Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 47

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, is the compression wave speed, E_r is the modulus of elasticity, and ρ_r is the density of the reservoir bottom materials. The equilibrium condition at the surface of the layer of reservoir bottom materials (y^′ = 0) is that the pressure in the fluid equals the normal stress; thus

(A2.6)

The solution of Eq. (A2.5) subject to the equilibrium condition of Eq. (A2.6) and the radiation condition in the negative y^′‐direction gives

(A2.7)

By definition, ; therefore, the compliance function for the reservoir bottom is given by

(A2.8)

The compliance function is imaginary‐valued for all excitation frequencies, so the reservoir bottom materials, as modeled, introduces an additional mechanism for energy loss. Because the thickness of the sediment layer is not recognized explicitly, this compliance function is applied at the surface of the underlying foundation (y = 0).

The substitution of Eqs. (A2.4) and (A2.8) into Eq. (A2.3) gives the boundary condition at the absorptive reservoir bottom:

(A2.9)

where the damping coefficient ξ = ρ/ρ_rC_r. This boundary condition for time‐harmonic motion takes the following form for transient motion:

(A2.10)

which is identical to Eq. (2.3.4a).

We next relate the wave reflection coefficient, α, which is the ratio of the amplitude of the reflected hydrodynamic pressure wave to the amplitude of a vertically propagating pressure wave incident on the reservoir bottom, to the damping coefficient ξ. Consider a downward traveling wave in the fluid domain that strikes the fluid‐sediment boundary. Hydrodynamic pressures are governed by the one‐dimensional version of Eq. (2.3.8):

(A2.11)

The general solution of this equation is

(A2.12)

where A(ω) is the amplitude of the hydrodynamic pressure wave incident to the reservoir bottom, and B(ω) is the amplitude of the reflected wave.

An equation for the ratio B(ω)/A(ω), termed the reflection coefficient, α, can be obtained by substituting Eq. (A2.12) into the boundary condition of Eq. (A2.9) with i.e. zero free‐field acceleration:

(A2.13)

which is independent of the excitation frequency.

The wave reflection coefficient α is a more physically meaningful description than is the damping coefficient ξ of the behavior of hydrodynamic pressure waves at the reservoir bottom. Although the wave reflection coefficient depends on a pressure wave's angle of incidence at the reservoir bottom, the value α for vertically incident waves, as given in Eq. (A2.13), is used here as a metric to characterize the absorptiveness of the reservoir bottom materials. The wave reflection coefficient, α, may range within the limiting values of 1 and −1. For rigid reservoir bottom materials, C_r = ∞ and ξ = 0, resulting in α = 1, i.e. full reflection or no absorption of hydrodynamic pressures waves. For very soft reservoir bottom materials, C_r approaches zero and ξ tends to ∞, resulting in α = − 1. It is believed that α values from 1 to 0 would cover the wide range of materials encountered at the bottom of actual reservoirs (Ghanaat and Redpath 1995); α = 0 indicates no reflection of hydrodynamic pressure waves.