Synopses & Reviews
Chapter I is devoted to an introduction of the concept of Yield Design, starting from historical landmarks and based upon field and laboratory observations of the collapse of mechanical systems. Compatibility between the equilibrium of the considered system subjected to prescribed loads and the resistance of its constituent material is set as the cornerstone of Yield Design analyses as it is apparent in recent construction codes implementing the Ultimate Limit State Design philosophy.
Chapter II presents the simple example of a truss structure in order to give an outline of the method introducing the concept of potential stability which is consistent with the restricted available data.
Since the general theory will be developed within the Continuum mechanics framework, Chapter III recalls the fundamentals of this model in its primal formulation leading to the classical equilibrium equations, and in the dual formulation with the theorem/principle of virtual (rate of) work.
Chapters IV to VI present the core of the theory.
In Chapter IV, after defining the concept of multi-parameter loading mode, the compatibility between equilibrium and resistance is first expressed in its primal form, on the basis of the equilibrium equations and the strength domain of the material defined by a convex strength criterion. The definition of the domain of potentially safe loads follows from the mathematical compatibility between the equilibrium equations and the convex strength condition. The domain is convex. It can be approached through the construction of statically admissible stress fields which comply with the strength condition and yield an interior estimate.
Chapters V and VI are devoted to the dual approach of the domain of potentially safe loads. Through the theorem/principle of virtual (rate of) work, it is possible to derive a necessary condition to be satisfied by the potentially safe loads, which does not refer to any stress field but uses kinematically admissible virtual velocity fields as test functions.
This leads to the kinematic exterior approach of the domain of potentially safe loads, where the material strength condition is expressed in its mathematical dual formulation of maximum resisting (rate of) work. It is essential to keep in mind that this formulation does not imply any constitutive law and is just the mathematical dualisation of the primal one.
Chapter VII is some kind of a return to Chapter I, which highlights the role played implicitly by the theory of Yield Design as the fundamental basis of the implementation of the Ultimate Limit State Design (ULSD) philosophy. It appears that the fundamental inequality of the kinematic exterior approach makes it possible to give an unambiguous quantified meaning to the symbolic inequality of ULSD.
With the explicit introduction of resistance parameters, Chapter VIII takes advantage of the symmetric roles played by the loads applied to a system on the one side and the resistance of its constituent materials on the other in the equations to be satisfied for potential stability. It introduces the concept of potentially safe dimensioning of a system under a given set of prescribed loads as the counterpart of potentially safe loads when the dimensioning of the system is given. Potentially safe dimensionings generate a convex domain for which interior and kinematic exterior approaches are derived from the general theory. Optimal dimensioning of the system results in minimising a given objective function. Also it is possible to account for the variability of the prescribed loads and for the physical scattering of the resistance parameters by giving a stochastic character to these data. From the definition of the domains of potentially safe loads and potentially safe dimensionings, there is no ambiguity in defining the concept of Probability of stability of a system. Again, the interior approach and, essentially, the kinematic exterior approach provide upper and lower bound estimates for this probability.
Chapter IX gets on to the Yield Design of structures. The curvilinear one-dimensional continuum model is first recalled with the concepts of wrench of forces and velocity distributor. The implementation of the Yield Design theory is straightforward provided the strength criteria of the constitutive elements, the joints and supports of the structure are correctly written.
In order to conclude with a concise presentation of the Yield Design analysis of plates, Chapter X is devoted to the construction of the corresponding two-dimensional model. The kinematics is defined by velocity distributor fields. The external forces are represented by force and moment densities, the internal forces are modelled by tensorial wrench fields.
In Chapter XI the implementation of the Yield Design theory is presented in the case when the considered system is subjected to pure bending, with strength criteria depending only on the internal moment tensor for metal plates and reinforced concrete slabs. The kinematic exterior approach appears as the most popular method, especially with relevant virtual motions based on the concept of hinge lines.
Review
Salençon presents the theory of yield design within the originalequilibrium/resistance framework without referring to the theories of plasticity or limit analysis. He develops the general theory forthe three-dimensional continuum model in a versatile form based on simple arguments from the mathematical theory of convexity. Then hesimply transposes to the one-dimensional curvilinear continuum for the yield design analysis of beams, and to the two-dimensional continuum model of plates and thin slabs subjected to bending.Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)
Synopsis
This book offers complete, practical guidance to all aspects yield design. It presents the core theory as the basis for the implementation of the Ultimate Limit State Design (ULSD) philosophy, emphasizing compatibility between equilibrium and resistance in yield design analyses. Developing the theory within the continuum mechanics framework, the book discusses the dual approach of the domain of potentially safe loads, resistance parameters, yield design of structures, models to illustrate the analysis of plates, the implementation of the theory using the popular kinematic exterior approach, and much more.
Synopsis
Since the middle of the 20th Century yield design approaches have been identified with the lower and upper bound theorem of limit analysis theory – a theory associated with perfect plasticity. This theory is very restrictive regarding the applicability of yield design approaches, which have been used for centuries for the stability of civil engineering structures.
This book presents a theory of yield design within the original “equilibrium/resistance” framework rather than referring to the theories of plasticity or limit analysis; expressing the compatibility between the equilibrium of the considered structure and the resistance of its constituent material through simple mathematical arguments of duality and convex analysis results in a general formulation, which encompasses the many aspects of its implementation to various stability analysis problems.
After a historic outline and an introductory example, the general theory is developed for the three-dimensional continuum model in a versatile form based upon simple arguments from the mathematical theory of convexity. It is then straightforwardly transposed to the one-dimensional curvilinear continuum, for the yield design analysis of beams, and the two-dimensional continuum model of plates and thin slabs subjected to bending. Field and laboratory observations of the collapse of mechanical systems are presented along with the defining concept of the multi-parameter loading mode. The compatibility of equilibrium and resistance is first expressed in its primal form, on the basis of the equilibrium equations and the strength domain of the material defined by a convex strength criterion along with the dual approach in the field of potentially safe loads, as is the highlighting of the role implicitly played by the theory of yield design as the fundamental basis of the implementation of the ultimate limit state design (ULSD) philosophy with the explicit introduction of resistance parameters.
Contents
1. Origins and Topicality of a Concept.
2. An Introductory Example of the Yield Design Approach.
3. The Continuum Mechanics Framework.
4. Primal Approach of the Theory of Yield Design.
5. Dual Approach of the Theory of Yield Design.
6. Kinematic Exterior Approach.
7. Ultimate Limit State Design from the Theory of Yield Design.
8. Optimality and Probability Approaches of Yield Design.
9. Yield Design of Structures.
10. Yield Design of Plates: the Model.
11. Yield Design of Plates Subjected to Pure Bending.
About the Authors
Jean Salençon is Emeritus Professor at École polytechnique and École des ponts et chaussées, ParisTech, France. Since 2009 he has been a member of the Administrative Board of CNRS (Paris, France). He has received many awards including the Légion d’Honneur (Commander), Ordre National du Mérite (Officer) and Palmes Académiques (Commander). His research interests include structure analysis, soil mechanics and continuum mechanics.
Table of Contents
Preface xiChapter 1. Origins and Topicality of a Concept 1
1.1. Historical milestones 1
1.2. Topicality of the yield design approach 8
1.3. Bibliography 11
Chapter 2. An Introductory Example of the Yield Design Approach 19
2.1. Setting the problem 19
2.2. Potential stability of the structure 22
2.3. To what extent potential stability is a relevant concept? 24
2.4. Bibliography 28
Chapter 3. The Continuum Mechanics Framework 29
3.1. Modeling the continuum 29
3.2. Dynamics 34
3.3. The theory of virtual work 41
3.4. Statically and kinematically admissible fields 46
3.5. Bibliography 48
Chapter 4. Primal Approach of the Theory of Yield Design 51
4.1. Settlement of the problem 51
4.2. Potentially safe loads 57
4.3. Comments 60
4.4. Some usual isotropic strength criteria 66
4.5. Bibliography 70
Chapter 5. Dual Approach of the Theory of Yield Design 73
5.1. A static exterior approach 73
5.2. A kinematic necessary condition 76
5.3. The π functions 78
5.4. π functions for usual isotropic strength criteria 84
5.5. Bibliography 88
Chapter 6. Kinematic Exterior Approach 91
6.1. Equation of the kinematic exterior approach 91
6.2. Relevant virtual velocity fields 94
6.3. One domain, two approaches 100
6.4. Bibliography 107
Chapter 7. Ultimate Limit State Design from the Theory of Yield Design 111
7.1. Basic principles of ultimate limit state design 111
7.2. Revisiting the yield design theory in the context of ULSD 113
7.3. The yield design theory applied to ULSD 114
7.4. Conclusion 117
7.5. Bibliography 118
Chapter 8. Optimality and Probability Approaches of Yield Design 119
8.1. Optimal dimensioning and probabilistic approach 119
8.2. Domain of potential stability 120
8.3. Optimal dimensioning 130
8.4. Probabilistic approach of yield design 133
8.5. Bibliography 141
Chapter 9. Yield Design of Structures 145
9.1. The curvilinear one-dimensional continuum 145
9.2. Implementation of the yield design theory 157
9.3. Typical strength criteria 164
9.4. Final comments 172
9.5. Bibliography 174
Chapter 10. Yield Design of Plates: the Model 177
10.1. Modeling plates as two-dimensional continua 177
10.2. Dynamics 182
10.3. Theorem/principle of virtual work 191
10.4. Plate model derived from the three-dimensional continuum 198
10.5. Bibliography 204
Chapter 11. Yield Design of Plates Subjected to Pure Bending 205
11.1. The yield design problem 205
11.2. Implementation of the yield design theory 208
11.3. Strength criteria and π functions 213
11.4. Final comments 226
11.5. Bibliography 234
Index 237