Proceedings of the First International Congress on Construction History, Madrid, 20th-24th January 2003, ed. S. Huerta, Madrid: I. Juan de Herrera, SEdHC, ETSAM, A. E. Benvenuto, COAM, F. Dragados, 2003. The masonry arch between «limit» and «elastic» analysis. A critical re-examination of Durand-Claye's method Federico Foce Danila Aita The historical evolution of the theory of the masonry arch is marked by two alternative structural philosophies corresponding to what we modernly call «limit» and «elastic» analysis. According to the first approach, global stability is the main question and the safety of the arch is guaranteed if any equilibrium condition preventing rigid mechanisms exists. This is the equilibrium approach followed in the first 18th century studies on the masonry arch and successfully developed in the first half of the 19thcentury on the basis of Coulomb' s method of maxima and minima. According to the second approach, local stress becomes the object of investigation and the safety of the arch is assured if actual stresses at each cross section are below the admissible strength of materials. This approach, which necessarily involves the complete set of equilibrium, compatibility and stress-strain equations, was the unavoidable result of the developments of structural mechanics starting from thc twenties of the 19th century, when «elasticity» and «strength» became the new passwords of the theoretical research. Nevertheless, the alternative between limit and elastic structural philosophies -stability versus stressis not so radical as it may seem at first sight (Foce and Sinopoli 200 1). As a matter of fact, it can be rationally removed by following an intermediate approach proposed by the French scholar Alfred Durand-Claye in 1867. This approach aims at verifying, as the author wrote, «s'il existe des solutions d'équilibre compatibles avec un effortlimite donné». This methodology preserves the nondeterministic character of the limit analysis and, at the same time, embodies the main aspect of the elastic analysis by imposing a restriction on the stress level. THE THEORY OF THE MASONRY ARCH IN THE TRADITION OF LIMIT ANALYSIS Aiming at discussing Durand-Claye's contribution between limit and elastic analysis, it is reasonable to provide a brief account of the main steps of the vault theory «before» and «after» it. The method proposed by the French scholar represents, in fact, the only rational procedure for conciliating the spirit of the limit approach with the new instances of the Résistance des matériaux. History shows that, before Durand-Claye' s contribution, any attempt of introducing the basic aspect of strength within the logic of collapse analysis resulted in fallacious procedures; on the other side, after it the uncontested success of the elastic methods of structural mechanics hid the profound sense of the ]imit approach applied to masonry vaulted structures, imposing the illusory research of the «unique» solution. As known, the limit analysis of rigid systems represents the way followed from the beginning of the mechanical research Oil the masonry arch. For the 896 F. Foce, D. Aita are speaking of the first 18th century studies on the subjectit was certainly the only possible, as it requires nothing but the laws of equilibrium. No matter of stress, strain and actual strength enters the problem of the arch, nor could it be otherwise. As asserted by Coulomb following Bossut, masonry is «composé de fibres roides, ou qui ne sont susceptibles ni de compression, ni d'alongement» (Coulomb 1773, 351). In short, the masonry arch is seen as an assemblage of rigid and infinitely resistant voussoirs subject to the constraint of impenetrability and to the friction resistance (cohesion of the mortar, if present, is usually considered as negligible). If the equilibrium of such an assemblage is impossible, this means that the voussoirs will take some relative movements which transform the arch into a mechanism. These movements may be both rotations around the edge of the joints or slidings along the surface of the joints. By properly combining rotations and slidings at different joints the complete spectre of the collapse modes can be derived. The main purpose of limit analysis is exact]y to assure that, for an arch of given geometry, load and friction coefficient, no collapse mechanism can occur. For at least one hundred and fifty years theoretical investigation dealt with the analysis of the collapse mechanisms and the consequent design of the structural elements (thickness of the arch, size of the piers etc.). Initially, this investigation was developed on the base of simplified or arbitrarious hypotheses about the location of the rupture joints, the types of collapse mechanisms, the role of friction. The socalled «wedge theory» formulated by La Rire in 1712 is probably the first example of this type. From a theoretical point of view, however, a real improvement of the limit analysis may be found only in the well-known Essai by Coulomb (] 773). The method of maxima and minima, introduced by Coulomb together with a rational way of quantifying the role of friction, makes the complete discussion of the collapse modes of a symmetric arch in terms of statically admissible thrusts possible. The conditions of stable equilibrium can then be expressed by disequalities between the extreme values of the crown thrust which prevent the rotational and translational movements of a voussoir. To tell the truth, Coulomb did not fuIJy develop his method, as he failed to analyse all the possible rotations of a voussoir in relation to the application time -we point of the thrust at the extrados and intrados of the crown joint. As recently shown (Foce 2002), the general treatment of the problem was given by Persy in 1825 and Navier in ] 826, almost fifty years after Coulomb, and the complete discussion of the eight collapse modes of a symmetric arch was given by Michon in 1857. The theoretical contribution of these authors can be summarized as follows. For a symmetric arch, let's consider the limit equilibrium of a generic voussoir -sub tended by the angle 8 and subject to its own weight W( 8) and the horizontal crown thrust (Figure 1)- with respect to the four rigid movements of downwards sliding, upwards sliding, rotation around the intrados edge, rotation around the extrados edge. The corresponding values of the thrust are: for the downwards sliding A = W/tang( qJ+ 8) for the upwards sliding Al = Wltang( qJ- 8) for rotation around the intrados edge for rotation around the extrados edge = W x/Yi B BI = W x/Y, where qJis the friction angle and Xi' X", Yi' Y" are the lever arms of weight and thrust from the intrados and extrados edges. ) I I , I I I :Ye I I Figure 1 According to the method of maxima and minima, with respect to 8 the greatest values of the thrust corresponding to the inwards movements of the The masonry arch between «¡¡mit" and «elastio, analysis voussoir -that is max(A) and max(B)and the smallest values corresponding to the outwards movements -that is min(A¡) and min(B¡)must be determined. Moreover, for the geometrical compatibility in the case of mechanisms with relative rotation at the crown joint, the thrust must be applied at the extrados or the intrados of the crown. Thus, conserving the notations max(B) and min(B) when the thrust is applied at the crown extrados, the new notations max(b) and min(b) must be introduced when it is applied at the crown intrados. The six extreme values of the thrust -max(A), max(B), max(b), min(A), min(B) and min(b)are associated with the six rupture joints [A], [B], lb], [AJ, [B) and [b). The relative location of the rupture joints makes it possible to deduce the complete spectre of the collapse mechanisms. As shown by Persy, Navier and Michon, there are eight collapse modes for a symmetric arch (Figure 2). According to the method of maxima and minima, none ofthem can occur if max(A, B) < min(A" B) for thrust applied at the crown extrados, and max(A, b) < min( b ,) for thrust applied at the crown intrados. A" Equilibrium becomes unstable -so that collapse can occurwhen one of the following conditions is fulfilled: 1 mode II mode III mode IV mode V mode VI mode VII mode VIII mode max(B) = min(B¡) = min(b) max(B) = min(A¡) max(B) = min(A) max(b) max(A) = min(A¡) max(A) = min(A¡) max(A) = min(b¡) = min(b¡) max(A) ELASTICITY AND STRENGTH: with with with with with with with with [B] [b] [B] [B] [A] [Al [A] [A] over [B) under [b) over [A) under [A) over [A¡] under [A) over [b) under [b) NEW PASSWORDS OF 19TH CENTURY THEORETlCAL 897 published his Le¡;:ons on the mathematical theory of elasticity, the first comprehensive book on the subject; in 1855 Saint-Venant completed his fundamental memoirs on the elastic flexure and torsion of beams. During this revolutionary development of structural mechanics in the name of «elasticity» and «strength», the research on the masonry arch followed its own way, in confirmation of the persistent distinction between construction en charpente and construction en mar;onnerie. Persy's Cuurs and Navier' s Résumé are good examples of this state of things. Both the treatises are mainly devoted to the new themes of the Résistance des matériaux and contain separate sections dealing with the stability of masonry arches and retaining walls in the tradition of limit analysis. Nevertheless, the signs of the new times were becoming more and more manifest. It is not by chance that the first clear hint towards the el as tic interpretation of the masonry arch may be found in the same Résumé des Ler;ons by Navier. As we have seen, this book contains the correct discussion of the stability conditions on the base ofCoulomb's method. After this discussion, however, Navier takes into account the actual strength of masonry and suggests an apparently small revision of the results of limit analysis. Examining the first rotational collapse mode, for instance, he interpretes the opening at the rupture joints as the incipient cracking corresponding to the triangular stress distribution on the joint surface. This interpretation is of liule consequence in quantitative terms, as it provides a slight variation of the extreme values of the crown thrusts calculated in accordance with the collapse approach. It represents, however, a radical turn from a qualitative point of view. Navier's idea, in fact, is declared in the statement that, RESEARCH The years of Persy, Navier and Michon's fundamental works are very significative not only for the theory of the masonry arch but also as they correspond to the period during which the theories of elasticity and strength of materials were being scientifically formalised. In the twenties Navier himself, Cauchy and Poisson laid the foundation of the general theory of the elastic bodies, followed some years later by Lamé, Clapeyron and Saint-Venant. In 1852 Lamé les voussoirs n' étant pas des corps parfaitement durs, on ne peut admettre que les pressions s' exercent ainsi contre des aretes. Cela n'empeche pas que l'on ne puisse calculer, avec une exactitude suffisante, l'équilibre des voutes d' apres les regles énoncées précédemment: mais il parait nécessaire d'avoi égard a l'élasticité de la matiere des voussoirs . . . Cette question serait un cas particulier d'une question plus générale. qui consiste a déterminer les effets qui se produisent dans un corps élastique de figure quelconque, soumis a ]'action de diverses forces. (Navier 1826, 164-165) 898 F. Foce. D. Aita [A¡J n mode Imode ma.x(B) = min(B¡) max(b) ID mode max(B) = min(A¡) N mode ma.x(B) = min(A¡) = min(b¡) [A] [Ad V mode ma.x(A) = min(A¡) vn mode lb¡ max(A) = min(b¡) Figure 2 The eight collapse modes of a symmetric arch (redrawn from Michon 1857) VI mode max(A) = min(A¡) VllI mode max(A) = min(b¡) The masonry arch between «limit» and «elastic» analysis As it is easy to understand from our modern viewpoint, Navier's hint towards the elastic treatment of the masonry arch requires one to abandon the logic of the collapse approach and to set up the proper methodology for solving elastic hyperstatic systems. This methodological turn was not immediate and Navier himself did not draw all the consequencies of his idea. As a matter of fact, history shows that from Navier's hint to its conscious acceptance almost fifty years passed during which the theory of the arch tried to coinciliate the traditional approach in terms of limit analysis with the news claims in the name of strength and elasticity. The first, and perhaps, most eloquent example of this contradictory state of things is Méry' s wellknown memoir of 1840. In spite of the great success of the so-called Méry' s rule -the only one still quoted in to-day' s textbooksMéry' s paper manifestly testifies this methodological incoherence. The first part of the memoir is an interesting discussion of the rotational collapse modes by means of the new graphic tool of the thrust line, introduced by Gerstner in 1833. In particular, Méry starts from Boistard' s experimental results and Audoy's theoretical analysis in order to «translate» the limit conditions of stability in terms of thrust line. This part of the memoir is intentionally developed «dans I'hypothese abstraite de matériaux infinitement résistants» and describe the «équilibre mathématique» corresponding to the collapse of the arch. In the second part of the paper Méry considers the actual strength of masonry. In this case the thrust line ne peut donc jamais atteindre l' extrémité des voussoirs, et el1e s'en éloignera d'autant plus que les matériaux serant plus mous. Par conséquent el1e sera renfermée dans des limites plus resserrées que ce11es que nous avons considéréesjusqu'a présent. (Méry 1840,64) These new limits are fixed by Méry on the basis of the very personal criterium for which the distance of the thrust line from the extrados and intrados «doit etre assez grand pour supporter les deux tiers de la pression totale», (Méryl840, 64). Méry's basic idea is summarized in the following reasonement (Figure 3): 899 lateral bands DO' and dd'] dépend de la pression et de la qualité des matériaux; 1'autre ¡the central band d'O'], dépend de la forme de la voúte. On pourra déterminer la premiére d'aprés cette condition que 1'on ne doit pas attendre des matériaux un effort permanent supérieur au dixiéme de celui qui l'écraiserait; quant a I'intervalle [d'O'], il doit offrir a la courbe des pressions assez de latitude pour qu'el1e y soit toujours renfermée. (Méry 1840,65) I I I I I I I I I I I I I I I I I I I I I I I L_-____------------- Figure 3 By this criterium, la question de ]a solidité des voútes est ramenée a celle de leur équilibre mathématique. Ainsi, il n'y aura qu'une seule courbe des pressions possible, si l'interval1e [of the central bandl est réduit au minimum nécessaire a la stabilité de la voúte; mais au contraire si, pour plus de securité. l'on rend cet interval1e un peu plus grand, il existera une infinité de positions admissibles a priori pour la courbe des pressions . . . (Méry 1840,65). Nous partageons par la pensée la longuer des joints en deux parties qui doivent étre soumis a des considérations As we can see, Méry saves the reference -also linguisticto the logic of limit analysis and improperly takes it as the basis for analysing a stable arch with finite compressive strength. This contradictory line of reasoning is a Leitmotiv of 19th century literature on the arch theory. Scheft1er, for instance, studies in detail the properties of the lines of maximum and minimum thrust under the assumption of infinitely resistant material. Dealing then with the tout a faít diíférentes; I'une [comprehending the two «Pressbarkeit des Materials», he states that in a stable 900 F. Foce, D. Aita arch the thrust line must run within a central strip whose thickness is half the arch thickness (Scheffler 1857,69). Another example from the American context is W oodbury' s treatise. Referring to tbe studies on the collapse analysis by Coulomb, Audoy and Petit, Woodbury gives a correct treatment 01' what he calls the !ine 01' «ultimate» thrust. Coming then to consider the actual strength 01' masonry, he states that for a well-proportioned arch «the curve 01' pressure should lie entirely between two other curves which divide the joints into three equal parts» (Woodbury 1858,329). Among the thrust lines which can be drawn within these two curves, he se1ects that one -called line 01' «actual» thrust- which passes through the limits already fixed; viz., at the key, 1/3 the length of the joint from the extrados, and at the reins, 1/3 of the joint from the intrados . . . We have here a perfectly distinct point of departure for a new ca1culation of the thrust of arches. (Woodbury 1858,332) For a strange way 01' reasoning, what should be good for the safety 01'the arch -that is the absence 01' crackingbecomes what actually occurs for the engineer's satisfaction. In spite 01' Woodbury's conviction 01' having discovered «an exact mechanical foundation for a theory of the actual thrust», we recognize here, once more, abad compromise between different approaches. This same compromise vitiates an anonimous Cours de ponts used in the sixties as textbook for the students 01'the École des Ponts et Chaussées at Paris. The idea that the actual thrust line must run within a central band is applied to different practical cases «d' apres la nature meme de la voúte». Thus, for a vault «a construire avec soin, dan s des conditions ordinaires», the central band will have a thickness 01' 1/3 01' the arch; in the case 01' «voútes faites avec des mortiers médiocres ou construites dans une mauvaise saisoD», the central band will be 1/4 01' the arch thickness; finally, «pour des voútes executées avec perfection» the thrust will be applied to the middle point 01' the joints at the crown and the haunch. Obviously, there is no reason for regarding these choises 01' the thrust line as the correct ones. They simply remove the statical indetermination when stable conditions 01'equilibrium are analysed. In other words, the central band may be useful for !imiting the part of the arch ring where the thrust line should preferably lie but, as clearly remarked with reference to Méry's thrust !ine, by Poncelet cette courbe elle-méme reste indéterminé, a moins de supposer fictivement la rupture de I'équilibre par rotation autour des arete s des plan s de joints limités aux nouveaux intrados et extrados, ce qui réclamerait des calculs ou tatonnements fort pénibles et peu justifiés en principe. (Poncelet 1852, 539-540) AIMS AND FEATURES OF DURAND-CLAYE'S METHOD The contradictory context above out]ined is the background in which Durand-Claye's contribution must be placed. As we have seen, this uncertainty is fundamentally connected to the meaning of the thrust line drawn in accordance with Méry's or similar methods, based on two a priori assumptions for removing the statical indetermination. Some reasonable doubts arise: for instance, do those assumptions lead to a statically admissible solution in terms of strength throughout the arch ring? Do they correspond to some type of limit condition ? Where are the weak joints of the arch? Before tackling these questions and exposing his method, Durand-Claye develops a deep discussion about the Principe de l'emploi des courbes de pression; valeur de la méthode. The point is the following: is the value of the results obtained by means of the thrust line comparable with that assured by the formulas 01' the strength of material s usually adopted for the analysis of metallic arches') To sol ve this point, says Durand-Claye, il suffit connues les arcs agir sur de se reporter un instant aux méthodes bien qui ont servi a établir les formules usitées pour métalliques: l'expérienee a montré qu'en faisant les matériaux des forces déterminées et limitées, il se produisait entre ces force s et les actions moléculaires un état d'équilibre . . . Mais dan s toutes ces expériences d'élasticité statique, on négligeait forcément l'étude du développement de ees forces moléeulaires; on constatait un résultat. Lorsque ensuite dan s les calculs d'une poutre ou d'un are métallique, on emploie ces formules qui ne sont que l' énoncé de certaines faits . . . on vérifie seulement que le systeme projeté admettra les solutions les formules; on d' équilibre auxquelles s' appliquent néglige les causes mal définies qui pouITont venir influer pendant la pose ou le décintrement sur le développement des forces moléculaires, comme elles ont pu intervenir The masonry arch between <<limit»and «elastic» analysis dans la mise en expériences des tiges ou des fils qui servaient a formuler les lois de l' élasticité statique. A vec les dimensions calculées. I'équilibre sera possible, mais d'une fa<;:on mathématique on ne peut affirmer sa nécessité, puisqu'on n'est pas parti de I'état initial du systeme. (Durand-Claye 1867,64) For stone or masonry arches the formulas of the strength of material s are abandoned in favour of the graphical tool of the thrust line. Now, continues the author, pour que le tracé de cette courbe ait quelque valeur et soit autre chose qu'une série de constructions géométriques, il convient de définir rationellement la poussée qui lui sert d' origine. Achaque valeur prise pour la poussée, achaque point d'application correspond une courbe de pression: on cherche si parmi ces courbes il s' en trouve une correspondant a une solution d'équilibre. Autrement dit, on cherche si l' ouvrage étudié avec ses charges, surcharges, profils, admet une combinaison de la poussée et des poids telle qu' en aucun point les matériaux n'atteignent la résistance limite qui leur est imposée. Ainsi recherche de la possibilité d'une solution d'équilibre, voila que! est le véritable sens de la construction et des tatonnements indiqués par M. Méry. Le résultat est le meme que pour l'arc métallique: l'ouvrage admettra au moins une solution d'équilibre . . . mais nous négligeons toujours les causes accessoires qui peuvent se développer au décintrement; de la possibilité de l'équilibre, nous concluons a la stabilité. (Durand-Claye 1867.65) 901 Which is, then, the right way for avoiding any arbitrarious hypothesis? Here is Durand-Claye's answer: Nous ne déterminerons plus, un peu au hasard, une solution quelconque d'équilibre: connaissant les profils, les charges, la résistance effective ou réduite imposée aux matériaux, nous déterminerons par des constructions géométriques toutes les solutions d'équilibre quc peut comporter une voúte. Nous ferons ainsi une application généralisée de la méthode des courbes de pression. Cette représentation complete. . . aura sur le tracé d'une courbe unique un double avantage: 10 elle permettra de suivre tous les modes de répartition qui pourront se produire tant que la voúte sera en équilibre; elle mettra en évidence l'influence de la résistance des matériaux et montrera les points réellement faibles; 2° elle pouna de plus s'appliquer également bien a tous les exemples; elle permettra par la constance des ses procédés d'effectuer une étude réellement comparative des divers types, en tenant compte des différences de résistance et non plus seulement des épaisseurs et des profils, conditions purement géométriques. (Durand-Claye 1867,66) il n' échappe a personne que le tracé habituel de ces courbes présente une sorte de vague et d'incertitude. On n'arrive a trouver une courbe d'équilibre. que seule mérite le nom de courbe de pression, que par tatonnements, en prenant arbitrairement deux points pour déterminer la courbc ou faisant varier la poussée en grandeur et en position, ce qui revient toujours a prendre au sentiment les deux conditions nécessaires pour définir la courbe. On n' a aucune idée du degré de stabilité de la voúte, puisqu'on s'arrete dés qu'une courbe d'équilibre est trouvée et que la voúte peut en comporter une infinité; l'influence de la résistance des matériaux n'est point suffisamment mise en évidence et les parties faibles du profil sont indiquées avec un certain vague. (Durand- The idea is clear. Instead of searching for a single thrust line regarding it as the «actuai» one, al! the admissible thrust lines have to be determined. Among them also the true one will be necessarily comprised in the case of a well-designed arch. This procedure overturns the deterministic logic of Méry's or similar methods (as well as the elastic methods for hyperstatic systems). Its value resides exactly in checking the existence of admissible solutions without requiring to find the true one: as Durand-Claye writes, «de la possibilité de l' équilibre, nous concluons a la stabilité». As it is easy to understand, we find here a particular form of what we modernly call the sate theorem of limit analysis. This theorem is implicit also in the stability conditions of Persy, Navier and Michon. The only difference is that now the range of admissible solutions is restricted in order to consider the actual strength of materials. In this sense DurandClaye's method is a rational mediation between limit and elastic approach. Obviously, the ultimate condition according to Durand-Claye is not the collapse condition of limit analysis, unless one takes an infinite value of the compressive strength and neglects the tensile strength. What we have said has been «translated» by Durand-Claye into an eloquent graphic construction Claye 1867, 65-66) based on the following steps: In other words, «le principe méme de J'emploi des courbes de pression suivant la méthode de M. Méry, semble donc avoir une valeur comparable a celle des théories admises dans la résistance des matériaux». However, 902 F. Foce, D. Aita Research of the admissible thrusts for the rotational and translational equilibrium of a generic voussoir Durand-Claye considers a symmetric arch and examines the ideal voussoir between the crown joint c(A, and a generic joint cd (Figure 4, left). Let's suppose that the resultant of the horizontal thrust and the weight W of the voussoir goes through the point a of cd. By varying the application point of the thrust from Coto do maintaining fixed point a, the thrusts will then be measured by horizontal segments drawn from the crown joint c{ldoto the line A(,Bo; for a coinciding with c and d, the two lines aof3o and Yo~)are similarly obtained, so that the area aof3oYrA, contains the extremes of the segments measuring the admissible thrusts for the rotationa] equilibrium of the voussoir. Consider now the normal component of the resultant on the joint cd. For values of the thrust corresponding to Ao and Bo' it is measured by segments bounded by the joint cd and the points A and B, respectiveIy; in a similar way, the pair of points ao' 130and YrJ'80 of the area aof3oYo8oat the crown joint correspond to the pairs of points a, 13 and To keep account of the translational equilibrium along cd, it is sufficient to draw at point a the friction cone, defined by the friction angle qJ. Let' s take the weight W applied at point {l and consider the two horizontal thrusts which give a resultant coinciding with the boundaries of the cone. These thrusts define two vertical lines at the crown between which the extremes of the segments representing the admissible thrusts for the transational equilibrium are comprised. To assure both the rotational and translational equilibrium at the joint cd, the admissible thrusts must be comprised within the intersection of the area aof3oYrJ80with the two vertical lines. By repeating this procedure for every joint Durand-Claye finds the area within which the extremes of the statically admissible thrusts must be included. Tf this area shrinks lO a single point, equilibrium becomes unstab]e and a collapse mechanism can occur. Up to this phase, the procedure graphicaIly translates the results already obtained in analytical terms by Persy, Navier and Michon. this Research of the admissible thrusts with respect to the strength of material at a generic joint area contains the extremes of the segments perpendicular to cd, which measure the normal components corresponding to the admissible thrusts for the rotational equilibrium. The innovation of Durand-Claye's method concerns the introduction of finite valucs of the compressive and shear strength. Let 0"" be the admissible value of y, 8 of the area af3y8 at the joint cd; therefore, d J~-J1I r 80 Bo Po Co ro Figure 4 Durand-Claye's method (redrawn from Durand-Claye 1867) 903 The masonry arch between «limit» and «elastic» analysis the compressive strength, N the normal force at the joint cd and e the eccentricity of its application point. Assuming the usual linear law for the normal stress distribution, the force N for which the value 0* is reached at the most compressed edge of a rectangular section of height h and width bis: (1) N = (j" * bh' for h + 61e1 (2) = N (j" ~ * b(~ - lel) for lels~ ~ 6 s lels 6 ~ 2 Figure 5 The «area 01' stability» 01' Durand-Claye's method (from Durand-Claye 1867) Equation (1) represents two symmetric hyperboles having as asymptotes the joint line and the straight lines e = :!: h/6, respectively. Equation (2) represents = :!: h/2 and tangent to the hyperboles for e = :!: h/6 (Figure 4, Research right). Thus, the area cúXI contain the admissible normal components with respect to the compressive strength. A similar area eowodo can be drawn for the admissible thrusts at the crown joint. With regard to the shear component, Durand-Claye represents the admissible shear force S on ed by means of the segments at, at'o Applying the weight W to a and drawing the perpendiculars to ed from t and t', two segments can be obtained by intersecting these perpendiculars with the horizontal straight line from the extremity of the weight. These segments define two vertical straight lines at the crown between which the admissible thrusts with respect to the shear force on ed are contained. Imagine now to repeat the preceding constructions for every joint. This operation defines, if it actually exists, the «area of stability» common to all the areas rspq. For a well-designed arch, the area of stability is a curvilinear quadrilateral of the type r¡sp¡q¡, in the sense that it corresponds to the joints c¡d¡ and e/r Its form and dimension immediately provide useful information about the safety of the arch, the position of the admissible thrust lines and the location of the most compressed joints. When the area of stability shrinks to a point, the limit condition is attained and a unique admissible thrust line exists. For example, if two straight lines with nil value for e Research equilibrium of the admissible and strength thrusts with respect at a generic to equilibrium the area of the admissible and strength r¡sp¡q¡ is reduced thrusts with respect throughout to the point to the arch r¡ = p¡' this means that the limit compression is reachetl) at the crown extrados, at the intrados of the joint ed"and at , the extrados of the joint cd. ' j I joint Consider once again the joint cd and examine the intersection p(j"(jJX of the areas af3y8 and ewd (Figure 5). The area p(j"(jJXhas its reciprocal in the area poo;,m;,x" at the crown joint. The intersection of this latter with the area cowA, provides the area rspq which, in its turn, defines the reciprocal area RSPQ at the joint cd. We conclude, then, that the admissible thrusts with respect to equilibrium and strength at the joint ed are represented by horizontal segments whose extremes are contained within the area rspq. Definition of the safety coefficient Let r¡sp¡q¡ be the area of stability for the admissible compressive strength (j"*.By decreasing this strength, the are a r¡sPjq¡ will correspondingly de crease and shrink to a point for a certain ultimate value (j". The ratio 0*/0"" provides the safety coefficient of th~ real arch according lo Durand-Claye's method. 904 Simplifications the method F. Foce, D. Aita and developments of The preceding construction of the area of stability has been simplified by Durand-Claye in his second memoir of 1868. This simplification is based on the fact that, by referring the line cmd to the crown joint, two families of hyperboles --caJled deforméescan be drawn, whose intersection with the line camado at the crown immediately gives the area of stability rspq. In particular, Durand-Claye applies this simplification to the case of metaJlic arches characterised by equal tensile and compressive strength. In this case, the admissible normal components at the generic joint cd are bounded by two hyperboles given by equation (1) for any value of the eccentricity. The deformées of these hyperboles at the crown are again two equilateral hyperboles having as asymptotes the joint line and the horizontal straight lines through the points of the middle third of the joint cd. In his third paper of 1878 Durand-Claye extends the method to masonry piers and domes. In the last paper of 1880 he studies Sto Peter's dome and finds that, for 0*= 10 Kg/cm2, an area of stability exists. This area shrinks to a point for 0;, = 5 Kglcm2, so that the safety coefficient for the chosen o*is 2. Besides Durand-Claye's memoirs, we may quote here the contributions of Cunq (1880), who suggested a simple way for drawing the deformées at the crown joint, and Ceradini (1873), who extended the method to non-symmetric load condition. «NEW THEORY» VERSUS «OLD METHODS»: Durand-Claye' s work is cited. In this connection, Résal expresses a particularly eulogistic opinion: Remarquons que. . . son raisonnement est absolument rigoureux et s'appuie exclusivement sur les principes de la mécanique rationelle et sur la loi du trapeze, considerée comme base de la résistance des matériaux. A ce point de vue, il se sépare completement des méthodes indiquées dans les articles précédents, méthodes dont la base fondamentale est toujours une hypothese plus ou moins discutable. (RésaI1887, 98) After these «old methods», Résal presents the «méthode nouveJle pour la détermination de la courbe des pressions». This «new method» is nothing but tbe theory ofthe elastic hyperstatic arch applied to masonry. The presentation ofthe «new method» reveals, however, some embarassment: «Nous avons réduit a deux le nombre des formules a employer pour déterminer la courbe des pressions . . . et nous avons cherché a rendre cette méthode suffisamment simple pour que son application ne parOt pas notarnment plus compliquée et plus laborieuse que la méthode de M. Méry, a laqueJle les constructeurs sont habitués» (Résal 1887, 103). Few years later this embarassement has totaJly desappeared, as Résal himselftestifies in his Cours de ponts of 1896. By this time the elastic appraach is considered as law, so that nous nous abstiendrons. . . d'exposer et méme de signaler les autres méthodes . . . ElIes ne peuvent inspirer de confiance, et comme elles exigent en général des constructions géométriques tres compliquées ou des calculs laborieux, el1es ne semblent pas étre sorties du domaine de la théorie pour entrer dans la pratique des constructions». (Résal 1896, 190) WHAT ABOUT DURAND-CLAYE? A glance at the historical evolution of the arch theory after Durand-Claye' s contribution can help in answering this question. To this aim we take a hint fram a brief report written by lean Résal in his treatise on masonry bridges (Résal, 1887). Résal distinguishes three categories of «old methods»: 1) Méthodes des courbes des pressions hypothétiques, among which the names of Méry, Dupuit, Scheffler, Laterrade are quoted; 2) Recherche du profU théorique des voutes le plus advantageux au point de vue de la stabilité, with contributions by Yvon ViJlarceau, Denfert-Rochereau and Saint-Guilhem; 3) Méthodes des aires de stabilité, among which And what about Durand-Clay's method which had been so highly appreciated just a few years before? In the panorama outlined by Résal, it no longer has a reason to existo Résal's words, in effect, resound the methodological position of structural engineering at the end of the 19thcentury. For what reason -he seems to ask- is it necessary to look for aJl of the admissible thrust lines when it is possible to determine the only one which actually occurs. Why renounce to the economy of thought expressed in the two compatibility equations useful for univocaJly deterrnining the thrust at the crown and its point of application, when the altemative is that of resorting to exhausting graphic constructions, which leave the solution indeterminate? 905 The masonry arch between «¡¡mit» and «elastic» analysis A contemparary of Résal could not have opposed real objections against these questions. If the problem is that of «designing» a new masonry arch it is, in fact, reasonable to operate in arder that the conditions for the applicability of the elastic analysis are likely fulfilled. Today, however, the perspective has radically overturned. The task of the structural engineer is no longer that of «designing» new masonry arches, but that of «assessing» structures which bear the signs of centuries of life. Here, then, is the question: which approach for a theory of the masonryarch'J Jacques Heyman has already provided a convincing reply, fully recuperating the structural philosophy of limit analysis since his fundamental study on the stone skeleton of 1966. From the modern viewpoint, the contraposition between <<new theory» and «old methods» has no longer the meaning of ~ lA . «right» versus «wrong». As George Fillmore Swain already wrote in 1927 anticipating Heyman' s position, the modern structural engineer works with the conviction that «simpler and less mathematical methods will give results quite accurate enough for practice, and perhaps as accurate as the elastic theory» (Swain 1927, 425). Among these methods there is also Durand-Claye's forgotten procedure. In effect, whenever the «deterministic» attitude of the elastic approach shows itself illusory, there is nothing but to embrace the alternative logic of limit analysis, eventually modified according to Durand-Claye. ApPLICATlONS METHOD The [ast remarks about the actuality of the methods based on the limit approach suggest a reappraisal of . Case of stable equilibrium K=l.] ~=OA8J f ! ,0 N [daN] ~ OF DURAND-CLAYE'S ~ 8=54° N' 8 54° 8=90° 18l,,'m 1 ft540) I .. Al(54") Figure 6 Al (90') . A(90') ~~ 906 F. Foce, D. Aita Durand-Claye's procedure in arder to consider any tensile and compressive strength. Without presenting the mathematics of the problem, we give here some results far [he simple case of a semicircular arch of constant thickness subject to its own weight (Sinopoli, Corradi and Foce 1997). The first series of results refers to the collapse analysis under the assumptions of infinite compressive strength (lo) = 8) and nil tensile strength (ur = O). In this case the boundaries of the are a c()wOdO degenerates in two straight Jines perpendicular to the joint codo (the same happens at each joint). Figure 6 shows the area of stabiJity for a case of stable equilibrium corresponding to the ratio K = 1.3 between the extrados and intrados radius and for friction coefficient Far K = 1.1136 J.l = 0.483. the area of stabiJity shrinks to point e (Figure 7, left) corresponding to the only admissible thrust max( B) = mine B /) and the first collapse mode can occur (Figure 7, right). By decreasing maintaining K the friction coefficient to J.l = 0.395 = a1so straight 1.1136, the line corresponding to [he admissible thrust for the upwards translational equilibrium on the springing joint (that is min(A/) ) goes through the point e (Figure 8, 1eft) and the third collapse mode can occur (Figure 8, right). Finally, when the range of admissible thrusts for the upwards and downwards translational equilibrium shrinks to a single value max(A) = min(A) (Figure 1eft), the fifth collapse mode can occur far K and J.l = 0.309 (Figure 9, = 1.2205 9, right). The second series of results refers to finite values of compressive and tensile strength. The ultimate compressive strength 0',,(10 is plotted in Figure 10 as a function of K and for three values of the tensile strength, that is O',= O, O',= 10',11l O and ur = luJ From the curves O',,(K) an immediate estimation of the safety coefficient of a real semicircular arch can be deduced by forming the ratio u*IO',,(K). max (B) = min (El) I f-' 8 1 Callapse made: max (B) = min (B 1) K=1.1I36 ¡'¡>O.395 f\) o ~ N [daN] 8 ~ 8 =0° >-' 8 8=54° e [cm] 8 =90° Figure 7 907 The masonry arch betwecn «limil» and «elastic» analysis max(B)=mín(B,) I~ -1 mm(A¡1 ¡ ]][ 8 Callapsemade max (B) = min (A 1) K=L1136 [1<0.395 N [daN] 8 -,-, 8 =0° 8=54° e =90° e [cm] 1A(5~') ~ A¡(YO') A¡(5~') ~ Figure 8 max (A) = mm (A 1) 1+ 1 V Callapsemade max (A) = min (Al) I bo N [daN] ~ ~o " G o ~ o K=)2205 [1=0309 gI 8=00 8=54° N 8 e [cm] A(2Y') . Figure9 A¡(2Y') ~ I ~ 908 F. Foce, D. Aita la,lldaNimm~J , " h Figure 10 NOTES 1. Incidentally, we observe that Méry' s criterium does not correspond to the so-called middle third rule, as usually asserted in the literature. REFERENCE LIST Ceradini, C. 1873. Sull'equilibrio delle volteo Giornale di scienze naturali ed economiche di Palermo, 9: 9-27. Coulomb, C. A. 1776. Essai sur une application des regle s de maximis et minimis a quelques problemes de statique, relatifs a l'architecture. Mémoires de mathématique & de physique, présentés a l'Académie Royale des Sciences par divers savans, 7 (1773): 343-382. Cours de ponts a l'École des Ponts et Chaussées, Biblioteca de la Escuela de Caminos, Madrid, Manuscript 5C 202. Cunq. 1880. Note sur la vérification de la stabilité des voutes. Annales des Ponts et Chaussées, 20 : 145-156. Résal, 1., Degrand, J. 1887. Ponts en mafonnerie. 1, Stabilité des voútes. Paris. Résal, J. 1896. Cours de ponts. Paris. Durand-Claye, A. 1867. Note sur la vérification de la stabilité des voutes en ma~onnerie et sur l' emploi des courbes de pression. Annales des Ponts et Chaussées, 63-93. 13: Durand-Claye, A. 1868. Note sur la vérification de la stabilité des arcs métalliques et sur l' emploi des courbes ] 5: de pression. Annales des Ponts et Chaussées, 109-144. Durand-Claye, A. 1878. Stabilité des voútes. Paris. Durand-Claye, A. 1880. Vérification de la stabilité des voutes et des ares. Appl ications aux voutes sphériques. Annales des Ponts et Chaussées, 19: 416-440. Foce, F. and Sinopoli, A. 200 1. Stability and strength of materials for static analysis 01' masonry arches: DurandClaye's method, in Arch 'Ol-Third International Corif'erence on Arch Bridge.I', edited by C. Abdunur, 437-443. Paris: Presses de I'École NationaJe des Ponts et Chaussées. Foce, F. 2002. Sulla teoria dell'arco murario. Una rilettura storico-critica, in A. Becchi, F. Foce, Degli archi e delle volteo Arte del costruire tra meccanica e stereotomia. Venezia: Marsilio. Méry, E. ] 840. Sur l' équilibre des voutes en berceau. Annales des Ponts et Chaussées, 19: 50-70. Michon, P. F. 1857. Instruction sur la stabilité des voútes et des murs de revetement, Metz: Lithographie de I'École d' Application. Navier, C.-L.-M.-H. 1826. Résumé des lefons données a l'École des Ponts et Chaussées sur l' application de la mécanique a l'établissement des constructions et des machines. Paris: Didot. Persy, N. ]825. Cours sur la stabilité des constructions, a l'usage des éleves de l'école roya le de l'Artillerie et du Génie. Metz: Lithographie de l'École Royale de l' Artillerie et du Génie. Poncelet, J.-V. ]852. Examen critique et historique des principales théories concernant I'équilibre des voutes. Comptes rendus, 35: 494-502; 531-540: 577-587. Scheffler, H. 1857 Theorie der Gewolbe, Futtermauern und eisernen Brücken. Braunschweig, Verlag der Schulbuchhandlung. Sinopoli, A., M. Corradi and F. Foce. 1997. A modern formulation for pre-elastic theories on masonry arches, Journal of engineering mechanics (ASCE), 123: 204-213. Swain, G. F. 1927. Structural Engineering. Vol. III Stresses, ¡iYaphical statics and masonry. New York McGraw-Hill. Woodbury, D. P. 1858. Treatise on the various elements of stability in the well-proportioned arch, with numerous tables of the ultimate and actual thrust. New York: Van N ostrand.