Introduction to Finite Element Analysis: A Textbook for Engineering Students

Foreword
Preface
Acknowledgements
Editorial Note
Contents
Editors and Contributors
1 Introduction
	1.1 History of Finite Element Method
	1.2 Finite Element Method
		1.2.1 Derive the Differential Equation and Obtain the Closed Form Solution
		1.2.2 Analytical Solution
		1.2.3 Finite Element Solution
	1.3 Applications of FEA
		1.3.1 Solid Mechanics
		1.3.2 Heat Conduction
		1.3.3 Fluid Flow Through Pipes
		1.3.4 Electrostatics
	1.4 Different Numerical Methods
		1.4.1 Mesh Methods
		1.4.2 Mesh Free Methods
		1.4.3 Mixed Method
		1.4.4 Advantages of Using FEM Over Other Methods
	1.5 FEASTSMT©
	1.6 Exercise Problems
	Bibliography
2 Linear Algebra
	2.1 Introduction
	2.2 Matrices
	2.3 Special Matrices
	2.4 Matrix Algebra
	2.5 Linear Dependence and Rank of a Matrix
		2.5.1 Properties of Rank
	2.6 Solution of System of Linear Simultaneous Equations
		2.6.1 Rouche’s Theorem
		2.6.2 Homogeneous System
	2.7 Eigenvalues and Eigenvectors
		2.7.1 Properties of Eigenvalues
		2.7.2 Properties of Eigenvectors
	2.8 Diagonalization of a Matrix
		2.8.1 Diagonalization by Similarity Transformation
		2.8.2 Diagonalization of a Real Symmetric Matrix by Orthogonal Reduction
	2.9 Quadratic Forms
	2.10 Solution Schemes of a System of Linear Simultaneous Equations
		2.10.1 Direct Methods
		2.10.2 Iterative Methods
	2.11 Worked Out Examples
	2.12 Exercise Problems
	Bibliography
3 Classical Approximate Solutions
	3.1 Introduction
		3.1.1 Brachistochrone
		3.1.2 Geodesic
		3.1.3 Isoperimetric
	3.2 Mathematical Preliminaries
	3.3 Classification of Partial Differential Equations
	3.4 Boundary and Initial Conditions
	3.5 Weak Formulation of Differential Equations and Rayleigh–Ritz Method
	3.6 Weighted Residual Technique and Galerkin’s Method
	3.7 Link Between Classical Approximate Solution and Finite Element Method
	3.8 Exercise
	Bibliography
4 Elementary Concepts in Elasticity
	4.1 Equilibrium Equations
	4.2 Kinematic Conditions
	4.3 Strain–Displacement Equations
	4.4 Constitutive Relations
	4.5 Exercise Problems
	Bibliography
5 Finite Element Formulations
	5.1 Introduction
	5.2 Functional Overview of a Finite Element Solver
	5.3 The Domain of Linear Structural Analysis
	5.4 Displacement-Based FE Formulations
	5.5 Classification of Structural Finite Elements
		5.5.1 Three-Dimensional Elements
		5.5.2 Two-Dimensional Elements
		5.5.3 One-Dimensional Elements
	5.6 Assumed Displacement Fields and Shape (Interpolation) Functions
	5.7 Natural Coordinate System
	5.8 Use of Principle of Minimum Potential Energy
	5.9 Numerical Integration for Stiffness Matrix Computation
	5.10 Transformation of Element Stiffness Matrices
	5.11 Isoparametric Finite Element Formulations
	5.12 Stiffness Matrix for Rod/Truss/Link Finite Element
	5.13 Stiffness Matrix for Three-Dimensional Solid Finite Element
	5.14 Stiffness Matrix for Planar Finite elements
	5.15 Stiffness Matrix for Axisymmetric Solid Element
	5.16 Stiffness Matrix for Beam Element
	5.17 Stiffness Matrix for Axisymmetric Shell Element
	5.18 Stiffness Matrix for Plate element
	5.19 Worked Out Problems
	5.20 Exercise Problems
	Bibliography
6 Boundary Conditions and Loads
	6.1 Introduction
	6.2 Types of Boundary Conditions
	6.3 Single Point Constraints (SPC)
	6.4 Multi Point Constraints (MPC)
	6.5 Numerical Examples of Boundary Conditions
	6.6 Load Vectors
		6.6.1 Load Vector for Rod/Truss/Link Element
		6.6.2 Load Vector for 3D Solid Element
		6.6.3 Load Vector for Planar Element
		6.6.4 Load Vector for Axi-Symmetric Solid Element
		6.6.5 Load Vector for Beam Element
		6.6.6 Load Vector for Axi-Symmetric Shell Element
		6.6.7 Load Vector for Plate Element
	6.7 Numerical Examples of Load Vector
	6.8 Exercises
	Bibliography
7 Finite Element Formulation for Heat Transfer
	7.1 Introduction
	7.2 Heat Transfer Essentials
		7.2.1 Conduction
		7.2.2 Convection
		7.2.3 Radiation
	7.3 Governing Equations
	7.4 Initial and Boundary Conditions
		7.4.1 Initial Conditions
		7.4.2 Boundary Conditions
		7.4.3 Dirichlet Boundary Condition
		7.4.4 Neumann Boundary Condition
		7.4.5 Robin Boundary Condition
	7.5 One-Dimensional Steady State Conduction
		7.5.1 Equivalence with Elasticity
		7.5.2 Method of Weighted Residuals for One-Dimensional Steady State Conduction
		7.5.3 Examples
	7.6 One-Dimensional Transient Conduction
		7.6.1 Method of Weighted Residuals for One-Dimensional Transient Conduction
		7.6.2 Examples
	7.7 Higher dimensional heat transfer problems
		7.7.1 Method of Weighted Residuals for Steady State Conduction
		7.7.2 Examples
	7.8 Exercises
	References
8 Dynamic Analysis
	8.1 Introduction
	8.2 Primer to Dynamics
	8.3 Work and Energy
	8.4 Conservative Force Field
	8.5 Single Degree of Freedom System
		8.5.1 First-Order System
		8.5.2 Second-Order System
	8.6 Equation of Motion for N Particles
	8.7 Lagrange’s Equations
	8.8 Motion of a Multi-degree-of-Freedom System Under Constraints
	8.9 Solution to Multi-degrees-of-Freedom System
		8.9.1 Expressions for System Parameters: Mass and Stiffness
	8.10 Free Vibration Analysis
	8.11 Time Domain Solution/Transient Analysis
		8.11.1 Modal Transient Method
		8.11.2 Direct Transient Method
	8.12 Frequency-Response Analysis
	8.13 Random Response Analysis
	Bibliography
9 Buckling of Column
	9.1 Introduction
	9.2 Derivation of Shape Function for Euler Column
	9.3 Governing Differential Equation and Finite Element Formulation Using Galerkin Method
	9.4 Finite Element Formulation Using Energy Method
	9.5 Closed Form Solutions
	9.6 Example 1
	9.7 Example 2
	9.8 Example 3
	9.9 Example 4
	9.10 Knockdown Factor
	9.11 Exercise
	Bibliography
10 Features of FEASTSMT Software
	10.1 Introduction
	10.2 Analysis Capabilities
	10.3 Graphical User Interface
	10.4 Model Viewing and Displaying
	10.5 Selection Options
	10.6 Building Geometry Model
	10.7 Mesh Generation
	10.8 Model Checking
	10.9 Element Library
	10.10 Material Data
	10.11 Solution Schemes
	10.12 Loads and Boundary Conditions
	10.13 Post-processing
	10.14 Structural Deflection/Mode Shapes
	10.15 Contours
	10.16 Arrow Plots
	10.17 X–Y Plots
	10.18 Animation
11 Modeling Techniques and Interpretation of Results
	11.1 Introduction
	11.2 Usage of Elements
		11.2.1 One-dimensional elements in FEASTSMT
		11.2.2 Two-Dimensional elements
		11.2.3 Three-Dimensional elements
		11.2.4 Zero-Dimensional elements
	11.3 Element Quality
	11.4 Boundary Conditions
		11.4.1 Exploiting Symmetries
	11.5 Stress Concentration
	11.6 Good Finite Element Modeling Practices
	11.7 Exercise Problems
	Bibliography
12 Linear Static Analysis Using FEASTSMT Software
	12.1 Static Analysis of an Earth Dam
		12.1.1 Pre-processor
		12.1.2 Solution
		12.1.3 Post-processing
	12.2 Static Analysis of a Portal Frame
		12.2.1 Pre-processor
		12.2.2 Solution
		12.2.3 Post-processing
	12.3 Static Analysis of a Truss Structure
		12.3.1 Pre-processor
		12.3.2 Solution
		12.3.3 Post-processing
	12.4 Static Analysis of a Rotating Disk
		12.4.1 Pre-processor
		12.4.2 Solution
		12.4.3 Post-processing
	12.5 Exercise Problems
13 Heat Transfer Analysis Using FEASTSMT Software
	13.1 Rocket Nozzle Heat Transfer
		13.1.1 Pre-processor
		13.1.2 Solution
		13.1.3 Post-processing
	13.2 Heat Transfer Analysis of a Printed Circuit Board
		13.2.1 Pre-processor
		13.2.2 Solution
		13.2.3 Processing
	13.3 Heat Transfer Analysis of Cooling Fins
		13.3.1 Pre-processor
		13.3.2 Solution
		13.3.3 Post-processing
	13.4 Exercise Problems
14 DynamicAnalysis Using FEASTSMT Software
	14.1 Free Vibration Analysis of a C-section Beam
		14.1.1 Pre-processor
		14.1.2 Solution
		14.1.3 Post-processing
	14.2 Frequency Response Analysis of a Bracket
		14.2.1 Pre-processor
		14.2.2 Solver
		14.2.3 Post-processing
	14.3 Transient Analysis of a Bridge
	14.4 Pre-processor
		14.4.1 Solver
		14.4.2 Post-processing
	14.5 Exercise Problems
15 Buckling Analysis Using FEASTSMT Software
	15.1 Buckling Analysis of a Cylinder with Stiffeners
	15.2 Pre-processor
	15.3 Solution
	15.4 Result Viewing Using Post-processing
	15.5 Exercise Problems
Index