In this work, a simple and effective Finite Element-Boundary Integral Equation (FE-BIE) approach for the static analysis of elastic beams and frames in bilateral frictionless contact with an elastic half-plane is extended to include the case of material nonlinearity. Elastic-perfectly plastic behavior is assumed for the supported structure and in particular for the foundation beam. Potential plastic hinges are placed along structural elements and modeled as nonlinear semi-rigid connections characterized by a rigid-plastic moment-rotation relationship. Incremental, load-controlled analyses of beams and frames resting on an elastic half-plane with several potential plastic hinges along the structural members are then performed. Numerical examples illustrate the effectiveness of the model in determining the stiffness degradation of the structure for increasing loads, together with ultimate loads and collapse mechanisms.