Distributed Plasticity Analysis of Frame Structures in Rate Form

Abstract

Distributed plasticity beam column elements are able to efficiently track hysteretic nonlinear behavior of structures under static or dynamic loading. This is accomplished by a refined discretization of the element in control sections along its length, each one being represented by a set of longitudinal fibers. The global response of the element results from a two level integration. In the first the non-linear stress of every fiber is integrated across the cross-sectional area to derive the constitutive relation of the control section and then integration along the element’s length is proved sufficient to yield the current state of the element. This work focuses on the formulation of both displacement and force based beam-column elements where the internal variables that describe the element’s state, namely fiber stresses or strains are expressed in rate form, herein using Bouc-Wen hysteretic models. Both formulations are derived from a unified approach based on the two field Hellinger-Reissner potential which highlights their differences. For simplicity reasons the methodology is applied on plane frame elements based on Euler–Bernoulli kinematics. The main advantage of expressing the evolution of each internal variable through a differential equation offers the ability to solve the entire set simultaneously with the global structure’s equations of motion in state space form. Accurate solutions are derived from proper implementation of an efficient numerical ODE solver.