Earthquake Engineering for Concrete Dams. Anil K. Chopra

Earthquake Engineering for Concrete Dams

ft, and damping ratio ζ₁ = 5%. The Young's modulus for mass concrete is varied over a wide range by specifying three values for the frequency ratio Ω_r = 0.80, 1.0, and 2.0, which, for the selected system properties, correspond to E_s = 3.94, 2.52, and 0.63 million psi, respectively. The first two cover the range of values representative of mass concrete, and the third is unrealistically low, chosen to help identify the condition under which water compressibility can be neglected. The unit weight of water is 62.4 lb/cu ft and the velocity of pressure waves in water is C = 4720 ft/sec. Two values for the depth of water are considered: an empty reservoir (H/H_s = 0) and a full reservoir (H/H_s = 1). The wave reflection coefficient is varied over a wide range; the values considered are: α = 1.0, 0.75, 0.50, 0.25, and 0.

The response of the dam to harmonic horizontal and vertical ground motion is determined by numerically evaluating Eq. (2.4.10) wherein the fundamental natural vibration frequency ω₁, and mode shape images were determined using a finite element idealization of the dam, and the integrals involved in M₁, images , images , and B₁(ω) were computed in discretized form.

The complex‐valued frequency response function, images , is a dimensionless response factor representing the ratio of horizontal acceleration at the dam crest to unit free‐field ground acceleration in the l(= x or y) direction. For each case mentioned above, the absolute value of this complex‐valued response factor is plotted against the normalized excitation frequency ω/ω₁. When presented in this form, the results apply to dams of all heights with the idealized triangular cross section, and chosen Ω_r, H/H_s, and α values.

2.5.3 Dam–Water Interaction Effects

Frequency response functions for dams subjected to horizontal and vertical ground motions are presented in Figures 2.5.1–2.5.4 for two selected values of Ω_r = 0.67 and 1.0. Each plot contains response curves for the dam with full reservoir for five values of α and the response curve for the dam alone, i.e. with an empty reservoir. The latter is the familiar response curve for a SDF system with frequency‐independent mass, stiffness, and damping parameters. However, dam–water interaction including water compressibility introduces frequency‐dependent terms in Eq. (2.4.10), resulting in complicated shapes for the response curves.

The frequency response function due to horizontal ground motion displays strongly resonant behavior with large amplification over an especially narrow frequency band because of dam–water interaction and water compressibility. The single resonant peak in the response of the dam without water may become two resonant peaks for a full reservoir if the reservoir bottom is non‐absorptive, a behavior that develops for systems with smaller Ω_r or stiffer dams (Figure 2.5.1). With increasing wave absorption at the reservoir bottom, i.e. decreasing α, the first resonant peak is reduced, whereas the second peak is increased, and for a small enough α the two peaks coalesce, resulting in a single resonant peak at an intermediate resonant frequency. For systems with the larger Ω_r value, or relatively flexible dams, only a single resonant peak develops for all values of α (Figure 2.5.3). For such systems, as α decreases, increased absorption of energy through the reservoir bottom further reduces the resonant amplitude, with little change in the resonant frequency. The fundamental resonant frequency of the dam including hydrodynamic effects is lower than both the natural frequency ω₁ of the dam alone and the fundamental natural frequency images of the impounded water.

Chart depicting the dam response to absolute value of harmonic horizontal acceleration at the dam crest.

Figure 2.5.1 Dam response to harmonic horizontal ground motion; frequency ratio, Ω_r = 0.67, i.e. E_s = 5.67 million psi; α = 1.0, 0.75, 0.50, 0.25, and 0; response of dam alone is also shown.

Chart depicting the dam response to absolute value of harmonic vertical ground motion at dam crest.

Figure 2.5.2 Dam response to harmonic vertical ground motion; frequency ratio, Ω_r = 0.67, i.e. E_s = 5.67 million psi; α = 1.0, 0.75, 0.50, 0.25, and 0; response of dam alone is also shown.

The response function due to horizontal ground motion is especially complicated if the pressure waves are fully reflected at the reservoir bottom, i.e. α = 1, because at excitation frequencies equal to images , the natural vibration frequencies of the impounded water, the added mass and force are both unbounded. When determined by a limiting process, however, the response function due to horizontal ground motion has bounded values at images (Chopra 1968), which appear as local dips in the response curve (Figures 2.5.1 and 2.5.3).

Figure 2.5.3 Dam response to harmonic horizontal ground motion; frequency ratio, Ω_r = 1.0, i.e. E_s= 2.52 million psi; α = 1.0, 0.75, 0.50, 0.25, and 0; response of dam alone is also shown.

Figure 2.5.4 Dam response to harmonic vertical ground motion; frequency ratio, Ω_r = 1.0, i.e. E_s = 2.52 million psi; α = 1.0, 0.75, 0.50, 0.25, and 0; response of dam alone is also shown.

The response function due to vertical ground motion also displays the first resonance at a frequency lower than the natural frequency ω₁ of the dam alone with complicated behavior in the frequency range between ω₁ and images that is dominated by the unbounded response values at excitation frequencies equal to images (Figures 2.5.2 and 2.5.4). These unbounded peaks are not the result of resonance in the usual sense, which is associated with the denominator in Eq. (2.4.10) attaining a minimum, but are caused by the unbounded added force.

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