Introduction to Finite Element Analysis for Engineers

✍ Scribed by Saad A. Ragab, Hassan E. Fayed

• Publisher: CRC Press
• Year: 2024
• Tongue: English
• Leaves: 773
• Edition: 2
• Category: Library

✦ Synopsis

Now in its second edition, Introduction to Finite Element Analysis for Engineers is an essential introduction to FEA as a method to solve differential equations. With many practical examples focusing on both solid mechanics and fluid mechanics, it includes problems for both applications.

Using a structure of classes of differential equations, the book also includes MATLAB® codes and aims to build a comprehensive understanding of FEA and its applications in modern engineering. New chapters present finite-element models of a system of partial differential equations in two or more independent variables typified by problems in theory of elasticity and plates. Chapter ten presents the finite element method for a nonlinear Mindlin-Reissner plate, and panel flutter is included as a typical example of fluid-structure interactions. The book demonstrates the power and versatility of FEA as a tool with a large number of examples of practical engineering problems. These problems range from those which can be solved without a computer, to those requiring MATLAB® or Python.

With applications in civil, mechanical, aerospace, and biomedical engineering, the textbook is ideal for senior undergraduate and first-year graduate students and also aligns with mathematics courses.

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Author’s Biography
Chapter 1: Introduction
1.1. Computational Sciences and Mechanics
1.2. Brief Mathematical Background: Linear Algebra
1.2.1. Vectors
1.2.2. Matrices
1.3. Brief Mathematical Background: Partial Differential Equations
1.3.1. Elliptic Equations: Poisson's Equation
1.3.2. Parabolic Equations: Heat Equation
1.3.3. Hyperbolic Equations: Wave Equation
Chapter 2: Second-Order Ordinary Differential Equations
2.1. Model Problem
2.1.1. Axial Deformation of a Bar
2.1.2. One-Dimensional Heat Conduction in Solids
2.2. A Motivational Example
2.3. The Method of Weighted Residuals
2.4. Weak Form
2.5. Global Basis Functions and Matrix Formulation
2.6. Finite Element Formulation Using Element Shape Functions
2.6.1. Linear Element
2.6.2. Quadratic Element
2.6.3. Higher-Order Elements and Lagrange Interpolation Functions
2.7. Thermoelastic Effects in One Dimension
2.8. Numerical Evaluations of Element Matrices
2.9. Biomedical Engineering Applications
2.9.1. Reaction-Diffusion: Oxygen Consumption in Flat Tissues
2.9.2. Reaction-Diffusion: Oxygen Consumption in Cylindrical Tissues and Spherical Cells
2.9.3. Pulsatile Blood Flow in Arteries: Womersley Problem
2.10. Problems
Chapter 3: Fourth-Order Ordinary Differential Equations
3.1. Euler-Bernoulli Beam Theory
3.2. Weak Form of the Beam Equation
3.3. Finite Element Method: Beam Element
3.4. Plane Frames
3.5. Plane Trusses
3.6. Principle of Virtual Displacements (Work)
3.7. Principle of Minimum Total Potential Energy
3.7.1. Rayleigh-Ritz Method
3.8. Problems
Chapter 4: Elliptic Equations: Equilibrium in Two Dimensions
4.1. Model Problem
4.2. Weak Form
4.3. Mesh Generation and Connectivity Matrix
4.4. Approximations and Element Shape Functions
4.4.1. Bilinear Triangular Element
4.4.2. Bilinear Rectangular Element
4.5. Element Equations and Matrices
4.5.1. Matrices for Bilinear Triangular Element
4.5.2. Matrices for Bilinear Rectangular Element
4.5.3. Matrices for Natural and Mixed Boundary Conditions
4.6. Elements Assembly and Global System
4.7. Applications
4.7.1. Heat Conduction in Solids
4.7.2. Fully Developed Laminar Flow in Noncircular Ducts
4.7.3. Torsion of Noncircular Sections
4.8. Isoparametric Elements and Numerical Integration
4.8.1. Shape Functions of Canonical Elements
4.8.2. Mapping
4.8.3. Approximations
4.8.4. Element Matrices
4.9. Applications
4.9.1. Potential Flow Around 2D Airfoils, Lift
4.9.2. Oxygen Transport and Consumption in Krogh Capillary-Tissue Cylinder
4.10. Problems
Chapter 5: Parabolic Equations: Time-Dependent Diffusion Problems
5.1. Model Problem
5.2. The Weak Form
5.3. The Weak Form for an Element
5.4. Approximations and Element Matrices
5.5. Temporal Approximation: Time Marching
5.6. Transient Heat Conduction
5.7. Biomedical Engineering Applications
5.7.1. Bioheat: Pennes' Heat Conduction Model
5.7.2. Unsteady Oxygen Consumption in Spherical and Cylindrical Domains
5.7.3. Transient Oxygen Uptake in a Krogh Cylinder Tissue
5.8. Appendix
5.9. Problems
Chapter 6: Hyperbolic Equations: Waves and Vibrations Problems
6.1. Model Problems
6.2. The Weak Forms and Finite Element Models
6.2.1. First Model Problem: Second-Order in Space and Time
6.2.2. Second Model Problem: Transverse Vibrations of Beams
6.3. Time Advancement Scheme: Newmark Method
6.4. Applications: Waves and Vibrations on Strings, Bars, and Beams
6.5. Problems
Chapter 7: Differential Eigenvalue Problems
7.1. Natural Frequencies of Longitudinal Vibration of Bars
7.2. Natural Frequencies of Beams and Frames
7.3. Effects of Axial Force on Beam Deflection
7.4. Hydrodynamic Stability: Orr-Sommerfeld Equation
7.4.1. Orr-Sommerfeld Equation
7.4.2. Weak Form of Orr-Sommerfeld Equation
7.5. Problems
Chapter 8: Plane Elasticity
8.1. Constitutive Equations for Linear Elasticity
8.2. Principle of Virtual Displacements: Plane Elasticity
8.3. Element Equations and Matrices
8.4. Elements Assembly and Global System
8.5. Uncoupled Linear Thermoelasticity
8.5.1. Constitutive Equations
8.5.2. Element Equations and Matrices, Thermal Loading
8.6. Problems
Chapter 9: Kirchhoff-Love and Reissner-Mindlin Plates
9.1. Classical Plate Theory, CPT
9.1.1. Principle of Virtual Displacements
9.1.2. Approximations and Element Shape Functions
9.1.3. Element Equations and Matrices
9.1.4. Isoparametric Hermite Elements
9.2. Reissner-Mindlin First-Order Shear Deformation Plate Theory
9.2.1. Principle of Virtual Displacements
9.2.2. Approximations and Element Equations
9.3. Dynamic Response of Plates
9.3.1. Equation of Motion for CPT
9.3.2. Equations of Motion for Reissner-Mindlin Plate
9.3.3. Finite Element Model
9.3.4. Free Vibrations: Natural Frequencies
9.4. Fluid-Structure Interaction: Linear Analysis
9.4.1. Coupled Fluid-Structural Model
9.4.2. Free Vibrations of Submerged Plates: NAVMI Factors
9.4.3. Linear Hydroelastic Stability
9.5. Problems
Chapter 10: Nonlinear Reissner-Mindlin Plate and Applications
10.1. Equations of Equilibrium of Finite Displacements
10.2. Principle of Virtual Displacements
10.3. Geometrically Nonlinear Reissner-Mindlin Plate
10.4. Finite Element Model
10.5. Static Deflection of Geometrically Nonlinear Plates
10.6. Buckling and Post-Buckling of Plates
10.6.1. Critical Loads of Plates
10.6.2. Post-Buckling of Plates
10.7. Thermal Buckling of Reissner-Mindlin Plate
10.8. Nonlinear Vibrations: Element Matrices
10.8.1. The Duffing Equation
10.8.2. Free Vibration of Plates: Harmonic Balance
10.9. Supersonic Panel Flutter: Limit Cycle Oscillations
10.10. Nonlinear Hydroelastic Stability of Panels
Appendix
References
Index