Earthquake Engineering for Concrete Dams. Anil K. Chopra
dam–water interaction. Thereafter, results for dam response are presented for a wide range of parameters that characterize the dam–water system. These results provide a basis to identify the effects of dam–water interaction and their influence on the vibration properties – natural vibration period and damping ratio – and on the response of concrete gravity dams to earthquake ground motion. Also investigated are the implications of neglecting compressibility of water, an approximation that enables representation of hydrodynamic effects by inertia forces associated with an added mass of water moving with the dam. Finally, we develop an equivalent single‐degree‐of‐freedom (SDF) system to model the response of dams including dam–water interaction that enables estimation of peak response directly from the earthquake response (or design) spectrum. Such an analysis is intended for preliminary design and safety evaluation of dams.
2.1 SYSTEM AND GROUND MOTION
The system considered consists of a monolith of a concrete gravity dam fixed (or clamped) to the horizontal surface of underlying rock, assumed to be rigid, and impounding a reservoir† of water with wave‐absorptive reservoir bottom (Figure 2.1.1). We will initially study the planar vibrations of an individual monolith due to earthquake excitation, a simplification that is supported by observations of monoliths vibrating somewhat independently during the earthquake response of Koyna Dam (Chopra and Chakrabarti 1972) and forced vibration tests of Pine Flat Dam (Rea et al. 1975); this simplification is discussed further in Section 5.1. The system is analyzed under the assumption of linear behavior.
Figure 2.1.1 Dam–water system.
The dam is idealized as a two‐dimensional finite element system, thus making it possible to consider arbitrary geometry and variation of material properties. However, certain restrictions on the geometry are imposed to permit a continuum solution of the hydrodynamic wave equation in the fluid domain. For the purpose of determining hydrodynamic effects, and only for this purpose, the upstream face of the dam is assumed to be vertical. This assumption is reasonable for most concrete gravity dams, because typically the upstream face is vertical or almost vertical for most of its height, and the hydrodynamic pressure on the dam face is insensitive to small departures of the face slope from being vertical, especially if these departures are in the lower part of the dam, which is usually the case. The impounded water in the reservoir is idealized by a fluid region of constant depth and infinite length in the upstream direction.
The bottom of a reservoir upstream from a dam is generally not rigid; its flexibility could arise from flexibility of the underlying foundation† or deposited sediments (Figure 2.1.1). The reservoir bottom is approximately modeled by a boundary that partially absorbs incident hydrodynamic pressure waves; see Appendix 2.A for a description of this model.
The excitation for the two‐dimensional dam–water system is defined by the two components of free‐field ground acceleration in the plane of the monolith (or cross section) of the dam: the horizontal component
2.2 DAM RESPONSE ANALYSIS
2.2.1 Frequency Response Function
The displacements of the dam – relative to its base – vibrating in its fundamental vibration mode due to the l‐component of ground motion (l = x and y represents horizontal and vertical components, respectively) can be expressed as
in which rx(x, y, t) and ry(x, y, t) are the horizontal and vertical components of displacement, respectively;
Under the approximation of Eq. (2.2.1), the equation of motion for a dam supported on rigid foundation with an empty reservoir is
in which the generalized mass
(2.2.3)
where the integration extends over the cross‐sectional area of the dam monolith; the mass density of the dam concrete mk(x, y) = m(x, y), k = x and y is considered separately for the horizontal and vertical components of dam motion for convenience later in expressing the hydrodynamic effects in terms of an added mass and added damping;
(2.2.4)
Equation (2.2.2) can be rewritten as
where
(2.2.6)
For harmonic free‐field ground acceleration