Week 9_Introduction to FEA

8/3/2019 Week 9_Introduction to FEA

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Introduction to FEA

Prof. M. Abdel WahabProfessor of Applied Mechanics

Ghent University, Laboratory Soete, Belgium


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Introduction to FEA

What is FE?

Numerical procedures for analysing structures

and continuum.Why do we need FE?

The problem is too complicated to be solvedanalytically (exact solution).

How does FE work?FE procedures produces simultaneous algebraicequations solved by digital computer.


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Introduction to FEA

What type of problems?

Static Thermal

Electro-Magnetic Transient

Acoustics


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Applications of FEA

FE procedures are used in the design of:

Civil Engineering constructions


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Applications of FEA

Mechanical engines Bio-medical


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Applications of FEA

Bridges


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Element type

Type Shape Unknowns

bar u, v, (w)

beam u, v, qz

, (w, qx

, qy

)

plane u, v

axisymmetric u, v

shell u, v, w, qx, q

y, (q

z)

solid u, v, w

x, u

z, w

y, v

qy

qz

qx


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Principle of FE

Element

Node

FE code determines the

displacement at every node [u]

that minimises the total

energy. From these

displacements the strains [e]and stresses [s] can be found.[e] = [B] [u][s

] = [D] [e] = [D] [B] [u]

From energy principles:

[K][u]=[F] FE solutions are always wrong!


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Finite Element Guidelines

Results can often be incorrect:

incorrect constraints

incorrect loading

incorrect finite element mesh

Check that the approximated results areacceptable


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Mesh density

refine where the changes of stress are highest

e.g. crack tip


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Convergence

mesh refinement requires more CPU

a compromise between accuracy of results and

computing time

convergence can be assessed from two analyses withincreasing mesh refinement


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Are the results correct?

Check deformed shape, do you expect that?

Check reaction forces, are they equal and opposite to the

applied load?

Sum all nodal forces, are they equal to zero?

Calculate a simple analytical solution, is it consistent withthe FE results?


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How does FEM work?

Partial differential equations

a)Writing the variational equivalent or weak

form

b) Discretisation of space

A system of simultaneous algebraic equations


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Finite Element Method (FEM)

A simple bar element

a)Weak form

b) Discretisation of space

FE equation

02

2

f

xuEA

0)(2

2

dxf

x

uEAx

UNu T

FUK


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STRESS ANALYSIS

Linear elastic

Hydraulic Manifold

3-D solid elementsIn-service pressure

68 MPa

FUK

UBe es D


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STRESS ANALYSIS

Non-linearity

Iterative solution

Incremental solution

Road sweeping brushLarge deformation

Penetration of 50 mm

FUUK )(

FUUK )(


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Material non-linearity

Autofrettage(manufacturing barrels of handguns

and cannons)

Mandrel residual hoop stress

2-D axsymmetric modelOverstrain

STRESS ANALYSIS

%100

i

im

R

RR


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HEAT TRANSFER ANALYSIS

Governing differential equation

FE equation (transient analysis)

Numerical integration

Steady state analysis

2

2

2

2

2

2

z

T

y

T

x

T

c

K

t

T

p

qTKt

TC

t

tTttT

t

T

)()(

qTK


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HEAT TRANSFER ANALYSIS

Steady state analysis

Railko Marine Bearing

Two layers: a backing and a liner

2-D plane heat transfer elements

Inside Ti=100o

C, outside To=30o

C


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Governing differential equation

Heat transfer

Lap strap CFRP

Transient analysis

DIFFUSION ANALYSIS

2

2

2

2

2

2

z

T

y

T

x

T

c

K

t

T

p

2

2

2

2

2

2

z

c

y

c

x

cDt

c

/sm103.9213aD

/sm106.3213CFRPD


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Multi-physics analysis

Thermal-stress analysis of the Railko bearing

COUPLED-FIELD ANALYSIS

Tt e

refTTT


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Modelling of macro-crackCrack faces should coincident

LEFM

EPFM

Composites

FRACTURE MECHANICS ANALYSIS

ru r1

sr

1

e

1

1

m

m

re

11

1

mr

s

ru 1 e r


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Fatigue crack initiationDamage variable D

Evolution of Din adhesive layer used for scarf joint

FATIGUE ANALYSIS

1

1

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mV

m

eqNRmAD

s


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Fatigue crack propagationA modified Paris law

FATIGUE ANALYSIS

2

1

max

max

max

1

1

n

c

n

th

n

G

G

G

G

DGdN

da


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Fatigue crack propagation

A single lap joint

Total strain energy release rate

(Gmax or GT)

Mode I strain energy release rate

(GI)

FATIGUE ANALYSIS


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FE equation - generalised equation of motion

Un-damped free vibration

Solution

FE equation becomes

Eigenvalue problem is solved using iterative numericalsolution

MODAL ANALYSIS

FUKUCUM

0 UKUM )sin( tUU nm

0][][ 2 mn UMK


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Free vibration of a pre-stressed concrete bridge,B14, between the villages Peutie and Melsbroekand crosses the highway E19

3-D finite element model

MODAL ANALYSIS


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FE equation - generalised equation of motion

Explicit direct numerical integration

TRANSIENT DYNAMICS ANALYSIS

FUKUCUM

)()(2

1)( ttUttU

ttU

)()(2)(1)(2

ttUtUttUt

tU


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A footbridge, Wilcott bridgeFRP composite

Shrewsbury, UK

3-D beam elements

Excited using a walking

Pedestrian (1.6 Hz)

BS5400: F(t)=180sin(.t) N

TRANSIENT DYNAMICS ANALYSIS


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Governing partial differential equation -LWR (Lighthill,Witham and Richards) model

FE equation

A simulation of a 5 km road

BC at x=0

TRAFFIC FLOW ANALYSIS

0

x

ku

t

ko

0

kBt

k

A


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The Finite Element Method is a generic technique

Many disciplines and applications

FEA is well established numerical technique

However, research and development into FEM is still

going on in order to improve accuracy, facilitate the userinteraction with the software and further implementation ofspecific applications.

CONCLUSIONS


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QUESTIONS

Week 9_Introduction to FEA

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