An arch is a curved structure that spans a space and may or may not support weight above it. Arch may be synonymous with vault, but a vault may be distinguished as a continuous arch forming a roof. Arches appeared as early as the 2nd millennium BC in Mesopotamian brick architecture, and their systematic use started with the Ancient Romans who were the first to apply the technique to a wide range of structures.
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.
...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.
Henry Adams, Mont Saint Michel and Chartres (1904) Chapter XV The Mystics
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.
Henry Adams, Mont Saint Michel and Chartres (1904) Chapter XVI "Saint Thomas Aquinas." The closing lines of the book. In a letter to William James (17 February 1908), Adams wrote with customary self-deprecation: "If you will read my Chartres,— the last chapter is the only thing I ever wrote that I almost think good." (J. C. Levinson et al. eds., The Letters of Henry Adams, Volume VI: 1906–1918. Cambridge, MA: Belknap Press, 1988, p. 121)
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).
Alex Bellos, Alex Through the Looking Glass: How Life Reflects Numbers, and Numbers Reflect Life (2014)
"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."
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....
John Cassian, The Conferences of John Cassian (1894)
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.
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.
Marco Frascari, Eleven Exercises in the Art of Architectural Drawing: Slow Food for the Architect's Imagination (2011)
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.
M.L. Gambhir, Fundamentals of Structural Mechanics and Analysis (2011)
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.
David Gregory, "The Properties of the Catenaria or Curve Line formed by a heavy and flexible Chain hanging freely from two Points of Suspension," Philisophical Transactions No. 231 (1697) p. 637, in The Philosophical Transactions Of The Royal Society Of London From Their Commencement In 1665 To The Year 1800Vol. 4From 1794 to 1702 p. 184.
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.
David Gregory, "Answer to the Animadversions concerning the Catenary," (1699) Philisophical Transactions No. 259 p. 419, in The Philosophical Transactions Of The Royal Society Of London From Their Commencement In 1665 To The Year 1800Vol. 4From 1794 to 1702 p. 456.
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.
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 parabola 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. 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 voussoir 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.
Jaques Heyman, Equilibrium of Shell Structures (1977)
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.
Tr: As hangs the flexible line, so but inverted will stand the rigid arch.
History fades into fable; fact becomes clouded with doubt and controversy; the inscription moulders from the tablet: the statue falls from the pedestal. Columns, arches, pyramids, what are they but heaps of sand; and their epitaphs, but characters written in the dust?
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.
Thomas Jefferson, Letter to Thomas Paine (Dec. 23, 1788) Memoirs, Correspondence and Private Papers of Thomas Jefferson, ed., Thomas Jefferson Randolph (1829) Vol. 2, pp. 418-419.
He talked to her endlessly about his love of horizontals: how they, the great levels of sky and land in Lincolnshire, 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 bending moment and shear forces. ...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.
Malcolm Millais, Building Structures: From Concepts to Design
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.
D.S. Prakash, Graphical Methods in Structural Analysis (1997)
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.
D.S. Robertson, Greek and Roman Architecture (1943) 2nd edn., p.231.
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.
William Rosen, Justinian's Flea: The First Great Plague and the End of the Roman Empire (2007)
Certainly the most striking contemporary example of a similar form [weighted catenary arch] is to be found in St. Louis' Gateway Arch... In its incredible scale and construction out of metal plates this structure also serves as a convenient reminder of the important developments of 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.
Bjorn N. Sandaker, Arne P. Eggen, Mark R. Cruvellier, The Structural Basis of Architecture (2013)
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;
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. 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.
B. Priestley Shires, "Domestic Architecture: England under the Edwards" (May 29, 1885) The Illustrated Carpenter and Builder, Vol. 16-17, " p. 355.
The Column and the Arch: Essays on Architectural History (1899)
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 lintels, 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 domes.
[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 Hadrian, after long subjection of Greece to Roman control, the arcaded conduit to the Tower of the Winds 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 voussoirs 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 abutment 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 , 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.
[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...
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