The shape that carries itself, and the arch that is its reflection
A chain hanging between two points has no choice about its shape. It cannot resist bending, so the only configuration available is the one in which every part of it is in pure tension along its own length — and that shape is determined entirely by the loads.
Turn the whole thing upside down. Every tension becomes a compression of the same magnitude, and the inverted shape carries the identical load in pure compression, with no bending anywhere. That is an arch, and the relationship is exact rather than analogical.
Why bending disappears
A cable has no bending stiffness. It cannot carry a moment because it would have to be stiff to do so, and it is not.
That sounds like a limitation and is the whole mechanism. Since the cable can only carry force along itself, the shape it settles into must be one in which the loads are balanced by axial force alone. If any bending were required at some point, the cable would move until none was — and where it stops moving is the funicular shape for that load.
So the cable is a form-finder. It does not resist the load in a shape chosen by a designer; it finds the shape in which the load can be resisted without bending, and adopts it.
The shape depends on the load, which is the part most often glossed. Under a load uniform along the horizontal — a suspension bridge deck — the shape is a parabola. Under a load uniform along the cable itself — a chain carrying only its own weight — it is a catenary, the hyperbolic cosine. Under a set of point loads it is a polygon with a vertex at each load.
The two curves are genuinely different functions and they are very hard to tell apart at ordinary sags. Galileo asserted the hanging chain was a parabola; it took Huygens, Leibniz and Johann Bernoulli, in 1691, to establish that it is not.
Inverting it
Hooke published the principle in 1675 as an anagram, having found it by drawing rather than by summing forces, which decoded reads: as hangs the flexible line, so but inverted will stand the rigid arch.
The inversion works because equilibrium does not care about sign. A set of forces in equilibrium remains in equilibrium if every force is reversed, so a shape that balances a load in tension balances the same load in compression when turned over. Nothing about the geometry changes.
That makes the hanging model a design instrument, and it was used as one. Poleni analysed the cracked dome of St Peter’s in 1748 by hanging a chain loaded to represent the dome’s weight and checking that the inverted curve lay within the masonry. Gaudí spent ten years on a hanging model of strings and small bags of shot for the Colònia Güell chapel, photographing it and turning the photographs upside down to obtain the elevations.
What the supports have to do
A cable pulls inward and downward on whatever holds it. An arch pushes outward and downward. Neither can be supported by a vertical reaction alone, and that horizontal component is the whole difficulty of both.
The horizontal force is the same everywhere along the cable — it has to be, since nothing horizontal is applied between the ends — and it is inversely proportional to the sag. A shallow cable carrying the same load has a large horizontal pull; a deep one has a small one.
for a uniform load over span with sag . Halving the sag doubles the thrust.
For an arch that thrust has to be resisted at the springing, and the history of arch construction is largely the history of what resists it: massive abutments cut into rock, flying buttresses transferring it down a pier, or a tie across the base turning the whole thing into a self-contained unit. A tied arch needs no horizontal reaction at all, at the cost of a member in tension across the span — which is a truss, approached from the other direction.
The thrust line, and a lower bound
The funicular idea gives masonry its assessment method, and the method is unusual in that it proves a structure is safe without knowing what it is actually doing.
Masonry has no useful tensile strength. It stands because compression finds a path through it. The thrust line is the funicular polygon for the actual loads: the line along which the compressive resultant travels from the crown down to the foundation.
If a thrust line can be drawn that stays inside the masonry everywhere, then a set of internal forces exists that is in equilibrium with the loads and requires no tension. That is enough to conclude the arch can stand. It does not say the arch is carrying load along that particular line — it almost certainly is not — only that at least one safe possibility exists.
That is the safe theorem of plasticity, and it is why Heyman’s analysis of Gothic cathedrals works with a straightedge. It also explains the middle-third rule: if the thrust line stays within the middle third of the section, no tension is required anywhere across it.
The corollary is more interesting. A masonry arch that has cracked has not failed — it has moved to a geometry in which a thrust line fits. Cracks in old arches are usually evidence of successful adjustment rather than of impending collapse, which is the opposite of the intuition.
What a funicular structure is not
The shape carries one load case with no bending. Change the load and the funicular shape changes, but the built structure cannot.
A suspension bridge shaped for its own dead weight is not funicular for a train crossing one half of it. The asymmetric load wants a different shape, and since the cable cannot take one, the deck has to resist the difference in bending. That is why suspension bridges have stiffening trusses, and why the Tacoma Narrows deck — which had a shallow plate girder instead — behaved as it did.
The same applies to arches. A masonry arch under asymmetric load has its thrust line move, and if it moves outside the masonry the arch hinges. Four hinges make a mechanism, and that is how arches actually collapse: not by crushing, but by turning into a four-bar linkage.
What the shape does to the supports
A funicular shape carries its load without bending and pays for it at the ends.
The thrust is , which is the same expression as a truss chord force with the sag in place of the depth. Halving the sag doubles the pull on the abutments, and for a masonry arch that is the difference between a buttress and a collapse.
Equilibrium at the supports is where the difference between a beam and an arch actually shows. A beam delivers a vertical reaction; an arch delivers a thrust, and the history of arch construction is the history of what resists it.
Where the model stops
No bending stiffness. True for a cable and false for every arch ever built — a real arch has a section with a second moment of area, which is what lets its thrust line wander. A real arch has some, which is what lets it tolerate a thrust line that wanders.
No self-weight, or self-weight only. The two clean solutions are the parabola and the catenary, and each assumes a specific load distribution. A real arch carries both its own weight and a deck, and its funicular is neither curve exactly.
Small deflections. A cable changes shape under load substantially, so the usual first-order assumption does not apply to it at all. Cable analysis is genuinely nonlinear, and the linear methods used elsewhere on this site do not apply to it.
Rigid supports. An arch’s thrust spreads its abutments, which flattens it, which increases the thrust — a load making itself worse, in masonry. Several medieval arches have this recorded in their geometry.
The figures carry a specific distortion worth naming: the sag is drawn at perhaps a fifth of the span, and a real suspension bridge cable sags about a tenth while a shallow arch is far flatter still. Since the thrust goes inversely with the sag, every figure on this page understates the horizontal force by a factor of two or more. The shape is honest; the consequence at the supports is drawn smaller than it is.
The ladder from here
Later rungs: the catenary derived, and why the hyperbolic cosine appears. The parabola for a uniform horizontal load. The funicular polygon and the pole diagram. Thrust lines in masonry and the middle-third rule. Heyman’s safe theorem. The three-pinned arch, determinate by construction. Tied arches. Suspension bridges and the stiffening truss. Cable nets and prestress. And form-finding by computer, which is the hanging model rewritten as an optimisation and used to design roofs nobody could have drawn.
Hooke’s anagram was published in 1675 and the solution only after his death. He had the principle, and he did not have the mathematics to find the curve.