An efficient solution methodology to study the response of a beam on viscoelastic and nonlinear unilateral foundation: Static response

Many engineering materials and foundations such as soils demonstrate nonlinear and viscoelastic behaviour. Yet, it is challenging to develop static and dynamic models of systems that include these materials and are able to predict the behaviour over a wide range of loading conditions. This research is focused on a specific example: a pinned–pinned beam interacting with polyurethane foam foundation. Two cases, when the foundation can react in tension and compression as well as only in compression, are considered. The model developed here is capable of predicting the response to static as well as dynamic forces, whether concentrated or distributed. Galerkin’s method is used to derive modal amplitude equations. In the tensionless foundation case, the contact region changes with beam motion and the estimation of the co-ordinates of the lift-off points is embedded into the solution procedure. An efficient solution technique is proposed that is capable of handling cases where there are multiple contact and non-contact regions. Depending on the loading profiles a high number of modes may need to be included in the solution and to speed up computation time, a convolution method is used to evaluate the integral terms in the model. The adaptability of the solution scheme to complicated loading patterns is demonstrated via examples. The solution approach proposed is applicable to dynamic loadings as well and in these cases the automated treatment of complicated response patterns makes the convolution approach particularly attractive. The influence of various parameters on the static response is discussed.