MEC 3500 - FINITE ELEMENT ANALYSIS I
Credits:
Lectures/Tut.:
Labs:
Prerequisite:
Leads to:
4
2hr/wk
3 sessions
MEC 2401, MEC 3402
MEC 4405
Syllabus
Part 1 (Exam – 50%):

Introduction to the theory of the finite element method – discretization of the problem, elements and
nodes, general approximations, symmetry and boundary conditions

Interpolation functions for different type of elements – 1-2-3 dimensional elements

Formulation and solution of the finite element system equations for elasticity problems, 1-2-3
dimensional elements, the axisymmetric case.
Part 2 (Continuous assessment - 50%):

Introduction to commercial finite element programs – pre-processing, material non-linearity, geometric
non-linearity, buckling problems, transient response problems, mesh generation, model validation,
boundary conditions, loading, solving the system equations, post-processing.

Practical examples in 1-2-3 dimensional problems in stress analysis, heat transfer, fluid mechanics,
dynamics.
Assessment
Exam 50%
Continuous assessment
50%
Reference texts:
Finite element analysis, theory and practice – M.J.Fagan
Finite element analysis, theory and application with Ansys – S.Moaveni
Basic principles of the finite element method – K.M.Entwistle
Concepts and applications of finite element analysis – R.D.Cook, D.S.Malkus, M.E.Plesha
The finite element method, volumes 1,2,3 – O.C.Zienkiewicz, R.L.Taylor
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
1
LECTURE 1
INTRODUCTION
WHAT IS FINITE ELEMENT
ANALYSIS ?
 A MATHEMATICAL TOOL TO
SOLVE PROBLEMS






STRUCTURAL
HEAT TRANSFER
FLUIDS
DYNAMICS
ELECTROMAGNETIC
ELECTRICAL CIRCUITS
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
2
 A NUMERICAL TECHNIQUE THERE ARE SOME APPROXIMATIONS
INVOLVED IN THE SOLUTIONS
 IN F.E.A. THE ACCURACY OF THE
SOLUTION DEPENDS ON THE
WAY THE OBJECT IS MODELLED
MATHEMATICALLY
 TYPE OF ELEMENT
 LOADING
 BOUNDARY CONDITIONS
WHAT IS REQUIRED TO USE
FINITE ELEMENT ANALYSIS ?
 SOUND ENGINEERING
KNOWLEDGE
 PROPER UNDERSTANDING OF
THE PROBLEM
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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IN THIS PART OF THE MODULE :
 WE WILL PUT PARTICULAR
EMPHASIS ON THE PRACTICAL
SIDE OF F.E.A.
 WE WILL NOT BE
PROGRAMMING F.E.A. FROM
SCRATCH BUT WILL BE USING
ANSYS AS OUR F.E.A. PACKAGE
 NO NOTES – JUST PAY
ATTENTION IN CLASS
 TEXTBOOK – Finite element analysis –
Theory and Practice – M.J.Fagan
 ANSYS help files
 GOOD
IDEA
TO
READ
THE
TEXTBOOK BEFORE ATTENDING
CLASS
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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AN ENGINEERING PROBLEM (e.g. Beam
under bending) NORMALLY REQUIRES
FINDING THE DISTRIBUTION OF AN
UNKNOWN VARIABLE – Temperature,
Displacement, stresses, etc.
MAIN STEPS IN F.E.A.
1. DISCRETISATION
OF
THE
PROBLEM
 Divide the model into elements –
different type of elements
(solid/beam/plate)
 Elements are connected at nodes
– we must have an appropriate
number and an appropriate
distribution of elements
 The unknown variable is
assumed to act over each
element in a predefined manner
– linear element v.s. quadratic
element – this leads to step 2
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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2. SELECTION OF THE
APROXIMATING FUNCTION
(e.g. for the displacement)
 In ANSYS we have elements of
different order for each type
(solid/beam/plate) of element.
3. APPLY
LOADS
AND
BOUNDARY CONDITIONS
4. SET
UP
THE
SYSTEM
EQUATION – GENERALLY IT IS
OF THE FORM [K]{u}={F}
 [K] is the stiffness matrix
 {u} is the vector of unknowns
 {F} is the vector of applied
nodal forces
5. SOLVE
THE
SYSTEM
EQUATION to obtain the unknown
variables at each node – In ANSYS
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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we have a choice of different
solvers
6. CALCULATE THE DERIVED
VARIABLES – strains, stresses,
heat flow
STEPS 1TO 4 – PRE-PROCESSING
STEP 5 – SOLUTION
STEP 6 – POSTPROCESSING
SAMPLE ANALYSIS
2-D CANTILEVER – Solid
elements, beam elements
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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HELP
FILE
TOUR
Ansys Command Reference – Explain some commands for keypoints, lines,
volumes, etc – show equivalent in menu system – use them to build a model.
Ansys Element Reference – Go through ‘Element input’, ‘Solution output’,
‘Co-ordinate systems’ folders & explain all terms there.
Element pictorial summary – go through some elements
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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 THE HELP FILES ARE YOUR
CONTINUOUS
POINT
OF
REFERENCE WHEN USING
ANSYS
 YOU CAN USE THE MENU
SYSTEM OR THE COMMAND
SYSTEM
 TO LEARN, START WITH
THE MENU SYSTEM BUT FOR
CLASS
TESTS
IT
IS
COMPULSARY TO USE THE
COMMAND LINE
WORK TO DO:
READ
 THE OPERATIONS GUIDE
(1.5 hrs)
 THE
BASIC
ANALYSIS
PROCEDURE GUIDE (5 hrs)
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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THE IDEA IS TO GET AN
OVERVIEW OF WHAT CAN BE
DONE – DO NOT GO INTO TOO
MUCH DETAIL. – TRY TO READ 1
HOUR EACH DAY UNTIL THE
NEXT LAB SESSION
Next lecture ask me on any difficulties
– we will then start on examples.
Contact Lab officer to get user account
so that you can start working from
today.
7 lab sessions – 14 hours
7 class lectures – 14 hours
1 group assignment -10% - 10 hours
3 tests – 40% - 26 hours private study
1 theoretical exam – 50% - 36 hours
private study
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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SOME MORE INSTRUCTIONS :
 ANSYS CREATES A NUMBER
OF FILES WHILE IT IS BEING
USED
 IT IS BEST TO MAKE ANSYS
WORK IN THE TEMP
DIRECTORY OF YOUR
WORKSTATION
 TEMP HAS READ & WRITE
PERMISSIONS FOR ALL
USERS.
 CREATE A SUBDIRECTORY
WITHIN THE TEMP FOLDER.
 CHANGE THE WORKING
DIRECTORY FROM THE
ANSYS LAUNCHER TO YOUR
DIRECTORY IN THE TEMP
FOLDER.
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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 WHEN YOU HAVE FINISHED
WORKING IN THE
TEMPORARY DIRECTORY
MOVE ALL IMPORTANT
FILES TO YOUR HOME
DIRECTORY ON THE
SERVER
 KEEP A BACKUP OF THE
INPUT TEXT FILE ON
FLOPPY DISCS.
 NORMALLY YOU DO NOT
NEED THE FOLLOWING
FILES AND YOU CAN
REMOVE THEM TO SAVE
SOME DISK SPACE:
file.err – error & warning messages
file.tri – triangularized stiffness
matrix
file.esav – element matrices
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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etc….
 RUN THE FOLLOWING
EXAMPLES AND TRY TO
BECOME FAMILIAR WITH
THE APDL COMMANDS USED.
 I ENCOURAGE YOU TO
EXPLORE DIFFERENT
OPTIONS AVAILABLE IN THE
POSTPROCESSOR IN ORDER
TO VIEW RESULTS.
 ADVENTOUROUS STUDENTS
CAN CHANGE THE GEOMETRY,
BOUNDARY CONDITIONS AND
LOADING AS THEY WISH OR
ELSE CARRY OUT SOME
DIFFERENT PROBLEM THAT
THEY CAN THINK ABOUT
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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EXAMPLE 1 – Cantilever using beam elements
/prep7 !enter preprocessor
!
!Parameter definition
E=200E3 !Young's modulus
NU=0.3 !Poisson's ratio
length=1000
depth=100
area=depth*1
izz=1*depth*depth*depth/12
force=-650
!
!Choose element type - plane stress unit thickness
et,1,beam3
r,1,area,izz,depth
ex,1,e
nuxy,1,nu
!
! Build f.e.model of cantilever
k,1,0,0
k,2,length,0
l,1,2
lesize,1,,,10
lmesh,all
!
!Apply boundary conditions & load
nsel,s,loc,x,0
d,all,all,0
nsel,s,loc,x,length
f,all,fy,force
nsel,all
!
/solu
solve
!
/post1
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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!
!plot bending moment diagram
etable,imoment,smisc,6
etable,jmoment,smisc,12
plls,imoment,jmoment
!
!plot bending stress variation at top surface
etable,imaxbs,ls,2
etable,jmaxbs,ls,5
plls,imaxbs,jmaxbs
EXAMPLE 2 – Cantilever using 2-D solid elements
/prep7 !enter preprocessor
!
!Parameter definition
E=200E3 !Young's modulus
NU=0.3 !Poisson's ratio
length=1000
depth=100
force=-650
!
!Choose element type - plane stress unit thickness
et,1,plane82
ex,1,e
nuxy,1,nu
!
! Build f.e.model of cantilever
k,1,0,0
k,2,length,0
k,3,length,depth
k,4,0,depth
l,1,2
l,2,3
l,3,4
l,4,1
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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al,1,2,3,4
lesize,2,,,8
lesize,4,,,8
lesize,1,,,20
lesize,3,,,20
amesh,all
!
!Apply boundary conditions & load
nsel,s,loc,x,0
d,all,all,0
nsel,s,loc,x,length
nsel,r,loc,y,depth/2
f,all,fy,force
nsel,all
!
/solu
solve
EXAMPLE 3 – Point load acting on plate
/prep7 !enter preprocessor
!
!Parameter definition
E=200E3 !Young's modulus
NU=0.0 !Poisson's ratio
width=200
hlength=500
force=-100
!
!Choose element type - plane stress unit thickness
et,1,plane82
ex,1,e
nuxy,1,nu
!
! Build f.e.model of cantilever
k,1,0,0
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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k,2,width,0
k,3,width,hlength
k,4,0,hlength
l,1,2
l,2,3
l,3,4
l,4,1
al,1,2,3,4
! The following lines can be used to vary the mesh density and do a mesh convergence
! study
!lesize,2,,,32
!lesize,4,,,32
!lesize,1,,,18
!lesize,3,,,18
amesh,all
!
!Apply boundary conditions & load
nsel,s,loc,y,0
d,all,uy,0
nsel,s,loc,y,hlength
nsel,r,loc,x,width/2
f,all,fy,force
nsel,all
!
/solu
solve
FEA 1 – Dr Martin Muscat 2003 © - Lecture 1
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