Earthquake Engineering for Concrete Dams. Anil K. Chopra

reduces the added force associated with both ground motion components and the added mass to bounded values at (Section 2.3.3). Consequently, the dips at in the response function due to horizontal ground motion are eliminated; and the unbounded values at in the response function due to vertical ground motion are reduced to bounded peaks, which disappear for the smaller values of α.

      2.5.4 Implications of Ignoring Water Compressibility

      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.

      2.5.5 Comparison of Responses to Horizontal and Vertical Ground Motions

      Comparing the response of the dam to horizontal and vertical ground motions (Figures 2.5.5 and 2.5.6) it is apparent – consistent with common view – that without impounded water the response to vertical ground motion is relatively small because the excitation term in Eq. (2.4.10) is much smaller for vertical ground motion (l = y) than for horizontal ground motion (l = x). With impounded water in the reservoir, the response is affected by the added hydrodynamic mass, damping, and force terms in Eq. (2.4.10). Because the added hydrodynamic mass and damping are independent of the excitation direction, the response functions due to horizontal and vertical ground motions display the same resonant frequency and effective damping. However, the added force is associated with hydrodynamic pressures acting in the horizontal direction on the vertical upstream face of a rigid dam, whether the ground motion is horizontal or vertical. In the latter case, is much larger than the small . As a result, hydrodynamic effects (including water compressibility) cause a larger increase in response to vertical ground motion than in response to horizontal ground motion. This is apparent by comparing the response curves for the dam without water and with water associated with the two excitations (Figures 2.5.1 versus 2.5.2 and Figures 2.5.3