ch03

3-1
CHAPTER 3
PROBLEM (3.1) A rigid bar BDE is supported by two links AB and CD as shown in Fig. P3.1.
After load P is applied, point E moves 2.4mm downward and the axial strain in bar AB equals -500
.
What is the axial strain in bar CD?
SOLUTION
The change in length of bar AB is
6
( 500 10 )(800) 0.4
AB AB AB
L mm
From triangles B’BF and EE’F:
0.4 2.4 , 214.3
1500 x mm xx
From triangles DD’F and EE’F:
285.7 1285.7 , 0.533
2.4 D
D
mm
Thus
0.533 533
1000
D
CD CD
L
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PROBLEM (3.2) A spherical balloon changes its diameter from 200 to 201 mm when
pressurized. Determine the average circumferential strain.
SOLUTION
0
0
15000
200
f
dd
d
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PROBLEM (3.3) A hollow cylinder is subjected to an internal pressure that increases its 200
mm inner diameter by 0.5 mm and its 400 mm outer diameter by 0.3 mm. Calculate:
(a) The maximum normal strain in the circumferential direction.
(b) The average normal strain in the radial direction.
SOLUTION
(a)
2 ( ) 2
2
cr r r r
rr
max 0.25
( ) ( ) 2500
100
c c i
0.15
( ) 750
200
co
1000 mm
F
2.4 mm
0.4 mm
D’
B’
D
E
B
x
E’
500 mm
D

3-2
(b)
0.25 0.15 1000
200 100
oi
oi
rr
rr
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PROBLEM (3.4) A prismatic bar of length L is subjected to an axial load P, as shown in Fig.
P3.4. Calculate the maximum strain
x
, if the displacement along the member varies as follows:
(a) u = (
2
x
/L)
3
10
.
(b) u = L(
3
10
) sin (
x/2L).
SOLUTION
(a)
2
1000 500
x
x du x
uL dx L
At x=L:
max
( ) 2000
x
(b)
sin cos
1000 2 2000 2
x
L x du x
uL dx L
At x=0:
max
( ) 1570
2000
x
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PROBLEM (3.5) A horizontal rod AB is supported and loaded by a force P as seen in Fig.
P3.5. Determine the permissible normal strains in the wires CE and DF, if the allowable vertical
displacement of end B is 1/8 in.
Assumption: Rod AB is rigid.
Geometry:
0.125
40 60 80
CE DF
Solving,
0.0625 . 0.09375 .
CE DF
in in
Normal Strains:
0.0625 1786
35
CE
CE CE
L
0.09375 1875
50
DF
DF DF
L
B
C
D
F
D
E
CE
A
B
Figure (a)
F
20 in.
35 in.
50 in.
20 in.
40 in.

3-3
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PROBLEM (3.6) The structure shown in Fig. P3.5 consists of a horizontal rod AB supported by
two vertical wires (CE and DE) and by a pin at A. What is the the allowable vertical displacement
of the end B, if the permissible normal strain in each wire is
all
= 1500
?
Assumption: Rod AB is rigid.
SOLUTION
Geometry [see: Figure (a) of Solution 3.5]:
40 60 80
CE DF B
Normal Strains:
0.0015(35) 0.0525 .
CE all CE
L in
82(0.0525) 0.105 .
4
B CE in
Also
0.0015(50) 0.075 .
DF all DE
L in
84
(0.075) 0.1 .
63
B DF all
in
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PROBLEM (*3.7) The assembly of the strut BC and rod AB is used to support a vertical load
P as depicted in Fig. P3.7. Determine:
(a) The normal stresses
AB
and
BC
in the rod and strut.
(b) The normal strain
AB
, if the rod elongates 0.05 in.
(c) The normal strain
BC
, if the strut shortens 0.025 in.
Given: P = 3.5 kips. The cross-sectional areas of the members:
BC
A
= 0.25
2
.in
and
BC
A
=0.4
2
.in
*SOLUTION
We have
22
60 45 75 .
AB
L in  
22
60 25 65 .
BC
L in  
Equilibrium:
4 12
0: 0
5 13
x AB BC
F F F   
35
0: 3.5 0
5 13
y AB BC
F F F   
B
C
D
E
A
F
50 in.
35 in.
20 in.
20 in.
40 in.
A
C
BC
F
AB
F
P
B
45
25
12
60
5
3
4
13
5

3-4
Solving,
3.25 ( ) 3.75 ( )
BC AB
F kips C F kips T
(a)
3.75 3.25
15 8.125
0.25 0.4
AB BC
ksi ksi
(b)
0.05 667
75
AB
(c)
0.025 385
65
BC
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PROBLEM (3.8) As a result of loading, the thin 40 mm by 20 mm rectangular plate of Fig. P3.8
deforms into a parallelogram in which sides AB and CD elongate 0.005 mm and rotate 1200
rad
clockwise, while sides AD and BC shorten 0.002 mm and rotate 400
rad counterclockwise.
Calculate the strain components in the xy plane.
SOLUTION
0.002 50
40
xu
x
0.005 250
20
yv
y
400 1200 1600
xy vu
xy
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PROBLEM (3.9) Solve Prob. 3.8, assuming that sides AD and BC elongate 0.001 mm and
rotate 1600
rad clockwise and the other sides have the same extension and rotation.
SOLUTION
0.001 25
40
xu
x
0.005 250
20
yv
y
1600 1200 400
xy vu
xy
40 mm
20 mm
A
D
y
C
B
x

3-5
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PROBLEM (3.10) A thin 8 in. by 6 in. rectangular plate (see Fig. P3.10) is acted upon by a
biaxial tensile loading resulting in the uniform strains
x
= 600
and
y
= 400
. Calculate the
change in length of diagonal AC.
SOLUTION
6
400(10 )(6)
AB
L
3
2.4 10 .in
6
600(10 )(8)
AD
L
3
4.8 10 .in
2 2 2
AC AB AD
L L L
2 2 2
AC AC AB AB AD AD
L L L L L L    
or
AB AD
AC AB AD
AC AC
LL
L L L
LL
(1)
Substituting the numerical values:
3
68
[ (2.4) (4.8)]10
10 10
AC
L
3
5.28 10 .in
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PROBLEM (3.11) Redo Prob.3.10, with the plate in biaxial compression for which
x
= -200
and
y
= - 100
.
SOLUTION
63
100(10 )6 0.6 10 .
AB
L in
63
200(10 )8 1.6 10 .
AD
L in
Using Eq. (1) of Solution of Prob. 3.8:
3
68
[ ( 0.6) ( 1.6)]10
10 10
AC
L
3
1.64 10 .in
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PROBLEM (3.12) Determine the normal strain in the members AB and CB of the pin-
connected plane structure shown in Fig. 3.12 if point B moves leftward 3 mm, after load P is applied.
Assumption: Axial deformation is uniform throughout the length of each member.
SOLUTION
3
1500
AB
AB AB
L
L
2000
0.003cos
2.5
BC
BC BC
L
L
8 in.
6 in.
A
D
y
C
B
x
10 in.
3 mm
2.5 m
A
C
B
1..5 m
2 m
BC
L