Analytical optimal designs for long and short statically determinate beam structures

Abstract

Analytical solutions for optimal beam design may serve as benchmarks for numerical studies and for basic understanding. The influence of load case, of boundary conditions and of cross sectional type is severe, so many cases are studied. Short beams based on Timoshenko theory are included in the present energy approach. With a given amount of material/volume the objective is to minimize compliance, and the necessary optimality criterion is to obtain the same gradient of elastic energy along the beam axis x, i.e., for all volumes A(x) dx, where the design function is the area A(x). The beams considered in the paper are geometrically unconstrained, i.e., no minimum and/or maximum constraints are specified for the variable cross sectional area function A(x) other than the volume constraint. The obtained explicitly described designs may be used for comparison with obtained distribution of volume densities for two and three dimensional numerical models, or in the sense of topology optimization for the distribution of the number of active design pixels/voxels. Also the presented decrease in compliance, relative to compliance for uniform beam design, show how much it is possible to obtain for these optimal compliance designs of statically determinate cases.