|
Article Excerpt
Abstract: Traditional unreinforced masonry architecture has disappeared from new building activity in the western world. Nevertheless, the architectural heritage of masonry must be preserved, and this involves structural analysis. The classical theory of structures does not apply well to such heterogeneous structures with unknown boundary conditions. Nevertheless, there exists a theory of masonry structures based in simple assumptions about the material: good compressive strength, almost no tensile strength and a constructive care to avoid sliding failure. The theory was born at the end of the 17th Century, developed during the 18th and was applied in the 19th Century. It was abandoned and eventually forgotten at the beginning of the 20th Century. After half a century, in the 1960s, Heyman incorporated the old theory within the frame of modern limit analysis with its implicit treasure of critical observation and experience. The safe theorem permits using equilibrium equations and simple material statements cited. No affirmation about boundary conditions, impossible to know and essentially changing, is made (other than the usual about strength and small displacements). In the first part of the paper, an outline of the old theory is summarised and discussed. In the second part, the main ideas and concepts of limit analysis of masonry structures are discussed.
Keywords: Arches, Architectural heritage, Architectural history, Building materials, Domes, Elastic analysis, Finite element method, Graphical equilibrium analysis, History, Limit analysis, Masonry architecture, Masonry structures, Safety, Stability, Statics, Structural behaviour, Theory of structures, Vaults
Introduction
Masonry was the main building material in the western world until the beginning of the 20th Century. Masonry vaults are no longer built, but we have to preserve our architectural heritage. In many cases, it is a matter of cosmetic maintenance. However, sometimes, the building presents, or seems to present, structural problems. It may happen, also, that the loads have increased and the safety of the structure should be checked for the new use. In these last situations, a structural analysis is needed.
The problem is that masonry structures are essentially different from modern structures of steel, reinforced concrete or laminated wood. The usual structural theory of framed, trussed or shell structures made of reinforced concrete or steel is of no use to study masonry architecture. In Figure 1a, a constructive section of a medieval building is represented. In the first place, no such linear elements as columns and beams are seen. The structural elements are two- or three-dimensional, but not linear, as in frame or truss theories, or thin as in the usual shell theory.
Let us consider now the material. Though from the outside, regular ashlar masonry would be seen, the internal structure is, in fact, much more irregular and complex. The wall, for example, consists of two skins of ashlar masonry and a nucleus made of rubble with some kind of mortar. The masonry material is, in itself, a structure. How to apply, then, the usual material assumptions: continuity, isotropy, definite elastic or non-elastic properties? Figure 1b shows a few of the usual masonry types. If we try to define, for example, an elastic modulus: Where? At the outer skin, in the inside skin, at the nucleus? Besides, it is a fact that masonry buildings present cracks, and these cracks run often through the whole thickness of walls and vaults. They may have been closed at the surface (by some plasterwork), but the crack remain inside, hidden and unknown. Finally, speaking of boundary conditions, it is well known that the foundations of most masonry buildings are superficial and present noticeable settlements: they are far from the rigid foundations of the structural textbooks. They are unknown, and essentially unknowable, as slight changes of the soil conditions, the sudden action of loads (e.g., storms or earthquakes) could alter the response to the loads.
[FIGURE 1 OMITTED]
The above remarks do not imply that masonry structures cannot be designed or analyzed. In fact, some kind of design was needed to build the Pantheon, Hagia Sophia, and the great Gothic cathedrals. It was not an amateur task, nor a matter of blind chance. There was, indeed, a traditional scientia of structures, based mainly in the geometrical study of safety. This pre-scientific theory has been discussed elsewhere (Huerta, 2004), but our concern is, now, how to analyze these buildings within the frame of the scientific theory of structures, based on the laws of statics and the strength of materials. We have briefly discussed the difficulties, even the impossibility, of applying directly the usual assumptions of modern common structural analysis. Another approach is needed.
The Theory of Masonry Structures
Fortunately, there exists a scientific theory of masonry structures. It was born at the end of the 17th Century, developed during the 18th and 19th Centuries, and was forgotten, and eventually disappeared from structural textbooks and curricula in the mid-20th Century. The elastic theory swept away the then so-called old theory of arches and vaults. Indeed, the old theory of vaults was of no use to design steel frames, suspended bridges, lattice iron arches, etc. However, this theory was used successfully for arch and bridge design all through the 19th Century and was applied, again successfully, to the study of masonry buildings, particularly to check the stability of the new neo-Gothic masonry buildings at the end of the 19th and during the 20th Century. The old masonry arch theory began to be rediscovered in the 1930s when engineers were forced to give concrete answers about the strength capacity of medieval bridges for new loads. Eventually, in the 1960s, the old theory of masonry structures was incorporated within the modern frame of Limit Analysis, mainly thanks to the work of Jacques Heyman of Cambridge (1966 ff). Heyman had participated actively in the developing of the plastic theory of steel frames within the famous team of John Baker in the 1940s and 1950s. He realized, crucially, that masonry was in fact a ductile material. Though the individual stones may be brittle, he saw clearly that the apparent shortcomings (from the classical elastic point of view) of the old theory, where not such, and that the correctness and the success of the old approach could be explained within the broader frame of modern Limit Analysis of masonry structures.
We should use this theoretical frame if we want to understand the behaviour of masonry buildings. However, the coming of computers, the exponential cheapening of calculation costs, the modern multi-purpose Finite Element or Discrete Element packages, with the implicit or explicit promise of being the universal method to be used for any kind of structure, have obscured the actual situation and the potentialities of the new theory of masonry.
Indeed, in the last decade, the number of papers and books concerned with the structural analysis of historical masonry architecture has increased exponentially. The matter which some twenty years ago interested only architects and engineers involved in restoration works has become the subject of numerous scientific and academic papers, of research projects, dissertations, congresses and seminars. The work in instrumentation and analytical aspects has been, also, enormous. All the machinery of the modern computational methods and the latest analytical techniques has been addressed to the study of old masonry buildings, many of them more than a millennia old. However, it does not seem that all this display has led to a better understanding of masonry structures. Quite often masterpieces of architecture are suffering invasive interventions with insertion of reinforced concrete, steel, glass fibre, Kevlar, epoxy, and so on. Almost any building under study is considered in danger and scrutinized in detail and, almost invariably, followed the usual insertion of modern reinforcement of one kind or another. No doubt, sometimes a historical building requires a structural consolidation, but what is remarkable is the extensive use of such modern techniques.
Within a historical perspective, it is a recent phenomenon. Until, say, 1900 buildings were restored and consolidated using the same materials and techniques with which they were built, following the tradition of masonry construction. Old buildings were looked at with confidence as they have withstood many proofs during their lives. There is nothing against the use of new materials and techniques when needed, but it may be that this extensive use is the expression not strictly of the bad state of the buildings, but of the bad state of the understanding of old construction. It often occurs that interventions tend to convert masonry to what it is not: a monolithic structure that can withstand tensions as well as compressions.
The present paper is intended to examine this crucial matter and to open a discussion and reflection in the way structural analysis and consolidation of old buildings is made today. The discussion will be centred in the theory and not in the computational techniques. Being the matter of historical construction, it may be justified to use a historical approach to obtain a more ample view.
In the first part of the paper, an outline of the "old" theory of masonry structures will be given. In the second part, the ideas and concepts of modern Limit Analysis will be exposed, and its main corollary, the equilibrium approach, will be discussed.
The Theory of the Arch: An Historical Outline
The main lines of the development of arch theory are well known. The works of Poncelet (1852) and Winkler (187-980) give a good review of the early theories from the 17th to the mid-19th century. Those theories refer to masonry arches (often called rigid in 19th Century engineering manuals). The theory of the elastic arch was developed during the 19th Century and was applied first to iron and wooden arches. After the 1880s, it was applied to any kind of arch. A detailed study of the history of the elastic theory may be found in Mairle (1933) and a good review of the (1941), Timoshenko (1953) and Charlton (1982). Heyman (1972, 1998b) has studied the evolution of masonry arch theory within the frame of limit analysis, and has placed it rigorously within the general frame of the modern theory of structures (Heyman, 1966, 1995a, 2008). Kurrer (1997) covered both the history of rigid (masonry) and elastic theories, and a overall view may be found in his books on the history of structural theory (Kurrer, 2002, 2008). Foce (2002) has contributed another historical review and, more important, has compiled a comprehensive bibliography of the primary sources. Finally, the present author has contributed jointly with Kurrer another historical outline of masonry arch and vault theory (Huerta & Kurrer, 2008).
England: The Equilibration Theory
It was Robert Hooke who, circa 1670 (see Hooke, 1676) posed the problem of the arch (Heyman 1972): What is the ideal form of the arch, and how large is the thrust of the arch against the abutments? He gave the solution in the form of an anagram in 1675 (cf Hooke, 1676), which, deciphered, says: As hangs the flexible line, so but inverted will stand the rigid arch (Figure 2a). The statics of arches and hanging chords is the same. This was Hooke's genial idea. The mathematician Gregory (1697), in an essay on the catenaria, gave the same statement as Hooke and added a fundamental remark: None but the catenary is the figure of a true legitimate arch, or fornix. Moreover, when an arch of any other figure is supported, it is because in its thickness some catenaria is included. Hooke's analysis gave a simple approach to understand and calculate masonry arches, and Gregory's comment freed the form of the arch from that of the catenary. However, Gregory's fundamental statement was ignored.
Following Hooke, English engineers tried to design the arch of the same form of the corresponding hanging chain, to build an equilibrated arch. The matter was tackled mathematically by many mathematicians and engineers in the second half of the 18th Century, for example, by Emerson (1754) and Hutton (1772). The arch of uniform thickness, which sustains only its own weight, is a mere theoretical problem. In a real arch, the load is defined by the lines of intrados and extrados. There were two basic problems: first, to find the curve of extrados for a given intrados, and, second, to find the curve of intrados for a given extrados (see Figure 2b). The solution may be obtained mathematically, but Robison (1801; Figure 2c) proposed a way to obtain the curve experimentally.
The physical interpretation of the equilibrated arch is a series of smooth voussoirs with the joints always normal to the curve of intrados. The approach lead to a certain fixed form for the transmission of the thrusts: the line of intrados. The theory gave no information about the thickness of the arch and did not explain common phenomena as the cracking of arches.
In fact, the brilliant analysis of Hooke and the crucial statement of Gregory did not lead to concrete methods for the analysis or design of real arches. Mathematicians and engineers loved the problems posed by the theory, but the incapacity of the theory to afford concrete answers was made manifest in 1801, when the Select Committee, composed of the best English engineers, scientists and practitioners, was unable to ascertain the feasibility of Telford's design for a great iron bridge over the Thames in London (see Figure 8). Quoting Peacock (1855): "The answers which were given were singularly humiliating to the pride of philosophy: they were not only altogether at variance with each other, but in very instance incomplete and unsatisfactory" (pp 422-423).
France: The Joint of Rupture Theory
In France, the first contribution to arch theory is contained in La Hire's Traite de mecanique, published in 1695. The problem he posed was what load should sustain a semicircular arch made of frictionless voussoirs to be in equilibrium. To solve the problem, he used the polygon of forces, and it turned out that for the arch to be stable, the load at the springings must be infinite. La Hire knew that, in practice, friction would prevent the sliding, but since then it was considered a good practice to increase the thickness of the arch from the crown to the springings.
[FIGURE 2 OMITTED]
The main contribution of La Hire (1712) was published almost two decades later. The work was not directed to the study of the form of the arch or of the load, but to obtain its thrust in order to calculate the depth of the abutments. La Hire noted that in a collapsed arch or barrel vault, the inferior part remains united to the abutment, marking the joint of rupture of the arch or barrel vault. The thrust must pass through this point and be tangent to the intrados. Once located, the point of rupture, the calculation of the thrust follows easily, establishing the equilibrium of the upper part (Figure 3a), and the depth of the...
|
