A Fibre Smart Displacement Based (FSDB) beam element for non linear analysis of r/c frames

Beam finite elements for non linear plastic analysis of beam-like structures are formulated according to Displacement Based (DB) or Force Based (FB) approaches. DB formulations rely on modelling the displacement field by means of displacement shape functions. Despite the greater simplicity of DB over FB approaches, the latter provide more accurate responses for those requiring a coarser mesh. In order to fill the existing gap between the two approaches, improvement of the DB formulation without the introduction of mesh refinement, is necessary. To this aim, the authors recently provided a contribution to the improvement of the DB approach by proposing new enriched, adaptive displacement shape functions, leading to the Smart Displacement Based (SDB) beam element. In this paper the SDB element is extended to include the axial force-bending moment interaction, crucial for the analysis of reinforced concrete (r/c) cross sections. The proposed extension requires the formulation of discontinuous axial displacement shape functions which are dependent on the diffusion of plastic deformations. The stiffness matrix of the extended smart element is provided explicitly and is dependent on the displacement shape functions updating. The axial force-bending moment interaction is approached by means of a fibre discretisation of the r/c cross section. The extended element, addressed as Fibre Smart Displacement Based (FSDB) beam element, is shown to be accurate furthermore, it is accompanied by an optional procedure proposal in order to verify the strong equilibrium of the axial force along the beam element, which is not usually accomplished by DB beam elements. Given a fixed mesh discretisation, the performance of the FSDB beam element is compared with the DB approach to show the higher degree of accuracy in the proposed element.