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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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Finite Element Guidelines
Mesh density
refine where the changes of stress are highest
e.g. crack tip
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Finite Element Guidelines
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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Finite Element Guidelines
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
2/111
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