Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 21

It is apparent from the preceding section that traditional seismic coefficient methods must be abandoned in favor of dynamic analysis procedures in order to reliably predict the earthquake‐induced demands on dams. Because of the versatility of the FEM in modeling arbitrary geometries and variations of material properties, this method is suited for formulating a computational model of a concrete dam. In fact, analysis of the dam alone (no impounded water) supported on rigid foundation to ground motion specified at the base would be a standard application of the FEM. However, analysis of concrete dams is greatly complicated by the fact that the structure interacts with the water impounded in the reservoir and with the deformable foundation that supports it, and because the fluid and foundation domains extend to large distances (Figures 1.2.1 and 1.2.2).

The interaction mechanisms may be modeled in a crude way by combining finite‐element models for a limited extent of the impounded water and of the foundation with a finite‐element model of the dam, thus reducing the “semi‐unbounded” system to a finite‐sized model with rigid boundaries, which, generally, do not exist at the site (Figure 1.6.1). Such a model does not allow for radiation of hydrodynamic pressure waves in the upstream direction or stress waves in the foundation because these waves are reflected back from the rigid boundaries, thus trapping the energy in the bounded system. Thus, a significant energy loss mechanism, referred to as radiation damping, is not represented in the bounded models of the fluid and foundation domains. Developing procedures for analysis of dam–water–foundation systems that recognize the semi‐unbounded geometry of the fluid and foundation domains was a major research objective during the 1970–1995 era. Research results on this challenging problem are featured prominently in this book.

While such research was in progress, an expedient solution was proposed by Clough (1980) that included in the finite‐element model a limited extent of the foundation, assumed to have no mass, and modeled hydrodynamic effects by an added mass of water moving with the dam; the design ground motion defined typically at the ground surface was applied at the bottom fixed boundary of the foundation domain; see Figure 1.6.2. This modeling approach became popular in actual projects because it was easy to implement in commercial finite‐element software. However, such a model solves a problem that is very different from the real problem on two counts: (i) the assumptions of massless rock and incompressible water – implied by the added mass water model – are unrealistic, as will be demonstrated in Chapters 6 and 9; and (ii) applying ground motion specified at the ground surface to the bottom boundary of the finite‐element model contradicts the recorded evidence that motions at depth generally differ significantly from surface motions.

Figure 1.6.1 Standard finite‐element analysis model with rigid, wave‐reflecting boundaries.

Figure 1.6.2 A popular finite‐element model that assumes foundation to have no mass and models hydrodynamic effects by an added mass of water moving with the dam.