Nonlinear viscoelastic behavior is a characteristic of many engineering materials including flexible polyurethane foam, yet it is difficult to develop dynamic models of systems that include these materials and are able to predict system behavior over a wide range of excitations. This research is focused on a specific example system in the form of a pinned-pinned beam interacting with a viscoelastic foundation. Two cases are considered: (1) the beam and the foundation are glued so that they are always in contact and the foundation can undergo both tension and compression, and (2) the beam is not glued to the foundation and the foundation reacts only in compression so that the contact region changes with beam motion. Static as well as dynamic transverse and axial forces act on the beam, and the Galerkin method is used to derive modal amplitude equations for the beam-foundation system. In the second case of the beam on tensionless foundation, loss of contact between the beam and the foundation can arise and determination of the loss-of-contact points is integrated into the solution procedure through a constraint equation. The static responses for both cases are examined as a function of the foundation nonlinearity and loading conditions. The steady-state response of the system subject to static and harmonic loads is studied by using numerical direct time-integration. Numerical challenges and the accuracy of this approach are discussed, and predictions of solutions by the three-mode and five-mode approximate models are compared to establish convergence of solutions. Frequency responses are studied for a range of foam nonlinearities and loading conditions.