In this post we’ll study the Zangar’s theory for dams and retaining walls, following the previous posts on the same subject where we went through the theories of Westergaard (“*Hydrodynamic effects during earthquakes on dams and retaining walls: Westergaard’s theory*“) and von Karman (“*Hydrodynamic effects during earthquakes on dams and retaining walls: von Karman’s theory*“).

Zanagar proposed an Electric analog with the problem studied by Westergaard (“*Electric analog indicates effect of horizontal earthquake shock on dams*“, Zangar, C. N. & Haefeli, R. J. – Civil Engineering, April 1952). Zanagar referred to the same assumptions made by Westergaard (rigid dam, small displacements, plane motion, infinite reservoir) by neglected the compressibility of the water (similarly to von Karman). He found that the differential equation for the pressure gets the shape of the Laplace’s equation, that relates to the flow of an electric current. Zangar found a solution through the nominal values of the equipotential lines intersecting the dam.

Zangar found that the pressures distribution is almost parabolic and for a generic depth y from the liquid surface the pressure can be obtained from:

Where:

– **Hi** is the total water depth;

– **Cz** is a numerical coefficient;

– **α** is the seismic coefficient that is the ratio between the maximum horizontal acceleration and the gravity acceleration.

For what regards Cz, it is the unknown quantity that defines the magnitude and distribution of the pressures which are determined by the equipotential lines in the flow. Cz is a function of the shape of the dam and reservoir and is unaffected by the intensity of the quake.

Zangar stated that the designer need only to select a reasonable value for **α** and use the propoer Cz values to determine the water pressures on any dam due to horizontal earthquake. The problem, therefore, is to determine the correct Cz. In his “*Hydrodynamic pressures on dams due to horizontal earthquake effects*“, published by the Engineering Monographs of the United States Department of the Interior Bureau of Reclamation (1952), Zangar provided pressures due to earthquake determined for several shapes of dams. Dams studied were those with constant upstream slopes of 0, 15, 30, 45, 60 and 75 degrees. The pressure at the base of the dam and the maximum pressure on the slope were depicted on a diagram, that is shown below:

The pressure coefficient varies almost linearly from 0.735 for a dam with vertical face (0 degrees) to a 0.165 (75 degrees). The distribution of pressure for these constant slopes is shown in the figure below:

To permit rapid use of these data by Engineers, the experimentally determined pressure curves from the figure 6 above were represented by a family of parabolas which closely approximate the experimental curves for constant slopes. The parabolic distribution is given by the equation:

Where cm is the maximum value of C obtained from figure 5, hence 0.735 for vertical dams.

The horizontal force Ve above any elevation y and the total overturning moment Me above y due to Pe may analytically be shown to be:

For any issues or questions, you can contact the author at:

*Eng. Onorio Francesco Salvatore*

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