Anil K. Chopra - Earthquake Engineering for Concrete Dams

(2.3.23)

where n 1= the minimum value of n such that μ n> ω / C or . Equations (2.3.23)and (2.3.22)are identical if n 1= 1, i.e. , because then the term involving sin ωt in Eq. (2.3.23)vanishes. Thus, Westergaard's solution is valid only if the excitation frequency ω is less than the fundamental frequency of the fluid domain (Chopra 1967).

Westergaard's classic paper introduced the concept that the hydrodynamic pressure acting on the upstream face of a rigid dam due to horizontal ground motion can be interpreted as the inertia forces associated with an added mass m aof water moving with the dam:

(2.3.24)

Comparing this with Eq. (2.3.22)and recalling that , the added mass is

(2.3.25)

Because Eq. (2.3.22)is valid only for , the added mass analogy is also restricted to the same range of frequencies. Note that the added mass of Eq. (2.3.25)depends on the excitation frequency and is relevant only for horizontal ground motion in the stream direction.

If the compressibility of water is neglected, the added mass is given by the limit of Eq. (2.3.25)as the wave speed C approaches infinity, resulting in

(2.3.26)

Observe from Eqs. (2.3.25)and (2.3.26)that the added mass is independent of the excitation frequency only when water compressibility is neglected. This added mass may then be visualized as the mass of a body of water of width

(2.3.27)

Moving with a rigid dam, the body of water defined by Eqs. (2.3.27)and (2.3.26)is shown in Figure 2.3.4. Also included is Westergaard's (1933) popular approximation,

(2.3.28)

Although the two results are close, neither of them is valid because they ignore compressibility of water that has an important influence on the response of dams, as will be demonstrated in Section 2.5.4. Before closing this section, we note that the above‐mentioned added mass concept was restricted to horizontal ground motion in the stream direction.

Figure 2.3.4Body of water, assumed to be incompressible, moving with a rigid dam subjected to horizontal ground acceleration. Two results are presented: Eqs. (2.3.27)and (2.3.2).

Equation (2.2.2), which governs the fundamental modal coordinate, is extended to include the hydrodynamic pressure, p l(0, y , t ), on the upstream face ( x = 0) of the dam, resulting in

(2.4.1)

The hydrodynamic pressure is generated by horizontal acceleration of the upstream face of the dam:

(2.4.2a)

and by vertical acceleration of the reservoir bottom:

(2.4.2b)

The normal pressure gradient at the vertical upstream face of the dam is proportional to the total acceleration of this boundary, leading to the boundary condition:

(2.4.3)

The boundary conditions at the reservoir bottom and free surface of water are given by Eqs. (2.3.4b)and (2.3.6), respectively. In addition to these boundary conditions, the hydrodynamic pressures must satisfy the radiation condition in the upstream direction.

The steady‐state response of the dam–water system to unit harmonic free‐field ground acceleration, , can be expressed in terms of complex‐valued frequency response functions. Thus the modal coordinate and hydrodynamic pressure are given by

(2.4.4)

(2.4.5)

and Eq. (2.4.1)can be expressed in terms of the frequency response functions:

(2.4.6)

Similarly, the wave Eq. (2.3.1), becomes the Helmholz Equation (2.3.8), and the boundary accelerations of Eq. (2.4.2)become

(2.4.7a)

(2.4.7b)

The frequency response function is governed by Eq. (2.3.8)subject to the boundary conditions of Eqs. transformed according to Eqs. (2.4.4)and (2.4.5):