This chapter deals with the elegant and especially significant applications of the least work principle described by Fränkel (1882). These applications almost certainly originate from his friendship and collaboration with Winkler. Thus, he begins with the relatively difficult problems of the elastic arch and suspension bridge. In so doing, it seems that he was mindful of Winkler's principle (Chapter 3) for the thrust-line of an arch (1879a). Indeed, that principle probably led to his search for a means of establishing a principle of least work for elastic structures generally, the successful outcome of which was marred by the discovery, just as his work was poised for publication, that he had been anticipated by Castigliano. Thus he acknowledges Winkler for that information and it may be judged by the fact that Winkler himself had then only recently become aware of Castigliano's work through the French (1880) edition of the original work.

It will be noted that, throughout application of the least work principle to an arch, Fränkel contrasted the results with those derived by Winkler. Having thus discussed the principle in relation to what he termed ‘the elegant work’ of Winkler, he turned his attention to what he identified as the closely related problem of the stiffened suspension bridge (incorporating inversion of the arch) and succeeded in determining the elastic theory in advance of Levy's celebrated treatment (Chapter 3).

The precise analysis of statically-indeterminate systems of bars, including trusses and pin-jointed frameworks generally, seems to be due to the famous French engineer, Navier. It was included in his lectures at l'Ecole des Ponts et Chaussées, which appeared in the form of his celebrated Leçons in 1826. According to Saint-Venant (Navier, 1864, p. 108) the method was part of the course as early as 1819. It was elaborated (1862) by the mathematician Clebsch in Germany; while, in Britain, Maxwell (1864b) who, it seems, was unaware of Navier's elegant and general method, published an original method of solving the problem. Levy, who was apparently aware of Navier's work, published a novel method in 1874 (Chapter 6). But it was not really until the German engineer, Mohr, published his analysis in the same year that the subject began to be appreciated by engineers (on the Continent at first and much later in Britain).

This chapter is concerned with those original contributions, in principle only: various sophistications and devices to increase their utility in engineering are considered in Chapters 8 and 10.

Navier, 1826

Navier's contribution to the analysis of statically-indeterminate pin-jointed systems is to be found essentially in the two articles of his Leçons (1826, art. 632, p. 296; 1833, art. 533, p. 345).

For the purpose of this chapter, secondary effects are understood to include dynamic stresses as well as those which arise from the nature of construction details, especially the rigidity of joints (connections) in triangulated trusses (bridge girders). The term secondary stress is usually associated with these latter, following the initiative of Professor Asimont of the Munich Polytechnikum in 1877. In that year a prize was offered by the Polytechnikum (as noted in Chapter 1) for a method of calculating those stresses in trusses (termed Sekundarspannung by Asimont, to distinguish them from the stresses due to the axial or primary forces in the bars, that is, Hauptspannung). Dynamic stresses, on the other hand, became the subject of research in the nineteenth century, due to the failure of a number of iron railway bridges, which were caused by the passage of trains.

Secondary stresses

According to Grimm (1908), Asimont formulated the problem of secondary stresses in rigidly-jointed trusses and suggested that, since the resultants no longer pass through the panel points, a solution might be afforded by ‘Euler's equation of the elastic line’. In the event, the prize was awarded in 1879 to Professor Manderla and his solution was published soon afterwards (1880), although an approximate solution by Engesser in which chords were treated as continuous and web members as pin-jointed, appeared a year earlier (1879).

Carl Culmann was born 10 July 1821 in Bergzabern, Rheinpfalz, and died in Zurich, 9 December 1881.

After completing his studies in Karlsruhe he ‘worked on railway construction in mountainous country and later (1848) was transferred to the office of the Royal Railways Commission in Munich.

In the summer of 1849 the Railways Commission sent him on a two-year study tour of the British Isles and the U.S.A. The period of this tour coincided with the completion of the wrought iron Britannia (tubular) Bridge by Robert Stephenson and with the end of a phase of intensive development of wooden bridge construction in the U.S.A. The substance of Culmann's report of the tour was published in Allgemeine Bauzeitung in 1851 under the title ‘A description of the latest advances in bridge, railway and river-boat construction in England and the United States of North America’. It aroused great interest and established Culmann's reputation as a young engineer with outstanding qualities of perception. Indeed, it seems to have been a material factor in his leaving the railway industry in 1855 to teach at the newly established Federal Polytechnic Institute at Zurich, where he believed he would have greater opportunities for combining theory and practice of engineering.

Culmann clearly recognised the urgent need to develop Navier's methods for application to the design of railway bridges and his report emphasised methods of calculating the forces in the new bridge forms to enable them to be exploited with confidence in their safety.

This chapter is concerned primarily with principles involving energy concepts, which were revived or formulated early in the nineteenth century within the science of statics and the theory of elastic structures. Principles relating to energy, which were implicitly available at the beginning of the century, included the principle of virtual work (known then as the principle of virtual velocities) and the supreme law of conservation of energy.

Early history

According to Dugas (1955), the use of the principle of virtual velocities can be traced to Jordanus of Nemore in the thirteenth century (and to Aristotle's law of powers of the fourth century b.c.). A revealing account of the principle is provided by Mach (1883) and it is remarkable that the principle, and its use, preceded explicit recognition of conservation of energy. Mach discusses the meaning of the terminology at some length: he ascribes it to John Bernoulli and notes that the word ‘virtual’ is used in the sense of something which is physically possible. Bernoulli's definition of vitesse virtuelle is said by Mach to be incorporated in Thomson & Tait's wording: ‘If the point of application of a force be displaced through a small space, the resolved part of the displacement in the direction of the force has been called its virtual velocity.’ It is believed that the truth of the principle was first noted explicitly by Stevinus, at the close of the sixteenth century, in connection with his research into the equilibrium of systems of pulleys.

After the theory of statically-indeterminate frameworks was established in Europe c. 1875, theory of structures advanced rapidly, especially by virtue of the property of a linear relationship between ‘cause’ and ‘effect’, which characterised engineering structures and which afforded the principle of superposition (Chapter 3) and the reciprocal theorem (Chapter 5). Dominant among the contributors to these advances were Mohr and Müller-Breslau in Germany, and much of this chapter is concerned with them and their work. Also, the Italian railway engineer, Crotti, deserves special mention for his unique contribution to the development of a general theory of elastic structures.

Mohr and Müller-Breslau

Mohr, born in 1835, was some sixteen years older than Müller-Breslau but nevertheless there seems to have been antagonism and rivalry between them, to judge, especially, from published comment by the latter, which is illuminating in various respects and is therefore included in this chapter. Müller-Breslau died in 1925, only seven years after Mohr. His later work was arranged for publication by his son who was a professor at the Breslau Polytechnikum. The final edition of part of his monumental work Graphische Statik der Baukonstruktionen anticipated important developments in theory of structures in the present century.

At the age of thirty-three, Mohr became professor of engineering mechanics at Stuttgart, having spent the early years of his career in railway construction in common with many distinguished civil engineers of the nineteenth century.

The state of knowledge of applied mechanics in Britain at the beginning of the nineteenth century is probably reflected in David Brewster's Robison's Mechanical Philosophy (1822) which is based mainly upon articles published by John Robison, professor of natural philosophy at Edinburgh University, in the fourth edition of the Encyclopaedia Britannica (1797). The first of the four volumes includes chapters on strength of materials, carpentry, roofs, construction of arches and construction of centres for bridges. With regard to strength of materials Robison refers (in general terms) to the experiments of Couplet, Pitot, De La Hire and Duhamel in relation to cohesion. He also refers specifically to elasticity and ductility and mentions plastic substance and properties. Robison is much concerned with cohesion in terms of attraction between particles, referring to the theories of Newton and Boscovich. Then he suggests that ‘connecting forces are proportional to the distances of the particles from their quiescent, neutral or inactive positions’. This ‘seems to have been first reviewed as a law of nature by the penetrating eye of Dr Robert Hooke’. Robison quotes what he describes as Hooke's cipher, ceiiinosssttu, for the law of elasticity (ut tensio sic vis) which bears his name and he records Hooke's anticipation – and rejection – of the facts used by John Bernoulli in support of Leibnitz's doctrine of vires vivae.

With the theories of flexure and bending-stress in beams, established in the eighteenth century by James (Jacob) Bernoulli and Euler (c. 1740) and Coulomb (1773) respectively, Navier developed the analysis of forces and deflexions of beams of varying degrees of complexity, with regard to support and restraint, as part of his extensive and unique researches in theory of elasticity. In those researches, evaluated by Saint-Venant and others (1864), he laid the foundations of modern technical theory of elasticity and anticipated important applications.

It had become well known in carpentry that continuity of beams over supports and building-in the ends of beams, contributed substantially to their strength or carrying capacity. Indeed, Robison had considered this subject in an elementary fashion toward the end of the eighteenth century (Brewster, 1822). Navier was clearly mindful of the common use of such statically-indeterminate construction in timber (to judge by the detail of his illustrations) when he embarked on the precise analysis of systems of that kind and, in the event, his analysis was timely with regard to the development of wrought iron beams and structures, which was stimulated by the needs of railway construction. It was, in fact, the statically-indeterminate beam (including, especially, the continuous beam) which dominated the development of the beam in the nineteenth century.

Navier, 1826

The analysis of encastré and continuous beams is believed to have been published for the first time in Navier's celebrated Leçons of 1826 (though Clapeyron refers to earlier lithographed notes).

At the beginning of the nineteenth century, to which elastic arch theory belongs, masonry was still the principal structural material. Many major structures, especially bridges, depended on the arch as a means of exploiting the strength of stone in compression. The origin of an explicit theory of the arch is variously ascribed to Hooke, De La Hire, Parent and David Gregory in the seventeenth century. Robison (Brewster, 1822) believed that Hooke suggested the inversion of the shape adopted by a suspended rope or chain, namely the catenary, as the statically correct form for an arch: others (Straub, 1952) ascribed that concept to David Gregory. In any event, it appeared to disregard a distribution of load different from that which would be due to a uniform voussoir arch. Heyman has reviewed the development of the theory of the arch in detail (1972) and leaves little doubt that it was highly developed in the eighteenth century. Coulomb's theory of 1773 (1776) of the distribution of force in loaded stone arches and their stability (ultimate load carrying capacity) was generally accepted by Navier, to judge by the contents of his Leçons (1826, 1833; in which, quite separately, elastic theory of the arch rib appears). But those youthful partners, Lamé & Clapeyron rediscovered (while in Russia) the theory of the ultimate strength of stone arches for themselves (1823), apparently ignorant of Coulomb's theory (or, indeed, that of Couplet which Heyman has described (1972)).

Bar frameworks for bridges and roofs, with timber as the material of construction, are of ancient origin. Indeed, the illustrations of such structures in early nineteenth-century works on theory of structures, for example Navier's Leçons (1826), include them within the scope of carpentry, and pictorial illustrations clearly indicate timberwork. The widespread adoption of iron framework for roof trusses, arches and bridge girders, coincided with the construction of railways (a direct consequence of the new iron age) and an urgent need for numerous bridges and large buildings. The first major iron lattice girder bridges as an alternative to solid and plate girders (by virtue of the economy afforded by reduction in self-weight) appeared, it is believed, soon after 1840. Dempsey (1864) gives an interesting account of the history of cast and wrought iron construction, noting that the first iron vessels (boats) were made in 1820–1 by Manby of Tipton. He believed that the rolled ‘I’ section was introduced by Kennedy and Vernon of Liverpool in 1844, angle section being of earlier origin, and that Fairbairn made plate girders as long ago as 1832.

Dempsey credits Smart with the invention of the diagonal lattice girder, referred to as the ‘patent iron bridge’, in 1824 but seems to believe that it was first used for a major railway bridge after 1840, on the Dublin and Drogheda Railway. That bridge was described by Hemans (1844).

Exploitation of the doctrine of energy, using energy functions and their derivatives in theory of structures, seems to have begun in earnest on the continent of Europe (by coincidence) soon after Cotterill's three important articles appeared in 1865. It was primarily due to the researches and principles of the Italian railway engineer Castigliano (1873, 1879), after Menabrea (1858). The implicit objective was to remedy deficiencies of statics by means of conditions of compatibility of elastic strain. Indeed, Castigliano's so-called principle of least work (terminology of Menabrea, 1884) was to become perhaps the best-known general method of structural analysis toward the end of the century. The contributions of Fränkel, Crotti and Engesser to the energy approach are, however, significant. But, in restrospect, Cotterill's priority over Castigliano and others seems unquestionable after careful study of his original articles. His obscurity, until comparatively recently, is undoubtedly due to the fashion in Britain to publish original work in both pure and applied science in journals devoted to natural philosophy, outstanding among which is the Philosophical Magazine in which Moseley and Maxwell, as well as Cotterill, published their contributions to theory of structures. Originally, like Moseley, a Cambridge mathematician of St John's College, Cotterill (who lived from 1836 until 1922) became professor of applied mathematics at the Royal Naval College.

The objective of this work is to provide an account, with appropriate detail, of some of the salient features of the development of the theory and analysis of engineering structures during the nineteenth century. There seemed to be two possible approaches to the subject: that whereby emphasis is on personalities and their contributions to the subject, or that whereby emphasis is primarily on subject development, with due acknowledgement of personalities and regard to the chronological aspect. Experience indicates that the former is conducive to some degree of repetition and confusion concerning the subject matter and, therefore, the latter approach is adopted (though personal names are used in subheadings to identify developments). But Chapter 6 is unique in being devoted to Levy's little-known, though highly-significant work on theory of frameworks. Free translation of original material is used extensively throughout the book in order to avoid misrepresentation or serious omission. Also, original notation for mathematical analyses are preserved as far as possible.

Chapters 4 and 11 differ from the others in being little more than brief reviews of topics which, though important, are peripheral to my purpose herein. The former embraces graphical analysis of simple frameworks which, together with the vast subject of graphical analysis of engineering problems generally, has very limited relevance to the features with which this work is concerned and which have determined the development of modern theory of structures.

The following is an abridged version of the author's free translation of Navier's Obituary Notice of 1837, by Prony, which is included in the 1864 edition of Navier's Leçons, edited and with additional notes by Saint-Venant.