Civil Engineering Reference
In-Depth Information
Evaluation of the integrals gives,
−
63
L
6
L
w
1
θ
1
w
2
θ
2
6
6
−
L
2
2
2
EI
L
q
12
2
L
−
3
LL
=
(2.26)
3
6
−
3
L
2
symmetrical
2
L
which recovers the standard “slope-deflection” equations for beam elements.
The above case is for a uniformly distributed load applied to the beam. For the case in
which loading is applied only at the nodes we have,
63
L
−
6
L
w
1
θ
1
w
2
θ
2
f
z
1
m
1
f
z
2
m
2
2
2
2
EI
L
2
L
−
3
LL
=
(2.27)
3
6
−
3
L
2
symmetrical
2
L
which represents the beam element stiffness relationship.
Hence, in matrix notation we again have,
[
k
m
]
{
w
} = {
f
}
(2.28)
Beam-column elements, in which axial and bending effects are combined from (2.11)
and (2.27), are described further in Chapter 4.
2.4.2 Beam element mass matrix
If the element in Figure 2.2 were vibrating transversely, it would be subjected to an addi-
tional restoring force
2
2
−
ρA(∂
w/∂t
)
. The matrix form, by analogy with (2.15), is just,
N
1
N
1
N
1
N
2
N
1
N
3
N
1
N
4
w
1
θ
1
w
2
θ
2
L
N
2
N
1
N
2
N
2
N
2
N
3
N
2
N
4
d
2
d
t
−
ρA
(2.29)
d
x
N
3
N
1
N
3
N
2
N
3
N
3
N
3
N
4
2
0
N
4
N
1
N
4
N
2
N
4
N
3
N
4
N
4
and evaluation of the integrals yields the beam element mass matrix given by,
−
156 22
L
54
13
L
ρAL
420
2
2
4
L
13
L
−
3
L
[
m
m
]
=
(2.30)
156
−
22
L
2
symmetrical
4
L
In this instance, the approximation of the consistent mass terms by lumped ones can
lead to large errors in the prediction of beam frequencies as shown by Leckie and Lindberg
(1963). Strategies for lumping the mass matrix of a beam element are described further in
Chapter 10.
