sm7 107 109

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PROBLEMS (7.107 through 7.109) As depicted in the cross section in Figs. P7.107 through
P7.109, a beam is acted upon by a moment M with its vector forming an angle
α
with the horizontal
axis. Calculate the stress at points A, B, and D.
SOLUTION (7.107)
y
364
80(120) 11.52 10
12
z
I
mm==×
364
120(80) 5.12 10
12
y
I
mm==×
mkNM o
z⋅== 72.225cos3
mkNM o
y⋅−=−= 27.125sin
Apply Eq.(7.45):
33
66
2.72 10 ( 0.06) 1.27 10 (0.04)
11.52 10 5.12 10
yA
zA
Azy
Mz
My
II
σ
−−
×− ×
=− + =− −
××
MPa28.492.92.14 =−=
MPa
B12.2492.92.14 =+=
σ
MPa
D12.2492.92.14 −=−−=
σ
SOLUTION (7.108)
33
1(100 100 ) 96 90
12
z
I=×−×
46
105.2 mm×=
3
3
2 90(4)
(5 100 )
12 12
y
I=×+
43
108.833 mm×=
mkNMmkNM o
y
o
z⋅==⋅== 732.160sin2,160cos2
33
66
1 10 ( 0.05) 1.732 10 ( 0.05)
2.5 10 0.83 10
yA
zA
Azy
Mz
My
II
σ
−−
×− ×−
=− + =− −
××
MPa9.12386.10320 =+=
MPa
B86.8386.10320 −=−=
σ
MPa
D86.8386.10320 =+−=
σ
z
120 mm
80 mm
D
C
A
M
z
B
M
y
y
4 mm
z
5 mm
m15 m
C
5 mm
M
y
100 m
m
100 mm
M
z
D
A
B
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.
Continued on next slide

SOLUTION (7.109)
33
1(120 80 100 60 )
12
z
I=×−×
y
M
46
1032.3 mm×=
33
1(80 120 60 100 )
12
y
I=×−×
46
1052.6 mm×=
mkNMmkNM o
y
o
z⋅==⋅−=−= 5.230sin5,33.430cos5
33
66
4.33 10 ( 0.04) 2.5 10 (0.06)
3.32 10 6.52 10
yA
zA
Azy
Mz
My
II
σ
−−
×− ×
=− + = +
××
52.17 23.0 29.2
M
Pa=− + =−
52.17 23.0 75.2
B
M
Pa
σ
=− − =−
52.17 23.0 75.2
D
M
Pa
σ
=+=
y
z
120 mm
80 mm
60 mm
C
100 mm
D
M
z
A
B
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.