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. Since discretization (subdivision) and the use of iterative schemes (like Newton-Raphson) are sometimes undesirable, especially for students, avoiding it can be didactically interesting. This work proposes the use of a new approach to perform a simplified nonlinear analysis using the two-cycle method and a tangent stiffness matrix obtained directly from the homogeneous solution of the problem's (beam-column) differential equation. The proposed approach is compared to the results obtained by the traditional two-cycle method which uses geometric and elastic stiffness matrices based on cubic (Hermitian) polynomials and a P-Delta approximation using the pseudo (fictitious) lateral load method.

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). 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.