Lecture 1 Basic Concepts of FEM

7/28/2019 Lecture 1 Basic Concepts of FEM

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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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[ ]

==

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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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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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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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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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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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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then the differentiation is defined by

=

dt

tdatA

dt

d ij )()(

and the integration by

= dttadttA ij )()(

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Lecture 1 Basic Concepts of FEM