Statically Equivalent Loads

The actual loads for a particular problem, which may be distributed over lines, areas or volumes, may be replaced, for example, as a set of point loads. If the resultant forces and moments of the replacement loads are identical to those of the actual loads then they are said to be statically equivalent. The replacement of distributed loads as point loads is necessary in finite element formulations for which the model may only be loaded at nodes. The transformation of actual to statically equivalent loads is not normally unique, i.e., there may be multiple statically equivalent loads that equilibrate the actual loads. The set of statically equivalent loads that does the same work as the actual loads over the assumed displacement field of the finite element model is, however, unique and these loads are called consistent nodal loads. Information regarding the actual nature of the loading is, potentially, lost in this transformation since forms of loading that are self-balancing, i.e., have no resultant forces or moments, will have statically equivalent loads that are zero. It is easy to see, therefore, that element formulations which require the loads at the static boundary to be applied in the form of consistent nodal loads can lead to a lack of equilibrium in a pointwise sense – this is one aspect of the weak equilibrium that can occur with CFE formulations. Other formulations, e.g., EFEs, do not normally require the use of statically equivalent loads with distributions of force being admissible with the FE formulation. In this case strong pointwise equilibrium can be achieved on, for example, the static boundary.

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