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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 (Figure 1.2) built in a wall (section AB) and subjected to a weight applied at the other extremity (section CD), he first defines the “absolute resistance to fracture as that offered to a longitudinal pull”. 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 “this resistance opposes the separation of the part BD lying outside the wall, from that portion lying inside”. The reasoning follows “it is clear that if the cylinder breaks, fracture will occur at the point B where the edge of the mortise acts as a fulcrum for the lever BC” [GAL 38a]. Introducing the second fundamental concept of the yield design approach, namely equilibrium, by writing the balance equation for the lever about B, Galileo finally relates the “absolute resistance of the prism BD” to its “ absolute resistance to fracture” through the ratio of the short lever arm BA/2 to the long lever arm BC.
Figure 1.2. Prism subjected to the transverse application of a weight (Galileo, Discorsi, 2nd day)
Galileo’s reasoning has been criticized, as shown in Figure 1.3, on the basis that the equilibrium of the cross-section BA is not satisfied.
Figure 1.3. Translator note [GALL 54, p. 115]
Staying within the framework of the yield design approach for the beam, the criticism amounts to pointing out that the global equilibrium equation for the horizontal resultant force has not been taken into consideration. As a matter of fact, by focusing his attention only on the moment equation for the global equilibrium of the beam, Galileo obtains a necessary condition for the beam to sustain the load in a model where the constituent material is considered at the mesoscale of the section, with its resistance determined through the longitudinal pull test, and not at a more local level such as the longitudinal fibers as the criticism in Figure 1.3 would require: this is consistent with the fact that resistance to compression is never referred.
1.1.2. Note on an application of the rules of maximum and minimum to some statical problems, relevant to architecture
The appearance of soil mechanics as an engineering science is often associated with Coulomb’s memoir [COU 73] presented to the French Academy of sciences in 1773 after Coulomb returned from his eight year period in Martinique as a lieutenant in the French military corps of engineers. This Essay was devoted to various problems that he had encountered when building the “Fort Bourbon”: stability of pillars, arches and vaults, calculation of earth pressure on retaining walls, etc. (Figure 1.4).
Figure 1.4. Figure plates in Coulomb’s Essay [COU 73]
The first guiding idea of Coulomb’s rationale in tackling these problems is making a clear distinction between the active forces, which are the prescribed loads acting on the structure under consideration, and the characteristics of resistance of the material, which set the bounds to the “coherence” forces that can be mobilized (Figure 1.5).
Figure 1.5. Defining friction and cohesion in Coulomb’s Essay [COU 73]
The second guiding idea is that the resistance forces are exerted locally along an assumed failure surface, anticipating, to a certain extent, the concept of the stress vector to be introduced some 50 years later. In the simple case of a stone column under a compressive load (Figure 1.4), Coulomb explains the principles of the analysis: the active force on the assumed fracture surface must be balanced by the “coherence” force; the fracture surface will be determined through a minimization process.
On the basis of the same principle, Coulomb’s stability analysis of a retaining wall is a fundamental landmark for the theory of yield design. Coulomb starts with the celebrated “Coulomb’s wedge” reasoning (Figures 1.4, 1.6), where he assumes the failure surface to be plane and states a condition for stability that the active forces on the assumed fracture surface Ba must be balanced by the “coherence” forces, from which he derives, through minimization and maximization processes, two bounds for the horizontal force that can be applied to CB so that the wall be stable. Because of its simplicity, this reasoning is often presented as the Coulomb analysis of the stability of a retaining wall. In fact, Coulomb, after showing how the friction along the wall could be taken into account, states that, to be complete, the analysis should look for the curve that produces the highest pressure on CB and sketches the process for this determination.
Figure 1.6. Coulomb’s wedge [COU 73]
1.1.3. Compatibility between equilibrium and resistance
It is not difficult to point out the common features of the analyses that have been briefly presented here.
– First, the concept of resistance is introduced as a mechanical characteristic of the constituent material. After having been determined through a given simple experiment, it is used in any other circumstances and sets the limits to the resisting forces that can be actually mobilized.
– Then, the idea that the resistance of a given structure – a result at the global level – can be derived from the knowledge of the resistance of its constituent material(s), which is a property at the local level.
– For this determination, the rationale is based upon the statement that equilibrium equations of the structure must be satisfied while complying with the limits imposed by the resistance of the constituent material(s). In other words, equilibrium and resistance must be mathematically compatible.
– The practical implementation of this statement is made through the choice or the assumption of some particularly crucial zone in the structure (cross-section in the first case and failure surface in the second case), where it is anticipated that compatibility between equilibrium and resistance should be checked.
As it is shown in Figure 1.3 in the case of Galileo’s analysis, it may be objected that such approaches do not take into account the behavior of the material, that is the fact that the material deforms under the forces it is subjected to. But it must be recalled that although the concept of linear elasticity was first introduced by Hooke in the 1660s, it was only in 1807 that Young’s recognized shear as an elastic deformation; three-dimensional linear elasticity itself was only really formalized in the 1820s (Navier, Cauchy and others) at the same time as the concept of the stress tensor. As noted before, the yield design approach implicitly embodies an anticipation of the concept of internal forces. This is not surprising since the intuition of internal forces is primarily linked to that of rupture being localized on surfaces or lines as observed on full-, reduced- or small-scale experiments (Figures 1.7 and 1.8).
Figure 1.7. “Slip line” pattern under a foundation in a purely cohesive material (medium-scale experiment) [HAB 84]
Figure 1.8. Bending of a reinforced plaster slab: evidence of hinge curves (M. Milicevic)
1.2. Topicality of the yield design approach
1.2.1. The Coulomb’s Essay legacy
Coulomb’s Memoir was at the origin of many methods used by engineers for the stability analyses of various types of structures. In the case of masonry vaults, the works by Méry [MER 40] and Durand-Claye [DUR 67, DUR 80] have been extensively studied by HEYMAN [HEY 66, HEY 69, HEY 72, HEY 80, HEY 82, HEY 98] and Delbecq [DEL 81, DEL 82]: it is interesting to note that they often combined Coulomb’s original reasoning with elastic arguments, thus losing its original theoretical meaning without any damage from the practical point of view.
Soil mechanics, which is sometimes considered as having found its very origin in Coulomb’s Memoir, exhibits numerous methods clearly related to it for the stability analysis of slopes, retaining walls, fills and earth dams or for the calculation of the bearing capacity of the surface foundations [BER 52, BIS 54, BØN 77, BRI 53, BU 93, CHA 07, CHE 69a, CHE 69b, CHE 70a, CHE 70b, CHE 73a, CHE 75a, CHE 75b, COU 79, JOS 80, DRU 52, GRE 49, HIL 50, HOU 82, KÖT 03, KÖT 09, LAU 11, MAN 72, MAR 05, MAR 09, MAS 99, MAT 79, MEY 51, MEY 53, MEY 63, MIC 98, MIC 09, PRA 55, REN 35, SAL 74, SAL 76, SAL 82, SAL 85, SAL 95a, SAL 95b, SAL 06, SOK 55, SOK 60, SOK 65, SAR 91, TAY 37a, TAY 37b, UKR 98], including the limit equilibrium methods and the slip line methods, which were also applied to solving metal forming problems. Finite element methods have also been developed and used extensively within this framework for applications to soil mechanics and to some related problems [AND 72, DEL 77, FRÉ 73, KAM 10, KRA 03, KRA 05, LYA 02a, LYA 02b, LYS 70, MAK 06, MAK 07, MAK 08, MAR 11, PAS 09].
Another field of application is the bearing capacity of metallic plates and reinforced concrete slabs through the yield hinges theory as developed by Johansen, Save, Massonnet and others [JOH 31, JOH 43, MAS 63, SAV 73, SAV 95, BRA 07].
Considerable attention has been devoted by Chen, Drucker and co-authors to applying the theorems of limit analysis to the determi...
