PROBLEM 7.1
For the given state of stress, determine the normal and shearing stresses exerted
on the oblique face of the shaded triangular element shown. Use a method of
analysis based on the equilibrium of that element, as was done in the derivations
of Sec. 7.1A.
SOLUTION
F
0:
A
8 A cos 20 cos 20
8cos 2 20
3cos 20 sin 20
3 A cos 20 sin 20
3sin 20 cos 20
3 A sin 20 cos 20
4sin 2 20
4 A sin 20 sin 20
0
0
9.46 ksi
F
0:
A
8 A cos 20 sin 20
8cos 20 sin 20
3(cos2 20
3 A cos 20 cos 20
sin 2 20 )
3 A sin 20 sin 20
4 A sin 20 cos 20
0
4sin 20 cos 20
1.013 ksi
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
1027
PROBLEM 7.4
For the given state of stress, determine the normal and shearing stresses exerted
on the oblique face of the shaded triangular element shown. Use a method of
analysis based on the equilibrium of that element, as was done in the derivations
of Sec. 7.1A.
SOLUTION
Stresses
F
0:
A
80 A cos 55 cos55
80 cos 2 55
F
0:
A
Areas
Forces
40 A sin 55 sin 55
40sin 2 55
80 A cos 55 sin 55
0
0.521 MPa
40 A sin 55 cos 55
56.4 MPa
120 cos 55 sin 55
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
1030
PROBLEM 7.5
For the given state of stress, determine (a) the principal planes, (b) the principal
stresses.
SOLUTION
60 MPa
x
(a)
tan 2
2
xy
p
x
2
p
y
(2)(35)
60 40
y
40 MPa
xy
35 MPa
3.50
74.05
37.0 , 53.0
p
2
(b)
x
max, min
y
x
2
60 40
2
y
2
xy
2
60 40
2
2
(35) 2
50 36.4 MPa
max
min
13.60 MPa
86.4 MPa
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
1031
PROBLEM 7.8
For the given state of stress, determine (a) the principal planes, (b) the principal
stresses.
SOLUTION
x
(a)
tan 2
18 ksi
2
p
xy
x
2
p
12 ksi
y
y
(2)(8)
18 12
xy
8 ksi
0.5333
28.07
14.0 ,104.0 ◄
p
2
(b)
max,min
x
y
x
2
18 12
2
y
2
18
12
2
2
xy
2
(8)2
3 17 ksi
max
min
20.0 ksi ◄
14.00 ksi ◄
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
1034
PROBLEM 7.22
Two members of uniform cross section 50 80 mm are glued together along plane
a-a that forms an angle of 25 with the horizontal. Knowing that the allowable
600 kPa, determine the largest
stresses for the glued joint are
800 kPa and
centric load P that can be applied.
SOLUTION
For plane a-a,
65 .
x
0,
x
P
0,
cos 2
x
y
y
P
A
sin 2
2
xy
sin
cos
0
(50 10 3 )(80 10 3 )(800 103 )
sin 2 65
A
sin 2 65
(
P
xy
y )sin
A
sin 65 cos 65
P 2
sin 65
A
0
3.90 103 N
P
sin 65 cos 65 0
A
(50 10 3 )(80 10 3 )(600 103 )
6.27 103 N
sin 65 cos 65
cos
xy (cos
Allowable value of P is the smaller one.
2
sin 2 )
P
3.90 kN
PROPRIETARY MATERIAL. Copyright © 2015 McGraw-Hill Education. This is proprietary material solely for authorized instructor use.
Not authorized for sale or distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
1048