Catenary: The Shape of a Hanging Chain

A chain that settles into balance at every point

Hang a flexible, inextensible chain or cable freely between two fixed points and let it come to rest under its own weight. Because a chain has no resistance to bending — it is pure tension, no stiffness — its final shape is entirely determined by one condition: every tiny segment of chain must be in force balance, with the tension pulling on it from each neighbouring segment and gravity pulling straight down, and nothing left over to accelerate it. Solve that condition along the whole length of the curve and the answer is the catenary, from the Latin catena, chain.

Not a parabola — a hyperbolic cosine

It is tempting to assume a hanging chain is a parabola, and Galileo himself made exactly that guess in the early 17th century. It is close for a shallow sag but wrong in general, and the correct derivation, worked out later by Leibniz, Huygens and Johann Bernoulli, gives a different curve entirely: the hyperbolic cosine.

y(x) = a * cosh(x / a) = a * (e^(x/a) + e^(-x/a)) / 2

a = H / w    H = horizontal tension component (constant along the chain)
             w = weight per unit length of chain

The parameter a sets how tightly the curve is drawn in: a large horizontal tension H relative to the chain's weight w gives a large a and a shallow, nearly flat sag; a small H gives a small a and a sharply curved, deeply drooping chain. Because a parabola is loaded uniformly per unit of horizontal span while a hanging chain is loaded uniformly per unit of its own arc length, the two curves genuinely differ — most noticeably near the supports, where the chain's arc length per horizontal distance grows fastest.

Tension varies, but always points along the curve

A subtlety worth making explicit: the tension in a hanging chain is not constant along its length. Its horizontal component H is constant everywhere — there is nothing else horizontal to balance it against — but the total tension grows toward the supports, because there the chain must also carry the accumulated weight of everything hanging below that point, added as a vertical component. At the very lowest point of the sag, the chain is momentarily horizontal, and its tension there is purely horizontal, equal to H; at the supports, the tension is largest and most steeply angled.

Turn it upside down and it becomes the strongest arch

The catenary's most striking practical property appears when you flip it. A hanging chain carries its own weight purely in tension, with zero bending anywhere along its length — every internal force lies exactly along the curve. Invert that same shape into an arch and gravity, now pushing down instead of pulling down, pushes along exactly the same lines the chain's tension once followed, converting every tension force into an equal compression force. The result is an inverted catenary arch that carries its own weight in pure compression, with no bending moment anywhere — the theoretically ideal, most material-efficient arch shape for a structure loaded only by its own weight. Robert Hooke stated the principle in 1675 as a cipher — "as hangs the flexible line, so but inverted will stand the rigid arch" — and Antoni Gaudí famously used hanging chain models loaded with weighted bags to design the compression-only stone arches and vaults of the Sagrada Família.

Where a parabola is the right answer instead

Not every hanging cable is a catenary — the shape depends on how the load is distributed. The main cables of a suspension bridge are hung with closely spaced vertical hangers carrying the roadway deck, and that deck's weight is essentially uniform per unit of horizontal span rather than per unit of the cable's own length. Solve the same force-balance condition for that loading law and the shape that comes out is a true parabola, not a catenary — a small but real distinction that is often blurred in casual descriptions of suspension bridges, but matters for anyone actually computing the cable's geometry.

Frequently asked questions

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