Earthquake Engineering for Concrete Dams. Anil K. Chopra
design – in the dynamics of arch dams. When dam–water interaction and water compressibility are properly considered, hydrodynamic effects result in significant increases in the earthquake‐induced stresses in arch dams, more so than for gravity dams. Similarly, when dam–foundation interaction including foundation mass and radiation damping are properly considered, this interaction mechanism generally has a profound influence on the earthquake‐induced stresses in arch dams just as in the case of gravity dams.
Figure 1.4.1 Distribution of seismic coefficients over the dam surface in the first two vibration modes of an arch dam.
Source: Adapted from Bureau of Reclamation (1977).
1.5 UNREALISTIC ESTIMATION OF SEISMIC DEMAND AND STRUCTURAL CAPACITY
Traditional design procedures greatly underestimate seismic demands imposed on both arch and gravity dams, as well as the capacity of these structures to resist these demands. The seismic forces associated with the mass of the dam and the hydrodynamic pressures are underestimated, as mentioned earlier. The tensile strength of concrete, which is not insignificant, is essentially ignored in the no‐tension requirement in the design criteria for gravity dams and by the small value allowed for arch dams. A progressive, systematic approach to computation of seismic demands and evaluation of structural capacity is presented in Chapter 12.
Methods for designing dams must be improved in at least two major ways: (i) the tensile strength of concrete should be determined by testing cylindrical cores that are large enough – diameter equal to three or four times the size of the coarse aggregate; and (ii) seismic demands should be computed by dynamic response analysis of the dam–water–foundation system. Development of such analysis procedures is one of the main thrusts of this book.
1.6 REASONS WHY STANDARD FINITE‐ELEMENT METHOD IS INADEQUATE
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.
1.7 RIGOROUS METHODS
Earthquake analysis of dams should include the following factors: (i) the semi‐unbounded extent of the impounded water and foundation domains; (ii) dam–foundation interaction considering mass, flexibility, and damping of rock; and (iii) dam–water interaction considering compressibility of water and the sediments that invariably deposit at the reservoir bottom. Two approaches exist for such rigorous analyses: the substructure method and a direct finite‐element method.
Presented in Chapters 5 and 8, the substructure method determines the response of idealized systems shown in Figures 1.7.1 and 1.7.2 to free‐field ground motion specified at the interface between the dam and foundation; this is the motion that would have existed in the absence of the dam and impounded water. The substructure method permits different types of models for the three substructures – dam, fluid domain, and foundation domain: finite‐element model for the dam; and “continuum” models for the fluid and foundation domains of semi‐unbounded geometry. The substructure concept permits modeling of the semi‐unbounded fluid and foundation domains without truncating them to finite size and specifying the earthquake excitation directly at the dam–foundation interface.