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## The Use of Symmetry in Natural Frequency and Buckling
Analysis

When modelling symmetric structures it is common practice to
reduce the size of the model by invoking the principle of
symmetry. Appropriate freedom conditions are applied on the plane
of symmetry so that the half of the structure modelled behaves as
though it is still attached to the other half of the structure.

The nature of a symmetry boundary condition means that a
structure must deform symmetrically about the plane of symmetry.
This normally means that in addition to the structural geometry
being symmetric the loading must also be symmetric. Whilst most
analysts are comfortable with the concept of symmetry in linear
static problems, experience shows that this is not necessarily
the case with regard to natural frequency and buckling analysis.

Symmetric half models can be used for buckling and natural
frequency analysis but this is not as straight forward as it is
for linear static analysis.

A symmetry model with symmetric boundary conditions will yield
the symmetric buckling and vibration modes only. To obtain the
antisymmetric modes it will be necessary to run the model a
second time with antisymmetric boundary conditions applied to the
geometric symmetry plane of the structure. For large models it
may be better to use the symmetry approach, since running the
half model twice will usually be faster than running the full
model once.

Antisymmetric boundary conditions are simply the opposite of
symmetric conditions - any degrees of freedom that were fixed in
the symmetric case are now free and those degrees of freedom that
were free are now fixed.

The following example shows this. Both a symmetric and an
antisymmetric model are run and the results compared with a full
model. The table contains the natural frequencies whilst the
figures show the first 10 modes.

Full Model |
Symmetric Model |
Antisymmetric Model |

17.24 |
125.04 |
17.34 |

56.94 |
0 |
56.94 |

125.04 |
125.04 |
0 |

175.18 |
175.18 |
0 |

195.97 |
0 |
195.97 |

254.95 |
254.95 |
0 |

258.08 |
258.08 |
0 |

291.29 |
291.29 |
0 |

457.03 |
0 |
457.03 |

560.78 |
560.78 |
0 |

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