978-0073398167 Chapter 11 Solution Manual Part 9

subject Type Homework Help
subject Pages 17
subject Words 1140
subject Authors David Mazurek, E. Johnston, Ferdinand Beer, John DeWolf

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page-pf1
PROBLEM 11.77
The couple M is applied to a beam of the cross section shown in a plane
forming an angle
β
with the vertical. Determine the stress at (a) point A,
(b) point B, (c) point D.
SOLUTION
page-pf2
PROBLEM 11.78
The couple M is applied to a beam of the cross section shown in a
plane forming an angle
β
with the vertical. Determine the stress at
(a) point A, (b) point B, (c) point D.
SOLUTION
2
42 4
43494
4
4 3 4 94
48
8 2 3 89
(0.109757)(20) 17.5611 10 mm 17.5611 10 m
(20) 62.832 10 mm 62.832 10 m
88
4 (4)(20) 8.4883 mm
33
20 8.4883 11.5117 mm
z
y
AD
B
A
r
Ir r r
Ir
r
yy
y
zz
ππ π
ππ
ππ
ππ
−−
−−
  
=−=−
  
  
= =×=×
===×=×
==−=− =−
=−=
= − 20 mm 0
100cos30 86.603 N m
100sin30 50 N m
DB
z
y
z
M
M
= =
= °= ⋅
= °= ⋅
(a)
33
99
(86.603)( 8.4883 10 ) (50)(20 10 )
17.5611 10 62.832 10
yA
zA
Azy
Mz
My
II
σ
−−
−−
−× ×
=−+ = +
××
6
57.8 10 Pa= ×
57.8 MPa
A
σ
=
(b)
3
99
(86.603)(11.5117 10 ) (50)(0)
17.5611 10 62.832 10
yB
zB
Bzy
Mz
My
II
σ
−−
×
=−+ = +
××
6
56.8 10 Pa=−×
56.8 MPa
B
σ
= −
(c)
33
99
(86.603)( 8.4883 10 ) (50)( 20 10 )
17.5611 10 62.832 10
yD
zD
Dzy
Mz
My
II
σ
−−
− × −×
=−+ = +
××
6
25.9 10 Pa= ×
25.9 MPa
D
=
σ
page-pf3
consent of McGraw-Hill Education.
PROBLEM 11.79
The couple M acts in a vertical plane and is applied to a beam
oriented as shown. Determine (a) the angle that the neutral axis forms
with the horizontal, (b) the maximum tensile stress in the beam.
SOLUTION
Bending moments:
35sin15 9.059 kip in
35cos15 33.807 kip in
y
z
M
M
=− °=−
=− °=
Moments of inertia:
3 34
3 34
11
(2.4)(4) (2 0.4)(2) 12.267 in
12 12
11
(2)(2.4) (2)(1.6) 2.987 in
12 12
y
z
I
I
= −× =
=+=
(a) Neutral axis:
4
4
2.987 in
tan tan tan( 15 ) 0.06525
12.267 in
z
y
I
I
φθ
= = − °=
3.73
φ
= °
15 15 3.73 11.27
αφ
= °− = °− °= °
11.3
α
= °
(b) Maximum tensile stress at D:
44
1.2 in. 2 in.
( 9.059 kip in)( 2 in.) (33.807 kip in)( 1.2 in.)
12.267 in 2.987 in
1.477 ksi 13.582 ksi 15.059 ksi
DD
yD zD
Dyz
yz
Mz My
II
s
′′
′′
=−=−
− ⋅− ⋅−
=−= −
=+=
15.06 ksi
D
s
=
page-pf4
consent of McGraw-Hill Education.
page-pf5
consent of McGraw-Hill Education.
PROBLEM 11.80
The couple M acts in a vertical plane and is applied to a beam oriented as shown.
Determine (a) the angle that the neutral axis forms with the horizontal, (b) the
maximum tensile stress in the beam.
SOLUTION
For
W310 38.7×
rolled steel shape,
6 4 64
6 4 64
33
33
85.1 10 mm 85.1 10 m
7.27 10 mm 7.27 10 m
1(310) 155 mm
2
1(165) 82.5 mm
2
(16 10 ) cos15 15.455 10 N m
(16 10 ) sin 15 4.1411 10 N m
=×=×
=×=×

= =−=−= =



= =−== =


= × °= ×
= × °= ×
z
y
AB D E
AE B D
z
y
I
I
yy y y
zz z z
M
M
(a) Angle of neutral axis.
6
6
85.1 10
tan tan tan15 3.1365
7.27 10
z
y
I
I
ϕθ
×
= = °=
×
72.3
72.3 15
ϕ
α
= °
= °− °
57.3
α
= °
3 3 33
66
(15.455 10 )( 155 10 ) (4.1411 10 )(82.5 10 )
85.1 10 7.27 10
σ
−−
−−
=−+
× −× × ×
=−+
××
yE
zE
Ezy
Mz
My
II
6
75.1 10 Pa= ×
75.1 MPa
E
σ
=
page-pf6
PROBLEM 11.81
The couple M acts in a vertical plane and is applied to a beam oriented
as shown. Determine (a) the angle that the neutral axis forms with the
horizontal, (b) the maximum tensile stress in the beam.
SOLUTION
6 4 64
page-pf7
PROBLEM 11.82
The couple M acts in a vertical plane and is applied to a beam oriented
as shown. Determine (a) the angle that the neutral axis forms with the
horizontal, (b) the maximum tensile stress in the beam.
SOLUTION
3 4 94
3 4 94
176.9 10 mm 176.9 10 m
281 10 mm 281 10 m
18.57 mm, 25 mm
400cos30 346.41 N m
400sin30 200 N m
z
y
EE
z
y
I
I
yz
M
M
=×=×
=×=×
=−=
= °= ⋅
= °= ⋅
(a)
9
9
176.9 10
tan tan tan30 0.36346
281 10
19.97
z
y
I
I
ϕθ
ϕ
×
= = ⋅ °=
×
= °
30 19.97
α
= °− °
10.03
α
= °
(b) Maximum tensile stress occurs at point E.
33
99
6 66
(346.41)( 18.57 10 ) (200)(25 10 )
176.9 10 281 10
36.36 10 17.79 10 54.2 10 Pa
yE
zE
Ezy
Mz
My
II
−−
−−
′′
−× ×
=−+ =− +
××
= ×+ ×= ×
σ
54.2 MPa
E
σ
=
page-pf8
consent of McGraw-Hill Education.
PROBLEM 11.83
The couple M acts in a vertical plane and is applied to a beam oriented
as shown. Determine (a) the angle that the neutral axis forms with the
horizontal, (b) the maximum tensile stress in the beam.
SOLUTION
44
21.4 in 6.74 in
0.859 in. 4 0.859 in. 3.141 in.
4 in. 4 in. 0.25 in.
15 sin 45 10.6066 kip in.
15 cos45 10.6066 kip in.
′′
′′
= =
′′ ′
= = =−+ =−
=−= =−
=− °=−
= °=
zy
AB D
A BD
y
z
II
zz z
y yy
M
M
(a) Angle of neutral axis:
21.4
tan tan tan( 45 ) 3.1751
6.74
z
y
I
I
ϕθ
= = − °=
72.5
72.5 45
ϕ
α
=−°
= °− °
27.5
α
= °
(10.6066)( 0.25) ( 10.6066)( 3.141)
21.4 6.74
yD
zD
Dzy
Mz
My
II
σ
−− −
=−+=− +
0.12391 4.9429= +
5.07 ksi
D
σ
=
page-pf9
consent of McGraw-Hill Education.
PROBLEM 11.84
The couple M acts in a vertical plane and is applied to a beam
oriented as shown. Determine (a) the angle that the neutral
axis forms with the horizontal, (b) the maximum tensile stress
in the beam.
SOLUTION
3 4 94
3 4 94
53.6 10 mm 53.6 10 m
14.77 10 mm 14.77 10 m
120 sin 70 112.763 N m
120 cos 70 41.042 N m
z
y
z
y
I
I
M
M
=×=×
=×=×
= °=
= °= ⋅
(a) Angle of neutral axis:
20= °
θ
9
9
53.6 10
tan tan tan 20 1.32084
14.77 10
52.871
52.871 20
z
y
I
I
×
= = °=
×
= °
= °− °
ϕθ
ϕ
a
32.9
a
= °
99
16 mm 0.016 m
10 mm 0.010 m
(112.763)( 0.016) (41.042)(0.010)
53.6 10 14.77 10
E
E
yE
zE
Ezy
y
z
Mz
My
II
σ
′′
−−
=−=
= =
=−+
=−+
××
61.4 MPa
E
σ
=
page-pfa
consent of McGraw-Hill Education.
PROBLEM 11.85
For the loading shown, determine (a) the stress at points A
and B, (b) the point where the neutral axis intersects line
ABD.
SOLUTION
Add y and z axes as shown.
2
34
34
(4)(1.8) 7.2 in
1(4)(1.8) 1.944 in
12
1(1.8)(4) 9.6 in
12
z
y
A
I
I
= =
= =
= =
Resultant force and bending couples:
150 500 250 900 lb
(150)(0.9) (500)(0.9) (250)(0.9) 540 lb in
(150)(2) (500)(2) (250)(2) 800 lb in
z
y
P
M
M
=++=
=−+ + =
=+−=⋅
900 (540)(0.9) (800)(2)
(a) 41.7 psi
7.2 1.944 9.6
yA
zA
A
zy
MZ
My
P
AI I
s
=− + −− + =
900 (540)(0.9) (800)( 2)
292 psi
7.2 1.944 9.6
yB
zB
B
zy
MZ
My
P
AI I
s
=− + −− + =
0 0.9 in., ?
900 (540)(0.9) (800)
O= 7.2 1.944 9.6
125 83.333 0 1.5 in.
y
z
zy
zz
My
My
Pz
AI I
zz
s
= = =
−+=− +
−+ = =
Intersects AB at 0.500 in. from A
PROBLEM 11.78
The couple M is applied to a beam of the cross section shown in a
plane forming an angle
β
with the vertical. Determine the stress at
(a) point A, (b) point B, (c) point D.
SOLUTION
2
42 4
43494
4
4 3 4 94
48
8 2 3 89
(0.109757)(20) 17.5611 10 mm 17.5611 10 m
(20) 62.832 10 mm 62.832 10 m
88
4 (4)(20) 8.4883 mm
33
20 8.4883 11.5117 mm
z
y
AD
B
A
r
Ir r r
Ir
r
yy
y
zz
ππ π
ππ
ππ
ππ
−−
−−
  
=−=−
  
  
= =×=×
===×=×
==−=− =−
=−=
= − 20 mm 0
100cos30 86.603 N m
100sin30 50 N m
DB
z
y
z
M
M
= =
= °= ⋅
= °= ⋅
(a)
33
99
(86.603)( 8.4883 10 ) (50)(20 10 )
17.5611 10 62.832 10
yA
zA
Azy
Mz
My
II
σ
−−
−−
−× ×
=−+ = +
××
6
57.8 10 Pa= ×
57.8 MPa
A
σ
=
(b)
3
99
(86.603)(11.5117 10 ) (50)(0)
17.5611 10 62.832 10
yB
zB
Bzy
Mz
My
II
σ
−−
×
=−+ = +
××
6
56.8 10 Pa=−×
56.8 MPa
B
σ
= −
(c)
33
99
(86.603)( 8.4883 10 ) (50)( 20 10 )
17.5611 10 62.832 10
yD
zD
Dzy
Mz
My
II
σ
−−
− × −×
=−+ = +
××
6
25.9 10 Pa= ×
25.9 MPa
D
=
σ
consent of McGraw-Hill Education.
PROBLEM 11.79
The couple M acts in a vertical plane and is applied to a beam
oriented as shown. Determine (a) the angle that the neutral axis forms
with the horizontal, (b) the maximum tensile stress in the beam.
SOLUTION
Bending moments:
35sin15 9.059 kip in
35cos15 33.807 kip in
y
z
M
M
=− °=−
=− °=
Moments of inertia:
3 34
3 34
11
(2.4)(4) (2 0.4)(2) 12.267 in
12 12
11
(2)(2.4) (2)(1.6) 2.987 in
12 12
y
z
I
I
= −× =
=+=
(a) Neutral axis:
4
4
2.987 in
tan tan tan( 15 ) 0.06525
12.267 in
z
y
I
I
φθ
= = − °=
3.73
φ
= °
15 15 3.73 11.27
αφ
= °− = °− °= °
11.3
α
= °
(b) Maximum tensile stress at D:
44
1.2 in. 2 in.
( 9.059 kip in)( 2 in.) (33.807 kip in)( 1.2 in.)
12.267 in 2.987 in
1.477 ksi 13.582 ksi 15.059 ksi
DD
yD zD
Dyz
yz
Mz My
II
s
′′
′′
=−=−
− ⋅− ⋅−
=−= −
=+=
15.06 ksi
D
s
=
consent of McGraw-Hill Education.
consent of McGraw-Hill Education.
PROBLEM 11.80
The couple M acts in a vertical plane and is applied to a beam oriented as shown.
Determine (a) the angle that the neutral axis forms with the horizontal, (b) the
maximum tensile stress in the beam.
SOLUTION
For
W310 38.7×
rolled steel shape,
6 4 64
6 4 64
33
33
85.1 10 mm 85.1 10 m
7.27 10 mm 7.27 10 m
1(310) 155 mm
2
1(165) 82.5 mm
2
(16 10 ) cos15 15.455 10 N m
(16 10 ) sin 15 4.1411 10 N m
=×=×
=×=×

= =−=−= =



= =−== =


= × °= ×
= × °= ×
z
y
AB D E
AE B D
z
y
I
I
yy y y
zz z z
M
M
(a) Angle of neutral axis.
6
6
85.1 10
tan tan tan15 3.1365
7.27 10
z
y
I
I
ϕθ
×
= = °=
×
72.3
72.3 15
ϕ
α
= °
= °− °
57.3
α
= °
3 3 33
66
(15.455 10 )( 155 10 ) (4.1411 10 )(82.5 10 )
85.1 10 7.27 10
σ
−−
−−
=−+
× −× × ×
=−+
××
yE
zE
Ezy
Mz
My
II
6
75.1 10 Pa= ×
75.1 MPa
E
σ
=
PROBLEM 11.81
The couple M acts in a vertical plane and is applied to a beam oriented
as shown. Determine (a) the angle that the neutral axis forms with the
horizontal, (b) the maximum tensile stress in the beam.
SOLUTION
6 4 64
PROBLEM 11.82
The couple M acts in a vertical plane and is applied to a beam oriented
as shown. Determine (a) the angle that the neutral axis forms with the
horizontal, (b) the maximum tensile stress in the beam.
SOLUTION
3 4 94
3 4 94
176.9 10 mm 176.9 10 m
281 10 mm 281 10 m
18.57 mm, 25 mm
400cos30 346.41 N m
400sin30 200 N m
z
y
EE
z
y
I
I
yz
M
M
=×=×
=×=×
=−=
= °= ⋅
= °= ⋅
(a)
9
9
176.9 10
tan tan tan30 0.36346
281 10
19.97
z
y
I
I
ϕθ
ϕ
×
= = ⋅ °=
×
= °
30 19.97
α
= °− °
10.03
α
= °
(b) Maximum tensile stress occurs at point E.
33
99
6 66
(346.41)( 18.57 10 ) (200)(25 10 )
176.9 10 281 10
36.36 10 17.79 10 54.2 10 Pa
yE
zE
Ezy
Mz
My
II
−−
−−
′′
−× ×
=−+ =− +
××
= ×+ ×= ×
σ
54.2 MPa
E
σ
=
consent of McGraw-Hill Education.
PROBLEM 11.83
The couple M acts in a vertical plane and is applied to a beam oriented
as shown. Determine (a) the angle that the neutral axis forms with the
horizontal, (b) the maximum tensile stress in the beam.
SOLUTION
44
21.4 in 6.74 in
0.859 in. 4 0.859 in. 3.141 in.
4 in. 4 in. 0.25 in.
15 sin 45 10.6066 kip in.
15 cos45 10.6066 kip in.
′′
′′
= =
′′ ′
= = =−+ =−
=−= =−
=− °=−
= °=
zy
AB D
A BD
y
z
II
zz z
y yy
M
M
(a) Angle of neutral axis:
21.4
tan tan tan( 45 ) 3.1751
6.74
z
y
I
I
ϕθ
= = − °=
72.5
72.5 45
ϕ
α
=−°
= °− °
27.5
α
= °
(10.6066)( 0.25) ( 10.6066)( 3.141)
21.4 6.74
yD
zD
Dzy
Mz
My
II
σ
−− −
=−+=− +
0.12391 4.9429= +
5.07 ksi
D
σ
=
consent of McGraw-Hill Education.
PROBLEM 11.84
The couple M acts in a vertical plane and is applied to a beam
oriented as shown. Determine (a) the angle that the neutral
axis forms with the horizontal, (b) the maximum tensile stress
in the beam.
SOLUTION
3 4 94
3 4 94
53.6 10 mm 53.6 10 m
14.77 10 mm 14.77 10 m
120 sin 70 112.763 N m
120 cos 70 41.042 N m
z
y
z
y
I
I
M
M
=×=×
=×=×
= °=
= °= ⋅
(a) Angle of neutral axis:
20= °
θ
9
9
53.6 10
tan tan tan 20 1.32084
14.77 10
52.871
52.871 20
z
y
I
I
×
= = °=
×
= °
= °− °
ϕθ
ϕ
a
32.9
a
= °
99
16 mm 0.016 m
10 mm 0.010 m
(112.763)( 0.016) (41.042)(0.010)
53.6 10 14.77 10
E
E
yE
zE
Ezy
y
z
Mz
My
II
σ
′′
−−
=−=
= =
=−+
=−+
××
61.4 MPa
E
σ
=
consent of McGraw-Hill Education.
PROBLEM 11.85
For the loading shown, determine (a) the stress at points A
and B, (b) the point where the neutral axis intersects line
ABD.
SOLUTION
Add y and z axes as shown.
2
34
34
(4)(1.8) 7.2 in
1(4)(1.8) 1.944 in
12
1(1.8)(4) 9.6 in
12
z
y
A
I
I
= =
= =
= =
Resultant force and bending couples:
150 500 250 900 lb
(150)(0.9) (500)(0.9) (250)(0.9) 540 lb in
(150)(2) (500)(2) (250)(2) 800 lb in
z
y
P
M
M
=++=
=−+ + =
=+−=⋅
900 (540)(0.9) (800)(2)
(a) 41.7 psi
7.2 1.944 9.6
yA
zA
A
zy
MZ
My
P
AI I
s
=− + −− + =
900 (540)(0.9) (800)( 2)
292 psi
7.2 1.944 9.6
yB
zB
B
zy
MZ
My
P
AI I
s
=− + −− + =
0 0.9 in., ?
900 (540)(0.9) (800)
O= 7.2 1.944 9.6
125 83.333 0 1.5 in.
y
z
zy
zz
My
My
Pz
AI I
zz
s
= = =
−+=− +
−+ = =
Intersects AB at 0.500 in. from A

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