6.1. IntroductionThe catastrophic consequences on life and property resulting from a failure of large dams have led engineers to design and built these structures to resist strong ground motion with no or only minor damages. This has provided a strong impetus for wide researchers, particularly in developing new methods of dynamic analysis for concrete gravity dams in the seismic region. Dams need to be functional under all possible combinations of loading as they cater to multiple functions and their catastrophic failure with a sudden release of reservoir water may lead to huge economic and human losses. An occurrence of earthquakes may be detrimental to the strength and stability of dam structure and hence earthquake resistant design of dam is given vital importance. The dynamic analysis of a concrete gravity dam is a reasonably complex problem. The response of a dam subjected to dynamic loading is a combined effect of the interaction among dam, reservoir, and foundation systems. Concrete dams are constructed as monoliths separated by transverse contraction joints, oriented normal to the dam axis. These vertical joints extended from the foundation to the top of the dam and from the upstream face to the downstream face. Certain assumptions are made in the analysis of gravity dam like materials in the foundation and body of a dam is isotropic and homogeneous. The occurrence of an earthquake imparts a quantity of energy to a structure called seismic energy or absorbed energy. During an earthquake, part of the absorbed energy is temporarily stored in the structure as kinetic energy and elastic strain energy; the remaining absorbed energy is dissipated throughout the structure’s components through damping and inelastic deformation. Eventually, all energy absorbed by the structure should dissipate. At the time of an earthquake, besides the normal forces, namely, self-weight, water pressure and silt, inertia and hydrodynamic forces act on a dam. The inertia force is the product of mass and acceleration and this force is acting in the direction opposite to that of ground motion. The horizontal inertia force acts from upstream to downstream as well as from downstream to upstream. Similarly, the vertical inertia force acts from downward to upward as well as from upward to a downward direction. Acceleration downward decreases the weight. Hence the dam is designed for the worst combination that is for the horizontal and vertical inertia forces. The inertia effect between the dam and reservoir causes the hydrodynamic forces on the dam. The direction of forces is opposite to the earthquake acceleration, but the acceleration changes sign in practice. Hence the force could be either pressure or suction.Hydrodynamic forces are worked out on the assumption that fluid is incompressible and dam is rigid and hence the same motion throughout its body as that of the base of the dam. In general, the dam reservoir system is evaluated by treating the dam and reservoir as a two coupled system, namely dynamic response of dam ignoring the effect of reservoir water and hydrodynamic pressure on the dam to represent the effect of reservoir water. It is invariably assumed that interaction effect between the dam and reservoir are small so that the solutions of the uncoupled system can be combined to obtain a complete solution for the response of the dam. For the determination of earthquake, forces code suggests the seismic coefficient method and Response spectrum an method. But the actual behavior of dams is not evaluated critically and a lot of research in this field is necessitated. However, analyzing dam for seismic force is not a simple problem. Like all other structure, concrete gravity dam requires nonlinear, dynamic and probabilistic study to evaluate the internal forces due to seismic loading. It is not always possible to obtain rigorous mathematical solutions for the engineering problem. In fact analytical solutions can be obtained only for certain simplified situations for the problems involving complex material properties, loading and boundary conditions, the engineer introduce assumptions and idealization deemed necessary to make the problem mathematically manageable, but still capable of providing sufficiently approximate solutions and the satisfactory results from point of view of safety and economy. The link between the real physical system and the mathematically feasible solutions is provided by the mathematical model which is the symbolic designations for the substitute idealized system including all the assumptions imposed on the physical problem. Dynamic analysis of buildings and dams are very complex phenomena. There are many factors that influence the dynamic response of gravity dam against earthquake, such as fluid-structure interaction, structure foundation interaction. In order to solve this complex problem, we use mathematical modal including all the assumptions imposed on the physical problem. The dam-reservoir interaction problem can be analyzed using different approaches like Added mass approach, Lagrangian approach & Eulerian approach. Hence in order to evaluate the safety of dams, it is necessary to study various aspects influencing the seismic response of concrete gravity dam.6.2 Methods and Modeling of Dam-Reservoir-Foundation Systems6.2.1GeneralMany problems in engineering and applied science are governed by differential equations or integral equations. The solution of these equations would provide an exact, closed-form result to the particular problem being studied. However, complexities in the geometry, properties and boundary conditions that are seen in most of the real world problems usually mean that an exact solution cannot be obtained in a reasonable amount of time. Current product design cycle times imply that the engineers must obtain design solutions that can be readily achieved in a reasonable time frame, and with reasonable effort. The FEM is a numerical procedure for obtaining approximate solutions to many of the problems encountered in engineering analysis.The finite element method (FEM) is a numerical method for the analysis of field problems. The field problems in engineering and physics, for example, the structural analysis problems, heat transfer problems, fluid flow problems, etc. whether in 1D 2D or 3D, the FEM can be used effectively. In FEM, a complex region defining a continuum is discretized into simple geometric shapes called elements. The properties and governing relationships are assumed over these elements and expressed mathematically in terms of unknown values at specific points in the elements called nodes. An assembly process is used to link the individual elements to the given system. When the effects of loads and boundary conditions a are considered, a set of linear or nonlinear algebraic equations is usually obtained. The solution of these equations gives the approximate behavior of the continuum or system. While the continuum is having an infinite number of degrees of freedom (DOF), the discretized model have finite DOFs. Usually, the problems assessed are too complicated to be solved satisfactorily by classical analysis methods. The results obtained by classical analysis are rarely exact.For problems involving complicated geometries, loading, and material properties, it is generally not possible to obtain the solution by classical analytical methods. Analytical solutions are those given by mathematical expressions that yield the value of desired unknown quantities at any location in a body and thus valid for infinite locations in the body. The analytical solutions generally require a solution of ordinary or partial differential equation which because of complicated geometries, loadings and material properties are usually not obtainable. Hence we need to rely on numerical methods such as FEM for the acceptable solution. Finite element formulation results in the system of simultaneous algebraic equations rather than differential equations.6.2.2 Basic Steps in AnalysisOne of the reasons for wide applications of the FEM is due to the availability of a number of general purpose analysis programs. The whole analysis procedure is well organized into three basic components: preprocessing, processing & post-processing.Finite element analyzes of practical problems require handling of a large amount of input data. Manual preparation of input data is the tedious, time-consuming and error-prone task, particularly for complex 3D analysis. In general, a preprocessor creates the finite element model and input data necessary for a finite element analysis program. The postprocessor accepts the results of the analysis and generates tables, diagrams, graphs etc. for interpretation of results obtained. The preprocessor accepts input from the user created finite element mesh and other data required for analysis, and displays the model for data check and correction, if any, to be made by the user in an interactive mode. Graphical post-processing of the results helps to visualize the physical consequences of the analysis. The basic steps involved in the finite element analysis are as given in Figure 6.1 below.