A new non-linear beam element based on smart displacement shape functions

Structures composed by beam elements exhibiting diffused inelastic behavior are analysed by means of a displacement(stiffness) approach or, alternatively, by adopting a force (flexibility) approach. The first is based on the formulation of displacement shape functions that are affected by approximations in the correct modeling of the beam element response during the nonlinear analysis as inelastic deformations spread along the element. Such approximations are mainly due to the fact that the displacement shape function are unaltered during the analysis and do not follow the beam stiffness degradation. Differently, the force based approach requires the adoption of exact force shape functions that are not affected by the inelastic deformation development. Despite the fact that the latter approach leads to more accurate results, it requires more complicated iterative solution strategies and additional calculations when the beam inelastic deformed configuration has to be retrieved. In this work, within the framework of the classical displacement based approach, the use of adaptive shape functions able to update together with the diffusion of inelastic deformations along the beam axis is proposed. The proposed shape functions are formulated by making use of generalised function able to account for a flexural stiffness degradation of the beam according to a multi-stepped beam model. The presented approach leads to results whose accuracy appears to be comparable to those obtained through a force based approach however requiring a reduced implementation effort and a more straightforward approach. Moreover, the number of beam element required in the structure discretisation is drastically reduced with respect to the classical displacement base finite element approach.