A total secant flexibility-based formulation for frame elements with physical and geometrical nonlinearities

This paper presents the formulation for a novel flexibility-based 1D frame element that captures material and geometrical nonlinearities. Decomposing the total strain to elastic and inelastic components a secant solution strategy based on a direct iterative scheme is introduced and the corresponding solution strategy is demonstrated. Concerning geometrical nonlinearities, the strain and slope of the deformed element are assumed to be small; however the equilibrium equations are satisfied for the deformed element that takes account of P–Δ effects. Using Simpson integration scheme along with piecewise interpolation of the curvature, the geometry of the deformed element is consistently updated. The implementation of the formulation in conjunction with a nonlocal damage model for capturing the response of element with softening such as reinforced concrete beam–columns is outlined and the formulation accuracy and efficiency is verified by some numerical examples on reinforced concrete frames.