ESA formulation for large displacement analysis of framed structures with elastic–plasticity

In this work, a spatial beam element for geometrically and materially non-linear analysis of framed structures is presented in this work. The equilibrium equations of a straight beam element are formulated using an updated Lagrangian (UL) incremental description. Internal moments are represented as the resultants of stresses calculated by engineering theories: Euler–Bernoulli–Navier theory for bending and Saint-Venant theory for torsion. Although the element developed can undergo large displacements and rotations, strains are assumed to be small. The non-linear cross-sectional displacement field including large rotation effects is introduced in the analysis, resulting in the geometric potential of bending and torsional moments which corresponds to that of semitangential behaviour. In such a way, the joint equilibrium of non-collinear elements is provided. For the force recovering, the external stiffness approach (ESA) is presented as an alternative to the common natural deformation approach (NDA). Material non-linearity is introduced for an elastic–perfectly plastic material through the plastic hinge formation at finite element nodes and for this a new plastic reduction matrix of the element is determined. The interaction of element forces at a hinge and the possibility of elastic unloading are taken into account. The effectiveness of the numerical algorithm discussed is validated through the test problem.