Robust Optimal Design of Beams Subject to Uncertain Loads

Optimality conditions are derived for the robust optimal design of beams subject to a combination of uncertain and deterministic transverse and boundary loads using a variational min-max approach. The potential energy of the beam is maximized to compute the worst case loading and minimized to determine the optimal cross-sectional shape which results in coupled nonlinear differential equations for the unknown functions except for the case of a variable width beam. The uncertain component of the transverse load acting on the beam is not known a priori resulting in load uncertainty subject only to an norm constraint. Similarly the optimal area function is subject to a volume constraint leading to an isoperimetric variational problem. The min-max approach leads to robust optimal designs which are not susceptible to unexpected load variations as it occurs under operational conditions. The solution methodology is illustrated for the variable width beam by obtaining analytical results for several cases. 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. It is observed that the optimal shapes are more than 70% efficient for the examples given in this study.