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Methods of structural analysis

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Statically admissible

A stress state satisfying equilibrium and static boundary conditions.

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Kinematically admissible

Kinematic relationships and kinematic boundary conditions are fulfilled.

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Lower bound (static) theorem

Every loading for which it is possible to specify a statically admissible stress state that does not infringe the yield condition is not greater than the ultimate load.

 

Guarantees safe design --> most design methods are therefore based on lower bound theorem:

  • strut-and-tie models and stress fields
  • equilibrium solutions for slabs (e.g. strip method)
  • yield condition for membrane elements and slabs
  • etc.
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Upper bound (kinematic) theorem

Every loading that results from equating the work of external forces for a kinematically admissible deformation state with the associated dissipation work is not less than the limit load.

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Compability theorem

A load for which a complete solution can be specified is equal to the ultimate load.

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Complete solution

Statically admissible state that does not infringe the yield condition and a compatible kinematically admissible state of deformation can be specified for that load.

 

Finding a complete solution is difficult (if not impossible, unless numerical approaches are used), but the advantage of the compability theorem is that it does not require performing neither a detailed compability check nor a plasticity verification for the mechanism found.

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Main consequences of the theorems of limit analysis

Residual stresses (Eigenspannung) and restraints have no influence on the ultimate load (as long as the resulting deformations remain infinitesimally small).
(This only applies to limit analysis methods; in elastic solutions and particularly in stability problems, the failure load depends on residual stresses and restraints)

Adding (subtracting) weightless material cannot increase (decrease) the ultimate load.

Raising (lowering) the yield limit of the material in any region of a system cannot increase (decrease the ultimate load.

The ultimate load that can be calculated with a yield surface circumscribing (inscribing) the effective yield surface forms an upper (lower) bound to the effective ultimate load.