Symmetry about a point is best described in terms of a spherical coordinate system having its Z axis on the line of symmetry. The structure, loading, and displacements are each said to be symmetric about a point if they do not vary with angular position about the point, i.e., they are independent of the angular coordinates SB and SA. Radial translation is the only displacement component that is permissible.
Enforce symmetry about a point using the Local Constraint as follows:
Model any spherical sector of the structure using any symmetric mesh of joints and elements.
Assign each joint a local coordinate system such that local axes 1, 2, and 3 correspond to the coordinate directions +SB, +SA, and +SR, respectively.
For each symmetric set of joints (i.e., having the same coordinate SR, but different coordinates SB and SA,) define a Local Constraint using only degree of freedom U3.
For all joints, restrain the degrees of freedom U1, U2, R1, R2, and R3.
Fully restrain any joints that lie at the point of symmetry.
It is also possible to define a case for symmetry about a point that is similar to cyclic symmetry around a line; for example, when each octant of the structure is identical.
See Also