Todays Agenda:
The Direct Stiness Method
Derive stiness matrix using solid mechanics principles
Focus will be on devising algorithms for assembling
element level matrices
Properties of the stiness matrix
Incorporation of Dirichlet/Displacement boundar

3
Fundamentals of FEA: Strong form,
weighted residual form, weak form and
the principle of minimum total
potential energy
3.1
Introduction
Earlier we studied the direct stiness method which involved applying principles from mechanics of solids to
derive t

AER501
Solutions to exercises1
DRAFT Version: Wednesday 16th September, 2015
19:05
Chapter 3
Problem 3.3.6.1
The residual error corresponding to the assumed approximation u
b(x) = c1 x(1 x2 ) is given by
R(x, c1 ) =
d2 u
b
+ 1000x2 = 6c1 x + 1000x2 .
dx2

7
Finite element analysis of linear
elasticity problems
So far we have considered only one-dimensional (line) elements that form structures such as trusses and frames.
Line elements have properties associated with the other two dimensions in their cross s

Outline
Review of linear elasticity governing equations
General derivation of Ke & f e
3-node constant strain triangle (CST)
4-node rectangular element
Finite element analysis of
linear elasticity problems
AER501
AER501
Finite element analysis of linear e

4
Finite element analysis of rods
We shall look at how the element stiness matrix of rod/bar structures can be evaluated using the weak/minimization
form. Consider an axially loaded rod (with Youngs modulus E and cross-section area A(x) of length 1 cantil

AER501
Topics to be covered
Introduction to structural optimization
Non-gradient optimization methods
Gradient-based optimization methods
Sensitivity analysis
Last 2 sessions course review and
problem solving
Requirements
The design spiral
CONCEPTUA

12
Sensitivity Analysis
12.1
Introduction
Sensitivity analysis is of fundamental importance to design based on computational approaches. It allows the use
of gradient descent methods, reveals when optimal designs have been produced and indicates which var

11
Introduction to gradient-based
numerical optimization
There are several excellent texts on numerical optimization; for example, Nocedal and Wright, Numerical
Optimization, Springer-Verlag, 1999, and C T Kelley, Iterative Methods for Optimization, SIAM,

8
Theoretical and convergence aspects of
finite element methods
8.1
Criteria for monotonic convergence
To ensure monotonic convergence, the elements must be complete and the elements and mesh must be compatible.
In other words
Compatibility + Completeness

H
An Outline of
MSA History
H1
Appendix H: AN OUTLINE OF MSA HISTORY
TABLE OF CONTENTS
Page
H.1.
H.2.
H.3.
H.4.
H.5.
H.6.
H.7.
H.8.
H.9.
H.10.
INTRODUCTION
Background and Terminology
Prolog - Victorian Artifacts: 1858-1930
Act I - Gestation and Birth: 193

6
Finite element analysis of dynamic
problems
6.1
Derivation of weak form
To illustrate how the weak form of time-dependent problems can be constructed, consider the equations governing the axial vibrations of a cantilevered one-dimensional rod structure

AER501: Midterm Course Review
1
Finite element analysis
1.1
1.3
In AER501, we have so far dealt with the strong form of the equations governing the static response of rods and beams. The strong
form includes the governing equations AND all the boundary co

5
Finite element analysis of beams and
frame structures
A beam is a bar-like structural component whose primary function is to support transverse loading and carry
it to the supports. A beam resists transverse loading by bending. Under transverse loading,

AER501: Advanced Mechanics of Structures
UTIAS
University of Toronto Institute for Aerospace Studies
http:/utias.utoronto.ca/~pbn
Assignment 1 (7 pts)
Due October 12, 2015
Write a MATLAB code for computing the displacements, stresses and strains
of two-di