A novel primal‐dual eigenstress‐driven method for shakedown analysis of structures

Abstract

The shakedown load of elastoplastic structures under multiple variable loading is an important factor in structural design and integrity analysis. In classical plasticity shakedown analysis is an essential and challenging problem. Most existing methods are based on the solution of super large‐scale mathematical programming, basis reduction or mechanics insight method which have their own limitations in practical engineering problem. In the present study, the proposed method explores the Karush–Kuhn–Tucker (KKT) conditions and the physical reinterpretation of its primal‐dual variables of the Melan's lower bound theorem, and establishes the relation of the primal‐dual variables in finite element formulation. Based on the primal‐dual theory, the primal‐dual eigenstress‐driven method is proposed which is a two‐level nested algorithm: the inner level is the eigenstress‐driven incremental algorithm for constructing the beneficial residual stress field and calculating the safe multiplier and the outer level is the load multiplier descent algorithm for searching the shakedown multiplier. For the purpose of algorithm's universality, the proposed algorithm is fully integrated with the commercial finite element software ANSYS APDL without special optimization solvers, which is well suited for large‐scale practical engineering structures. Several numerical examples are provided to demonstrate the accuracy and efficiency of the proposed algorithm.

Structural topology optimization of elastoplastic continuum under shakedown theory

“…In order to solve the above minimization problem (10), we state the optimality conditions characterizing the solution of the static shakedown problem, which are obtained 30 by constructing the Lagrangian functional of the constrained minimization problem (10) as $$$$ L\left(\lambda, {\boldsymbol{\upsigma}}^s,{\boldsymbol{\upalpha}}_1,{\boldsymbol{\upalpha}}_2\right)=-\lambda +\sum \limits_{i=1}^{NV}\sum \limits_{j=1}^{NE}{\boldsymbol{\upalpha}}_{1, ji}{f}_{ji}\left(\lambda {\boldsymbol{\upsigma}}_{ji}^E+{\boldsymbol{\upsigma}}_j^s\right)-{{\boldsymbol{\upalpha}}_2}^T\sum \limits_{j=1}^{NE}{C_j}^T{\boldsymbol{\upsigma}}_j^s, $$$$ where $$$ {\boldsymbol{\upalpha}}_1 $$$ and $$$ {\boldsymbol{\upalpha}}_2 $$$ are the Lagrangian multipliers related to the yield condition and the self‐equilibrated condition respectively, which are also called the dual variables in the optimization theory. Accordingly, $$$ \lambda, {\boldsymbol{\upsigma}}^s $$$ are the primal variables.…”

“…To solve the optimality conditions (13) of shakedown analysis, we utilize a recently novel approach called primal‐dual eigenstress‐driven method (PEM) 30 . The PEM approach proposes an efficient iterative strategy for determining the quantity of primal and dual variables $$$ \lambda, {\boldsymbol{\upsigma}}^s,{\boldsymbol{\upalpha}}_1,{\boldsymbol{\upalpha}}_2 $$$ that do not need mathematical programming solvers and elastoplastic incremental analysis.…”

“…The Equation (27) is satisfied naturally in the KKT conditions (13a,b) of shakedown analysis when the analysis procedure with the PEM algorithm 30 is performed. Therefore, $$$ {\hat{\boldsymbol{\upalpha}}}_1={\boldsymbol{\upalpha}}_1 $$$ can be obtained through the shakedown analysis.…”

“…where (𝛔 i ) M,j is von-Mises stress of the jth element at ith loading vertex, 𝜎 y,j is the jth element yield stress. In order to solve the above minimization problem (10), we state the optimality conditions characterizing the solution of the static shakedown problem, which are obtained 30 by constructing the Lagrangian functional of the constrained minimization problem (10) as…”

“…[22][23][24] In addition, based on the classical shakedown theory of elastic-perfectly plastic systems, many extensions of shakedown theory are developed for covering various engineering concerns, such as geometric nonlinearities, [25][26][27] creeping effect, 28 frictional contact 6,29 and so forth. In this article, the recently developed PEM method 30 is applied for shakedown analysis of elastic-perfectly plastic continuum. The PEM method provides an efficient iterative strategy that does not need mathematical programming solvers and elastoplastic incremental analysis to determine the shakedown load limit and the corresponding adjoint variables needed in the sensitivity analysis.…”