Civil Engineering Reference
In-Depth Information
Combining the mass matrix with previously obtained results for the stiffness
matrix and force vector, the finite element equations of motion for a beam ele-
ment are
v
1
¨
1
¨
¨
v
1
1
v
2
2
−
V
1
(
t
)
L
m
(
e
)
+
k
(
e
)
−
M
1
(
t
)
V
2
(
t
)
M
2
(
t
)
[
N
]
T
q
(
x
,
t
)d
x
=−
+
(10.79)
v
2
¨
2
0
and all quantities are as previously defined. In the dynamic case, the nodal shear
forces and bending moments may be time dependent, as indicated.
Assembly procedures for the beam element including the mass matrix are
identical to those for the static equilibrium case. The global mass matrix is directly
assembled, using the individual element mass matrices in conjunction with the
element-to-global displacement relations. While system assembly is procedurally
straightforward, the process is tedious when carried out by hand. Consequently, a
complex example is not attempted. Instead, a relatively simple example of natural
frequency determination is examined.
EXAMPLE 10.5
Using a single finite element, determine the natural circular frequencies of vibration of a
cantilevered beam of length
L
, assuming constant values of
,
E
, and
A
.
■
Solution
The beam is depicted in Figure 10.10, with node 1 at the fixed support such that the bound-
ary (constraint) conditions are
v
1
=
1
=
0
. For free vibration, applied force and bending
moment at the free end (node 2) are
V
2
=
M
2
=
0
and there is no applied distributed load.
Under these conditions, the first two equations represented by Equation 10.79 are con-
straint equations and not of interest. Using the constraint conditions and the known applied
forces, the last two equations are
AL
420
156
v
2
¨
2
12
4
L
2
v
2
0
0
−
22
L
EI
z
L
3
−
6
L
+
=
−
22
L
4
L
2
−
6
L
2
For computational convenience, the equations are rewritten as
156
v
2
¨
2
12
4
L
2
v
2
0
0
−
22
L
420
EI
z
mL
3
−
6
L
+
=
−
22
L
4
L
2
−
6
L
2
y
E
,
I
z
1
2
x
L
Figure 10.10
The cantilevered beam of
Example 10.5 modeled as one element.
