Civil Engineering Reference
In-Depth Information
as the
reduced
system equations (this is the partitioned set of matrix equations,
written explicitly for the active displacements). In this example,
F
1
,
F
2
,
F
3
, and
F
4
are the components of the reaction forces at constrained nodes 1 and 2, while
F
5
and
F
6
are global components of applied external force at node 3. Given the
external force components, the last two of Equations 3.44 can be explicitly solved
for displacements
U
5
and
U
6
. The values obtained for these two displacements
are then substituted into the constraint equations (the first four of Equations 3.44)
and the reaction force components computed.
A more general approach to application of boundary conditions and compu-
tation of reactions is as follows. Letting the subscript
c
denote constrained
displacements and subscript
a
denote unconstrained (active) displacements, the
system equations can be partitioned (Appendix A) to obtain
K
cc
U
c
U
a
F
c
F
a
K
ca
=
(3.45)
K
ac
K
aa
where the values of the constrained displacements
U
c
are known (but not neces-
sarily zero), as are the applied external forces
F
a
. Thus, the unknown, active
displacements are obtained via the lower partition as
[
K
ac
]
{
U
c
}+
[
K
aa
]
{
U
a
}={
F
a
}
(3.46a)
[
K
aa
]
−
1
(
{
U
a
}=
{
F
a
}−
[
K
ac
]
{
U
c
}
)
(3.46b)
where we have assumed that the specified displacements {
U
c
} are not necessar-
ily zero, although that is usually the case in a truss structure. (Again, note that, for
numerical efficiency, methods other than matrix inversion are applied to obtain
the solutions formally represented by Equations 3.46.) Given the displacement
solution of Equations 3.46, the reactions are obtained using the upper partition of
matrix Equation 3.45 as
{
F
c
}=
[
K
cc
]
{
U
c
}+
[
K
ca
]
{
U
a
}
(3.47)
where [
K
ca
]
=
[
K
ac
]
T
by the symmetry property of the stiffness matrix.
3.6 ELEMENT STRAIN AND STRESS
The final computational step in finite element analysis of a truss structure is to
utilize the global displacements obtained in the solution step to determine the
strain and stress in each element of the truss. For an element connecting nodes
i
and
j
, the element nodal displacements
in the element coordinate system
are
given by Equations 3.19 and 3.20 as
u
(
e
1
U
(
e
1
cos
U
(
e
2
sin
=
+
(3.48)
u
(
e
2
U
(
e
3
cos
U
(
e
4
sin
=
+
