Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 29

The displacements of the dam – relative to its base – vibrating in its fundamental vibration mode due to the l‐component of ground motion (l = x and y represents horizontal and vertical components, respectively) can be expressed as

(2.2.1)

in which r^x(x, y, t) and r^y(x, y, t) are the horizontal and vertical components of displacement, respectively; and are the horizontal and vertical components, respectively, of the shape of the fundamental (or first) natural vibration mode of the dam fixed (or clamped) at its base to a rigid foundation with an empty reservoir; and is the modal coordinate associated with this vibration mode.

Under the approximation of Eq. (2.2.1), the equation of motion for a dam supported on rigid foundation with an empty reservoir is

(2.2.2)

in which the generalized mass

(2.2.3)

where the integration extends over the cross‐sectional area of the dam monolith; the mass density of the dam concrete m_k(x, y) = m(x, y), k = x and y is considered separately for the horizontal and vertical components of dam motion for convenience later in expressing the hydrodynamic effects in terms of an added mass and added damping; ; ω₁, and ζ₁ are the fundamental natural frequency and the viscous damping ratio of the dam alone;

(2.2.4)

Equation (2.2.2) can be rewritten as

(2.2.5)

where

(2.2.6)

For harmonic free‐field ground acceleration , where ω is the exciting frequency, the modal coordinate can be expressed in terms of its complex‐valued frequency response function, . Upon substitution into Eq. (2.2.2) and canceling e^iωt on both sides gives

(2.2.7)

We will later extend Eq. (2.2.7) to include dam–water interaction (Section 2.4) and dam–foundation interaction (Section 3.2.4).