Civil Engineering Reference
In-Depth Information
The general set-up for the stiffness method of analysis represents a sys-
tem of linear equations, where the displacement vector is the unknown.
Except for those designated as supports, there are six unknown joint dis-
placement components for each joint in the structure in a 3-D structure.
Each displacement released at a support is still an unknown displacement
component to the system. There is an equation for each degree of freedom
of the structure. Each non-related component at a support has a displace-
ment that is set identically equal to zero, and as far as the system of equa-
tions is concerned, this particular equation may be omitted, along with
any coefficient in the other equations which corresponds to the dropped
displacement.
Sometimes the system solution is handled in six by six blocks of
coefficients or six rows of equations at a time, where each block repre-
senting the accumulated stiffness for a joint in the case of the diagonal,
or the carry-over effects from other joints in the case of off-diagonals.
In this case, unless the support joint is fully restrained, its correspond-
ing row of six by six blocks is maintained intact and a number of suf-
ficient sizes to simulate “infinite stiffness” in the restrained direction
are added to the diagonal of the diagonal block in the master stiffness
matrix.
The building of the global joint stiffness matrix consists of various
stages of operations. First, the member stiffness matrix is defined in its
own system, giving due consideration to member end releases, for each
member in the structure. This will be discussed in Chapter 5. The member
stiffness matrix can be considered as four separate six by six blocks. These
represent the forces at the ends due to the motions at the end and were
discussed in Sections 4.10 and 4.11.
]
=
KK
KK
[
ii
ij
K
m
ji
jj
The four separate components of the global member stiffness should be
placed in the global joint stiffness matrix. The diagonal terms will be added
to other terms representing the stiffness of other members connected to
that joint. The off-diagonal terms will simply be placed in their appropri-
ate position. The transformation of the local member system to the global
joint system was shown in Equation 4.34 and is derived as follows starting
with the local stiffness equation:
[
][
=
[]
K
P
mm m
∆
