The paper addresses the inelastic quasi-static response of sandwich beams and panels with foam core to localized loads. The plane and axisymmetric formulations for local indentation or local low-velocity impact by a rigid body are considered; no overall bending is assumed. The governing equations for the face are derived using the Kirchhoff-Love static theory under the assumption that the core crushing follows elastic-ideally-plastic behavior. Analytical solutions are constructed on the basis of the principle of minimum work. The solutions allow predicting the face deflection, size of crushed core area and contact force. In general, the solutions are in good agreement with experimental data and finite element analysis.