Modal Response Contribution

In order to optimize structure calculation, it is inconceivable to miss the understanding of the modal response contribution and truncation error. This chapter enlightens the reader on the subject by dealing with certain points, namely the determination of the elastic forces modal contribution, modal participation factors and truncation error, and static correction procedure. At the end of the chapter, examples will be treated in order to bring clarity to the reader on the points cited before.

Free Vibration of Single Degree of Freedom Linear Oscillator

This chapter is dedicated to understanding and studying a didactic case represented by a free vibration of a linear oscillator with a single degree of freedom. Mathematical equations of the problem will be detailed as well as the solution that goes with single degree of freedom oscillator for translational vibration for all cases: free undamped oscillator, as well as free damped oscillator, and torsional free undamped vibration passing by critical, subcritical, and over damping system. At the end of the chapter, some examples will be treated.

Passive Dampers

Civil structures are subjected to various types of loading, which induce severe damage to the structures. Many techniques have been developed for structural rehabilitation; one of the emerging technologies is the use of energy dissipation systems such as fluid viscous dampers (referred to hereafter by FVD). In this chapter, the effect of these devices on the dynamic behavior of an RC building is investigated, with an optimal choice of the linear FVD parameter (i.e., damping coefficient), using a simplified and effective approach. It was found that the maximum inter-story drift of the analyzed retrofitted structures can be significantly reduced compared to the original ones.

Numerical Methods for Solving Dynamic Equilibrium Equations

Once the number of degrees of freedom exceeds a certain number, it would be impossible to solve the dynamic equilibrium equation manually, hence the need to switch to a numerical resolution, whose general principle is to convert a dynamic equation into a static one. We are interested, for the dynamic analysis of the structures and the continuous media, in “one-step” algorithms rather than “multi-step” one. It is mainly because the systems to be solved are of large size and that it is important to minimize the number of operations and value to be memorized to the detriment, if necessary, of precision. A “one-step” algorithm, like that of Newmark, makes it possible to calculate the solution at time tn+1, starting from the solution at time tn. In addition to the disadvantage of requiring the storage of several steps, the “multi-step” algorithms such as that of Houbolt requires a startup procedure. This chapter allows the reader to enumerate and understand different numerical method with different examples.

Seismic Analysis

An earthquake is caused by sudden motion of the earth's crust. Every year, tens of thousands of earthquakes of all sizes occur all over the world. Some cause tiny or major tremors, others occur in remote areas where no one lives. This chapter allow readers to find out more about the earth structure as well as earthquake nature. Therefore, to detail the definition and construction of a spectrum, a presentation of earthquake analysis is given. In order to become familiar with this analysis, two applications are presented at the end of the chapter with a detailed solution.

Single Degree of Freedom Forced Vibration System

This chapter concerns the study of forced vibration of a single degree of freedom system, treating undamped and damped system under harmonic, periodic, and arbitrary loading with different cases and examples. Passing by all components of the general solution of an undamped forced system, which are a transient solution, depends only on initial conditions, transient solution due to the load at the end the stationary solution. In this chapter, a study of the dynamic influence factor depends on the ration between load frequency and structure one is presented.

Multi-Degrees of Freedom System and Hydrodynamic Principle

A system with one degree of freedom is far from reality, because we do not take into account all the degrees of freedom. In order to be close to the reality, it is necessary to use a system with several degrees of freedom. Efforts in this chapter are concentrated to the study of multi-degrees of freedom system, whether for a free undamped and forced damped system, by detailing the modal superposition method as well as a coupled coordinates. We finish the chapter with hydrodynamic study using Hozner method as well as some applications.

Extracting Eigenvalues and Eigenvectors Technique

In the process of solving dynamics problems at a given stage, one needs to have the dynamic characteristic of the structure in order to be able to push the calculation (resolution). Further, these characteristics are the eigenvalues and eigenvectors. In this chapter, some important points are dealt with, namely problem with standard eigenvalues, property of eigenvalues and eigenvectors, Rayleigh quotient, offset of eigenvalues, and different methods for extracting eigenvalues and eigenvectors are presented. At the end, some examples are exposed.

Nonlinear Static Analysis Method

This chapter presents the nonlinear static methods of analyses for seismic design of structures considered by Eurocode 8. The first method is the nonlinear pushover procedure, which is based on the N2 method. The second method is the classical nonlinear time history analysis. The first method is studied in more detail, because the second method is a well-established procedure whose only drawback is the time necessary for the analyses. Nonlinear solvers and procedure in program Z_Soil are described. After a simple nonlinear SDOF application, a test-bed application consisting of an existing two-story reinforced concrete building in Bonefro, Italy is used to compare the two nonlinear procedures. The selected building is representative of typical residential building construction in Italy in the 1970s and 1980s. The aim of this chapter section is to compare 2D and 3D procedures implemented in Z_Soil software. The second example is a 14-story reinforced concrete building designed according to the Algerian code using Sap2000 software.

Nonlinear Transient Analysis

In this chapter, the author begin by presenting the main causes of non-linearity, which are geometric and material source, change in boundary condition. The last presented source is pretensions. Then they go to the physical understanding of non-linear behavior by presenting the different phases of hysteresis curve sequence of a reinforced concrete structure. In this chapter, readers pass over various numerical formulation, which allow them to deal with non-linearity, namely Lagrange and Euler formulation, total Lagrangian formulation, Piola-Kirchhoff 2, and corotational formulation. Some examples are exposed at the end of the chapter.