# Calendar

Lec # Topics key dates

Part 1: Review of the Equations of Linear Elasticity

1

Introduction

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

4

Kinematics

Strain at a Point

Transformation of Stress Components

Compatibility Conditions

5

Thermodynamic Principles

The First Law of Thermodynamics: Energy Equation

The Second Law of Thermodynamics

6

Constitutive Equations

Generalized Hooke's Law

Strain Energy Density Function

Elastic Symmetry

Thermoelastic Constitutive Equations

7

Boundary Value Problems of Elasticity

Summary of Equations

Classification of Boundary Value Problems

Existence and Uniqueness of Solutions

Assignment 1 due

Part 2: Energy and Variational Principles

8-9

Preliminary Concepts

Introduction

Work and Energy

Strain and Complementary Strain Energy

Virtual Work

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

Assignment 2 due in lecture 11

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

Assignment 3 due in lecture 14

15

Energy Theorems of Structural Mechanics

Castigliano's First Theorem

Castigliano's Second Theorem

Betti's and Maxwell's Reciprocity Theorems

16

Some Preliminaries

17-18

The Ritz Method

Description of the Method

Matrix Form of the Ritz Equations

One Dimensional Examples

Assignment 4 due in lecture 17

19

Weighted Residual Methods

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

Assignment 5 due

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

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

24

Isoparametric Derivation of Bar Element Stiffness Matrix

25-27

Formulation of Continuum Elements

Quadrilateral Elements

Triangular Elements

Convergence Considerations

Element Matrices in Global Coordinate System

Assignment 6 due in lecture 25

28-29

Formulation of Structural Elements

Beam Elements and Axisymmetric Shell Elements

Plate and Shell Elements

Assignment 7 due in lecture 28

30

Numerical Integration

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

Assignment 8 due in lecture 33

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

Assignment 9 due in lecture 36

Term Project due in lecture 36

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

Approach to Design for Longevity

Material Selection Example