Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 38

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 ζ₁ = 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 ω₁, and mode shape were determined using a finite element idealization of the dam, and the integrals involved in M₁, , , and B₁(ω) were computed in discretized form.

The complex‐valued frequency response function, , is a dimensionless response factor representing the ratio of horizontal acceleration at the dam crest to unit free‐field ground acceleration in the l(= x or y) direction. For each case mentioned above, the absolute value of this complex‐valued response factor is plotted against the normalized excitation frequency ω/ω₁. When presented in this form, the results apply to dams of all heights with the idealized triangular cross section, and chosen Ω_r, H/H_s, and α values.