Lecture Notes

Lec # Topics lecture notes

Part 1: Review of the Equations of Linear Elasticity



Introduction (PDF)



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)



Strain at a Point

Transformation of Stress Components

Compatibility Conditions

Kinematics of Deformation (PDF)


Thermodynamic Principles

The First Law of Thermodynamics: Energy Equation

The Second Law of Thermodynamics

Thermodynamics Principles (PDF)


Constitutive Equations

Generalized Hooke's Law

Strain Energy Density Function

Elastic Symmetry

Thermoelastic Constitutive Equations

Constitutive Equations (PDF)


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


Preliminary Concepts


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)


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)


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



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)


Some Preliminaries



The Ritz Method

Description of the Method

Matrix Form of the Ritz Equations

One Dimensional Examples

Approximate Methods (PDF)

The Ritz Method Cont. (PDF)


Weighted Residual Methods

A Brief Description of Galerkin, Least-squares and Collocation Methods



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)


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)


Isoparametric Derivation of Bar Element Stiffness Matrix

Formulation of Isoparametric Elements (PDF)


Formulation of Continuum Elements

Quadrilateral Elements

Triangular Elements

Convergence Considerations

Element Matrices in Global Coordinate System



Formulation of Structural Elements

Beam Elements and Axisymmetric Shell Elements

Plate and Shell Elements



Numerical Integration

Numerical Integration (PDF)


Direct Solution of Linear System of Equations



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)


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



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