Anil K. Chopra - Earthquake Engineering for Concrete Dams

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 rincreases 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 rresults 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 rdecreases (i.e. E sincreases, 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 2.6.3also shows that the wave reflection coefficient, α , has a larger effect on the added damping for smaller values of Ω rthan for larger Ω r. If the reservoir bottom is absorptive ( α < 1), the added damping ratio ζ rincreases as Ω rdecreases, with the rate of increase becoming smaller as α decreases.

Figure 2.6.2Comparison 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 rand the wave reflection coefficient α .

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

Figure 2.6.4Damping ratio of the equivalent SDF system representing dams on rigid foundation with impounded water; ζ 1= 2%.

Considering that is less than ω 1, 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 rdue to reservoir bottom absorption, the overall damping ratio, ζ r, will be less than ζ 1( Figure 2.6.4).

Because the equivalent SDF system accurately predicts the response of dam–water system to horizontal ground motion over a complete range of excitation frequencies and for a wide range of parameters characterizing the properties of the dam, water, and reservoir bottom materials, it is applicable to the analysis of dam response to arbitrary ground motion. In particular, the peak displacements of the dam and equivalent static lateral forces can be expressed by appropriately modifying Eqs. (2.2.12)and (2.2.14), specialized for horizontal ground motion:

(2.6.14)

(2.6.15)

In these equations, ; and are the ordinates of the deformation and pseudo‐acceleration response spectra at the natural vibration period and damping ratio of the equivalent SDF system, and with defined by Eqs. (2.6.5a)and (2.6.6). Substituting these equations in Eq. (2.6.15)gives the final expression for the equivalent static lateral forces (Chopra 1978):

(2.6.16)

Comparing Eq. (2.6.16)with Eq. (2.2.12), we note that dam–water interaction introduces hydrodynamic pressures at the upstream face, increases Γ 1, and modifies the period and damping ratio where the spectral ordinate is determined.

The normal pressure gradient at the horizontal bottom of the reservoir is proportional to the vertical acceleration of the boundary. This is the specified free‐field excitation if the boundary is rigid, resulting in Eq. (2.3.3), repeated here for convenience: