Architecture

"The exact shape of funicular (sometimes called linear or theoretical) arch[es] that carry all applied loads by axial compression only, may be developed by the same methods as used for finding the shape of cables. ...If the actual shape of the arch is different from funicular shape, the bending moment at any section of an arch is proportional to the ordinate or intercept between the given arch and funicular arch... This principle is called Eddy's Theorem."

"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."

"Centres, or centre-pieces of wood, are put by builders under an arch of stone while it is in the process of construction till the key-stone is put in. Just such is the use Satan makes of pleasures to construct evil habits upon; the pleasure lasts till the habit is fully formed; but that done, the habit may stand eternal. The pleasures are sent for firewood, and the hell begins in this life."

"Wherefore a monk's whole attention should thus be fixed on one point, and the rise and circle of all his thoughts be vigorously restricted to it; viz., to the recollection of God, as when a man, who is anxious to raise on high a vault of a round arch, must constantly draw a line round from its exact centre, and in accordance with the sure standard it gives discover by the laws of building all the evenness and roundness required...."

"Why do you speak to me of the stones? It is only the arch that matters to me." Polo answers: "Without stones there is no arch."

"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)."

"Granted a Church, Saint Thomas's Church was the most expressive that man has made, and the great gothic Cathedrals were its most complete expression. Perhaps the best proof of it is their apparent instability. Of all the elaborate symbolism which has been suggested for the gothic Cathedral, the most vital and most perfect may be that the slender nervure, the springing motion of the broken arch, the leap downwards of the flying buttress,— the visible effort to throw off a visible strain,— never let us forget that Faith alone supports it, and that, if Faith fails, Heaven is lost. The equilibrium is visibly delicate beyond the line of safety; danger lurks in every stone. The peril of the heavy tower, of the restless vault, of the vagrant buttress; the uncertainty of logic, the inequalities of the syllogism, the irregularities of the mental mirror,— all these haunting nightmares of the Church are expressed as strongly by the gothic Cathedral as though it had been the cry of human suffering, and as no emotion had ever been expressed before or is likely to find expression again. The delight of its aspirations is flung up to the sky. The pathos of its self-distrust and anguish of doubt, is buried in the earth as its last secret. You can read out of it whatever else pleases your youth and confidence; to me, this is all."

"...we must go to the poets to see what they all meant by it; but the sum is an emotion — clear and strong as love and much clearer than logic,— whose charm lies in its unstable balance. The Transition is the equilibrium between the Love of God,— which is Faith, and the Logic of God,— which is Reason; between the round arch and the pointed. One may not be sure which pleases most, but one need not be harsh towards people who think that the moment of balance is exquisite. The last and highest moment is seen at Chartres where, in 1200, the charm depends on the constant doubt whether emotion or science is uppermost."

"Like all great churches, that are not mere store-houses of theology, Chartres expressed, besides whatever else it meant, an emotion, the deepest man ever felt,— the struggle of his own littleness to grasp the infinite. You may, if you like, figure in it a mathematic formula of infinity,— the broken arch, our finite idea of space; the spire, pointing, with its converging lines, to Unity beyond space; the sleepless, restless thrust of the vaults, telling the unsatisfied, incomplete, overstrained effort of man to rival the energy, intelligence and purpose of God. Thomas Aquinas and the schoolmen tried to put it in words, but their church is another chapter. In act, all man's work ends there;— mathematics, physics, chemistry, dynamics, optics, every sort of machinery science may invent,— to this favor come at last, as religion and philosophy did before science was born."

"[B]uilding stones and concrete exhibit the properties of elasticity, although not so perfectly as to permit a constant ratio between stress and strain... For compression within the limits of safe stress, however, such proportionality may be assumed without any considerable error so that a constant coefficient of elasticity may be used... The theory of the elastic arch may therefore be applied to arches of all classes of masonry, including monolithic arches of concrete, provided no tensile stresses or only very small tensile stresses are allowed to occur. The same theory also forms the basis for the design of reinforced concrete arches..."

"He talked to her endlessly about his love of horizontals: how they, the great levels of sky and land in , meant to him the eternality of the will, just as the bowed Norman arches of the church, repeating themselves, meant the dogged leaping forward of the persistent human soul, on and on, nobody knows where; in contradiction to the perpendicular lines and to the Gothic arch, which, he said, leapt up at heaven and touched the ecstasy and lost itself in the divine."

"Like two cathedral towers these stately pines Uplift their fretted summits tipped with cones; The arch beneath them is not built with stones, Not Art but Nature traced these lovely lines, And carved this graceful arabasque of vines; No organ but the wind here sighs and moans, No sepulchre conceals a martyr's bones, No marble bishop on his tomb reclines. Enter! the pavement, carpeted with leaves, Gives back a softened echo to thy tread! Listen! the choir is singing; all the birds, In leafy galleries beneath the eaves, Are singing! listen, ere the sound be fled, And learn there may be worship without words."

"There is [a].. type of structural behaviour which is not beam-like, truss-like or frame-like but funicular. ...from the Latin for rope - funis. ...the cable is flexible and can only have internal forces of axial tension. ...a cable is not a frame ...Like any spanning structure it has to carry the overall and s. ...If a cable is loaded with a uniformly distributed load, the cable will take up a parabolic shape. ...the funicular shape for this load pattern. ...Because... cables are in direct tension, if they were turned upside down they would be in direct compression. ...this would not be possible for a cable but if the structure could carry compression then the funicular shape obtained from the hanging cable gives the correct shape for an arch that is in direct compression everywhere. ...the idea of inverting cables to find arch shapes was only stated in 1675 by ...Robert Hooke... G. Poleni in 1748 as part of his investigation into the structural behaviour of the dome at St. Peter's... used a correctly loaded chain to determine the funicular shape... If...the loading changes or the arch is built to the wrong shape and the funicular line moves outside the arch, then the arch will have to maintain its shape by frame action or collapse."

"So counsel'd he, and both together went Into the thickest wood; there soon they chose The fig-tree, not that kind for fruit renowned, But such as at this day to Indians known In Malabar or Decan spreads her arms, Branching so broad and long, that in the ground The bended twigs take root, and daughters grow About the mother tree, a pillar'd shade High overarch'd, and echoing walks between."

"Arches form a distinct class of two-dimensional structural elements that resist external loads through their profile (form). Compared to a beam element of the same span and subjected to the same load, the B.M. in an arch will be much smaller because of the negative B.M. due to the horizontal thrust at the supports (abutments). Graphical solution of arches is much simpler than the analytical solution, and is of adequate accuracy for practical purposes. The solution is based on Eddy's theorem on B.M. in arches and the concept of pressure (thrust) lines. ...In case the structure has the profile of the force polygon, the B.M. at any section will be zero... The structure, in such a case, will be subjected only to axial compression. Such a profile along the length of a beam or frame is known as the pressure line or line of thrust. ...the profile for a given system of forces, which would induce only compressive forces. The profile of a pressure line resembles an arch with linear segments; the profile is sometimes known as a linear arch."

"The Romans were the first builders in Europe, perhaps the first in the world, fully to appreciate the advantages of the arch, the vault and the dome."

"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."

"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."

"French architects and engineers in the 16th, 17th, and 18th centuries occupied themselves a good deal with roofs with curved ribs, and two systems of constructing the rib were worked out. In the most modern of them, that invented by Colonel Emy, the ribs were constructed of a series of thicknesses of bent timber, one on the back of another, and held together by bolts. In the older system that of Philibert de l'Orme, the ribs were also built up, but the pieces composing them are placed side by side, and either form a polygon approaching a semicircle or are cut to bring them to a curve.&bnsp;thumb|Bourse de commerce (dome of the Paris Corn Market)In fact, the ribs are very much such as... used for the great dome of the Paris Corn Market. There is, however, a great difference between a dome—the strongest of all forms—and one permitting the introduction of as many rings of ties as may be desired; and a roof over an ordinary oblong space, where no such binding together is admissible, and where straight rafters may have to be used, which loads the rib at certain points only. In the latter case, a good many precautions have, generally speaking, to be taken to prevent the rib from being unequally loaded, and so either spreading or losing its shape in some other way. The rib made of unbent timber, side by side, on De l'Orme's plan, is admitted to be stronger than the one made of bent timbers laid one on the back of the other; but both have been largely used, and good examples of both may be met with, even if we confine ourselves to English ones alone, and leave the French ones unnoticed. thumb|Chatsworth - Great Conservatory in the 19th centuryA very fine roof with ribs, one on which the load (though light) is borne without a rafter solely by the rib, is the one erected over the great conservatory built by His Grace the Duke of Devonshire, at Chatsworth. ...It consists of a wide and lofty central portion, with a kind of broad aisle at the sides, roofed at a lower level. The central roof here is of the section of a pointed arch and hipped at both ends, and is entirely covered with glass. It is carried by timber ribs, and the glazing is on the ridge and furrow principle. The low aisle referred to forms to some extent an abutment for the ribs, and the ridge-and-furrow glazing helps, no doubt, to fortify them, but still the greater part of the strength is derived from the ribs themselves."

"All building naturally divides into two classes, the architecture of the beam and that of the arch, which have been called trabeated and arcuated—according to the means which it uses for covering openings and spaces—the first being that which covers them by beams or s, the second that which uses arches. The logical development of the two classes leads in the one to flat ceilings and straight roofs, in the other to vaults and s."

"[A]long with the order, the architecture of Rome had inherited from the Etruscans the arch, despised and rejected by the Greeks... It was probably the child of the bricklayer, who has no other means of bridging an opening; at least we find it first in alluvial Mesopotamia, where the Chaldees, who had no stone to build with, raised their great pyramids and built their palaces of bricks, and where the Assyrian conquerors who appropriated their civilization and art, as the Romans did the Greek, adopted it from them and used it on a great scale. Born in the oriental brick-fields, it came to the Greeks with all the associations of ignoble material, profane uses, and hated sponsors. Every influence of religious association, conservatism, and respect for the Egyptian example, from which they had learned much, bound them to their trabeated style. Still more, the instinct for harmony of form which dominated both Egyptians and Greeks could but warn them that the use of the arch not only implied a change of their constructive system, but was at war with their whole architectural scheme of lines, proportions, and monumental effect. Even as late as the time of , after long subjection of Greece to Roman control, the arcaded conduit to the at Athens seems to show the persistent resistance of Greek workmen on their own soil to the very principle of the arch, for the arches are cut through solid slabs of stone instead of being built up in the fashion of the true arch."

"The recognized leading position of the author in this field of structural design and the extensive use of his system of arch construction in all parts of the world should be sufficient justification for presenting this book... The present work constitutes one of the most thorough treatments of reinforced concrete arches in any language. ...After a brief discussion of the fundamental principles of arches and a simple though comprehensive treatment of the stresses in reinforced concrete sections, there are presented analytic and graphic methods for the complete design of all types of concrete arches occurring in practice. The graphic methods which are given, permitting the use of influence lines, will be found very practical... The effects of temperature, of yielding abutments and of nonvertical loads are separately considered. ...In addition to the exact treatments, simple approximations and short cuts are introduced which will prove highly useful for preliminary and less exacting designs. Easily-applied formula are developed for determining in advance the best curve for an arch and the required dimensions and reinforcement. ...The principle of the Melan system of arch construction is fully explained and its inherent economy concretely demonstrated."

"On the arch strip... we have certain forces acting; usually... forces of gravity... vertical loading. ...the dead weight of the arch and superstructure together with the useful or live load. Now consider the archstrip (Fig 1) divided into separate segments or s by joint-planes or sections 1, 2, 3, ..., with the end planes A and B resting against fixed abutments. With the... assumption of no shears in the head planes, each voussoir, e.g., 1-2, is subjected to the action of three forces, namely, the given external load P2 and the two resultant pressures R1 and R2 in the abutting joints. These forces must hold each other in equilibrium and may therefore be represented by a force triangle 1-2-O; in this way either joint-pressure may be determined when the other is known. Consequently all of the joint-pressures may be found as soon as any one of these forces is given in its magnitude, direction, and point of application. The forces acting at the end planes A and B are the pressures, and their opposing forces are the abutment reactions K1 and K2. The application of these reactions takes the place of the abutments, so that the arch may be considered an independent system in equilibrium under the forces \sum P, K1 and K2."

"The action and reaction between adjacent voussoirs occur along the lines of the forces R, therefore the polygon composed of these forces is appropriately named the Line of Resistance. It may be obtained as the funicular polygon of the forces P with the end-reactions K as the terminal sides."

"And Jacob rose up early in the morning, and took the stone that he had put for his pillows, and set it up for a pillar, and poured oil upon the top of it. And he called the name of that place Beth–el: but the name of that city was called Luz at the first: that is, The house of God. And Jacob vowed a vow, saying, If God will be with me, and will keep me in this way that I go, and will give me bread to eat, and raiment to put on, So that I come again to my father's house in peace; then shall the Lord be my God: And this stone, which I have set for a pillar, shall be God's house: and of all that thou shalt give me I will surely give the tenth unto thee."

"Fairbairn has deduced the following formula from experiments made mostly on very thin cylindrical tubes of various lengths and diameters—viz., for wrought-iron cylindrical tubes let l = the length, d = the diameter, and t the thickness of the shell, all expressed in the same unit of measure, and let p = the collapsing pressure in pounds per unit of area; thenp = 9,672,000 \frac{t^{2.19}}{l d}. { IV.}In case a tube is stiffened by T-iron rings or by flanges, l represents the distance between two such adjacent rings or flanges."

"Resistance of Cylindrical Shells to an External Fluid Pressure.—Thin hollow cylinders exposed to an external fluid pressure never give way by direct crushing, but by collapsing; it may be assumed that, other things equal, the resistance of tubes to collapsing is greater as their form is more truly cylindrical and their shell more perfectly homogeneous."

"To find the value of p which would split the cylinder [of unity length] from end to end... The force tending to rupture such a ring at the sections formed by any diametrical plane is given by formula:F = 2rp,and the area of these sections byS = 2t.The bursting pressure is, therefore, found from the equation:2rp = 2tk; hence p = \frac{tk}{R}...only half as great as... equation { II.}"

"The force, F, producing the first-named tension is represented by the formula:F = \pi r^2 p;and the sectional area, S, of a thin shell resisting this force may be represented with sufficient accuracy, as in the case of thin spherical shells, by the formula:S = 2 \pi rt.The value of p, when it becomes the bursting pressure, is found from the equation,r^2 \pi p = 2 \pi rtk; hence p = \frac{2tk}{r} { II.}the same as that of a spherical shell of equal radius and thickness."

"Resistance of Cylindrical Shells to an Internal Fluid Pressure.—The tension produced in a cylindrical shell by an internal fluid pressure may be considered as being of two different kinds—viz., first, a tension acting in a longitudinal direction, tending to pull the ends of the cylinder apart; and, secondly, a tension acting in a diametrical direction, tending to split the cylinder from end to end."

"The area of a [circular] diametrical section, S, of a thin spherical shell is very nearly given by formula:S = 2 \pi rtwhen t represents the thickness, and r = the inner radius of the shell, and t is supposed to be very small compared with r. The whole force, F, to be resisted by the tenacity of section S is equal to the excess of the internal fluid pressure per unit of area over the external pressure, into the area of the plane passing through this section, orF = \pi r^2 pAssuming that every portion of section S is equally strained by F, and designating by k the coefficient of the ultimate tenacity of the material of which the shell is composed, the bursting pressure will be found from the equation: \pi r^2 p = 2 \pi rtk; hencep = \frac{2tk}{r} { I.}and the proper ratio of the thickness to the radius of a thin hollow sphere is given by the formula:\frac{t}{r} = \frac{p}{2k}..."

"The hollow sphere encloses the largest space in proportion to the superficial area of its shell, and all vessels that are not spherical, exposed to an internal fluid pressure, experience distortion on account of their tendency to assume the spherical form. A hollow sphere, having a shell of uniform thickness composed of a homogeneous material, experiences the same tension at all sections of metal formed by diametrical planes."

"Resistance of Spherical Shells to an Internal Fluid Pressure.—An elastic fluid contained in a closed vessel presses each unit of area of the surrounding walls with equal force. The resistance offered by the walls depends on their superficial area, their form, their thickness, and the coefficient of resistance of the material."

"Shells under compressive loading investigated under the assumption of perfect properties may be considered to be optimal structures. Their load carrying capacity is significantly larger compared to shells which show deviations in geometry, material behaviour, loading and boundary conditions. ...Unfortunately, comparatively little quantitative information exists about the initial imperfections in actual structures... One possibility to improve this situation is to perform systematic numerical simulations... Classical numerical concepts of the load carrying capacity of imperfect structures focus on the model of a perfect shell configuration and on the analytical estimation of unstable, postcritical equilibrium paths. This was first demonstrated by Koiter, whose postbuckling theory describes the nonlinear static load carrying behaviour of structures in the initial stages of buckling. ...[I]nitial unfavourable imperfections will lead to a reduction in load carrying capacity. This approach has certain restrictions as the results are evaluated by linearisation around the bifurcation point of the perfect shell. For the numerical simulation of the load carrying behaviour of imperfect shells it is commonly assumed that the initial geometric imperfections have the shape of the lowest bifurcation mode of the respective shell. ...In the cases of high imperfection sensitive shells [with] multi-mode-buckling... the lowest bifurcation mode is not always the ”worst” imperfection shape. Recently, a specific concept employing finite element procedures... directly evaluates the ”worst” imperfection shape and [is based upon] analysis of the imperfect shell space."

"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..."

"[T]he basic assumption in the linear theory of shells that the displacements of the shell are considered to be small in comparison to the thickness is abandoned in the present nonlinear analysis of shells. A shell is called thin if the maximum value of the ratio h/R, where h is the thickness of the shell and R is the principal radius of curvature of the middle surface... is less than or equal to 1/20 ...beyond this range... the shell is regarded as thick. ...in a large number of practical applications the ratio... lies in the range between 1/50 and 1/1000, making the theory of thin shells of great practical importance. In this chapter the nonlinear equations... in terms of orthoganal are derived assuming the material... is isotropic, homogeneous, and elastic. An important simplification based on the assumption that second invariant of the median surface strains in the expression for the extensional ... can be neglected, originally made by [Noah] Burger, is introduced to derive a simplified set of nonlinear... differential equations."

"It was the great nineteenth century mathematician, Carl Gauss who proved mathematically that any curved surface, natural or man-made, can be characterized as only one of three different possible shapes: as -like, -like, or saddle-like. All three of these geometric shaped can be used as the basis for thin-shell structures."

"There is... [a] need to code cheaper and accessible programs in line with using sustainable methods to better the livelihood of mankind. To address this issue a theory is formulated based on the Euler-Bernoulli beam model. This model is applicable to thin elements which include plate and membrane elements. This paper illustrates a finite element theory to calculate the master stiffness of a curved plate. The master takes into account the stiffness, the geometry and the loading of the element. The of this is established from which the load which is unknown in the matrix is evaluated by the principle of bifurcation."

"Like Candela, Isler has concentrated his practice on thin s... Isler derives his forms not from analytical geometry (as were Candela's hypars) but directly from physical and funicular models - flexible membranes that assume the least energy, or minimal surface, for a specific boundary and force patterns. In the mid-1950s Isler invented two new form-making techniques, the first by using pneumatic models and the second by experimenting with hanging cloth models sprayed with water and put out to freeze in wintertime. Later, in 1965, he added a third technique that made shapes "by the flow method, by... the advancing velocity of a liquid inside a tube... At the wall, velocity is zero because of friction, whereas in the center there is maximum velocity... and forms a dome shape." While the frozen cloths conform to the funicular shape given by gravity, the other methods, pneumatic and flow, are hydraulic."

"Cement Hall, Swiss National Exhibition, Zurich (1939)... was built by architect and engineer Hans Leuzinger and ... to demonstrate the potential of thin shells. ...The width of this parabolic shell is 50.32 ft... built by the Gunite method, [it] is only 2.36 in thick. ...A sculptural piece by Robert Maillart entititled "Endless Ribbon" [1935-1936]... is a thin shell built of reinforced concrete. This... can be classified within Constructivism or... constructive spatial art. [I]t represents... the direct correlation between the language of sculpture and that of modern thin-shell architecture that moves the sensitive engineer such as Maillart to express himself also as a sculptor. Maillart's structural shells as a whole testify to this similarity of language... He was the father of flared columns connecting with floor slabs to eliminate supporting beams and designed unique bridge forms. Maillart's name remains associated with innovative concrete forms that extend to the structural virtuosity of thin shells."

"My invention shows a new product which helps to replace timber where it is endangered by wetness, as in wood flooring, water containers, plant pots, etc. The new substance consists of a metal net of wire or sticks which are connected or formed like a flexible woven mat. I give this net a form which looks in the best possible way, similar to the articles I want to create. Then I put in hydraulic cement or similar bitumen tar or mix, to fill up the joints."

"The meandor compression spring... or corrugated spring... is a promising design variant for compression axle springs of composite material... The middle surface of the meander spring is the... "." ...[T]he middle surface ...is the ...[i.e.,] can be unrolled onto a flat plane without stretching or tearing it. ...[A] developable surface ...in three-dimensional space and ...complete ...is necessarily ruled. This property is of primary importance for manufacturability... The meander spring... is modeled for mechanical analysis as a thin shell."

"Galankin... writes that Shukhov used to make calculations in a unique way... so laconic... that other specialists found them difficult to understand. In spite their brevity... if Shukhov was asked about load, rod stress, rod profile and section, rivet quantity, material weight, temperature impact or any other specifications, he always had an answer, because his concise calculations covered all these aspects, but nothing that was irrelevant..."

"In 2010, the journal Detail published an analysis of Shukhov’s constructions, calling his approach to design 'an early example of ’..."

"Shukhov’s lattice-suspended and vaulted structures represented a carrying surface, which could be shaped in any form. ...The density of the grid made it possible to attach it to the shell without additional structures. ...[T]he grids were two to three times lighter than roofs with conventional frames..."

"Suspension lattice structures... were based on tension, the most advantageous type of stress for metal constructions... designed according to Shukhov’s elaborate investigations of material properties. ...They were called ‘roofs without trusses’... and the clear, extremely simple suspension structure system and the easy-to-perform node conjunction made on-site construction fast and straightforward."

"According to the formula, the minimal covering weight could be achieved only if the construction had no s and if the distance between es was equal to the distance between the missing purlins. The answer to this riddle was the spatial lattice structure, where trusses and purlins were the same, and the distances between trusses and purlins were equal. In 1895 Shukhov obtained a patent for the invention. The new structures were first presented to the general public at the All-Russian Industrial Art Exhibition in Nizhniy Novgorod in 1896... where Shukhov designed a number of objects using three types of lattice structure: suspension, vaulted and rigid spatial shell."

"One of Shukhov’s most significant architectural inventions was the thin metal lattice (or ' shell') structure. This was developed after intense research into the most rational type of rafters which weighed and cost the least and which could be assembled quickly. Shukhov suggested a formula to define the proportional relationships between structural elements, which at first sight seemed senseless:α = e = c'where a- length of panels, e - minimal distance between frames, and с - distance between two purlins, dependent on the actual situation"

"Looking back to the past century, there were various visionary designers such as Isler, Otto, Candela and Shukhov using design methods comparable to today’s computational design, operating in a pre-computational parametric set-up. They applied mathematical algorithms and principles observed in nature to develop pioneering buildings... as milestones of architectural production. Vladimir Shukhov... was one of the first to embrace such an innovative approach."

"Now what got me into shells... was a building in South America... and it absolutely enchanted me. It was in ... a baseball stadium... doubly curved... I saw this thing, and all I'd ever known... the buildings that were being built in those days were boxes, very uninspired things, and here I see this picture, this gorgeous shell... and it just enchanted me. I thought, "My God, what a beautiful structure, but how on earth does it stand up? How did the guy do it?" ..[T]hat is what really made me decide, "I've got to get the answer to this. How is this done?" [T]hat's what set me off on years and years of searching before I finally found a way to do it. ...[W]hen I finally did a building... I searched around into the theory of shells until I understood the theory... [T]hen I started looking for ways of solving these things, and I found out that they did not exist in the... architectural structural engineering profession... [F]ortunately at that time... we had here a very huge aerospace industry, and I had friends at Lockheed... I got in touch with a guy at Lockheed... and they had ways of solving very difficult problems because they were building the rockets... and so I found this way of doing it... [I]t was a very tedious, complex way of doing it, but you would get an actual, true answer... [I]t was done... the same way that it's done today with computers... But you could do it my way... You took two guys... I was one of them... they go through it step by step, and then [periodically]... compare the answer... to make sure that they don't go off. ...You could get an answer. You don't get it down to the forth of fifth decimal... The smallest I could get, chunks of the shell to work with, would be about [a theoretical] 2 feet square... Today, with a computer, you can get down to 2/10 of an inch... But it was such a tedious, horrible way of working that I never used the system again. By then I learned enough that I knew ways of short-cutting that were not as accurate... but that were close enough... [B]y this time I had developed a pretty good feel... and so I'd cut shells in pieces in my mind, put them back together again, and put them in equilibrium... From this, I finally found my way of doing things. ...As far as I know, nobody had ever done it that way... They'd done aerospace stuff this way... but there they'd put ten guys on it, not two. ...Several times I did things that I thought were the first, but I just said that "It's not universally used, or known..." or something like that. ...You're doing something that you're not really aware of yourself, which is, you're developing a very sophisticated, intuitive way of working with these things, where you can just... take them and you can say it doesn't look right... which was the way architecture was done all through the Middle Ages... in fact, in Antiquity... they didn't have engineers, all they had were architects."