Dynamic Response Analysis

Abstract

The dynamic response characteristics of a one-mass-spring-damper system are outlined. The resonant vibration, damping, and natural periods of the dynamic system are explained. Multi-masses-springs-dampers systems for the dynamic response analysis of buildings and other structures, and three methods of solving the differential equations governing the motions of the system: the direct integration in the time domain, the integration in frequency domain (Fourier transformation), and the modal analysis method are explained. Furthermore, the basic concept of the Finite Element Method (FEM), which is frequently used for the dynamic response analysis of reactor buildings/facilities, and foundation ground/surrounding slopes is introduced.

Appendix 14.1: Basic Concept of Finite Element Method

The Finite Element Method (FEM) is often used to examine the seismic stability of the reactor and turbine buildings, and foundation grounds and surrounding slopes. This numerical method was developed to analyze the deformation, stress, and strain of a continuum body. The FEM analysis process is shown in Fig. 14.10.

Fig. 14.10

Analysis by the finite element method

• Step (1) The structures and the ground are divided into finite number of elements. For two-dimensional analysis, triangle and square elements are used. For the three-dimensional analysis, triangular pyramid elements and cube elements are used.

• Step (2) The relationship between the external forces acting at nodal points of the elements and the displacements of the nodal points is obtained based on the elastic theory. Because the stress and strain are assumed to be uniform inside the elements in most cases, division into small elements at the points of stress concentration is required.

• Step (3) Equilibrium equations are obtained at the nodal point by considering the external force and inertia forces. The number of equilibrium equations is the product of the degree of freedom at the nodal points (displacements in x, y, and z directions, as well as rotation of the nodal points) with the number of the nodal points.

• Step (4) Boundary conditions (for example, fixed boundary and forced displacements at the nodal points) are taken into consideration when developing the total vibration equation.

• Steps (5), (6), and (7) For a static problem, the equilibrium equation is solved, and for a dynamic problem, the vibration equation is solved by the numerical method shown in Sect. 14.2. The stress and strain in each element are calculated from the displacement of the nodal points.

Examples of the FEM model for the reinforced-concrete containment vessel, and the underground duct and ground are shown in Figs. 4.16 and 6.6, respectively.