| Part 1: Review of the Equations of Linear Elasticity |
| 1 |
Introduction |
Introduction (PDF) |
| 2-3 |
Kinetics
Stress at a Point
Stress Tensor and the Cauchy Formula
Transformation of Stress Components
Principal Stresses and Principal Planes
Equations of Motion
Symmetry of the Stress Tensor |
Stress and Momentum Balance (PDF)
Mathematical Aside: Vectors, Indicial Notation and Summation Convention (PDF) |
| 4 |
Kinematics
Strain at a Point
Transformation of Stress Components
Compatibility Conditions |
Kinematics of Deformation (PDF) |
| 5 |
Thermodynamic Principles
The First Law of Thermodynamics: Energy Equation
The Second Law of Thermodynamics |
Thermodynamics Principles (PDF) |
| 6 |
Constitutive Equations
Generalized Hooke's Law
Strain Energy Density Function
Elastic Symmetry
Thermoelastic Constitutive Equations |
Constitutive Equations (PDF) |
| 7 |
Boundary Value Problems of Elasticity
Summary of Equations
Classification of Boundary Value Problems
Existence and Uniqueness of Solutions |
Boundary Value Problems of Linear Elasticity (PDF) |
| Part 2: Energy and Variational Principles |
| 8-9 |
Preliminary Concepts
Introduction
Work and Energy
Strain and Complementary Strain Energy
Virtual Work |
Concepts of Work and Energy (PDF)
Strain Energy and Potential Energy of a Beam (PDF)
Principles of Virtual Displacements (PDF)
Principles of Virtual Forces (PDF)
|
| 10-11 |
Concepts of Calculus of Variations
Concept of a Functional
The Variational Operator
The First Variation of a Functional
Extremum of a Functional
The Euler Equations
Natural and Essential Boundary Conditions
A More General Functional
Minimization with Linear Equality Constraints |
Calculus of Variations (PDF) |
| 12-14 |
Virtual Work and Energy Principles
Principle of Virtual Displacements
Unit Dummy Displacement Method
Principle of Total Potential Energy
Principle of Virtual Forces and Complementary Potential
Energy
Unit Dummy Load Method |
|
| 15 |
Energy Theorems of Structural Mechanics
Castigliano's First Theorem
Castigliano's Second Theorem
Betti's and Maxwell's Reciprocity Theorems |
Principle of Minimum Potential Energy and Castigliano's First Theorem (PDF) |
| 16 |
Some Preliminaries |
|
| 17-18 |
The Ritz Method
Description of the Method
Matrix Form of the Ritz Equations
One Dimensional Examples |
Approximate Methods (PDF)
The Ritz Method Cont. (PDF) |
| 19 |
Weighted Residual Methods
A Brief Description of Galerkin, Least-squares and Collocation Methods |
|
| 20-22 |
Formulation of the Displacement Based Finite Element Method
General Derivation of Finite Element Equilibrium Equations
Imposition of Displacement Boundary Conditions
Generalized Coordinate Models for Specific Problems
Lumping of Structure Properties and Loads |
The Finite Element Method (PDF)
The Finite Element Method II (PDF)
The Finite Element Method III (PDF)
The Finite Element Method IV: Imposition of Boundary Conditions (PDF)
Finite Element Model of a Beam (PDF) |
| 23 |
Convergence of Analysis Results
Definition of Convergence
Properties of the Finite Element Solution
Rate of Convergence
Calculation of Stresses and the Assessment of Error |
The Finite Element Method V: For Three-Dimensional Elasticity Problems (PDF) |
| 24 |
Isoparametric Derivation of Bar Element Stiffness Matrix |
Formulation of Isoparametric Elements (PDF) |
| 25-27 |
Formulation of Continuum Elements
Quadrilateral Elements
Triangular Elements
Convergence Considerations
Element Matrices in Global Coordinate System |
|
| 28-29 |
Formulation of Structural Elements
Beam Elements and Axisymmetric Shell Elements
Plate and Shell Elements |
|
| 30 |
Numerical Integration |
Numerical Integration (PDF) |
| 31 |
Direct Solution of Linear System of Equations |
|
| 32-33 |
Types of Structural Failure
Yield Stress and Ultimate Stress
Maximum Normal Stress Theory
Tresca Condition, Hydraulic Stress, von Mises Criterion, Distortion Energy Interpretation
Graphical Representation of Failure Regions
Extension to Orthotropic Materials, Hill Criterion, Hoffman Criterion
Nature of Failure Criteria, Functional Forms
General Failure Analysis Procedure
Application to Pressure Tank |
Failure, Fracture, and Fatigue (PDF - 2.4 MB) |
| 34-37 |
Fracture Mechanics
Description of Phenomena and Importance
Energy Approach to Crack Growth, Energy Consumed by Crack Growth, Griffith's Experiment and Formula
Definition of Stress Intensity Factor
Stresses at Crack Tip, Mode I, II and III Cracks
Solutions of Linear Elastic Fracture Mechanics, Geometry Effects
Combined Loading; Material Selection Example |
|
| 38-42 |
Fatigue and Longevity
Terminology, SN Diagrams, Goodman Diagrams
Effects of R Value, Stress Concentrations
Ground-Air-Ground Cycle, Miner's Rule
Micromechanical Effects
Paris' Law
Fatigue Life Prediction
R Effects and Forman's Law, Sequencing Effects
Approached to Design for Longevity
Material Selection Example |
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