AAE 221 – Structures II – Spring 2004
Chapter 2.3 Beam shearing
1. First the location of the centroid and the moments of inertia must be found
(
)
(
)
(
)
t
t
t
t
t
t
A
z
A
y
i
i
cm
62
.
6
2
48
5
.
12
25
2
98
1
1
2
2
2
=
+
=
=
The moments of inertia are
4
2
2
3
2
2
3
2
4
3
2
2
3
2
6135
)
48
(
)
12
.
6
(
12
)
(
48
)
25
(
)
88
.
5
(
12
)
25
(
2
39233
12
)
48
(
)
25
(
)
5
.
24
(
12
25
2
t
t
t
t
t
t
t
t
A
y
I
I
t
t
t
t
t
t
t
A
z
I
I
i
i
zi
z
i
i
yi
y
=
+
⋅
+
+
⋅
=
+
=
=
⋅
+
+
⋅
=
+
=
The axial displacement is
0
)
(
=
¢
¢
x
u
EA
with the boundary conditions
F
L
u
EA
u
=
¢
=
)
(
0
)
0
(
The bending displacements are given by
0
)
(
0
)
(
=
¢
¢
¢
¢
=
¢
¢
¢
¢
x
w
EI
x
v
EI
y
z
with the boundary conditions
0
)
(
38
.
18
(
)
(
0
)
0
(
)
0
(
=
¢
¢
¢

=
¢
¢
=
¢
=
L
v
t
F
L
v
EI
v
o
v
F
L
w
EI
F
t
L
w
EI
w
w
=
¢
¢
¢
=
¢
¢
=
¢
=
)
(
)
25
(
)
(
0
)
0
(
0
)
0
(
By integrating and applying the boundary conditions, the displacements can be shown to
be:
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y
y
z
z
EI
Fx
EI
x
L
t
F
x
w
EI
x
M
x
v
EA
Fx
x
u
3
2
2
6
1
)
25
(
2
1
)
(
2
1
)
(
)
(
+

=

=
=
The rotation is found from
0
)
(
=
¢
¢
x
J
f
m
o
The boundary conditions are
t
M
L
J
=
¢
=
)
(
0
)
0
(
f
m
f
with
4
3
3
3
98
)
98
(
3
1
3
1
)
38
.
18
(
t
t
d
b
J
t
F
M
t
=
=
=
=
x
x
Integrating and applying boundary conditions,
Fx
t
x
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 Spring '04
 Lambros
 Trigraph, EO Eo, Boundary conditions

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