limit states of materials and...

Limit States of Materials and Structures

Dieter Weichert · Alan PonterEditors

Limit States of Materialsand Structures

Direct Methods

123

EditorsDieter WeichertRWTH AachenInst. Allgemeine MechanikTemplergraben 6452062 [email protected]

Alan PonterUniversity of LeicesterDept. EngineeringUniversity RoadLeicesterUnited Kingdom LE1 [email protected]

ISBN: 978-1-4020-9633-4 e-ISBN: 978-1-4020-9634-1

DOI 10.1007/978-1-4020-9634-1

Library of Congress Control Number: 2008941180

c© Springer Science+Business Media B.V. 2009No part of this work may be reproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical, photocopying, microfilming, recordingor otherwise, without written permission from the Publisher, with the exceptionof any material supplied specifically for the purpose of being enteredand executed on a computer system, for exclusive use by the purchaser of the work.

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Preface

To predict loading limits for structures and structural elements is one ofthe oldest and most important tasks of engineers. Among the theoretical andnumerical methods available for this purpose, so-called “Direct Methods”, em-bracing Limit- and Shakedown Analysis, play an eminent role due to the factthat they allow rapid access to the requested information in mathematicallyconstructive manners.

The collection of papers in this book is the outcome of a workshop heldat Aachen University of Technology in November 2007. The individual con-tributions stem in particular from the areas of new numerical developmentsrendering the methods more attractive for industrial design, extensions of thegeneral methodology to new horizons of application, probabilistic approachesand concrete technological applications.

The papers are arranged according to the order of the presentations in theworkshop and give an excellent insight into state-of-the-art developments inthis broad and growing field of research.

The editors warmly thank all the scientists, who have contributed by theiroutstanding papers to the quality of this edition. Special thanks go to JaanSimon for his great help in putting together the manuscript to its final shape.We hope you may enjoy reading it!

Aachen and Leicester, Dieter WeichertSeptember 2008 Alan Ponter

v

Contents

The Linear Matching Method for Limit Loads, ShakedownLimits and Ratchet Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1A.R.S. Ponter

Large Problems in Numerical Limit Analysis: ADecomposition Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23F. Pastor, Z. Kammoun, E. Loute, J. Pastor, and H. Smaoui

Gurson Model for Porous Pressure Sensitive Materials . . . . . . . 45J. Pastor and Ph. Thoré

A Direct Method for the Determination of Effective StrengthDomains for Periodic Elastic-Plastic Media . . . . . . . . . . . . . . . . . . . 67S. Bourgeois, H. Magoariec, and O. Débordes

Stochastic Limit Load Analysis of Elasto-Plastic Plane Frames 87K. Marti

Limit Load Analysis of Plane Frames Under StochasticUncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113S. Zier and K. Marti

Reliability Analysis of Inelastic Shell Structures UnderVariable Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135T.N. Trần, P.T. Pha.m, Ð.K. Vũ, and M. Staat

Static Shakedown Theorem for Solids with Temperature-Dependent Elastic Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157A. Oueslati and G. de Saxcé

On Shakedown of Structures Under Variable Loads with aKinematic Non-linear and Non-associated Hardening Rule . . . . 179C. Bouby, G. de Saxcé, and J.-B. Tritsch

vii

viii Contents

Limit Analysis of Orthotropic Laminates by Linear MatchingMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197P. Fuschi, A. A. Pisano, and O. Barrera

Mechanical Surface Treatments and Life Improvement . . . . . . . . 221G. Inglebert, I. Caron, T. Da Silva Botelho, and M. Quillien

Force Method – Based Procedures in the Limit EquilibriumAnalysis of Framed Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233K.V. Spiliopoulos

Shakedown Analysis of Composite Materials Based onNon-linear Mathematical Programming . . . . . . . . . . . . . . . . . . . . . . . 253H.X. Li and H.S. Yu

A Non-linear Programming Approach to Shakedown Analysisfor a General Yield Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271H.S. Yu and H.X. Li

Application of Shakedown Analysis to Large-Scale Problemswith Selective Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289A. Hachemi, S. Mouhtamid, A.D. Nguyen and D. Weichert

Contributors

Barrera, OlgaDepartment dastec, University ‘Mediterranea’ of Reggio Calabria, viaMelissari I-89124, Reggio Calabria, Italy, e-mail: [email protected]

Bouby, Célinelemta, Nancy-Université, CNRS, 2, rue Jean Lamour, Vandoeuvre LesNancy Cedex, F-54519, France, e-mail: [email protected]

Bourgeois, StéphaneLaboratoire de Mécanique et d’Acoustique (UPR 7051) & Ecole Centrale deMarseille, Technopole de Chateau-Gombert, 38 rue de Joliot Curie, 13451Marseille cedex 20, France, e-mail: [email protected]

Caron, IsabelleLaboratoire d’Ingénierie des Systèmes Mécaniques et des Matériaux, EA2336,Institut Supérieur de Mécanique de Paris (supmeca), 3 rue Fernand Hainaut,93407 Saint-Ouen cedex, France, e-mail: [email protected]

Da Silva Botelho, TonyLaboratoire d’Ingénierie des Systèmes Mécaniques et des Matériaux, EA2336,Institut Supérieur de Mécanique de Paris (supmeca), 3 rue Fernand Hainaut,93407 Saint-Ouen cedex, France, e-mail: [email protected]

Débordes, OlivierLaboratoire de Mécanique et d’Acoustique (UPR 7051) & Ecole Centrale deMarseille, Technopole de Chateau-Gombert, 38 rue de Joliot Curie, 13451Marseille cedex 20, France, e-mail: [email protected]

De Saxcé, GéryLaboratoire de Mécanique de Lille, UMR CNRS 8107, Université desSciences et Technologies de Lille, Cité Scientifique, F-59655 Villeneuved’Ascq cedex, France, e-mail: [email protected]

ix

x Contributors

Fuschi, PaoloDepartment dastec, University ‘Mediterranea’ of Reggio Calabria, viaMelissari I-89124, Reggio Calabria, Italy, e-mail: [email protected]

Hachemi, AbdelkaderRWTH Aachen, Institut für Allgemeine Mechanik, Templergraben 64, 52062Aachen, Germany, e-mail: [email protected]

Inglebert, GenevièveLaboratoire d’Ingénierie des Systèmes Mécaniques et des Matériaux, EA2336,Institut Supérieur de Mécanique de Paris (supmeca), 3 rue Fernand Hainaut,93407 Saint-Ouen cedex, France, e-mail: [email protected]

Kammoun, ZiedEcole Polytech. de Tunisie, Laboratoire de Systèmes et de MécaniqueAppliquée, e-mail: [email protected]

Loute, EtienneFacultés universitaires Saint-Louis, Bld. Jardin Botanique 43, 1000 Brussels,Belgium, e-mail: [email protected]

Li, Hua-XiangDepartment of Engineering Science, University of Oxford, Oxford OX1 3PJ,UK, e-mail: [email protected]

Magoariec, HélèneLaboratoire de Tribologie et Dynamique des Systèmes (UMR 5513) & EcoleCentrale de Lyon, Bâtiment G8, 36 avenue Guy de Collongue, 69134 Ecullycedex, France, e-mail: [email protected]

Marti, KurtUniversität der Bundeswehr München (UniBwM), Institut für Mathe-matik und Rechneranwendung, Werner-Heisenberg-Weg 39, 85577Neubiberg/München, Germany, e-mail: [email protected]

Mouhtamid, SaidRWTH Aachen, Institut für Allgemeine Mechanik, Templergraben 64, 52062Aachen, Germany, e-mail: [email protected]

Nguyen, An DanhRWTH Aachen, Institut für Allgemeine Mechanik, Templergraben 64, 52062Aachen, Germany, e-mail: [email protected]

Oueslati, AbdelbacetLaboratoire de Mécanique de Lille, UMR CNRS 8107, Université desSciences et Technologies de Lille, Cité Scientifique, F-59655 VilleneuvedŠAscq cedex, France, e-mail: [email protected]

Contributors xi

Pastor, FranckUniversité Catholique de Louvain, Laboratoire cesame, AvenueGeorges Lemaître 4, B1348 Louvain-La-Neuve, Belgium, e-mail:[email protected]

Pastor, JosephUniversité de Savoie, Laboratoire locie, Campus de Savoie-Technolac,73376 Le Bourget du Lac cedex, France, e-mail: [email protected]

Pha.m, Phú TìnhFachhochschule Aachen (Campus Jülich), Labor Biomechanik, Ginsterweg1, 52428 Jülich, Germany, e-mail: [email protected]

Pisano, Aurora A.Department dastec, University ‘Mediterranea’ of Reggio Calabria, viaMelissari I-89124, Reggio Calabria, Italy, e-mail: [email protected]

Ponter, Alan R.S.University of Leicester, Department of Engineering, University Road,Leicester LE1 7RH, UK, e-mail: [email protected]

Quillien, MurielLaboratoire d’Ingénierie des Systèmes Mécaniques et des Matériaux, EA2336,Institut Supérieur de Mécanique de Paris (supmeca), 3 rue Fernand Hainaut,93407 Saint-Ouen cedex, France, e-mail: [email protected]

Smaoui, HichemEcole Nationale d’Ingénieurs de Tunis (enit), BP 37, Le Belvedere Tunis1002, Tunisie, e-mail: [email protected]

Spiliopoulos, Konstantinos V.National Technical University of Athens, School of Civil Engineering, HeroonPolytechniou 9, Zografou 15780 Athens, Greece,e-mail: [email protected]

Staat, ManfredFachhochschule Aachen (Campus Jülich), Labor Biomechanik, Ginsterweg1, 52428 Jülich, Germany, e-mail: [email protected]

Thoré, PhilippeUniversité de Savoie, Laboratoire locie, Campus de Savoie-Technolac, 73376Le Bourget du Lac cedex, France, e-mail: [email protected]

Tr`̂an, Thanh Ngo.cFachhochschule Aachen (Campus Jülich), Labor Biomechanik, Ginsterweg1, 52428 Jülich, Germany, e-mail: [email protected]

Tritsch, Jean-BernardLaboratoire de Mécanique de Lille, UMR CNRS 8107, Université desSciences et Technologies de Lille, Cité Scientifique, F-59655 Villeneuved’Ascq cedex, France, e-mail: [email protected]

xii Contributors

Vũ, Dúc KhôiUniversity of Erlangen-Nuremberg, Institute of Applied Mechanics,Egerlandstrasse 5, 91058 Erlangen, Germany, e-mail: [email protected]

Weichert, DieterRWTH Aachen, Institut für Allgemeine Mechanik, Templergraben 64, 52062Aachen, Germany, e-mail: [email protected]

Yu, Hai-SuiUniversity of Nottingham, School of Civil Engineering, University Park,Coates Building, Room B65, NG 7 2RD Nottingham, UK, e-mail: [email protected]

Zier, SimoneUniversität der Bundeswehr München (UniBwM), Institut für Mathe-matik und Rechneranwendung, Werner-Heisenberg-Weg 39, 85577Neubiberg/München, Germany, e-mail: [email protected]

The Linear Matching Methodfor Limit Loads, ShakedownLimits and Ratchet Limits

A.R.S. Ponter

Abstract The paper describes the application of the Linear MatchingMethod to the direct evaluation of limits associated with an elastic-perfectlyplastic body subjected to cyclic loading. Methods for limit load and shake-down limit are followed by ratchet limits. The method is distinguished fromother programming methods by ensuring that equilibrium and compatibilityare satisfied at each stage. The method has been extended beyond the range ofmost other direct methods by including ratchet limits and high temperaturematerial behaviour. Implementation is possible within the user routines ofcommercial finite element codes. The paper emphasise the theoretical char-acteristics of the method and discusses significant aspects of convergence,both theoretical and numerical. The application of the method to industrialLife Assessment problems and to geotechnical problems is summarized.

1 Introduction

Classically, numerical methods for Direct Method have relied upon the appli-cation of mathematical programming methods to limit load and shakedownlimit theorems of plasticity and this is strongly reflected in other papers inthis volume. Either upper or lower bound methods are possible dependingon whether the approximating continuum descriptions correspond to equilib-rium stress fields or compatible strain fields. The objective function, a loadparameter, is then either maximized or minimized according the upper andlower bounds of classical plasticity.

This approach has both advantages and disadvantages. Mathematical pro-gramming procedures have progressed significantly in recent years and highly

Alan R.S. PonterDepartment of Engineering, University of Leicester, Leicester, UKe-mail: [email protected]

D. Weichert, A. Ponter (eds.), Limit States of Materials and Structures, 1DOI 10.1007/978-1-4020-9634-1_1, c© Springer Science+Business Media B.V. 2009

2 A.R.S. Ponter

efficient solution methods are widely available producing fast and reliablesolution methods. However, the general methodology relies upon the classi-cal theorems of plasticity and extensions do not exist to behaviour outsideshakedown or to other material behaviour, time dependent creep behaviourin particular. For this reason, alternative approaches have been consideredthat provide a more flexible approach to the formulation of direct meth-ods. A number of methods discussed within the design community [6, 13,21]have proved to have a common theme, the representation of stress and strainfields through linear problems with spatially varying moduli. This first oc-curred through Marriot’s [21] observation that very good lower bound limitloads could be found by systematically decreasing the Young’s modulus inregions of high stress in a standard linear elastic solution. This has the effectof reducing the maximum stress and hence increasing the load at which allstresses lie within yield. This simple procedure can be very successful in com-puting safe lower bounds but, to the present time, has not been developedinto a method for evaluation the maximum lower bound, i.e. the limit load.

The Linear Matching Method [8, 23, 25, 26, 28] adopts the basic assump-tion that limit state solutions may be developed from linear solutions withspatially varying moduli and builds it into a programming method. In thispaper, the method is described for an elastic-perfectly plastic body for theevaluation of all the possible limits; limit load, shakedown limit and ratchetlimit. It is shown for shakedown that the method becomes a convergent up-per bound method. Each iteration provides both a kinematically admissiblestrain rate history and an equilibrium distribution of residual stress (in theGalerkin sense), both upper and lower bounds are generated that becomeequal to the minimum upper bound at convergence. This lower bound gener-ally does not monotonically increase. A dual method, based upon equilibriumstresses and the lower bound theorem also exists but appears not to convergefor perfect plasticity [23].

The extension of the method to the ratchet limit is then discussed, basedupon a general minimum theorem for perfect plasticity [14, 25]. This is fol-lowed by a summary of various applications to industrial problems.

2 Definition of the Problem

2.1 External Loads

We consider the problem of a body with volume V subjected to a cyclichistory of load λPi(xi, t) over part, namely ST , of the surface S, and a cyclichistory of temperature λθ(xi, t) within V. On the remainder of S, namelySU , the displacement rate u̇i = 0. Both load and temperature history havethe same cycle time Δt and, in the following, we are concerned with thebehaviour of the body in a typical cycle 0 ≤ t ≤ Δt in a cyclic state. The(positive) load parameter λ allows us to consider a class of loading histories.

Linear Matching Method 3

2.2 Material Behaviour

Consider such a body composed of a solid under conditions of small strainswhere the total strain is the sum of a linearly elastic and perfectly plasticcomponent,

ε̇ij = ε̇eij + ε̇pij , (1)

where the plastic strains are associated with a strictly convex yield conditionf(σij) = 0

ε̇pij = 0, f < 0,

ε̇pij = α̇∂f

∂σij, f = 0. (2)

The elastic moduli are assumed to be independent of temperature.

2.3 Structure of the Asymptotic Cyclic Solution

For the problem defined above the stresses and strain rates will asymptoteto a cyclic state, where

σij(t) = σij(t+ Δt), ε̇ij(t) = ε̇ij(t+ Δt). (3)

In the following we are concerned with the properties of this asymptoticsolution.

Consider a typical cycle, 0 ≤ t ≤ Δt. The cyclic solution may be expressedin terms of three components, the elastic solution, a transient solution accu-mulated up to the beginning of the cycle and a residual solution that repre-sents the remaining changes within the cycle. The linear elastic solution (i.e.ε̇pij = 0) is denoted by λσ̂ij and λε̂ij . The general form of the stress solutionis given by

σij(xi, t) = λσ̂ij(xi, t) + ρ̄ij(xi) + ρrij(xi, t), (4)

where ρ̄ij denotes a constant residual stress field in equilibrium with zerosurface traction on ST and corresponds to the residual state of stress at thebeginning and end of the cycle. The history ρrij is the change in the residualstress during the cycle and satisfies

ρrij(xi, 0) = ρrij(xi,Δt) = 0. (5)

The total plastic strain is similarly subdivided into two parts

εpij = εpTij (xi) + ε

prij (xi, t), (6)

4 A.R.S. Ponter

where εpTij denotes the accumulation of plastic strain at the beginning of thecycle and εprij denotes the additional plastic strain during the cycle.

The relationship between the transient and residual quantities is given by

εTij = Cijklρ̄ij + εpTij , (7)

εrij = Cijklρrij + ε

prij , (8)

where both εTij and εrij are compatible strain fields. Cijkl denotes the linearelastic compliance tensor. The cyclic solution is always non-unique to the ex-tent that the transient plastic strain εpTij may contain an arbitrary additionalcompatible component. Note that

εprij (xi,Δt) − εprij (xi, 0) = Δε

prij =

Δt∫

0

εprij dt (9)

the accumulated plastic strain during a cycle which, for consistency may beadded to εTij . Hence as both εTij and εTij+Δε

prij are compatible then Δε

prij is also

compatible with a displacement field Δui, the accumulation of displacementper cycle.

This argument emphasises the crucial role of the residual plastic strainrate history ε̇prij in defining the cyclic state. If ε̇

prij were known for 0 ≤ t ≤ Δt

then ρrij(xi, t) is uniquely defined by the solution of the initial strain rateproblem defined by the time derivative of (8),

ε̇rij = Cijklρ̇rij + ε̇

prij (10)

and the initial condition (5). The cyclic stress history (4) is then knownexcept for ρ̄ij . However the additional requirement of the compatibility ofΔεprij is sufficient to define ρ̄ij and hence the entire history of stress in thecycle is known and the final condition of (5) is always satisfied. It is natural,therefore, to concentrate on a class of inelastic strain rate histories that havethe same properties as the solution ε̇prij . We will refer to all strain rate historiesas ε̇cij that accumulate over a cycle to a compatible strain increment Δεcij askinematically admissible (ka).

3 Shakedown Theorems

In this section we consider the case of shakedown. The shakedown limit can beseen to be that range of load multiplier λ ≤ λs for which the changing residualstress ρrij is zero, where λs is the shakedown limit. In the following we statethe shakedown theorems and then convert the upper bound theorem into aform naturally aligned to the subsequent discussion of the Linear MatchingMethod.

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