CIV5136 Chap.3 Symmetry and Antisymmetry
Symmetry and Antisymmetry
Symmetry reduction is valid only when geometry, supports, properties and loading transform consistently about an axis. Symmetric and antisymmetric response impose different centreline kinematic conditions.
A half-model must therefore be derived from the reflected deformation, then reconstructed to verify signs, reactions and the original total loading.
Begin with four reflection tests: geometry, restraints, member properties and loading. A single different release or flexural rigidity can destroy response symmetry even when the structural outline looks mirrored.
When the underlying structure is symmetric but the loads are not, a linear analysis may decompose the loading into symmetric and antisymmetric parts. Each part still needs its own boundary condition at the cut.
The centreline rule comes from deformation parity. For a typical vertically mirrored frame under symmetric loading, horizontal movement reverses under reflection and must vanish on the axis.
Rotation also changes sign and vanishes, while vertical translation may remain. Antisymmetric response changes which quantities have odd parity. Reusing the symmetric restraints for the antisymmetric case creates a different structure that can still produce a plausible-looking answer.
Load accounting and reconstruction complete the reduction.
A load located on the axis may need sharing between half-models, while an off-axis load is retained only on the selected half. After solution, reflect displacements and actions with the correct signs, rebuild the complete reaction set and verify the original total load. This final step exposes accidental doubling and an incorrect centreline condition.
What this chapter covers
- 01
Geometry, stiffness and support reflection tests
- 02
Symmetric displacement parity
- 03
Antisymmetric displacement parity
- 04
Centreline translations and rotations
- 05
Load splitting on a cut boundary
- 06
Symmetric and antisymmetric load decomposition
- 07
Full-model reconstruction
- 08
False symmetry caused by properties or releases
Worked example · free
Set the boundary of a symmetric half-frame
- +1Sketch the mirrored deflected shape before cutting the model.
- +1Horizontal translation changes sign under reflection and is zero on the axis.
- +1Rotation changes sign and is zero at the symmetric centreline.
- +1Vertical translation has even parity and may remain active.
- +1Reflect the solved half and confirm equal vertical responses and opposite horizontal actions where appropriate.
Key terms
- Symmetric response
- A displacement or action field that reproduces itself under reflection with the appropriate vector signs.
- Antisymmetric response
- A field whose reflected quantities reverse according to antisymmetric parity.
- Centreline boundary
- The restraints or releases placed on a reduced model to reproduce the omitted mirrored half.
- Load decomposition
- Separation of an asymmetric load pair into symmetric and antisymmetric parts for linear superposition.
- Parity check
- A test of whether a displacement, rotation or action is even or odd under reflection.
Symmetry and Antisymmetry FAQ
Does symmetric geometry always allow a half-model?
No. Supports, member stiffnesses, releases and loading must also mirror correctly. Any unpaired difference can make the response asymmetric even when the outline looks perfectly balanced. Test those four ingredients before replacing the omitted half with centreline conditions. Geometry is only the first test. A reflected deformation sketch confirms the correct zero movement.
Is displacement on a symmetry line always zero?
Only the components with odd reflection parity are zero. For a typical vertically symmetric frame under symmetric loading, horizontal movement and rotation vanish while vertical movement can remain.
When can load decomposition be used?
A linear structure with symmetric geometry and properties can analyse symmetric and antisymmetric load parts separately, each with its correct centreline condition, then superpose the reconstructed responses.
How is a reduced model verified?
Reflect its displacements and member actions to rebuild the full structure. Mirrored values must have the expected signs, and the reconstructed reactions must balance the original unsplit loading.
What happens to a load on the mirror axis?
Account for it so the two reconstructed halves reproduce the original full load exactly. Depending on the reduced equilibrium convention, an axis load may be shared between half-models rather than applied in full to each. Write the split beside the cut and check the total after reflection to prevent accidental doubling.
Can unequal mirrored loads still use symmetry ideas?
For a linear structure whose geometry, supports and properties remain symmetric, resolve the load pair into a symmetric part and an antisymmetric part. Analyse each with its own centreline condition, reconstruct both full responses and superpose them. Nonlinear behaviour invalidates that simple response superposition.
Exam move
For every symmetry question, make a four-row audit: geometry, supports, properties and loading. If all pass, draw the deformed shape of the complete structure and determine the reflection parity of each centreline movement. Only then replace the removed half with boundary conditions. Practise one symmetric and one antisymmetric case side by side because the wrong habit is to reuse the same centreline restraint.
End by reconstructing the complete response and checking the original load total. Add arrows at a mirrored pair of joints and write equal or opposite beside each displacement component. Then apply the same parity language to reactions and member end actions. This forces the half-model boundary to follow the physical reflection rather than a memorised support symbol.
