Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 36

Equation (2.2.2), which governs the fundamental modal coordinate, is extended to include the hydrodynamic pressure, p^l(0, y, t), on the upstream face (x = 0) of the dam, resulting in

(2.4.1)

The hydrodynamic pressure is generated by horizontal acceleration of the upstream face of the dam:

(2.4.2a)

and by vertical acceleration of the reservoir bottom:

(2.4.2b)

The normal pressure gradient at the vertical upstream face of the dam is proportional to the total acceleration of this boundary, leading to the boundary condition:

(2.4.3)

The boundary conditions at the reservoir bottom and free surface of water are given by Eqs. (2.3.4b) and (2.3.6), respectively. In addition to these boundary conditions, the hydrodynamic pressures must satisfy the radiation condition in the upstream direction.

The steady‐state response of the dam–water system to unit harmonic free‐field ground acceleration, , can be expressed in terms of complex‐valued frequency response functions. Thus the modal coordinate and hydrodynamic pressure are given by

(2.4.4)

(2.4.5)

and Eq. (2.4.1) can be expressed in terms of the frequency response functions:

(2.4.6)

Similarly, the wave Eq. (2.3.1), becomes the Helmholz Equation (2.3.8), and the boundary accelerations of Eq. (2.4.2) become

(2.4.7a)

(2.4.7b)

The frequency response function is governed by Eq. (2.3.8) subject to the boundary conditions of Eqs. transformed according to Eqs. (2.4.4) and (2.4.5):

(2.4.8a)

(2.4.8b)

(2.4.8c)

Note that the terms multiplying −ρ on the right side of Eqs. (2.4.8a) and (2.4.8b) are the amplitudes of the boundary accelerations given by Eq. (2.4.7).

Using the principle of superposition, which is applicable because the governing equations and boundary conditions are linear, the frequency response function for hydrodynamic pressure can be expressed as

(2.4.9)

where the frequency response functions and were presented in Eq. (2.3.12).

Substituting Eq. (2.4.9) with into Eq. (2.4.6) leads to the frequency response function for the fundamental modal coordinate when the dam is subjected to the l‐component of ground motion (l = x, y):

(2.4.10)

in which

(2.4.11a)

(2.4.11b)

Equation (2.4.10) may be expressed in terms of the natural vibration frequency ω₁ and damping ratio ζ₁ of the dam alone:

(2.4.12)

A comparison of Eq. (2.4.10) with Eq. (2.2.7) shows that the effects of dam–water interaction and reservoir bottom absorption are contained in the frequency‐dependent hydrodynamic terms B₀(ω) and B₁(ω). The hydrodynamic effects can be interpreted as introducing an added force , and modifying the properties of the dam by an added mass represented by the real component of B₁(ω), and an added damping represented by the imaginary component B₁(ω). The added mass arises from the portion of the impounded water that reacts in phase with the motion of the dam, and the added damping arises from radiation of pressure waves in the upstream direction and from their refraction into the absorptive reservoir bottom.