Earthquake Engineering for Concrete Dams. Anil K. Chopra

Figure 2.3.1 Acceleration excitations causing hydrodynamic pressures on the dam defined by frequency response functions: (a) ; (b) ; and (c) .

      The normal pressure gradient at the vertical upstream face of the dam is proportional to the horizontal acceleration of this boundary, resulting in the boundary condition for excitation cases (i) and (ii), respectively:

(2.3.2a)

      (2.3.2b)

      where ρ is the density of water, and δ_kl is the Kronecker delta function (δ_xx = δ_yy = 1, δ_xy = δ_yx = 0) and, contrary to the usual convention, summation is not implied when repeated indices appear.

      Similarly, the normal pressure gradient at the horizontal bottom of the reservoir is proportional to the vertical acceleration of this boundary:

(2.3.3)

      which is valid only if hydrodynamic waves are fully reflected at the boundary. This boundary condition is generalized to account for the influence of sediments at the reservoir bottom or of foundation flexibility on hydrodynamic pressures (Appendix 2)

(2.3.4a)

      or

(2.3.4b)

where is the compression wave velocity, E_r is the Young's modulus, and ρ_r is the density of the reservoir bottom materials. The second term on the right side in Eq. (2.3.4b) represents the modification of the vertical free‐field ground acceleration due to flexibility at the reservoir bottom. Because this interactive acceleration is proportional to the time derivative of the hydrodynamic pressure, the reservoir‐bottom flexibility produces a damping effect associated with partial refraction of hydrodynamic pressure waves at the reservoir bottom; ξ may be interpreted as a damping coefficient. For reservoir bottom that is rigid, C_r = ∞, ξ = 0, and the second term on the right‐side of Eq. (2.3.4b) is zero, giving the boundary condition for a fully reflective reservoir bottom [Eq. (2.3.3)].

      The wave reflection coefficient α, defined as the ratio of the amplitude of the reflected hydrodynamic pressure wave to the amplitude of a vertically propagating pressure wave incident on the reservoir bottom, is related to the damping coefficient, ξ (Appendix 2; Rosenblueth 1968; Hall and Chopra 1982) by

(2.3.5)

      The material properties of the sedimentary deposits at the reservoir bottom are highly variable and difficult to characterize. In contrast, the properties of the underlying rock can be better defined. Substituting them in Eq. (2.3.5) gives the corresponding value of α. For a realistic range of properties of rock, α would generally vary between 0.5 and 0.85. Researchers have attempted to measure α in the field (Ghanaat and Redpath 1995).

      Neglecting the effects of waves at the free surface of water, an assumption discussed in Chopra (1967), leads to the boundary condition

(2.3.6)

The hydrodynamic pressures must satisfy the boundary conditions of Eqs. (2.3.2), (2.3.4b), and (2.3.6), and the radiation condition in the upstream direction.

      2.3.2 Solutions to Boundary Value Problems

      The steady state hydrodynamic pressure due to unit harmonic free‐field ground acceleration can be expressed in terms of its complex frequency response function

(2.3.7)

      Substituting this in Eq. (2.2.1) leads to the Helmholtz equation:

(2.3.8)

The frequency response function, , for the hydrodynamic pressure when the excitation is the horizontal ground acceleration and the dam is rigid (Figure 2.3.1a), is the solution of Eq. (2.3.8) subject to the boundary conditions of Eqs. transformed according to Eq. (2.3.7):

      (2.3.9)

The frequency response function for the hydrodynamic pressure, when the excitation is the vertical ground acceleration and the dam is rigid (Figure 2.3.1b), is the solution of Eq. (2.3.8) subject to the boundary conditions of Eqs. transformed according to Eq. (2.3.7):

      (2.3.10)

      The frequency response function for the hydrodynamic pressure