講義文稿分析教案fundamentals of finite element analysis david hutton

1、Hutton: Fundamentals of Finite Element AnalysisFront MatterPreface© The McGrawHill Companies, 2004PREFACEFundamentals of Finite Element Analysis is intended to be the text for a senior-level nite element course in engineering programs. The most appropriate major programs are civil engineering,

2、engineering mechan-ics, and mechanical engineering. The nite element method is such a widely usedanalysis-and-design technique that it is essential that undergraduate engineering s have a basic knowledge oTow course omer of 1992. Tone-third use of comthat time, the course has mechanical engineering

3、progthe developmental process for the cowas used, and we tried many. I found thethe other; namely, essentially no theory and alland no software application. The former approach training in using computer programs, while the latter rep study. I have written this text to seek a middle ground.Pedagogic

4、ally, I believe that training undergraduate engineeringuse a particular software package without providing knowledge of the unde theory is a disservice to theers. While I am acutely aware that most engineering progr nite element software package availabtext theten this text to be software-independen

5、t. I emphasize the basic theory of the nite element method, in a context that can be understood by uningAs the text is intended for an undergraduate course, the prerequisites required are statics, dynamics, mechanics of materials, and calculus through ordinary dif- ferential equations. Of necessity,

6、 partial differential equations are introduced but in a manner that should be understood based on the stated prerequisites. Applications of the nite element method to heat transfer and uid mechanics are included, but the necessary derivations are such that previous coursework inthose topics is not r

7、equired. Manys will have taken heat transfer and uidmechanics courses, and the instructor can expand the topics based on the stu- dents background.Chapter 1 is a general introduction to the nite element method and in-cludes a description of the basic concept of dividing a domain into nite-size subdo

8、mains. The nite difference method is introduced for comparison to thexiHutton: Fundamentals of Finite Element AnalysisFront MatterPreface© The McGrawHill Companies, 2004xiiPrefacenite element method. A general procedure in the sequence of mdenition,solution, and interpretation of results is dis

9、cussed and related to terallyaccepted terms of preprocessing, solution, and postprocessing. A brief history ofthe nite element method is included, as are a few examples illustrating applica- tion of the method.Chapter 2 introduces the concept of a nite element stiffness matrix and associated displac

10、ement equation, in terms of interpolation functions, using the linear spring as a nite element. The linear spring is known to most undergradu-ates so the mechanics should not be new. However, representation ofthe spring as a nite element is new but provides a simple, concise example ofthe nite eleme

11、nt method. The premise of spring element formulation is ex- tended to the bar element, and energy methods are introduced. The rst theorem of Castigliano is applied, as is the principle of minimum potential energy.Castiglianos theorem is a simple method to introduce the undergraduate to minimum princ

12、iples without use of variational calculus.Chapter 3 uses the bar element of Chapter 2 to illustrate assembly of equilibrium equations for a structure composed of many nite elements. T formation from element coordinates to global coordinates is developeandillustrated with both two- and three-dimensio

13、nal examples. The dir method is utilized and two methods for global matrix assemb Application of boundary conditions and solution of the restions is discussed. Use of the basic displacement solu and stress is shown as a postprocessing operatiChapter 4 introduces the beam/e requirements for higher-or

14、der elem requires an adjustment to the aNodal load vectors are d using the method oChapters ing in tess ed.a- nin the early part of the course in which the text is used can be on the nite ele- ment method without introduction of new physical concepts. The bar and beam elements can be used to give th

15、esolution using available nite elemethe bar and beam elements (in the two-dimen tively simple framework forusing basic programChapter 5 is the springboard t analysis. The method of w technique used in texclusively graduateso inclined,material in this chapter repeats the bar and beam developments and

16、 extends the nite element concept to one-dimensional heat transfer. Application to the barHutton: Fundamentals of Finite Element AnalysisFront MatterPreface© The McGrawHill Companies, 2004Prefacexiiiand beam elements illustrates that the method is in agreement with the basic me- chanics approac

17、h of Chapters 24. Introduction of heat transfer exposes the stu- dent to additional applications of the nite element method that are, most likely, new to theChapter 6 is a stand-alone description of the requirements of interpolationfunctions used in develonite element mContinuity and completeness re

18、quirements are delineated. Natural (serendipity) coordinates, triangular coordinates, and volume coordinates are dened and used to develop interpolation functions for several element types in two- and three-dimensions. The concept of isoparametric mapis introduced in the context ofthe plane quadrila

19、teral element. As a precursor to following chapters, numerical integration using Gaussian quadrature is covered and several examples included.The use of two-dimensional elements to m problems is included.Chapter 7 uses Galerkins nite element method to develop the nment equations for several commonly

20、 encountered situations in heat transfer. One-, two- and three-dimensional formulations are discussed for conduction and convection. Radiation is not included, as that phenomenon introduces a nonlin-earity that undergraduates are not prepared to deal with at the intendedlevel of the text. Heat trans

21、fer with mass transport is included. The nite differ- ence method in conjunction with the nite element method is utilized to present methods of solving time-dependent heat transfer problems.Chapter 8 introduces nite element applications to uid mechanics. Thegeneral equations governing uid ow are so

22、complex and nonlinear that the topic is introduced via ideal ow. The stream function and velocity potential function are illustrated and the applicable restrictions noted. Example problems are included that note the analogy with heat transfer and use heat transfer nite element solutions to solve ide

23、al ow problems. A brief discussion of viscous ow shows the nonlinearities that arise when nonideal ows are considered.Chapter 9 applies the nite element method to problems in solid mechanics with the proviso that the material response is linearly elastic and small deection. Both plane stress and pla

24、ne strain are dened and the nite element formulations developed for each case. General three-dimensional states of stress and axisym- metric stress are included. A moped using the Prandtl stress function. The purpose of the torsion section is tomake theaware that all torsionally loaded objects are n

25、ot circular and theanalysis methods must be adjusted to suit geometry.Chapter 10 introduces the concept of dynamic motion of structures. It is notpresumed that thehas taken a course in mechanical vibrations; as a re-sult, this chapter includes a primer on basic vibration theory. Most of this mater-

26、ial is drawn from my previously published text Applied Mechanical Vibrations.The concept of the mass or inertia matrix is developed by examples of simple spring-mass systems and then extended to continuous bodies. Both lumped and consistent mass matrices are dened and used in examples. Modal analysi

27、s is the basic method espoused for dynamic response; hence, a considerable amount ofHutton: Fundamentals of Finite Element AnalysisFront MatterPreface© The McGrawHill Companies, 2004xivPrefacetext material is devoted to determination of natural modes, orthogonality, and modal superposition. Com

28、bination of nite difference and nite element meth- ods for solving transient dynamic structural problems is included.The appendices are included in order to provide thethat might be new or may be “rusty” in thes mind.Appendix A is a review of matrix algebra and should be known to the stu-dent from a

29、 course in linear algebra.Appendix B states ta homogeneous, isotropic, elastic material. I have found over the years that un-dergraduate engineerings do not have a rm grasp of these relations. Ingeneral, the dimensional eqAppendix C c Somehas been exposedons.methods. I include the appe derlying the

30、softwaAppendix D describes the baware of the algorithms un-software. The FEPC (FEPnite elemen was developed by the late Dr. Charles Knand State University and is used in conjunctionhis estate. Dr. Knights programs allow analysis o using bar, beam, and plane stress elements. The appeterms the capabil

31、ities and limitations of the software. T available to theAppendix E includes problems for several ch solved via commercial nite element softable ANSYS, ALGOR, COSMOS,having many degrees ofms n generalEPC program ishat should be has avail-stemsi-tional problems of this sort will be added to the websi

32、te on a continuing basis.The textbook features a Web site (ment analysis links and the FEPC program. At this site, instructors wil access tocan access these tools by contacting their local McGraw-Hill sales representative for password information.I thank Raghu Agarwal, Rong Y. Chen, Nels Madsen, Rob

33、ert L. Rankin, Joseph J. Rencis, Stephen R. Swanson, and Lonny L. Thompson, who reviewed some or all of the manuscript and provided constructive suggestions and criti- cisms that have helped improve the book.I am grateful to all the staff at McGraw-Hill who have labored to make this project a realit

34、y. I especially acknowledge the patient encouragement and pro- fessionalism of Jonathan Plant, Senior Editor, Lisa Kalner Williams, Develop- mental Editor, and Kay Brimeyer, Senior Project Manager.David V. Hutton Pullman, WAHutton: Fundamentals of Finite Element Analysis1. Basic Concepts of the Fini

35、te Element MethodText© The McGrawHill Companies, 20041C H A P T E RBasic Concepts of the Finite Element Method1.1 INTRODUCTIONThe nite element method (FEM), sometimes referred to as nite element analysis (FEA), is a computational technique used to obtain approximate solu- tions of boundary valu

36、e problems in engineering. Simply stated, a boundary value problem is a mathematical problem in which one or more dependent vari- ables must satisfy a differential equation everywhere within a known domain of independent variables and satisfy specic conditions on the boundary of the domain. Boundary

37、 value problems are also sometimes called eld problems. The eld is the domain of interest and most often represents a physical structure. The eld variables are the dependent variables of interest governed by the dif- ferential equation. The boundary conditions are the specied values of the eld varia

38、bles (or related variables such as derivatives) on the boundaries of the eld. Depending on the type of physical problem being analyzed, the eld variables may include physical displacement, temperature, heat ux, and uid velocity to name only a few.1.2HOW DOES THE FINITE ELEMENT METHOD WORK?eral techn

39、iques and terminology of nite element analysis will be intro-Tduced with reference to Figure 1.1. The gure depicts a volume of some material or materials having known physical properties. The volume represents the domain of a boundary value problem to be solved. For simplicity, at this point, we ass

40、ume a two-dimensional case with a single eld variable (x, y) to be determined at every point P(x, y) such that a known governing equation (or equa- tions) is satised exactly at every such point. Note that this implies an exact1Hutton: Fundamentals of Finite Element Analysis1. Basic Concepts of the F

41、inite Element MethodText© The McGrawHill Companies, 20042CHAPTER 1Basic Concepts of the Finite Element Method3P(x, y)21(a)(b)(c)Figure 1.1(a) A general two-dimensional domain of eld variable (x, y).(b) A three-node nite element dened in the domain. (c) Additional elements showing a partial nite

42、 element mesh of the domain.mathematical solution is obtained; that is, the solution is ad-form algebraicexpression of the independent variables. In practical problems, the domain may be geometrically complex as is, often, the governing equation and the likelihoodof obtaining an exactd-form solution

43、 is very low. Therefore, approximatesolutions based on numerical techniques and digital computation are mostoften obtained in engineering analyses of complex problems. Finite element analysis is a powerful technique for obtaining such approximate solutions with good accuracy.A small triangular eleme

44、nt that enof interest is shown in Figure 1.1b. That this element is not a differential elementof size dx × dy makes this a nite element. As we treat this example as a two-dimensional problem, it is assumed that the thickness in the z direction is con- stant and z dependency is not indicated in

45、the differential equation. The vertices of the triangular element are numbered to indicate that these points are nodes. A node is a specic point in the nite element at which the value of the eld vari- able is to be explicitly calculated. Exterior nodes are located on the boundaries of the nite eleme

46、nt and may be used to connect an element to adjacent nite elements. Nodes that do not lie on element boundaries are interior nodes and cannot be connected to any other element. The triangular element of Figure 1.1b has only exterior nodes.Hutton: Fundamentals of Finite Element Analysis1. Basic Conce

47、pts of the Finite Element MethodText© The McGrawHill Companies, 20041.2 How Does the Finite Element Method Work?3If the values of the eld variable are computed only at nodes, how are values obtained at other points within a nite element? The answer contains the crux of the nite element method:

48、The values of the eld variable computed at the nodes are used to approximate the values at nonnodal points (that is, in the element interior) by interpolation of the nodal values. For the three-node triangle exam- ple, the nodes are all exterior and, at any other point within the element, the eld va

49、riable is described by the approximate relation (x , y) = N1(x , y) 1 + N2(x , y) 2 + N3(x , y) 3(1.1)where 1, 2, and 3 are the values of the eld variable at the nodes, and N1, N2, and N3 are the interpolation functions, also known as shape functions or blend- ing functions. In the nite element appr

50、oach, the nodal values of the eld vari- able are treated as unknown constants that are to be determined. The interpola- tion functions are most often polynomial forms of the independent variables, derived to satisfy certain required conditions at the nodes. These conditions arediscussed inin subsequ

51、ent chapters. The major point to behere isthat the interpolation functions are predetermined, known functions of the inde-pendent variables; and these functions describe the variation of the eld variable within the nite element.The triangular element described by Equation 1.1 is said to have 3 degre

52、es ofdom, as three nodal values of the eld variable are rthe eld vavariable represents a scalar(Chapter 7). If the domain of Figure 1.1 reproplane stress (Chapter 9), the eld variable becomes the displacement vector andthe values of two components must be computed at the three-node triangular elemen

53、t hasber of degrees ofcase,of the number of nodes and the number of values of the eld variable (and pos- sibly its derivatives) that must be computed at each node.How does this element-based approach work over the entire domain of in- terest? As depicted in Figure 1.1c, every element is connected at

54、 its exterior nodes to other elements. The nite element equations are formulated such that, at the nodal connections, the value of the eld variable at any connection is the same for each element connected to the node. Thus, continuity of the eld vari- able at the nodes is ensured. In fact, nite elem

55、ent formulations are such that continuity of the eld variable across interelement boundaries is also ensured. This feature avoids the physically unacceptable possibility of gaps or voids oc- curring in the domain. In structural problems, such gaps would represent physi- cal separation of the materia

56、l. In heat transfer, a “gap” would manifest itself in the form of different temperatures at the same physical point.Although continuity of the eld variable from element to element is inherent to the nite element formulation, interelement continuity of gradients (i.e., de- rivatives) of the eld varia

57、ble does not generally exist. This is a critical observa- tion. In most cases, such derivatives are of more interest than are eld variable values. For example, in structural problems, the eld variable is displacement butHutton: Fundamentals of Finite Element Analysis1. Basic Concepts of the Finite Element MethodText© The McGrawHill Companies, 20044CHAPTER 1 Basic Concepts of the Finite Element Methodthe true interest is more often in strain and stress. As strain is dened in terms of rst derivatives of dis