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should be no translation normal to the plane of symmetry and no rotations about the axis
of symmetry and other axes orthogonal to the axis of symmetry. For the antisymmetric
loading systems, there should be no translation in the plane of symmetry and no rotation
about an axis normal to the plane of symmetry at the new boundary. Figures 5.30d and e
show the boundary conditions for the symmetric and the antisymmetric load systems of
Figs. 5.30b and c. Since the structure lies in one plane and the plane of symmetry is the
YOZ
plane, the translation in the
X
direction and the rotations about the
Y
and
Z
axes are
constrained for the symmetric load system. Rotation about the
X
axis is not shown because
there is no deformation out of the
XOZ
plane. For the antisymmetric loading, translation
in the
Z
direction and rotation about the
Y
axis are zero. These boundary conditions can be
identified by predicting the displacement patterns of the structures in Figs. 5.30b and c.
After the symmetry of the structure is recognized, the loads processed and boundary con-
ditions imposed, the structure can be analyzed separately for symmetric and antisymmetric
load systems and then the principle of superposition can be used to obtain the response of
the whole structure.
5.3.16 Reanalysis
The objective of structural reanalysis is to compute the responses of a modified structure
efficiently by utilizing the response of the original (unmodified) structure. Typically, these
modifications are a result of proposed changes in design. Reanalysis methodology applies
to structures with either a modest number of localized changes of arbitrary magnitude or a
widely distributed change of limited magnitude. The former case, which is treated in this
section, is normally handled with an exact reanalysis and the latter case with an approximate
reanalysis [Pilkey and Wang, 1988]. Both cases are formulated for solution as problems of
much lower order than the original problem. Thus, reanalysis technology avoids the cost
of a complete analysis of a structure that has been modified. This economy is achieved by
exploiting the linearity properties of the structure, and by expressing the responses of the
structure as functions of the modifications. The availability of reanalysis technology is often
important in the effective implementation of iterative structural optimization.
Begin with a linear structural system described by the equation
KV
=
P
(5.105)
Suppose the system is modified so that
K
=
K
+
K
(5.106)
The governing equation for the modified system is
K
V
=
P
(5.107)
where
V
is the solution of the modified system. Substitute Eq. (5.106) into Eq. (5.107) and
rearrange
KV
=
KV
P
−
(5.108)
Premultiply Eq. (5.108) by
K
−
1
,
V
=
K
−
1
P
K
−
1
KV
−
(5.109)
For local modifications,
K
will be a sparse matrix, that is,
K
contains a small number
K
be a submatrix of
of non-zero columns and rows. Let
K
which contains only the
.
...
non-zero columns of
K
Furthermore, let the non-zero columns of
K
be the
i, j, k,
,
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