Professional engineers will benefit from the introduction to the many useful applications of finite element analysis, and will gain a better understanding of its limitations and special uses.
New to this edition: Examples and applications in Matlab, Ansys, ,and Abaqus Structured problem solving approach in all worked examples New discussions throughout, including the direct method of deriving finite element equations, use of strong and weak form formulations, complete treatment of dynamic analysis, and detailed analysis of heat transfer problems
More examples and exercises All figures revised and redrawn for clarity
The Finite Element Method in Engineering, Fifth Edition, provides a complete introduction to finite element methods with applications to solid mechanics, fluid mechanics, and heat transfer. Written by bestselling author S.S. Rao, this book provides students with a thorough grounding of the mathematical principles for setting up finite element solutions in civil, mechanical, and aerospace engineering applications. The new edition of this textbook includes examples using modern computer tools such as MatLab, Ansys, Nastran, and Abaqus.This book discusses a wide range of topics, including discretization of the domain; interpolation models; higher order and isoparametric elements; derivation of element matrices and vectors; assembly of element matrices and vectors and derivation of system equations; numerical solution of finite element equations; basic equations of fluid mechanics; inviscid and irrotational flows; solution of quasi-harmonic equations; and solutions of Helmhotz and Reynolds equations. New to this edition are examples and applications in Matlab, Ansys, and Abaqus; structured problem solving approach in all worked examples; and new discussions throughout, including the direct method of deriving finite element equations, use of strong and weak form formulations, complete treatment of dynamic analysis, and detailed analysis of heat transfer problems. All figures are revised and redrawn for clarity.This book will benefit professional engineers, practicing engineers learning finite element methods, and students in mechanical, structural, civil, and aerospace engineering. Examples and applications in Matlab, Ansys, and Abaqus Structured problem solving approach in all worked examples New discussions throughout, including the direct method of deriving finite element equations, use of strong and weak form formulations, complete treatment of dynamic analysis, and detailed analysis of heat transfer problems More examples and exercises All figures revised and redrawn for clarity
Front Cover 1
The Finite Element Method in Engineering 4
Copyright 5
Dedication 6
Table of Contents 8
Preface 14
Approach of the Book 14
New to this Edition 14
Organization 15
Resources for Instructors 16
Acknowledgments 16
Part 1: Introduction 18
Chapter 1. Overview of Finite Element Method 20
1.1 Basic Concept 20
1.2 Historical Background 21
1.3 General Applicability of the Method 24
1.4 Engineering Applications of the Finite Element Method 26
1.5 General Description of the Finite Element Method 26
1.6 One-Dimensional Problems with Linear Interpolation Model 29
1.7 One-Dimensional Problems with Cubic Interpolation Model 41
1.8 Derivation of Finite Element Equations Using a Direct Approach 45
1.9 Commercial Finite Element Program Packages 57
1.10 Solutions Using Finite Element Software 57
References 59
Problems 60
Part 2: Basic Procedure 68
Chapter 2. Discretization of the Domain 70
2.1 Introduction 70
2.2 Basic Element Shapes 70
2.3 Discretization Process 73
2.4 Node Numbering Scheme 80
2.5 Automatic Mesh Generation 82
References 85
Problems 86
Chapter 3. Interpolation Models 92
3.1 Introduction 92
3.2 Polynomial Form of Interpolation Functions 94
3.3 Simplex, Complex, and Multiplex Elements 95
3.4 Interpolation Polynomial in Terms of Nodal Degrees of Freedom 95
3.5 Selection of the Order of the Interpolation Polynomial 97
3.6 Convergence Requirements 99
3.7 Linear Interpolation Polynomials in Terms of Global Coordinates 102
3.8 Interpolation Polynomials for Vector Quantities 113
3.9 Linear Interpolation Polynomials in Terms of Local Coordinates 116
3.10 Integration of Functions of Natural Coordinates 125
3.11 Patch Test 126
References 128
Problems 129
Chapter 4. Higher Order and Isoparametric Elements 136
4.1 Introduction 137
4.2 Higher Order One-Dimensional Elements 137
4.3 Higher Order Elements in Terms of Natural Coordinates 138
4.4 Higher Order Elements in Terms of Classical Interpolation Polynomials 147
4.5 One-Dimensional Elements Using Classical Interpolation Polynomials 151
4.6 Two-Dimensional (Rectangular) Elements Using Classical Interpolation Polynomials 152
4.7 Continuity Conditions 154
4.8 Comparative Study of Elements 156
4.9 Isoparametric Elements 157
4.10 Numerical Integration 165
References 168
Problems 169
Chapter 5. Derivation of Element Matrices and Vectors 174
5.1 Introduction 175
5.2 Variational Approach 175
5.3 Solution of Equilibrium Problems Using Variational (Rayleigh-Ritz) Method 180
5.4 Solution of Eigenvalue Problems Using Variational (Rayleigh-Ritz) Method 184
5.5 Solution of Propagation Problems Using Variational (Rayleigh-Ritz) Method 185
5.6 Equivalence of Finite Element and Variational (Rayleigh-Ritz) Methods 186
5.7 Derivation of Finite Element Equations Using Variational (Rayleigh-Ritz) Approach 186
5.8 Weighted Residual Approach 192
5.9 Solution of Eigenvalue Problems Using Weighted Residual Method 199
5.10 Solution of Propagation Problems Using Weighted Residual Method 200
5.11 Derivation of Finite Element Equations Using Weighted Residual (Galerkin) Approach 201
5.12 Derivation of Finite Element Equations Using Weighted Residual (Least Squares) Approach 204
5.13 Strong and Weak Form Formulations 206
References 208
Problems 209
Chapter 6. Assembly of Element Matrices and Vectors and Derivation of System Equations 216
6.1 Coordinate Transformation 216
6.2 Assemblage of Element Equations 221
6.3 Incorporation of Boundary Conditions 228
6.4 Penalty Method 236
6.5 Multipoint Constraints—Penalty Method 240
6.6 Symmetry Conditions—Penalty Method 243
6.7 Rigid Elements 245
References 249
Problems 249
Chapter 7. Numerical Solution of Finite Element Equations 258
7.1 Introduction 258
7.2 Solution of Equilibrium Problems 259
7.3 Solution of Eigenvalue Problems 268
7.4 Solution of Propagation Problems 279
7.5 Parallel Processing in Finite Element Analysis 285
References 286
Problems 287
Part 3: Application to Solid Mechanics Problems 292
Chapter 8. Basic Equations and Solution Procedure 294
8.1 Introduction 294
8.2 Basic Equations of Solid Mechanics 294
8.3 Formulations of Solid and Structural Mechanics 311
8.4 Formulation of Finite Element Equations (Static Analysis) 316
8.5 Nature of Finite Element Solutions 320
References 321
Problems 321
Chapter 9. Analysis of Trusses, Beams, and Frames 328
9.1 Introduction 328
9.2 Space Truss Element 329
9.3 Beam Element 340
9.4 Space Frame Element 345
9.5 Characteristics of Stiffness Matrices 355
References 356
Problems 357
Chapter 10. Analysis of Plates 372
10.1 Introduction 372
10.2 Triangular Membrane Element 373
10.3 Numerical Results with Membrane Element 384
10.4 Quadratic Triangle Element 386
10.5 Rectangular Plate Element (In-plane Forces) 389
10.6 Bending Behavior of Plates 393
10.7 Finite Element Analysis of Plates in Bending 396
10.8 Triangular Plate Bending Element 396
10.9 Numerical Results with Bending Elements 400
10.10 Analysis of Three-Dimensional Structures Using Plate Elements 403
References 406
Problems 406
Chapter 11. Analysis of Three-Dimensional Problems 418
11.1 Introduction 418
11.2 Tetrahedron Element 418
11.3 Hexahedron Element 426
11.4 Analysis of Solids of Revolution 430
References 438
Problems 439
Chapter 12. Dynamic Analysis 444
12.1 Dynamic Equations of Motion 444
12.2 Consistent and Lumped Mass Matrices 447
12.3 Consistent Mass Matrices in a Global Coordinate System 456
12.4 Free Vibration Analysis 457
12.5 Dynamic Response Using Finite Element Method 469
12.6 Nonconservative Stability and Flutter Problems 477
12.7 Substructures Method 478
References 479
Problems 479
Part 4: Application to Heat Transfer Problems 488
Chapter 13. Formulation and Solution Procedure 490
13.1 Introduction 490
13.2 Basic Equations of Heat Transfer 490
13.3 Governing Equation for Three-Dimensional Bodies 492
13.4 Statement of the Problem 496
13.5 Derivation of Finite Element Equations 497
References 501
Problems 501
Chapter 14. One-Dimensional Problems 506
14.1 Introduction 506
14.2 Straight Uniform Fin Analysis 506
14.3 Convection Loss from End Surface of Fin 509
14.3 Tapered Fin Analysis 513
14.4 Analysis of Uniform Fins Using Quadratic Elements 516
14.5 Unsteady State Problems 519
14.6 Heat Transfer Problems with Radiation 524
References 528
Problems 528
Chapter 15. Two-Dimensional Problems 534
15.1 Introduction 534
15.2 Solution 534
15.3 Unsteady State Problems 543
References 543
Problems 543
Chapter 16. Three-Dimensional Problems 548
16.1 Introduction 548
16.2 Axisymmetric Problems 548
16.3 Three-Dimensional Heat Transfer Problems 553
16.4 Unsteady State Problems 558
References 559
Problems 559
Part 5: Application to Fluid Mechanics Problems 564
Chapter 17. Basic Equations of Fluid Mechanics 566
17.1 Introduction 566
17.2 Basic Characteristics of Fluids 566
17.3 Methods of Describing the Motion of a Fluid 567
17.4 Continuity Equation 568
17.5 Equations of Motion or Momentum Equations 569
17.6 Energy, State, and Viscosity Equations 573
17.7 Solution Procedure 574
17.8 Inviscid Fluid Flow 576
17.9 Irrotational Flow 577
17.10 Velocity Potential 578
17.11 Stream Function 579
17.12 Bernoulli Equation 581
References 583
Problems 583
Chapter 18. Inviscid and Incompressible Flows 588
18.1 Introduction 588
18.2 Potential Function Formulation 590
18.3 Finite Element Solution Using the Galerkin Approach 590
18.4 Stream Function Formulation 601
References 603
Problems 603
Chapter 19. Viscous and Non-Newtonian Flows 608
19.1 Introduction 608
19.2 Stream Function Formulation (Using Variational Approach) 609
19.3 Velocity–Pressure Formulation (Using Galerkin Approach) 613
19.4 Solution of Navier–Stokes Equations 615
19.5 Stream Function–Vorticity Formulation 617
19.6 Flow of Non-Newtonian Fluids 619
19.7 Other Developments 624
References 625
Problems 625
Part 6: Solution and Applications of Quasi-Harmonic Equations 628
Chapter 20. Solution of Quasi-Harmonic Equations 630
20.1 Introduction 630
20.2 Finite Element Equations for Steady-State Problems 632
20.3 Solution of Poisson’s Equation 632
20.4 Transient Field Problems 639
References 641
Problems 641
Part 7: ABAQUS and ANSYS Software and MATLAB® Programs for Finite Element Analysis 646
Chapter 21. Finite Element Analysis Using ABAQUS 648
21.1 Introduction 648
21.2 Examples 649
Problems 679
Chapter 22. Finite Element Analysis Using ANSYS 680
22.1 Introduction 680
22.2 GUI Layout in ANSYS 681
22.3 Terminology 681
22.4 Finite Element Discretization 682
22.5 System of Units 684
22.6 Stages in Solution 684
Problems 698
Chapter 23. MATLAB Programs for Finite Element Analysis 700
23.1 Solution of Linear System of Equations Using Choleski Method 701
23.2 Incorporation of Boundary Conditions 703
23.3 Analysis of Space Trusses 704
23.4 Analysis of Plates Subjected to In-plane Loads Using CST Elements 708
23.5 Analysis of Three-Dimensional Structures Using CST Elements 711
23.6 Temperature Distribution in One-Dimensional Fins 714
23.7 Temperature Distribution in One-Dimensional Fins Including Radiation Heat Transfer 715
23.8 Two-Dimensional Heat Transfer Analysis 716
23.9 Confined Fluid Flow around a Cylinder Using Potential Function Approach 718
23.10 Torsion Analysis of Shafts 719
Problems 720
Appendix: Green-Gauss Theorem (Integration by Parts in Two and Three Dimensions) 722
Index 724
Overview of Finite Element Method
Chapter Outline
1.3 General Applicability of the Method 7
1.3.1 One-Dimensional Heat Transfer 7
1.3.2 One-Dimensional Fluid Flow 8
1.3.3 Solid Bar under Axial Load 9
1.4 Engineering Applications of the Finite Element Method 9
1.5 General Description of the Finite Element Method 9
1.6 One-Dimensional Problems with Linear Interpolation Model 12
1.7 One-Dimensional Problems with Cubic Interpolation Model 24
1.8 Derivation of Finite Element Equations Using a Direct Approach 28
1.8.1 Bar Element under Axial Load 29
1.8.3 Line Element for Heat Flow 30
1.8.4 Pipe Element (Fluid Flow) 32
1.8.5 Electrical Resistor Element (Line Element for Current Flow) 33
1.1 Basic Concept
The basic idea in the finite element method is to find the solution of a complicated problem by replacing it by a simpler one. Since the actual problem is replaced by a simpler one in finding the solution, we will be able to find only an approximate solution rather than the exact solution. The existing mathematical tools will not be sufficient to find the exact solution (and sometimes, even an approximate solution) of most of the practical problems. Thus, in the absence of any other convenient method to find even the approximate solution of a given problem, we have to prefer the finite element method. Moreover, in the finite element method, it will often be possible to improve or refine the approximate solution by spending more computational effort.
In the finite element method, the solution region is considered as built up of many small, interconnected subregions called finite elements. As an example of how a finite element model might be used to represent a complex geometrical shape, consider the milling machine structure shown in Figure 1.1(a). Since it is very difficult to find the exact response (like stresses and displacements) of the machine under any specified cutting (loading) condition, this structure is approximated as composed of several pieces as shown in Figure 1.1(b) in the finite element method. In each piece or element, a convenient approximate solution is assumed and the conditions of overall equilibrium of the structure are derived. The satisfaction of these conditions will yield an approximate solution for the displacements and stresses. Figure 1.2 shows the finite element idealization of a fighter aircraft.
Figure 1.1 Representation of a Milling Machine Structure by Finite Elements.
Figure 1.2 Finite Element Mesh of a Fighter Aircraft. (Reprinted with permission from Anamet Laboratories, Inc.)
1.2 Historical Background
Although the name of the finite element method was given recently, the concept dates back for several centuries. For example, ancient mathematicians found the circumference of a circle by approximating it by the perimeter of a polygon as shown in Figure 1.3. In terms of the present-day notation, each side of the polygon can be called a “finite element.” By considering the approximating polygon inscribed or circumscribed, one can obtain a lower bound S(l) or an upper bound S(u) for the true circumference S. Furthermore, as the number of sides of the polygon is increased, the approximate values converge to the true value. These characteristics, as will be seen later, will hold true in any general finite element application.
Figure 1.3 Lower and Upper Bounds to the Circumference of a Circle.
To find the differential equation of a surface of minimum area bounded by a specified closed curve, Schellback discretized the surface into several triangles and used a finite difference expression to find the total discretized area in 1851 [1.37]. In the current finite element method, a differential equation is solved by replacing it by a set of algebraic equations. Since the early 1900s, the behavior of structural frameworks, composed of several bars arranged in a regular pattern, has been approximated by that of an isotropic elastic body [1.38]. In 1943, Courant presented a method of determining the torsional rigidity of a hollow shaft by dividing the cross section into several triangles and using a linear variation of the stress function ϕ over each triangle in terms of the values of ϕ at net points (called nodes in the present day finite element terminology) [1.1]. This work is considered by some to be the origin of the present-day finite element method. Since mid-1950s, engineers in aircraft industry have worked on developing approximate methods for the prediction of stresses induced in aircraft wings. In 1956, Turner, Cough, Martin, and Topp [1.2] presented a method for modeling the wing skin using three-node triangles. At about the same time, Argyris and Kelsey presented several papers outlining matrix procedures, which contained some of the finite element ideas, for the solution of structural analysis problems [1.3]. Reference [1.2] is considered as one of the key contributions in the development of the finite element method.
The name finite element was coined, for the first time, by Clough in 1960 [1.4]. Although the finite element method was originally developed mostly based on intuition and physical argument, the method was recognized as a form of the classical Rayleigh-Ritz method in the early 1960s. Once the mathematical basis of the method was recognized, the developments of new finite elements for different types of problems and the popularity of the method started to grow almost exponentially [1.39–1.41]. The digital computer provided a rapid means of performing the many calculations involved in the finite element analysis and made the method practically viable. Along with the development of high-speed digital computers, the application of the finite element method also progressed at a very impressive rate. Zienkiewicz and Cheung [1.6] presented the broad interpretation of the method and its applicability to any general field problem. The book by Przemieniecki [1.5] presents the finite element method as applied to the solution of stress analysis problems.
With this broad interpretation of the finite element method, it has been found that the finite element equations can also be derived by using a weighted residual method such as Galerkin method or the least squares approach. This led to widespread interest among applied mathematicians in applying the finite element method for the solution of linear and nonlinear differential equations. It is to be noted that traditionally, mathematicians developed techniques such as matrix theory and solution methods for differential equations, and engineers used those methods to solve engineering analysis problems. Only in the case of finite element method, engineers developed and perfected the technique and applied mathematicians use the method for the solution of complex ordinary and partial differential equations. Today, it has become an industry standard to solve practical engineering problems using the finite element method. Millions of degrees of freedom (dof) are being used in the solution of some important practical problems.
A brief history of the beginning of the finite element method was presented by Gupta and Meek [1.7]. Books that deal with the basic theory, mathematical foundations, mechanical design, structural, fluid flow, heat transfer, electromagnetics and manufacturing applications, and computer programming aspects are given at the end of the chapter [1.10–1.32]. The rapid progress of the finite element method can be seen by noting that, annually about 3800 papers were being published with a total of about 56,000 papers and 380 books and 400 conference proceedings published as estimated in 1995 [1.42]. With all the progress, today the finite element method is considered one of the well-established and convenient analysis tools by engineers and applied scientists.
Example 1.1
The circumference of a circle (S) is approximated by the perimeters of inscribed and circumscribed n-sided polygons as shown in Figure 1.3. Prove the following:
n→∞S(l)=S and limn→∞S(u)=S
where S(l) and S(u) denote the perimeters of the inscribed and circumscribed polygons, respectively.
Solution
Approach: Express the perimeters of polygons in terms of the radius of the circle R and the number of sides of the polygons n and find their limiting values as →∞.
If the radius of the circle is R, each side of the inscribed and the circumscribed polygon (Figure...
| Erscheint lt. Verlag | 20.12.2010 |
|---|---|
| Sprache | englisch |
| Themenwelt | Sachbuch/Ratgeber |
| Mathematik / Informatik ► Mathematik ► Analysis | |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| Naturwissenschaften ► Chemie | |
| Naturwissenschaften ► Physik / Astronomie | |
| Technik ► Bauwesen | |
| Technik ► Fahrzeugbau / Schiffbau | |
| Technik ► Luft- / Raumfahrttechnik | |
| Technik ► Maschinenbau | |
| ISBN-10 | 0-08-095204-6 / 0080952046 |
| ISBN-13 | 978-0-08-095204-8 / 9780080952048 |
| Haben Sie eine Frage zum Produkt? |
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