Hutton - Fundamentals of Finite Element Analysis

Hutton: Fundamentals ofFinite Element AnalysisFront MatterPreface© The McGraw−HillCompanies, 2004PREFACEFundamentals of Finite Element Analysis is intended to be the text for asenior-level finite element course in engineering programs. The mostappropriate major programs are civil engineering, engineering mechanics, and mechanical engineering.

The finite element method is such a widely usedanalysis-and-design technique that it is essential that undergraduate engineeringstudents have a basic knowledge of the theory and applications of the technique.Toward that objective, I developed and taught an undergraduate “special topics”course on the finite element method at Washington State University in the summer of 1992. The course was composed of approximately two-thirds theory andone-third use of commercial software in solving finite element problems. Sincethat time, the course has become a regularly offered technical elective in themechanical engineering program and is generally in high demand. Duringthe developmental process for the course, I was never satisfied with any text thatwas used, and we tried many. I found the available texts to be at one extreme orthe other; namely, essentially no theory and all software application, or all theoryand no software application.

The former approach, in my opinion, representstraining in using computer programs, while the latter represents graduate-levelstudy. I have written this text to seek a middle ground.Pedagogically, I believe that training undergraduate engineering students touse a particular software package without providing knowledge of the underlyingtheory is a disservice to the student and can be dangerous for their future employers. While I am acutely aware that most engineering programs have a specificfinite element software package available for student use, I do not believe that thetext the students use should be tied only to that software. Therefore, I have written this text to be software-independent.

I emphasize the basic theory of the finiteelement method, in a context that can be understood by undergraduate engineering students, and leave the software-specific portions to the instructor.As the text is intended for an undergraduate course, the prerequisites requiredare statics, dynamics, mechanics of materials, and calculus through ordinary differential equations. Of necessity, partial differential equations are introducedbut in a manner that should be understood based on the stated prerequisites.Applications of the finite element method to heat transfer and fluid mechanics areincluded, but the necessary derivations are such that previous coursework inthose topics is not required.

Many students will have taken heat transfer and fluidmechanics courses, and the instructor can expand the topics based on the students’ background.Chapter 1 is a general introduction to the finite element method and includes a description of the basic concept of dividing a domain into finite-sizesubdomains. The finite difference method is introduced for comparison to thexiHutton: Fundamentals ofFinite Element AnalysisxiiFront MatterPreface© The McGraw−HillCompanies, 2004Prefacefinite element method. A general procedure in the sequence of model definition,solution, and interpretation of results is discussed and related to the generallyaccepted terms of preprocessing, solution, and postprocessing. A brief history ofthe finite element method is included, as are a few examples illustrating application of the method.Chapter 2 introduces the concept of a finite element stiffness matrix andassociated displacement equation, in terms of interpolation functions, using thelinear spring as a finite element.

The linear spring is known to most undergraduate students so the mechanics should not be new. However, representation ofthe spring as a finite element is new but provides a simple, concise example ofthe finite element method. The premise of spring element formulation is extended to the bar element, and energy methods are introduced. The first theoremof Castigliano is applied, as is the principle of minimum potential energy.Castigliano’s theorem is a simple method to introduce the undergraduate studentto minimum principles without use of variational calculus.Chapter 3 uses the bar element of Chapter 2 to illustrate assembly of globalequilibrium equations for a structure composed of many finite elements.

Transformation from element coordinates to global coordinates is developed andillustrated with both two- and three-dimensional examples. The direct stiffnessmethod is utilized and two methods for global matrix assembly are presented.Application of boundary conditions and solution of the resultant constraint equations is discussed. Use of the basic displacement solution to obtain element strainand stress is shown as a postprocessing operation.Chapter 4 introduces the beam/flexure element as a bridge to continuityrequirements for higher-order elements.

Slope continuity is introduced and thisrequires an adjustment to the assumed interpolation functions to insure continuity.Nodal load vectors are discussed in the context of discrete and distributed loads,using the method of work equivalence.Chapters 2, 3, and 4 introduce the basic procedures of finite-element modeling in the context of simple structural elements that should be well-known to thestudent from the prerequisite mechanics of materials course.

Thus the emphasisin the early part of the course in which the text is used can be on the finite element method without introduction of new physical concepts. The bar and beamelements can be used to give the student practical truss and frame problems forsolution using available finite element software. If the instructor is so inclined,the bar and beam elements (in the two-dimensional context) also provide a relatively simple framework for student development of finite element softwareusing basic programming languages.Chapter 5 is the springboard to more advanced concepts of finite elementanalysis.

The method of weighted residuals is introduced as the fundamentaltechnique used in the remainder of the text. The Galerkin method is utilizedexclusively since I have found this method is both understandable for undergraduate students and is amenable to a wide range of engineering problems. Thematerial in this chapter repeats the bar and beam developments and extends thefinite element concept to one-dimensional heat transfer. Application to the barHutton: Fundamentals ofFinite Element AnalysisFront MatterPreface© The McGraw−HillCompanies, 2004Prefaceand beam elements illustrates that the method is in agreement with the basic mechanics approach of Chapters 2–4.

Introduction of heat transfer exposes the student to additional applications of the finite element method that are, most likely,new to the student.Chapter 6 is a stand-alone description of the requirements of interpolationfunctions used in developing finite element models for any physical problem.Continuity and completeness requirements are delineated. Natural (serendipity)coordinates, triangular coordinates, and volume coordinates are defined and usedto develop interpolation functions for several element types in two- and threedimensions.

The concept of isoparametric mapping is introduced in the context ofthe plane quadrilateral element. As a precursor to following chapters, numericalintegration using Gaussian quadrature is covered and several examples included.The use of two-dimensional elements to model three-dimensional axisymmetricproblems is included.Chapter 7 uses Galerkin’s finite element method to develop the finite element equations for several commonly encountered situations in heat transfer.One-, two- and three-dimensional formulations are discussed for conduction andconvection. Radiation is not included, as that phenomenon introduces a nonlinearity that undergraduate students are not prepared to deal with at the intendedlevel of the text. Heat transfer with mass transport is included.

The finite difference method in conjunction with the finite element method is utilized to presentmethods of solving time-dependent heat transfer problems.Chapter 8 introduces finite element applications to fluid mechanics. Thegeneral equations governing fluid flow are so complex and nonlinear that thetopic is introduced via ideal flow. The stream function and velocity potentialfunction are illustrated and the applicable restrictions noted. Example problemsare included that note the analogy with heat transfer and use heat transfer finiteelement solutions to solve ideal flow problems. A brief discussion of viscousflow shows the nonlinearities that arise when nonideal flows are considered.Chapter 9 applies the finite element method to problems in solid mechanicswith the proviso that the material response is linearly elastic and small deflection.Both plane stress and plane strain are defined and the finite element formulationsdeveloped for each case.