. Determine the
largest allowable force P that can be applied to rod A if its deflection is not
to exceed 2.50 mm.

SOLUTION

Let r be a radial coordinate. Over the hollow rubber cylinder,
12
.
RrR≤≤
Shearing stress
τ acting on a cylindrical surface of radius r is

2
P P
Arh
τ
π==
The shearing strain is

2
P
GGhr
τ
γ
π
==
Shearing deformation over radial length dr,

2
d
dr
Pdr
ddr
Gh r
δ
γ
δγ
π
=
==

Total deformation
.

22
11
2
1
21
2
121
2
ln (ln ln )
22
2
ln or
2ln(/)
RR
RR
R
R Pdr
d
Gh r
PP
rRR
Gh Gh
RPGh
P
Gh R R R
δδ
π
ππ
πδ
π==
==−
==


12
Data: 10 mm 0.010 m, 25 mm 0.025 m, 80 mm 0.080 mRRh== == ==

63
63
3
12 10 Pa 2.50 10 m
(2 )(12 10 )(0.080)(2.50 10 )
16.46 10 N
ln (0.025/0.010)
G
P δ
π
−
−
=× = ×
××
==×
16.46 kN 

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PROBLEM 2.88
A vibration isolation support consists of a rod A of radius R 1 and a tube B of
inner radius R
2 bonded to a 80-mm-long hollow rubber cylinder with a
modulus of rigidity G = 10.93 MPa. Determine the required value of the ratio
R
2/R1 if a 10-kN force P is to cause a 2-mm deflection of rod A.

SOLUTION
Let r be a radial coordinate. Over the hollow rubber cylinder,
12
.RrR≤≤
Shearing stress
τ acting on a cylindrical surface of radius r is

2
P P
Arh
τ
π==
The shearing strain is

2
P
GGhr
τ
γ
π
==
Shearing deformation over radial length dr,

2
δ
γ
δγ
δ
π
=
=
=
d
dr
ddr
Pdr
dr
Gh r

Total deformation
.

22
11
2
1
21
2
1
2
ln (ln ln )
22
ln
2
RR
RR
R
R Pdr
d
Gh r
PP
rRR
Gh Gh
RP
Gh R
δδ
π
ππ
π==
==−
=



6
2
3
1
2
1
2 (2 )(10.93 10 )(0.080)(0.002)
ln 1.0988
10.10
exp(1.0988) 3.00
R Gh
RP
R
Rπδ π ×
== =
==

21
/3.00=RR 

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PROBLEM 2.89
∗

The material constants E, G, k, and v are related by Eqs. (2.33) and (2.43). Show that any one of these
constants may be expressed in terms of any other two constants. For example, show that
(a) k = GE/(9G − 3E) and (b) v = (3k – 2G)/(6k + 2G).

SOLUTION
and
3(1 2 ) 2(1 )
EE
kG
vv
==
−+
(a)
1or1
22
EE
vv
GG
+= = −

22
3[2 2 4 ] 18 6
31 2 1
2
===
−+ −
−−


E EG EG
k
GEG GEE
G

96
=
−
EG
k
GE

(b)
2(1 )
3(1 2 )
kv
Gv
+
=
−


36 2 2
32 2 6
−=+
−=+
kkvGGv
kGGk

32
62
−
=
+
kG
v
kG


101. PROPRIETARY MATERIAL. © 2012 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,
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PROBLEM 2.90
∗

Show that for any given material, the ratio G/E of the modulus of rigidity over the modulus of elasticity is
always less than
1
2
but more than
1
3
. [Hint: Refer to Eq. (2.43) and to Sec. 2.13.]

SOLUTION
or 2(1 )
2(1 )
EE
Gv
vG
==+
+
Assume
0v> for almost all materials, and
1
2
<v for a positive bulk modulus.
Applying the bounds,
1
2<21 3
2
E
G

≤+=



Taking the reciprocals,
11
23
G
E
>>

or
11
32
G
E
<<


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PROBLEM 2.91
∗

A composite cube with 40-mm sides and the properties shown is
made with glass polymer fibers aligned in the x direction. The cube
is constrained against deformations in the y and z directions and is
subjected to a tensile load of 65 kN in the x direction. Determine
(a) the change in the length of the cube in the x direction, (b) the
stresses
,,
xy
σσ and .
z
σ

50 GPa 0.254
15.2 GPa 0.254
15.2 GPa 0.428
xxz
yxy
zzy
Ev
Ev
Ev
==
==
==


SOLUTION
Stress-to-strain equations are

yx yx zx z
x
xy z
v v
EE Eσσσ
ε
=− − (1)

xyx y zyz
y
xyz
vv
EEEσσ σ
ε
=− + − (2)

yz yxz x z
z
x yz
vv
E EE
σσ σ
ε
=− − + (3)

xyyx
x y
vv
EE
=
(4)

yzzy
y z
vv
EE
=
(5)

zxxz
z x
vv
EE
=
(6)
The constraint conditions
are 0 and 0.
yz
εε==
Using (2) and (3) with the constraint conditions gives

1 zy xy
y zx
yzx
vv
EEE
σσσ−= (7)

1yz xz
y zx
yzx
v V
EEE
σσσ−+= (8)

1 0.428 0.254
or 0.428 0.077216
15.2 15.2 50
0.428 1 0.254
or 0.428 0.077216
15.2 15.2 50
yz xyz x
y zx yz x
σσσσσ σ
σσ σ σσ σ−= −=
−+= −+=

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without permission.

PROBLEM 2.91
∗
(Continued)

Solving simultaneously,
0.134993
y zx
σσ σ==
Using (4) and (5) in (1),
1 xy xz
x xyz
xx
v v
EE E
εσσσ=−−

262
3
6
6
3
6
9
1
[1 (0.254)(0.134993) (0.254)(0.134993)]
0.93142
(40)(40) 1600 mm 1600 10 m
65 10
40.625 10 Pa
1600 10
(0.93142)(40.625 10 )
756.78 10
50 10
x x
x
x
x
x
x
E
E
E
A
P
A σ
σ
σ
ε
−
−
−
=− −
=
== =×
×
== = ×
×
×
==×
×

(a)
6
(40 mm)(756.78 10 )δε
−
== ×
xxx
L 0.0303 mmδ=
x

(b)
6
40.625 10 Paσ=×
x
40.6 MPaσ=
x


66
(0.134993)(40.625 10 ) 5.48 10 Paσσ== × = ×
yz  5.48 MPaσσ==
yz 

104. PROPRIETARY MATERIAL. © 2012 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,
reproduced, or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited
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without permission.


PROBLEM 2.92
∗

The composite cube of Prob. 2.91 is constrained against
deformation in the z direction and elongated in the x direction by
0.035 mm due to a tensile load in the x direction. Determine (a) the
stresses
, ,
x y
σσ and ,
z
σ (b) the change in the dimension in the
y direction.

50 GPa 0.254
15.2 GPa 0.254
15.2 GPa 0.428
xxz
yxy
zzy
Ev
Ev
Ev
==
==
==


SOLUTION

yx yx zx z
x
xy z
v v
EE Eσσσ
ε
=− − (1)

xyx y zyz
y
xyz
vv
EEEσσ σ
ε
=− + − (2)

yz yxz x z
z
x yz
vv
E EE
σσ σ
ε
=− − + (3)

xy yx
xy
vv
EE
=
(4)

yzzy
y z
vv
EE
=
(5)

zxxz
z x
vv
EE
=
(6)
Constraint condition: 0
Load condition : 0ε
σ=
=
z
y

From Equation (3),
1
0
xz x z
xz
v
EE
σσ=− +

(0.254)(15.2)
0.077216
50
xz z
z xx
x
vE
E
σσ σ== =

105. PROPRIETARY MATERIAL. © 2012 The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed,
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without permission.

PROBLEM 2.92
∗
(Continued)

From Equation (1) with
0,
y
σ=

11
11
[ 0.254 ] [1 (0.254)(0.077216)]
0.98039
0.98039
zx xz
xxzxz
xz xx
x zx
xx
x
x
xx
x
vv
EEEE
EE
E
E
εσσσσ
σσ σ
σ
ε
σ=−=−
=− =−
=
=

But,
60.035 mm
875 10
40 mm
x
x
x
L
δ
ε
−
== = ×
(a)
96
3
(50 10 )(875 10 )
44.625 10 Pa
0.98039
σ
−
××
==×
x
44.6 MPaσ=
x


0
y
σ=

66
(0.077216)(44.625 10 ) 3.446 10 Paσ=×=×
z
3.45 MPaσ=
z

From (2),
66
99
6
1
(0.254)(44.625 10 ) (0.428)(3.446 10 )
0
50 10 15.2 10
323.73 10xy zy
yxyz
xyz
vv
EEE
εσσσ
−
=+−
××
=− + −
××
=− ×
(b)
6
(40 mm)( 323.73 10 )δε
−
== − ×
yyy
L 0.0129 mmδ=−
y 

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PROBLEM 2.93
Two holes have been drilled through a long steel bar that is subjected to a centric
axial load as shown. For
6.5P= kips, determine the maximum value of the
stress (a) at A, (b) at B.

SOLUTION
(a) At hole A:
11 1
in.
22 4
r=⋅=


1
32.50 in.
2
=− =d


2
net1
(2.50) 1.25 in
2
== =


Adt

non
net
6.5
5.2 ksi
1.25
P
A
σ===

()
1
4
22
0.1667
3
r
D
==

From Fig. 2.60a,
2.56K=

max non
(2.56)(5.2)σσ==K
max
13.31 ksiσ= 
(b) At hole B:
1
(1.5) 0.75, 3 1.5 1.5 in.
2
rd== =−=


2
net non
net16.5
(1.5) 0.75 in , 8.667 ksi
20.75
σ

== = = = =


P
Adt
A


22(0.75)
0.5
3
r
D
==

From Fig. 2.60a,
2.16K=

max non
(2.16)(8.667)σσ==K
max
18.72 ksiσ= 

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PROBLEM 2.94
Knowing that
all
16 ksi,σ= determine the maximum allowable value of the
centric axial load P.

SOLUTION
At hole A:
11 1
in.
22 4
=⋅=r


1
32.50 in.
2
=− =d


2
net1
(2.50) 1.25 in
2
== =


Adt

()
1
4
22
0.1667
3
r
D
==

From Fig. 2.60a,
2.56K=

net max
max
net
(1.25)(16)
7.81 kips
2.56
KP A
P
AKσ
σ
=∴= = =
At hole B:
1
(1.5) 0.75 in, 3 1.5 1.5 in.
2
== =−=rd


2
net1
(1.5) 0.75 in ,
2
== =


Adt

22(0.75)
0.5
3
r
D
==

From Fig. 2.60a,
2.16K=

net max
(0.75)(16)
5.56 kips
2.16
A
P
Kσ
== =
Smaller value for P controls.
5.56 kipsP= 

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PROBLEM 2.95
Knowing that the hole has a diameter of 9 mm, determine (a) the radius
r
f of the fillets for which the same maximum stress occurs at the hole A
and at the fillets, (b) the corresponding maximum allowable load P if
the allowable stress is 100 MPa.

SOLUTION
For the circular hole,
1
(9) 4.5 mm
2
r
==




22(4.5)
96 9 87 mm 0.09375
96
r
d
D
=−= = =


62
net
(0.087 m)(0.009 m) 783 10 mAdt
−
== = ×
From Fig. 2.60a,
hole
2.72K=

hole
max
net
KP
A
σ=

66
3net max
hole
(783 10 )(100 10 )
28.787 10 N
2.72
A
P
Kσ
−
××
== =×

(a) For fillet,
96 mm, 60 mmDd==

96
1.60
60
D
d
==


62
min
(0.060 m)(0.009 m) 540 10 mAdt
−
== = ×

66
fillet min max
max fillet 3
min
(5.40 10 )(100 10 )
28.787 10
1.876
KP A
AP
K
σ
σ
−
××
===
×
=
∴

From Fig. 2.60b,
0.19 0.19 0.19(60)
f
f
r
rd
d
≈∴≈ =
11.4 mm
f
r= 
(b)
28.8 kNP= 

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PROBLEM 2.96
For 100 kN,P= determine the minimum plate thickness t required if
the allowable stress is 125 MPa.


SOLUTION
At the hole: 20 mm 88 40 48 mm
AA
rd==−=

2 2(20)
0.455
88
A
A
r
D
==

From Fig. 2.60a,
2.20K=

max
net max
3
3
6
(2.20)(100 10 N)
36.7 10 m 36.7 mm
(0.048 m)(125 10 Pa)
AA
KP KP KP
t
Adt d
t
σ
σ
−
== ∴=
×
==×=
×

At the fillet:
88
88 mm, 64 mm 1.375
64
B
B
D
Dd
d
====


15
15 mm 0.2344
64
B
B
B
r
r
d
===

From Fig. 2.60b,
1.70K=

max
min B
KPKP
Adt
σ==

3
3
6
max
(1.70)(100 10 N)
21.25 10 m 21.25 mm
(0.064 m)(125 10 Pa)
B
KP
t
d
σ
−×
== =×=
×

The larger value is the required minimum plate thickness.

36.7 mmt= 

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PROBLEM 2.97
The aluminum test specimen shown is subjected to two equal and opposite
centric axial forces of magnitude P. (a) Knowing that E = 70 GPa
and
all
σ=200 MPa, determine the maximum allowable value of P and the
corresponding total elongation of the specimen. (b) Solve part a, assuming that
the specimen has been replaced by an aluminum bar of the same length and a
uniform 60 × 15-mm rectangular cross section.

SOLUTION

69
all
262
min
20010Pa 7010Pa
(60 mm)(15 mm) 900 mm 900 10 m
E
Aσ
−
=× =×
===×

(a) Test specimen
. 75 mm, 60 mm, 6 mmDdr===

75 6
1.25 0.10
60 60
Dr
dd
== ==

From Fig. 2.60b
max
1.95
P
K K
Aσ==

66
3max
(900 10 )(200 10 )
92.308 10 N
1.95
A
P
Kσ
−
××
== =×
92.3 kNP= 
Wide area
*232
(75 mm)(15 mm) 1125 mm 1.125 10 mA
−
===×

3
9363
6
92.308 10 0.150 0.300 0.150
70 10 1.125 10 900 10 1.125 10
7.91 10 m
ii i
ii i
PL LP
AE E A
δ
−− −
−
×
=Σ = Σ = + +

××××
=× 0.791 mmδ= 
(b) Uniform bar
.

66 3
all
(900 10 )(200 10 ) 180 10 NPAσ
−
==× ×=× 180.0 kNP= 

3
3
69
(180 10 )(0.600)
1.714 10 m
(900 10 )(70 10 )
PL
AE
δ
−
−×
== = ×
××
 1.714 mmδ= 

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PROBLEM 2.98
For the test specimen of Prob. 2.97, determine the maximum value of the
normal stress corresponding to a total elongation of 0.75 mm.
PROBLEM 2.97 The aluminum test specimen shown is subjected to two equal
and opposite centric axial forces of magnitude P. (a) Knowing that E = 70 GPa
and
all
200σ= MPa, determine the maximum allowable value of P and the
corresponding total elongation of the specimen. (b) Solve part a, assuming that
the specimen has been replaced by an aluminum bar of the same length and a
uniform
60 15-mm× rectangular cross section.

SOLUTION

3
13 2
232
13
262
2
0.75 10 m
150 mm 0.150 m, 300 mm 0.300 m
(75 mm)(15 mm) 1125 mm 1.125 10 m
(60 mm)(15 mm) 900 mm 900 10 m
ii i
ii i
PL LP
EA E A
LL L
AA
A
δδ
−
−
−
=Σ = Σ = ×
== = = =
== = = ×
===×


1
36 3
93
30.150 0.300 0.150
600 m
1.125 10 900 10 1.125 10
(70 10 )(0.75 10 )
87.5 10 N
600
δ
−
−− −
−
Σ= + + =
×× ×
××
== =×
Σ
i
i
i
i
L
A
E
P
L
A

Stress concentration
. 75 mm, 60 mm, 6 mmDdr===

75 6
1.25 0.10
60 60
Dr
dd
== ==

From Fig. 2.60b
1.95K=

3
6
max 6
min
(1.95)(87.5 10 )
189.6 10 Pa
900 10
P
K
A
σ
−
×
== =×
×

max
189.6 MPaσ= 
Note that
max all
.σσ< 

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PROBLEM 2.99
A hole is to be drilled in the plate at A. The diameters of the bits
available to drill the hole range from
1
2
to 1
1
/2 in. in
1
4
-in.
increments. If the allowable stress in the plate is 21 ksi,
determine (a) the diameter d of the largest bit that can be used if
the allowable load P at the hole is to exceed that at the fillets, (b)
the corresponding allowable load P.

SOLUTION
At the fillets:
4.6875 0.375
1.5 0.12
3.125 3.125
Dr
dd
== ==

From Fig. 2.60b,
2.10K=

2
min
all
max all
min
min all
all
(3.125)(0.5) 1.5625 in
(1.5625)(21)
15.625 kips
2.10
σσ
σ
==
==
== =
A
P
K
A
A
P
K

At the hole:
net
( 2 ) , from Fig. 2.60 =−ADrtK a

net all
max all all
net
σ
σσ
==∴=
PA
KP
AK
with
all
4.6875 in. 0.5 in. 21 ksiDt σ===

Hole diam. r
2dD r=− 2/rD K
net
A
all
P
0.5 in. 0.25 in. 4.1875 in. 0.107 2.68 2.0938 in
2
16.41 kips
0.75 in. 0.375 in. 3.9375 in. 0.16 2.58 1.96875 in
2
16.02 kips
1 in. 0.5 in. 3.6875 in. 0.213 2.49 1.84375 in
2
15.55 kips
1.25 in. 0.625 in. 3.4375 in. 0.267 2.41 1.71875 in
2
14.98 kips
1.5 in. 0.75 in. 3.1875 in. 0.32 2.34 1.59375 in
2
14.30 kips

(a) Largest hole with
all
15.625 kipsP> is the
3
4
-in. diameter hole. 
(b) Allowable load
all
15.63 kipsP= 

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PROBLEM 2.100
(a) For 13 kipsP= and
1
2
in.,d= determine the maximum stress
in the plate shown. (b) Solve part a, assuming that the hole at A is
not drilled.

SOLUTION
Maximum stress at hole:
Use Fig. 2.60a for values of K.

2
net
max
net
20.5
0.017, 2.68
4.6875
(0.5)(4.6875 0.5) 2.0938 in
(2.68)(13)
16.64 ksi,
2.0938
σ
== =
=−=
== =
r
K
D
A
P
K
A

Maximum stress at fillets:
Use Fig. 2.60b for values of K.

0.375 4.6875
0.12 1.5
3.125 3.125rD
dd
== = =
2.10K=

2
min
max
min
(0.5)(3.125) 1.5625 in
(2.10)(13)
17.47 ksi
1.5625
σ
==
== =
A
P
K
A

(a) With hole and fillets: 17.47 ksi 
(b) Without hole: 17.47 ksi 

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PROBLEM 2.101
Rod ABC consists of two cylindrical portions AB and BC; it is made of a mild
steel that is assumed to be elastoplastic with
200 GPaE= and 250 MPa.
Y
σ=
A force P is applied to the rod and then removed to give it a permanent set
2
p
δ= mm. Determine the maximum value of the force P and the maximum
amount
m
δ by which the rod should be stretched to give it the desired permanent
set.

SOLUTION

22 62
232 32
66 3
max min
(30) 706.86 mm 706.86 10 m
4
(40) 1.25664 10 mm 1.25664 10 m
4
(706.86 10 )(250 10 ) 176.715 10 N
AB
BC
Y
A
A
PA
π
π
σ
−
−
−
== =×
==× =×
==× ×=×


max
176.7 kNP= 

33
969 3
3
(176.715 10 )(0.8) (176.715 10 )(1.2)
(200 10 )(706.86 10 ) (200 10 )(1.25664 10 )
1.84375 10 m 1.84375 mm
BCAB
AB BC
PLPL
EA EA
δ
−−
−
′′ ××
′=+= +
×× × ×
=×=


or 2 1.84375
pm mp
δδδ δδδ′′=− =+=+ 3.84 mm
m
δ= 

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PROBLEM 2.102
Rod ABC consists of two cylindrical portions AB and BC; it is made of a mild
steel that is assumed to be elastoplastic with
200 GPaE= and 250 MPa.
Y
σ=
A force P is applied to the rod until its end A has moved down by an amount
5mm.
m
δ= Determine the maximum value of the force P and the permanent
set of the rod after the force has been removed.

SOLUTION

22 62
232 32
66 3
max min
(30) 706.86 mm 706.86 10 m
4
(40) 1.25664 10 mm 1.25644 10 m
4
(706.86 10 )(250 10 ) 176.715 10 N
AB
BC
Y
A
A
PA
π
π
σ
−
−
−
== =×
==× =×
==× ×=×


max
176.7 kNP= 

33
969 3
3
(176.715 10 )(0.8) (176.715 10 )(1.2)
(200 10 )(706.68 10 ) (200 10 )(1.25664 10 )
1.84375 10 m 1.84375 mm
BCAB
AB BC
PLPL
EA EA
δ
−−
−
′′ ××
′=+= +
×× × ×
=×=


5 1.84375 3.16 mm
pm
δδδ ′=−=− = 3.16 mm
p
δ= 

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PROBLEM 2.103
The 30-mm square bar AB has a length 2.2 m;L= it is made of a mild steel that is
assumed to be elastoplastic with
200 GPaE= and 345 MPa.
Y
σ= A force P is applied
to the bar until end A has moved down by an amount
.
m
δ Determine the maximum
value of the force P and the permanent set of the bar after the force has been removed,
knowing that (a)
4.5 mm,
m
δ= (b) 8 mm.
m
δ=

SOLUTION

262
6
3
9
(30)(30) 900 mm 900 10 m
(2.2)(345 10 )
3.795 10 3.795 mm
200 10
Y
YY
A
L
L
E
σ
δε
−
−
== =×
×
== = = ×=
×

If
66 3
, (900 10 )(345 10 ) 310.5 10 Nδδ σ
−
≥==× ×=×
mYm Y
PA
Unloading
: 3.795 mm
m Y
Y
PL L
AE Eσ
δδ
′====

Pm
δδδ ′=−
(a)
3
4.5 mm 310.5 10 Nδδ=>=×
mYm
P 310.5 kNδ=
m


perm
4.5 mm 3.795 mmδ=−
perm
0.705 mmδ= 
(b)
3
8 mm 310.5 10 Nδδ=> =×
mYm
P 310.5 kNδ=
m


perm
8.0 mm 3.795 mmδ=−
perm
4.205 mmδ= 

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PROBLEM 2.104
The 30-mm square bar AB has a length 2.5m;L= it is made of mild steel that is assumed
to be elastoplastic with
200 GPaE= and 345 MPa.
Y
σ= A force P is applied to the bar
and then removed to give it a permanent set
.
p
δ Determine the maximum value of the
force P and the maximum amount
m
δ by which the bar should be stretched if the desired
value of
p
δ is (a) 3.5 mm, (b) 6.5 mm.

SOLUTION

262
6
3
9
(30)(30) 900 mm 900 10 m
(2.5)(345 10 )
4.3125 10 m 4.3125 mm
200 10
Y
YY
A
L
L
E
σ
δε
−
== =×
×
== = = × =
×

When
m
δ exceeds ,
Y
δ thus producing a permanent stretch of ,
p
δthe maximum force is

66 3
(900 10 )(345 10 ) 310.5 10 Nσ
−
==× ×= ×
mY
PA 310.5 kN= 

δδδδδ δδδ′=−=− ∴ =+
p mmYmpY
(a)
3.5 mm 3.5 mm 4.3125 mmδδ==+
pm 7.81 mm= 
(b)
6.5 mm 6.5 mm 4.3125 mmδδ==+
pm 10.81 mm= 

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PROBLEM 2.105
Rod AB is made of a mild steel that is assumed to be elastoplastic
with
6
29 10 psiE=× and 36 ksi.
Y
σ= After the rod has been attached
to the rigid lever CD, it is found that end C is
3
8
in. too high. A vertical
force Q is then applied at C until this point has moved to position
.C′
Determine the required magnitude of Q and the deflection
1
δ if the
lever is to snap back to a horizontal position after Q is removed.

SOLUTION
Since the rod AB is to be stretched permanently, the peak force in the rod is ,
Y
PP= where

2
3
(36) 3.976 kips
48
YY
PA
π
σ
== =



Referring to the free body diagram of lever CD,


0: 33 22 0
22 (22)(3.976)
2.65 kips
33 33
Σ= − =
== =
D
MQP
QP
2.65 kips=Q 
During unloading, the spring back at B is

3
6
(60)(36 10 )
0.0745 in.
29 10
AB Y
BABY
L
L
Eσ
δε ×
== = =
×

From the deformation diagram,
Slope:
33
0.1117 in.
22 33 22
CB
CB
δδ
θδδ
== ∴= =
− 0.1117 in.δ=
C


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PROBLEM 2.106
Solve Prob. 2.105, assuming that the yield point of the mild steel is
50 ksi.
PROBLEM 2.105 Rod AB is made of a mild steel that is assumed to
be elastoplastic with
6
29 10 psiE=× and 36 ksi.
Y
σ= After the rod
has been attached to the rigid lever CD, it is found that end C is
3
8
in.
too high. A vertical force Q is then applied at C until this point has
moved to position C′. Determine the required magnitude of Q and the
deflection
1
δ if the lever is to snap back to a horizontal position after
Q is removed.

SOLUTION
Since the rod AB is to be stretched permanently, the peak force in the rod is ,
Y
PP= where

2
3
(50) 5.522 kips
48
YY
PA
π
σ
== =



Referring to the free body diagram of lever CD,




0: 33 22 0
D
MQPΣ= − =

22 (22)(5.522)
3.68 kips
33 33
== =QP
3.68 kips=Q 
During unloading, the spring back at B is

3
6
(60)(50 10 )
0.1034 in.
29 10σ
δε ×
== = =
×
AB Y
BABY
L
L
E

From the deformation diagram,
Slope:
33

22 33 22
CB
CB
δδ
θδδ
== ∴ = 0.1552 in.δ=
C


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PROBLEM 2.107
Each cable has a cross-sectional area of 100 mm
2
and is made of an
elastoplastic material for which
345 MPa
Y
σ= and 200 GPa.E= A force
Q is applied at C to the rigid bar ABC and is gradually increased from 0 to
50 kN and then reduced to zero. Knowing that the cables were initially taut,
determine (a) the maximum stress that occurs in cable BD, (b) the
maximum deflection of point C, (c) the final displacement of point C.
(Hint: In Part c, cable CE is not taut.)

SOLUTION
Elongation constraints for taut cables.
Let
rotation angle of rigid bar .θ= ABC

CEBD
AB AC
L L
δδ
θ
==

1
2
δδδ==
AB
BD CE CE
AC
L
L
(1)
Equilibrium of bar ABC
.




0: 0
AABBDACCEAC
MLFLFLQ=+−=

1
2
AB
CE BD CE BD
AC
L
QF F F F
L
=+ = +
(2)
Assume cable CE is yielded.
66 3
(100 10 )(345 10 ) 34.5 10 N
CE Y
FAσ
−
==× ×=×
From (2),
33 3
2( ) (2)(50 10 34.5 10 ) 31.0 10 N
BD CE
FQF=− = ×−×=×
Since
3
< 34.5 10 N,
BD Y
FAσ=× cable BD is elastic when 50 kN.Q=

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PROBLEM 2.107 (Continued)

(a) Maximum stresses
. 345 MPa
CE Y
σσ==

3
6
6
31.0 10
310 10 Pa
100 10
BD
BD
F
A
σ
−
×
== =×
×
310 MPa
BD
σ= 
(b) Maximum of deflection of point C
.

3
3
96
(31.0 10 )(2)
3.1 10 m
(200 10 )(100 10 )
BD BD
BD
FL
EA
δ
−
−×
== =×
××

From (1),
3
26.210m
CCE BD
δδ δ
−
== =× 6.20 mm↓ 
Permanent elongation of cable CE:
()()
σ
δδ
=−
YCE
CE p CE
L
E


max max
6
33
9
()() ()
(345 10 )(2)
6.20 10 2.75 10 m
200 10
CE CE Y CE
CE P CE CE
FL L
EA E σ
δδ δ
−−
=−=−
×
=×− =×
×

(c) Unloading
. Cable CE is slack (0)
CE
F= at 0.Q=
From (2),
2( ) 2(0 0) 0
BD CE
FQF=− =−=
Since cable BD remained elastic,
0.
BD BD
BD
FL
EA
δ== 

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PROBLEM 2.108
Solve Prob. 2.107, assuming that the cables are replaced by rods of the
same cross-sectional area and material. Further assume that the rods are
braced so that they can carry compressive forces.
PROBLEM 2.107 Each cable has a cross-sectional area of 100 mm
2

and is made of an elastoplastic material for which
345 MPa
Y
σ= and
200 GPa.E= A force Q is applied at C to the rigid bar ABC and is
gradually increased from 0 to 50 kN and then reduced to zero. Knowing
that the cables were initially taut, determine (a) the maximum stress
that occurs in cable BD, (b) the maximum deflection of point C, (c) the
final displacement of point C. (Hint: In Part c, cable CE is not taut.)

SOLUTION
Elongation constraints.
Let
rotation angle of rigid bar .ABCθ=

δδ
θ
==
BC CE
AB AC
L L


1
2
δδδ==
AB
BD CE CE
AC
L
L
(1)
Equilibrium of bar ABC
.




0: 0=+−=
AABBDACCEAC
MLFLFLQ

1
2
AB
CE BD CE BD
AC
L
QF F F F
L
=+ =+
(2)
Assume cable CE is yielded.
66 3
(100 10 )(345 10 ) 34.5 10 N
CE Y
FAσ
−
==× ×=×
From (2),
33 3
2( ) (2)(50 10 34.5 10 ) 31.0 10 N
BD CE
FQF=− = ×−×=×
Since
3
34.5 10 N,
BD Y
FAσ<=× cable BD is elastic when 50 kN.Q=

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PROBLEM 2.108 (Continued)

(a) Maximum stresses
. 345 MPa
CE Y
σσ==

3
6
6
31.0 10
310 10 Pa
100 10
BD
BD
F
A
σ
−
×
== =×
×
310 MPa
BD
σ= 
(b) Maximum of deflection of point C
.

3
3
96
(31.0 10 )(2)
3.1 10 m
(200 10 )(100 10 )
BD BD
BD
FL
EA
δ
−
−×
== =×
××

From (1),
3
26.210m
CCE BD
δδ δ
−
== =× 6.20 mm↓ 
Unloading
.
3
50 10 N,
CE C
Q δδ′′′=× =
From (1),
1
2BD C
δδ′′=
Elastic
96 1
62
(200 10 )(100 10 )( )
510
2
CBD
BD C
BDEA
F
L δδ
δ
−
′××′
′′ ′== =×


96
6
(200 10 )(100 10 )( )
10 10
2
CE C
CE C
CE
EA
F
Lδδ
δ
−
′′ ××
′′== =×

From (2),
61
2
12.5 10
CE BD C
QF F δ′′ ′ ′=+ =×
Equating expressions for
,′Q
63
12.5 10 50 10
C
δ′×=×

3
410 m
C
δ
−
′=×
(c) Final displacement
.
33 3
( ) 6.2 10 4 10 2.2 10 m
CCmC
δδ δ
−− −
′=−=×−×=× 2.20 mm↓ 

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PROBLEM 2.109
Rod AB consists of two cylindrical portions AC and BC, each with a cross-sectional
area of 1750 mm
2
. Portion AC is made of a mild steel with 200 GPaE= and
250 MPa,
Y
σ= and portion CB is made of a high-strength steel with 200 GPaE=
and
345 MPa.
Y
σ= A load P is applied at C as shown. Assuming both steels to be
elastoplastic, determine (a) the maximum deflection of C if P is gradually increased
from zero to 975 kN and then reduced back to zero, (b) the maximum stress in each
portion of the rod, (c) the permanent deflection of C.

SOLUTION
Displacement at C to cause yielding of AC.

6
, 3
,, 9
(0.190)(250 10 )
0.2375 10 m
200 10σ
δε
−×
== = =×
×AC Y AC
CY AC Y AC
L
L
E

Corresponding force
.
66 3
,
(1750 10 )(250 10 ) 437.5 10 Nσ
−
==× ×=×
AC Y AC
FA

96 3
3
(200 10 )(1750 10 )(0.2375 10 )
437.5 10 N
0.190
C
CB
CB
EA
F
Lδ
−−
×× ×
=− =− =− ×

For equilibrium of element at C,

3
()0 87510N
AC CB Y Y AC CB
FFP PFF−+= =−=×
Since applied load
33
975 10 N 875 10 N,P=× >× portion AC yields.

33 3
437.5 10 975 10 N 537.5 10 N
CB AC
FFP=−= ×−× =− ×
(a)
3
3
96
(537.5 10 )(0.190)
0.29179 10 m
(200 10 )(1750 10 )
CB CD
C
FL
EA
δ
−
−×
=− = = ×
××


0.292 mm
(b) Maximum stresses
:
,
250 MPaσσ==
AC Y AC
250 MPa 

3
6
6
537.5 10
307.14 10 Pa 307 MPa
1750 10
σ
−
×
==− =− × =−
×
BC
BC
F
A
307 MPa− 
(c) Deflection and forces for unloading
.

33
3
3
96

975 10 2 487.5 10 N
(487.5 10 )(0.190)
0.26464 10 m
(200 10 )(1750 10 )
AC AC CB CB AC
CB AC AC
AB
AC CB AC AC
PL PL L
PP P
EA EA L
PPPPP
δ
δ
−
−
′′
′′′′= =− ∴ =− =−
′′′′′=×= −= = ×
×
′==×
××


33
3
0.29179 10 0.26464 10
0.02715 10 mδδδ
−−
−
′=−= × − ×
=×
pm
0.0272 mm 

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PROBLEM 2.110
For the composite rod of Prob. 2.109, if P is gradually increased from zero until the
deflection of point C reaches a maximum value of
0.3 mm
m
δ= and then decreased
back to zero, determine (a) the maximum value of P, (b) the maximum stress in each
portion of the rod, (c) the permanent deflection of C after the load is removed.
PROBLEM 2.109 Rod AB consists of two cylindrical portions AC and BC, each with a
cross-sectional area of 1750 mm
2
. Portion AC is made of a mild steel with 200E=GPa
and
250 MPa,
Y
σ= and portion CB is made of a high-strength steel with 200=EGPa
and
345 MPa.
Y
σ= A load P is applied at C as shown. Assuming both steels to be
elastoplastic, determine (a) the maximum deflection of C if P is gradually increased
from zero to 975 kN and then reduced back to zero, (b) the maximum stress in each
portion of the rod, (c) the permanent deflection of C.

SOLUTION
Displacement at C is 0.30 mm.
m
δ= The corresponding strains are

3
30.30 mm
1.5789 10
190 mm
0.30 mm
1.5789 10
190 mm
m
AC
AC
m
CB
CB
L
L
δ
ε
δ
ε
−
−
== = ×
=− =− =− ×

Strains at initial yielding:

6
, 3
, 9
6
, 3
, 9
250 10
1.25 10 (yielding)
200 10
345 10
1.725 10 (elastic)
200 10σ
ε
σ
ε
−
−×
== =×
×
×
==− =−×
×YAC
YAC
YBC
YCB
E
E

(a) Forces
:
66 3
(1750 10 )(250 10 ) 437.5 10 N
AC Y
FAσ
−−
== × ×= ×

96 3 3
(200 10 )(1750 10 )( 1.5789 10 ) 552.6 10 N
CB CB
FEAε
−− −
==× ×−×=−×
For equilibrium of element at C,
0
AC CB
FFP−−=

333
437.5 10 552.6 10 990.1 10 N=−= ×+ ×= ×
AC CD
PF F 990 kN= 
(b) Stresses
:
,
:σσ=
AC Y AC
AC 250 MPa= 

3
6
6
552.6 10
: 316 10 Pa
1750 10
σ
−
×
==− =−×
×
CB
CB
F
CB
A
316 MPa− 

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PROBLEM 2.110 (Continued)

(c) Deflection and forces for unloading
.

33
3
3
96

2 990.1 10 N 495.05 10 N
(495.05 10 )(0.190)
0.26874 10 m 0.26874 mm
(200 10 )(1750 10 )
AC AC CB CB AC
CB AC AC
AB
AC CB AC AC
PL PL L
PP P
EA EA L
PP P P P
δ
δ
−
−
′′
′′′= =− ∴ =− =−
′′ ′ ′ ′=−= = × ∴ = ×
×
′==×=
××


0.30 mm 0.26874 mmδδδ ′=−= −
pm
0.031mm 

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PROBLEM 2.111
Two tempered-steel bars, each
3
16
-in. thick, are bonded to a
1
2
-in. mild-steel
bar. This composite bar is subjected as shown to a centric axial load of
magnitude P. Both steels are elastoplastic with
6
29 10E=× and with yield
strengths equal to 100 ksi and 50 ksi, respectively, for the tempered and mild
steel. The load P is gradually increased from zero until the deformation of
the bar reaches a maximum value
0.04 in.
m
δ= and then decreased back to
zero. Determine (a) the maximum value of P, (b) the maximum stress in the
tempered-steel bars, (c) the permanent set after the load is removed.

SOLUTION
For the mild steel,
2
11
(2) 1.00 in
2
==


A
3
1
1 6
(14)(50 10 )
0.024138 in.
29 10σ
δ ×
== =
×
Y
Y
L
E

For the tempered steel,
2
23
2 (2) 0.75 in
16
==


A
3
2
2 3
(14)(100 10 )
0.048276 in.
29 10σ
δ ×
== =
×
Y
Y
L
E

Total area:
2
12
1.75 in=+ =AA A
12
.
YmY
δδδ<< The mild steel yields. Tempered steel is elastic.
(a) Forces
:
33
111
(1.00)(50 10 ) 50 10 lbσ== ×=×
Y
PA

3
32
2
(29 10 )(0.75)(0.04)
62.14 10 lb
14δ ×
== =×
m
EA
P
L


3
12
112.14 10 lb 112.1kips=+ = × =PPP 112.1 kips=P 
(b) Stresses
:
31
11
1
50 10 psi 50 ksi
Y
P
A
σσ== =× =

3
32
2
2
62.14 10
82.86 10 psi 82.86 ksi
0.75
σ
×
== = × =
P
A
82.86 ksi 
Unloading
:
3
6
(112.14 10 )(14)
0.03094 in.
(29 10 )(1.75)
δ
×
′== =
×
PL
EA

(c) Permanent set
: 0.04 0.03094 0.00906 in.
pm
δδδ ′=−= − = 0.00906in. 

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PROBLEM 2.112
For the composite bar of Prob. 2.111, if P is gradually increased from zero to
98 kips and then decreased back to zero, determine (a) the maximum
deformation of the bar, (b) the maximum stress in the tempered-steel bars,
(c) the permanent set after the load is removed.
PROBLEM 2.111 Two tempered-steel bars, each
3
16
-in. thick, are bonded to
a
1
2
-in. mild-steel bar. This composite bar is subjected as shown to a centric
axial load of magnitude P. Both steels are elastoplastic with
6
29 10E=× psi
and with yield strengths equal to 100 ksi and 50 ksi, respectively, for the
tempered and mild steel.

SOLUTION
Areas: Mild steel:
2
11
(2) 1.00 in
2
==


A
Tempered steel:
2
23
2 (2) 0.75 in
16
==


A
Total:
2
12
1.75 in=+ =AA A
Total force to yield the mild steel:

33
11
(1.75)(50 10 ) 87.50 10 lb
Y
YYY
P
PA
A
σσ=∴= = ×= ×
,
Y
PP>therefore, mild steel yields.
Let
1
force carried by mild steel.P=

2
force carried by tempered steel.P=

33
111
(1.00)(50 10 ) 50 10 lbσ== ×=×PA

33 3
12 2 1
,981050104810lbPP PP PP+= =−=× −× =×
(a)
3
2
6
2
(48 10 )(14)
(29 10 )(0.75)
δ
×
==
×
m
PL
EA
0.03090in.= 
(b)
3
32
2
2
48 10
64 10 psi
0.75
σ
×
== =×
P
A
64ksi= 
Unloading
:
3
6
(98 10 )(14)
0.02703 in.
(29 10 )(1.75)
δ
×
′== =
×
PL
EA

(c)
0.03090 0.02703δδδ ′=−= −
Pm
0.00387in.= 

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PROBLEM 2.113
The rigid bar ABC is supported by two links, AD and BE, of uniform
37.5 6-mm× rectangular cross section and made of a mild steel that is
assumed to be elastoplastic with
200 GPaE= and 250 MPa.
Y
σ=
The magnitude of the force Q applied at B is gradually increased from
zero to 260 kN. Knowing that
0.640 m,a= determine (a) the value of
the normal stress in each link, (b) the maximum deflection of point B.

SOLUTION
Statics: 0 : 0.640( ) 2.64 0
CBEAD
MQPPΣ= − − =
Deformation: 2.64 , 0.640
AB
aδθδθθ===
Elastic analysis:

262
96
6
66
9
96
6
66
(37.5)(6) 225 mm 225 10 m
(200 10 )(225 10 )
26.47 10
1.7
(26.47 10 )(2.64 ) 69.88 10
310.6 10
(200 10 )(225 10 )
45 10
1.0
(45 10 )(0.640 ) 28.80 10
AD A A A
AD
AD
AD
BE B B B
BE
A
EA
P
L
P
A
EA
P
L
δδδ
θθ
σθ
δδδ
θθ
σ
−
−
−
== =×
××
== =×
=× =×
== ×
××
== =×
=× = ×
9
128 10
BE
BE
P
A
θ==×

From Statics,
2.64
4.125
0.640
BE AD BE AD
QP P P P=+ =+

666
[28.80 10 (4.125)(69.88 10 ] 317.06 10θθ=×+ ×= ×
Y
θat yielding of link AD:
69
250 10 310.6 10
AD Y
σσ θ== ×= ×

6
663
804.89 10
(317.06 10 )(804.89 10 ) 255.2 10 N
Y
Y
Q
θ
−
−
=×
=× ×=×

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PROBLEM 2.113 (Continued)

(a) Since
3
260 10 ,
Y
QQ=×> link AD yields. 250 MPa
AD
σ= 

66 3
(225 10 )(250 10 ) 56.25 10 N
AD Y
PAσ
−−
==× × = ×
From Statics,
33
4.125 260 10 (4.125)(56.25 10 )
BE AD
PQ P=− = × − ×

3
3
6
6
27.97 10 N
27.97 10
124.3 10 Pa
225 10
σ
−
=×
×
== = ×
×
BE
BE
BE
P
P
A
124.3 MPaσ=
BE

(b)
3
6
96
(27.97 10 )(1.0)
621.53 10 m
(200 10 )(225 10 )
BE BE
B
PL
EA
δ
−
−×
== =×
××
0.622 mm
B
δ=↓ 

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PROBLEM 2.114
Solve Prob. 2.113, knowing that a = 1.76 m and that the magnitude of
the force Q applied at B is gradually increased from zero to 135 kN.
PROBLEM 2.113 The rigid bar ABC is supported by two links, AD
and BE, of uniform
37.5 6-mm× rectangular cross section and made
of a mild steel that is assumed to be elastoplastic with
200 GPaE=
and
250 MPa.
Y
σ= The magnitude of the force Q applied at B is
gradually increased from zero to 260 kN. Knowing that
0.640 m,a=
determine (a) the value of the normal stress in each link, (b) the
maximum deflection of point B.

SOLUTION
Statics: 0 : 1.76( ) 2.64 0
CBEAD
MQPPΣ= − − =
Deformation: 2.64 , 1.76
AB
δθδθ==
Elastic Analysis:

262
96
6
66
9
96
6
66
(37.5)(6) 225 mm 225 10 m
(200 10 )(225 10 )
26.47 10
1.7
(26.47 10 )(2.64 ) 69.88 10
310.6 10
(200 10 )(225 10 )
45 10
1.0
(4510)(1.76) 79.210
AD A A A
AD
AD
AD
BE B B B
BE
BE
A
EA
P
L
P
A
EA
P
L
δδδ
θθ
σθ
δδδ
θθ
σ
−
−
−
== =×
××
== =×
=× =×
== ×
××
== =×
=× = ×
9
352 10
BE
P
A
θ==×

From Statics,
2.64
1.500
1.76
BE AD BE AD
QP P P P=+ =+

666
[73.8 10 (1.500)(69.88 10 ] 178.62 10θθ=×+ × = ×
Y
θat yielding of link BE:
69
250 10 352 10
BE Y Y
σσ θ== ×= ×

6
66 3
710.23 10
(178.62 10 )(710.23 10 ) 126.86 10 N
Y
Y
Q
θ
−
−
=×
=× ×=×
Since
3
135 10 N ,
Y
QQ=× > link BE yields. 250 MPa
BE Y
σσ== 

66 3
(225 10 )(250 10 ) 56.25 10 Nσ
−
==× ×= ×
BE Y
PA


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PROBLEM 2.114 (Continued)

From Statics,
31
( ) 52.5 10 N
1.500
AD BE
PQP=−=×
(a)
3
6
6
52.5 10
233.3 10
225 10
σ
−
×
== = ×
×
AD
AD
P
A
233 MPaσ=
AD

From elastic analysis of AD,
3
6
751.29 10 rad
69.88 10θ
−
==×
×
AD
P

(b)
3
1.76 1.322 10 m
B
δθ
−
==× 1.322 mm
B
δ=↓ 

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PROBLEM 2.115
∗

Solve Prob. 2.113, assuming that the magnitude of the force Q
applied at B is gradually increased from zero to 260 kN and then
decreased back to zero. Knowing that
0.640 m,a= determine (a)
the residual stress in each link, (b) the final deflection of point B.
Assume that the links are braced so that they can carry compressive
forces without buckling.
PROBLEM 2.113 The rigid bar ABC is supported by two links,
AD and BE, of uniform
37.5 6-mm× rectangular cross section and
made of a mild steel that is assumed to be elastoplastic with
200 GPaE= and 250 MPa.
Y
σ= The magnitude of the force Q
applied at B is gradually increased from zero to 260 kN. Knowing
that
0.640 m,a= determine (a) the value of the normal stress in
each link, (b) the maximum deflection of point B.

SOLUTION
See solution to Problem 2.113 for the normal stresses in each link and the deflection of Point B after loading.

6
6
6
250 10 Pa
124.3 10 Pa
621.53 10 m
AD
BE
B
σ
σ
δ
−
=×
=×
=×

The elastic analysis given in the solution to Problem 2.113 applies to the unloading.

6
3
6
66
317.06 10
260 10
820.03 10
317.06 10 317.06 10
Q
Q
Q θ
−
′′=×
×
′===×
××


9966
99 6 6
6
310.6 10 (310.6 10 )(820.03 10 ) 254.70 10 Pa
128 10 (128 10 )(820.03 10 ) 104.96 10 Pa
0.640 524.82 10 m
AD
BE
B
σθ
σθ
δθ
−
−
−
′=×= × ×= ×
′=× = × × = ×
′′==×

(a) Residual stresses
.

666
,res
250 10 254.70 10 4.70 10 Paσσσ
−
′=−=×− ×=−×
AD AD AD 4.70MPa=− 

666
,res
124.3 10 104.96 10 19.34 10 Paσσσ ′=−= ×− ×= ×
BE BE BE 19.34 MPa= 
(b)
666
,
621.53 10 524.82 10 96.71 10 mδδδ
−−−
′=−= × − × = ×
BP B B
0.0967 mm=↓ 

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PROBLEM 2.116
A uniform steel rod of cross-sectional area A is attached to rigid supports
and is unstressed at a temperature of
45 F.° The steel is assumed to be
elastoplastic with
36
Y
σ= ksi and
6
29 10 psi.E=× Knowing that
6
6.5 10 / F,α
−
=× ° determine the stress in the bar (a) when the temperature
is raised to
320 F,° (b) after the temperature has returned to 45 F°.

SOLUTION
Let P be the compressive force in the rod.
Determine temperature change to cause yielding.

3
66
() () 0
36 10
( ) 190.98 F
(29 10 )(6.5 10 )
Y
Y
Y
Y
LPL
LT LT
AE E
T
Eσ
δα α
σ
α
−
=− + Δ =− + Δ =
×
Δ= = = °
××

But
320 45 275 F ( )
Y
TTΔ= − = °>Δ
(a) Yielding occurs.
36 ksi
Y
σσ=− =− 
Cooling
:
66 3
(T) 275F
()0
()
(29 10 )(6.5 10 )(275) 51.8375 10 psi
PT
PL
LT
AE
P
ET
A
δδ δ α
σα
−
′Δ= °
′
′′ ′ ′===− + Δ=
′
′′==− Δ
=− × × =− ×
(b) Residual stress
:

33
res
36 10 51.8375 10 15.84 10 psiσσσ ′=− − =− × + × = ×
Y
15.84 ksi 

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PROBLEM 2.117
The steel rod ABC is attached to rigid supports and is unstressed at a
temperature of
25 C.° The steel is assumed elastoplastic, with E = 200
GPa and
250 MPa.
y
σ= The temperature of both portions of the rod is
then raised to
150 C° . Knowing that α=
6
11.7 10 / C,
−
×° determine
(a) the stress in both portions of the rod, (b) the deflection of point C.

SOLUTION

62
62
500 10 m 0.150 m
300 10 m 0.250 m
−
−
=× =
=× =
AC AC
CB CB
AL
AL

Constraint
: 0
PT
δδ+=
Determine ΔT to cause yielding in portion CB.

()
AC CB
AB
AC CB
AC CB
AB AC CB
PL PL
L T
EA EA
LLP
T
LE A Aα
α−−=Δ

Δ= + 


At yielding,
663
(300 10 )(2.50 10 ) 75 10 N
YCBY
PP A σ
−
== = × × =×

3
96 6 6
()
75 10 0.150 0.250
90.812 C
(0.400)(200 10 )(11.7 10 ) 500 10 300 10
α
−−−

Δ= + 

× 
=+=°

×× × × 
AC CBY
Y
AB AC CB
LLP
T
LE A A

Actual ΔT:
150 C 25 C 125 C ( )
Y
T°− °= °>Δ
Yielding occurs
. For
3
(), 7510N
YY
TT PPΔ>Δ = = ×
(a)
3
6
6
75 10
150 10 Pa
500 10
Y
AC
AC
P
A
σ
−
−×
=− =− =− ×
×
150 MPa
AC
σ=− 

σσ=− =−
Y
CB Y
CB
P
A
250 MPa
CB
σ=− 
(b) For
( ) , portion remains elastic.
Y
TT ACΔ>Δ

/
3
66
96
()
(75 10 )(0.150)
(0.150)(11.7 10 )(125) 106.9 10 m
(200 10 )(500 10 )
YAC
CA AC
AC
PL
LT
EA
δα
−−
−
=− + Δ
×
=− + × = ×
××

Since Point A is stationary,
6
/
106.9 10 m
CCA
δδ
−
== × 0.1069 mm
C
δ=→ 

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PROBLEM 2.118
∗

Solve Prob. 2.117, assuming that the temperature of the rod is raised
to 150°C and then returned to
25 C.°
PROBLEM 2.117 The steel rod ABC is attached to rigid supports
and is unstressed at a temperature of
25 C.° The steel is assumed
elastoplastic, with
200E= GPa and 250
Y
σ= MPa. The
temperature of both portions of the rod is then raised to
150 C.°
Knowing that
6
11.7 10 / C,α
−
=× ° determine (a) the stress in both
portions of the rod, (b) the deflection of point C.

SOLUTION

62 62
500 10 m 0.150 m 300 10 m 0.250 m
−−
=× = =× =
AC AC CB CB
ALA L
Constraint: 0
PT
δδ+=
Determine ΔT to cause yielding in portion CB.

()
AC CB
AB
AC CB
AC CB
AB AC CB
PL PL
L T
EA EA
LLP
T
LE A Aα
α−−=Δ

Δ= + 


At yielding,
66 3
(300 10 )(250 10 ) 75 10 N
YCBY
PP A σ
−
== = × × =×
3
96 6 6
75 10 0.150 0.250
()
(0.400)(200 10 )(11.7 10 ) 500 10 300 10
90.812 C
α
−−−
 × 
Δ= + = +  
×× × × 
=°
AC CBY
Y
AB AC CB
LLP
T
LE A A

Actual
:150C 25C 125C ( )Δ°−°=°>Δ
Y
TT
Yielding occurs
. For
3
( ) 75 10 N
YY
TT PPΔ>Δ = = ×
Cooling:
()
()
()125Cα ′Δ
′′Δ= ° =
+
AC CB
AC CB
AB
LL
AA
ELT
TP
66
96
3
0.150 0.250
500 10 300 10
(200 10 )(0.400)(11.7 10 )(125)
103.235 10 N
−−
−
××
××
==×
+

Residual force:
33 3
res
103.235 10 75 10 28.235 10 N (tension)′=−= × −× = ×
Y
PPP

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PROBLEM 2.118
∗
(Continued)

(a) Residual stresses
.
3
res
6
28.235 10
500 10
AC
AC
P
A
σ
−
×
==
×
56.5 MPa
AC
σ= 

3
res
6
28.235 10
300 10
CB
CB
P
A
σ
−
×
==
×
9.41 MPaσ=
CB

(b) Permanent deflection of point C
.
res
δ=
AC
C
AC
PL
EA
0.0424 mm
C
δ=→ 

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PROBLEM 2.119
∗

For the composite bar of Prob. 2.111, determine the residual stresses in
the tempered-steel bars if P is gradually increased from zero to
98 kips and then decreased back to zero.
PROBLEM 2.111 Two tempered-steel bars, each
3
16
-in. thick, are
bonded to a
1
2
-in. mild-steel bar. This composite bar is subjected as
shown to a centric axial load of magnitude P. Both steels are
elastoplastic with
6
29 10E=× psi and with yield strengths equal to
100 ksi and 50 ksi, respectively, for the tempered and mild steel. The
load P is gradually increased from zero until the deformation of the bar
reaches a maximum value
0.04
m
δ= in. and then decreased back to
zero. Determine (a) the maximum value of P, (b) the maximum stress
in the tempered-steel bars, (c) the permanent set after the load is
removed.

SOLUTION
Areas: Mild steel:
2
11
(2) 1.00 in
2
==


A
Tempered steel:
2
23
(2) (2) 0.75 in
16
==


A
Total:
2
12
1.75 in=+ =AA A
Total force to yield the mild steel:
33
11
(1.75)(50 10 ) 87.50 10 lbσσ=∴= = ×= ×
Y
YYY
P
PA
A

P > P
Y; therefore, mild steel yields.
Let
1
2
33
111
33 3
12 2 1
3
32
2
2
force carried by mild steel.
force carried by tempered steel.
(1.00)(50 10 ) 50 10 lb
,981050104810lb
48 10
64 10 psi
0.75
σ
σ
=
=
== ×=×
+= =−=× −× =×
×
== =×
Y
P
P
PA
PP PP PP
P
A
Unloading
:
3
3
98 10
56 10 psi
1.75
P
A
σ
×
′== =×

Residual stresses
.
Mild steel:
33 3
1, res 1
50 10 56 10 6 10 psiσσσ
−
′=−=× −× =−× 6 ksi=−
Tempered steel:
333
2,res 2 1
64 10 56 10 8 10 psiσσσ=−=× −× =× 8.00 ksi 

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PROBLEM 2.120
∗

For the composite bar in Prob. 2.111, determine the residual stresses in
the tempered-steel bars if P is gradually increased from zero until the
deformation of the bar reaches a maximum value
0.04
m
δ= in. and is
then decreased back to zero.
PROBLEM 2.111 Two tempered-steel bars, each
3
16
-in. thick, are bonded
to a
1
2
-in. mild-steel bar. This composite bar is subjected as shown to a
centric axial load of magnitude P. Both steels are elastoplastic
with
6
29 10E=× psi and with yield strengths equal to 100 ksi and 50 ksi,
respectively, for the tempered and mild steel. The load P is gradually
increased from zero until the deformation of the bar reaches a maximum
value
0.04
m
δ= in. and then decreased back to zero. Determine (a) the
maximum value of P, (b) the maximum stress in the tempered-steel bars,
(c) the permanent set after the load is removed.

SOLUTION
For the mild steel,
3
2 1
11 6
1 (14)(50 10 )
(2) 1.00 in 0.024138 in
2 29 10 δ
δ ×
== == =

×
Y
Y
L
A
E

For the tempered steel,
3
2 2
22 6
3 (14)(100 10 )
2 (2) 0.75 in 0.048276 in.
16 29 10 δ
δ ×
== == =

×
Y
Y
L
A
E

Total area:
2
12
1.75 in=+=AA A

12
δδδ<<
YmY

The mild steel yields. Tempered steel is elastic.
Forces
:
33
111
6
32
2
(1.00)(50 10 ) 50 10 lb
(29 10 )(0.75)(0.04)
62.14 10 lb
14δ
δ== ×=×
×
== =×
Y
m
PA
EA
P
L

Stresses:
3
3312
11 2
12
62.14 10
50 10 psi 82.86 10 psi
0.75
σδ σ
×
===× == = ×
Y
PP
AA

Unloading
:
3112.14
64.08 10 psi
1.75
P
A
σ′== = ×
Residual stresses
.
33 3
1, res 1
50 10 64.08 10 14.08 10 psi 14.08 ksiσσσ ′=−=× − × =− × =−

333
2,res 2
82.86 10 64.08 10 18.78 10 psi 18.78 ksiσσσ ′=−= × − × = × = 

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PROBLEM 2.121
∗

Narrow bars of aluminum are bonded to the two sides of a thick steel plate as
shown. Initially, at
1
70 F,T=° all stresses are zero. Knowing that the
temperature will be slowly raised to T
2 and then reduced to T 1, determine
(a) the highest temperature T
2 that does not result in residual stresses, (b) the
temperature T
2 that will result in a residual stress in the aluminum equal to
58 ksi. Assume
6
12.8 10 / Fα
−
=× °
a
for the aluminum and
6
6.5 10 / Fα
−
=× °
s

for the steel. Further assume that the aluminum is elastoplastic, with
E
6
10.9 10=× psi and 58
Y
σ= ksi. (Hint: Neglect the small stresses in the
plate.)

SOLUTION
Determine temperature change to cause yielding.

3
66
() ()
()()
58 10
( ) 844.62 F
() (10.9 10 )(12.8 6.5)(10 )
aY sY
as Y Y
Y
Y
as
PL
LTLT
EA
P
ET
A
T
E
δα α
σαα σ
σ
αα
−
=+ Δ= Δ
==− − Δ =−
×
Δ= = = °
− ×−

(a)
21
( ) 70 844.62 915 F=+Δ = + = °
YY
TT T 915 F° 
After yielding,

() ()
Y
as
LL TL T
E
σ
δαα
=+Δ=Δ
Cooling
:
() ()
as
PL
L TL T
AEδαα
′
′′′=+ Δ=Δ

The residual stress
is

res
()()
YYas
P
E T
Aσσ σ αα
′
=−=− − Δ

Set
res
()()
Y
YY as
E T
σσ
σσ αα=−
−= − − Δ

3
66
2 (2)(58 10 )
1689 F
() (10.9 10 )(12.8 6.5)(10 )σ
αα
−
×
Δ= = = °
− ×−
Y
as
T
E
(b)
21
70 1689 1759 F=+Δ= + = °TT T 1759 F° 
If
2
1759 F,T>° the aluminum bar will most likely yield in compression.

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PROBLEM 2.122
∗

Bar AB has a cross-sectional area of 1200 mm
2
and is made of a steel
that is assumed to be elastoplastic with
200 GPaE= and
250 MPa.
Y
σ= Knowing that the force F increases from 0 to 520 kN
and then decreases to zero, determine (a) the permanent deflection of
point C, (b) the residual stress in the bar.

SOLUTION


262
1200 mm 1200 10 mA
−
==×
Force to yield portion AC
:
66
3
(1200 10 )(250 10 )
300 10 N
AC Y
PAσ
−
== × ×
=×



For equilibrium,
0.+−=
CB AC
FP P

33
3
300 10 520 10
220 10 N
=−=×−×
=− ×
CB AC
PPF


3
96
3
3
6
6
(220 10 )(0.440 0.120)
(200 10 )(1200 10 )
0.293333 10 m
220 10
1200 10
183.333 10 Pa
δ
σ
−
−
−
×−
=− =
××
=×
×
==
×
=− ×
CB CB
C
CB
CB
PL
EA
P
A

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PROBLEM 2.122
∗
(Continued)

Unloading
:

()
δ
′′ ′ −
′==−=

′ +=


AC AC CB CB AC CB
C
AC BC CB
AC
PLPLFPL
EA EA EA
LL FL
P
EA EA EA


3
3
33 3
3
6
6
3
6
6
(520 10 )(0.440 0.120)
378.182 10 N
0.440
378.182 10 520 10 141.818 10 N
378.182 10
315.152 10 Pa
1200 10
141.818 10
118.182 10 Pa
1200 10
(3
σ
σ
δ
−
−
×−
′== =×
+
′′=−= ×−×=− ×
′ ×
′== = ×
×
′ ×
′==− =− ×
×
′=
CB
AC
AC CB
CB AC
AC
AC
BC
BC
C
FL
P
LL
PPF
P
A
P
A
3
9678.182)(0.120)
0.189091 10 m
(200 10 )(1200 10 )
−
=×
××

(a)
333
,
0.293333 10 0.189091 10 0.1042 10 mδδδ
−−−
′=−= × − × = ×
Cp C C
0.1042 mm= 
(b)
666
,res
250 10 315.152 10 65.2 10 Paσσσ ′=− = ×− × =− ×
AC Y AC 65.2 MPa=− 

666
,res
183.333 10 118.182 10 65.2 10 Paσσσ ′=−=− ×+ ×=−×
CB CB CB 65.2 MPa=− 

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PROBLEM 2.123
∗

Solve Prob. 2.122, assuming that 180a= mm.
PROBLEM 2.122 Bar AB has a cross-sectional area of 1200 mm
2
and
is made of a steel that is assumed to be elastoplastic with
200E= GPa
and
250
Y
σ= MPa. Knowing that the force F increases from 0 to
520 kN and then decreases to zero, determine (a) the permanent
deflection of point C, (b) the residual stress in the bar.

SOLUTION

262
1200 mm 1200 10 mA
−
==×
Force to yield portion AC:
66
3
(1200 10 )(250 10 )
300 10 N
AC Y
PAσ
−
== × ×
=×
For equilibrium,
0.+−=
CB AC
FP P




33
3
300 10 520 10
220 10 N
CB AC
PPF=−=×−×
=− ×


3
96
3
3
6
6
(220 10 )(0.440 0.180)
(200 10 )(1200 10 )
0.238333 10 m
220 10
1200 10
183.333 10 Pa
δ
σ
−
−
−
×−
=− =
××
=×
×
==−
×
=− ×
CB CB
C
CB
CB
PL
EA
P
A

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PROBLEM 2.123
∗
(Continued)

Unloading
:

()
AC AC CB CB AC CB
C
AC BC CB
AC
PLPLFPL
EA EA EA
LL FL
P
EA EA EA
δ
′′ ′ −
′==−=

′=+=




3
3
33 3
3
3
96
3
6
(520 10 )(0.440 0.180)
307.273 10 N
0.440
307.273 10 520 10 212.727 10 N
(307.273 10 )(0.180)
0.230455 10 m
(200 10 )(1200 10 )
307.273 10
256.061
1200 10
CB
AC
AC CB
CB AC
C
AC
AC
FL
P
LL
PPF
P
A
δ
σ
−
−
−
×−
′== =×
+
′′=−= ×−×=− ×
×
′==×
××
′ ×
′== =
×
6
3
6
6
10 Pa
212.727 10
177.273 10 P
1200 10
CB
CB
P
A
σ
−
×
′−×
′== =− ×
×

(a)
333
,
0.238333 10 0.230455 10 0.00788 10 mδδδ
−−−
′=−= × − × = ×
Cp C C 0.00788 mm= 
(b)
666
,res
250 10 256.061 10 6.06 10 Pa
AC AC AC
σσσ ′=−=×− ×=−× 6.06 MPa=− 

666
,res
183.333 10 177.273 10 6.06 10 Pa
CB CB CB
σσσ ′=−=− ×+ ×=−× 6.06 MPa=− 


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PROBLEM 2.124
Rod BD is made of steel
6
(2910psi)E=× and is used to brace the axially
compressed member ABC. The maximum force that can be developed in member
BD is 0.02P. If the stress must not exceed 18 ksi and the maximum change in
length of BD must not exceed 0.001 times the length of ABC, determine the
smallest-diameter rod that can be used for member BD.

SOLUTION

3
0.02 (0.02)(130) 2.6 kips 2.6 10 lb== = =×
BD
FP
Considering stress:
3
18 ksi 18 10 psiσ==×

22.6
0.14444 in
18
σ
σ=∴===
BD BD
FF
A
A

Considering deformation:
(0.001)(144) 0.144 in.δ==

3
2
6
(2.6 10 )(54)
0.03362 in
(29 10 )(0.144)
δ
δ
×
=∴== =
×
BD BD BD BD
FL FL
A
AE E

Larger area governs.
2
0.14444 in=A

2 4 (4)(0.14444)
4π
ππ
=∴==
A
Ad d 0.429in.=d 

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PROBLEM 2.125
Two solid cylindrical rods are joined at B and loaded as shown. Rod AB is
made of steel
( 200 GPa)E= and rod BC of brass ( 105 GPa).E= Determine
(a) the total deformation of the composite rod ABC, (b) the deflection of
point B.

SOLUTION
Rod AB:
3
30 10 N
AB
FP=− =− ×

9
22 62
3
6
96
0.250 m
200 10 GPa
(30) 706.85 mm 706.85 10 m
4
(30 10 )(0.250)
53.052 10 m
(200 10 )(706.85 10 )
π
δ
−
−
−
=
=×
== =×
×
==− =−×
××
AB
AB
AB
AB AB
AB
AB AB
L
E
A
FL
EA

Rod BC
:
3
30 40 70 kN 70 10 N
BC
F=+= =×

9
232 32
3
6
93
0.300 m
105 10 Pa
(50) 1.9635 10 mm 1.9635 10 m
4
(70 10 )(0.300)
101.859 10 m
(105 10 )(1.9635 10 )
π
δ
−
−
−
=
=×
==× =×
×
==− =−×
××
BC
BC
BC
BC BC
BC
BC BC
L
E
A
FL
EA

(a) Total deformation
:
6
tot
154.9 10 mδδδ
−
=+=− ×
AB BC
0.1549 mm=− 
(b) Deflection of Point B
.
BBC
δδ= 0.1019 mm
B
δ=↓ 

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PROBLEM 2.126
Two solid cylindrical rods are joined at B and loaded as shown. Rod AB is
made of steel
6
(2910=×E psi), and rod BC of brass
6
(1510 psi).E=×
Determine (a) the total deformation of the composite rod ABC, (b) the
deflection of point B.

SOLUTION
Portion AB:
3
22 2
6
3
3
6
40 10 lb
40 in.
2 in.
(2) 3.1416 in
44
29 10 psi
(40 10 )(40)
17.5619 10 in.
(29 10 )(3.1416)
AB
AB
AB
AB
AB AB
AB
AB AB
P
L
d
Ad
E
PL
EA
ππ
δ
−
=×
=
=
== =
=×
×
== =×
×
Portion BC
:
3
22 2
6
3
3
6
20 10 lb
30 in.
3 in.
(3) 7.0686 in
44
15 10 psi
( 20 10 )(30)
5.6588 10 in.
(15 10 )(7.0686)
BC
BC
BC
BC
BC BC
BC
BC BC
P
L
d
Ad
E
PL
EA
ππ
δ
−
=− ×
=
=
== =
=×
−×
== =−×
×
(a)
66
17.5619 10 5.6588 10
AB BC
δδ δ
−−
=+= × − ×
3
11.90 10 in.δ
−
=× ↓ 
(b)
BBC
δδ=−
3
5.66 10 in.
B
δ
−
=× ↑ 

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PROBLEM 2.127
The uniform wire ABC, of unstretched length 2l, is attached to the
supports shown and a vertical load P is applied at the midpoint B.
Denoting by A the cross-sectional area of the wire and by E the
modulus of elasticity, show that, for
,lδ the deflection at the
midpoint B is

3δ=
P
l
AE

SOLUTION
Use approximation.
sin tan
l
δ
θθ
≈≈
Statics
: 0: 2 sin 0θΣ= −=
YAB
FPP

2sin 2θδ
=≈
AB
P Pl
P

Elongation
:
2
2
AB
AB
Pl
Pl
AE AE
δ
δ==
Deflection
:
From the right triangle,

22 2
22
()
AB
ll
lδδ
δ+=+
=
22
2
AB AB
llδδ++−
3
1
21 2
2
AB
AB AB
ll
l
Pl
AE
δ
δδ
δ
=+≈


≈


3
3
3
PlP
l
AE AE
δδ≈∴≈ 

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PROBLEM 2.128
The brass strip AB has been attached to a fixed support
at A and rests on a rough support at B. Knowing that
the coefficient of friction is 0.60 between the strip and
the support at B, determine the decrease in temperature
for which slipping will impend.

SOLUTION
Brass strip:

6
105 GPa
20 10 / C
E
α
−
=
=× °



0: 0Σ= −= =
y
F NW NW
0: 0
()0μμμ
μ
δα
ααΣ= − = = =
=− + Δ = Δ = =
x
FPNPWmg
PLPmg
LT T
EAEAEA

Data:
262
2
9
0.60
(20)(3) 60 mm 60 10 m
100 kg
9.81 m/s
105 10 Pa
A
m
g
E
μ
−
= == =×
=
=
=×

966
(0.60)(100)(9.81)
(105 10 )(60 10 )(20 10 )
T
−
Δ=
×× × 4.67 CTΔ= °

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PROBLEM 2.129
Members AB and CD are
1
8
1 -in.-diameter steel rods, and members BC and AD
are
7
8
-in.-diameter steel rods. When the turnbuckle is tightened, the diagonal
member AC is put in tension. Knowing that
6
29 10E=× psi and 4=hft,
determine the largest allowable tension in AC so that the deformations in
members AB and CD do not exceed 0.04 in.

SOLUTION

22 2
6
3
0.04 in.
4ft 48 in.
(1.125) 0.99402 in
44
(29 10 )(0.99402)(0.04)
24.022 10 lb
48
AB CD
CD
CD
CD CD
CD
CD
CD CD
CD
CD
hL
Ad
FL
EA
EA
F
L
δδ
ππ
δ
δ==
== =
== =
=
×
== =×

Use joint C as a free body.

45
0: 0
54
Σ= − = ∴ =
y CD AC AC CD
F FF FF

335
(24.022 10 ) 30.0 10 lb
4
=×=×
AC
F 30.0 kips=
AC
F 

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PROBLEM 2.130
The 1.5-m concrete post is reinforced with six steel bars, each with a 28-mm
diameter. Knowing that
200
s
E= GPa and 25
c
E= GPa, determine the
normal stresses in the steel and in the concrete when a 1550-kN axial centric
force P is applied to the post.

SOLUTION
Let portion of axial force carried by concrete.
portion carried by the six steel rods.
=
=
c
s
P
P

()
ccc
c
cc
sss
s
ss
cs cc ss
cc ss
PL E A
P
EA L
PL E A
P
EA L
PP P EA EA
L
P
LEAEA
δ
δ
δ
δ
δ
δ
ε
==
==
=+= +
−
==
+


22 32
32
223
32
32
3
6
93936
6 (28) 3.6945 10 mm
44
3.6945 10 m
(450) 3.6945 10
44
155.349 10 mm
153.349 10 m
1.5 m
1550 10
335.31 10
(25 10 )(155.349 10 ) (200 10 )(3.6945 10 )
ss
ccs
Ad
AdA
L
ππ
ππ
ε
−
−
−
−−
== =×
=×
=−= − ×
=×
=×
=
×
==×
××+××


966
(200 10 )(335.31 10 ) 67.1 10 Paσε
−
== × × =×
ss
E σ
s
= 67.1 MPa 

966
(25 10 )( 335.31 10 ) 8.38 10 Paσε
−
==× − × =×
cc
E σ
c
= 8.38 MPa 

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PROBLEM 2.131
The brass shell
6
(11.610/F)α
−
=× °
b
is fully bonded to the steel core
6
(6.510/F).α
−
=× °
s
Determine the largest allowable increase in
temperature if the stress in the steel core is not to exceed 8 ksi.

SOLUTION
Let axial force developed in the steel core.
s
P=
For equilibrium with zero total force, the compressive force in the brass shell is
.
s
P
Strains: ()
()
s
ss
ss
s
bb
bb
P
T
EA
P
T
EA
εα
εα=+Δ
=− + Δ
Matching:
s b
εε=
() ()
ss
sb
ss bb
PP
TT
EA EA
αα+Δ=− +Δ

11
()()
sbs
ss bb
P T
EA EAαα

+=−Δ

(1)

2
2
(1.5)(1.5) (1.0)(1.0) 1.25 in
(1.0)(1.0) 1.0 in
=−=
==
b
s
A
A


6
33
5.1 10 / F
(8 10 )(1.0) 8 10 lb
bs
sss
PA
αα
σ
−
−=× °
==× =×


91
6611 1 1
87.816 10 lb
(29 10 )(1.0) (15 10 )(1.25)
ss bb
EA EA
−−
+= + = ×
××
From (1),
93 6
(87.816 10 )(8 10 ) (5.1 10 )( )T
−−
××=×Δ

137.8 FΔ= °T 

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PROBLEM 2.132
A fabric used in air-inflated structures is subjected to a biaxial loading
that results in normal stresses
120
x
σ= MPa and 160
z
σ= MPa.
Knowing that the properties of the fabric can be approximated as
E = 87 GPa and v = 0.34, determine the change in length of (a) side AB,
(b) side BC, (c) diagonal AC.

SOLUTION

6
6
666
9
66 3
9
120 10 Pa,
0,
160 10 Pa
11
( ) [120 10 (0.34)(160 10 )] 754.02 10
87 10
11
( ) [ (0.34)(120 10 ) 160 10 ] 1.3701 10
87 10σ
σ
σ
εσσσ
εσσσ
−
−
=×
=
=×
=−−= ×− ×= ×
×
=− − + = − × + × = ×
×
x
y
z
xxyz
zxyz
vv
E
vv
E

(a)
6
( ) (100 mm)(754.02 10 )δε
−
== ×
AB x
AB 0.0754 mm= 
(b)
6
( ) (75 mm)(1.3701 10 )δε
−
== ×
BC z
BC 0.1028 mm= 
Label sides of right triangle ABC as a, b, and c.

222
cab=+
Obtain differentials by calculus.

222=+
=+
cdc ada bdb
ab
dc da db
cc

But 22
100 mm,
75 mm,
(100 75 ) 125 mm
0.0754 mm
0.1370 mm
δ
δ
=
=
=+=
==
==
AB
BC
a
b
c
da
db
(c)
100 75
(0.0754) (0.1028)
125 125
δ== +
AC
dc 0.1220 mm= 

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PROBLEM 2.133
An elastomeric bearing (0.9G= MPa) is used to support a bridge
girder as shown to provide flexibility during earthquakes. The beam
must not displace more than 10 mm when a 22-kN lateral load is
applied as shown. Knowing that the maximum allowable shearing
stress is 420 kPa, determine (a) the smallest allowable dimension b,
(b) the smallest required thickness a.

SOLUTION
Shearing force:
3
22 10 NP=×
Shearing stress:
3
420 10 Paτ=×

3
32
3
32
22 10
52.381 10 m
420 10
52.381 10 mm
(200 mm)( )
PP
A
A
Ab
τ
τ
−
=∴=
×
==×
×
=×
=


(a)
3
52.381 10
262 mm
200 200
A
b
×
== =
262 mm=b 

3
3
6
420 10
466.67 10
0.9 10Gτ
γ
−×
== = ×
×

(b) But
3
10 mm
21.4 mm
466.67 10
a
aδδ
γ
γ
−
=∴== =
× 21.4 mm=a 

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PROBLEM 2.134
Knowing that 10 kips,P= determine the maximum stress when (a) 0.50 in.,=r
(b)
0.625 in.r=

SOLUTION

3
10 10 lb 5.0 in. 2.50 in.
5.0
2.00
2.50
PDd
D
d
=× = =
==


2
min3
(2.50) 1.875 in
4
Adt

== =



(a)
0.50
0.50 in. 0.20
2.50
r
r
d
===

From Fig. 2.60b,
1.94K=

3
3
max
min
(1.94) (10 10 )
10.35 10 psi
1.875
KP
A
σ
×
== =×


10.35 ksi 
(b)
3
3
max
min
0.625
0.625 in. 0.25 1.82
2.50
(1.82) (10 10 )
9.71 10 psi
1.875
r
rK
d
KP
A
σ
====
×
== =×

9.71 ksi 

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PROBLEM 2.135
The uniform rod BC has a cross-sectional area A and is made of a
mild steel that can be assumed to be elastoplastic with a modulus of
elasticity E and a yield strength
.
y
σ Using the block-and-spring
system shown, it is desired to simulate the deflection of end C of the
rod as the axial force P is gradually applied and removed, that is, the
deflection of points C and
C′ should be the same for all values of P.
Denoting by
μ the coefficient of friction between the block and the
horizontal surface, derive an expression for (a) the required mass m
of the block, (b) the required constant k of the spring.

SOLUTION
Force-deflection diagram for Point C or rod BC.
For
max
σ
δδ
σ<=
==
==
YY
CC
YY
PP A
PL EA
P
EA L
PPA
Force-deflection diagram for Point C′ of block-and-spring system
.
0: 0Σ= − = =
y
F Nmg Nmg
0: 0Σ= − = =
x ff
F PF PF
If block does not move, i.e.,
or ,μμ μ<= <
f
F Nmg Pmg
then or
cc
P
Pk
Kδδ′′==
If P =
μ mg, then slip at occurs.μ==
m
PF mg
If the force P is the removed, the spring returns to its initial length.



(a) Equating P
Y and F max,
σ
σμ
μ
==
Y
Y
A
Amgm g

(b) Equating slopes,
EA
k
L
= 