Designing by Geometry. Rankine's Theorems of Transformation of Structures.

William John Macquorn Rankine (1820-1872) was one of the main figures in establishing engineering science in the second half of the 19th. Century. His Manual of Applied Mechanics (1858) gathers most of his contributions to strength of materials and structural theory. A few additions are to be found in his Manual of Civil Engineering (1862). The book is based in his Lectures on Engineering delivered in the Glasgow University, and formed part of his intention of converting engineering science in a university degree (Channell 1982, Buchanan 1985). Both in plan and in content the book shows and enormous rigour and originality. It is difficult to read. As remarked by Timoshenko (1953, 198): "In his work Rankine prefers to treat each problem first in its most general form and only later does he consider various particular cases which may be of some practical interest. Rankine's adoption of this method of writing makes his books difficult to read, and they demand considerable concentration of the reader." Besides, Rankine does not repeat any demonstration or formula, and sometimes the reader must trace back the complete development through four or five previous paragraphs. The method is that of a mathematician. However, the Manual had 21 editions (the last in 1921) an exerted a considerable influence both in England and America. In this article we will concentrate only in one of the more originals contributions of Rankine in the field of structural theory, his Theorems of Transformation of Structures. These theorems have deserved no attention either to his contemporaries or to modern historians of structural theory. It appears that the only exception is Timoshenko (1953,198-200) who cited the general statement and described briefly its applications to arches. The present author has studied the application of the Theorems to masonry structures (Huerta and Aroca 1989; Huerta 1990, 2004, 2007). Rankine discovered the Theorems during the preparation of his Lectures for his Chair of Engineering in the University of Glasgow . He considered it very important, as he published it in a short note communicated to the Royal Society in 1856 (Rankine 1856). He included it, also, in his article "Mechanics (applied)" for the 8th edition of the Encyclopaedia Britannica (Rankine 1857). Eventually, the Theorems were incoroporated in the Manual of applied mechanics and applied to frames, cables, rib arches and masonry structures. The theorems were also included in his Manual of civil engineering (1862), generally in a shortened way, but with some additions

The geometry and construction of Byzantine vaults: the fundamental contribution of Auguste Choisy

In 1883 Auguste Choisy published his book L=art de bâtir chez les Byzantins. In it he explained, for the first time, all the details of the geometry and construction of byzantine vaults. The main source was the direct study of the monuments, interpreting his observations in the light of traditional vaulting techniques. He is explicit about this: *ma seule ressource était d'interroger les monuments eux mêmes, ou mieux encore de rapprocher les uns des autres les faits anciens et les traditions contemporaines+ (Choisy 1883, 3). Choisy concentrated his attention on the vaults, as he was convinced that the vault governs the whole architectural system: *Toutes les circonstances de la construction découlent ainsi de la nature de la voûte byzantine; et j'ai cru qu'il convenait de ranger les faits autour de cet élément fondamental du système+ (4). The other fundamental principle is the economy of construction, as the vaults *. . .s'y subordonnent dans l'économie générale des édifices+. The observations were made during a six month mission of the Adminiatration des des Ponts et Chaussées the year 1875 (Mandoul 2008, 29). The next year he published a *Note sur la construction des voûtes sans cintrage pendant la période byzantine+ (Choisy 1876), were he resumed the main results concerning the technique of vaulting without centring. The book had an enormous impact on contemporary historians of byzantine architecture. It was cited and praised by the new light it threw to the constructive aspects, for its clarity and rigour of exposition, and for their superb plates. Eventually, his theories were incorporated in the manuals and histories of Byzantine architecture. The book of Choisy concentrated on *l=art de bâtir+. The interest on the technical aspects of architecture almost disappeared after the First World War, maybe due to the coming of the modern architecture and the new materials (iron, steel and reinforced concrete). As a concequence Choisy=s works on *l=art de bâtir+ were almost systematically ignored. The first specifical study of Byzantine construction after the Second World Ward was written by Ward-Perkins (1958) and it has been considered, since then, the standard reference for Byzantine construction. Ward-Perkins ignore the work of Choisy making a passing criticism of his geometrical theories of Byzantine vaults. However, the detailed description of wall construction made by Ward-perkins coincides pretty well with that of Choisy (7-13). He apparently was unaware that the whole theory of Byzantine vaulting without theory centring is Choisy=s. Besides, he attributes to Giovanonni the detailed description of the use brick ribs in vaults construction. In all, it appears that Ward-Perkins did not read carefully Choisy=s book on Byzantine construction nor was familiar with the history of vault construction. The consquence was that subsequent authors didn=t take seriously Choisy=s work or simply ignored. Sanpaolesi (1971) in a work with the suggestive title *Strutture a cupola autoportanti+ simply ignore him. To Mango (1975), author of one of the standard manuals on Byzantine architecture, Choisy is superseded; Krautheimer (1984) did not consider Choisy in treating, summarily, the vaulting problems. Robert Ousterhout author of a book on the Master Builders of Byzantium (1998) considers Choisy *outdated+, being *more than a century old+. Even in detailed archeological studies of vaulted structures his work is ignored (Deichmann 1979). There are some exceptions in specialised studies on vault construction: Besenval (1984), Cejka (1978) and Storz (1994). It must be said from the beginning, that Choisy=s L=art de bâtir chez les Byzantins is still the best source for anyone interested in understanding the geometry, construction and structural behaviour of Byzantine vaulted buildings. In what follows, we will try to demonstrate that this assertion is true

Galileo was wrong: The geometrical design of masonry arches

Since antiquity master builders have used always simple geometrical rules for designing arches. Typically, for a certain form, the thickness is a fraction of the span. This is a proportional design independent of the scale: the same ratio thickness/span applies for spans of 10 m or 100 m. The same kind of rules was also used for more complex problems, like the design of a buttress for a spatial cross-vault. Galileo attacked this kind of proportional design in his Dialogues. He stated the so-called square-cube law: internal stresses grow linearly with scale and therefore the elements of the structures must become thicker in proportion. This law has been accepted many times uncritically for engineering historians, who have considered the traditional geometrical design as unscientific and incorrect. In fact, Galileo’s law applies only to strength problems. Stability problems, such as the masonry arch problem, are governed by geometry. Therefore, Galileo was wrong in applying his reasoning to masonry buildings

Technical Challenges in the Construction of Gothic Vaults: The Gothic Theory of Structural Design

The construction of a Gothic vault implied the solution of several technical challenges. The literature on Gothic vault construction is quite large and its growth continues steadily. The main challenge of any structure is that, during and after construction, it must be "safe", that is, it must not collapse. Indeed, it must be amply safe, able to support different loads for long periods of time. Masonry architecture has shown its structural safety for centuries or millennia. The Pantheon of Rome stands today after almost 2,000 years without having needed any structural reinforcement (of course, the survival of any building implies continuous maintenance) . Hagia Sophia in Istanbul, finished in the 6th century AD, has withstood not only the dead loads but also many severe earthquakes . Finally, the Gothic cathedrals, with their appearance of weakness, are• more than a half millennium old. The question arises of what the source of this amazing strength is and how the illiterate master masons were able to design such daring and safe structures . This question is usually evaded in manuals of Gothic architecture. This is quite surprising, the structure being a fundamental part of Gothic buildings. The present article aims to give such an explanation, which has been studied in detail elsewhere. In the first part, the Gothic design methods "V ill be discussed. In the second part, the validity of these methods wi11 be verified within the frame of the modern theory of masonry structures . References have been reduced to a minimum to make the text simpler and more direct

The medieval ‘scientia' of structures: the rules of Rodrigo Gil de Hontañón

Medieval builders didn't have a scientific structural theory, however gothic cathedrals were not build without a theory. Gothic masters had a ‘scientia', a body of knowledge which permitted the safe design of their buildings. The nature of this theory has not only a historical or erudite interest; perhaps something could be learned from the true masters of masonry architecture. Literary sources from the gothic period are scarce; in almost all we find structural rules to design the principal structural elements: walls, vaults (ribs and keystones) and, above all, buttresses. These rules (arithmetical or geometrical) conduced in most cases to a certain proportions independently of size, to geometrically similar designs (for example, the depth of a buttress is a fraction of the span). Very rarely, and this is the case with Rodrigo Gil, appeared arithmetical rules which lead to non-proportional designs (following Rodrigo's rule the buttresses become more slender in relation to the span as the size increases). These rules were a means to register stable forms. Proportional rules are, as has pointed Professor Heyman, essentially correct. It is a problem of stability and not of strength. Non-proportional rules express a finer adjustment to some non-proportional design problems: buttress design for the thin domical cross vaults (bóvedas baídas), boss design for the vaults themselves, and wall design for towers. The rules were deduced empirically, give correct dimensions, but above all draw our attention to some significant facts of design which so far have remained unnoticed

Elliptic harbor wave model with perfectly matched layer and exterior bathymetry effects

Standard strategies for dealing with the Sommerfeld condition in elliptic mild-slope models require strong assumptions on the wave field in the region exterior to the computational domain. More precisely, constant bathymetry along (and beyond) the open boundary, and parabolic approximations–based boundary conditions are usually imposed. Generally, these restrictions require large computational domains, implying higher costs for the numerical solver. An alternative method for coastal/harbor applications is proposed here. This approach is based on a perfectly matched layer (PML) that incorporates the effects of the exterior bathymetry. The model only requires constant exterior depth in the alongshore direction, a common approach used for idealizing the exterior bathymetry in elliptic models. In opposition to standard open boundary conditions for mild-slope models, the features of the proposed PML approach include (1) completely noncollinear coastlines, (2) better representation of the real unbounded domain using two different lateral sections to define the exterior bathymetry, and (3) the generation of reliable solutions for any incoming wave direction in a small computational domain. Numerical results of synthetic tests demonstrate that solutions are not significantly perturbed when open boundaries are placed close to the area of interest. In more complex problems, this provides important performance improvements in computational time, as shown for a real application of harbor agitation.Peer ReviewedPostprint (author's final draft

El proyecto de estructuras en la obra de Gaudí

La obra de Gaudí integra todos los aspectos del proyecto de arquitectura. En el presente artículo se estudia el análisis y proyecto de arcos y bóvedas de fábrica. Es bien conocido que Gaudí empleó modelos colgantes y también métodos de estática gráfica. Estos métodos no son originales y se remontan a finales del siglo XVII. Tampoco es original el empleo en el proyecto de formas equilibradas (catenarias). Lo que es completamente original y supuso en «giro copernicano» es plantear todo el proyecto, desde el comienzo, buscando formas equilibradas. Por otra parte, Gaudí emplea bóvedas de formas inusuales, compuestas de superficies regladas, mostrando una gran intuición estructural. Finalmente, en la Sagrada Familia busca esqueletos arbóreos de equilibrio. En el presente artículo se estudian en detalle el origen y naturaleza de los métodos de equilibrio de Gaudí, resaltando su validez basada, en última instancia, en los Teoremas Fundamentales del Análisis Límite. The work of Gaudi embraces all the facets of architectural design. The present paper studies the analysis and design of masonry arches and vaults. It is well known that Gaudi used hanging models and graphics methods as design tools. These methods can be traced back to the end of the XVIIth century. Also, it was not original the use of equilibrated, catenarian, forms. What was completely original was the idea of basing all the structural design in considerations of equilibrium. Gaudí, also, employed unusual geometrical forms for some of his vaults, ruled surfaces, showing a deep structural insight. Finally, he designed tree-forms of equilibrium for the supports of the vaults in the Sagrada Familia. In the present paper Gaudí's equilibrium methods are studied with some detail, stressing their validity within the frame of Limit Analysis

Ripartendo dalla colonna (Rev. del libro de R. Gargiani "La colonne. N o u velle histoire de la construction" Lausanne 2008)

Una storia della costruzione che mette in luce un rinnovato interesse disciplinare, riconsiderando aspetti tecnici ed evoluzione delle forme architettonich

Thomas Young's theory of the arch: His analysis of Telford's design for an arch of 600 feet span over the Thames in London

Thomas Young's theory of the arch: His analysis of Telford's design for an arch of 600 feet span over the Thames in Londo

Structural Analysis of Thin Tile Vaults and Domes: The Inner Oval Dome of the Basilica de los Desamparados in Valencia

The inner oval dome of the Basílica de la Virgen los Desamparados, built in 1701, is one of the most slender masonry vaults ever built. It is a tile dome with a total thickness of 80 mm and a main span of 18.50 m. It was built without centering with great ingenuity and economy of means, thirty three years after the termination of the building in 1667. The dome is in contact with the external dome only in the inferior part with the projecting ribs of the intrados, the lunettes of the windows, and, in the upper part, through 126 inclined iron bars. This unique construction was revealed in the 1990's in the studies previous to the restoration of the Basílica, and has given rise to different theories about the mode of construction and the structural behaviour and safety of the dome. The present contribution aims to provide a plausible hypothesis about the mode of construction and to explain the safety of the inner dome which has stood, without need of repairs or reinforcement, for 300 hundred years