# DYNAMIC ANALYSIS AND  EARTHQUAKE RESISTANT DESIGN

To carry out the dynamic analysis of structures, sometimes including ground, first a model of the system is prepared and then its dynamic properties such as natural frequencies and corresponding vibration modes etc. are explored.

The response of the model to an external force is then studied and the results investigated to ascertain whether the deformation and/ or the induced stresses etc. are within the specified limits or not. With the development of computers, the methods of dynamic analysis have also considerably developed.

Dynamic behavior of the system is obtained by solving the equation of motion, of which external force is the inertial force. Accordingly, to study the dynamic response of the system, we must evaluate the inertial force.

We can consider the continuous mass system wherein the inertial force is expressed in a distributed fashion or discrete mass system wherein the distribution of inertial force is considered concentrated at some points on the basis of some engineering assumption. In a continuous mass system the general solution of dynamic response is analytically obtained for each structural element and these solutions then combined in order to satisfy the boundary conditions between the elements. In the case of a discrete mass system, the governing equation is written after considering the resultant equilibrium of the inertial force, restoring force and external force on the basis of the degrees of freedom for that point mass. In general, the material constants of the restoring force are obtained on the basis of static evaluation in calculation.

The representative methods include the spring-mass system, which is quite fundamental, the finite .element method (FEM), or the boundary element method (BEM), and hybrid methods which are combinations of the aforesaid. Each method has its own merits. In the case of a complex vibration system which consists of many subsystems, it may not be possible to analyze all the subsystems with the same accuracy by just one method. It may be necessary to use more optimized modeling and select more appropriate methods of analysis depending on the structural properties or the importance of the object of analysis. For example, if we want to study the overall structural behavior of different types of civil structures or buildings, analysis using the FEM based on beam elements may be simpler and better, but it is necessary to use other appropriate elements modeling the structure to analyze the induced localized stresses. Also, one has to decide from engineering considerations whether the behavior of an object should be considered a two-dimensional plane problem or a three-dimensional one. For seismic response analysis of the ground, the traveling wave is often restricted and the governing equation analyzed by the calculus of finite differences method. On the other hand, the shear column model, which is analyzed by converting the ground into a lumped mass system, can take nonlinearity into consideration and hence is also widely used. Among the methods based on the wave theory, the transfer matrix method, which assumes the ground as a layered structure, can be grouped in a continuous mass system. If, however, we assume that each layer is a thin layer and further assume a displacement distribution pattern, then a thinlayer method can be developed as a discrete system. In a soil-foundation interaction system, some area exists wherein the parameters change in a complex manner both physically and geometrically. The FEM is suitable in this case. The BEM has also been applied in analyzing the dynamic soil-structure interaction system by treating the problem as a complex boundary value one. In this method the boundary condition of the ground at an infinite distance is included in the fundamental solution (Green function) and hence it is suitable for studying the scatter of waves toward an infinite distance. A hybrid combination of FEM and BEM was recently developed for the purpose of treating the geometrical and material irregularities of structure of the model as well as the infinite boundary problem.

Models of the beam as a structural element, based on the continuous mass system and the lumped mass system, are discussed in Sec. 1.2; the equations of a two-dimensional and a three-dimensional continuous body based on the FEM are discussed in Sec. 1.3; Sec. 1.4 discusses the FEM to solve the governing equation for the dynamic field while the equation of a continuous body, based on the BEM of dynamic analysis, is discussed in Sec. 1.5 for dynamic analysis of a soil-structure system.