Table of Contents

Preface

Chapter 1. Origins and Topicality of a Concept

1.1. Historical milestones

1.2. Topicality of the yield design approach

1.3. Bibliography

Chapter 2. An Introductory Example of the Yield Design Approach

2.1. Setting the problem

2.2. Potential stability of the structure

2.3. To what extent potential stability is a relevant concept?

2.4. Bibliography

Chapter 3. The Continuum Mechanics Framework

3.1. Modeling the continuum

3.2. Dynamics

3.3. The theory of virtual work

3.4. Statically and kinematically admissible fields

3.5. Bibliography

Chapter 4. Primal Approach of the Theory of Yield Design

4.1. Settlement of the problem

4.2. Potentially safe loads

4.3. Comments

4.4. Some usual isotropic strength criteria

4.5. Bibliography

Chapter 5. Dual Approach of the Theory of Yield Design

5.1. A static exterior approach

5.2. A kinematic necessary condition

5.3. The π functions

5.4. π functions for usual isotropic strength criteria

5.5. Bibliography

Chapter 6. Kinematic Exterior Approach

6.1. Equation of the kinematic exterior approach

6.2. Relevant virtual velocity fields

6.3. One domain, two approaches

6.4. Bibliography

Chapter 7. Ultimate Limit State Design from the Theory of Yield Design

7.1. Basic principles of ultimate limit state design

7.2. Revisiting the yield design theory in the context of ULSD

7.3. The yield design theory applied to ULSD

7.4. Conclusion

7.5. Bibliography

Chapter 8. Optimality and Probability Approaches of Yield Design

8.1. Optimal dimensioning and probabilistic approach

8.2. Domain of potential stability

8.3. Optimal dimensioning

8.4. Probabilistic approach of yield design

8.5. Bibliography

Chapter 9. Yield Design of Structures

9.1. The curvilinear one-dimensional continuum

9.2. Implementation of the yield design theory

9.3. Typical strength criteria

9.4. Final comments

9.5. Bibliography

Chapter 10. Yield Design of Plates: the Model

10.1. Modeling plates as two-dimensional continua

10.2. Dynamics

10.3. Theorem/principle of virtual work

10.4. Plate model derived from the three-dimensional continuum

10.5. Bibliography

Chapter 11. Yield Design of Plates Subjected to Pure Bending

11.1. The yield design problem

11.2. Implementation of the yield design theory

11.3. Strength criteria and π functions

11.4. Final comments

11.5. Bibliography

Index

First published 2013 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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ISBN: 978-1-84821-540-5

Preface

“One of the principal objects of theoretical research in any department of knowledge is to find the point of view from which the subject appears in its greatest simplicity”¹

This book originates from the lecture notes for a course on yield design taught at Hong Kong City University during recent years. It is presented in the form of a survey of the theory of yield design, which brings together and summarizes the books and lecture notes I published in French on that topic when teaching at École Nationale des Ponts et Chausssées and École Polytechnique (Paris, France).

The terminology “yield design” has been chosen as a counterpart and translation of the French “Calcul à la rupture” or “Analyse à la rupture” which has been used for a long time by civil engineers and others to refer to stability analyses of structures where only the concepts of equilibrium and resistance are taken into account.

In an explicit form, such analyses have been carried out for nearly four centuries, if we take Galileo’s Discorsi as a starting point of the story, but they were overshadowed by the achievements of the theory of elasticity in the 19th Century.

To make a long story short, we may jump to the mid-20th Century when we observe a renewal of interest in the yield design methods with the development of the theory of plasticity. At that time, within the framework of the perfectly plastic model with associated flow rule for the constituent material, the lower and upper bound theorems of limit analysis and the theory of limit loads were established, which provide the traditional yield design approaches with sound theoretical bases. In particular, the upper bound theorem of limit analysis refers to the kinematic approach, where the rate of work by the external forces is compared with the plastic dissipation rate. Also, after several unconvincing attempts based on the concept of a rigid perfectly plastic material, the status of limit loads was definitely settled in the 1970s through the mathematical theorem of existence and uniqueness of the solution to the elastoplastic evolution problem: under the assumption of elastic and perfectly plastic behavior with associated flow rule, these loads are the maximum loads that can actually be sustained by the system considered in a given geometry.

This is a happy ending to the story from the theoretical point of view but, since it is dependent on the assumption of a perfectly plastic behavior with associated flow rule, it may appear as substantiating the idea that the yield design approach loses all interest when this assumption is not valid (which is often the case for practical problems, e.g. stability analyses of earth structures in civil engineering).

As a matter of fact, the lower and upper bound theorems are only the consequences of the sole assumption that the resistance of the constituent material is defined by a convex domain assigned to the internal forces. In particular, the upper bound theorem is derived from the dual definition of this domain without referring to a flow rule or constitutive equation. Therefore, these theorems hold as the lower and upper bound theorems for the extreme loads in the yield design theory, encompassing the many aspects of its implementation to various stability analysis problems. From the theoretical viewpoint, the status of the extreme loads is now restricted to that of upper bounds for the stability or load carrying capacity of the system. This does not make any difference in what concerns the application of the method to practice since practical validation is the general rule come what may.

Therefore, the purpose of this book is to present a theory of yield design within the original “equilibrium/resistance” framework without referring to the theories of plasticity or limit analysis. The general theory is developed for the three-dimensional continuum model in a versatile form based on 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 to the two-dimensional continuum model of plates and thin slabs subjected to bending.

The book is structured as follows:

– Chapter 1 gives an introduction of the concept of yield design, starting from historical landmarks and based on 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 is apparent in recent construction codes implementing the ultimate limit state design (ULSD) philosophy.

– Chapter 2 presents the simple example of a truss structure in order to give an outline of the method introducing the concept of potential stability.

– Since the general theory will be developed within the continuum mechanics framework, Chapter 3 recalls the fundamentals of this model in its primal formulation, leading to the classical equilibrium equations, and its dual formulation with the theorem/principle of virtual (rate of) work.

Chapters 4 – 6 present the core of the theory:

– In Chapter 4, 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 condition. The definition of the domain of potentially safe loads follows from the mathematical compatibility between the equilibrium equations and the strength condition. As a consequence of the convexity of the strength condition, the domain of potentially safe loads is convex, which makes it possible to obtain convenient interior estimates through the construction of statically admissible stress fields that comply with the strength condition.

– Chapters 5 and 6 discuss 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 dualization of the primal one.

– Chapter 7 is a kind of a return to Chapter 1 since it highlights the role played implicitly by the theory of yield design as the fundamental basis of the 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.

– Chapter 8, with the explicit introduction of resistance parameters, 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 dimensioning generates a convex domain for which interior and kinematic exterior approaches are derived from the general theory. Optimal dimensioning of the system results in minimizing 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 lower and upper bound estimates for this probability.

– Chapter 9 looks at 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 that the strength criteria of the constitutive elements, the joints and supports of the structure are correctly written.

– To conclude with a concise presentation of the yield design analysis of plates and thin slabs, Chapter 10 analyzes 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 and the internal forces are modeled by tensorial wrench fields.

– Chapter 11 presents the implementation of the yield design theory to metal plates and reinforced concrete slabs subjected to pure bending with strength criteria depending only on the internal moment tensor. The kinematic exterior approach appears as the most popular method, especially with relevant virtual motions based on the concept of hinge lines.

Acknowledgment

The author wishes to express his gratitude to Professors Habibou Maitournam, Noël Challamel and Pierre Suquet for their friendly comments and suggestions that contributed to the improvement of this book.

Jean SALENÇON
March 2013

1 GIBBS J.W., Proceedings of the American Academy of Arts and Sciences, May 1880 – June 1881, XVI, VIII, Boston, University Press/John Wilson & Co., pp. 420–421, 1881.

Chapter 1

Origins and Topicality of a Concept

Limit state design is, to some extent, a familiar terminology within the syllabuses of civil engineers’ education, as it appears explicitly in the stability analyses of various types of structures or is present “anonymously” in the methods used for such analyses. Nevertheless, the variety of the corresponding approaches often makes it difficult to recognize that they proceed from the same fundamental principles, which are now the basis of the ultimate limit state design (ULSD) approach to the safety analysis of structures. As an introduction to the theory, this chapter will both present some famous historical milestones and the topicality of the subject referring to the principles of ULSD.

1.1. Historical milestones

1.1.1. Dialogs concerning two new sciences

The fundamental concept to be acknowledged first is that of yield strength as introduced by Galileo in his Discorsi [GAL 38a] on the simple experiment of a specimen in pure tension (Figure 1.1).

Figure 1.1. Longitudinal pull test (Galileo, Discorsi, 1st day [GAL 38a])

Galileo uses this first characterization of the tenacity and coherence (tenacità e coerenza) of the material to explain the difficulty he finds in breaking a rod or a beam in tension while it is far easier to break it in bending: “A prism or solid cylinder of glass, steel, wood or other breakable material which is capable of sustaining a very heavy weight when applied longitudinally is, as previously remarked, easily broken by the transverse application of a weight which may be much smaller in proportion as the length of the cylinder exceeds its thickness”. Considering a cantilever beam () built in a wall (section ) and subjected to a weight applied at the other extremity (section ), he first defines the “”. Then, he assumes that this resistance to tension will be localized in the section of the beam where it is fastened to the wall and that “”. The reasoning follows “” [GAL 38a]. Introducing the second fundamental concept of the yield design approach, namely , by writing the balance equation for the lever about , Galileo finally relates the “” to its “ ” through the ratio of the short lever arm 2 to the long lever arm .

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