Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 45

Figure 2.6.2 demonstrates that dam–water interaction lengthens the vibration period, with this effect being especially small for H/H_s, less than 0.5, but increasing rapidly with water depth (Chakrabarti and Chopra 1974). Furthermore, the vibration period ratio, , increases as the frequency ratio, Ω_r, decreases (i.e. the modulus of elasticity, E_s, of the concrete increases) because of interaction between the closely‐spaced fundamental vibration frequencies of the dam and water (Section 2.5.3); these observations first appeared in Chopra (1968). As the reservoir bottom becomes more absorptive, i.e. as the wave reflection coefficient α decreases, the fundamental resonant period is reduced from its value for a non‐absorptive reservoir bottom. This occurs because reservoir bottom absorption reduces the hydrodynamic terms (Section 2.3.3), thus reducing the value of the added mass. The wave reflection coefficient, α, has little influence on the fundamental resonant period for larger values of Ω_r, i.e. smaller values of E_s. However, the ratio is relatively insensitive to E_s if the reservoir bottom is absorptive with α ≤ 0.5.

The effects of reservoir bottom absorption on the added damping ratio ζ_r (Figure 2.6.3), and thus, on the damping ratio, , of the equivalent SDF system (Figure 2.6.4), are more complicated than its effects on the vibration period. As the wave reflection coefficient, α, decreases from unity, ζ_r increases monotonically from zero for larger values of Ω_r, i.e. smaller values of E_s, but the trends are more complicated for smaller values of Ω_r, i.e. larger values of E_s. This latter, unexpected behavior in ζ_r results from the previously observed effects of reservoir bottom absorption on the natural vibration frequency, , of the equivalent SDF system (Eq. 2.6.11), which is the frequency at which the added damping, ζ_r, is evaluated (Eq. 2.6.13). The added damping ratio depends on the relative values of and ; recall that the latter is the fundamental natural vibration frequency of the impounded water. As Ω_r decreases (i.e. E_s increases, implying that the dam becomes stiffer), approaches , and the imaginary‐valued component of the hydrodynamic term, , increases as α decreases from unity to zero, thus increasing ζ_r. Figure 2.6.3 also shows that the wave reflection coefficient, α, has a larger effect on the added damping for smaller values of Ω_r than for larger Ω_r. If the reservoir bottom is absorptive (α < 1), the added damping ratio ζ_r increases as Ω_r decreases, with the rate of increase becoming smaller as α decreases.

Figure 2.6.2 Comparison of exact and approximate (equivalent SDF system) values of the ratio of fundamental vibration periods of the dam on rigid foundation with and without impounded water. Results presented are for various values of the frequency ratio Ω_r and the wave reflection coefficient α.

Figure 2.6.3 Added damping ratio ζ_r due to dam–water interaction and reservoir bottom absorption. Results presented are for various values of the frequency ratio Ω_r and the wave reflection coefficient α.

Figure 2.6.4 Damping ratio of the equivalent SDF system representing dams on rigid foundation with impounded water; ζ₁ = 2%.

Considering that is less than ω₁, Eq. (2.6.12) indicates that dam–water interaction reduces the effectiveness of the structural damping. Unless this reduction is compensated by the added damping ζ_r due to reservoir bottom absorption, the overall damping ratio, ζ_r, will be less than ζ₁ (Figure 2.6.4).