Hutton - Fundamentals of Finite Element Analysis (523155), страница 5
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As will be seen as weproceed, the term displacement is quite general in the finite element method andcan represent physical displacement, temperature, or fluid velocity, for example.The term finite element was first used by Clough [6] in 1960 in the context ofplane stress analysis and has been in common usage since that time.During the decades of the 1960s and 1970s, the finite element method wasextended to applications in plate bending, shell bending, pressure vessels, andgeneral three-dimensional problems in elastic structural analysis [7–11] as wellas to fluid flow and heat transfer [12, 13]. Further extension of the method tolarge deflections and dynamic analysis also occurred during this time period[14 , 15]. An excellent history of the finite element method and detailed bibliography is given by Noor [16].The finite element method is computationally intensive, owing to the requiredoperations on very large matrices.
In the early years, applications were performedusing mainframe computers, which, at the time, were considered to be very powerful, high-speed tools for use in engineering analysis. During the 1960s, the finiteelement software code NASTRAN [17] was developed in conjunction with thespace exploration program of the United States. NASTRAN was the first majorfinite element software code. It was, and still is, capable of hundreds of thousandsof degrees of freedom (nodal field variable computations).
In the years since thedevelopment of NASTRAN, many commercial software packages have been introduced for finite element analysis. Among these are ANSYS [18], ALGOR [19],and COSMOS/M [20]. In today’s computational environment, most of thesepackages can be used on desktop computers and engineering workstations toobtain solutions to large problems in static and dynamic structural analysis, heattransfer, fluid flow, electromagnetics, and seismic response. In this text, we do notutilize or champion a particular code. Rather, we develop the fundamentals forunderstanding of finite element analysis to enable the reader to use such softwarepackages with an educated understanding.1.5 EXAMPLES OF FINITE ELEMENTANALYSISWe now present, briefly, a few examples of the types of problems that can beanalyzed via the finite element method.
Figure 1.7 depicts a rectangular regionwith a central hole. The area has been “meshed” with a finite element grid of twodimensional elements assumed to have a constant thickness in the z direction.Note that the mesh of elements is irregular: The element shapes (triangles andquadrilaterals) and sizes vary. In particular, note that around the geometric discontinuity of the hole, the elements are of smaller size. This represents not onlyHutton: Fundamentals ofFinite Element Analysis1.
Basic Concepts of theFinite Element MethodText© The McGraw−HillCompanies, 20041.5 Examples of Finite Element AnalysisFigure 1.7A mesh of finite elements over a rectangular region having acentral hole.an improvement in geometric accuracy in the vicinity of the discontinuity butalso solution accuracy, as is discussed in subsequent chapters.The geometry depicted in Figure 1.7 could represent the finite elementmodel of several physical problems. For plane stress analysis, the geometrywould represent a thin plate with a central hole subjected to edge loading in theplane depicted. In this case, the finite element solution would be used to examine stress concentration effects in the vicinity of the hole.
The element meshshown could also represent the case of fluid flow around a circular cylinder. Inyet another application, the model shown could depict a heat transfer fin attached to a pipe (the hole) from which heat is transferred to the fin for dissipation to the surroundings. In each case, the formulation of the equations governing physical behavior of the elements in response to external influences is quitedifferent.Figure 1.8a shows a truss module that was at one time considered abuilding-block element for space station construction [21]. Designed to fold inaccordion fashion into a small volume for transport into orbit, the module, whendeployed, extends to overall dimensions 1.4 m × 1.4 m × 2.8 m.
By attachingsuch modules end-to-end, a truss of essentially any length could be obtained.The structure was analyzed via the finite element method to determine thevibration characteristics as the number of modules, thus overall length, wasvaried. As the connections between the various structural members are pin orball-and-socket joints, a simple axial tension-compression element (Chapter 2)was used in the model. The finite element model of one module was composedof 33 elements. A sample vibration shape of a five-module truss is shown inFigure 1.8b.The truss example just described involves a rather large structure modeledby a small number of relatively large finite elements. In contrast, Figure 1.9shows the finite element model of a very thin tube designed for use in heat13Hutton: Fundamentals ofFinite Element Analysis1.
Basic Concepts of theFinite Element MethodText© The McGraw−HillCompanies, 2004XGYGZG(a)(b)Figure 1.8(a) Deployable truss module showing details of folding joints.(b) A sample vibration-mode shape of a five-module truss as obtainedvia finite element analysis. (Courtesy: AIAA)14Hutton: Fundamentals ofFinite Element Analysis1. Basic Concepts of theFinite Element MethodText© The McGraw−HillCompanies, 20041.5 Examples of Finite Element Analysis0.00197ZX0.4880.25Figure 1.9Finite element model of a thin-walledheat exchanger tube.transfer in a spacecraft application. The tube has inside diameter of 0.976 in.
andwall thickness 0.00197 in. and overall length 36 in. Materials considered forconstruction of the tube were copper and titanium alloys. Owing to the wallthickness, prototype tubes were found to be very fragile and difficult to handlewithout damage. The objectives of the finite element analysis were to examinethe bending, torsional, and buckling loads allowable. The figure shows the finiteelement mesh used to model a section of the tube only 0.25 in. in length. Thismodel contains 1920 three-dimensional solid elements, each having eight nodeswith 3 degrees of freedom at each node. Such a large number of elements wasrequired for a small structure in consideration of computational accuracy.
Theconcern here was the so-called aspect ratio of the elements, as is defined anddiscussed in subsequent chapters.As a final example, Figure 1.10a represents the finite element model of themain load-carrying component of a prosthetic device. The device is intended tobe a hand attachment to an artificial arm. In use, the hand would allow a lowerarm amputee to engage in weight lifting as part of a physical fitness program.The finite element model was used to determine the stress distribution in thecomponent in terms of the range of weight loading anticipated, so as to properlysize the component and select the material. Figure 1.10b shows a prototype of thecompleted hand design.15Hutton: Fundamentals ofFinite Element Analysis161.
Basic Concepts of theFinite Element MethodCHAPTER 1Text© The McGraw−HillCompanies, 2004Basic Concepts of the Finite Element Method(a)(b)Figure 1.10(a) A finite element model of a prosthetic hand for weightlifting. (b) Completedprototype of a prosthetic hand, attached to a bar.(Courtesy of Payam Sadat. All rights reserved.)1.6 OBJECTIVES OF THE TEXTI wrote Fundamentals of Finite Element Analysis for use in senior-level finiteelement courses in engineering programs. The majority of available textbookson the finite element method are written for graduate-level courses. Thesetexts are heavy on the theory of finite element analysis and rely on mathematicaltechniques (notably, variational calculus) that are not usually in the repertoire ofundergraduate engineering students.
Knowledge of advanced mathematical techniques is not required for successful use of this text. The prerequisite study isbased on the undergraduate coursework common to most engineering programs:linear algebra, calculus through differential equations, and the usual series ofstatics, dynamics, and mechanics of materials. Although not required, prior studyof fluid mechanics and heat transfer is helpful. Given this assumed background,the finite element method is developed on the basis of physical laws (equilibrium, conservation of mass, and the like), the principle of minimum potential energy (Chapter 2), and Galerkin’s finite element method (introduced and developed in Chapter 5).Hutton: Fundamentals ofFinite Element Analysis1. Basic Concepts of theFinite Element MethodText© The McGraw−HillCompanies, 2004ReferencesAs the reader progresses through the text, he or she will discern that wecover a significant amount of finite element theory in addition to applicationexamples.
Given the availability of many powerful and sophisticated finiteelement software packages, why study the theory? The finite element method isa tool, and like any other tool, using it without proper instruction can be quitedangerous. My premise is that the proper instruction in this context includesunderstanding the basic theory underlying formulation of finite element modelsof physical problems. As stated previously, critical analysis of the results of afinite element model computation is essential, since those results may eventuallybecome the basis for design.
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