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Share Embed Donate. No part of this publication may be reproduced or distributed in any form or by any means, without the prior written permission of Computers and Structures, Inc. Copies of this publication may be obtained from: Computers and Structures, Inc. Chapter 22 has been written on the direct use of absolute earthquake displacement loading acting at the base of the structure.
Several new types of numerical errors for absolute displacement loading have been identified. First, the fundamental nature of displacement loading is significantly different from the base acceleration loading traditionally used in earthquake engineering. Second, a smaller integration time step is required to define the earthquake displacement and to solve the dynamic equilibrium equations.
Third, a large number of modes are required for absolute displacement loading to obtain the same accuracy as produced when base acceleration is used as the loading. Fourth, the 90 percent mass participation rule, intended to assure accuracy of the analysis, does not apply for absolute displacement loading.
Finally, the effective modal damping for displacement loading is larger than when acceleration loading is used. To reduce those errors associated with displacement loading, a higher order integration method based on a cubic variation of loads within a time step is introduced in Chapter In addition, static and dynamic participation factors have been defined that allow the structural engineer to minimize the errors associated with displacement type loading.
In addition, Chapter 19 on viscous damping has been expanded to illustrate the physical effects of modal damping on the results of a dynamic analysis. Appendix H, on the speed of modern personal computers, has been updated. Several other additions and modifications have been made in this printing. It is assumed that the reader has an understanding of statics, mechanics of solids, and elementary structural analysis.
The level of knowledge expected is equal to that of an individual with an undergraduate degree in Civil or Mechanical Engineering. Elementary matrix and vector notations are defined in the Appendices and are used extensively. A background in tensor notation and complex variables is not required. All equations are developed using a physical approach, because this book is written for the student and professional engineer and not for my academic colleagues.
Threedimensional structural analysis is relatively simple because of the high speed of the modern computer. Therefore, all equations are presented in three-dimensional form and anisotropic material properties are automatically included. A computer programming background is not necessary to use a computer program intelligently. However, detailed numerical algorithms are given so that the readers completely understand the computational methods that are summarized in this book.
The Appendices contain an elementary summary of the numerical methods used; therefore, it should not be necessary to spend additional time reading theoretical research papers to understand the theory presented in this book. The author has developed and published many computational techniques for the static and dynamic analysis of structures.
It has been personally satisfying that many members of the engineering profession have found these computational methods useful. Therefore, one reason for compiling this theoretical and application book is to consolidate in one publication this research and development.
In addition, the recently developed Fast Nonlinear Analysis FNA method and other numerical methods are presented in detail for the first time. The fundamental physical laws that are the basis of the static and dynamic analysis of structures are over years old.
Therefore, anyone who believes they have discovered a new fundamental principle of mechanics is a victim of their own ignorance. This book contains computational tricks that the author has found to be effective for the development of structural analysis programs.
The static and dynamic analysis of structures has been automated to a large degree because of the existence of inexpensive personal computers. However, the field of structural engineering, in my opinion, will never be automated.
The idea that an expertsystem computer program, with artificial intelligence, will replace a creative human is an insult to all structural engineers. The material in this book has evolved over the past thirty-five years with the help of my former students and professional colleagues. Their contributions are acknowledged. They have provided the motivation for this publication. The material presented in the first edition of Three Dimensional Dynamic Analysis of Structures is included and updated in this book.
I am looking forward to additional comments and questions from the readers in order to expand the material in future editions of the book.
Edward L. Material Properties 1. Summary References Energy and Work 3. References Incompatible Elements 6. Geometric Stiffness and P-Delta Effects References Nonlinear Elements First, the stress-strain relationship contains the material property information that must be evaluated by laboratory or field experiments.
Second, the total structure, each element, and each infinitesimal particle within each element must be in force equilibrium in their deformed position. Third, displacement compatibility conditions must be satisfied. If all three equations are satisfied at all points in time, other conditions will automatically be satisfied. For example, at any point in time the total work done by the external loads must equal the kinetic and strain energy stored within the structural system plus any energy that has been dissipated by the system.
Virtual work and variational principles are of significant value in the mathematical derivation of certain equations; however, they are not fundamental equations of mechanics.
Before the development of the finite element method, most analytical solutions in solid mechanics were restricted to materials that were isotropic equal properties in all directions and homogeneous same properties at all points in the solid.
Since the introduction of the finite element method, this limitation no longer exists. Hence, it is reasonable to start with a definition of anisotropic materials, which may be different in every element in a structure. The positive definition of stresses, in reference to an orthogonal system, is shown in Figure 1.
Each column of the C matrix represents the strains caused by the application of a unit stress. The a matrix indicates the strains caused by a unit temperature increase. Basic energy principles require that the C matrix for linear material be symmetrical. Therefore, these experimental values are normally averaged so that symmetrical values can be used in the analyses. Therefore, it is not necessary to calculate the E matrix in analytical form as indicated in many classical books on solid mechanics.
In addition, the initial thermal stresses are numerically evaluated within the computer program. Consequently, for the most general anisotropic material, the basic computer input data will be twenty-one elastic constants, plus six coefficients of thermal expansion. Initial stresses, in addition to thermal stresses, may exist for many different types of structural systems. These initial stresses may be the result of the fabrication or construction history of the structure.
If these initial stresses are known, they may be added directly to Equation 1. For this special case, the material is defined as orthotropic and Equation 1. This type of material property is very common.
For example, rocks, concrete, wood and many fiber reinforced materials exhibit orthotropic behavior. It should be pointed out, however, that laboratory tests indicate that Equation 1. For isotropic materials, Equation 1. It can easily be shown that the application of a pure shear stress should result in pure tension and compression strains on the element if it is rotated 45 degrees. Most computer programs use Equation 1. For this case the compliance matrix is reduced to a 3 x 3 array.
The cross-sections of many dams, tunnels, and solids with a near infinite dimension along the 3-axis can be considered in a state of plane strain for constant loading in the plane. However, from Equation 1. These real properties exist for a nearly incompressible material with a relatively low shear modulus. For this case the stress-strain matrix is reduced to a 3 x 3 array.
The membrane behavior of thin plates and shear wall structures can be considered in a state of plane strain for constant loading in the plane. These materials are often referred to as nearly incompressible solids. The incompressible terminology is very misleading because the compressibility, or bulk modulus, of these materials is normally lower than other solids. From Equation 1. For isotropic materials, the bulk modulus and shear modulus are known as Lame's elastic constants and are considered to be fundamental material properties for both solids and fluids.
Table 1. It is apparent that the major difference between liquids and solids is that liquids have a very small shear modulus compared to the bulk modulus, and liquids are not incompressible. Therefore, it is possible to calculate all of the other elastic properties for isotropic materials from these equations.
It is apparent that shear waves cannot propagate in fluids since the shear modulus is zero. Many axisymmetric structures have anisotropic materials. For the case of axisymmetric solids subjected to nonaxisymmetric loads, the compliance matrix, as defined by Equation 1. The solution of this special case of a three-dimensional solid can be accomplished by expressing the node point displacements and loads in a series of harmonic functions.
The solution is then expressed as a summation of the results of a series of two-dimensional, axisymmetric problems. However, for one-dimensional elements in structural engineering, we often rewrite these equations in terms of forces and deformations. Also, L Equation 1. It is AE important to note that the stiffness and flexibility terms are not a function of the load and are only the material and geometric properties of the member.
Also, the inverse L of the torsional stiffness is the torsional flexibility.
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