Yield Design

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.