A force-based mathematical programming method for the incremental analysis of 3D frames with non-holonomic hardening plastic hinges

In this work, a fully automated, step-by-step, 1st order matrix force method is developed for the analysis of planar (2D) and spatial (3D) structural frames made of elastic-hardening plastic material. Plasticity (ideal and hardening) is modelled using plastic hinges of zero-length. The proposed automation techniques utilize Lagrange multipliers to model all forms of discontinuities in a simple and efficient way, within the framework of mathematical programming: internal discontinuities (e.g. articulations) as well as element eccentricities are taken into account. The problem may be solved using any quadratic/non-linear optimization algorithm with linear constraints; the redundant forces/moments serve as the primary unknowns. A load-controlled numerical strategy that is suitable for any predefined loading scenario analysis is proposed; its efficiency is demonstrated via a set of examples which are compared with existing results from the literature or output from commercial software based on the equivalent direct stiffness method.