Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 40

Earthquake analysis of dams is greatly simplified if compressibility of water is ignored, because then hydrodynamic effects can be modeled by a frequency‐independent added mass (Section 2.3.4). Here, we answer the important question: can water compressibility be ignored in the earthquake analysis of concrete gravity dams?

Frequency response curves of the dam without water, when presented using normalized scales as in Figures 2.5.5 and 2.5.6, are independent of the modulus of elasticity, E_s, of concrete. Similarly, the response curves including hydrodynamic effects do not vary with E_s if water compressibility is ignored (Chopra 1968). However, the Ω_r parameter, or correspondingly the E_s value, affects the response functions when water compressibility is included.

The response of the dam is greatly influenced by the frequency ratio, Ω_r. i.e. the relative frequencies of the two interacting systems, as demonstrated by Figures 2.5.1 and 2.5.3. The percentage decrease in the fundamental resonant frequency of the dam due to dam–water interaction is larger for the smaller values of Ω_r, i.e. larger values of E_s. The response value at resonance as well as the shape of the response curve in the neighborhood of the natural frequencies of the dam and of the impounded water also depend significantly on the frequency ratio Ω_r.

With increasing Ω_r, or decreasing E_s, the effects of water compressibility on response become smaller and the response curve approaches the result for incompressible water. For systems with Ω_r = 2.0, the effects of water compressibility are insignificant in the response to horizontal ground motion (Figures 2.5.5 and 2.5.7) but are still noticeable in the response to vertical ground motion (Figures 2.5.6 and 2.5.8). These results confirm the earlier conclusion that the effects of water compressibility become insignificant for systems with Ω_r > 2 or (Chopra 1968), which for the system considered implies E_s < 0.63 million psi. However, E_s for mass concrete is generally in the range of 2–5 million psi. Consequently, errors in response results will be significant if water compressibility is ignored. This conclusion will be reinforced further in the context of response to earthquake excitation (Chapter 6).

Figure 2.5.5 Influence of frequency ratio, Ω_r, on dam response to harmonic horizontal ground motion. Results are presented (1) including water compressibility with α = 1 for Ω_r = 0.67, 0.80, 1.0, and 2.0 (E_s = 5.67, 3.94, 2.52, and 0.63 million psi); and (2) assuming water to be incompressible.

Figure 2.5.6 Influence of frequency ratio, Ω_r, on dam response to harmonic vertical ground motion. Results are presented (1) including water compressibility with α = 1 for Ω_r = 0.67, 0.80, 1.0, and 2.0 (E_s = 5.67, 3.94, 2.52, and 0.63 million psi); and (2) assuming water to be incompressible.

Figure 2.5.7 Influence of frequency ratio, Ω_r, on dam response to harmonic horizontal ground motion. Results are presented (1) including water compressibility with α = 0.75 for Ω_r = 0.67, 0.80, 1.0, and 2.0 (E_s = 5.67, 3.94, 2.52, and 0.63 million psi); and (2) assuming water to be incompressible.

Figure 2.5.8 Influence of frequency ratio, Ω_r, on dam response to harmonic vertical ground motion. Results are presented (1) including water compressibility with α = 0.75 for Ω_r = 0.67, 0.80, 1.0, and 2.0 (E_s = 5.67, 3.94, 2.52, and 0.63 million psi); and (2) assuming water to be incompressible.