Lecture 1 Basic Concepts of FEM
Transcript of Lecture 1 Basic Concepts of FEM
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19/07/2007 1
Lecture 1 The Basic Concept of
the Finite Element Method
Yan Zhuge
CIVE 3011 Structural Analysis andComputer Applications
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FEM Thefinite element method(FEM) is a computer
based procedure that can be used to analysestructures and continua.
FEM is based on the idea of building a
complicated object with simple blocks, ordividing a complicated object into small andmanageable pieces.
Common applications include static, dynamicand thermal behaviour of physical systems, andtheir components.
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FEM The results obtained with a finite element
analysis are rarely "exact." Nevertheless, a veryaccurate solution can be obtained if a proper
finite element model, based on principles of finite
element analysis, is used.
Example:
i
R
Approximation
of the area of a
circle
h
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Engineering Application we are concerned with the effects of forcing
functions (loads, fluid pressure etc.) on systems in several instances the problem addressed is too
complicated to be solved satisfactorily byclassical analytical methods (due to irregulargeometry, non-homogeneous media and arbitraryloading conditions etc.)
The finite element method, which is based on the
concept of discretisation finds use here Finite element method is probably the most
widely used form of computer-based engineering
analysis.
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Applications of FEM in
Engineering
Mechanical/Aerospace/Civil Engineering
Structure analysis (Static/dynamic,linear/nonlinear)
Thermal/fluid flows Geomechanics
Biomechanics
....
Examples:
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Curved Beam
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Building
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Beach Chair
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Picnic Table
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Bridge Maximum Deflection: 17. 6513mm in y-direction
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Wind Loads Maximum Deflection: 1.5847mmin z-direction
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Suspension Bridge
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The Bus Shelter
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Computer Simulation of 9/11
Attack
http://www.youtube.com/watch?v=gH02Eh44yU
g Structural engineers need to know from a
scientific perspective what happened to the
buildings during the terrorist attacks in order toprevent future failures.
The search for answers continues with the help of
a state-of-the-art animated visualization createdby researchers at Purdue University.
What is your comment?
http://www.youtube.com/watch?v=gH02Eh44yUghttp://www.youtube.com/watch?v=gH02Eh44yUghttp://www.youtube.com/watch?v=gH02Eh44yUghttp://www.youtube.com/watch?v=gH02Eh44yUg -
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A Brief History of the FEM 1943 - Courant (Variation methods)
1956 - Turner, Clough, Martin and Topp(Stiffness)
1960 - Clough ("Finite Element", plane
problems) 1970s - Applications on mainframe computers
1980s - Microcomputers, pre- andpostprocessors
1990s - Analysis of large structural systems
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Steps of Finite Element Analysis
Define the type of analysis. Many programs
provide modulus for different types of analysis,for instance, static or dynamic analysis.
Define the type/types of elements to be used inthe analysis. Typical element types are truss,
beam, plane stress, plane strain, plate and shellelement.
Define the location of each node in a global
coordinate. Connect the elements at the nodes to form an
approximate system for the whole structure.
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Steps of Finite Element Analysis
Define the boundary conditions of the problem.
Apply the loads on the structures. A wide varietyof loading conditions can be applied to astructure.
Assign material properties. Again, more than onematerial property may be used in a finite elementmodel.
Execute the input file and to produce the results.
Post-process results.
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Objectives of the Course Understand the basic theory of the FEM
Know the behaviour and usage of eachtype of elements covered in this course
Have some hand on experiences in solvingvarious simple engineering problems byFEM
Can interpret and evaluate the quality ofthe results
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Elements and NodesFinite elementsresemble fragments of
the structures. Nodesappear on elementboundaries and serve asconnectors that fastenelements together. AllElements that share anode have the samedisplacementcomponents at that node
for frame and truss structures,
elements and nodes are more
or less natural.
for elastic continuum, such as a deep beam or a plate /shell
structure, such a natural subdivision does not exist and we have toartificially divide the continuum into a number of elements.
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Additional NodesAdditional nodes may be inserted:
when we require results at more locations or at locations inbetween the member ends
When members are not prismatic
1
2 3 4 5 6
1 2 3 4 5
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Artificial Elements These artificial elements, called finite element, are
usually either triangular or rectangular in shape as shown
below:
Superficially, it appears that a FE structure can beproduced by sawing the actual structure apart and then
pinning it back together at nodes.
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Responsibility of the User FE computer programs have become
widely available, easier to use, and candisplay results with attractive graphics. It
is hard to disbelieve FE results because ofthe effort needed to get them and the polish
of their presentation. However, smooth
and colourful stress contours can beproduced by any model, good or bad.
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Responsibility of the User Responsibility for results produced is taken by the
engineerwho uses the software, not the software vendor,even if results are affected by errors in the software.
FE modelling requires that the physical action of theproblem be understood well enough to choose suitable
kinds of elements, and enough of them, to represent thephysical action adequately. When the computer has donethe calculations, we must always check the results to seeif they are reasonable. Modelling and errors will befurther discussed in the following lectures.
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An Example
Examine the computed
displacements first
FE calculates nodaldisplacements, then uses the
displacement information to
calculate strains and finally
stresses
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A recent example - ArupThe problem - cracking and excessive deflection in
a new 3 storey concrete-framed structure
The design was based on a 3D computer analysis package
the causes of the problems
Torsion in Concrete were not considered
Application of Loading the forces were not compatible with thebehaviour of the actual structure
Construction sequence Construction process must be considered
Member properties - the effects of cracking and creep must beconsidered
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The procedure for FE analysis:
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Review of Matrix Algebra
Linear System of Algebraic Equations
11212111 ... bxaxaxa nn =+++
22222121 ... bxaxaxa nn =+++
nnnnnn bxaxaxa =+++ ...2211
...
where x1, x2,....., xn, are the unknows.
In matrix form:
Ax = b
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Review of Matrix Algebra
[ ]
==
nnnn
n
n
ij
aaa
aaa
aaa
aA
...
............
...
...
21
22221
11211
{ }
==
n
i
x
x
x
xx:
2
1
{ }
==
n
i
b
b
b
bb:
2
1
where
A is called a n x n (square) matrix, andx andb
are (column) vectors of dimensions n.
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Review of Matrix Algebra
Row and Column Vectors
[ ]321 vvvv =
=
3
2
1
w
w
w
w
Matrix Addition and SubtractionFor two matrices A andB, both of the same size (m x n),
the addition and subtraction are defined by
C = A + B with cij = aij + bij
D = A B with d ij
= aij
bij
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Review of Matrix Algebra
Scalar Multiplication
ijaA =
Matrix Multiplication
For two matrices A (of size l x m) andB (of size m x n),
the product ofAB is defined by
C = AB
=
=m
k
kjikij bac1
with
where i = 1,2,...,l; j = 1,2,...,n.
In general, AB BA, but (AB)C = A(BC)
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Review of Matrix Algebra
Transpose of a Matrix
IfA = [aij], then the transpose ofA is A
T = [aji]
Note that (AB)T = BTAT
Symmetric Matrix
A square (n x n) matrix A is called symmetric, if
A = AT or aij = aji
Unit (Identity) Matrix
=
1...00
............
0...10
0...01
INote that AI
= A, Ix = x
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Review of Matrix Algebra
Singular Matrix
A square matrix A is singularif det A = 0, whichindicates problems in the systems
Matrix Inversion
For a square and nonsingular matrix A (det A 0), itsinverse A-1 is constructed in such a way that
AA-1 = A-1A = I We can show that (AB)-1 = B-1A-1
If det A = 0 (A is singular), then A-1 does not exist!
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Review of Matrix Algebra
Positive Definite Matrix
A square (n x n) matrix A is said to bepositive definite,if for any nonzero vectorx of dimension n,
xTAx> 0
Note that positive definite matrixes are nonsingular
Differentiation and Integration of a Matrix
Let )()( tatA ij=
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Review of Matrix Algebra
then the differentiation is defined by
=
dt
tdatA
dt
d ij )()(
and the integration by
= dttadttA ij )()(