Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 42

It was shown in Section 2.4 that including the interaction between the dam and compressible water results in the following complex‐valued frequency response function for the modal coordinate due to horizontal ground motion (Eq. (2.4.10) with the superscript l = x dropped):

(2.6.1)

in which the hydrodynamic terms B₀(ω) and B₁(ω) are equivalent to those in Eq. (2.4.10) but are now defined slightly differently:

(2.6.2a)

(2.6.2b)

where and are frequency response functions for the hydrodynamic pressure on the upstream face due to horizontal ground acceleration of a rigid dam, and acceleration of a dam in its fundamental mode of vibration, respectively:

(2.6.3)

where

(2.6.4)

in which f₀(y) = 1 and . For convenience later in defining the added hydrodynamic mass, the preceding pressure functions are defined for the positive x‐direction upstream, thus giving algebraic signs opposite those of the corresponding equations in Section 2.3, where the positive x‐direction was downstream.

Including dam–water interaction, the frequency response function for the modal coordinate associated with the fundamental vibration mode of the dam, Eq. (2.6.1), is a complicated function of excitation frequency ω that contains frequency‐dependent hydrodynamic terms. To develop a simplified analytical procedure, the dam–water system will be modeled by an equivalent SDF system with frequency‐independent values for the hydrodynamic terms. Such a procedure, developed by Chopra (1978) for dam–water systems with non‐absorptive reservoir bottom and later extended to include reservoir bottom absorption (Fenves and Chopra 1985c), is presented next.