The objective of this work is to investigate the quasi-static plastic behavior of a fully clamped metal foam core sandwich beam transversely loaded by a flat punch. A rigid-plastic beam-on-foundation model is extended to study the local denting deformation of a metal foam core sandwich beam. The effects of local denting and core strength on the overall deformation are incorporated in the analysis. Analytical solutions are derived for three different regimes of post-yield deformation mechanisms. Additionally, finite element results are obtained. Comparisons of the present analytical predictions with numerical, previous experimental, and analytical results are presented, respectively. It is shown that local denting has a significant effect on the finite deflection response of the metal foam core sandwich structure. The load-carrying and energy absorption capacities of sandwich beams may be overestimated if the effect of local denting is neglected in analysis. It is demonstrated that the present analytial model can reasonably predict the behaviors of post-yield deformation of sandwich beams. Moreover, the present analytical method can be extended to predict the low velocity/energy impact problems of sandwich structures.