Optimal Design of a Beam under Uncertain Loads

Optimal design of beams subject to a combination of uncertain and deterministic transverse loads is presented using a min-max approach. The compliance of the beam is maximized to compute the worst case loading and minimized to determine the optimal cross-sectional shape. The uncertain component of the transverse load acting on the beam is not known a priori resulting in load uncertainty subject only to the constraint that its norm is finite. The minmax approach leads to robust optimal designs which are not susceptible to unexpected load variations as it occurs under operational conditions. The optimality conditions in the form of coupled differential equations are derived with respect to load and the shape functions. The resulting equations are solved analytically and the results are given for several cases to illustrate the method and to study the behavior of the optimal shapes and the worst case loadings. The efficiency of the optimal designs is computed with respect to a uniform beam under worst case loading taking the maximum deflection as the quantity for comparison.