TS 4466 2 Credits
Finite Element Method
Instructor:
Wong Foek Tjong, Ph.D.
Course description

The course aims to enable the students to
understand the basic concepts and
procedures of the finite element method
(FEM) and to apply the FEM by using a
commercial software
 Teaches
understanding of how finite element
methods work rather than how to use a
software
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The instructor

Graduated from Universitas Parahyangan, Bandung in
March1994
Final project: Dynamic Analysis of Multi-degree-ofFreedom Structures Subjected to Ergodic Random Excitation

Graduated from Institut Teknologi Bandung in April1998
Master thesis: Active Vibration Control of Structures by Using
Artificial Neural Network Observer

Graduated from Asian Institute of Technology, Thailand
in May 2009
Dissertation: Kriging-based Finite Element Method for Plates
and Shells

Contact: wftjong@peter.petra.ac.id
P Building, Room P402B
Tel. 62-31-298-3391
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Course outline
1.
2.
Overview of the FEM
The direct stiffness method


3.
One-dimensional elements


4.

6.
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Bar, beam, torsional bar elements
Frame element in 3D space
Two-dimensional elements for plane-strain/planestress problems

5.
Spring and bar systems
Truss structures
Constant strain triangle element
Bilinear isoparametric quadrilateral element
Introduction to plate and shell elements
Applications of the FEM using SAP2000
4
References

D.L. Logan (2007)
A First Course in the Finite Element Method
the 4th Ed., Toronto, Nelson

D.V. Hutton (2004)
Fundamentals of Finite Element Analysis
New York, McGraw-Hill

R. D. Cook, D.S. Malkus, M.E. Plesha and R.J. Witt (2002)
Concepts and Applications of Finite Element Analysis
4th Ed., John Wiley and Sons

W. Weaver, Jr. and P.R. Johnston (1984)
Finite Elements for Structural Analysis
New Jersey, Prentice-Hall
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References (cont’d)

Computers and Structures, Inc. (2006)
CSI Analysis Reference Manual, Berkeley, CSI

C. Felippa (2008)
Introduction to Finite Element Methods
http://www.colorado.edu/engineering/cas/cours
es.d/IFEM.d/
R. Krisnakumar (2010)
Introduction to Finite Element Methods
http://www.youtube.com
(Video of lecture series on FEMs)

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Softwares

MATLAB Ver. 6.5
Strongly recommended software for matrix
computation and programming

SAP 2000 Ver. 11.0.0
For applications
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Grading weights
Homework assignments
 Mid-semester exam
 Take home test
 Final exam-- project

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15%
35%
15%
35%
8
Late coming to the class



The tolerance for coming late to the class is 20
minutes.
Those who come late more than 20 minutes are
NOT allowed to attend the class.
Please refer to the “FEM Lecture Plan” for more
academic norms
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Any question about the course
before we begin with the
Overview of the FEM?
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Discussion: the task of a structural
engineer

Let take a look on a typical job vacancy announcement
that you may read once you graduate from your study
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Discussion (cont’d)


Why do you think a design engineer is required to
master a structural analysis and design software?
An engineer needs to understand the behavior of a
structure so that he/she can make judicious decisions in
design, retrofitting, or rehabilitation of the structure
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Behavior of a
Real Structure
Simulation
Simplifications and
assumptions of the
real structure
Mathematical
Model
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Experiment
Replicate conditions of the
structure (possibly on a
smaller scale) and observe
the behavior of the model
Physical
Model
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An example of the FEM
applications
Real experiment
It is often expensive or
dangerous
FE simulation
It replicates conditions of
the real experiment
Source: W.J. Barry (2003), “FEM Lecture Slides”, AIT Thailand
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The need for modeling



A real structure cannot be analyzed, it
can only be “load tested” to determine
the responses
We can only analyze a “model” of the
structure (perform simulation)
We need to model the structure as
close as possible to represent the
behavior of the real structure
Source: W. Kanok-Nukulchai
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The idealization process for a
simple structure
Source: C. Felippa
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Mathematical
Models
Analytical Solution
Techniques
Closed-form
Solutions
Only possible for
simple geometries and
boundary conditions
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Numerical Solution
Techniques
•Finite difference methods
•Finite element methods
•Boundary element methods
•Mesh-free methods
•etc.
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Finite element method (1)


It is a computational technique used to obtain
approximate solutions of engineering problems.
In the context of structural analyzes, it may be regarded
as a generalized direct stiffness method.


The direct stiffness method you studied in MK 4215 Structural
Analysis III is actually the application of the FEM to frame
structures
It is originated as a method of structural analysis but is
now widely used in various disciplines such as heat
transfer, fluid flow, seepage, electricity and magnetism,
and others.
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Finite element method (2)

Modern FEM were first developed and applied by
aeronautical engineers, i.e. M.J. Turner et al., at Boeing
company in the period 1950s.

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1956: The first engineering FEM paper
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Finite element method (3)


The name “finite element method” was
coined by R.W. Clough in 1960. It is
called “finite” in order to distinguish with
“infinitesimal element” in Calculus.
1967: First FEM book by O.C.
Zienkiewicz
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Finite element method (4)


The computation is carried out automatically
using a computer or a network of computers.
The results are generally not exact.
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Example of applications in
structural engineering
1. Framed structures
(b) Grid
(a) Truss
Source: Weaver and Johnston, 1984
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Example of applications in
structural engineering (cont’d)
1. Framed structures (cont’d)
(c) Frame
(d) Arch
Source: Weaver and Johnston, 1984
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Example of applications in
structural engineering (cont’d)
2. Two-dimensional continua
(b) Plane strain
(a) Plane stress
Source: Weaver and Johnston, 1984
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Example of applications in
structural engineering (cont’d)
3. Three-dimensional continua
(a) General solid
(b) Axisymmetric solid
Source: Weaver and Johnston, 1984
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Example of applications in
structural engineering (cont’d)
4. Plate in bending
Source: Weaver and Johnston, 1984
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Example of applications in
structural engineering (cont’d)
5. Shells
(a) General shell
(b) Axis symmetric
shell
Source: Weaver and Johnston, 1984
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Example of applications in
structural engineering (cont’d)
The analysis and
design of buildings
The analysis of a double
curvature dam taking into
account soil-structure
interactions effects
Source: http://gid.cimne.upc.es/gidinpractice/gp01.html
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Example of applications in
structural engineering (cont’d)
The structural analysis
of an F-16 aircraft
The analysis of the Cathedral of
Barcelona using 3D solid elements.
(courtesy of Barcelona Cathedral)
Source: http://gid.cimne.upc.es/gidinpractice/gp01.html
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Discretization (1)


Fundamental concept is discretization, i.e. dividing a
continuum (continuous body, structural system) into a
finite number of smaller and simple elements whose
union approximates the geometry of the continuum.
Mesh generation programs, called preprocessors, help
the user in doing this work

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GiD, a software for pre and post processor
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Some basic element shapes
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Some 1st order (linear) elements
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Some 2nd order (quadratic)
elements
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Discretization (2)


One suggestion on performing discretization is
to divide structural regions with high stress
concentration into finer division e.g. in the
vicinity of the support and around the hole(s).
The accuracy of the results can be improved by
using a finer mesh (h-refinement) or using a
higher order elements (p-refinement).
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Examples of discretization (1)
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Examples of discretization (2)
z
y
x
h
D
Clamped
D=100, D/h =100
E = 2 x 106 ; ν = 0.3; k = 5/6
Load: uniform q = -1E-6
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76 nodes, 119 elements
172 active DOF
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Examples of discretization (3)
Cooling Tower– Nuclear Power Plant (taken
from a FEM Course Project of Doddy and
Andre, Dec 2008)
150 m
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Structural Model and Its Example
of the Analysis Results
The structure is divided into
smaller parts called “element”
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Membrane force contour in the
circumferential direction
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The FE Model with
a Finer Mesh
The structure is modeled with
a finer mesh
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The result is now better
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Examples of FEM software





For General purposes:
NASTRAN, ANSYS, ADINA, ABAQUS, etc.
For structural analysis, particularly in Civil Engineering:
SANS, SAP, STAAD, GT STRUDL, etc.
For building structures:
ETABS, BATS etc.
For geotechnical design:
PLAXIS
For conducting researches on earthquake engineering:
DRAIN-2D, DRAIN-3D, RUAOMOKO, OpenSees etc.
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Typical capabilities of a FE program

Data generation


Element types


Linear-elastic, nonlinear
Load types


E.g. SAP2000: Frame, Cable, Shell, Plane, Asolid, Solid, etc.
Material behavior


Automatic generation of nodes, elements, and restraints
Force, displacement, thermal, time-varying excitation
Plotting results

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Original and deformed geometry, stress contours
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Why do we need to study the basic
theory of FEM?


Cook, Malkus, and Plesha (1989, pp.6)
Concepts and assumptions behind the computer codes
(FEM software) should be mastered. Engineers are
expected to be able to use the software to gain better
advantages and will less likely misuse them.
SAP2000 disclaimer
The user accepts and understands that no warranty is
expressed or implied by the developers or the
distributors on the accuracy or reliability of the program.
The user must explicitly understand the assumptions of
the program and must independently verify the results.
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Any question before we proceed
to computational steps of the
FEM?
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Computational steps of the FEMthe direct stiffness method

Discretize the structure (problem domain)



Divide the structure or continuum into finite elements
Once the structure has been discretized, the
computational steps faithfully follow the steps in the
direct stiffness method.
The direct stiffness method:

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The global stiffness matrix of the discrete structure are obtained
by superimposing (assembling) the stiffness matrices of the
element in a direct manner.
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Computational steps… (cont’d)




Generate element stiffness matrix and element
force matrix for each element.
Assemble the element matrices to obtain the
global stiffness equation of the structure.
Apply the known nodal loads.
Specify how the structure is supported:
 Set
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several nodal displacements to known values.
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General steps of the FEM (cont’d)
Solve simultaneous linear algebraic
equation.
The nodal parameters (displacements)
are obtained.
 Calculate element stresses or stress
resultants (internal forces).

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Any question before we
continue to a brief introduction
to MATLAB?
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Example


Suppose you want to calculate the natural
frequency (Hz) of a SDOF system with the
mass m=100 kg and stiffness k=5 KN/m
The formula is

1
f 

2 2
k
m
Type in the Command Window:
>>m=100
>>k=5*1000
>> f=1/(2*pi)*sqrt(k/m)

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Edit/Debug Window
Click this icon to open a
new Edit/Debug window
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Array and matrix operations
Array operations are operations performed
between arrays on an element-by-element
basis, e.g.
>> A=[ 1 2; 3 4], B=[-1 3; -2 1]
>> A+B, A+4
 Common array operations:

 Array
multiplication (A .* B)
 Array right division (A ./ B)
 Array left division (A .\ B)  B in the numerator
 Array exponential (A .^ B)
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Matrix operations follow the normal rules of
linear algebra, e.g.
>> A=[ 1 2; 3 4]
>> B=[-1 3; -2 1]
>> A*B
 What is the different between A.^3 and A^3?

Be careful to distinguish between array operations and
matrix operations in your MATLAB code
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Solving a set of linear algebraic
equations
a11 x1  a12 x2 
 a1n xn  b1
a21 x1  a22 x2 
 a2 n xn  b2
an1 x1  an 2 x2 
 ann xn  bn
The equations can be written in matrix form
as follows:
Ax  b
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Solving a set of linear algebraic
equations (cont’d)
MATLAB command to solve the
equations:
>> x= A \ B (left division operator)
 Other commands related to linear
algebra:
>> det (A)
>> rank (A)
>> inv (A)

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Please look at the Matlab
Tutorial folder to learn more
about Matlab
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Homework (due date next class)
1.
Write an essay explaining (approx. 500
words):

What is finite element method?
 Why do you interested to take this course
(TS4466 Finite Element Method)? What do
you expect?
2.
Divide the following continuum into finite
elements:
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Homework (2)
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Homework (3)
3.
Divide this equilateral triangle into several quadrilateral
elements. You are not allowed to use a triangular
element.
4.
Solve the following simultaneous algebraic equations
using Matlab.
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a.
b.
500 x  250 y  250 z  88
250 x  800 y  150 z  66
400 x  150 y  400 z  44
500 x  250 y  250 z  88
250 x  800 y  150 z  66
125 x  400 y  75 z  44
Explain why the second equation (No. b) does not have
a unique solution? (Connect this fact with the
determinant and the rank of the coefficient matrix)
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