Hutton - Fundamentals of Finite Element Analysis, страница 2

General three-dimensional states of stress and axisymmetric stress are included. A model for torsion of noncircular sections is developed using the Prandtl stress function. The purpose of the torsion section is tomake the student aware that all torsionally loaded objects are not circular and theanalysis methods must be adjusted to suit geometry.Chapter 10 introduces the concept of dynamic motion of structures.

It is notpresumed that the student has taken a course in mechanical vibrations; as a result, this chapter includes a primer on basic vibration theory. Most of this material is drawn from my previously published text Applied Mechanical Vibrations.The concept of the mass or inertia matrix is developed by examples of simplespring-mass systems and then extended to continuous bodies. Both lumped andconsistent mass matrices are defined and used in examples. Modal analysis is thebasic method espoused for dynamic response; hence, a considerable amount ofxiiiHutton: Fundamentals ofFinite Element AnalysisxivFront MatterPreface© The McGraw−HillCompanies, 2004Prefacetext material is devoted to determination of natural modes, orthogonality, andmodal superposition.

Combination of finite difference and finite element methods for solving transient dynamic structural problems is included.The appendices are included in order to provide the student with materialthat might be new or may be “rusty” in the student’s mind.Appendix A is a review of matrix algebra and should be known to the student from a course in linear algebra.Appendix B states the general three-dimensional constitutive relations fora homogeneous, isotropic, elastic material. I have found over the years that undergraduate engineering students do not have a firm grasp of these relations. Ingeneral, the student has been exposed to so many special cases that the threedimensional equations are not truly understood.Appendix C covers three methods for solving linear algebraic equations.Some students may use this material as an outline for programming solutionmethods.

I include the appendix only so the reader is aware of the algorithms underlying the software he/she will use in solving finite element problems.Appendix D describes the basic computational capabilities of the FEPCsoftware. The FEPC (FEPfinite element program for the PCpersonal computer)was developed by the late Dr. Charles Knight of Virginia Polytechnic Instituteand State University and is used in conjunction with this text with permission ofhis estate. Dr. Knight’s programs allow analysis of two-dimensional programsusing bar, beam, and plane stress elements. The appendix describes in generalterms the capabilities and limitations of the software.

The FEPC program isavailable to the student at www.mhhe.com/hutton.Appendix E includes problems for several chapters of the text that should besolved via commercial finite element software. Whether the instructor has available ANSYS, ALGOR, COSMOS, etc., these problems are oriented to systemshaving many degrees of freedom and not amenable to hand calculation. Additional problems of this sort will be added to the website on a continuing basis.The textbook features a Web site (www.mhhe.com/hutton) with finite element analysis links and the FEPC program. At this site, instructors will haveaccess to PowerPoint images and an Instructors’ Solutions Manual.

Instructorscan access these tools by contacting their local McGraw-Hill sales representativefor password information.I thank Raghu Agarwal, Rong Y. Chen, Nels Madsen, Robert L. Rankin,Joseph J. Rencis, Stephen R. Swanson, and Lonny L. Thompson, who reviewedsome or all of the manuscript and provided constructive suggestions and criticisms that have helped improve the book.I am grateful to all the staff at McGraw-Hill who have labored to make thisproject a reality. I especially acknowledge the patient encouragement and professionalism of Jonathan Plant, Senior Editor, Lisa Kalner Williams, Developmental Editor, and Kay Brimeyer, Senior Project Manager.David V.

HuttonPullman, WAHutton: Fundamentals ofFinite Element Analysis1. Basic Concepts of theFinite Element MethodText© The McGraw−HillCompanies, 2004C H A P T E R1Basic Concepts of theFinite Element Method1.1 INTRODUCTIONThe finite element method (FEM), sometimes referred to as finite elementanalysis (FEA), is a computational technique used to obtain approximate solutions of boundary value problems in engineering. Simply stated, a boundaryvalue problem is a mathematical problem in which one or more dependent variables must satisfy a differential equation everywhere within a known domain ofindependent variables and satisfy specific conditions on the boundary of thedomain. Boundary value problems are also sometimes called field problems.

Thefield is the domain of interest and most often represents a physical structure.The field variables are the dependent variables of interest governed by the differential equation. The boundary conditions are the specified values of the fieldvariables (or related variables such as derivatives) on the boundaries of the field.Depending on the type of physical problem being analyzed, the field variablesmay include physical displacement, temperature, heat flux, and fluid velocity toname only a few.1.2 HOW DOES THE FINITE ELEMENTMETHOD WORK?The general techniques and terminology of finite element analysis will be introduced with reference to Figure 1.1.

The figure depicts a volume of some materialor materials having known physical properties. The volume represents thedomain of a boundary value problem to be solved. For simplicity, at this point,we assume a two-dimensional case with a single field variable (x, y) to bedetermined at every point P(x, y) such that a known governing equation (or equations) is satisfied exactly at every such point. Note that this implies an exact1Hutton: Fundamentals ofFinite Element Analysis21. Basic Concepts of theFinite Element MethodCHAPTER 1Text© The McGraw−HillCompanies, 2004Basic Concepts of the Finite Element Method3P(x, y)21(a)(b)(c)Figure 1.1(a) A general two-dimensional domain of field variable (x, y).(b) A three-node finite element defined in the domain.

(c) Additionalelements showing a partial finite element mesh of the domain.mathematical solution is obtained; that is, the solution is a closed-form algebraicexpression of the independent variables. In practical problems, the domain maybe geometrically complex as is, often, the governing equation and the likelihoodof obtaining an exact closed-form solution is very low. Therefore, approximatesolutions based on numerical techniques and digital computation are mostoften obtained in engineering analyses of complex problems.

Finite elementanalysis is a powerful technique for obtaining such approximate solutions withgood accuracy.A small triangular element that encloses a finite-sized subdomain of the areaof interest is shown in Figure 1.1b. That this element is not a differential elementof size dx × dy makes this a finite element. As we treat this example as a twodimensional problem, it is assumed that the thickness in the z direction is constant and z dependency is not indicated in the differential equation.

The verticesof the triangular element are numbered to indicate that these points are nodes. Anode is a specific point in the finite element at which the value of the field variable is to be explicitly calculated. Exterior nodes are located on the boundariesof the finite element and may be used to connect an element to adjacent finiteelements. Nodes that do not lie on element boundaries are interior nodes andcannot be connected to any other element. The triangular element of Figure 1.1bhas only exterior nodes.Hutton: Fundamentals ofFinite Element Analysis1.

Basic Concepts of theFinite Element MethodText© The McGraw−HillCompanies, 20041.2 How Does the Finite Element Method Work?If the values of the field variable are computed only at nodes, how are valuesobtained at other points within a finite element? The answer contains the crux ofthe finite element method: The values of the field variable computed at the nodesare used to approximate the values at nonnodal points (that is, in the elementinterior) by interpolation of the nodal values. For the three-node triangle example, the nodes are all exterior and, at any other point within the element, the fieldvariable is described by the approximate relation(x , y) = N 1 (x , y)1 + N 2 (x , y)2 + N 3 (x , y)3(1.1)where 1 , 2 , and 3 are the values of the field variable at the nodes, and N1, N2,and N3 are the interpolation functions, also known as shape functions or blending functions.

In the finite element approach, the nodal values of the field variable are treated as unknown constants that are to be determined. The interpolation functions are most often polynomial forms of the independent variables,derived to satisfy certain required conditions at the nodes. These conditions arediscussed in detail in subsequent chapters. The major point to be made here isthat the interpolation functions are predetermined, known functions of the independent variables; and these functions describe the variation of the field variablewithin the finite element.The triangular element described by Equation 1.1 is said to have 3 degreesof freedom, as three nodal values of the field variable are required to describethe field variable everywhere in the element.

This would be the case if the fieldvariable represents a scalar field, such as temperature in a heat transfer problem(Chapter 7). If the domain of Figure 1.1 represents a thin, solid body subjected toplane stress (Chapter 9), the field variable becomes the displacement vector andthe values of two components must be computed at each node. In the latter case,the three-node triangular element has 6 degrees of freedom. In general, the number of degrees of freedom associated with a finite element is equal to the productof the number of nodes and the number of values of the field variable (and possibly its derivatives) that must be computed at each node.How does this element-based approach work over the entire domain of interest? As depicted in Figure 1.1c, every element is connected at its exteriornodes to other elements. The finite element equations are formulated such that, atthe nodal connections, the value of the field variable at any connection is thesame for each element connected to the node.