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The concept of substructuring can also be used to manipulate the stiffness equations for
individual elements which contain interior nodes, that is, elements that are not common
with the nodes of other elements. The displacements of the interior nodes can be condensed
out.
The effectiveness of substructuring analysis can be improved by defining higher levels
of substructures. At each level, the stiffness equations for the substructure are formed and
condensed, and then assembled to form the stiffness equations of the next level substruc-
ture. Multi level substructuring can greatly reduce the dimensions of the global stiffness
equations for the complete structure.
5.3.9
Summary of the Solution Procedure
The computation implementation of the displacement method proceeds as follows:
1. Idealize the structure by subdividing it conceptually into elements. Define global
coordinates, the number and location of the nodes, and the nodal variables
V, P
. Read
in the loading cases and the number of elements (members). Construct an incidence
table.
2. For each element
i
, input properties, e.g.,
A
,
E
,
I
, and
J
for a beam element. Com-
pute the element stiffness matrix
k
i
in local coordinates. Establish the global to local
coordinate transformation
T
i
, and calculate the element stiffness matrix in global
coordinates using
k
i
k
i
T
i
.
3. In accordance with the incidence table, assemble the element matrices
k
i
into a sys-
tem stiffness m
at
rix
K
, summing all submatrices of like subscript. Also assemble the
loading vector
P
.
4. In essence, the system equations
KV
T
iT
=
P
have been established. Incorporate the
boundary conditions by removing each column that corresponds to a displacement
that is specified to be zero. Also, temporarily discard the rows corresponding to the
unknown support reactions. In practice, the corresponding rows and columns are not
established at all, as their absence can be accounted for in the incidence table. The
result
wi
ll be a square, non singular stiffness matrix. What remains of the applied load
vector
P
should contain the values of applied forces including zeros corresponding
to DOF where no loads are applied.
5. Solve the set of equa
t
ions
KV
=
=
P
for the system nodal displacements
V
for each case
of applied loading,
P
.
6. Considerable information can now be obtained as a postprocessing computation. The
support reactions can be calculated using the equations discarded in Step 4. For each
member, the end displacements are calculated using the compatibility conditions
(
kv
. Member displacements and
forces can be transformed from global to local coordinates. If desired, in-span displace-
ments and forces can be computed using, for example, the transfer matrix method.
For beams, cross-sectional stresses can now be computed using the formulas e.g.,
σ
=
v
=
aV
)
, and the end forces are found from
p
=
I
, that relate stresses to forces.
7. It is essential that the results of these computations be carefully scrutinized. At the
outset, the plausibility of the responses should be studied. Further controls are pro-
vided by checking the conditions of equilibrium for the whole system, for particular
nodes, and for particular parts. In the case of elements, the compatibility conditions
should be verified.
Mz
/
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