Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 33

The steady state hydrodynamic pressure due to unit harmonic free‐field ground acceleration can be expressed in terms of its complex frequency response function

(2.3.7)

Substituting this in Eq. (2.2.1) leads to the Helmholtz equation:

(2.3.8)

The frequency response function, , for the hydrodynamic pressure when the excitation is the horizontal ground acceleration and the dam is rigid (Figure 2.3.1a), is the solution of Eq. (2.3.8) subject to the boundary conditions of Eqs. transformed according to Eq. (2.3.7):

(2.3.9)

The frequency response function for the hydrodynamic pressure, when the excitation is the vertical ground acceleration and the dam is rigid (Figure 2.3.1b), is the solution of Eq. (2.3.8) subject to the boundary conditions of Eqs. transformed according to Eq. (2.3.7):

(2.3.10)

The frequency response function for the hydrodynamic pressure due to horizontal acceleration of the dam vibrating in its fundamental natural vibration mode (Figure 2.3.1c) is the solution of Eq. (2.3.8) subject to boundary conditions of Eqs. transformed according to Eq. (2.3.7):

(2.3.11)

The complex‐valued frequency response functions and are obtained using standard solution methods for boundary value problems. Specialized for the upstream face of the dam, these functions are (Fenves and Chopra 1984a)

(2.3.12a)

(2.3.12b)

(2.3.12c)

where

(2.3.13)

The frequency response functions for hydrodynamic pressure due to horizontal motions of the upstream face of the dam, given by Eqs. (2.3.12a) and (2.3.12c) are the sum of the contributions of an infinite number of natural vibration modes of the impounded water. The complex‐valued, frequency‐dependent eigenvalues μ_n(ω) satisfy Eq. (2.3.14) and the eigenfunctions ϒ_n(y, ω) are defined by Eq. (2.3.15):

(2.3.14)

(2.3.15)

If the ground motion is vertical, pressure waves do not propagate upstream resulting in the much simpler frequency response function (Eq. (2.3.12b)), which is independent of the x‐coordinate.

Non‐absorptive Reservoir Bottom For a rigid, non‐absorptive reservoir bottom, as mentioned earlier, ξ = 0 and α = 1; the eigenvalues μ_n(ω) and eigenfunctions ϒ_n(y, ω) are real‐valued and independent of the excitation frequency:

(2.3.16)

where are the natural vibration frequencies of the impounded water with rigid non‐absorptive reservoir bottom, and

(2.3.17)

Then Eqs. (2.3.12a) and (2.3.12b) specialize to

(2.3.18)

(2.3.19)

These are the same as the equations defining hydrodynamic pressures on the upstream face of a rigid dam for a non‐absorptive reservoir bottom presented in Chopra (1967).

Incompressible Water Neglecting compressibility of water is equivalent to assuming the speed C of the hydrodynamic pressure waves to be infinite. The limits of Eqs. (2.3.18) and (2.3.19) as C → ∞ result in

(2.3.20)

(2.3.21)

Observe that the hydrodynamic pressure functions and are now independent of the excitation frequency, and is equal to the hydrostatic pressure.