Civil Engineering Reference
In-Depth Information
The element kinetic energy expressed in terms of nodal velocities and interpola-
tion functions is then written as
(
1
2
T
(
e
)
T
[
N
]
T
[
N
]
T
[
N
]
T
[
N
]
=
{˙
u
}
{˙
u
}+{˙
v
}
{˙
v
}
V
(
e
)
T
[
N
]
T
[
N
]
d
V
(
e
)
+{˙
w
}
{˙
w
}
)
(10.89)
Denoting the nodal velocities as
{˙
u
}
{
˙
}=
}
{˙
{˙
v
(10.90)
w
}
a
3
M
×
1
column matrix, the kinetic energy is expressed as
T
[
N
]
T
[
N
]
0
0
1
2
{
˙
}
{
˙
}
T
(
e
)
d
V
(
e
)
=
[
N
]
T
[
N
]
0
0
[
N
]
T
[
N
]
0
0
V
(
e
)
T
m
(
e
)
{
˙
}
1
2
{
˙
}
=
(10.91)
and the element mass matrix is thus identified as
[
N
]
T
[
N
]
0
0
m
(
e
)
=
d
V
(
e
)
(10.92)
[
N
]
T
[
N
]
0
0
[
N
]
T
[
N
]
0
0
V
(
e
)
Note that, in Equation 10.92, the zero terms actually represent
M
×
M
null
matrices. Therefore, the mass matrix as derived is a
3
M
×
3
M
matrix, which is
also readily shown to be symmetric. Also note that the mass matrix of Equa-
tion 10.92 is a
consistent
mass matrix. The following example illustrates the
computations for a two-dimensional element.
EXAMPLE 10.6
Formulate the mass matrix for the two-dimensional rectangular element depicted in Fig-
ure 10.12. The element has uniform thickness 5 mm and density
=
7
.
83
×
10
−
6
kg/mm
3
.
4
(10, 30)
(40, 30)
s
r
1
(10, 10)
(40, 10)
y
x
Figure 10.12
The rectangular element
of Example 10.6.
