Civil Engineering Reference
In-Depth Information
As in the case of the bar element, transverse beam deflection is discretized
using the same interpolation functions previously developed for the beam func-
tion. Now, however, the nodal displacements are assumed to be time dependent.
Hence,
v
(
x
,
t
)
2
(
t
)
(10.73)
and the interpolation functions are as given in Equation 4.26 or 4.29. Application
of Galerkin's method to Equation 10.72 for a finite element of length
L
results in
the residual equations
L
=
N
1
(
x
)
v
1
(
t
)
+
N
2
(
x
)
1
(
t
)
+
N
3
(
x
)
v
2
(
t
)
+
N
4
(
x
)
N
i
(
x
)
q
2
v
4
v
A
∂
EI
z
∂
+
+
=
0
i
=
1, 4
(10.74)
t
2
x
4
∂
∂
0
As the last two terms of the integrand are the same as treated in Equation 5.42,
development of the stiffness matrix and nodal force vector are not repeated here.
Instead, we focus on the first term of the integrand, which represents the terms of
the mass matrix.
For each of the four equations represented by Equation 10.74, the first integral
term becomes
v
1
¨
1
¨
¨
L
L
N
2
¨
1
+
N
4
¨
2
)d
x
A
N
i
(
N
1
¨
v
1
+
N
3
¨
v
2
+
=
A
N
i
[
N
]d
x
i
=
1, 4
v
2
¨
2
0
0
(10.75)
and, when all four equations are expressed in matrix form, the inertia terms
become
v
1
¨
1
¨
¨
¨
v
1
¨
1
¨
L
=
m
(
e
)
[
N
]
T
[
N
]d
x
A
(10.76)
v
2
¨
2
v
2
¨
2
0
The consistent mass matrix for a two-dimensional beam element is given by
L
m
(
e
)
=
[
N
]
T
[
N
]d
x
A
(10.77)
0
Substitution for the interpolation functions and performing the required integra-
tions gives the mass matrix as
156
22
L
54
−
13
L
m
(
e
)
=
AL
420
4
L
2
3
L
2
22
L
13
L
−
(10.78)
54
13
L
156
−
22
L
3
L
2
4
L
2
−
13
L
−
−
22
L
and it is to be noted that we have assumed constant cross-sectional area in this
development.
