Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 27

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 transverse to the longitudinal axis of the dam (i.e. in the stream direction) and the vertical component .