Why an arch stands

A stone arch is a pile of loose blocks — no glue that matters, no steel, no tension anywhere. It stands for a thousand years anyway, and the reason is a single line you can draw through it.

Blocks that only push

Stone is very strong when squeezed and very weak when pulled: enormous loads in compression, cracks at a small fraction of that in tension. Mortar is weaker still. So the masonry builder's one rule: every part of the structure must be pushing on its neighbours, never pulling. An arch is the shape that lets a row of blocks cross a gap under that rule.

Why it matters

Almost every large building before the nineteenth century — bridges, cathedrals, aqueducts, domes — was built this way, and many still carry traffic. Designed without any theory of stress, they are safe for a reason unrelated to stone strength: in a typical masonry arch the actual stress is a small fraction of what would crush the stone. What decides whether it stands is its shape, which you can check with a drawing.

The line of thrust

Take one block. Its weight pulls it down; its two neighbours push on its faces. Those pushes add up to one force, entering one face and leaving the other at a definite point. Do this for every block and join the points: that is the line of thrust, the path the squeeze takes from the crown down to the ground. The thrust itself is the outward push the arch delivers to its supports — the push flying buttresses exist to catch.

Now the rule about pulling becomes a rule about geometry. Where the line passes through the middle of a block, the whole face is squeezed. As it drifts toward one edge the squeeze concentrates there; when it reaches the edge, that edge carries everything and the far edge nothing — a joint about to open. If the line passed outside the stone, the joint would have to pull, and stone cannot. So the line of thrust must stay inside the masonry, everywhere, or the arch cannot be in equilibrium.

For a given arch and load there is not one line of thrust but a whole family, some steeper and some flatter, because the blocks can press on each other a little harder or softer. Jacques Heyman turned this into a rule in 1966, with three simplifying assumptions: the stone has no tensile strength, it cannot be crushed, and blocks do not slide. The safe theorem then says: if any line of thrust fits inside the stone and balances the loads, the arch stands. You need not know which line the real arch has chosen; one will do.

How it fails

Thin the arch, or load one shoulder heavily, and the family of possible lines narrows until only one is left. That last line touches the stone at four places (or five, in the symmetric case). At each a joint opens on the far side — a hinge, a crack the structure can rotate about. Three hinges still stand; a fourth turns the ring into a mechanism, and it folds. Under those assumptions masonry does not fail by breaking but by turning into a machine.

In short

An arch carries its load by squeezing, because squeezing is the only thing stone does well. The squeeze follows the line of thrust, and the arch stands if some such line fits inside the masonry. When none fits, hinges open and the blocks rotate apart — not because the stone gave up, but because the shape ran out of room.