Structural Analysis Methods II: Structural Systems - Mechanics of Structures: Variational and Computational Methods - page 300

FIGURE 5.29
Results for a frame with distributed loading. Example 5.7.
the loading in order to be able to exploit the symmetry, and (3) Designation of appropriate
boundary conditions on the portion of the structure to be analyzed.
Structural symmetry means that a structure is symmetric around one or more planes or
axes of geometry, material properties and boundary conditions. Apart from the applied
loading, the structure in Fig. 5.30a is symmetric about the Z axis. It can be seen that by
rotating the right hand side of the structure about the Z axis by 180 degrees, the left-
and right-hand sides of the structure can be made to coincide. In general, if one part of
the structure coincides with another part after being rotated around one or more axes or
planes, the structure is symmetric.
In analyzing symmetrical structures, the concept of symmetric and antisymmetric loads
is important. If, when one part of a symmetric structure is rotated about the axis or plane
of symmetry to coincide with another part, the loads on the two parts coincide, the system
of loads is said to be symmetric. After the rotation, if the loads on the two parts have the
same magnitudes but reversed directions, the load system is called antisymmetric. A load
system which is neither symmetric nor antisymmetric on a symmetric structure can be
transformed into a superposition of symmetric and antisymmetric load systems. Figures
5.30b and c illustrate how the unsymmetric loading on the symmetric structure of Fig. 5.30a
can be converted into the superposition of symmetric and antisymmetric load systems. The
load system on Fig. 5.30c is antisymmetric because by rotating the right part of the structure
by 180 degrees around the Z axis, the loads on the right part will have the same magnitudes
but reversed directions as those on the left part of the structure. Analyze the symmetric
structure with these symmetric and antisymmetric load systems and use the principle of
superposition to obtain the response of the whole structure.