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

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

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

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

The determinants of bank margins revisited: A note on the effects of diversification

Most of the theoretical and empirical literature on bank margins has dealt solely with interest margins. Applying the seminal Ho-Saunders model (JFQA, 1981) to a multi-output framework, we show that the relationship between bank margins and market power (controlling for risk) varies significantly across bank specializations. Using a set of both accounting margins and New Empirical Industrial Organization (NEIO) margins, we find that market power rises significantly with output diversification towards non-traditional activities. These results contribute to explain the paradoxical coexistence of decreasing interest margins and higher market power found in previous studies.bank margins, specialization, market structure.

The finance-growth nexus: a regional perspective

Recent cross-country studies suggest that finance and economic growth are significantly related. The characteristics and geographical scope of this relationship has become central to explain differences in economic development. Two concepts are highly relevant in this context. Firstly, financial deepening, that involves the development of traditional and non-traditional financial services in these territories. Secondly, bank dependence, which makes households and firms rely heavily on banks. Employing dynamic causality and panel data techniques on a sample of Spanish banks during 1993-1999 we find that -at a regional level- economic growth predicted financial deepening in this period which suggests that banks locate and distribute new financial products mostly in well developed territories. Regarding bank dependence, bank lending specialization appears to be a key issue in financing firms and households compared to other bank specializations. According to previous studies, lending dependence confer banks a special role in promoting regional economic growth in bank-based financial systems.Banks, financial deepening, bank dependence, regional growth.

New evidence of scope economies among lending,deposit-taking, loan commitments and mutual fund activities.

Financial innovation and technology affect bank cost, revenue and profits. Most of the previous empirical studies have not found significant cost, profit or revenue scope economies or output pair complementarities either between traditional and non-traditional banking products or between traditional activities themselves. We study scope economies and output pair complementarities in a ‘broad banking’ environment: the Spanish banking sector. The results indicate that after including off-balance sheet business in the output mix, cost and profit scope economies rise and are statistically significant. Besides, consumer valuation of financial services is only detected when the off-balance sheet business is added to the output definition.banking, scope economies, off-balance sheet.

Scope Economies and Competition Beyond the Balance Sheet: a ‘broad banking’ Experience

There is a recent trend in banking research aiming to assess how financial innovation and technology are affecting bank cost, revenue and profits. Most of the studies have not found significant cost, profit or revenue complementarities either between traditional and non-traditional banking products or between traditional activities themselves. We study complementarities in a ‘broad banking’ environment: the Spanish banking sector. The results indicate that after including off-balance sheet business in the output mix, cost and profit scope economies increase significantly. Besides, consumer valuation of financial services is only detected when the off-balance sheet business is added to the output definition.banking, cost, profit, scope economies, off-balance sheet.