During an earthquake, there are dynamic effects on dams and retaining walls that can be highly variable. They can go from no damages at all up to massive destruction.

One of the first studies on the evaluation of the hydrodynamic actions on a massive gravity dam during horizontal solicitations was given by H. M. Westergaard with his “*Water pressures on dams during earthquakes*“, Transactions of American Society of Civil Engineering, November 1993.

Westergaard studied a dam with vertical surface under horizontal seismic accelerations (normal to the surface), with sinusoidal law. The assumptions were:

– plane motion;

– small displacement, both horizontal and vertical, of the liquid particles;

– compressible liquid;

– rigid dam;

– infinite reservoir in direction normal to dam.

In this way, he determined the motion equation imposing the equivalence between internal and external forces acting on the liquid element of volume *dx dy dz*:

Where:

– **σ** is the unit stress in the water due to the dynamic action;

– **ξ** and **μ** are the horizontal and vertical displacements of water particle;

– **γ** is the specific weight of water.

Additionally, he assumed as linear the law that links the stress to the increase in volume strain on unit of volume. In this way, he defined the elastic equation:

Where:

– **Ea** = bulk modulus of water.

To define the distribution of pressures, he assumed also that the dam will move during an earthquake with a simple harmonic motion having period *T* and maximum acceleration *αg* (with *α* being the seismic coefficient, that is the ratio between the maximum horizontal acceleration and the gravity acceleration).

The following boundary conditions were defined:

– no stress on the free surface (*σ = 0* for *y = 0*);

– no vertical displacement on the bottom (μ* = 0* for *y = Hi*);

– horizontal displacement of the wall equal to the horizontal displacement of the ground;

– stresses going to reduce when the distance from the dam increases (*σ -> 0* for *x = ∞*).

With the above, Westergaard reached to determine the equation for the maximum distribution of the hydrodynamic pressures, that we have obviously for *x = 0* (that means on the wall of the dam) for the instants t = 0, T, 2T, …, etc. with T being the period of the seismic motion.

The pressure distribution has a parabolic shape, with horizontal tangent on the surface (y = 0) and vertical on the bottom (y = Hi). The parabolic axis is located on the wall.

With the above outcome, Westergaard defined the equations for the calculation of the thrust (to be added to the hydrostatic one) and the bending moment on the dam, for each vertical depth y.

From the above equations (exact, but complex), he reached more practical ones, approximated but easier to use. With the approximate solution the parabolic distribution is not vertical anymore on the bottom.

The **pressure at generic depth y (kg/m²)** is defined as:

The **thrust at generic depth y (kg/m)** is defined as:

The **bending moment at generic depth y (km/m)** is defined as:

*C* is a numerical coefficient that, with the height in *m* and the period in *s*, can be written as:

Obviously, the maximum values for pressure, thrust and moment are located at the bottom, hence at *y = Hi*.

The **pressure at bottom (kg/m²)** is:

The **thrust at the bottom (kg/m)** is:

The **bending moment at the bottom (kgm/m)** is:

With a parabolic distribution, the arm of the thrust referred to the bottom of the dam is:

Of course, we need to bear in mind that the above formulas are related to an approximate parabolic distribution, having the tangent not vertical at the bottom. Anyway, the error is quite low if the dam is not very high. It was noticed that with the above equations the total thrust S is always bigger than the exact one, but with differences of just a couple of percentage units if the height of the dam is **Hs ≤ 250 m**. With dams bigger (**Hs > 300 m**) the differences become quite bigger.

In the graph below we can see both the distributions exact and approximate for a gravity dam of 100 m, seismic period of 1″ and α = 0.1.

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

*Eng. Onorio Francesco Salvatore*

[…] the Westergaard’s theory for the analysis of gravity dams under seismic actions (“Hydrodynamic effects during earthquakes on dams and retaining wall: Westergaard’s theory“). In this post we’ll see the von Karman’s theory of the same […]

[…] 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: […]

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