Catenary

In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends. The curve has a U-like shape, superficially similar in appearance to a parabola, but it is not a parabola: it is a (scaled, rotated) graph of the hyperbolic cosine function. The mathematical properties of the catenary curve were first studied by Robert Hooke in the 1670s, and its equation was derived by Gottfried Leibniz, Christiaan Huygens and Johann Bernoulli in 1691. Catenaries and related curves are used in architecture and engineering, in the design of bridges and arches, so that forces do not produce a bending moment. The curve appears as a cross section of the surface of revolution of the catenary curve, the catenoid—the shape assumed by a soap film bounded by two parallel circular rings and a minimal surface, specifically a minimal surface of revolution.
The catenary is also called the alysoid, chainette, or, particularly in the material sciences, funicular.

Antoni Gaudí's catenary model
at Casa Milà

Quotes

• 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)

• In 1690... Jacob Bernoulli brought up the problem of the catenary in a memoir... in the Acta eruditorum...Huygens' solution represents the past... a complex, though skillful, geometrical method. Leibniz, using his new [infinitesimal calculus] reaches a correct analytical formula...$y/a=(b^{\frac {x}{a}}+b^{\frac {-x}{a}})/2$ where $a$ is [a] segment... and $b$... corresponds to... e... Johann Bernoulli ...supplied two correct constructions ...presents valid statistical arguments and... new and important... equations of equilibrium in differential form. ...In 1697-1698, Jacob Bernoulli was the first to derive the general equations that not only solved the problem, but also permitted the treatment of the more general theme of the equilibrium of a flexible rope, subject to any distribution of tangential ($f_{t}$) and normal ($f_{n}$) forces. Bernoulli's equations are...${\frac {dT}{ds}}+f_{t}=0,\qquad {\frac {T}{r}}+f_{n}=0$where $T$ is the tension, $s$ the curvilinear abscissa, and $r$ the radius of curvature.
  • Edoardo Benvenuto, An Introduction to the History of Structural Mechanics (1991) Part 1. Statics and Resistance of Solids, pp. 271-273.

• The largest catenary structure of masonry is the Great Hall of the Palace of Taq Kisra, at Qesiphon, then the capital of Persia.
  • Henry J. Cowan, A History of Masonry and Concrete Domes in Building Construction (1977)

• The trolley-wire must... be suspended with only a very small sag, and to obtain this result without excessive tension in the wire the span must be relatively short, i.e. of the order of 10 ft. to 15 ft. It is obvious that for these short spans the method of construction adopted in tramway practice would be unsuitable, both from the mechanical as well as the electrical standpoint. However, by adopting the catenary system—that is, supporting the trolley-wire from another wire, suspended with considerable sag between supports of moderate span—we are able to obtain a level trolley-wire with a relatively small number of supporting structures. The wire from which the trolley-wire is supported is called the "catenary" or "messenger" wire, and by insulating this wire from the supporting structures there is no necessity for insulated hangers on the trolley-wire.
  • Alfred Thomas Dover, Electric Traction: A Treatise on the Application of Electric Power to Tramways and Railways (1917)

• 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)

• 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 1800 Vol. 4 From 1794 to 1702 p. 184.

• 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.
  • 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 1800 Vol. 4 From 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.
  • Joseph Gwilt, A Treatise on the Equilibrium of Arches (1839)

• 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.
  • Jaques Heyman, Equilibrium of Shell Structures (1977)

• 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.
  • Robert Hooke, cypher at the end of his A Description of Helioscopes, and Some Other Instruments (1676) p. 31, as quoted in "The Life of Dr. Robert Hooke," The Posthumous Works of Robert Hooke, Richard Waller (1705) English translation in Ted Ruddock, Arch Bridges and Their Builders 1735-1835 (1979) p. 46

• 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.

• 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 Parabola. 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 Joachim Jungius, in 1669, ascertained, by actual experiment, that the Catenary is neither a Parabola nor an Hyperbola. 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.
  • Sir John Leslie, Geometrical Analysis, and Geometry of Curve Lines: being volume second of a course of mathematics, and designed as an introduction to the study of natural philosophy (1821)

• 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.
  • Sir John Leslie, Geometrical Analysis, and Geometry of Curve Lines: being volume second of a course of mathematics, and designed as an introduction to the study of natural philosophy (1821)

• 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.
  • Nikolai Malsch, as quoted by Alex Bellos, The Grapes of Math (2014)

• I love the catenary because it tells the story of holding up the roof.
  • Nikolai Malsch, as quoted by Alex Bellos, The Grapes of Math (2014)

• 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)

• 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... Robert Maillart... Pier Luigi Nervi... Felix Candela... Eladio Dieste... Heinz Isler... 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... Tensile structures... 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 Catalan vaulting preceded the works of Candela... looked to nature and natural forms—an approach today called biomimicry...
  • R. Schmitz, "Is there a future for fabric-formed concrete structures?", Structures and Architecture: Beyond their Limits (2016) ed., Paulo J. da Sousa Cruz, pp. 1087-1088.

Square wheel rolling over
inverted catenaries

• One can roll noncircular wheels over appropriate road surfaces. The most striking example of this is the fact that a square wheel 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 Exploratorium...
  • Stan Wagon, Mathematica® in Action: Problem Solving Through Visualization and Computation (2010)

External links

• "Catenary" MacTutor History of Mathematics archive, University of St Andrews.
• "A General Form of the Suspension Bridge Catenary", Bulletin of the AMS 31, (1925) pp. 420-424.