Simultaneous Optimization of Beams and Their Elastic Foundations for Minimum Compliance

Nonuniform beams on nonuniform elastic foundations are considered. The beams have sandwich cross-sections with cores of negligible stiffness and are subjected to a uniformly distributed load. The total volume of the beam and the total stiffness of the foundation (which may include elastic supports) are specified. Both the cross-sectional area of the beam and the stiffness distribution of the foundation are design functions. They are chosen to minimize the compliance (or, equivalently, the area displaced by the beam deflection function). The calculus of variations is used to derive optimality conditions, and results are obtained for cantilevers and pinned-pinned beams. Several types of solutions are found, involving a single elastic support or a region of uniform foundation bordered at internal locations by elastic supports. In comparison to a reference uniform beam with uniform foundation, the decrease in compliance is significant.