Civil Engineering Reference
In-Depth Information
and summing moments about node 1,
M
3
PL
16
−
P
L
2
+
5
P
16
=
L
=
0
Thus, the finite element solution satisfies global equilibrium conditions.
The astute reader may wish to compare the results of Example 4.1 with those
given in many standard beam deflection tables, in which case it will be found that
the results are in exact agreement with elementary beam theory. In general, the
finite element method is an approximate method, but in the case of the flexure
element, the results are exact in certain cases. In this example, the deflection
equation of the neutral surface is a cubic equation and, since the interpolation
functions are cubic, the results are exact. When distributed loads exist, however,
the results are not necessarily exact, as will be discussed next.
4.6 WORK EQUIVALENCE
FOR DISTRIBUTED LOADS
The restriction that loads be applied only at element nodes for the flexure ele-
ment must be dealt with if a distributed load is present. The usual approach is to
replace the distributed load with nodal forces and moments such that the me-
chanical work done by the nodal load system is equivalent to that done by the
distributed load. Referring to Figure 4.1, the mechanical work performed by the
distributed load can be expressed as
L
=
(4.51)
W
q
(
x
)
v
(
x
)d
x
0
The objective here is to determine the equivalent nodal loads so that the work
expressed in Equation 4.51 is the same as
L
=
=
F
1
q
v
1
+
M
1
q
1
+
F
2
q
v
2
+
M
2
q
2
(4.52)
W
q
(
x
)
v
(
x
)d
x
0
where
F
1
q
,
F
2
q
are the equivalent forces at nodes 1 and 2, respectively, and
M
1
q
and
M
2
q
are the equivalent nodal moments. Substituting the discretized dis-
placement function given by Equation 4.27, the work integral becomes
L
=
q
(
x
)[
N
1
(
x
)
v
1
+
1
+
N
3
(
x
)
v
2
+
2
]d
x
(4.53)
W
N
2
(
x
)
N
4
(
x
)
0
