Civil Engineering Reference
In-Depth Information
U
4
v
2
u
2
U
3
Figure 10.18
The relation
of element and global dis-
placements at a single node.
As the same relation holds at the other element node, the complete transforma-
tion is
u
1
v
1
u
2
v
2
cos
sin
0
0
U
1
U
2
U
3
U
4
−
sin
cos
0
0
=
=
[
R
]
{
U
}
(10.176)
0
0
cos
sin
0
0
−
sin
cos
Since the nodal velocities are related by the same transformation, substitution
into the kinetic energy expression shows that the mass matrix in the global coor-
dinate system is
[
R
]
T
m
(
e
2
[
R
]
(10.177)
where we again use the subscript to indicate that the mass matrix is applicable to
two-dimensional structures.
If the matrix multiplications indicated in Equation 10.177 are performed for
an
arbitrary
angle, the resulting global mass matrix for a bar element is found to be
M
(
e
2
=
2010
0201
1020
0102
M
(
e
2
=
AL
6
(10.178)
and the result is exactly the same as the mass matrix in the element coordinate sys-
tem
regardless
of element orientation in the global system. This phenomenon
should come as no surprise. Mass is an absolute scalar property and therefore in-
dependent of coordinate system. A similar development leads to the same conclu-
sion when a bar element is used in modeling three-dimensional truss structures.
The complication described for including the additional transverse inertia
effects of the bar element are also applicable to the one-dimensional beam (flex-
ure) element. The mass matrix for the beam element given by Equation 10.78
is applicable only in a one-dimensional model. If the flexure element is used in
modeling two- or three-dimensional frame structures, additional consideration
must be given to formulation of the element mass matrix owing to axial inertia
