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, Hs. 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 Ωr – hence of the concrete modulus, Es – 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.
2.6.3 Hydrodynamic Effects on Natural Frequency and Damping Ratio
Figure 2.6.2 demonstrates that dam–water interaction lengthens the vibration period, with this effect being especially small for H/Hs, less than 0.5, but increasing rapidly with water depth (Chakrabarti and Chopra 1974). Furthermore, the vibration period ratio, , increases as the frequency ratio, Ωr, decreases (i.e. the modulus of elasticity, Es, 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 Ωr, i.e. smaller values of Es. However, the ratio is relatively insensitive to Es if the reservoir bottom is absorptive with α ≤ 0.5.
The effects of reservoir bottom absorption on the added damping ratio ζr (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, ζr increases monotonically from zero for larger values of Ωr, i.e. smaller values of Es, but the trends are more complicated for smaller values of Ωr, i.e. larger values of Es. This latter, unexpected behavior in ζr 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, ζr, 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 Ωr decreases (i.e. Es 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 ζr. Figure 2.6.3 also shows that the wave reflection coefficient, α, has a larger effect on the added damping for smaller values of Ωr than for larger Ωr. If the reservoir bottom is absorptive (α < 1), the added damping ratio ζr increases as Ωr 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 Ωr and the wave reflection coefficient α.
Figure 2.6.3 Added damping ratio ζr due to dam–water interaction and reservoir bottom absorption. Results presented are for various values of the frequency ratio Ωr and the wave reflection coefficient α.
Figure
