A hanging chain settles into the one shape where every link simply pulls on its neighbors. Turn that shape upside down and every stone simply pushes on its neighbors, which is why arches stand. Add triangles and cables and you have most of the bridges ever built. Three labs: the chain, the arch, and the truss and cable.
Hold a chain by its two ends and let it hang. The curve it makes is called a catenary (from the Latin for chain). Below, a chain of 41 small weights joined by 40 rigid links falls under gravity and settles. Drag either end, or grab the chain anywhere and let go. The gold dashed curve is the exact mathematical catenary for the same length and supports. The blue curve is a parabola, the shape people often guess.
How to read this: the green chain is simulated, and the dashed curves are two math guesses at its shape. The little arrow triangle shows how hard the chain pulls at one spot (move it with the Probe slider): a sideways pull (pale) plus an upward pull (amber) add up to the total pull (gold T). The bottom strip shows the pull at every point along the chain.
Try this: slide Probe toward an end and watch the amber arrow grow while the pale one stays the same.
Cut out any piece of chain. Three forces act on it: the pull of the chain on its left end, the pull on its right end, and its own weight. Weight points straight down, so the sideways parts of the two end pulls must cancel. That holds for every piece, so the horizontal part of the tension, H, is the same all along the chain.
The vertical part has to hold up the chain hanging between the lowest point and wherever you are, so it grows with that length s: V = w·s, where w is the weight per meter (N/m). The slope is V/H = s/a. A curve whose slope grows in step with its own length is a hyperbolic cosine. A parabola's slope instead grows with horizontal distance, which is why the two curves agree near the bottom and part ways up the sides.
The exact curve was published in 1691 by Leibniz, Huygens and Johann Bernoulli, answering a challenge set by Jacob Bernoulli.
Stone is strong when squeezed and weak when pulled. An arch of loose blocks (no mortar, no glue) can only stand if every joint is pressed shut. Follow the force as it passes from block to block down to the ground: its path is the thrust line. If some thrust line fits inside the stone everywhere, the arch can stand (green). If every possible thrust line pokes out of the stone somewhere, a joint would have to pull, it opens instead, and the arch falls (red). The inverted hanging chain is exactly that path for a thin arch.
Try this: pick Circle, then drag Thickness down until a red joint appears. Press Best fit to see if any line still fits. Thrust sets how hard the arch pushes sideways on its feet.
In a hanging chain every link is pulled along its own length. Flip the picture and every pull becomes a push of the same size along the same line, so an arch shaped like the flipped chain carries its own weight in pure compression. Robert Hooke put it this way in the 1670s: as hangs the flexible line, so but inverted will stand the rigid arch.
The page tries many thrust lines (different crossing points at the base and the top) and keeps the one that stays furthest from the faces; u marks where a line crosses a joint, from 0 on the inner face to 1 on the outer face. That any line fits is enough: this is Jacques Heyman's safe theorem for masonry. Model: idealized, rigid blocks deep, stone at 2,400 kg/m³, no sliding, no tension, unlimited crushing strength. A thin semicircle needs a thickness of about 0.106 of its radius in this model, close to the classical result.
The Gateway Arch in St. Louis, tall and as wide, is a weighted catenary. Its legs get thicker toward the ground, so its shape is the hanging curve of a chain that is heavier near its ends: flatter at the top and steeper at the legs than a plain catenary.
A plain beam carries a load by bending: its top gets squeezed and its bottom gets stretched. A truss does the same job with straight bars joined in triangles, so each bar is only pulled (tension, green) or pushed (compression, amber). Drive a truck across and watch the force in every bar, solved joint by joint. Click any joint to check that its forces add up to zero. Below it, a suspension cable carries a deck.
How to read this: each bar is colored by whether it's being stretched (green) or squashed (amber). Thicker means harder, and the number is the force in kilonewtons (1 kN is about the weight of ). The gold hump below shows how much a plain beam would bend under the same truck.
Try this: pause the truck in the middle, then switch Pratt to Howe and watch the diagonals change color.
How to read this: at true scale the two curves sit almost on top of each other; press Gap ×100 to magnify the difference. The gold arrow T is the cable's pull on the tower.
Each pin joint must not accelerate, so the bar forces meeting there, plus any load or support, add to zero in both directions. That gives two equations per joint. This truss has exactly as many unknowns (one per bar plus three support reactions) as equations, so it has one answer, found here by solving them all at once.
Cut the truss anywhere and the top and bottom chords form a pair of opposite forces h apart, resisting the same bending moment M a plain beam would feel (gold curve). Deeper trusses need smaller chord forces. Model: idealized pin joints, the truck as one point load shared by the two nearest floor beams, bar weights ignored. Moments as in E03 · Levers, pulleys & gears.
Pratt diagonals slope down toward the middle and end up in tension; Howe diagonals slope the other way and end up in compression. Steel is happy in tension, and long thin bars buckle when pushed, which is why the Pratt layout suits steel bridges.
A suspension bridge deck hangs from the cable by evenly spaced vertical hangers, so the load is spread evenly across the span. A bare chain's weight is spread evenly along its curved length. The first load gives a parabola and the second a catenary. Here ℓ is the span, f the sag and w the deck weight per meter (N/m).
A real main cable carries the deck plus its own weight, so its shape lies between the two, very close to the parabola because the deck is much heavier. Halve the sag and the pull doubles: flat cables are expensive.
A different kind of bridge problem started graph theory: 002 · Graphs covers the seven bridges of Königsberg.