Earthquake Engineering for Concrete Dams. Anil K. Chopra

      Putting this issue in historical context, the response of concrete gravity dams to the vertical component of ground motion was overestimated in the earliest studies based on the assumption of non‐absorptive reservoir bottom (Chopra and Chakarabarti 1973, 1974). An absorptive reservoir bottom gives a more plausible estimate of the response to vertical ground motion and its smaller contribution to the total earthquake response of the dam (Fenves and Chopra 1983). Although not negligible, this contribution is of secondary importance. Thus, the response to only the horizontal component of ground motion is considered in developing the procedure for simplified analysis of two‐dimensional dam–water–foundation systems (Sections 2.6, 3.4, and 3.5); this procedure is intended for preliminary analysis and design of dams.

2.6 EQUIVALENT SDF SYSTEM: HORIZONTAL GROUND MOTION

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.

      

      2.6.1 Modified Natural Frequency and Damping Ratio

      The properties of the equivalent SDF system are defined as those of the dam with an empty reservoir modified by an added mass and an added damping that represent the hydrodynamic effects of the impounded water. The mass density , of the equivalent SDF system is defined as

(2.6.5a)

      (2.6.5b)

where δ(x) is the Dirac delta function. Because the hydrodynamic pressure on a vertical upstream face acts in the horizontal direction, the “added mass” m_a(y) applies only to the horizontal displacement of the dam and is concentrated at the upstream face of the dam. The frequency‐independent “added mass” is defined as

(2.6.6)

      where the natural vibration frequency of the equivalent SDF system approximates the fundamental resonant frequency of the dam–water system. If the reservoir bottom is absorptive, m_a(y) is complex valued. Thus, it is not a mass quantity in the usual sense; only its real‐valued component contributes to an added mass, whereas the imaginary‐valued component implies an added damping. Furthermore, this added mass representing hydrodynamic effects on a flexible dam differs from the one in Eq. (2.3.25), which was determined assuming the dam to be rigid. It is inappropriate to use the latter in dynamic analysis of dams because they are flexible structures.

      The complex‐valued frequency response function for the modal coordinate of the equivalent SDF system will be of the same form as Eq. (2.2.7), but L₁, M₁, and C₁ will be different; thus

(2.6.7)

wherein the superscript l = x has been dropped for notational simplicity, and the over‐tilde is included to indicate that the equivalent SDF system models the dam response including dam–water interaction; , , and are obtained by substituting Eq. (2.6.5) in the standard definitions of generalized mass, damping, and force (Chopra 2017, Chapter 17):

(2.6.8a)