In a plan view a gravity dam looks like a beam holding back the water.  But for any but a very narrow dam this would not work.

It is very important that all water be excluded from underneath a gravity dam.  A boat can in principle float in a cavity that exceeds its own dimensions by only a minute distance.

The pressure of water on the surface of an object can cancel some of its weight.  In a dam this could result in uplift and overturning.

It is very important for any object that the line of thrust should meet the ground well within the base area.  Otherwise there is a danger near one edge that the pressure on the ground might be reduced to nothing.  In the case of a dam, cracks would let water in.  A dam should satisfy this condition throughout the filling of the reservoir.

Let's look into dams a little more.  A gravity dam is a wall to hold back water.  Many gardens on slopes have walls to hold back soil at a higher level.  These walls are dams for soil.  The soil exerts a pressure, rather like water, though it is not a fluid.  Soil mechanics therefore differs from fluid mechanics.  

The diagram below represents five walls built with differing amounts of tilt.  Based on the centre of gravity lying over the base, the three at left are stable, one is balanced on a corner, and the right-hand one would fall over.

This criterion is not actually adequate.  We can sense that by looking at the diagram.  What is the physical factor that our senses are responding to?

The pressure distribution at left is uniform.  For the fourth wall which is poised on one corner, there is no pressure at all, except at this corner.  In practice, of course, there wouldn't be an infinite pressure on that point.  The ground would give a little, resulting in a narrow strip on which the pressure would fall.  In practice the wall would fall over, because the depression in the ground at the corner only would allow the wall to tilt past the angle of stability.

The two triangles represent the intermediate cases.  In the second wall from the left, the centre of gravity is one third of the distance from one end of the base.  The pressure falls to zero at the other end.  In the middle diagram, only half the base feels any pressure at all.  The rest is in tension.  You can feel such effects on your feet if you lean forwards, backwards or sideways.

Now we have a criterion for stability.  There should be pressure at all points on the base, otherwise a part of the connection will be in tension, and may separate.  So the centre of pressure must lie within the middle third of the base, a restrictive rule.  In fact, this rule restricts the centre of pressure to one ninth of the base area.

Now we can look at a wall or dam that has to restrain soil or water.

We could build a leaning wall and then allow the soil to push against it.  Without the soil we would have a pressure distribution like the white triangle, and if we did our sums right we could end up with a uniform pressure distribution like the one at right.  

The wall would be rather dangerous, and the variable properties of soil, including different amounts of water, would render exact calculation impossible.

It is better  to build a wall having one face vertical and one with batter, as shown below.

Here is a simple retaining wall, connected to a foundation with a keel under the toe of the wall.

This wall is stable during construction, and does the job well afterwards.

But a large gravity dam costing a great deal of money must be designed using stress calculations, so that the cheapest reliable structure can be built.

Let's look at the design of a dam, using as little maths as we can get away with.

We know that the liquid pressure at the top is zero, so the dam can come to a point.  It obviously has be thicker further down.  Let's take the simplest case of a wall with a straight slope with batter.

What else do we know?  If the liquid has zero density the dam needs no thickness.  If the liquid has infinite density the dam has to be infinitely thick.  What if the density of liquid and concrete are the same?  We might guess that the slope would be 45 degrees, but we could easily be wrong.  We could even guess that if the ratio of densities of concrete and water is R, the angle A of the dam is given by R = tan(A).

The other consideration is that the dam must be stable at all stages of construction and filling, so let's see it before the water has started to enter.

The centre of gravity of the triangle is at one third of the base line.

So the dam is right on the criterion for good practice, as the centre of force is on the edge of the middle third.  The variation of pressure is shown by the white triangle below.

In order to place the centre of pressure within the middle third, we can add a rectangular block to the face of the dam.  There is a bonus.  It will be possible to build a road across the top of the dam, providing a free bridge across the valley.  At the very least it will provide a crossing for workers at the dam, which might incorporate a hydro-electric power plant.

Let's continue for the moment with the simple triangle, and try to find out how deep it must be when the reservoir is full.

At the very least, the dam must be thick enough to obtain the pressure condition shown by the white triangle at the bottom.  The centre of pressure has retreated to the back end of the middle third.

In practice we would make the dam a little thicker to provide a safety factor.

What is the critical thickness?  Can we find out without doing any maths?

We will assume that the concrete is 2.3 times as dense as water.

In the minimal configuration, the pressure of the water, represented by the thin triangle, has to change the upper white triangle into the lower one.  

The distribution of pressure is completely reversed.  The position of maximum pressure moves from the heel to the toe of the dam, just as would happen to a person who is pushed from the back.

An actual dam includes extra material as a safety factor, so that nowhere does the base pressure become zero at any time during filling.

If the density of the concrete is 2.3 times the density of water, then the effective density of the undercut part is only 2.3 - 1 = 1.3, which is 1.3 / 2.3 of the original, that is 0.57.  This is symbolised in the diagram by the missing parallelogram.

These simple ideas show that a gravity dam and its environment must be prepared with the utmost care.  The consequences of a failed dam can be catastrophic.

Arch dams and cupola dams derive their stability by thrusting against the sides of the valley.  Effectively they use the the surrounding rock as a part of the structure.  Such dams are usually lighter than gravity dams.  A similar point is made in the pages about arch bridges and suspension bridges, where they are compared with bridges which only rest on the ground, making no horizontal demands upon it.

It is not easy to find photographs of dams that are close to failure, so here is a picture of a wall that is failing under soil pressure.  Soil behaves in a more complicated way than either liquids or solids, though at least we don't have to worry about turbulence and leakage.  What do you think is the angle of the wall?  Using a protractor it seems to be about 11°.  Does it look more or less than this?

Soil is not much of a structural material, yet children of all ages manage to build quite intricate things, including bridges, from soil's least promising manifestation, sand, which works quite well when damp.  Precision moulds can be made from sand.

This is a cellular retaining wall with batter.  It is around four metres high.  Such a wall provides drainage, and shelter for plants and small animals.