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Occasionally, for elements such as beams and plates, the mass associated with the
rotational DOF is taken into account. For a beam element with two nodes, consider a
“disk” of length
dx
, at coordinate
x
from node
a
, as shown in Fig. 10.3b. The rotary
moment of inertia of this “disk” about node
a
is
=
ρ(
/
+
x
2
)
dm
r
I
A
dx
where
is the mass density (mass per unit length) of the element, and
A
is the area of the
beam cross-section. Because the rotary inertia of a half of an element is lumped at each
node, we have
ρ
I
r
y
+
/
2
2
2
dm
r
=
ρ
2
A
+
=
ρ
2
m
ra
=
m
rb
=
(10.11c)
12
12
0
The mass matrix to supplement Eq. (10.11b) becomes
0
0
0
0
I
2
=
ρ
2
A
+
m
ra
1
m
r
=
(10.11d)
0
0
12
0
0
m
rb
1
For the two- or three-dimensional solid elements, the formulation of the lumped mass
matrix is not as straightforward. It is common in finite element computer programs to
perform the integration involved in
m
i
of Eq. (10.5) by using numerical integration schemes,
such as the Newton-Cotes or Gauss quadratures described in Chapter 6, Section 6.6. See,
for example, Fergusson and Pilkey (1992). Usually, the numerical integration procedure,
including the number and location of integration points, employed for the integration
needed for the stiffness matrix [
k
i
of Eq. (10.5)] is used for the integration of
m
i
of Eq.
(10.5). In all cases, the numerical integration leads to a discrete mass matrix
m
i
and in
some cases numerical integration can provide the same consistent mass matrices formed
by analytical integration. See Problems 10.4, 10.5, and 10.6.
An alternative to using the same numerical integration for
m
i
as employed for
k
i
,
is to
form the mass matrix by using only the element nodes as the integration points. This is
referred to as nodal quadrature. As shown in Chapter 6, the interpolation (shape) functions
N
in
m
i
of Eq. (10.5) exhibit the property
0
i
=
k
N
i
(
x
k
,y
k
,z
k
)
=
k
.
1
i
=
Hence, if the nodes are the only integration points, a diagonal mass matrix is obtained
because the value of the product of the shape functions
N
i
and
N
j
(
=
)
i
j
at any nodal point
is zero. The main diagonal elements in this lumped mass matrix are
n
W
(
n
)
k
N
i
(
W
(
n
)
i
N
i
(
m
ii
=
x
k
,y
k
,z
k
)γ (
x
k
,y
k
,z
k
)
=
x
i
,y
i
,z
i
)γ (
x
i
,y
i
,z
i
)
k
W
(
n
)
i
=
γ(
x
i
,y
i
,z
i
)
(10.12a)
where
n
is the number of nodes in the element;
W
(
n
)
k
are the weights; and
x
k
,y
k
,
and
z
k
are
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