Architecture

"[P]icture... a circular piece of a plate which has an approximately spherical curvature at its center point O. Pass a plane XY tangent to the surface at O and let OZ be normal to it at O. Then if a be applied about OX it will not only make the curvature of the plate greater in the plane ZOY but at the same time it will make its curvature less in the plane ZOX to an equal amount as is evident by experiment on a shell of any elastic material, and as is proven in Gauss' theorem of the curvature of thin shells. Since the force applied to produce the given bending moment must produce both these equal changes of curvature simultaneously by producing elongations and compressions in twice as much material as in a plane plate of equal cross section, each of them is only half as great as would be produced in a plane plate in a single direction by this same moment. Hence it. appears that the deformations produced by an applied moment are not more than half as great in a spherical dish shaped plate as in a plane plate..."

"[P]rinciples as developed by Kelvin and by Love show that it is impossible to bend a nearly flat dish shaped shell about one horizontal axis without at the same time bending it in the opposite direction about a second horizontal axis at right angles to the first."

"Lord Rayleigh... concluded from physical reasoning that the middle-surface of a vibrating shell remains unstretched, and determined the character of the displacement of a point of the middle-surface in accordance with this condition. The direct application of the Kirchhoff-Gehring method led to a formula for the potential energy of the same form as Aron's and to equations of motion and boundary conditions which were difficult to reconcile with Lord Rayleigh's theory. Later investigations have shown that the extensional strain which was thus proved to be a necessary concomitant of the vibrations may be practically confined to a narrow region near the edge of the shell, but that, in this region, it may be so adjusted as to secure the satisfaction of the boundary conditions while the greater part of the shell vibrates according to Lord Rayleigh's type."

"E. Mathieu adapted to the problem [of curved plates or shells ] the method which Poisson had used for plates. He observed that the modes of vibration possible to a shell do not fall into classes characterized respectively by normal and tangential displacements, and he adopted equations of motion that could be deduced from Aron's formula for the by retaining the terms that depend on the stretching of the middle-surface only."

"The problem of curved plates or shells was first attacked from the point of view of the general equations of Elasticity by H. Aron. He expressed the geometry of the middle-surface by means of two parameters after the manner of Gauss, and he adapted to the problem the method which Clebsch had used for plates. He arrived at an expression for the potential energy of the strained shell which is of the same form as that obtained by Kirchhoff for plates, but the quantities that define the curvature of the middle-surface were replaced by the differences of their values in the strained and unstrained states."

"In an article in the Annales d'u Génie civil, March, 1879, on the “Resistance of Tubes subjected to an External Pressure,” by Théodore Belpaire, an attempt has been made to deduce a new formula for the collapsing strength of tubes. ...The writer ... considers the case of a tube with ends rigidly fixed, and supposes that under an external pressure it changes its form in such a manner that its generatrix becomes the arc of a circle, the centre of which lies on a perpendicular erected in the centre of the generatrix; and, neglecting, the elastic forces due to flexure or elongation of the fibres—which are very small as long as the curvature is slight—he investigates the shearing stresses; these attain their greatest value at the fixed ends. Calling S the greatest shearing stress, p the pressure in pounds per square inch, t the thickness of the tube in inches, L the length of the tube in inches, he deduces the following approximate formula for the external pressure which a given tube can bear with a degree of safety depending on the value attributed to S—viz.:p = \frac{2tS}{L}. { VI.}The writer deduces then a general value S from two experiments made by Fairbairn with elliptical tubes, because the uncertain and variable elements of strength due to the cylindrical form and to homogeneity of the material do not enter here. When the factor of safety in the foregoing equation is to be four, the value of S becomesS = 428,394 \frac{t}{D} - 7,111,550 (\frac{t}{D})^2;...With reference to those cases where the factor of safety exceeded four greatly, the writer claims that the high pressures necessary to produce collapse indicate merely the great increase of strength derived in the particular instances from the uncertain element of circular form."

"Fairbairn finds that the collapsing pressure of... an elliptic form of cross-section is found approximately by substituting... for d the [following]... let a be the greater and b the less semi-axis of the ellipse; then we are to maked = \frac{2 a^2}{b}. { V.}"

"The true Mathematical and Mechanical Form of all manner of es for building with the true butment necessary to each of them, a Problem which no Architectonick Writer hath ever yet attempted, much less perform'd. ...Ut pendet continaum flexile, sic stabit contiguum rigidum, which is the Linea Catenaria."

"Shell-like structures are familiar enough in nature but the use of such structures as containers, aircraft fuselages, submarine hulls and roofing structures is only of recent origin. That the inherent strength of shells... has not been utilised much in the past is probably due to the difficulty in obtaining suitable material... [S]hell structures in general are these days constructed of such varied materials as steel, light alloy, plastics, wood and reinforced concrete. ...[T]o simplify analysis it will be assumed that the material... is homogeneous, isotropic and perfectly elastic. ...[A]lthough reinforced concrete behaves in a reasonably elastic manner only in the lower stress ranges the majority of reinforced concrete shell roofs that are constructed in practice are designed as elastic structures."

"[[w:Membrane theory of shells|[M]embrane theory]]... is the theory of shells whose bending rigidity may be neglected. The spectacular simplification... makes it possible to examine a wide variety of shapes and support conditions. In particular, the stress problems of tanks and shell roofs... There is, of course a heavy penalty... [T]he inadequacies... can be discovered by a critical inspection of the... solutions, without any need for... solving the bending problem—a task which is often out of the reach of the practical engineer and even of the research worker. On the other hand, membrane theory is more than a first approximation... If a shell is so shaped and so supported that it can carry its load with a membrane stress system, it will be thin, light, and stiff and, therefore, the most desirable solution to a design problem. Membrane theory will guide the shell designer toward such structure."

"The title of first pioneer of the HP hyperbolic paraboloid or hypar construction] in concrete in the 1930s belongs to Fernand Aimond for the projects that he constructed.., for the formulation of the theoretical structural membrane model, and for his influence both on [Giorgio] Baroni in Italy in the late 1930s and on Candela in Mexico in the 1950s."

"The design of shells... implicates the design of internal stress fields of form dependent shapes... meeting the compatabilites of all boundary conditions..."

"Nature does not apply the construction principle of a beam supported by two s. Forms developed by nature are following the rational attempt to achieve distinct functionalities with the smallest possible material - and energy consumption. An impressive example is the phenomena of egg shells... The shell principle is adopted by humans... in building construction, in order to achieve wide spanning and material saving 'slender' structures."

"The emergence of lightweight structures can be traced back to the second half of the nineteenth Century. This period witnessed the advent of new material technologies such as steel, , resistant glass and, later, fabric membrane. Together with advances in analysis and design tools, engineers and architects have been challenged to build increasingly lighter structures. ...[A] pioneering structure was the lattice tower by... in 1896. In the 1920s, Anton Tedesko first introduced reinforced concrete thin shells in the United States. This expansion was pursued... by , and André Paduart... The limit of lightness was achieved with tensile structures constructed of prestressed cable nets and fabric membranes; the strength coming from the anticlastic curvature of the geometric surface. ...Nowadays, lightweight structures should be designed... by including the multitude of design contraints. This will result in hybrid systems lying at the boundary of different typologies."

"The resistant virtues of the structure that we make depend on their form; it is through their form that they are stable and not because of an awkward accumulation of materials. There is nothing more noble and elegant from an intellectual viewpoint than this; resistance through form."

"At the beginning, the was not covered by any type of roof. During the reconstruction... in 1964... the roof covering... was created. ...reinforced by cotton threads and covered by rubber layers. The material underwent large rheological displacements. In... 1968... catastrophe occurred, caused by wind and high humidity. In the eighties... a polyester fabric, intended only [for] seasonal application, was used. ...age and... repeated disassembling of the membrane caused its gradual destruction. ...In 2007 ...complete rebuilding ...assumed the roof to be a permanent structure. ...[V]erification of ...internal forces was conducted in 2012 ...by team."

"At the time of construction of the Wyss shell, three-dimensional computer software was not available and it would have been extremely difficult to convey, using only normal engineering drawings, the required form of the concrete at the feet of the shell... To overcome the problems... proposed that, rather than making sketches, drawings, or even a model of the detail, they should resort to modelling it at full-scale on site."

"For around 2000 years single and double curved shells structures, such as barrel and vault s, have been used to cover large spans in buildings. Until the twentieth century these were generally constructed either from masonry or some form of unreinforced concrete, materials strong in compression but relatively weak in tension. Well known examples such as the Pantheon... ... Santa Maria del Fiore... and St. Peter's Basilica... have a span to thickness ratio of less than 50 to 1, which is relatively thicker than a... typical hen's egg. ...[T]he stone vaulting of... medieval Gothic cathedrals... demonstrate the mason's art in the construction of... complex masonry shells. With the advent of reinforced concrete... strong in both compression and tension, it became possible... to construct thin shells with much higher span to thickness ratios... commonly... in the region of 500 to 1."

"The essential ingredients of a shell structure... are continuity and curvature. ...[S]hells are structurally continuous in the sense that they can transmit forces in a number of different directions in the surface of the shell... These have a quite different mode of action from skeletal structures... only capable of transmitting forces along discrete structural members. ...There seems to be a principle that closed surfaces are rigid. This principle is used in many areas of engineering construction. ...[A]lthough the ideas of 'closed' and 'open' shells... are fairly clear, it is difficult to quanitify intermediate cases into which... the majority of actual shell structures fall. ...There is a theorem, due to Cauchy, which states that a convex polyhedron is rigid. ...[N]on-convexity may produce deformability. ...While rigidity and strength are in many cases desirable attributes of shell structures, there are some important difficulties which can occur... [involving] unavoidable rigidity. ...[A] second broad principle... may be stated thus: efficient structures may fail catastrophically. Here I use the term 'efficient' to describe the consequences of employing the first principle. By designing a structure as... closed... we may be able to use thinner sheet material, and hence produce an economical, or efficient, design."

"In 1958, Felix Candela completed his most significant work, the Los Manantiales Restaurant shell, in Xochimilco, Mexico City. ...[He] was taking a risk... The form was original, unexplored, and impossible to analyze precisely. Candela’s career, however, habitually flew in the face of precise analysis. In his first acclaimed shell, the Cosmic Ray Pavilion of the University of Mexico City, he also designed an unprecedented form, using almost no calculation... Candela’s subsequent designs relied increasingly on structural understanding and practical experience. As a designer-contractor, he had the unique responsibility of building his own solutions. By closely observing his buildings, and using smaller projects to test new ideas, he developed an acute sense of concrete shell behavior."

"Shells were not being done in the United States at all, and I started seeing pictures of these buildings coming out of South America... Italy and Spain... they happened to be all Latin countries. ...I thought, "God, how do I do this?" you know, these three-dimensional curved structures... [T]he greatest of them all... was Nervi... I found out that I couldn't find out how to design... them. ...I realized after many years of striving that a lot of them really didn't know how to design them. They were just doing it intuitively, and that was not satisfactory to an engineer. I needed to know a rational way of doing it, and that sent me back to school and... to studying... [I]t took me years and years of very hard intellectual work to find ways to do this. ...[W]hen I finally did it, I was one of the few guys in the country that had really made that much effort, and so I became a pioneer..."

"The Hershey Arena... Tedesko designed a thin reinforced concrete, barrel-shell roof, three and one-half inches thick, supported across its width by eight arches. ...A roof posed a different problem than a bridge or dam. On a bridge, live load from traffic is significant, and a dam must resist the live load of water on its upstream face. On a long-span concrete roof, live load (mainly rain and snow) is a small fraction of the dead load of the structure itself. Tedesko realized that the supporting arches did not need to be of uniform depth. ...[he] designed the arches to be able to... support the entire roof load, including the thin shell and all of the live load. He also made calculations to show that the thin shell could carry its own weight and the live load without help from the arches, except near their lowest edges. It was thus a conservative design..."

"The idea of form over mass also developed in Europe in the pioneering work of Dyckerhoff and Widmann... in Weisbaden, Germany. Working in reinforced concrete, the firm experimented with new ways to cover large spaces in the 1920s. The firm built domes and cylindrical "barrel" shells to serve as large roofs of extraordinary thinness. The possibilities... fascinated an Austrian civil engineering student, Anton Tedesko... who joined the firm in 1930."

"Torroja was a specialist in stress analysis... and he wrote a... book on the mathematical theory of elasticity. This... led him to see a connection at Algeciras between the stresses in the shell and the reinforcement... but not to express those stresses in... visually evident ribs. We contrast... Nervi's Little Sports Palace... whereas Nervi sees shells as ribbed, Torroja sees them as ribless... since domes tend to spread, Nervi designed ribbed buttresses... whereas Torroja avoids buttresses by connecting vertical supporting columns with a... polygonal ring of horizontal ties... prestressed to counteract dead load and to lift the shell slightly off its scaffold... probably the first application of prestressing to a doubly curved shell. In the Nervi dome... the buttresses are supported below ground on a ring which carries the horizontal thrust and... transmits the vertical weight to the ground. ...[These] choices related to the [respective] local traditions in Italy and Spain."

"... all through school had a reputation of working alone and of doing his work in an unusual way. ...in 1950 he graduated with a degree in civil engineering. For his final-year design project, he chose to study thin shells... Following graduation... he helped [Pierre] Lardy with teaching, and also worked on the many cases of structural failure [both at his alma mater, the Federal Technical Institute]... When Isler left his position... he considered... [a] career as a painter, but challenged by shell design problems... while doing free-lance engineering work.. in late 1954, he designed a pneumatic form, thin shell factory for the Trösch Company. It was the first work in which he set the form completely on his own. In 1955, at an international congress in Amsterdam, he presented publicly for the first time his new designs..."

"The method of geometric modeling of multi-shell roofs depends mostly on the surface's properties forming the shell; their curvature, as well as continuity between them. ...s play a specific role, due to their characteristics. Catalan surfaces are s... They are oblique ruled surfaces which can be divided into two groups... second order—hyperbolic paraboloid... [and] of more than second order—s, cylindroids... The difference between hyperbolic paraboloid, conoid, and cylindroid results from different path of movement of a surface's ruling during formation. In all cases of Catalan surfaces' creation... each ruling is parallel to the fixed plane (not containing the surface's directrices)."

"[M]ost of Candela's structures are almost complete in themselves... the forms and proportions bear witness to his artistic sensibility. ...[B]alanced perfection ...makes a... structure into a work of art. ...[T]he whole must take precedence over any of its parts."

"Candela] is not just an engineer, or an architect, or a contractor and constructor, but all three... [W]hen he thinks out a new scheme, the method of construction and its economy is constantly in his mind. ...He prefers to obtain his economy by using his inventive skills as an engineer to reduce... material... [H]e recognizes the value of ... but he is also very conscious of its limitations. Especially is he skeptical about the value of the theory of elasticity as applied to concrete... of... calculations suggesting an accuracy which is purely fictitious... Designing... proceeds from a structural feeling acquired by experience and guided by rough calculations, a refinement of design... further analysis... and so on. ...[A] flair for making the right guess yields quicker and better results than a lot of mathematics... this is no reason for despising theoretical analysis... But one cannot design by theory..."

"Candela did not invent the concrete shell; nor is he the first to make use of the hyperbolic paraboloid... Other people... have contributed more to the theoretical analysis of shell structures. But nobody else can claim credit for such an exciting variety of shell structures... [H]e has concentrated his effort in one particular sphere: the construction of light concrete roofs."

"has designed some of the most striking thin shells in reinforced concrete of the second half of the twentieth century. He creates thin shells by hanging small membranes in tension and creating smooth curving surfaces that are then inverted and scaled up to create large-scale structures in compression. ...Within the constraint of economy, he discovered new forms from purely structural considerations and demonstrated the unlimited possibilities for thin compression shells to be found in hanging models."

"In the twentieth century, lightly reinforced brick shells were inspired by timbrel vaulting, a common building method in the Mediterranean. ... upon emigrating to the United States from in 1881, introduced the method with great success. His son, Rafael Guastavino, Jr. ...appears to be the first to have introduced steel reinforcing to thin brick shells. ...[T]wo patents... [1910, 1913] documented this system, which is a precursor to the thin shells of reinforced concrete developed widely in the ensuing decades."

"[T]he early work of Lambot... was one of the first applications of , but [was] also... a form of . His patent on wire-reinforced boats that was issued in 1847... This was the birth of reinforced concrete, but subsequent development differed from Lambot’s concept. The technology of the period could not accommodate the time and effort needed to make mesh of thousands of wires. Instead, large rods were used to make what is now called conventional reinforced concrete, and the concept of ferrocement was almost forgotten for 100 years."

"Thin shells — Three-dimensional spatial structures made up of one or more curved slabs or folded plates whose thicknesses are small compared to their other dimensions. Thin shells are characterized by their three-dimensional load-carrying behavior, which is determined by the geometry of their forms, by the manner in which they are supported, and by the nature of the applied load."

"Gaudí was drawn not just to the aesthetics of the catenary but also to what it represented mathematically. His use of catenaries made the structural mechanics of a building a principal feature of its design. Gaudí realized that the entire architecture of a building could be drafted using a model of hanging chains... when he was commissioned to design a church the Colònia Güell... he made an upside-down skeleton of the project. Instead of using metal chains, he used string weighed down by hundreds of sachets containing lead shot. The weight of each sachet on the string created a mesh of 'transformed' catenary curves. The arches of these transformed catenaries were the most stable curves to withstand a corresponding weight at the same position (such as the roof, or building materials)."

"One can roll noncircular wheels over appropriate road surfaces. The most striking example of this is the fact that a can roll on a road that consists of linked catenaries (the catenaries are defined by y = - cosh x) ...(the ride is not smooth in the sense that the center does not move forward at a constant rate of rotation, but... the actual ride... feels quite smooth). This animation was inspired by an exhibit at San Francisco's ..."

"Concrete being such a fluid and dynamic material... finds its identity once it is contained. ...A few... who used the forming materials at hand [were]... Antoni Gaudi... ... ... Felix Candela... ... ... Miguel Fisac... Many of these early innovators pushed the computational envelope... Some, like Antoni Gaudi, looked to nature for inspiration. The question... Do we need to "reinvent forming" or just draw from nature, i.e., gravity—catenary action? as Gaudi did. Alan Chandler in fabric framework notes "...for Felix Candela and Christopher Alexander fabric acted as a permanent shutter (framework)..." Chandler speaks of the family of fabric construction that includes... s... Pneumatic structures... Hydrostatic structures and... Shell structures derived from membrane form-finding. When faced with extremely complicated and complex shapes Heinz Isler and Antoni Gaudi used fabric as a modeling tool. These visionaries recognized that hanging chains and fabrics, forming catenaries, are in pure tension and when inverted are in pure compression and very stable. Gaudi, whose ing preceded the works of Candela... looked to nature and natural forms—an approach today called biomimicry..."

"Certainly the most striking contemporary example of a similar form [ arch] is to be found in St. Louis' ... In its incredible scale and construction out of metal plates this structure also serves as a convenient reminder of the important developments of the production of iron and steel that took place during the Industrial Revolution and that have so significantly affected arches as well as all other types of structural forms for the past 150 years."

"The arch is one of those brilliant innovations... Spanning... with horizontal beams is a losing game. ...By converting all the stress that fractures the middle of... stone beams—technically tension—into compression on stone piers larger... spaces could be spanned. ...But shift the pressure even slightly off center, and the pillar is likely to collapse. ...In their early incarnations, the limitations of both arch and dome was the ability of craftsmen to shape the stones carefully enough to create blocks precisely in the wedge shapes needed for a particular arch. Despite their mathematical sophistication in most other respects, the architects of antiquity lacked a proper geometric solution to the ideal form of the arch. (It was not until 1675 that the English polymath Robert Hooke described mathematically the shape of an arch loaded in pure compression, that is, with no tension, by showing how it describes an upside-down version of the catenary curve of a hanging chain.) As a result, the only way they could design an arch, and its component stones, was completely by eye, and... such tolerances commanded high prices. Rome overcame this drawback with typical ingenuity, first replacing stones and mortar... and expensive stonecutters with relatively cheap bricklayers. Even more ingeniously, some anonymous Roman builder found how to combine the mortar—in Latin pulvis puteoli—with lime, sand, and gravel to make the first concrete. ...The concrete domes of Rome were not surpassed until the age of steel."

"I love the catenary because it tells the story of holding up the roof."

"A non-catenary curve might be perfectly doable, but it takes more material, it has bigger beam sections, and overall it is much more complicated to construct... Even if the cladding falls out, the interiors and everything else falls away and the whole thing turns to dust and rubble and sand, [the catenary] should still stand."

"The most difficult properties of the Catenary were revealed before the close of the seventeenth century. This curve is entitled to particular attention, not only because it throws light on the theory of arches, but because it applies directly to the construction of suspended bridges, which are now deservedly coming into repute."

"The Catenary has its name from Catena, a chain; being the curve which a regular and very flexible chain will assume, if suspended loosely from both ends. It seems to have been first noticed by the famous Galileo, who proposed it as the figure of an arch of equilibration, but unfortunately mistook it for a . In fact, the Catenary, near its vertex, differs insensibly from that curve, but afterwards deviates more considerably. ...the Parabola diverges faster from its axis than the Catenary. The error of Galileo in confounding those two curves was not perceived till , in 1669, ascertained, by actual experiment, that the Catenary is neither a Parabola nor an . It was in 1691, that the penetrating genius of James Bernoulli discovered the true nature of the catenarian curve. A similar investigation was soon produced by John Bernoulli, by Huygens, and by Leibnitz. This latter philosopher, whose powers of invention and stores of learning were alike transcendant, discovered the fine relation of the Catenary to the Logarithmic Curve."

"I will begin with the subject of your bridge... and it is with great pleasure that I learn... that the execution of the arch of experiment exceeds your expectations. ...You hesitate between the catenary and portion of a circle. I have lately received from Italy a treatise on the equilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium. ...I would propose that you make your middle rail an exact catenary, and the interior and exterior rails parallels to that. It is true, they will not be exact catenaries, but they will depart very little from it; much less than portions of circles will."

"The true Mathematical and Mechanical Form of all manner of Arches for building with the true butment necessary to each of them, a Problem which no Architectonick Writer hath ever yet attempted, much less perform'd. ...Ut pendet continaum flexile, sic stabit contiguum rigidum, which is the Linea Catenaria."

"As Hooke saw in 1675 with his ut pendet continuum flexile, sic stabit contiguum rigidum inversum, ...a hanging chain may be inverted to give a satisfactory arch to carry the same loads, but working in compression rather than tension. The compressive arch, however, if of vanishingly small thickness, would be in unstable equilibrium, and stability is conferred in practice by making the arch ring of finite depth. Now if purely compressive forces, without bending, are to be transmitted from one portion of the arch to the next (as purely tensile forces are transmitted in the chain), then the arch centre line can accept only a single type of loading. Thus a parabolic arch can carry only a uniformly distributed horizontal load (although the magnitude of the load is arbitrary). It is the depth in a real arch which enables the arch to carry wider ranges of loading; a large number of different idealized centre-line arches can be contained within a given practical profile. ...[T]his must be so, or no mediaeval bridge would have survived its decentering."

"The flexible chain, hanging under the action of applied force, will assume a certain shape, namely the catenary if the chain is subjected only to its own weight, or a if the load is uniformly distributed horizontally. Whatever the load, there will be a corresponding shape, and the structural action in all cases is the same; purely tensile forces are transmitted along the centre line of the chain."

"It was comparatively late that the theory of arches attracted the notice of mathematicians. Dr. Hooke gave the hint, that the figure of a perfectly flexible cord or chain, suspended from two points, was the proper form for an arch. Galileo considered the catenary as a parabolic curve, and John Bernouilli appears to have been the first who discovered its nature. Dr. Gregory (Phil. Trans. 1697) published an investigation of its properties, and observes that the inverted catenary is the best form for an arch on account of its lightness. This is true so long as it is not pressed by an extraneous weight. It is not, however, capable of bearing a load on any part, much less of being filled up on the spandrels, which must be the case in practice. Other considerations must be involved before it can be fitted to receive a roadway or other weight, either upon its crown or haunches."

"What has been objected by an anonymous author, in the Leipsic Acts of Feb. 1699, in his animadversions on my demonstrations concerning the catenary, is this: that I have undertaken to demonstrate, after my manner, a matter found out and published by others seven years ago. This is true, and I cannot find any thing in it that is blame worthy. Those great men Huygens, Leibnitz, and Bernouilli, have discovered and communicated many properties of the catenaria, but without demonstration. I have contrived demonstrations, which was the thing I undertook to do. But was this matter that is the nature and primary properties of the catenaria all found out and published by others? ...From all ages architects have made use of arches in public buildings, as well for strength as beauty. Yet what was the true geometrical figure of an arch was not known before my demonstrations came out."

"Corol. 6.—In a vertical plane, but in an inverted situation, the chain will preserve its figure without falling, and therefore will constitute a very thin arch or fornix: that is, infinitely small, rigid, polished spheres, disposed in an inverted curve of a catenaria, will form an arch no part of which will be thrust outwards or inwards by other parts, but, the lowest parts remaining firm, it will support itself by means of its figure... none but the catenaria is the figure of the true and legitimate arch or fornix. And when the arches of other figures is supported, it is because in their thickness some catenaria is included. ...From Corol. 5... it may be collected, by what force an arch or buttress presses a wall outwardly, to which it is applied. For this is the same with that part of the force sustaining the chain, which draws according to a horizontal direction. For the force which in the chain draws inwards, in an arch equal to the chain drives outwards. All other circumstances, concerning the strength of walls to which arches are applied, may be geometrically determined from this theory, which are the chief things in the construction of edifices."

"Within a shed erected on the construction site of the church of the Sagrada family... Gaudí... made an upside-down model using lightweight cables to represent the structural lines of the future church—a model based on the structural notion of the inverted catenary. ...Analogically represented by little pouches filled with lead pellets the action of the stresses has been done ...The resulting chain configurations are used to determine the geometrical shapes and structural profiles of columns, pillars, arches, and vaults. ...Vicens Vilarrubias i Valls took photos of the model ...Gaudí used these photos upside-down to draw over them the external and internal elevations, studies of details and sections of the building."