Earthquake Engineering for Concrete Dams. Anil K. Chopra

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):

(2.4.8a)

(2.4.8b)

      (2.4.8c)

Note that the terms multiplying −ρ on the right side of Eqs. (2.4.8a) and (2.4.8b) are the amplitudes of the boundary accelerations given by Eq. (2.4.7).

Using the principle of superposition, which is applicable because the governing equations and boundary conditions are linear, the frequency response function for hydrodynamic pressure can be expressed as

(2.4.9)

      where the frequency response functions and were presented in Eq. (2.3.12).

Substituting Eq. (2.4.9) with into Eq. (2.4.6) leads to the frequency response function for the fundamental modal coordinate when the dam is subjected to the l‐component of ground motion (l = x, y):

(2.4.10)

      in which

      (2.4.11a)

      (2.4.11b)

Equation (2.4.10) may be expressed in terms of the natural vibration frequency ω₁ and damping ratio ζ₁ of the dam alone:

      (2.4.12)

      A comparison of Eq. (2.4.10) with Eq. (2.2.7) shows that the effects of dam–water interaction and reservoir bottom absorption are contained in the frequency‐dependent hydrodynamic terms B₀(ω) and B₁(ω). The hydrodynamic effects can be interpreted as introducing an added force , and modifying the properties of the dam by an added mass represented by the real component of B₁(ω), and an added damping represented by the imaginary component B₁(ω). The added mass arises from the portion of the impounded water that reacts in phase with the motion of the dam, and the added damping arises from radiation of pressure waves in the upstream direction and from their refraction into the absorptive reservoir bottom.

      

2.5 DAM RESPONSE

      2.5.1 System Parameters

      The frequency response function for a dam with a fixed cross‐sectional geometry and Poisson's ratio, when expressed as a function of the normalized excitation frequency ω/ω₁, depends on three system parameters: , the ratio of the fundamental natural vibration frequency of the impounded water to that of the dam alone; H/H_s, the ratio of water depth to the dam height; and α, the wave reflection coefficient at the reservoir bottom (Chopra 1968). We know that (Eq. (2.3.16)), and it can be shown that ω₁ = γC_s/H_s, where γ is a dimensionless factor that depends on the cross‐sectional shape of the dam monolith and the Poisson's ratio of the concrete in the dam, , E_s is the Young's modulus, and ρ_s is the density of concrete. Therefore

      (2.5.1)

For fixed values of γ, C, ρ_s, and H/H_s, the frequency ratio Ω_r, is proportional to . Thus Ω_r decreases with increasing E_s or dam stiffness, and vice versa.

      If the reservoir is empty or water is assumed to be incompressible, , when expressed as a function of ω/ω₁, is independent of E_s, and α; the incompressible case implies C = ∞ and thus Ω_r = ∞.

      2.5.2 System and Cases Analyzed

      The idealized monolith considered has a triangular cross section with a vertical upstream face and a downstream face with a slope of 0.8 horizontal to 1.0 vertical. The dam is assumed to be homogeneous and isotropic with linearly elastic properties for mass concrete: