Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 37

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 = ∞.