Stability and optimization of a clamped beam elastically restrained against translation on one end resting on Winkler foundation

•The characteristic equation for determining critical loads of the beam is derived.•The influence of spring support parameter on critical loads is analyzed.•The optimal shapes for beams is found for different values of parameters.•In all cases of optimally designed beam we have bimodal optimization.•In all cases of optimally designed beam we have saving materials.