Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 44

The effectiveness of the equivalent SDF system in representing the fundamental mode response of dams with impounded water is demonstrated in Figure 2.6.1. The exact and equivalent‐SDF‐system responses of an idealized concrete gravity dam monolith with the triangular cross section (described in Section 2.5.2) to harmonic horizontal ground motion were computed by numerically evaluating Eqs. (2.6.1) and (2.6.7), respectively. The absolute value of the complex‐valued frequency response function for horizontal acceleration at the dam crest is plotted against the normalized excitation frequency parameter, ω/ω₁, so the results are valid for dams of any height, H_s. Figure 2.6.1 demonstrates that the equivalent SDF system provides a good approximation of the fundamental mode response of the dam with impounded water for a wide range of values of the frequency ratio Ω_r – hence of the concrete modulus, E_s – and of the wave reflection coefficient, α, at the reservoir bottom. The approximation of the frequency bandwidth of the resonant peak is more accurate if the reservoir bottom is absorptive because additional energy is lost at this boundary, which eliminates the sharp peaks in the response curves.

Figure 2.6.1 Comparison of exact and equivalent SDF system response of dams on rigid foundation with impounded water due to harmonic horizontal ground motion; ζ₁ = 2%.

The exact value of the fundamental resonant period, obtained from the resonant peak of , computed from Eq. (2.6.1), is compared in Figure 2.6.2 with the natural vibration period, , of the equivalent SDF system in which is computed from Eq. (2.6.11). It is apparent that the natural vibration period of the equivalent SDF system provides a very accurate approximation of the fundamental resonant period of the dam with impounded water if the reservoir bottom is non‐absorptive, but it is slightly less accurate if the reservoir bottom is absorptive.