Evaluation of the P-Δ (P-Delta) effect in columns and frames using the two-cycle method based on the solution of the beam-column differential equation.

P-Delta is a nonlinear phenomenon that results from the consideration of axial loads acting on the deformed configuration of a member of the structure, usually a beam-column. This effect is especially significant in slender members, which can undergo large transversal displacements which tend to increase the bending moment caused by an axial load P acting upon them. The P-delta effect can be computed through a geometrically nonlinear analysis, usually employing the Finite Element Method, which subdivides each bar of the frame in finite segments known as elements.

Complete tangent stiffness matrix considering higher-order terms in the strain tensor and large rotations for a Euler Bernoulli - Timoshenko space beam-column element.

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).