Civil Engineering Reference
In-Depth Information
effects. For beam elements, most finite software packages include axial effects
(i.e., the beam element is a combination of the bar element and the two-
dimensional flexure element) and all appropriate inertia effects are included in
formulation of the consistent mass matrix.
EXAMPLE 10.10
As a complete example of modal analysis, we return to the truss structure of Section 3.7,
repeated here as Figure 10.19. Note that, for the current example, the static loads applied
in the earlier example have been removed. As we are interested here in the free-vibration
response of the structure, the static loads are of no consequence in the dynamic analysis.
With the additional specification that material density is
=
6(10)
−
4
lb-s
2
/in.
4
,we
solve the eigenvalue problem to determine the natural circular frequencies and modal
amplitude vectors for free vibration of the structure.
As the global stiffness matrix has already been assembled, the procedure is not
repeated here. We must, however, assemble the global mass matrix using the element
numbers and global node numbers as shown. The element and global mass matrices for
the bar element in two dimensions are given by Equation 10.178 as
2
.
2010
0201
1020
0102
m
(
e
)
=
AL
6
As elements 1, 3, 4, 5, 7, and 8 have the same length, area, and density, we have
M
(1)
=
M
(3)
=
M
(4)
=
M
(5)
=
M
(7)
=
M
(8)
2010
0201
1020
0102
(2
.
6)(10)
−
4
(1
.
5)(40)
6
=
5
60
0
.
20
.
6
2
.
60
.
20
0
.
60
.
2
.
202
.
(10)
−
3
lb-s
2
/in.
=
while for elements 2 and 6
2010
0201
1020
0102
2
.
6(10)
−
4
(1
.
5)(40
√
2)
6
M
(2)
=
M
(6)
=
7
.
36
0
3
.
68
0
07
.
36
0
3
.
68
(10)
−
3
lb-s
2
/in.
=
3
.
68
0
7
.
36
0
0
.
68
0
7
.
36
