Arches & Funicular Form: How a Hanging Chain Becomes an Arch
Hang a chain between two points and it settles into a smooth curve. That shape isn’t arbitrary — it’s the only shape in which the chain carries its load with pure tension, no bending. In 1675 Robert Hooke captured the consequence in a single line: “as hangs the flexible line, so but inverted, will stand the rigid arch.” Flip the hanging shape upside down and you get the ideal arch — one that stands in pure compression.
Hang a chain and it finds a shape of pure tension; flip it and the same shape stands as an arch in pure compression. At this sag the horizontal thrust is H ≈ 4.30 — deepen the sag and the thrust drops.
Catenary is the funicular for self-weight (uniform along the cable); a parabola is the funicular for load uniform along the span (a loaded deck). Thrust H = w·L²/(8·sag) for the parabola, w·a for the catenary — a deeper arch carries less thrust but rises higher. Real arches must keep this thrust line inside the stone and be buttressed. A teaching model — units are illustrative.
The funicular idea
A “funicular” shape is the shape a cable takes for a given load — and, inverted, the shape an arch should take to avoid bending. Bending is what makes beams big; remove it and you can span far with very little material. That’s why arches, vaults, and cable structures are so efficient.
Catenary vs. parabola
The funicular shape depends on how the load is distributed:
- Catenary — the load is the structure’s own weight, spread evenly along
its length. The math is
y = a·cosh(x/a). This is the shape of a free-hanging chain and of the St. Louis Gateway Arch. - Parabola — the load is spread evenly along the horizontal span (think of a suspension bridge, where the heavy deck hangs from the cable). The shape is a parabola.
They look similar, and at shallow sags they’re nearly identical — toggle the model above to compare.
The thrust trade-off
An arch doesn’t just push straight down on its supports; it pushes outward. That horizontal thrust is what cracks unbuttressed arches and spreads the feet of a frame. Its size depends on the sag (or rise):
H = w·L² / (8·sag) for a parabola; H = w·a for a catenary.
A shallow arch (small sag) carries a large thrust; a deep arch carries less, but rises higher and uses more material. Designing an arch is largely choosing that balance — and then making sure the thrust line stays inside the masonry and that the supports can resist the outward push (buttresses, ties, or the ground).
A teaching model for intuition — real arches and shells need a structural engineer, real loads, and material limits.
Further learning (elsewhere)
Hand-picked free resources — we link out rather than re-create what already exists well.