Читать книгу Earthquake Engineering for Concrete Dams - Anil K. Chopra - Страница 22

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.

Figure 1.7.1 Gravity dam–water–foundation system.

Figure 1.7.2 Arch dam–water–foundation system.

Formulated in the frequency domain, this method is restricted to linear analysis, and requires special purpose computer programs, e.g. EAGD‐84 for two‐dimensional analysis of gravity dams and EACD‐3D‐1996 for three‐dimensional analysis of arch dams. These freely available programs were developed by graduate students at the University of California, Berkeley, as a part of their research for the doctoral degree, not as commercial software programs. Thus, they lack user‐friendly interfaces to facilitate input of data to define the system to be analyzed and to process response results. Despite these limitations, the aforementioned programs have been employed for seismic design of a few new dams and for seismic evaluation of several existing dams.

Figure 1.7.3 Finite‐element model of a dam–water–foundation system with wave‐absorbing boundaries.

Although linear analyses have provided great insight into the earthquake response of concrete dams, it is evident that a reliable estimate of the seismic safety of a dam can be obtained only by a nonlinear analysis if the earthquake damage is expected to be significant. The nonlinear model must recognize the possibility that the reservoir may extend to great distances upstream of the dam and the supporting rock extend to large depths and large distances in horizontal directions (Figures 1.2.1 and 1.2.2). The Direct FEM, presented in Chapter 11, overcomes the limitations of the standard FEM by introducing wave‐absorbing (or non‐reflecting) boundaries at two locations: (i) upstream end of the fluid domain to model its essentially semi‐infinite length; and (ii) the bottom and side boundaries of the foundation domain to model its semi‐unbounded geometry (Figure 1.7.3). The finite‐element model of the fluid domain now includes water compressibility and reservoir bottom sediments, and the finite‐element model of the foundation domain includes mass, stiffness, and material damping appropriate for the rock; water–foundation interaction is also included. Thus, the untenable assumptions of massless rock and incompressible water in the popular FEM are eliminated.

The earthquake excitation also is more realistically defined in the Direct FEM compared to the popular FEM. The excitation defined at the bottom and side boundaries of the foundation domain is determined by deconvolution of the design ground motion, typically specified on level ground at the elevation of the abutments (Figure 1.7.3). The resulting spatially varying motions cannot be input directly at wave‐absorbing boundaries; instead, tractions determined from the motions are specified.

Presented in Chapter 11, the direct FEM has the great advantage over the substructure method in that is it applicable to nonlinear systems, thus permitting modeling of concrete cracking, as well as sliding and separation at contraction joints, lift joints, dam–foundation interface, and fissures in rock; however, it has the disadvantage in that it requires truncation of fluid and foundation domains, thus requiring absorbing boundaries to simulate their semi‐unbounded size. This method has been developed in a form that can be implemented in any commercial finite‐element code; thus, it is applicable to 3D models of all types of concrete dams: gravity, arch, and buttress. Validation of the Direct FEM applied to linear systems against the substructure method is also included in Chapter 11.