introduction to fem-mongi ben ouezdou

7/27/2019 Introduction to Fem-mongi Ben Ouezdou

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M. Ben Ouezdou, University of Nizwa, 2011 1

INTRODUCTION TO FINITE

ELEMENT METHOD

Professor Dr. Mongi Ben Ouezdou


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Basic Concept

Building a complicated object with simpleblocks,

or

dividing a complicated object into small

and manageable pieces.

M. Ben Ouezdou, University of Nizwa, 2011


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The structure is considered as anassemblage of a finite number of individualstructural components called elements.

These elements can be put together in a

number of ways,represent complex geometry.

Basic Concept



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FEM based on Principle of discretization

=

procedure in which a complex problem of

large extent is divided (discretized) intosmaller equivalent units.

Basic Concept



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Application

Application of this idea can be found everywhere in

everyday life and in engineering.

Examples:

Lego (kids play) aircraft



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Examples

beam bridge



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Steel frames

Concrete building

Application to buildings



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Advantages of the FEM

1- Incorporate diff properties of each element.2- No restriction for the shape of the medium;

hence arbitrary and irregular geometries

cause no difficulty.

3- Accommodation of any type of BC.

4- Handle non-linearities, time-dependant Pb.5- Valid for any engineering Pb.



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Some History of the FEM 1943: Courant (Variational Methods);

1956: Turner, Clough, Martin and Trop (Stiffness);

1960: Clough (Finite Element, plane problems);

1970s:Applications on mainframe computers;

1980s: Micocomputers, pre- and postprocessors;

1990s:Analysis of large structural systems.



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Some Terminology FE: small elements (1D, 2D, 3D) obtained by

subdividing the given domain to be analyzed.

Nodes or nodal points: intersections of the sidesof the elements.

Nodal lines and nodal planes: interfaces

between elements. Linear elements: FE with straight sides.

Higher order elements: FE with curved sides.

Primary unknowns: nodal displacements Secondary unknowns: strains, stresses,

moments, shear forces, etc.



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Basic procedure

Step 1: Discretise the continuum: subdivide into elements: generate a mesh;

Step 2: Select element displacement functions;

Step 3: Calculate element properties: stiffness matrix [k].

Step 4: obtain element load vector [F];

Step 5: Assemble element properties (element stiffnesses global stiffness,

load vector).

Step 6: Incorporate B.C. (set the element to the ground so disp = 0 or finite):the stiffness matrix developed in step 5 will be modified to realize the

condition that disp of some coordinates = 0 or finite.

[F] = [K] {u} and {u}=[K]-1{F}. [K]: global stiffness matrix, [F]: vector of known forces

and {u}: displacements. Step 7: Determine displacements, strains and stresses

Step 8: Check and iterate to eliminate precision errors if present.



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Discretization

=

Process of separating the length, area or volumeinto discrete (separate) parts or elements.

structure

1-D elements 2-D elements 3-D elements Axisymmetric

elements



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1-D elements

Used for beams or frames

Basic element

Node 1 Node 2

H-element

1 23

Curved element

3

1

2



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2-D elements

Used ,for plane-wall, diaphragm, slab, shell, etc.

1- Triangular elements

3 nodes 6 nodes

1 2

3

14 2

3

6 5

1

4

2

3

5

6

6- nodes curved triangle

- Triangular elements are the most used ones

- Curved elements for 2-D domain with curved boundaries



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2-D elements

2- Quadrilateral elements



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Used for the analysis of solid bodies (stresses underfoundation, contact stress under point loads, etc).

3-D elements

Tetrahedron Hexahedron Curved 3D element

Problem: Complex visualization and stiffness matrices size can be enormous.



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Axisymmetric elements

Used in problems that are axisymmetric in nature.

Can achieve huge simplification in axisymmetric problems.



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Guidelines for discretization Discretization is a major decision making step in FEM.

Simple structures: no problems. Most real structures: difficulties in

processing of subdividing the structure;

Numbering the nodes;

Assigning coordinates to each node;

Relating the structure coordinate numbers to elements numbers and

their coordinate number.

In most FEM Software: discretization is handled automaticallyby the preprocessing module of the software.



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Examples of discretization

1 2 3 4

1 2 3 4

Point loads:

5 nodes and 4 elements

Stepped beam:

5 nodes and 4 elements

1 2 3 4

1 23 45 5



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Examples of discretization

Nodal line

Change in loading

Material 1

Material 2

Change in material



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Size of elements when discretize

In recent past, the number of elements was limited due to

capabilities of both hardware and software. But nowadays a

sufficient fine discretization of a whole structure can easily beproduced very Quickly by graphical preprocessors.

Most new FE softwares provide automatic mesh generation;

But this tool should not be used in an uncaring manner:engineering knowledge is still required.

An inadequate modeling of apparently irrelevant details (e.g. small

cantilever slab or opening in a slab) can lead to faulty result and

unsafe design. A sufficiently FE mesh should be used in regions of high deformation

pr stress gradients.



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FE Equation



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Element stiffness matrix of a prismatic bar:

Direct method

[F] = [k] [u]

Load vector stiffness displacement

Node 1 Node 2

1, u12, u2A,E

x

=

2

1

1

1

1111

u

uEA

l



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Element stiffness matrix of a prismatic bar:

Formal approach

[F] = [k] [u]

Node 1 Node 2

1, u12, u2A,E

x

( )=V

TdVBEBk : element stiffness matrix, with

=

ll

11B

[ ]

= 11

11

l

EA

k

Same result !

Use conservation of energy: strain energy = work done by nodal forces

where



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Global stiffness matrix for a prismatic bar

1 3

E, A

x

E, 2A

P21 2

Example:

Solution: Use two 1-D bar elements

Element 1:

Find stresses in the 2 bars

[ ]

=11112

1l

EAk

u1 u2

Element 2: [ ]

=1111

2l

EAk

u2 u3

Global stiffness matrix [ ]

=

110

132

022

l

EAK

u1 u2 u3



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=

3

2

1

3

2

1

110132

022

FF

F

u

u

uEA

l

Global FE equation

Use B.C. and Loads condition: u1 = u3 = 0. and F2 = P.

=

0

0

110

132

022

F

P

F

2

3

1

uEA

l

{ } [ ] { }23 uEA

Pl

=EA

Pu

32

l=

A

P

EA

PEuu

EE 303

12

11=

=

==

l

ll

A

P

EA

PEuuEE

3302322 =

=

==

l

ll


0


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2211 vuvu

[ ]

2

2

1

1

22

22

22

22

v

u

v

u

scsscs

csccsc

scsscs

csccsc

EAk

= l

c = cos

s= sin

Bar element in 2-D

in global coordinates

l

x, u

Y, v

1

2

x, u02

u01

E, A

y, v01

v02

Element stiffness matrix in

global coordinates


El t tiff t i f b


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M1, 1

2211 vv

l

=

2

2

1

1

22

3

2

2

1

1

4626612612

2646

612612

v

v

EI

M

F

M

F

llll

ll

llll

ll

l

1 2

F1, v1 F2, v2

M2, 2

x

E, I

Element stiffness matrix of a beam:

Direct method

Element stiffness equation:


El t tiff t i f b


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Element stiffness matrix of a beam:

Formal approachStrain energy stored in, the beam element

Stiffness matrix for the simple beam element: k

=l

0

T BBk dxEI

with the strain-displacement matrix B:

+++=

232232

6212664126B

llllllll

xxxx

=

llll

ll

llll

ll

l

4626

612612

2646

61261222

3

EIkobtain the same result


Local Stiffness matri of a general 2 D beam element


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Local Stiffness matrix of a general 2-D beam element:

=

llll

llll

ll

llll

llll

ll

EIEIEIEI

EIEIEIEI

EAEA

EIEIEIEI

EIEIEIEI

EAEA

k

460

260

612

0

612

0

0000

260

460

6120

6120

0000

22

2323

22

2323

222111 vuvu



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M2, 2

Element Stiffness matrix of a general 2-D beam

element in a global coordinate system:

l

1

2

x, u

y, v x0, u02

u01

E, A

y0, v01

v02

M1,

1



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+

+

+

+

++++

+

++++

+

+

++

=

llllll

llllllllll

llllllllll

lllllll

llllllllll

llllllllll

EIc

EIs

EIEIc

EIs

EI

cEI

cEI

sEA

csEI

csEA

cEI

cEI

sEA

csEI

csEA

sEIcsEIcsEAsEIcEAsEIcsEIcsEAsEIcEA

EIs

EIc

EAs

EIEIc

EIs

EI

cEI

csEI

sEA

csEI

csEA

cEI

cEI

sEA

csEI

csEA

sEI

csEI

csEA

sEI

cEA

sEI

csEI

csEA

sEI

cEA

k

466266

6121261212

612661212

266466

6121261212

6121261212

2222

2

2

3

2

32

2

3

2

3

23

2

2

2

23

2

3

2

2

2

222

232

322

32

3

23

2

3

2

33

2

3

2

222111

vuvu



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Computation of nodal displacements

{F} = [k] {u} {u} = [k]-1 {F}

This step needs the use of the computer (mainly if stiffness matrix exceeds 5 x 5)

because it needs to invert the matrix.

Solution: enforcing 0 displacement BC and solve by:

- Unit diagonal method;

- Large diagonal method;

- Row column delete method.



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Features of the assembled stiffness matrix

Calculation of primary unknowns

1) The stiffness matrix has its non-zeros terms along its

main diagonal (terms distant from the diagonal are 0).

2) Stiffness matrices are symmetric: advantage in storing

the matrices.

=

534000

376500

469320

053474

002763

000435

k

Half band width

=

005

037469

534

276435

k

Values to be stored

-Reduction of the

required storage memory

-Reduction of the

solution time


C l l ti f i k


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Calculation of primary unknowns

Solution of equations {u} = [k]-1 {F}

direct scheme:

Linear problems

(Gaussian elimination)

iterative scheme:

Non Linear problems

(Jacobi, Gauss Seidal)

The most known methods of solutions are:

- Choleskys square root methods;

- Halfband Gauss elimination solution technique;

- Skyline technqiue

- Frontal solution technique.



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Calculation of secondary unknowns

nodal displacement {u}

stresses {} : (Hookes law)

strain {}

Standard software give the output in a tabularform and on graphical form.

Results include:

- Refined colored graphics;- Direct stresses x, y;

-Shear stresses xy; maximum shear, etc.


Summary:


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Computer implementations

Preprocessing (build FE model, loads andconstraints);

FE Analysis solver (assemble and solve the

system of equations); Postprocessing (sort and display the results).

Summary:



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Conclusions:

Procedures of FEM in Structural Analysis

1) Divide structure into elements with nodes;

2) Describe the behavior of the physical quantities on eachelement;

3) Assemble (connect) the elements at the nodes to form anapproximate system of equations for the whole structure;

4) Solve the system of equations involving unknown quantitiesat the nodes (e.g., displacements);

5) Calculate desired quantities (e.g., strains and stresses) atthe selected elements.

N.B.: be aware of the limitations of the FEM: such as loadsapplication is imposed (no moving loads), and do notmisuse the FEM (it is a numerical tool).



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References1- Yijun Liu, Introduction to Finite Element Method,

Lecture notes, University of Cincinnati, Ohio, USA,1998.

2- R. Vaidyanathan, P. Perumal, ComprehensiveStructural Analysis, 2nd ed., Laxmi Publication ed.,

New Delhi, 2008.


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