A unified approach to the Timoshenko 3D beam-column element tangent stiffness matrix considering higher-order terms in the strain tensor and large rotations

•Interpolation functions based on the differential equation problem.•Timoshenko theory to predict the behaviour of beam-columns.•Higher-order terms in strain tensor improves the analysis performance.•Stiffness matrix adjustment to consider the finite rotations. A structural geometric nonlinear analysis using the finite element method (FEM) depends on the consideration of five aspects: the interpolation (shape) functions, the bending theory, the kinematic description, the strain–displacement relations, and the nonlinear solution scheme. As the FEM provides a numerical solution, the structure discretization has a great influence on the analysis response. However, when applying interpolation functions calculated from the homogenous solution of the differential equation of the problem, a numerical solution closer to the analytical response of the structure is obtained, and the level of discretization could be reduced, as in the case of linear analysis. Thus, to reduce this influence and allow a minimal discretization of the structure for a geometric nonlinearity problem, this work uses interpolation functions obtained directly from the solution of the equilibrium differential equation of a deformed infinitesimal element, which includes the influence of axial forces. These shape functions are used to develop a complete tangent stiffness matrix in an updated Lagrangian formulation, which also integrates the Timoshenko beam theory, to consider shear deformation and higher-order terms in the strain tensor. This formulation was implemented, and its results for minimal discretization were compared with those from conventional formulations, analytical solutions, and Mastan2 v3.5 software. The results clearly show the efficiency of the developed formulation to predict the critical load of plane and spatial structures using a minimum discretization.