Introduction to Finite Element Analysis: A Textbook for Engineering Students

✍ Scribed by S. Unnikrishnan Nair (editor), S. Somanath (editor)

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 415
• Category: Library

✦ Synopsis

This textbook covers the basic concepts and applications of finite element analysis. It is specifically aimed at introducing this advanced topic to undergraduate-level engineering students and practicing engineers in a lucid manner. It also introduces a structural and heat transfer analysis software FEAST^SMT which has wide applications in civil, mechanical, nuclear and automobile engineering domains. This software has been developed by generations of scientists and engineers of Vikram Sarabhai Space Centre and Indian Space Research Organisation. Supported with many illustrative examples, the textbook covers the classical methods of estimating solutions of mathematical models. The book is written in an easy-to-understand manner. This textbook also contains numeral exercise problems to aid self-learning of the students. The solutions to these problems are demonstrated using finite element software. Furthermore, the textbook contains several tutorials and associated onlineresources on usage of the FEAST^SMT software. Given the contents, this textbook is highly useful for the undergraduate students of various disciplines of engineering. It is also a good reference book for the practicing engineers.

✦ Table of Contents

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