Earthquake Engineering for Concrete Dams. Anil K. Chopra

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).

      2.6.4 Peak Response

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 Γ₁, and modifies the period and damping ratio where the spectral ordinate is determined.

APPENDIX 2: WAVE-ABSORPTIVE RESERVOIR BOTTOM

      A2.1 Reservoir Bottom Sediments

      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:

(A2.1)

      This boundary condition is modified in this appendix to include the flexibility of sediments deposited at the reservoir bottom.

Flexibility of the reservoir bottom modifies the free‐field acceleration by an unknown interactive acceleration , and the boundary condition of Eq. (A2.1) becomes

(A2.2)

For steady‐state harmonic excitation due to , Eq. (A2.2) becomes

(A2.3)

      where and are complex frequency response functions for p(x, 0, t) and v(x, 0, t), respectively.

      The reservoir bottom is represented approximately by a one‐dimensional model – independent of the x‐coordinate – that does not explicitly consider the thickness of the sediment layer. The frequency response function for the vertical displacement of the reservoir bottom (i.e. the surface of the sediment layer) due to interaction between the impounded water and the reservoir bottom materials can be expressed in terms of the hydrodynamic pressure at the reservoir bottom:

(A2.4)

      The compliance function for the reservoir bottom is defined as the harmonic displacement at the reservoir bottom due to unit harmonic pressure p(x, 0, t) = 1e^iωt at the reservoir bottom.

      The compliance function can be derived by solving the one‐dimensional Helmholtz equation:

(A2.5)

that governs the steady‐state vibration of the model of the reservoir bottom materials, where is the frequency response function for vertical displacement in the layer of reservoir bottom materials,