Civil Engineering Reference
In-Depth Information
4
2
y
∂
∂
(
)
EI
m
y
e
0
+
−
⋅
θ
=
z
y
4
2
∂
x
∂
t
4
2
z
z
∂
∂
EI
m
0
+
=
y
z
4
2
x
t
∂
∂
2
4
2
2
y
∂
θ
∂
θ
∂
∂
θ
GI
EI
m
e
m
0
−
+
⋅
−
=
t
w
y
θ
2
4
2
2
x
x
t
t
∂
∂
∂
∂
where
EI
z
and
EI
y
are cross sectional stiffness with respect to bending in the
y
and
z
directions,
GI
t
and
EI
w
are the corresponding torsion stiffness associated with St.Venant's torsion and warping,
m
y
and
m
z
are translational mass (per unit length),
m
θ
is rotational mass (with respect to the shear
centre) and
e
is the vertical distance from the shear centre to the centroid. Obviously
m
y
=
m
z
(for
simplicity they are both set equal to
m
) and
2
mI
e
ρ=+
p
θ
Fig. 4.2
Simply supported beam with channel type of cross section
These equations are satisfied over the entire span for the following displacement
functions
(
)
T
yxt
z xt
,
,
⎧
⎡
⎤
a
=
⎣
⎡
aaa
nx
⎤
⎦
⎪
⎨
⎪
y
z
θ
⎢
⎥
(
)
(
)
=⋅
a
f xt
,
where
⎢
⎥
π
(
)
(
)
f
xt
,
sin
p
i t
=
⋅
ω
⎢
(
)
⎥
xt
,
θ
⎣
⎦
L
⎩
and
. Introducing this into the differential equations above, the following eigen-
value type of problem is obtained:
n
=
1,2,.......,
N
(
)
2
KMa
, where:
−
ω
⋅
=
⎡
4
⎤
n
⎛ ⎞
π
⎢
EI
0
0
⎥
⎜
⎝ ⎠
z
L
⎢
⎥
m
0
m e
⎡
−⋅
⎤
⎢
⎥
4
n
π
⎢
⎥
⎛ ⎞
⎢
⎥
and
M
=
⎢
0
m
0
K
=
0
EI
0
⎜ ⎟
⎥
⎢
y
⎥
L
⎝ ⎠
⎢
me
0
m
θ
⎥
−⋅
⎢
⎥
⎣
⎦
2
⎢
22
2
⎥
n
⎛
n
I
⎞
π
π
⎛ ⎞
w
0
0
GI
⎢
⎜
+
⎟
⎥
⎜ ⎟
⎜
t
⎟
L
L
⎝ ⎠
⎢
⎥
⎝
⎠
⎣
⎦
