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5.3.4
Assembly of a System Stiffness Matrix by a Summation Process
a
T
ka
is not normally utilized
explicitly to assemble the system stiffness matrix. Rather, this matrix should be recognized
as being a judiciously formed superposition of element stiffness matrices and, as such,
should be calculated by a summation process. This can be accomplished with the aid of
an
incidence table
which replaces the connectivity or incidence matrix
a
. In contrast to
a
which, in describing the topology of a system, is burdened by the frequent occurrence of
zero coefficients, the incidence table contains no zero coefficients. It simply identifies the
end nodes of each element with the corresponding global nodes. For the two-bar element,
three-node system of Fig. 5.12, the system nodes are numbered
a
,
b
, and
c
. An incidence
table would then appear as
As indicated previously, the congruent transformation
K
=
Global Node Numbers Corresponding
to Element End Numbers
Element
Element Begins at
Element Ends at
(5.64)
No
.
System Node No.
System Node No.
1
a
b
2
b
c
Since for each element (bar member) beginning and end system nodes are indicated, an
incidence table provides a sense of direction, a feature that can be important in interpreting
the responses that are calculated. Sometimes the beginning node of a bar element is said to
be the
+
+
−
−
1 node.
The incidence table is used to associate each end of each element with a particular system
node. Thus, this table tags each element end with a system number to indicate where it
should be placed in
K
. After all of the element ends are assigned to their correct system
nodes, all of the element stiffness coefficients, which are now associated with a particular
set of system node numbers, are summed to provide the corresponding system stiffness
coefficient.
It may be helpful to describe this stiffness matrix assembly process in terms of particular
stiffness coefficients. Suppose the stiffness matrix for the
i
th (1 or 2 in the case of Fig. 5.12)
element in terms of submatrices is of the form
or
1 node, and the end node is said to be the
or
k
jj
i
k
jk
k
i
=
(5.65)
k
kj
k
kk
where
j
and
k
are system node numbers (
a
,
b
,or
c
in the case of Fig. 5.12) provided by the
incidence table for element
i
. The contribution of the
i
th element to the system stiffness
matrix is
Column Index of
System Matrix
j
k
(5.66)
Row Index of
System Matrix
k
i
jj
k
i
jk
j
k
i
kj
k
i
kk
k
All element coefficients (or submatrices) thus identified as belonging to the same location
of the system matrix are then summed. In practial terms, this means that stiffness matrix
coefficients (or submatrices) of like subscripts are summed to form the corresponding (same
subscripts) coefficient of the global stiffness matrix.
This process of forming
K
is equivalent to a loop summation calculation over all elements
i
using
k
i
jk
K
jk
←
K
jk
+
(5.67)
where
j
and
k
are taken from the incidence table for each element
i
.
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