| Lec # | Topics | lecture notes |
|---|---|---|
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 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 | |