A dam is thin at the top and fat at the bottom, and from the outside that looks like a style choice. It isn't. The water behind it pushes *much* harder low down than high up, and the dam has to out-wrestle that push with nothing but its own weight.
Water pressure comes from the weight of the water stacked above you. Each metre of depth adds about 9.8 kilopascals, so every 10 metres adds roughly one more atmosphere. At the surface the water pushes on the wall with nothing at all; at the bottom of a 100 m lake it pushes with about ten atmospheres on top of the air's. Strangely, the width of the lake does not matter: a pond and a reservoir of the same depth push on a wall with the same pressure.
Draw that pressure against the wall and you get a triangle: zero at the top, biggest at the bottom. The total push is the area of that triangle, which is half the depth times the pressure at the bottom. But the pressure at the bottom is itself proportional to depth. So the push grows with depth squared: double the water and you get four times the force.
A dam rarely slides away; the danger is that it tips forward about its downstream bottom corner, the toe. What matters then is not just the force but its turning effect, force times height. The push of a triangular load acts one third of the way up from the bottom, so the tipping moment is the force times a third of the depth. That makes it grow with depth cubed: double the depth and the tipping effect is eight times bigger.
Drag the water surface (or use the slider). The dam is 100 m tall. Make it tip.
Pushing against the tipping is the dam's own weight, acting down through its centre of mass. Its holding moment is that weight times the horizontal distance from the toe to its centre. This is why width at the base is so powerful: widen the base and you add weight and lengthen the lever at once, so the holding moment grows with the square of the width. The ratio of holding to tipping is the safety factor. Below 1 the dam tips; engineers want a comfortable margin above it, not a bare 1.0.
Try it. With the wedge and the water at 100 m, the base has to be about 46 m wide just to reach a safety factor of 1. Lower the water to half and the same dam gains a huge margin, because halving the depth cuts the tipping to an eighth.
Switch between the shapes at a safety factor of about 1. A plain wall needs only a base of about 38 m to resist the full 100 m of water, since all its weight sits far from the toe. The wedge needs a wider base, about 46 m. Yet the wall uses roughly 3,700 square metres of concrete per metre of crest, and the wedge only about 2,300, around 40% less. The reason is that the wall carries thick concrete right at the top, where there is almost no water to resist. The wedge puts its material where the pressure is.
This is a simplified model, but it shows the logic. Real gravity dams are shaped like the load: broad at the base where the push piles up, and slim at the crest where little is asked of them.
Hoover Dam is 726 ft (221 m) tall, 660 ft (200 m) thick at its base and just 45 ft (14 m) across the crest. Lake Mead is up to about 590 ft (180 m) deep at the dam. Plug that depth into the same arithmetic and the water presses at the bottom with about 1.8 megapascals, roughly 17 atmospheres more than air. Each metre of the dam's length takes a push of about 160 meganewtons, the weight of some 16,000 tonnes, acting a third of the way up from the base.
| Water depth | Push per metre | Tipping effect |
|---|---|---|
| 45 m | 10 MN | 1/64 of full |
| 90 m | 40 MN | 1/8 of full |
| 180 m | 159 MN | 1 (full) |
Hoover is not purely a gravity dam, though. It is an arch-gravity dam: it is curved upstream so that part of the load is steered sideways into the canyon walls. Its mass and its curve work together.
Real engineers also check that the dam does not slide along its foundation, that water seeping underneath does not push upward and lighten it (called uplift), that no part of the concrete is pulled apart, and what happens in an earthquake or flood. The two ideas here, push growing with depth squared and tipping with depth cubed, are only the first page of a long design checklist. But they explain the shape you see: a dam is fat at the bottom because that is where the water is deep.