Complete tangent stiffness matrix considering higher-order terms in the strain tensor and large rotations for a Euler...

This paper presents a unified method developed by Rodrigues et al. [1] to obtain a complete tangent stiffness matrix for spatial geometric nonlinear analysis using minimal discretization. The formulation presents four distinct important aspects to a complete analysis: interpolation (shape) functions, higher-order terms in the strain tensor and in the finite rotations, an updated Lagrangian kinematic description, and shear deformation effect (Timoshenko beam theory). Thus, the tangent stiffness matrix is calculated from the differential equation solution of deformed infinitesimal element equilibrium, considering the axial load and the shear deformation in this relation. This solution provides interpolation functions that are used in an updated Lagrangian formulation to construct the spatial tangent stiffness matrix considering higher-order terms in the strain tensor and in the finite rotations. The method provides an efficient formulation to perform geometric nonlinear analyses and predict the critical buckling load for spatial structures with moderate slenderness and with the interaction between axial and torsion effects, considering just one element in each member or a reduced discretization.•Complete expressions for a geometric nonlinear analyses considering one element per member•Spatial analyses considering higher-order terms in the strain tensor and large rotations•Shear deformation influence included.