The dam retains a mixture of water (to a height hw) and soil (to height hs). The soil is considerably weaker than the underlying rock.

The total horizontal thrust P on the back of the dam is given by:

where yw and Ys are the weight densities of water and the soil and Ka is the soil's active earth pressure coefficient, which is assumed here (for simplicity) to be given by: 1 - sin|

with 9 the soil's angle of shearing resistance.

The destabilizing (i.e. overturning) moment that the earth and water pressures produce about the dam's toe O is:

Counteracting these actions is the dam's weight W, which provides a maximum (i.e. ultimate) sliding resistance Pult:

where yc is the concrete's weight density and 5 the angle of interface friction between the dam and the underlying rock. The stabilizing (i.e. restoring) moment of W about the dam toe O defines the limiting (i.e. ultimate) value of M:

Traditionally, the stability of the dam against rigid-body motion would be checked using appropriate factors of safety Fs for sliding and Fo for overturning. The allowable thrust Pa and allowable overturning moment Ma are then given by:

If we substitute the equations for P, Pult, M, and Mult into these inequalities and assume the water level rises to the top of the dam (i.e. hw = H), we obtain, after re-arrangement:

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