Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 43

The properties of the equivalent SDF system are defined as those of the dam with an empty reservoir modified by an added mass and an added damping that represent the hydrodynamic effects of the impounded water. The mass density , of the equivalent SDF system is defined as

(2.6.5a)

(2.6.5b)

where δ(x) is the Dirac delta function. Because the hydrodynamic pressure on a vertical upstream face acts in the horizontal direction, the “added mass” m_a(y) applies only to the horizontal displacement of the dam and is concentrated at the upstream face of the dam. The frequency‐independent “added mass” is defined as

(2.6.6)

where the natural vibration frequency of the equivalent SDF system approximates the fundamental resonant frequency of the dam–water system. If the reservoir bottom is absorptive, m_a(y) is complex valued. Thus, it is not a mass quantity in the usual sense; only its real‐valued component contributes to an added mass, whereas the imaginary‐valued component implies an added damping. Furthermore, this added mass representing hydrodynamic effects on a flexible dam differs from the one in Eq. (2.3.25), which was determined assuming the dam to be rigid. It is inappropriate to use the latter in dynamic analysis of dams because they are flexible structures.

The complex‐valued frequency response function for the modal coordinate of the equivalent SDF system will be of the same form as Eq. (2.2.7), but L₁, M₁, and C₁ will be different; thus

(2.6.7)

wherein the superscript l = x has been dropped for notational simplicity, and the over‐tilde is included to indicate that the equivalent SDF system models the dam response including dam–water interaction; , , and are obtained by substituting Eq. (2.6.5) in the standard definitions of generalized mass, damping, and force (Chopra 2017, Chapter 17):

(2.6.8a)

(2.6.8b)

(2.6.8c)

and M₁, C₁, and L₁ for the dam alone were defined in Section 2.2.1. Substituting Eq. (2.6.6) into Eq. (2.6.8) leads to

(2.6.9a)

(2.6.9b)

(2.6.9c)

We will demonstrate that if the excitation frequency ω that appears in is replaced by the resonant frequency , Eq. (2.6.7) will give the same resonant response as the rigorous Eq. (2.6.1). Using the solutions for p₀(y, ω) and p₁(y, ω) in Eq. (2.6.3), replacing ω by in C₁, and recalling Eq. (2.6.2) for the definition of B₀(ω) and B₁(ω), Eq. (2.6.9) becomes

(2.6.10a)

(2.6.10b)

(2.6.10c)

with the generalized mass, damping and force of the equivalent SDF system given by Eq. (2.6.10), a comparison of Eqs. (2.6.7) and (2.6.1) shows that , i.e. the equivalent SDF system gives the exact value for the fundamental resonant response of the dam–water system.

The natural vibration frequency, , of the equivalent SDF system is ; substituting and Eq. (2.6.10a) for , gives

(2.6.11)

Because appears on both sides, this equation must be solved iteratively for . Hydrodynamic effects always reduce the natural vibration frequency because Re[B₁(ω)] > 0 for all excitation frequencies. Does Eq. (2.6.11), determined from the properties of the equivalent SDF system, give the correct reduction in frequency of the dam due to dam–water interaction? The fundamental resonant frequency of the dam–water system is approximately given by the excitation frequency that makes the real‐valued component of the denominator in the exact fundamental mode response, Eq. (2.6.1), equal to zero. This argument also leads to Eq. (2.6.11), which demonstrates that the mass of the equivalent SDF system defined in Eqs. (2.6.5) and (2.6.6) reduces the fundamental resonant frequency of the dam due to hydrodynamic effects by the exact amount.

The damping ratio of the equivalent SDF system is determined by substituting Eqs. (2.6.10b) and (2.6.10a) and utilizing Eq. (2.6.11):

(2.6.12)

in which the added damping due to upstream propagation of hydrodynamic waves and their absorption at the reservoir bottom is represented by the frequency‐independent damping ratio, ζ_r, defined as

(2.6.13)

The damping ratio ζ_r is non‐negative because Im[B₁(ω)] ≤ 0 for all excitation frequencies.

For dam–water systems with non‐absorptive reservoir bottom (α = 1), is real‐valued (Section 2.3). Thus, m_a(y) is real‐valued and Eq. (2.6.11) reduces to the earlier result (Chopra 1978) for the natural vibration frequency of the equivalent SDF system; and the added damping ratio, ζ_r, is zero, so Eq. (2.6.13) reduces to the earlier expression (Chopra 1978) for the damping ratio of the equivalent SDF system. An absorptive reservoir bottom (α < 1) results in complex‐valued m_a(y), which modifies the resonant frequency and increases the damping because now hydrodynamic pressure waves propagate upstream and refract into the absorptive reservoir bottom at the excitation frequency, .