_______________________________________________________________________
PROBLEMS (8.116 through 120) The state of strain at a point in a thin steel plate is given in the
following table. Calculate:
(a) The in-plane principal strains and the maximum in-plane shear strain.
(b) The absolute or true maximum shearing strain (
ν
= 0.3).
Sketch the results found in part (a) on properly oriented deformed elements.
---------------------------------------------------------------------------------------
Problem
x
ε
y
ε
x
y
γ
---------------------------------------------------------------------------------------
8.116 400
μ
100
μ
200
μ
8.117 -900
μ
400
μ
-200
μ
8.118 -720
μ
0
μ
300
μ
8.119 200
μ
600
μ
600
μ
8.120 500
μ
-100
μ
150
μ
----------------------------------------------------------------------------------------
SOLUTION (116)
(a) 400 100
' 250
2
ε
μ
+
==
1
110
' tan 16.8
215 o
p
θ
−
==
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instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is unlawful.
" 16.8 45 61.8o
s
θ
=+=
22
100 150 180.3R
μ
=+=
max 2 361R
γ
μ
=
=
μ
ε
4303.180250
1
=
+
=
μ
ε
703.180250
2
=
−
=
Continued on next slide
430
μ
61.8o
x’
x
y’
70
μ
250
μ
250
μ
16.8o
x’
x
y’
360
μ
2
ε
R
ε
(
μ
)
1
ε
γ
(μ)
400, -100)
2
(
O
C
2'
p
θ
x
y
'
ε
(b) Using Eq. (8.2)
30.3(400 100)
0.7
z
εε
==− +
μ
214
−
=
max
( ) 430 214
a
γ
=+
μ
644=
SOLUTION (117)
(a) 900 400
' 250
2
ε
μ
−
+
==−
1
110
"tan 4.4
265o
p
θ
−
==
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instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is unlawful.
"4.44549
o
s
θ
=+=
.4
2
γ
(μ
22
100 650 657.6R
μ
=+=
max 2 1315R
γ
μ
=
=
μ
ε
4082506.657
1
=
−
=
μ
ε
9082506.657
3
−
=
−
−
=
(b) Using Eq. (8.2):
20.3( 900 400) 214
0.7
z
ε
εμ
==− − + =
Thus,
max
( ) 1315
a
γ
μ
=
Continued on next slide
2
ε
R
)
900, -100)
ε
(
μ
)
1
ε
(-
O
C
2"
p
θ
x
'
ε
y
y’
49.4o
408
μ
250
μ
4.4o
x’
x
1316
μ
908
μ
250
x’
μ
x
SOLUTION (8.118)
(a) 720
' 360
2
ε
μ
−
==−
1
115
" tan 11.3
236 o
p
θ
−
==
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instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is unlawful.
' 45 11.3 33.7o
s
θ
=− =
22
150 360 390R
μ
=+=
μ
γ
7802
max
=
=
r
μ
ε
30360390
2
=
−
=
μ
ε
750360390
3
−
=
−
−
=
(b) 10.3( 720) 308
0.7
z
ε
εμ
==− − =
Thus,
max
( ) 308 750 1058
a
γ
μ
=+=
Continued on next slide
3
ε
R
ε
(
μ
)
2
ε
γ
2(μ)
720, -150)
(-
O
C
2"
p
θ
x
'
ε
y
y
33.7o
30
μ
360
μ
11.3o
x’
x
780
μ
360
μ
x’
x
y’
750
μ
SOLUTION (8.119)
(a) 200 600
' 400
2
ε
μ
+
==
2
γ
(μ)
22
200 300 361R
μ
=+=
1
13
" tan 28.2
22 o
p
θ
−
==
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or
instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is unlawful.
" 45 28.2 73.2o
s
θ
=+ =
max 2 722R
γ
μ
=
=
1400 361 761
ε
μ
=
+=
μ
ε
39361400
3
=
−
=
(b) 30.3 (500 100) 343
0.7
z
ε
εμ
==− − =−
Thus,
max
( ) 761 343 1104
a
γ
μ
=+=
Continued on next slide
2
ε
R
ε
(
μ
)
1
ε
(200, -300)
O
C
2"
p
θ
x
y
'
ε
x’
73.2o
39
μ
400
μ
28.2o
x’
x
722
μ
400
μ
y’
x
y’
761
μ
SOLUTION (8.120)
(a) 500 100
' 200
2
ε
μ
−
==
1
175
'tan 7
2 300 o
p
θ
−
=
=
Excerpts from this work may be reproduced by instructors for distribution on a not-for-profit basis for testing or
instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
reproduction or translation of this work beyond that permitted by Sections 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner is unlawful.
"4575
o
s
θ
=+=
2
22
75 300 309R
μ
=+=
max 2 618R
γ
μ
=
=
1200 309 509
ε
μ
=
+=
2390 200 109
ε
μ
=
−+ =−
(b) 30.3 (500 100) 172
0.7
z
ε
εμ
==− − =−
Thus,
max
( ) 509 172 681
a
γ
μ
=+=
2
γ
(μ
2
ε
R
ε
(
μ
)
1
ε
)
500, -75)
(
O
C
2'
p
θ
x
'
ε
y
509
μ
x’
y’
52o
109
μ
220
μ
7o
x’
x
200
μ
618
μ
x
1
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5
100%