Kinematic limit analysis of frictional materials using nonlinear programming

Li, H.X.; Yu, Hai‐Sui

doi:10.1016/j.ijsolstr.2004.11.023

Cited by 22 publications

(19 citation statements)

References 24 publications

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“…In Equation (22), the first two terms are the J functional, where the discontinuity velocity term in the dissipated energy rate expression is cancelled out, since it is discarded by the finite element formulation. The last two terms introduce the compatibility constraint (8), present in problem (21), in the objective function. Now, taking into consideration the finite element discretization of the domain and the field approximations (19)- (20), the augmented Lagrangian definition is re-written in the following expanded way: …”

Section: Numerical Formulationmentioning

confidence: 99%

A non‐linear programming method approach for upper bound limit analysis

Silva

Antão

2007

Numerical Meth Engineering

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SUMMARYThis paper presents a finite element model based on mathematical non-linear programming in order to determine upper bounds of colapse loads of a mechanical structure.The proposed formulation is derived within a kinematical approach framework, employing two simultaneous and independent field approximations for the velocity and strain rate fields. The augmented Lagrangian is used to establish the compatibility between these two fields. In this model, only continuous velocity fields are used.Uzawa's minimization algorithm is applied to determine the optimal kinematical field that minimizes the difference between external and dissipated work rate. The use of this technique allows to bypass the complexity of the non-linear aspects of the problem, since non-linearity is addressed as a set of small local subproblems of optimization for each finite element.The obtained model is quite versatile and suitable for solving a wide range of collapse problems. This paper studies 3D strut-and-tie structures, 2D plane strain/stress and 3D solid problems.

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Section: Numerical Formulationmentioning

confidence: 99%

A non‐linear programming method approach for upper bound limit analysis

Silva

Antão

2007

Numerical Meth Engineering

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“…The more detailed description about how to deduce the plastic dissipation power for a general yield criterion can be found in the research of Li and Yu (2005). As a result, the kinematic shakedown analysis of a structure modelled by the general yield criterion can be formulated as the following nonlinear mathematical programming problem:…”

Section: Plastic Dissipation Power For a General Yield Criterionmentioning

confidence: 99%

“…(27), which was constructed to perform limit and shakedown analyses for the von Mises criterion (Zhang et al, 1991;Zhang and Lu, 1995;Liu et al, 1995;Li et al, 2003), was overcome by means of an iterative algorithm (Zhang et al, 1991), where a technique based on distinguishing rigid/plastic areas was developed. Based on this technique, Li and Yu (2005) developed a general iterative algorithm to solve the nonlinear programming problem for limit analysis of frictional materials. This developed algorithm can be extended to solve the nonlinear mathematical programming problem (27).…”

Section: Iterative Solution Algorithmmentioning

confidence: 99%

“…Recently, Li and Yu (2005) developed a novel nonlinear numerical approach to perform limit analysis for a general yield criterion by means of nonlinear mathematical programming technique and the finite element method. Kinematic limit analysis was finally constructed as a nonlinear mathematical programming problem and an upper bound to the limit load of a structure subjected to static loads can be calculated.…”

Section: Introductionmentioning

confidence: 99%

“…The purpose of this paper is to extend the nonlinear numerical technique of Li and Yu (2005) for limit analysis to shakedown analysis so that it can be used to evaluate the shakedown limit and shakedown condition of frictional materials under repeated/cyclic loading. A general yield criterion defined by a polynomial with both first and second-order terms is used, which makes the developed method suitable for most of frequently-used yield criteria.…”

Section: Introductionmentioning

confidence: 99%

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A nonlinear programming approach to kinematic shakedown analysis of frictional materials

2006

International Journal of Solids and Structures

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This paper develops a novel nonlinear numerical method to perform shakedown analysis of structures subjected to variable loads by means of nonlinear programming techniques and the displacement-based finite element method. The analysis is based on a general yield function which can take the form of most soil yield criteria (e.g. the Mohr-Coulomb or Drucker-Prager criterion). Using an associated flow rule, a general yield criterion can be directly introduced into the kinematic theorem of shakedown analysis without linearization. The plastic dissipation power can then be expressed in terms of the kinematically admissible velocity and a nonlinear formulation is obtained. By means of nonlinear mathematical programming techniques and the finite element method, a numerical model for kinematic shakedown analysis is developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load can be calculated. An effective, direct iterative algorithm is then proposed to solve the resulting nonlinear programming problem. The calculation is based on the kinematically admissible velocity with one-step calculation of the elastic stress field. Only a small number of equality constraints are introduced and the computational effort is very modest. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples.

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Upper bound limit analysis using simplex strain elements and second‐order cone programming

Makrodimopoulos

Martin

2006

Num Anal Meth Geomechanics

265

182

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SUMMARYIn geomechanics, limit analysis provides a useful method for assessing the capacity of structures such as footings and retaining walls, and the stability of slopes and excavations. This paper presents a finite element implementation of the kinematic (or upper bound) theorem that is novel in two main respects. First, it is shown that conventional linear strain elements (6-node triangle, 10-node tetrahedron) are suitable for obtaining strict upper bounds even in the case of cohesive-frictional materials, provided that the element sides are straight (or the faces planar) such that the strain field varies as a simplex. This is important because until now, the only way to obtain rigorous upper bounds has been to use constant strain elements combined with a discontinuous displacement field. It is well known (and confirmed here) that the accuracy of the latter approach is highly dependent on the alignment of the discontinuities, such that it can perform poorly if an unstructured mesh is employed. Second, the optimization of the displacement field is formulated as a standard second-order cone programming (SOCP) problem. Using a state-of-the-art SOCP code developed by researchers in mathematical programming, very large example problems are solved with outstanding speed. The examples concern plane strain and the Mohr-Coulomb criterion, but the same approach can be used in 3D with the Drucker-Prager criterion, and can readily be extended to other yield criteria having a similar conic quadratic form.

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Kinematic limit analysis of frictional materials using nonlinear programming

Cited by 22 publications

References 24 publications

A non‐linear programming method approach for upper bound limit analysis

A non‐linear programming method approach for upper bound limit analysis

A nonlinear programming approach to kinematic shakedown analysis of frictional materials

Upper bound limit analysis using simplex strain elements and second‐order cone programming

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