Anil K. Chopra - Earthquake Engineering for Concrete Dams

(2.2.14)

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

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.

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).

Figure 2.3.1Acceleration 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 klis 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 ris the Young's modulus, and ρ ris 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.

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)