Earthquake Engineering for Concrete Dams. Anil K. Chopra

(2.2.7) to include dam–water interaction (Section 2.4) and dam–foundation interaction (Section 3.2.4).

      2.2.2 Earthquake Response: Horizontal Ground Motion

      In preparation for response spectrum analysis of the dam including dam–water–foundation interaction subjected only to horizontal ground motion (to be developed in Chapters 3 and 4), such analysis for the dam alone is presented first.

      The response history of the modal coordinate due to arbitrary ground acceleration in the x‐direction can be computed from dam response to harmonic ground motion, characterized by the frequency response function (Eq. (2.2.7)), using standard Fourier synthesis techniques. Alternatively, it can be expressed in terms of D₁(t), the deformation response of the first‐mode single‐degree‐of‐freedom (SDF) system, an SDF system with vibration properties – natural frequency ω₁ and damping ratio ζ₁ – of the first vibration mode of the dam. The equation of motion of this SDF system subjected to ground acceleration is given by

(2.2.8)

Having temporarily limited the earthquake response analysis to the x‐component of ground motion, the superscript x may be dropped from , , and . Comparing Eq. (2.2.5) to Eq. (2.2.8) gives the relation between q₁ and D₁:

(2.2.9)

where D₁(t) can be determined by numerically solving Eq. (2.2.8). Substituting Eq. (2.2.9) in Eq. (2.2.1) gives the displacement history of the dam

      (2.2.10)

      We will be especially interested in the peak value of response, or for brevity, peak response, defined as the maximum over time of the absolute value of the response quantity:

      (2.2.11)

      where the subscript “o” attached to a response quantity denotes its peak value. The peak displacements can then be expressed as

(2.2.12)

      where is the ordinate of the deformation response (or design) spectrum for the x‐component of ground motion evaluated at period T₁ = 2π/ω₁, and damping ratio ζ₁; the subscript “o” that denotes peak value will subsequently be dropped to simplify notation.

The equivalent static forces associated with the peak displacements [Eq. (2.2.12)] are given by (Chopra 2017: Section 17.7)

(2.2.13)

in which is the ordinate of the pseudo‐acceleration response (or design) spectrum, and w_k(x, y) = gm_k(x, y). Because the vertical (k = y) component of displacements in the fundamental vibration mode, , is much smaller than their horizontal (k = x) component, the associated vertical forces may be dropped, leaving only the horizontal (k = x) component of forces in Eq. (2.2.13):

(2.2.14)

      wherein the superscript x has been dropped from for simplicity of notation.

2.3 HYDRODYNAMIC PRESSURES

In this section we will present results for hydrodynamic pressures on the upstream face of the dam for two cases: (i) a rigid dam excited by x and y components of ground motion; and (ii) a flexible dam undergoing motion in its first mode of vibration; the three excitations are shown schematically in Figure 2.3.1. All three of these results will be utilized in deriving the fundamental mode response of the dam–water system (Section 2.4), and the hydrodynamic pressures on a rigid dam will be compared with classical solutions.

      2.3.1 Governing Equation and Boundary Conditions

      Assuming water to be linearly compressible and neglecting its internal viscosity, irrotational motion of the water is governed by the two‐dimensional wave equation

(2.3.1)

where p(x, y, t) is the hydrodynamic pressure (in excess of hydrostatic pressure), and C is the speed of pressure waves in water; C = 4720 fps or 1480 mps. The hydrodynamic pressure is generated by horizontal motion of the vertical upstream face of the dam and by vertical motion of the horizontal reservoir bottom. The boundary conditions for Eq. (2.3.1) governing the pressure are expressed in Eqs. (2.3.2)–(2.3.5).