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It follows that the unknown displacements are given by
K
−
1
22
V
y
=
(
P
−
K
21
V
)
(5.71)
and the unknown forces (reactions) are
P
v
=
K
11
V
+
K
12
V
y
(5.72)
However, this form of the solution is only formal, since rearrangement and inversion
operations are usually circumvented. That is, rather than expressing the unknown dis-
placements in the form of Eq. (5.71), the system of equations
K
22
V
y
=
P
−
K
21
V
(5.73)
is solved for
V
y
.
A straightforward approach to the introduction of boundary conditions is to ignore
those columns in the system matrix that correspond to zero (prescribed) displacements and
those rows for the corresponding unknown reactions. Then solve the remaining equations
for the unknown nodal displacements and compute the reactions
P
v
through Eq. (5.72). This
is the technique employed in most of the example solutions of the displacement method in
this chapter.
Schemes have been developed whereby the boundary conditions are applied to the
element stiffness matrices prior to assembly of the global matrices (see, for example, Bathe
(1996) and Cook, et al. (1989)).
5.3.6 Internal Forces, Stress Resultants, and Stresses
The internal force, stress resultant, or stress distribution in an element has to be computed
using an additional procedure, since only nodal displacements
V
are calculated directly.
For beams, the nodal forces can be obtained using
p
i
k
i
v
i
=
(5.74)
where the displacement vector
v
i
follows from
V
through the compatibility relationships
v
aV
of Eq. (5.37). The nodal forces of Eq. (5.74) are component forces along the global co-
ordinate axes which can be transformed into local components by relations such as Eq. (5.22),
after which desired responses such as stresses can be calculated.
If the distribution of response variables along an element is needed, it is often convenient
to use the transfer matrix method for performing the calculations. Since both end displace-
ments
v
i
and forces
p
i
are obtained by postprocessing the results of a global displacement
analysis, i.e., the state vector is known at the ends of an element, it is a simple task to use
transfer matrices to compute the distribution of these variables along a member.
=
5.3.7 Some Characteristics of Stiffness Matrices
Several properties of stiffness matrices were developed in Chapter 3. As indicated in
Chapter 3, both element and global stiffness matrices are symmetric and positive defi-
nite. The symmetry property is important in practice, since only terms on and to one side
of the main diagonal need to be generated and retained in a computer program. The pos-
itive definite property applies only to a stiffness matrix to which constraints have been
applied, e.g., the rigid body motion has been removed. Singular matrices can be positive
semi-definite, but not positive definite.
Observe the structure of the stiffness equations. For example, consider a particular global
stiffness matrix
K
, and note that each row of the matrix corresponds to a force at a node.
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