Cable Catenary

The shape a chain decides for itself

Take a chain, hold the two ends at the same height, let it dangle. The curve it traces is a catenary — from catena, Latin for chain. The defining property is that the chain carries only tension along its own tangent: no shear, no bending moment, no compression. Each link is in equilibrium under three forces — its own weight, and the tensions pulling up-and-out along the chain at either end of the link. The configuration that satisfies that local balance everywhere along the chain is the catenary, and its equation in the natural coordinates (origin at the lowest point, y axis vertical, x axis horizontal) is

y = a cosh(x/a) − a            (often written y = a cosh(x/a) with origin shifted)
  = (a/2)·(e^(x/a) + e^(−x/a)) − a

The parameter a has the dimensions of length and is

a = H / (w · λ)

H = horizontal component of cable tension (constant along the cable)
w = gravitational acceleration × (mass per unit length)   [N/m]
λ = a normalization, typically taken as 1 — sometimes the linear weight density itself

So a is the ratio of horizontal pull to weight-per-length. A large a — strong pull, light cable — gives a shallow, nearly straight curve. A small a gives a deep U. At the lowest point, the radius of curvature is exactly a. Far from the lowest point the catenary asymptotes to two straight lines y = ±(x − x₀) with slope 1, so the curve looks like a V at distance and a smooth bowl up close. Every catenary is a scaled copy of every other — the family has only the single parameter a.

Galileo's mistake, Jungius's correction, Bernoulli's victory

The story is one of the cleanest sequences of error and repair in the history of mathematics. In the 1638 Discorsi, Galileo writes, almost in passing, that a hanging chain takes the shape of a parabola. He is wrong, but it was a natural error: a chain looks like a parabola, particularly when shallow. The German polymath Joachim Jungius, working through the geometry carefully, proved by 1639 that the parabola was not in fact the right answer — but he failed to derive the actual curve and his notes were not published until after his death in 1657.

The problem sat open for half a century. In May 1690 Jakob Bernoulli, having heard the story, posed the catenary as an open challenge in the Leipzig journal Acta Eruditorum. The challenge ran almost exactly a year before three solutions arrived simultaneously in the June 1691 issue: from his younger brother Johann Bernoulli, from Gottfried Wilhelm Leibniz, and from Christiaan Huygens. All three solutions are correct; all three use slightly different methods. Huygens coined the name catenaria. Johann Bernoulli later wrote, in a letter to a friend, that he had solved the problem in one night while his brother Jakob (who had set it) was still wrestling with it. The brothers had a feud, and the catenary was one of its battlefields.

It is striking how recent this is. Galileo died in 1642; Newton was born the same year. The calculus had only just been invented when the catenary was solved — Leibniz's notation paper appeared in 1684, six years before he wrote the curve down.

A short derivation

Cut a short piece of cable between horizontal position 0 (the low point) and x. Let s(x) be the arc length along the cable, and let H be the horizontal tension at the low point. By symmetry the tension at the low point is horizontal. At the other end of the piece, the tension is a vector T pointing along the cable tangent.

Three forces balance: horizontal pull H at the low end, tangent tension T at the upper end, and the weight w·s pulling straight down (where w is the weight per unit arc length). The horizontal component of T must equal H, so T cos θ = H. The vertical component must equal the weight, so T sin θ = w·s. Dividing,

tan θ = dy/dx = w·s / H

Differentiate with respect to x. Since (ds/dx)² = 1 + (dy/dx)²,

d²y/dx² = (w/H) · √(1 + (dy/dx)²)
       = (1/a) · √(1 + y'²)        with a = H/w

This ODE has the closed-form solution

y(x) = a cosh(x/a)   (with y(0) = a at the low point, and y'(0) = 0)

From this falls everything else:

y'(x)  = sinh(x/a)
s(x)   = a · sinh(x/a)               arc length from low point
T(x)   = H · cosh(x/a) = H + w·(y − a)   ≈   H + w·y     (shifting origin)
T_horizontal(x) = H                     constant along the cable
T_vertical(x)   = w·s = w·a·sinh(x/a)   grows with arc length

Note the cleanest consequence: the tension at any height y is just H + w·y. Add the weight of the cable above you, push horizontally, and that's the tension. This is why the support tensions are so much larger than the horizontal tension at the bottom — they carry the full self-weight of the cable above.

Catenary versus parabola — when does it matter

The parabola y = (x²/(2a)) and the catenary y = a·(cosh(x/a) − 1) agree to second order around the low point. Series-expanding:

a·(cosh(x/a) − 1) = x²/(2a)   +   x⁴/(24 a³)   +   x⁶/(720 a⁵)   + …
parabola y         = x²/(2a)

The first deviation is the x⁴ term. So as long as x ≪ a — meaning span/sag is large — the two curves are visually identical. The crossover where engineers begin to care is around sag/span ≈ 1/10:

Sag / Span	Max parabola–catenary deviation	Practical regime
1/20	≈ 0.05 %	Suspension-bridge main cable; safe to call it a parabola
1/10	≈ 0.4 %	Edge of indistinguishability
1/5	≈ 1.7 %	Power lines on a hot day — measurable
1/2	≈ 13 %	Decorative chain; catenary clearly visible
1 (vertical)	diverges	Mooring line dropping near-vertically into the seabed

The other distinction is mechanical, not just visual. A cable that carries a load uniform in horizontal distance — most famously a suspension-bridge deck — is genuinely a parabola; it is not an approximation. A cable that carries only its own weight is genuinely a catenary. The "approximation" is when we treat a main suspension cable as a parabola because the deck dominates the loading and the cable's own weight is negligible. The same physical span, drained of its deck, would re-settle as a catenary.

Worked example: power line between two towers

Consider an aluminum-conductor steel-reinforced (ACSR) transmission cable of mass 1.6 kg/m, hung between towers 300 m apart at the same height, with a designed sag of 9 m at midspan.

Self-weight per unit length:

w = m · g = 1.6 kg/m × 9.81 m/s² = 15.7 N/m

The shape parameter a comes from solving sag = a·(cosh(L/(2a)) − 1) with sag = 9 m and L/2 = 150 m. For shallow sag we have the approximation

sag ≈ (L/2)² / (2 a)    →    a ≈ (L/2)² / (2 · sag) = 150² / 18 = 1250 m

So the horizontal tension is

H = w · a = 15.7 N/m × 1250 m ≈ 19.6 kN

Arc length over the full 300 m span:

S = 2 a sinh(L/(2a)) = 2 · 1250 · sinh(0.12) ≈ 300.7 m

so the cable is 70 cm longer than the horizontal distance between the supports. Tension at each tower:

T_support = H + w · sag = 19.6 kN + 15.7 · 9 / 1000 kN ≈ 19.7 kN

Less than one percent above the horizontal tension — that is how shallow this curve is. Now warm the cable by 40 K (a hot summer afternoon plus heavy load current). Aluminum has a coefficient of linear thermal expansion of about 23 × 10⁻⁶ /K, so the cable lengthens by

ΔS = α · ΔT · S = 23e−6 · 40 · 300.7 ≈ 0.28 m

Solving for the new sag at fixed span (S becomes 300.98 m) gives sag ≈ 10.5 m — a meter and a half lower. That meter and a half is exactly the kind of margin that gets eaten in a long hot summer, and is exactly the kind of margin that, eaten through, started the August 2003 Northeast blackout when an Ohio 345 kV line sagged into a tree.

Inverted catenary: the ideal arch

In December 1675 Robert Hooke published an anagram in his book A Description of Helioscopes: abcccddeeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux. It was meant to stake priority on a result he wasn't yet ready to publish in full. After his death in 1703, the decoded sentence appeared in print:

Ut pendet continuum flexile, sic stabit contiguum rigidum inversum — As hangs the flexible line, so but inverted will stand the rigid arch.

The argument is direct. In a hanging cable, every infinitesimal element is in pure tension along the tangent and balances three forces: gravity and two tensions. Flip the whole curve vertically. Now every element is in pure compression along the tangent and balances gravity plus two compressions. The geometry is identical; only the sign of the internal stress is reversed. Stone, brick and concrete are very strong in compression and weak in tension — so a masonry arch built along an inverted catenary is the geometry that minimises bending stress in the masonry.

The famous modern example is the Gateway Arch in St. Louis, designed by Eero Saarinen and structural engineer Hannskarl Bandel, completed in 1965. It is 192 m tall and 192 m wide — almost a square — and its shape is not a pure catenary but a weighted catenary, also called a flattened catenary, whose centerline satisfies

y = A·(cosh(C·x) − 1)
A = 68.7672 ft,  C = 0.0100333 / ft   (Saarinen's documented parameters)

The weighting accounts for the fact that the arch's cross-section tapers from 16 m × 16 m at the base to 5 m × 5 m at the crown, so the linear weight density is not constant. A weighted catenary is the curve that keeps every horizontal slice in pure compression given the actual non-uniform self-weight. It is slightly steeper near the base than a pure catenary of the same span and height.

Suspension bridges — parabolas and catenaries side by side

A suspension-bridge main cable is both. Hanging by itself during construction, before the deck is hung, it is a catenary. Once the deck is attached, the deck's weight dominates the cable's self-weight by a factor of typically 10 or 20 — and the deck loads the cable uniformly per unit horizontal distance, not per unit arc length. The cable then settles into a parabola.

The Golden Gate Bridge main cables span 1,280 m between tower tops. The sag at midspan is roughly 144 m, giving a sag/span ratio close to 1/9. At that geometry the difference between a parabola and a catenary at the same three points (towers and midpoint) is fractions of a percent — too small to see in any picture. But the cable does carry both regimes during its life: during the original 1937 cable spinning, before the deck was hung, the cable hung as a catenary; once the deck was loaded onto its hangers, the cable's profile changed measurably as the loading shifted from arc-length-uniform to horizontally-uniform. Bridge engineers track this profile change explicitly during construction to set the correct hanger lengths.

Applications

• Suspension bridge main cables. Carry uniform deck load → parabola, but the cable's own self-weight makes the true profile a slight catenary correction. The horizontal tension H at midspan equals the wind-and-traffic anchorage pull at each end; tower tops bear the vertical reaction.
• Overhead transmission lines. Carry only their own self-weight → pure catenary. Sag and horizontal tension are tightly coupled: more pull means shallower sag means less ground clearance margin on hot days. NESC and IEC standards set the maximum permissible sag at design temperature, and utilities monitor real-time conductor temperature on critical spans.
• Mooring lines. A heavy chain anchored to the seabed, hanging in a catenary toward a buoy or vessel above, is one of the oldest engineering applications. The catenary acts as a shock absorber: when the vessel moves, the chain lifts off the seabed before transferring force to the anchor, so peak loads on the anchor stay low. Offshore oil platforms use catenary risers (catenary-shaped pipes) for the same reason.
• Power-line de-icing and the catenary trolley. Electric trains pick up power from an overhead contact wire that must stay close to horizontal as it passes under each support; a second cable above — the messenger — hangs in a catenary, and short droppers between the messenger and the contact wire keep the lower wire flat. The system is colloquially called a "catenary," though it's the messenger above that takes the catenary shape, not the contact wire.
• Suspended chairlifts and aerial tramways. The haul cable runs over tower sheaves under high tension; it is a catenary between towers, and the chair-load adds approximately uniform horizontal-distance loading similar to a suspension bridge.
• Masonry arches and bridges. An arch shaped as an inverted catenary stays in pure compression under its own weight — Gaudí's Casa Milà attic uses unweighted catenary arches; the Sagrada Família's columns and arches use weighted ones derived from the inverted hanging-string models in his workshop.
• Anchor catenary mooring. A ship's anchor chain hangs in a catenary that drapes some length flat on the seabed at the anchor. The horizontal pull at the seabed is what holds the anchor; the chain in between converts vertical pull at the bow into horizontal pull at the seabed via its catenary shape. The longer the scope (chain length / water depth), the closer the pull is to horizontal and the more holding force the anchor can develop.

Variants and related curves

Curve	Equation	Loading	Where
Catenary	y = a cosh(x/a)	Uniform per arc length (self-weight)	Chain, mooring, power line
Parabola	y = x²/(2a)	Uniform per horizontal length	Suspension-bridge main cable
Weighted catenary	y = A(cosh(Cx) − 1)	Non-uniform self-weight (tapered cross-section)	Gateway Arch, Gaudí columns
Tractrix	y = a sech(x/a)	Drag normal to motion	Pursuit curve; involute of catenary
Elastica	EI d²θ/ds² = wx	Bending stiffness matters	Stiff beams, flexible spline curves
Catenoid	r = a cosh(z/a) (revolved)	Surface tension	Soap film between two rings
Cycloid	x = a(θ − sin θ)	—	Brachistochrone, tautochrone (different problem)

The catenoid — the surface of revolution of a catenary about a horizontal axis — is the only minimal surface of revolution other than the plane. A soap film stretched between two coaxial rings forms a catenoid, because the soap film minimises area at fixed boundary, and the variational problem reduces to the catenary in one dimension. Euler proved this in 1744.

Common pitfalls

• Calling every hanging curve a parabola. Galileo's mistake is repeated often. Suspension-bridge main cables under deck load are parabolas; chains under their own weight are not.
• Confusing the parameter a with the sag. The sag is a·(cosh(L/(2a)) − 1), not a itself. Conflating them gives nonsense at moderate sag fractions.
• Forgetting the radius of curvature at the low point. It equals a exactly — a useful identity for sizing pulleys and roller saddles where a cable changes direction.
• Ignoring elastic stretch. Real cables elongate under load; the elastic catenary (1773, Routh) couples Hooke's law to the geometric catenary equation and is essential for design where Young's modulus times area is comparable to the tension. Most textbook catenaries assume an inextensible cable.
• Building an arch as a pure circle. A semicircular arch under its own self-weight develops bending stress, because its shape is not the (inverted) catenary that self-weight demands. Roman semicircular arches survive because their abutments are so massive — and because masonry tolerates the bending stress with the help of compressive prestress from the dead load above.
• Letting power-line tension drop too far to "reduce stress." Lower tension means deeper sag; sag eats clearance; clearance loss can mean flashover to ground or vegetation. The conductor tension/temperature/sag relationship is a hard design constraint, not a knob to ease.