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2004 | Buch | 5. Auflage

Undamped Single Degree-of-Freedom System Kapitel lesen Erstes Kapitel lesen

Structural Dynamics

Theory and Computation

verfasst von: Mario Paz, William Leigh

Verlag: Springer US

Enthalten in: Professional Book Archive

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Über dieses Buch

"The Fifth Edition of Structural Dynamics: Theory and Computation is the complete and comprehensive text in the field. It presents modern methods of analysis and techniques adaptable to computer programming clearly and easily. The book is ideal as a text for advanced undergraduates or graduate students taking a first course in structural dynamics. It is arranged in such a way that it can be used for a one- or two-semester course, or span the undergraduate and graduate levels. In addition, this text will serve the practicing engineer as a primary reference.
The text differs from the standard approach of other presentations in which topics are ordered by their mathematical complexity. This text is organized by the type of structural modeling. The author simplifies the subject by presenting a single degree-of-freedom system in the first chapters, then moves to systems with many degrees-of-freedom in the following chapters. Finally, the text moves to applications of the first chapters and special topics in structural dynamics.
New in this Edition:Problems reworked for SAP2000®.Step-by-step examples of how to use SAP2000® for every application of structural dynamics.Inclusion of companion Web site (extras.springer.com/2004) with three learning aids: SAP2000® student version; source code for the author’s educational programs in structural dynamics, so that the results of changed parameters can be seen step-by-step; and the compiler (executable files) for the author’s educational programs.Three earthquake engineering chapters updated to the latest ICC® building codes.Materials rearranged so that theory and dynamic analysis precede applications and special topics, facilitating using the book sequentially.Complete instructions provided to advanced topics as foundation for further study.This text is essential for civil engineering students. Professional civil engineers will find it an ideal reference."


Inhaltsverzeichnis

Frontmatter

Structures Modeled as a Single-Degree of Freedom System

Frontmatter
1. Undamped Single Degree-of-Freedom System
Abstract
The analysis and design of structures to resist the effect produced by time dependent forces or motions requires conceptual idealizations and simplifying assumptions through which the physical system is represented by an idealized system known as the analytical or mathematical model. These idealizations or simplifying assumptions may be classified in the following three groups:
1.
Material assumptions. These assumptions or simplifications include material properties such as homogeneity or isotrophy and material behaviors such as linearity or elasticity.
 
2.
Loading assumptions. Some common loading assumptions are to consider concentrated forces to be applied at a geometric point, to assume forces suddenly applied, or to assume external forces to be constant or periodic.
 
3.
Geometric Assumptions. A general assumption for beams, frames and trusses is to consider these structures to be formed by unidirectional elements. Another common assumption is to assume that some structures such as plates are two-dimensional systems with relatively small thicknesses. Of greater importance is to assume that continuous structures may be analyzed as discrete systems by specifying locations (nodes) and directions for displacements (nodal coordinates) in the structures as described in the following section.
 
Mario Paz, William Leigh
2. Damped Single Degree-of-Freedom System
Abstract
We have seen in the preceding chapter that the simple oscillator under idealized conditions of no damping, once excited, will oscillate indefinitely with a constant amplitude at its natural frequency. However, experience shows that it is not possible to have a device that vibrates under these ideal conditions. Forces designated as frictional or damping forces are always present in any physical system undergoing motion. These forces dissipate energy; more precisely, the unavoidable presence of these frictional forces constitute a mechanism through which the mechanical energy of the system, kinetic or potential energy, is transformed to other forms of energy such as heat. The mechanism of this energy transformation or dissipation is quite complex and is not completely understood at this time. In order to account for these dissipative forces in the analysis of dynamic systems, it is necessary to make some assumptions about these forces, on the basis of experience.
Mario Paz, William Leigh
3. Response of One-Degree-of-Freedom System to Harmonic Loading
Abstract
In this chapter, we will study the motion of structures idealized as single-degree-of-freedom systems excited harmonically, that is, structures subjected to forces or displacements whose magnitudes may be represented by a sine or cosine function of time. This type of excitation results in one of the most important motions in the study of mechanical vibrations as well as in applications to structural dynamics. Structures are very often subjected to the dynamic action of rotating machinery which produces harmonic excitations due to the unavoidable presence of mass eccentricities in the rotating parts of such machinery. Furthermore, even in those cases when the excitation is not a harmonic function, the response of the structure may be obtained using the Fourier Method, as the superposition of individual responses to the harmonic components of external excitation. This approach will be dealt with in Chapter 20 as a special topic.
Mario Paz, William Leigh
4. Response to General Dynamic Loading
Abstract
In the preceding chapter we studied the response of a single degree-of-freedom system with harmonic loading. Through this type of loading is important, real structures are often subjected to loads that are not harmonic. In this chapter we shall study the response of the single degree-of-freedom system to a general type of force. We shall see that the response can be obtained in terms of an integral that for some simple load functions can be evaluated analytically. For the general case, however, it will be necessary to resort to a numerical integration procedure.
Mario Paz, William Leigh
5. Response Spectra
Abstract
In this chapter, we introduce the concept of response spectrum, which in recent years has gained wide acceptance in structural dynamic practice, particularly in earthquake engineering design. Stated brief, the response spectrum is a plot of the maximum response (maximum displacement, velocity, acceleration, or any other quantity of interest) to a specified load function for all possible single-degree-of-freedom systems. The abscissa of the spectrum is the natural frequency (or period) of the system, and the ordinate the maximum response. A plot of this type is shown in Fig. 5.1, in which a one-story building is subject to a ground displacement indicated by the functiou s (t). The response spectral curve shown in Fig. 5.1 (a) gives, for any single-degree-of-freedom system, the maximum displacement of the response from an available spectral chart, for a specified excitation, we need only to know the natural frequency of the system.
Mario Paz, William Leigh
6. Nonlinear Structural Response
Abstract
In discussing the dynamic behavior of single-degree-of-freedom systems, we assumed that in the model representing the structure, the restoring force was proportional to the displacement. We also assumed the dissipation of energy through a viscous damping mechanism in which the damping force was proportional to the velocity. In addition, the mass in the model was always considered to be unchanging with time. As a consequence of these assumptions, the equation of motion for such a system resulted in a linear, second order ordinary differential equation with constant coefficients, namely,
$$m\ddot{u} + c\dot{u} + ku = F(t)$$
(6.1)
In the previous chapters it was illustrated that for particular forcing functions such as harmonic functions, it was relatively simple to solve this equation (6.1) and that a general solution always existed in terms of Duhamel’s integral. Equation (6.1) thus represents the dynamic behavior of many structures modeled as a single-degree-of-freedom system. There are, however, physical situations for which this linear model does not adequately represent the dynamic characteristics of the structure. The analysis in such cases requires the introduction of a model in which the spring force or the damping force may not remain proportional, respectively, to the displacement or to the velocity. Consequently, the resulting equation of motion will no longer be linear and its mathematical solution, in general, will have a much greater complexity, often requiring a numerical procedure for its integration.
Mario Paz, William Leigh

Structures Modeled as Shear Buildings

Frontmatter
7. Free Vibration of a Shear Building
Abstract
In Part I we analyzed and obtained the dynamic response for structures modeled as a single-degree-of-freedom system. Only if the structure can assume a unique shape during its motion will the single-degree model provide the exact dynamic response. Otherwise, when the structure takes more than one possible shape during motion, the solution obtained from a single-degree model will be at best, only an approximation to the true dynamic behavior.
Mario Paz, William Leigh
8. Forced Motion of Shear Buildings
Abstract
In the preceding chapter, we have shown that the free motion of a dynamic system may be expressed in terms of the normal modes of vibration. Our present interest is to demonstrate that the forced motion of such a system may also be expressed in terms of the normal modes of vibration and that the total response may be obtained as the superposition of the solution of independent modal equations. In other words, our aim in this chapter is to show that the normal modes may be used to transform the system of coupled differential equations into a set of uncoupled differential equations in which each equation contains only one dependent variable. Thus the modal superposition method reduced the problem of finding the response of a multi-degree-of-freedom system to the determination of the response of single-degree-of-freedom systems.
Mario Paz, William Leigh
9. Reduction of Dynamic Matrices
Abstract
In the discretization process it is sometimes necessary to divide a structure into a large number of elements because of changes in geometry, loading, or material properties. When the elements are assembled for the entire structure, the number of unknown displacements, that is, the number of degrees of freedom, may be quite large. As a consequence, the stiffness, mass, and damping matrices will be of large dimensions. The solution of the corresponding eigenproblem to determine natural frequencies and modal shapes will be difficult and, in addition, expensive. In such cases it is desirable to reduce the size of these matrices in order to make the solution of the eigenproblem more manageable and economical. Such reduction is referred to as condensation.
Mario Paz, William Leigh

Framed Structures Modeled as Discrete Multi-Degree-of-Freedom Systems

Frontmatter
10. Dynamic Analysis of Beams
Abstract
In this chapter, we shall study the dynamic behavior of structures designated as beams, that is, structures that carry loads that are mainly transverse to the longitudinal direction, thus producing flexural stresses and lateral displacements. We begin by establishing the static characteristics for a beam segment; and then introduce the dynamic effects produced by the inertial forces. Two approximate methods are presented to take into account the inertial effect in the structure: 1) the lumped mass method in which the distributed mass is assigned to point masses, and 2) the consistent mass method in which the assignment to point masses includes rotational effects. The latter method is consistent with the static deflections of the beam. In Part IV on Special Topics the exact theory for dynamics of beams considering the elastic and inertial distributed properties will be presented.
Mario Paz, William Leigh
11. Dynamic Analysis of Plane Frames
Abstract
The dynamic analysis using the stiffness matrix method for structures modeled as beams was presented in Chapter 10. This method of analysis when applied to beams requires the calculation of element matrices (stiffness, mass, and damping matrices), the assemblage from these matrices of the corresponding system matrices, the formation of the force vector, and the solution of the resultant equations of motion. These equations, as we have seen, may be solved in general by the modal superposition method or by numerical integration of the differential equations of motion. In this chapter and in the following chapters, the dynamic analysis of structures modeled as frames is presented.
Mario Paz, William Leigh
12. Dynamic Analysis of Grid Frames
Abstract
In Chapter 11 consideration was given to the dynamic analysis of the plane frame when subjected to forces acting on the plane of the structure. When the planar structural system is subjected to loads applied normally to its plane, the structure is referred to as a grid frame. This structure can also be treated as a special case of the three-dimensional frame to be presented in Chapter 13. The reason for considering the planar frame, whether loaded in its plane or normal to its plane, as a special case, is the immediate reduction of unknown nodal coordinates for an element of these special structures, hence a considerable reduction in the number of unknown displacements for the structural system.
Mario Paz, William Leigh
13. Dynamic Analysis of Three-dimensional Frames
Abstract
The stiffess method for dynamic analysis of frames presented in Chapter 11 for plane frames and in Chapter 12 for grid frames can readily be expanded for the analysis of three-dimensional space frames. Although for the plane frame or for the grid there were only three nodal coordinates at each joint, the three-dimensional frame has a total of six possible nodal displacements at each unconstrained joint: three translation components along the x, y, z axes and three rotational components about these axes. Consequently, a beam element of a three-dimensional frame or a space frame has for its two joints a total of 12 nodal coordinates; hence the resulting element matrices will be of dimension 12 x 12.
Mario Paz, William Leigh
14. Dynamic Analysis of Trusses
Abstract
The static analysis of trusses whose members are pin-connected reduces to the problem of determining the bar forces due to a set of loads applied at the joints. When the same trusses are subjected to the action of dynamics forces, the simple situation of only axial stresses in the members is no longer present. The inertial forces developed along the members of the truss will, in general, produce flexural bending in addition to axial forces. The bending moments at the ends of the truss members will still remain zero in the absence of external joint moments. The dynamic stiffness method for the analysis of trusses is developed as in the case of framed structures by establishing the basic relations between external forces, elastic forces, damping forces, inertial forces and the resulting displacements, velocities, and accelerations at the nodal coordinates, that is, by determining the stiffness, damping, and mass matrices for a member of the truss. The assemblage of system stiffness, damping, and mass matrices of the truss as well as the solution for the displacements at the nodal coordinates follows along the standard method presented in the preceding chapters for framed structures.
Mario Paz, William Leigh
15. Dynamic Analysis of Structures Using the Finite Element Method
Abstract
In the preceding chapters, we considered the dynamic analysis of structures modeled as beams, frames, or trusses. The elements of all these types of structures are described by a single coordinate along their longitudinal axis; that is, these are structures with unidirectional elements, called, “skeletal structures.” They, in general, consist of individual members or elements connected at points designated as “nodal points” or “joints.” For these types of structures, the behavior of each element is first considered independently through the calculation of the element stiffness matrix and the element mass matrix. These matrices are then assembled into the system stiffness matrix and the system mass matrix in such a way that the equilibrium of forces and the compatibility of displacements are satisfied at each nodal point. The analysis of such structures is commonly known as the Matrix Structural Method and could be applied equally to static and dynamic problems.
Mario Paz, William Leigh
16. Time History Response of Multi-Degree-of-Freedom Systems
Abstract
The nonlinear analysis of a single-degree-of-freedom system using the step-by-step linear acceleration method was presented in Chapter 6. The extension of this method with a modification known as the Wilson-θ method, for the solution of structures modeled as multidegree-of-freedom systems is developed in this chapter. The modification introduced in the method by Wilson et al. 1973 serves to assure the numerical stability of the solution process regardless of the magnitude selected for the time step; for this reason, such a method is said to be unconditionally stable. On the other hand, without Wilson’s modification, the step-by-step linear acceleration method is only conditionally stable and for numerical stability of the solution it may require such an extremely small time step as to make the method impractical if not impossible. The development of the necessary algorithm for the linear and nonlinear multidegree-of-freedom systems by the step-by-step linear acceleration method parallels the presentation for the single-degree-of-freedom system in Chapter 6.
Mario Paz, William Leigh

Structures Modeled with Distributed Properties

Frontmatter
17. Dynamic Analysis of Systems with Distributed Properties
Abstract
The dynamic analysis of structures, modeled as lumped parameter systems with discrete coordinates, was presented in Part I for single-degree-of-freedom systems and in Parts II and III for multidegree-of-freedom systems. Modeling structures with discrete coordinates provides a practical approach for the analysis of structures subjected to dynamic loads. However, the results obtained from these discrete models can only give approximate solutions to the actual behavior of dynamic systems which have continuous distributed properties and, consequently, an infinite number of degrees of freedom.
Mario Paz, William Leigh
18. Discretization of Continuous Systems
Abstract
The modal superposition method of analysis was applied in the preceding chapter to some simple structures having distributed properties. The determination of the response by this method requires the evaluation of several natural frequencies and corresponding mode shapes. The calculation of these dynamic properties is rather laborious, as we have seen, even for simple structures such as one-span uniform beams. The problem becomes increasingly more complicated and unmanageable as this method of solution is applied to more complex structures. However, the analysis of such structures becomes relatively simple if for each segment or element of the structure the properties are expressed in terms of dynamic coefficients much in the same manner as done previously when static deflection functions were used as an approximation to dynamic deflections in determining stiffness, mass, and other coefficients.
Mario Paz, William Leigh

Special Topics: Fourier Analysis Absolute Damping Generalized Coordinates

Frontmatter
19. Fourier Analysis and Response in the Frequency Domain
Abstract
This chapter presents the application of Fourier series to determine: (1) the response of a system to periodic forces, and (2) the response of a system to nonperiodic forces in the frequency domain as an alternate approach to the usual analysis in the time domain. In either case, the calculations require the evaluation of integrals that, except for some relatively simple loading functions, employ numerical methods for their computation. Thus, in general, to make practical use of the Fourier method, it is necessary to replace the integrations with finite sums.
Mario Paz, William Leigh
20. Evaluation of Absolute Damping from Modal Damping Ratios
Abstract
In the previous chapters, we determined the natural frequencies and modal shapes for undamped structures when modeled as shear buildings. We also determined the response of these structures using the modal superposition method. In this method, as we have seen, the differential equations of motion are uncoupled by means of a transformation of coordinates that incorporates the orthogonality property of the modal shapes.
Mario Paz, William Leigh
21. Generalized Coordinates and Rayleigh’s Method
Abstract
In the preceding chapters we concentrated our efforts in obtaining the response to dynamic loads of structures modeled by the simple oscillator, that is, structures that may be analyzed as a damped or undamped spring-mass system. Our plan in the present chapter is to discuss the conditions under which a structural system consisting of multiple interconnected rigid bodies or having distributed mass and elasticity can still be modeled as a one-degree of-freedom system. We begin by presenting an alternative method to the direct application of Newton’s Law of Motion, the principle of virtual work.
Mario Paz, William Leigh

Random Vibration

Frontmatter
22. Random Vibration
Abstract
The previous chapters of this book have dealt with the dynamic analysis of structures subjected to excitations which were known as a function of time. Such an analysis is said to be deterministic. When an excitation function applied to a structure has an irregular shape that is described indirectly by statistical means, we speak of a random vibration. Such a function is usually described as a continuous or discrete function of the exciting frequencies, in a manner similar to the description of a function by Fourier series. In structural dynamics, the random excitations most often encountered are either motion transmitted through the foundation or acoustic pressure. Both of these types of loading are usually generated by explosions occurring in the vicinity of the structure. Common sources of these explosions are construction work and mining. Other types of loading, such as earthquake excitation, may also be considered a random function of time. In these cases the structural response is obtained in probabilistic terms using random vibration theory.
Mario Paz, William Leigh

Earthquake Engineering

Frontmatter
23. Uniform Building Code 1997: Equivalent Lateral Force Method
Abstract
Several seismic building codes are currently in use in different regions of the United States. The International Building Code-2000 (IBC-2000) and The Uniform Building Code—1997 (UBC-1997), published by the International Conference of Building Officials, are the building codes most extensively used, particularly in the western part of the country. In addition to these building codes, two other major building codes are used: 1) The BOCA, or National Building Code [Building Officials and Code Administrators International 1999]; and 2) The Standard Building Code, of the Southern Building Code Congress International (1999).
Mario Paz, William Leigh
24. Uniform Building Code 1997 Dynamic Method
Abstract
In Chapter 5, we introduced the concept of response spectrum as a plot of the maximum response (spectral displacement, spectral velocity, or spectral acceleration) versus the natural frequency or natural period of a single degree-of-freedom system subjected to a specific excitation. In the present chapter, we will use seismic response spectra for earthquake resistant design of buildings modeled as discrete systems with concentrated masses at each level of the building.
Mario Paz, William Leigh
25. International Building Code IBC-2000
Abstract
The IBC-2000 is similar to the UBC-97, but it contains some significant differences. An important difference is that the IBC-2000 includes a set of maps to obtain seismic response spectral values that will result in the same level of risk at any given geographic location in the United States. This code also introduces the concept of Seismic Use Group, which is somewhat analogous to the Importance Factor in the UBC-97. In addition, the IBC-2000 classifies every building in a Seismic Design Category which determines the analysis procedure to be used, the maximum allowed height and drift limitations.
Mario Paz, William Leigh
Backmatter
Metadaten
Titel
Structural Dynamics
verfasst von
Mario Paz
William Leigh
Copyright-Jahr
2004
Verlag
Springer US
Electronic ISBN
978-1-4615-0481-8
Print ISBN
978-1-4613-5098-9
DOI
https://doi.org/10.1007/978-1-4615-0481-8