Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 35

In 1933 Westergaard derived an equation for the hydrodynamic pressure on the upstream face of a rigid dam due to time‐harmonic horizontal ground motion, a result that for several decades profoundly influenced the treatment of hydrodynamic effects in dam analysis. The range of validity of this result will be identified in this section. His result for hydrodynamic pressure on the upstream face of the dam due to , expressed in the Cartesian coordinate system and notation adopted herein, is

(2.3.22)

To evaluate this classical result, we substitute Eq. (2.3.18) in Eq. (2.3.7), and separate the real part to obtain the hydrodynamic pressure due to the excitation :

(2.3.23)

where n₁ = the minimum value of n such that μ_n > ω/C or . Equations (2.3.23) and (2.3.22) are identical if n₁ = 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_a of 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 2.3.4 Body 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).