Analogy for Beam‐Foundation Elastic Systems

Abstract

An analogy between two systems, each consisting of an elastic beam on elastic foundation, is given. Through this analogy, forces and displacements of one of two beams correspond to the displacements and forces, respectively, of the conjugate one, and field equilibrium and mechanical boundary conditions of one beam correspond to field compatibility equations and kinematical boundary conditions of the other beam. As the beam and foundation stiffnesses of the real system correspond to the foundation and beam compliances of the conjugate one, the problems of statically indeterminate beams not lying on foundation, having uniform or nonuniform cross section can be solved by using simple equilibrium equations on the conjugate beam. The analogy can be applied also when the beam on elastic foundation experiences inelastic deformations both distributed and concentrated. This enables us to solve several practical problems, e.g., temperature change effects, influence lines, secondary moments on statically indeterminate prestressed structures, etc. Applications to cylindrical tanks subjected to rise of temperature and influence lines of nonprismatic shear beams are reported.