Ch 1 Introduction 09

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MS5019 – FEM 1

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MS5019 – FEM 2

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1.1. Introduction

The Finite Element Method (FEM) is an versatile

and powerful mathematical tool that has wide

applications in a multitude of physical problems

such as stress analysis, fluid flow, heat transfer,

acoustics, aero-elasticity, micro-fluidics, MEMS

(Micro-Electro-Mechanical Systems), electricaland magnetic fields, electrostatic coupling and

many others.

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A. Formal Definition of FEA:

An approximate mathematical analysis tool to study

the behavior of a continua (or a system) to an external

influence such as stress or strain, heat, pressure,temperature, fluid velocity, magnetic field, etc.

This involves generating a mathematical formulation

of the physical process followed by a numerical

solution of the mathematics model.



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B. History of FEA:

− Force method

− Displacement method

1955

1956

Argyrys-Denke

Argyris-Turner

Modern FEM

1953Levy & Garvey Matrix method:

Force method in aircraft industry

1940Courant Approximation by “finite elements”

1908

1915

Ritz

Galerkin

Approximation method

1864

1878

Maxwell

Castigliano

Energy theorem

1819 Navier Hyper-static structure

Figure 1-1(a) Historical background to modern FEM, after J.F. Imbert [2]

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Engineers Mathematicians

Trial functions Finite differences

Variational

methods

Weighted

residualsRayleigh 1870

Ritz 1909

Gauss 1795

Galerkin 1915

Biezeno-Koch 1923

Richardson 1910

Liebman 1918

Southwell 1940

Structural analogue

substitution

Piecewise continous

trial functionCourant 1943

Prager-Synge 1947Hrenikoff 1941

McHenry 1943

Newmark 1949

Direct continuum

elements

Variational finite

differencesArgyris 1955

Turner et al1. 1956

Varga 1962

Modern FEM

Figure 1-1(b) Historical background to modern FEM, after O.C. Zienkiewics [3]



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C. Basic Concept:

Division of a given domain into a set of simple sub-

domains called finite elements accompanied with

polynomial approximations of solution over each

element in terms of nodal values.

Assembly of element equation with inter-element

continuity of solution and balance of force isconsidered.

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1.2. Basic Illustration

1. FE Discretization• Each line segment is an element

• Collection of these line segments is called a “mesh”

• Element are connected at nodes

2. Element equations

R

22 sin( )e θ H R=

R

θ

S eA. Circumference:

Rθ

H e



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3. Assembly of equations and solution

1

2For , 2 sin( ), 2 sin( )

n

e

e

e

P H

π π π θ H R P nR

n n n

=

=

= = =

∑

0 0

As , 2

1 sin( )If 2

0

sin( ) cos( )2 2 2

1lim lim x x

n P π R

π x x P R

n x

n x

π x π x R π R π R

x→ →

• → ∞ =

• = ⇒ =

• → ∞ ⇒ →

⎡ ⎤ ⎢ ⎥⎛ ⎞ ⎛ ⎞• ∴ = =⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥

⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

( )

Error, 2 sin

Total Error 2

e e e

e

π π E S H R

n n

nE π R P

⎡ ⎤⎛ ⎞= − = − ⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦

= = −

4. Assembly of equations and solution

5. Error Estimation

1,03354E-076,2831910000

1,03354E-056,283171000

0,0010334926,28215100

0,102845426,1803410

6,2831853072,5E-161

nEePn

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B. Frame Structure:

Figure 1-2 Example of discretization of a frame structure by FEM

(a) Real structure (b) Discretized structure



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C. Continuous problem:

Figure 1-3 Descritization of an elasticity 2D continuous problem by FEM

(a) Continuous problem

(b) Discrete model

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1.3. General Step in the FEM

Derive the element stiffness

matrix and equations

Define the strain-

displacement and stress-

strain relationship

Select a displacement

function

Discretize and Select

Element Types

Based on the concept of stiffness influence

coefficients (direct equilibrium method, work or

energy method, weighted residual method.)

Step 4

Both relationships are necessary for deriving the

equations for each element.

Step 3

Choosing a displacement function within each

element

Step 2

Dividing the body into an equivalent system of

finite elements with associated nodes and

choosing the most appropriate element type.

Step 1



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Interpret the results

Solve for the element strains

and stresses

Solve for the unknown

degrees of freedom (or

generalized displacements)

Assemble the elementequations to obtain the

global equations and

introduce boundary

conditions

The final goal is to interpret and analyse the

results for use in the design/analysis process.

Step 8

For the structural stress-analysis problem, strains

and stress (or moment and force) can be

obtained.

Step 7

Global equations obtained from step 5 is a set of

simultaneous algebric equations. These

equations can be solved by using an elimination

method (Gauss’s method) or an iterative method

(Gauss-Seidel, etc.)

Step 6

Individual element equations generated in step 4is added together using a method of

superposition (called the direct stiffness

method ).

Step 5

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n : total number of nodes

Element

Global

Level of

FormulationNodal

Displ.

Defor.

Energy

Work of

Ext. forcesStiffness

Matrix

Nodal

Forces

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=

k

j

i

d

d

d

d

⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

⎬

⎫

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

⋅

⋅

=

n

k

j

i

d

d

d

d

d

d

M

M

1

eeeeU dKd

T

21= eee Fd

T

=ℑ

Virtual Work Principle

FddKd

U

d

TT

0

δ δ

δ δ

δ

=

ℑ=

≠∀

eK eF

AssemblageK F

Linear Equation

System

FdK =

dSolution

ei

k

j

ei

k

j u j

v j

⎭⎬⎫

⎩⎨⎧

= j

j

j v

ud

KddT U 21= FdT =ℑ



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Pointelement

0D

Frameelement

Trusselement1D

(LineEle-

ment)

GeometryNameClass

Figure 1-4 (a) Different type of elements

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Coque

Bendingplate

Elasticity 2D(tin) shell

2D(PlanEle-

ment)

GeometryNameClass

Figure 1-4 (b) Different type of elements



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Coqueaxisymetric

Torusaxisymetric

Axi-symet

ric

GeometryNameClass

Figure 1-4 (c) Different type of elements

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Gap element that have stiffnessonly for compression direction.

Special Element

Thick Coque

Volume3D

(Volu

meEle-ment)

GeometryNameClass

Figure 1-4 (d) Different type of elements



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1.4. Analysis Type

Non-linear dynamic

Direct integration step by step

Modal

Dynamic response

− Modal superposition

− Direct integration step by step

Dynamic

Static Non-linear

Non-linear stability

Linear static

Initial stabilityStatic

Non-linearLinear Analysis

FKq =

[ ]G λ+ =K K X F

[ ]

2, λ λ ω− = =K M X 0

( )t + + =Mq Cq Kq F&& &

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1.5. Computer Code

Start

Input Data

FE modeling

Element

Characteristics

Ke, Fe

• Assemblage

• Restraints

K, F

Solution LES

q

Element’s stress

calculation

End

Sub program for

matrix calculation

Element’s

Library

Print Result

Figure 1-7

Simplified flowchart

for static analysis

(displacement method)



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1.6. Application Structural areas:

Stress analysis, including truss and frame analysis both for

structural and non-structural concentration problems typically

associated with holes, fillets, or other changes in geometry in

a body.

Buckling problem

Vibration analysis

Non-structural problems: Heat transfer

Fluid flow, including seepage thtough porous media

Distribution of electric or magnetic potemtial

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References:

1. Logan, D.L., 1992, A First Course in the Finite ElementMethod, PWS-KENT Publishing Co., Boston.

2. Imbert, J.F.,1984, Analyse des Structures par

Elements Finis , 2nd Ed., Cepadues.3. Zienkiewics, O.C., 1977, The Finite Eelement Method,

3rd ed., McGraw-Hill, London.

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Ch 1 Introduction 09