Civil Engineering Reference
In-Depth Information
value, since any lower-order derivative involves the spatial variable. However, if
we examine the conditions under which the element undergoes rigid body trans-
lation, for example, we find that the nodal forces must be of equal magnitude and
the same sense and the applied nodal moments must be zero. Also, for rigid body
translation, the slopes at the nodes of the element are zero. In such case, the sec-
ond derivative of deflection, directly proportional to bending moment, is zero
and the shear force, directly related to the third derivative of deflection, is con-
stant. (Simply recall the shear force and bending moment relations from the
mechanics of materials theory.) Therefore, the field variable representation as a
cubic polynomial allows rigid body translation. In the case of the beam element,
we must also verify the possibility of rigid body rotation. This consideration, as
well as those of constant bending moment and shear force, is left for end-of-
chapter problems.
6.3.1 Higher-Order One-Dimensional Elements
In formulating the truss element and the one-dimensional heat conduction ele-
ment (Chapter 5), only line elements having a single degree of freedom at each
of two nodes are considered. While quite appropriate for the problems consid-
ered, the linear element is by no means the only one-dimensional element that
can be formulated for a given problem type. Figure 6.2 depicts a three-node line
element in which node 2 is an
interior
node. As mentioned briefly in Chapter 1,
an interior node is
not
connected to any other node in any other element in the
model. Inclusion of the interior node is a mathematical tool to increase the order
of approximation of the field variable. Assuming that we deal with only 1 degree
of freedom at each node, the appropriate polynomial representation of the field
variable is
a
2
x
2
(
x
)
=
a
0
+
a
1
x
+
(6.17)
and the nodal conditions are
(
x
=
0)
=
1
x
L
2
=
=
2
(6.18)
(
x
=
L
)
=
3
L
2
L
2
x
1
2
3
Figure 6.2
A three-node
line element. Node 2 is an
interior node.
