# Tools

The following files are provided to give background in the mathematical aspects of the course.

Lec # Topics tools

Part 1: Review of the Equations of Linear Elasticity

1

Introduction

Math Tutorial (NB)

4

Kinematics

Strain at a Point

Transformation of Stress Components

Compatibility Conditions

Influence of Nonlinear Strains (NB)

Part 2: Energy and Variational Principles

15

Energy Theorems of Structural Mechanics

Castigliano's First Theorem

Castigliano's Second Theorem

Betti's and Maxwell's Reciprocity Theorems

Castigliano's Theorem Example (NB)

17-18

The Ritz Method

Description of the Method

Matrix Form of the Ritz Equations

One Dimensional Examples

The Ritz Method: Beam Example (NB)

Space Elevators (NB)

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

Notes on The Finite Element Method II (NB)

Fixes to the Ritz Method (NB)

Application to Rods (NB)

The Beam Model (NB)

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

2nd Plane Stress Element (NB)

Isoparametric 4-node 2nd Element (NB)

30

Numerical Integration