Virtual loads in response sensitivity analysis

Abstract

The objective of this work is to expand the significance and the use of virtual loads as a design response sensitivity analysis tool. The literature on the development of sensitivity analysis methods for structural design is reviewed along with the relevant equations. It is emphasized that the virtual load equations are based on the response rather than its derivative. The relation between the virtual load and the adjoint variable methods is then clarified through differentiation of the former and comparison. It is shown that the virtual load method is as widely applicable as other methods, with the additional advantage of giving the finite element contributions to the response. The mathematical basis for the method is reduced to the concept of vector projection. Examples of displacement, stress component, strain energy, and eigenvalue sensitivity are given to illustrate the arguments. The virtual load for principal stresses is derived and examined along with that for the von Mises equivalent stress. The findings are then generalized to all functions in linear response. Several uses of the finite element contributions to response are listed. The indications are toward benefits through crossfertilization among methods for wider applicability, including also nonlinear problems.