sm7 130 132

________________________________________________________________________
PROBLEM (*7.130 through 7.132) Determine the distance e that locates the shear center S of
a thin-walled beam of constant thickness t for the cross section shown in Figs. P7.130 through P7.132.
SOLUTION (*7.130)
From A to B:
y
s
B
2
()
22 22
AB sts
t==
Qs
2
22
Bta
=
Q
2
22
yB y
Bzz
VQ aV
It
I
τ ==
3
162
y
AB
z
atV
FAt
I
==
(See: Table, B.6)
From B to D:
2 () (2)
222 2222
BD B as ta ts
QQst as=+ − = + −
2
[(2)]
22 22
yBD y
BD zz
VQ V as
as
It I
τ == + −
2
2
y
D
z
aV
I
τ =
3
5
2()
362
y
BD B D B
z
ta V
Fat at
I
τττ =+ − =
22
22
0: () ()
22 22 2
AB BD BD
sFF
aa
2F
M
ee=+=+=
∑e
or
22
a
e =
Continued on next slide
Vy
E
a
D
a
A
z
S
e
s
t
C
2
23()
DB
A
a
τ
τ =−
A
F
B
τ
B
τ
D
τ
BD
F
1
13B
Aa
τ
=
AB
F
FBD
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instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
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SOLUTION (7.131)
y
(2)
yz y
AB zz
VQ bth V
I
tIt
τ ==
2y
z
bh V
I
=
2
24
AB
AB y
z
btbht
FV
I
τ
==
2
48
D
EyDE yBG
zz
bh b ht
VVF
II
ττ == y
V=
)2(:0 hFhFeVM DEABys +==
∑
or
22 2
25
()
44 16
zz
tb h b ht
eh
I
I
=+=
where,
322
2[() ()]
12 2 4
zth h h
Ibtbt≈+ +
32
15
()
12 2
thbh
=−
SOLUTION (7.132)
AB y
z
bhV
I
τ =
2
D
By
z
bh V
I
τ =
1(2 )
2
AB AB
Fb τ =t
2
y
z
bh
=
t
V
I
Continued on next slide
z
S C
t
AB
F
e
A
h
h/2
b
B
A
F
FBG
D
E
F
D
FEVy
D
G
y
S
t
e
A
F
y=BE
BD
F
AB BD
Vy
VF
F F
B
D
A
E
b
z
2b
h/2
h/2
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instructional purposes only to students enrolled in courses for which the textbook has been adopted. Any other
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2
1
24
BD BD y
z
bht
Fbt
I
τ ==V
BD
0:
xyAB
M
Ve F h F h==−
∑
or
22
223
(4 )
44
zz
ht bht
2
beb
I
I
=−=
Where,
32
12(2 ) ( )
12 2
zh
Ithbbt=++
2
1(18
12 hth 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
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