# Upper bound limit and shakedown analysis of elastic plastic bounded linearly kinematic hardening structures

• Obere Schranken der Traglast- und Einspielanalyse von elastisch-plastisch, beschränkt linear kinematisch verfestigenden Strukturen

Pham, Phu Tinh; Weichert, Dieter (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2011)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2011

Abstract

This thesis develops a new FEM based algorithm for shakedown analysis of structures made of elastic plastic bounded linearly kinematic hardening material. Its concept can be briefly described as: Hardening law is simulated using a two-surface plastic model. One yield surface is the initial surface, defined by yield stress sigma_y, and the other one is the bounding surface, defined by ultimate strength sigma_u. The initial surface can translate inside the bounding surface without changing its shape and size. The subsequent yield surface is bounded by one of the two following conditions: (1) it always stays inside the bounding surface, or (2) its centre cannot move outside the back-stress surface, where the back-stress surface is defined by pi = sigma_u - sigma_y. Both ways of bounding are equivalent. The subsequent yield surface may touch the bounding surface, it means ratchetting occurs and the benefit of hardening is quite clear; or it may not touch the bounding surface, it that means alternating plasticity occurs, and there is no effect of hardening. If sigma_y = sigma_u, the two-surface model becomes perfectly plastic model, and if sigma_u >= 2*sigma_y, the model becomes an unbounded kinematic hardening model. Since the two-surface model bases only on yield stress and ultimate strength, so it does not depend on the hardening curve, consequently it is a linearly kinematic hardening model. Direct methods lead to plastic limit and shakedown bounds directly. They help to reduce considerably computing costs and numerical errors, and make the solution simpler. Mathematically, the shakedown problem is considered as a nonlinear programming problem. Starting from upper bound theorem, shakedown bound is the minimum of the plastic dissipation function, which is based on the von Mises yield criterion, subjected to compatibility, incompressibility and normalized constraints. This constraint nonlinear optimization problem is solved by combined penalty function and Lagrange multiplier methods.