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Non‐linear finite element formulation of kinematic limit analysis
Author(s) -
Jiang GongLiang
Publication year - 1995
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620381607
Subject(s) - limit analysis , limit load , mathematics , augmented lagrangian method , limit (mathematics) , finite element method , linear programming , discontinuity (linguistics) , kinematics , dissipation , mathematical optimization , mathematical analysis , classical mechanics , physics , upper and lower bounds , thermodynamics
The objective of the research presented in this paper was to develop a general computational method for kinematic limit analysis problems that involve the determination of an optimal kinematically admissible velocity field of the studied structure under external loads. With the failure velocity field available, we can obtain a numerical estimate of the limit load exerted to the structure. The general kinematic limit analysis is characterized by the minimization of the plastic potential power dissipation functional of the mechanical system. This minimization problem was approached previously by linear and non‐linear programming schemes. In this paper, a new non‐linear solution scheme is described which is based on previous research on the regularized method. The regularized functional is a convex and non‐linear functional whose minimum represents a viscoplastic potential power dissipation, the velocity which minimizes this functional was proved to be kinematically admissible and can be obtained by the augmented Lagrangian method. This paper demonstrates that the non‐linear programming scheme is applicable to direct limit analysis. The basic idea of using this optimization method is to tranform the functional with a non‐linear term of the first‐order derivatives of the velocity to a functional in which this term is uncoupled with the strain rates, and in consequence the augmented Lagrangian (transformed functional) can be solved more easily. Some special problems related to incompressibility and discontinuity are discussed. A simple and accurate scheme is proposed to deal with the incompressibility problem and the problem of the linear Mohr–Coulomb yield surface in the principal stress space. Examples of plane stress and plane strain problems are given for von Mises and Mohr–Coulomb materials. Numerical results are provided in graphical form which serves to illustrate the kinematic admissibility. The limit loads obtained agreed well with the analytical results, which demonstrates the efficiency and accuracy of the non‐linear computational method presented.