Civil Engineering Reference
In-Depth Information
Expressing the nodal velocities as
u
1
˙
˙
v
1
˙
{
˙
}=
(10.171)
u
2
˙
v
2
the kinetic energy expression can be rewritten in the form
T
m
(
e
2
{}
1
2
{
˙
}
T
=
N
1
0
N
1
N
2
0
L
N
1
1
0
0
N
1
N
2
2
{
˙
}
T
=
A
d
x
{}
(10.172)
N
2
N
1
N
2
0
0
0
N
2
0
N
1
N
2
0
From Equation 10.172, the mass matrix of the bar element in two dimensions is
identified as
N
1
0
N
1
N
2
0
2010
0201
1020
0102
L
N
1
m
(
e
2
=
0
0
N
1
N
2
=
AL
6
A
d
x
N
2
N
1
N
2
0
0
0
N
2
0
N
1
N
2
0
(10.173)
The mass matrix defined by Equation 10.173 is described in the element
(local) coordinate system, since the axial and transverse directions are defined in
terms of the axis of the element. How, then, is this mass matrix transformed to
the global coordinate system of a structure? Recall that, in Chapter 3, the element
axial displacements are expressed in terms of global displacements via a rotation
transformation of the element
x
axis. To reiterate, the transverse displacements
were not considered, as no stiffness is associated with the transverse motion.
Now, however, the transverse displacements must be included in the transforma-
tion to global coordinates because of the associated mass and kinetic energy.
Figure 10.18 depicts a single node of a bar element oriented at angle
rela-
tive to the
X
axis of a global coordinate system. Nodal displacements in the
element frame are
u
2
,
v
2
and corresponding global displacements are
U
3
,
U
4
,
respectively. As the displacement in the two coordinate systems must be the
same, we have
u
2
=
U
3
cos
+
U
4
sin
(10.174)
v
2
=−
U
3
sin
+
U
4
cos
or
u
2
v
2
cos
U
3
U
4
sin
=
(10.175)
−
sin
cos
