## Solution Of The Reservoir Equation

Westergard's classic work (1933) on the hydrodynamic water pressure on dams during the earthquake started a new area for the researcher in this field. Westergaard's solution to wave equation for rigid dams during earthquakes was obtained based on the assumptions of dam with rectangular reservoir subjected to horizontal earthquake. In the solution, the reservoir extended to infinity in the upstream direction and the effect of surface wave was neglected. The system was subjected to the horizontal ground acceleration which was perpendicular to the dam axis. The motion was assumed to be small amplitude motion and water was taken inviscid fluid.

The pressure equation for the system shown in figure 2.18 is found under the following boundary conditions: p = 0 at y = h uy = 0 at y = 0 ux= ag cos() at x = 0 p = 0when x ^ro

Figure 2.18: Rigid dam-infinite reservoir system

Figure 2.18: Rigid dam-infinite reservoir system

The hydrodynamic pressure for the above mentioned boundary conditions is as follows:

n2c e sin I

in equation 2.39 qn and cnare defined as followings:

nncnx qn =

16h2 n2C 2T 2

The maximum pressure at x = 0 occurs when t = 0,T, 2T,... and can be written as:

n2 \n2cn

The maximum pressure at a given time on the upstream face (x = 0) happens at the bottom of the reservoir. The shape of the pressure diagram is such that the curve has a horizontal tangent at the top and a vertical tangent at the bottom.

In an approximation, the maximum pressure (equation 2.40 ) can be replaced by a parabolic curve. The maximum hydrodynamic pressure to be added to the hydrostatic pressure is as follows:

To further simplify the hydrodynamic pressure on the dam, one may consider the hydrodynamic pressure as a certain body of water moves with dam. This is called as added mass approach. This is possible because of the fact that the hydrodynamic pressure obtained (equation 2.39) has same phase compared to ground acceleration. Thus, we need to find the mass of water that attached to the dam and moves with it. For water with unit weight of 0.03125ton — ft3 the width of water along the x-direction is as follows(figure 2.19):

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