Extremum principles for bounds to shakedown loads

Zouain, Néstor; Silveira, José Luı́s

doi:10.1016/s0997-7538(99)00125-4

Cited by 17 publications

(8 citation statements)

References 13 publications

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“…In the following the elastic and plastic shakedown theorems are briefly discussed. For a more in-depth treatment we refer to the papers of Zouain and Silveira [36,37].…”

Section: Shakedown Analysismentioning

confidence: 99%

“…The plastic shakedown theorem follows by simply neglecting the equilibrium constraints in (1) [24,36,37]. Thus, for any scalar α which satisfies…”

Section: Plastic Shakedownmentioning

confidence: 99%

“…In the present, however, the static theorems are of primary interest and the kinematic theorems will not be treated any further. For a comprehensive review on statickinematic duality theory for shakedown analysis we refer [36,37].…”

Section: Kinematic Theoremsmentioning

confidence: 99%

“…[24,36,37], its potential significance is generally not appreciated. In this paper we show that for a number of typical problems of purely deviatoric plasticity the elastic shakedown multiplier coincides either with the plastic shakedown multiplier or with the classical plastic collapse multiplier.…”

Section: Introductionmentioning

confidence: 99%

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Bounds to Shakedown Loads for a Class of Deviatoric Plasticity Models

2006

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The problem of estimating bounds to shakedown loads for problems governed by a class of deviatoric plasticity models including those of Hill, von Mises, and Tresca is addressed. Assuming that an exact elastic solution is available, an upper bound to the elastic shakedown multiplier can be obtained relatively easily using the plastic shakedown theorem. A procedure for computing this upper bound for arbitrary load domains is presented. A number of problems are then examined and it is found that the elastic shakedown factor is given as the minimum of the plastic shakedown factor and the classical limit load factor. Finally, some exact solutions to a number of two dimensional problems are given.

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“…In the following the elastic and plastic shakedown theorems are briefly discussed. For a more in-depth treatment we refer to the papers of Zouain and Silveira [36,37].…”

Section: Shakedown Analysismentioning

confidence: 99%

“…The plastic shakedown theorem follows by simply neglecting the equilibrium constraints in (1) [24,36,37]. Thus, for any scalar α which satisfies…”

Section: Plastic Shakedownmentioning

confidence: 99%

Section: Kinematic Theoremsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bounds to Shakedown Loads for a Class of Deviatoric Plasticity Models

2006

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“…Using the kinematic formulation for incremental collapse, they obtained a static dual form. Further studies concerning dual bounds of shakedown loads were carried out by Zouain and Silveira (1999).…”

Section: Introductionmentioning

confidence: 99%

A dual form for discretized kinematic formulation in shakedown analysis

Khoi

Yan

Hung

2004

International Journal of Solids and Structures

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Shakedown and Safety Assessment

Zouain

2004

Encyclopedia of Computational Mechanics

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The main subjects of this chapter are the variational formulations for the computation of a safety factor in shakedown analysis, the numerical procedures available to implement this analysis, and an overview on the wide field of applications and conceptual extensions. In structural engineering and solid mechanics, the word shakedown , introduced in the 1940s by W. Prager, became, since then, synonymous with elastic adaptation in the presence of variable loadings, a safe stabilized phenomenon, often due to some limited plastic deformation (or equivalently, due to the residual stress distribution associated with inelastic strains). Concerning the failure analysis of ductile structures, the terms alternating plasticity and incremental collapse , besides plastic collapse , are also widely used to identify failure modes under variable loadings. A structure undergoes alternating plasticity when the fluctuating loading program produces, after any arbitrarily large time, some plastic deformation as well as a subsequent vanishing of the net plastic deformation. This induces failure due to low‐cycle fatigue. Likewise, the structure fails by incremental collapse when plastic deformations accumulate in the form of a compatible strain distribution that leads to excessive inelastic deformation. Shakedown analysis allows working under the realistic assumption that only the range of variable loadings is known, unlike the usual prescription of a particular loading history. That is, we deal with direct methods that are based solely on the knowledge of a range of load variations (or a reference loading, in limit analysis) in this chapter. It is worth emphasizing that the differences between incremental and direct methods go beyond the computational advantages frequently obtained with the latter ones. In fact, the fundamental questions about whether or not critical loads or cycles do exist, independently from loading histories, can only be answered by the theories of direct methods. The mathematical foundations of shakedown theory strongly rest on optimization theory and the related areas of convex analysis and mathematical programming. The present chapter contains simplified derivations of the statical, kinematical, and mixed variational principles of shakedown theory by means of the basic techniques of convex analysis. Moreover, numerical procedures suitable to solve the shakedown analysis problem are discussed here in the framework of mathematical programming techniques combined with common procedures in the field of finite element methods. Applications in fundamental examples are then presented, namely, (a) a restrained block under thermo‐mechanical loadings, (b) the classical Bree problem of a tube under variable pressure and temperature, with a more realistic model and including an exact solution, (c) a plate with a circular hole subjected to variable tractions, and (d) a plate with simulated imperfections showing critical mechanisms with strongly localized deformations.

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Extremum principles for bounds to shakedown loads

Cited by 17 publications

References 13 publications

Bounds to Shakedown Loads for a Class of Deviatoric Plasticity Models

Bounds to Shakedown Loads for a Class of Deviatoric Plasticity Models

A dual form for discretized kinematic formulation in shakedown analysis

Shakedown and Safety Assessment

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