Anil K. Chopra - Earthquake Engineering for Concrete Dams

(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 ω 1and damping ratio ζ 1of 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 B 0( ω ) and B 1( 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 B 1( ω ), and an added damping represented by the imaginary component B 1( 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.

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 / ω 1, 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 ω 1= 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 sis the Young's modulus, and ρ sis 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 rdecreases with increasing E sor dam stiffness, and vice versa.

If the reservoir is empty or water is assumed to be incompressible, , when expressed as a function of ω / ω 1, is independent of E s, and α ; the incompressible case implies C = ∞ and thus Ω r= ∞.

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: Poisson's ratio = 0.2, unit weight = 155 lb/cu ft, and damping ratio ζ 1= 5%. The Young's modulus for mass concrete is varied over a wide range by specifying three values for the frequency ratio Ω r= 0.80, 1.0, and 2.0, which, for the selected system properties, correspond to E s= 3.94, 2.52, and 0.63 million psi, respectively. The first two cover the range of values representative of mass concrete, and the third is unrealistically low, chosen to help identify the condition under which water compressibility can be neglected. The unit weight of water is 62.4 lb/cu ft and the velocity of pressure waves in water is C = 4720 ft/sec. Two values for the depth of water are considered: an empty reservoir ( H / H s= 0) and a full reservoir ( H / H s= 1). The wave reflection coefficient is varied over a wide range; the values considered are: α = 1.0, 0.75, 0.50, 0.25, and 0.

The response of the dam to harmonic horizontal and vertical ground motion is determined by numerically evaluating Eq. (2.4.10)wherein the fundamental natural vibration frequency ω 1, and mode shape were determined using a finite element idealization of the dam, and the integrals involved in M 1, , , and B 1( ω ) were computed in discretized form.