Introduction to Finite Elements in Engineering

by Chandrupatla, Tirupathi R.; Belegundu, Ashok D.

• ISBN13: 9780130615916
• ISBN10: 0130615919
• Edition: 3rd
• Format: Hardcover
• Copyright: 2002-01-01
• Publisher: Prentice Hall

Summary

This book provides an integrated approach to finite element methodologies. The development of finite element theory is combined with examples and exercises involving engineering applications. The steps used in the development of the theory are implemented in complete, self-contained computer programs. While the strategy and philosophy of the previous editions has been retained, the Third Edition has been updated and improved to include new material on additional topics.Chapter topics cover fundamental concepts, matrix algebra and gaussian elimination, one-dimensional problems, trusses, two-dimensional problems using constant strain triangles, axisymmetric solids subjected to axisymmetric loading, two-dimensional isoparametric elements and numerical integration, beams and frames, three-dimensional problems in stress analysis, scalar field problems, dynamic considerations, and preprocessing and postprocessing.For practicing engineers as a valuable learning resource.

Table of Contents

Preface	xv
Fundamental Concepts	1	(21)
Introduction	1	(1)
Historical Background	1	(1)
Outline of Presentation	2	(1)
Stresses and Equilibrium	2	(2)
Boundary Conditions	4	(1)
Strain-Displacement Relations	4	(2)
Stress-Strain Relations	6	(2)
Special Cases	7	(1)
Temperature Effects	8	(1)
Potential Energy and Equilibrium; The Rayleigh-Ritz Method	9	(4)
Potential Energy II	9	(2)
Rayleigh-Ritz Method	11	(2)
Galerkin's Method	13	(3)
Saint Venant's Principle	16	(1)
Von Mises Stress	17	(1)
Computer Programs	17	(1)
Conclusion	18	(4)
Historical References	18	(1)
Problems	18	(4)
Matrix Algebra and Gaussian Elimination	22	(23)
Matrix Algebra	22	(7)
Row and Column Vectors	23	(1)
Addition and Subtraction	23	(1)
Multiplication by a Scalar	23	(1)
Matrix Multiplication	23	(1)
Transposition	24	(1)
Differentiation and Integration	24	(1)
Square Matrix	25	(1)
Diagonal Matrix	25	(1)
Identity Matrix	25	(1)
Symmetric Matrix	25	(1)
Upper Triangular Matrix	26	(1)
Determinant of a Matrix	26	(1)
Matrix Inversion	26	(1)
Eigenvalues and Eigenvectors	27	(1)
Positive Definite Matrix	28	(1)
Cholesky Decomposition	29	(1)
Gaussian Elimination	29	(10)
General Algorithm for Gaussian Elimination	30	(3)
Symmetric Matrix	33	(1)
Symmetric Banded Matrices	33	(2)
Solution with Multiple Right Sides	35	(1)
Gaussian Elimination with Column Reduction	36	(2)
Skyline Solution	38	(1)
Frontal Solution	39	(1)
Conjugate Gradient Method for Equation Solving	39	(6)
Conjugate Gradient Algorithm	40	(1)
Problems	41	(2)
Program Listings	43	(2)
One-Dimensional Problems	45	(58)
Introduction	45	(1)
Finite Element Modeling	46	(2)
Element Division	46	(1)
Numbering Scheme	47	(1)
Coordinates and Shape Functions	48	(4)
The Potential-Energy Approach	52	(4)
Element Stiffness Matrix	53	(1)
Force Terms	54	(2)
The Galerkin Approach	56	(2)
Element Stiffness	56	(1)
Force Terms	57	(1)
Assembly of the Global Stiffness Matrix and Load Vector	58	(3)
Properties of K	61	(1)
The Finite Element Equations; Treatment of Boundary Conditions	62	(16)
Types of Boundary Conditions	62	(1)
Elimination Approach	63	(6)
Penalty Approach	69	(5)
Multipoint Constraints	74	(4)
Quadratic Shape Functions	78	(6)
Temperature Effects	84	(19)
Input Data File	88	(1)
Problems	88	(10)
Program Listing	98	(5)
Trusses	103	(27)
Introduction	103	(1)
Plane Trusses	104	(10)
Local and Global Coordinate Systems	104	(1)
Formulas for Calculating l and m	105	(1)
Element Stiffness Matrix	106	(1)
Stress Calculations	107	(4)
Temperature Effects	111	(3)
Three-Dimensional Trusses	114	(2)
Assembly of Global Stiffness Matrix for the Banded and Skyline Solutions	116	(14)
Assembly for Banded Solution	116	(3)
Input Data File	119	(1)
Problems	120	(8)
Program Listing	128	(2)
Two-Dimensional Problems Using Constant Strain Triangles	130	(48)
Introduction	130	(1)
Finite Element Modeling	131	(2)
Constant-Strain Triangle (CST)	133	(19)
Isoparametric Representation	135	(4)
Potential-Energy Approach	139	(1)
Element Stiffness	140	(1)
Force Terms	141	(5)
Galerkin Approach	146	(2)
Stress Calculations	148	(2)
Temperature Effects	150	(2)
Problem Modeling and Boundary Conditions	152	(2)
Some General Comments on Dividing into Elements	154	(1)
Orthotropic Materials	154	(24)
Temperature Effects	157	(3)
Input Data File	160	(2)
Problems	162	(12)
Program Listing	174	(4)
Axisymmetric Solids Subjected to Axisymmetric Loading	178	(30)
Introduction	178	(1)
Axisymmetric Formulation	179	(2)
Finite Element Modeling: Triangular Element	181	(10)
Potential-Energy Approach	183	(1)
Body Force Term	184	(1)
Rotating Flywheel	185	(1)
Surface Traction	185	(2)
Galerkin Approach	187	(3)
Stress Calculations	190	(1)
Temperature Effects	191	(1)
Problem Modeling and Boundary Conditions	191	(17)
Cylinder Subjected to Internal Pressure	191	(1)
Infinite Cylinder	192	(1)
Press Fit on a Rigid Shaft	192	(1)
Press Fit on an Elastic Shaft	193	(1)
Belleville Spring	194	(1)
Thermal Stress Problem	195	(2)
Input Data File	197	(1)
Problems	198	(7)
Program Listing	205	(3)
Two-Dimensional Isoparametric Elements and Numerical Integration	208	(29)
Introduction	208	(1)
The Four-Node Quadrilateral	208	(6)
Shape Functions	208	(3)
Element Stiffness Matrix	211	(2)
Element Force Vectors	213	(1)
Numerical Integration	214	(6)
Two-Dimensional Integrals	217	(1)
Stiffness Integration	217	(1)
Stress Calculations	218	(2)
Higher Order Elements	220	(5)
Nine-Node Quadrilateral	220	(2)
Eight-Node Quadrilateral	222	(1)
Six-Node Triangle	223	(2)
Four-Node Quadrilateral for Axisymmetric Problems	225	(1)
Conjugate Gradient Implementation of the Quadrilateral Element	226	(11)
Concluding Note	227	(1)
Input Data File	228	(2)
Problems	230	(3)
Program Listings	233	(4)
Beams and Frames	237	(38)
Introduction	237	(3)
Potential-Energy Approach	238	(1)
Galerkin Approach	239	(1)
Finite Element Formulation	240	(3)
Load Vector	243	(1)
Boundary Considerations	244	(1)
Shear Force and Bending Moment	245	(2)
Beams on Elastic Supports	247	(1)
Plane Frames	248	(5)
Three-Dimensional Frames	253	(4)
Some Comments	257	(18)
Input Data File	258	(3)
Problems	261	(6)
Program Listings	267	(8)
Three-Dimensional Problems in Stress Analysis	275	(31)
Introduction	275	(1)
Finite Element Formulation	276	(4)
Element Stiffness	279	(1)
Force Terms	280	(1)
Stress Calculations	280	(1)
Mesh Preparation	281	(4)
Hexahedral Elements and Higher Order Elements	285	(2)
Problem Modeling	287	(2)
Frontal Method for Finite Element Matrices	289	(17)
Connectivity and Prefront Routine	290	(1)
Element Assembly and Consideration of Specified dof	290	(1)
Elimination of Completed dof	291	(1)
Backsubstitution	291	(1)
Consideration of Multipoint Constraints	291	(1)
Input Data File	292	(1)
Problems	293	(4)
Program Listings	297	(9)
Scalar Field Problems	306	(61)
Introduction	306	(2)
Steady State Heat Transfer	308	(23)
One-Dimensional Heat Conduction	309	(7)
One-Dimensional Heat Transfer in Thin Fins	316	(4)
Two-Dimensional Steady-State Heat Conduction	320	(9)
Two-Dimensional Fins	329	(1)
Preprocessing for Program Heat2D	330	(1)
Torsion	331	(5)
Triangular Element	332	(1)
Galerkin Approach	333	(3)
Potential Flow, Seepage, Electric and Magnetic Fields, and Fluid Flow in Ducts	336	(12)
Potential Flow	336	(2)
Seepage	338	(1)
Electrical and Magnetic Field Problems	339	(2)
Fluid Flow in Ducts	341	(2)
Acoustics	343	(1)
Boundary Conditions	344	(1)
One-Dimensional Acoustics	344	(1)
1-D Axial Vibrations	345	(3)
Two-Dimensional Acoustics	348	(1)
Conclusion	348	(19)
Input Data File	349	(1)
Problems	350	(11)
Program Listings	361	(6)
Dynamic Considerations	367	(44)
Introduction	367	(1)
Formulation	367	(3)
Solid Body with Distributed Mass	368	(2)
Element Mass Matrices	370	(5)
Evaluation of Eigenvalues and Eigenvectors	375	(16)
Properties of Eigenvectors	376	(1)
Eigenvalue-Eigenvector Evaluation	376	(6)
Generalized Jacobi Method	382	(4)
Tridiagonalization and Implicit Shift Approach	386	(1)
Bringing Generalized Problem to Standard Form	386	(1)
Tridiagonalization	387	(3)
Implicit Symmetric QR Step with Wilkinson Shift for Diagonalization	390	(1)
Interfacing with Previous Finite Element Programs and a Program for Determining Critical Speeds of Shafts	391	(1)
Guyan Reduction	392	(2)
Rigid Body Modes	394	(2)
Conclusion	396	(15)
Input Data File	397	(2)
Problems	399	(5)
Program Listings	404	(7)
Preprocessing and Postprocessing	411	(29)
Introduction	411	(1)
Mesh Generation	411	(8)
Region and Block Representation	411	(1)
Block Corner Nodes, Sides, and Subdivisions	412	(7)
Postprocessing	419	(5)
Deformed Configuration and Mode Shape	419	(1)
Contour Plotting	420	(1)
Nodal Values from Known Constant Element Values for a Triangle	421	(2)
Least Squares Fit for a Four-Noded Quadrilateral	423	(1)
Conclusion	424	(16)
Input Data File	425	(1)
Problems	425	(2)
Program Listings	427	(13)
Appendix Proof of dA = det J dξ dη	440	(3)
Bibliography	443	(4)
Answers to Selected Problems	447	(2)
Index	449

Excerpts

The first edition of this book appeared over 10 years ago and the second edition followed a few years later. We received positive feedback from professors who taught from the book and from students and practicing engineers who used the book. We also benefited from the feedback received from the students in our courses for the past 20 years. We have incorporated several suggestions in this edition. The underlying philosophy of the book is to provide a clear presentation of theory, modeling, and implementation into computer programs. The pedagogy of earlier editions has been retained in this edition.New material has been introduced in several chapters. Worked examples and exercise problems have been added to supplement the learning process. Exercise problems stress both fundamental understanding and practical considerations. Theory and computer programs have been added to cover acoustics, axisymmetric quadrilateral elements, conjugate gradient approach, and eigenvalue evaluation. Three additional programs have now been introduced in this edition. All the programs have been developed to work in the Windows environment. The programs have a common structure that should enable the users to follow the development easily. The programs have been provided in Visual Basic, Microsoft Excel/Visual Basic, MATLAB, together with those provided earlier in QBASIC, FORTRAN and C. The Solutions Manual has also been updated.Chapter 1 gives a brief historical background and develops the fundamental concepts. Equations of equilibrium, stress-strain relations, strain-displacement relations, and the principles of potential energy are reviewed. The concept of Galerkin's method is introduced.Properties of matrices and determinants are reviewed in Chapter 2. The Gaussian elimination method is presented, and its relationship to the solution of symmetric banded matrix equations and the skyline solution is discussed. Cholesky decomposition and conjugate gradient method are discussed.Chapter 3 develops the key concepts of finite element formulation by considering one-dimensional problems. The steps include development of shape functions, derivation of element stiffness, formation of global stiffness, treatment of boundary conditions, solution of equations, and stress calculations. Both the potential energy approach and Galerkin's formulations are presented. Consideration of temperature effects is included.Finite element formulation for plane and three-dimensional trusses is developed in Chapter 4. The assembly of global stiffness in banded and skyline forms is explained. Computer programs for both banded and skyline solutions are given.Chapter 5 introduces the finite element formulation for two-dimensional plane stress and plane strain problems using constant strain triangle (CST) elements. Problem modeling and treatment of boundary conditions are presented in detail. Formulation for orthotropic materials is provided. Chapter 6 treats the modeling aspects of axisymmetric solids subjected to axisymmetric loading. Formulation using triangular elements is presented. Several real-world problems are included in this chapter.Chapter 7 introduces the concepts of isoparametric quadrilateral and higher order elements and numerical integration using Gaussian quadrature. Formulation for axisymmetric quadrilateral element and implementation of conjugate gradient method for quadrilateral element are given.Beams and application of Hermite shape functions are presented in Chapter 8. The chapter covers two-dimensional and three-dimensional frames.Chapter 9 presents three-dimensional stress analysis. Tetrahedral and hexahedral elements are presented. The frontal method and its implementation aspects are discussed.Scalar field problems are treated in detail in Chapter 10. While Galerkin as well as energy approaches have been used in every chapter, with equal importance, only Galerkin's approach is used i