Four Point Bending From Theory to Practice and StandardsFour Point Bending From Theory to Practice...
Transcript of Four Point Bending From Theory to Practice and StandardsFour Point Bending From Theory to Practice...
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Four Point Bending
From Theory to
Practice and Standards
Ad Pronk
University of Delft; The Netherlands
2nd European 4PB Workshop
University of Minho, Guimarães
24-25 September 2009
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
CONTENTS• “”QUIZ””• 4PB Theory• CEN Standard
(Shear - Calibration - Damping - Device Stiffness)
• 4PB Practice (Calibration - Stiffness Modulus - Fatigue)
• Fatigue Damage Model – Partial Healing• Healing – Endurance Limit ?• Temperature
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
The following slides of a table are an idea of my colleague and friend Rien Huurman.
He presented these slides already on the 1st
4PB workshop which was held in 2007 at the Delft University of Technology.
In my view this example is an excellent “tool” to visualize the “mistakes” we can make.
I have extended his example a bit.
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LOAD, GEOMETRY & MATERIALThis is a top view of a table.What is the force in each leg?This is not a joke!
Rien Huurman1st 4PB Workshop
Ad Pronk ; Delft University of Technology
2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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LOAD, GEOMETRY & MATERIAL
This is the 20 cm x 20 cm load area. The load is 100 kg.
Ad Pronk ; Delft University of Technology
2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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LOAD, GEOMETRY & MATERIAL
There are
4 legs.
2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
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LOAD, GEOMETRY & MATERIAL
Indication of thedeformations and the stress within the material.
2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
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LOAD, GEOMETRY & MATERIAL
The materials!
Steel
Rubber
2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
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LOAD, GEOMETRY & MATERIAL
“HOLY TRINITY”: LOAD GEOMETRY (structure) MATERIAL Including Boundary/Time Conditions
2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
1) Load 2) Geometry
Response3) Material+ Boundary conditions
⎫⎪⎪⎪ ⇒⎬⎪⎪⎪⎭
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
???
1) Load 2) Geometry
Correct Material Model+
Response
⎫⎪⎪→⎬⎪⎪⎭
QUESTION
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
-1
-0,5
0
0,5
1
Uni-axial Push-Pull (UPP) Test
Force
Deflection
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Huet-Sayegh
Model
This model covers
already a wide
frequency range
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2nd European 4PB Workshop, 26/27-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
00,20,40,60,8
1 Force
Response
Permanent
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Modified
Huet-Sayegh
Model
Linear Dashpot in series
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Another Example: 4PB Fatigue testing with Haversine Deflection Signals
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
0
0.5
1
1.5
2Haversine
Force
Deflection
You want to have this loading
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
0
0.5
1
1.5
2
Time
Def
lect
ion
-1
-0.5
0
0.5
1
Forc
eDeflection
Force
But this is what you get !!!!
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Why don’t you get what you want?
REASON
Asphalt is a viscous-elastic-plastic material
There are two causes why you can’t remain the desired wave shape for deflection & force
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
1) Any static component in the applied loadwill lead to permanent deformation.
The Result: Within a few cycles you have a a bended beam and the force becomes a sine.In order to get back to a zero deflection you need to pull/push back the bended beam.
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
2) When a constant deflection is applied
(as in a Haversine signal), the Force (or better the stress in the material)to obtain this deflection level willdisappear (relaxation)
The Result: A STRESS FREE BENDED BEAM
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
ASTM “Constant Strain Haversine”
Using the total strain (peak-peak value) an endurance limit of around 100 μm/m or more is calculated !!!!.
But, according to me, this is not correct and the endurance limit is half this value: 50 μm/m
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Four Point Bending: THEORY
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
A prismatic beam resting on 4 supports
The inner supports can move up and down
When the top of the beam is in tension the bottom will be in compression and vice versa
The centre line of the beam will keep itsoriginal length
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Boundary Conditions for Beam Theory
• Free rotation at supports
• Free horizontal translation at supports
• No vertical deflection at outer supports
• Neutral “fiber” halfway the beam
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Boundary Conditions
•Deflection V = 0 at x = 0 & x = L
•Moment M = 0 at x = - Δ & x = L + Δ
•Shear Force D = 0 at x = - Δ & x = L + Δ
•1st derative of V = 0 at x = L/2
•Only Steady State Solution
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Applied Load
Sinusoidal Point Load at the two inner supports: F = Fo .Sin(ω t)
Response is a delayed deflection:
V.Sin(ω t + ϕ )
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
At this point my first thoughts were to deal with the background of the differential equations for the bending of a slender beam: Thimoshenko.
But I’m afraid I will loose your attention.
Nevertheless you will see quite a lot of slides with equations. But I will go through them quickly and only point out some details.
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
2 2 2 4*
2 2 2 2 2{ , } { , } { , }b b bE I V x t B H V x t I V x tx x t x t
ρ ρ⎧ ⎫∂ ∂ ∂ ∂
+ − +⎨ ⎬∂ ∂ ∂ ∂ ∂⎩ ⎭
Ψ
4 2 2*
* 4 2 2{ , } { , } { , }{ , } b bB H I V x t E I V x t Q x t
G B H t t xρ
ρ⎡ ⎤⎧ ⎫∂ ∂ ∂
+ − =⎨ ⎬⎢ ⎥∂ ∂ ∂⎩ ⎭⎣ ⎦
Ψ2 * 2 2 2*
2 2 2 2
{ , } { , }sG B HI E V x t
B Ht x t xρ
ρ⎡ ⎤ ⎡ ⎤∂ ∂ ∂ ∂
− − +⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦
Ψ2 2 2
* *2 2 2{ , } { , } { , }s
IG B H V x t E Q x tB Ht t x
ρρ
⎡ ⎤∂ ∂ ∂+ = −⎢ ⎥∂ ∂ ∂⎣ ⎦
General Differential Equations
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
b bE I V x t bh V x t Q x tx x t
2 2 2*
2 2 2( , ) ( , ) ( , )ρ⎡ ⎤∂ ∂ ∂
+ =⎢ ⎥∂ ∂ ∂⎣ ⎦
* * iE E .e ϕ=
BENDING DEFLECTION Vb
{ } { }2
2mass total massF t Mass V x ,tt
∂=
∂
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
( ) { }
Ψ
Ψ
sG b h V x t Q x tx
EG b h b h
2*
2
**
{ , } ( , ) ( , )
& ,2 1
αμ
∂= −
∂
= =+
SHEAR DEFLECTION Vs
{ } { }2
2mass total massF t Mass V x ,tt
∂=
∂
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
( ) ( )
( )
( ) ( )
( )
Δ Δ
Δ Δ
t t
nt
t
t t
nt
t
ASin n Sin nL LF
Q x t Sin tL xSin n
L
ACos n Cos nL LF
Q x t SinL xCos n
L
0
1
0
1
2 1 2 12
( , ) ( )
2 1
2 22
( , ) (
2
π π
ω
π
π π
ω
π
∞
=
∞
=
⎡ ⎤⎧ ⎫⎛ ⎞ ⎛ ⎞+⎪ ⎪− − − ×⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎢ ⎥⎝ ⎠ ⎝ ⎠⎩ ⎭= ⎢ ⎥
⎛ ⎞⎢ ⎥× −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤⎧ ⎫⎛ ⎞ ⎛ ⎞+⎪ ⎪− ×⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎢ ⎥⎝ ⎠ ⎝ ⎠⎩ ⎭= ⎢ ⎥
⎛ ⎞⎢ ⎥× ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
∑
∑ t)
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
b bE I V x t bh V x t Q x tx x t
2 2 2*
2 2 2( , ) ( , ) ( , )ρ⎡ ⎤∂ ∂ ∂
+ =⎢ ⎥∂ ∂ ∂⎣ ⎦
* * iE E .e ϕ=
BENDING DEFLECTION Vb
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
{ } { } { } { }( )= + +i t i tb a c dV x e V x V x V x eω ω
Solutions Homogenous dif. Equation
Particular Solution satisfying the load condition
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
( ) ( )
( ) ( )
( )
( )
( )
Δ Δ
n
t t
n nt
an
Sini Arctg
Cos
t
ASin n Sin nL L
n Cos
F LV x
E I
xSin n eL
2
22 1
2 44
2 42 1 2 13
04 *
1
2 1 2 1
2 1 1 2
2{ }
2 1
ϕ
ωϕξ
π π
ω ωϕ
ξ ξ
π
π −
− −∞
= ⎛ ⎞⎜ ⎟⎜ ⎟
− ⎜ ⎟⎜ ⎟−⎜ ⎟⎝ ⎠
⎡ ⎤⎛ ⎞ ⎛ ⎞+− − −⎢ ⎥⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠⎢ ⎥×⎢ ⎥⎢ ⎥− − +⎢ ⎥⎢ ⎥
= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
⎛ ⎞⎢ ⎥× − ×⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥
⎣ ⎦
∑
( )4 422 1 4
2 1−
−=
*
nt
E In.
L bhπ
ξρ
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
{ }
( ) ( )
( )
( )
{ } ( ) ( ) ( )Δ Δ
nia n n
n
tn
n n
n
n
nt t t
V x A T x e
F LA
E I n Cos
SinArctg
Cos
A xT x Sin n Sin n Sin nL L L
*2 1
2 1 2 11
30
2 1 2 444 *
2 42 1 2 1
*2 1 2
22 1
2 1
{ }
2
2 1 1 2
2 1 2 1 . 2 1
ϕ
ω ωπ ϕ
ξ ξ
ϕϕ
ωϕξ
π π π
−
∞−
− −=
−
− −
−
−
−
⎡ ⎤= ⎣ ⎦
=
− − +
⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟
−⎜ ⎟⎝ ⎠
⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞+= − − − −⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠
∑
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
( ) i
c n tn
V x C Cos x L e 42 1 0
1{ } 2
ϕ
β∞ −
−=
⎛ ⎞= −⎜ ⎟
⎝ ⎠∑
( ) i
d n tn
V x D Cosh x L e 42 1 0
1{ } 2
ϕ
β∞ −
−=
⎛ ⎞= −⎜ ⎟
⎝ ⎠∑
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
( ) ( )i i
n t n tC Cos L e D Cosh L e4 42 1 0 2 1 02 2
ϕ ϕ
β β− −
− −
⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
{ } ( )
( )
Δ Δ
Δ
nii
n n n t
i
n t
A T e C Cos L e
D Cosh L e
*2 1 4
2 1 2 1 2 1 0
42 1 0
2
2 0
ϕϕ
ϕ
β
β
−−−
− − −
−
−
⎛ ⎞+ −⎜ ⎟
⎝ ⎠⎛ ⎞
+ − =⎜ ⎟⎝ ⎠
Boundary Conditions
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
{ } { }
( )
( )
nib a n n
n
i
c n tn
i
d n tn
V x V x A T x e
V x C Cos x L e
V x D Cosh x L e
*2 1
2 1 2 11
42 1 0
1
42 1 0
1
{ }
{ } 2
{ } 2
ϕ
ϕ
ϕ
β
β
−
∞−
− −=
∞ −
−=
∞ −
−=
⎡ ⎤= = +⎣ ⎦
⎛ ⎞+ = −⎜ ⎟
⎝ ⎠
⎛ ⎞+ = −⎜ ⎟
⎝ ⎠
∑
∑
∑
Complete solution without extra mass forces
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
( ) { }
Ψ
Ψ
sG b h V x t Q x tx
EG b h b h
2*
2
**
{ , } ( , ) ( , )
& ,2 1
αμ
∂= −
∂
= =+
SHEAR DEFLECTION Vs
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
( ) ( )
( )
( ) ( )
( )
Δ Δ
Δ Δ
t t
nt
t
t t
nt
t
ASin n Sin nL LF
Q x t Sin tL xSin n
L
ACos n Cos nL LF
Q x t SinL xCos n
L
0
1
0
1
2 1 2 12
( , ) ( )
2 1
2 22
( , ) (
2
π π
ω
π
π π
ω
π
∞
=
∞
=
⎡ ⎤⎧ ⎫⎛ ⎞ ⎛ ⎞+⎪ ⎪− − − ×⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎢ ⎥⎝ ⎠ ⎝ ⎠⎩ ⎭= ⎢ ⎥
⎛ ⎞⎢ ⎥× −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤⎧ ⎫⎛ ⎞ ⎛ ⎞+⎪ ⎪− ×⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎢ ⎥⎝ ⎠ ⎝ ⎠⎩ ⎭= ⎢ ⎥
⎛ ⎞⎢ ⎥× ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
∑
∑ t)
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
{ }
( )( )
{ } ( ) ( ) ( )Δ Δ
*2
*2
2 21
2*0
2 222 *
2
{ }
2 12
2
2 2 . 2
n
n
ig n n
n
tin ni
t
nt t t
V x H U x e
LFH e
L E e b h n
A xU x Cos n Cos n Cos nL L L
ϕ
ϕ
ϕ
μϕ ϕ
π α
π π π
∞−
=
−
+
⎡ ⎤= ⎣ ⎦
+= ⇒ =
⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞+= −⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠
∑ { } 0,eV x t H=
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Applied Solution for Back Calculation
• First order approximation for Vb
• Neglect overhanging beam ends (Δ=0)
• Neglect in first instance the deflection Vs
→
Solution is similar to the solution of a viscous- elastic spring mass system with complex spring constant K and an equivalent mass Mequiv
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Applied Solution for Back Calculation
{ }{ }
( )
( )
{ }
{ }{ } { }
{ }
*0 13
0
4
* 2 2 40 0
*
3
2
,1 2
i t
b
equiv
R xA xSin
F L eV x tR x E I Cos
R x
Sin
E IM xK x
L
LK x
L
ω ϕ
ϕ ζ ω
π
π
ζ ω
ζ
π
−
=⎛
=− +
= =
⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Replacing R{x} by R*{x} of the pseudo-static
solution leads to smaller deviations.
{ }*2 2
12 1
3 3
LR xA x x A
L L L
=⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Back Calculation
{ } { } { }( ) { } { }( )
{ }( ){ }( ) { }
{ } { } { }
{ } { }{ } { }
{ }( )
32* *0
*
*
*
2. 0
0
*,*
. 24
,
1 2b
bequiv
i mass ibeamequiv
mass i
F LE Cos x Z x Z x
V x R x I
Sin x
Cos x Z x
V xZ x M x
F
R A M xMM x R x
R x
ϕγ
ϕϕ
ϕ
ω
π
= + +
=+
=
⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥= +⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠
∑
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
CEN Standards
“Don’t shoot the pianist, he is doing his best”
I wasn’t allowed to finish the ‘job’
• Deflection Vs due to shear
•
Calibration - Reference beams - Damping Term
• 4PB Device Stiffness
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
In the EN standards of CEN only the deflection Vb
is used for the determination of Smix . The deflection
due to shear forces is not taken into account.
The effect of extra mass inertia forces is “dealt” with
but the coupling with the shear deflection is ignored.
How to deal with this ?
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Using the complete analytical solution
is not practical for routine back calculations.
So, another solution (patch) is needed.
Proposal: Use the pseudo-static solution as a
starting point.
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Online correction for deflection Vs due to shear
( )( )
( )
b t
LVs HL L AVb
L HA ProblemL
L LV V
2
2 2
2
4 123 4
2
36 1???
3 23
0.962 2
μα
υ αα
⎧ ⎫⎨ ⎬ +⎩ ⎭ =⎧ ⎫ −⎨ ⎬⎩ ⎭
+ ⎛ ⎞= → =⎜ ⎟⎝ ⎠
⎛ ⎞ ⎧ ⎫≈ ⎨ ⎬⎜ ⎟⎝ ⎠ ⎩ ⎭
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Procedure for the estimation of α
Compare the analytical solution with 3D
finite element calculations.
Problem: How to “load” a 3D beam
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Constant Deflection Amplitude
The constraints are due to ‘touching’ the beam
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Is it possible to bend a 3D beam without
physically touching it?
Answer: Yes, by applying shear forces in
the interfaces at the locations of the
supports (“fooling” ABACUS) and by
using the principle of St. Venant
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2nd European Workshop on 4PB, 24-25 September 2009, University of Minho, Guimarães, Portugal
1st Step 2nd Step
Deviations
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2nd European Workshop on 4PB, 24-25 September 2009, University of Minho, Guimarães, Portugal
-25.0
-16.7
-8.3
0.0
8.3
16.7
25.0
0.00 0.05 0.10 0.15Shear stress [MPa]
Hei
ght [
mm
]
mid clamp response
1st response at clamp
2nd response at clamp
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2nd European Workshop on 4PB, 24-25 September 2009, University of Minho, Guimarães, Portugal
Cowper 1966 (10+10μ)/(12+11μ) 0,8517
Timoshenko 1974 5/6 0,8333
Timoshenko 1922 (5+5μ)/(6+5μ) 0,8710
Olson 1935 (20+20μ)/(24+15μ) 0,9231
Pickett 1945 24,612(1+μ)/(29,538+5,942μ+64,077μ2) 0,8419
Tanji 1972 (6+12μ+6μ2)/(7+12μ+4μ2) 0,9354
Pai 1999 5/(6+(μ/(1+μ))2(H/B)4[1-
(90/π4)Σ{Tanh(nπB/H)/(n5(π(B/H))}])
0,8269
Hutchinson 2001 5(1+μ)/(6+5μ-(μ/(1+μ))2(H/B)4[1-
(90/π4)Σ{Tanh(nπB/H)/(n5(π(B/H))}])
0,8763
ABACUSABACUS 20092009 Comparison analytical solutions and finite element Comparison analytical solutions and finite element calculationscalculations
0,8590,859
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
In the former slides I have dealt with most
of the theoretical aspects of the 4PB test.
Is that enough for the practical application
of the 4PB test?
In theory YES but in practice NO.
Calibration of the 4PB device is required !!
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
At the moment an European calibration
4PB test is “running” and it turns out to
be very complex and time consuming.
For calibration you need a reference material with a KNOWN stiffness modulus
E.g. Aluminum: 71.3 GPa
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
• The product of E.I is the relevant parameter.
• Calibrate over a wide range of EI values.
If calibration fails due to not traceable errors
• Try to model your device with FEM
•
Introduce an extra term: Damping correction coefficient.
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Lesson I learned:
Never underestimate calibration efforts
Sometimes calibration seems to be overdone in view of the natural variation in material properties of asphalt mixes.
But it is, at least in my view, necessarily to carry out high quality calibrations in order to exchange results between laboratories and to investigate (new) material models.
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
In a glance I showed you some theoretical aspects of the 4PB device and how to deal with these aspects.
When I was writing the EN standards I tried to ‘help’ 4PB device owners by introducing a so called damping term. Sometimes you have checked “everything” but still a deviation is present in the system.
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
The damping term is meant for these cases.
E.g. You back calculate too high E values in your calibration tests. In spite of all efforts the ‘error’ is not traceable but you have the idea that there is too much friction at the supports.
For such a case I introduced the damping term which, in my view, is allowed after extensive testing (different frequencies etc.)
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Too low back calculated E values
In principle you can use a ‘negative’ damping term, but I advise you to check the clamping forces and “device stiffness”.
Commonly a default clamping force is used in all cases. However, I think you should verify this using reference beams with different beam stiffness's E.I.
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
4PB Device Stiffness
The bending beam theory is based on the assumption that you have an infinitely stiff bending device. But of course this is not true.
Using a relative deflection measurement (supports resting on the beam) this wouldn’t matter. But in case of an absolute deflection measurement you should correct for the device stiffness by using a very, very stiff beam.
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
“PRACTICE”
• Calibration Protocol
• Stiffness Modulus Measurements
• Presentation of Fatigue Results
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Calibration Protocol
•
Perform at least once a year a complete calibration test with at least three reference beams (representing E values for asphalt ranging from 6 to 12 GPa).
•
Perform in front of each 4PB project a routine stiffness test with a “DELRIN” beam and start a data base of the obtained figures.
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Calibration Protocol (Extra)
Even the E value of a reference beam is not known ‘exactly’. Therefore, given the small applied strains, a very good indication (or verification) can be obtained by tests like:
• Pulse velocity measurement
• Resonance frequency measurement
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Stiffness Modulus - EN Standards
• Frequency sweep
• Limit number of cycles & strain amplitude
• Practice Experience: ε < 50 μm/m
• No indication of fatigue damage
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Frequency Sweep Test
•
If possible perform a frequency sweep test in front of each 4PB (fatigue) test.
•Test the beam in controlled deflection mode.
• Limit the strain amplitude to 40 – 50 μm/m.
•
End the frequency sweep test with the start frequency (e.g. 1-2-4-6-8-10-1 Hz)
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Fatigue Measurements – EN Standards
• Why only a ε6 strain value?
•
Why are fatigue lives Nf,50 different when measured with different devices?
→ Need for Fatigue Damage Models
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
For Type Testing only the strain value ε6 is required in the EN standard.
This is strain amplitude for which the life time Nf,50 is 106 cycles.
So, the whole Fatigue characteristics are ‘combined in a single data point !!!
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Personally I felt this a big omission.
At least the slope of the Wohler curve should be given as well.
Furthermore 2 temperatures should be used.
A confidence interval should be included.
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
In the EN standards different devices are allowed for the determination of the stiffness
and fatigue characteristics of asphalt.
Stiffness is not a problem but Fatigue is !
Nf,50 is different for different devices.
How to solve the problem?
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Fatigue Damage Models
Starting point has to be a MATERIALMATERIALMATERIAL model
But in practice you measure the response of a SPECIMENSPECIMENSPECIMEN
Conclusion: The geometry of the specimen plays a large role
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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Uni-Axial Push-Pull Test
UPP
Herve DiBenedetto
In theory the response of the specimen is equal to that of the material
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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Force
Maximum
Strain
Two Point Bending Test
2PBLCPC, France
OCW, Belgium
Specimen Response
is not equal to
Material Response
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
The same is true for 4PB testing
•
Fatigue damage is less in outer sections (smaller strains and stresses).
•
Fatigue damage decreases from top and bottom inwards the beam and is theoretically nil at the ‘neutral’ fiber.
⇒ MATERIAL DAMAGE MODEL
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Partial Healing (PH) Model•
It is a MATERIALMATERIAL model. After integration over the stress/strain field in the specimen the response for the SPECIMENSPECIMEN is obtained
•
Equations for the SPECIMEN response are not the same as the PH equations and can only be determined in a numerical way
•
Equations of PH model are used for a fit on the SPECIMEN response in order to get SEED values for the numerical determination
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
However,
The shape for the evolutions of the phase lag and weighed stiffness modulus of the specimen using the PH material model parameters in combination with a numerical integration over the specimen is nearly the same as the one obtained by a regression using the PH model equations directly for the specimen response.
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t dQ tF t F e do d{ } .( ){ } . .1 10
τ β τα γ ττ
− −⎡ ⎤= − +∫ ⎢ ⎥⎣ ⎦
t dQ tG t G e do d{ } .( ){ } . .2 20
τ β τα γ ττ
− −⎡ ⎤= − +∫ ⎢ ⎥⎣ ⎦
Storage Modulus G{t}=Smix .Cos(ϕ)
Loss Modulus F{t}=Smix .Sin(ϕ)
disdis
Wd dQ t W t f F tdt dt T
π εΔ 2
0{ } { } { }=⎡ ⎤⎣ ⎦2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Healing – Low Endurance Limit
Can the PH model explains Healing?
Answer: Only Partially!
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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Beam 26-02
2000
3000
4000
5000
6000
7000
8000
0 40000 80000 120000 160000Number of Load cycles
Stiff
ness
mod
ulus
[MPa
]
Measured "Predicted"
Fitting on 1st
Load period only
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
The PH model underestimate the amount of Healing.
Given the “Strength/Power” of the PH equations what will be the evolution if the first 2 load periods are used for the
fitting?
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
3000
4000
5000
6000
7000
8000
0 40000 80000 120000 160000Number of Load cycles
Stiff
ness
mod
ulus
[MPa
]
Measured "Predicted"
Fitting on 1st & 2nd
Load period
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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A Healing project is submitted to the Centre of Transport and Shipping [DVS] (former DWW) of the Ministry for Transport, Public Works and Water Management.
The project includes 5 asphalt mixes which will be tested at 20 oC and 30 Hz using a rest- load ratio of 10 and different load periods.
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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What might be a reason that the PH model underestimates the amount of healing?
1. There is at least a second reversible damage term with a small β value. It can’t be determined in continuous tests.
2. Irreversible Fatigue damage occurs above a certain strain/stress level → Low Endurance Limit
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
LOW ENDURANCE LIMIT
“Does it exists?”
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
Does the PH model includes
an endurance limit?
In the first version: NO !
Regardless the value for the applied strain
amplitude permanent damage will occur
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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In the (original) PH model the following assumption is made:
The reversible and irreversible fatigue damages per cycle can be written as the
products of CONSTANTS times the Dissipated Energy per cycle.
→ α1 , α2 , β, γ1 , and γ2 are constants
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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How can we investigate the validity of this assumption:
One possibility is carrying out UPP Tests.
E13D 180-17 α1 α2 β γ1 γ2Individual fit 83 1187 51440 44 127
E13D 80-14 α1 α2 β γ1 γ2
Individual fit 82 850 50120 3 17
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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0
20
40
60
80
100
120
140
0 50 100 150 200
Strain amplitude [μm/m]
Para
met
er v
alue
γ1 &
γ2
Endurance Limit ?
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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Consequences of an Endurance Limit
If an Endurance Limit exists it may explain the underestimation of the healing effect.
Also the damage will be more concentrated in the top and bottom of the mid span of the beam. It has to be investigate what the influence of this non-homogeneity will be on the assumption: constant deflection means constant strain !!!
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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TEMPERATURE
The viscous dissipated energy per cycle Wdis = π σ ε Sin(ϕ) in a fatigue test is ‘completely’ transformed into Heat.
Due to this the temperature will increase and as a consequence the Stiffness modulus will decrease. How serious is this?
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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E13D 80-14
0,00,10,20,30,40,50,60,7
0 1000000 2000000 3000000 4000000 5000000
N
Tem
pera
ture
incr
ease
[o
C]
Centre oCEdge oC
Temperature Increase During Fatigue
UPP Test ε = 82 μm/m
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
E13D 180-17γ =0.604 ; δ=− 1.43 Ε(−05)
0,0
0,5
1,0
1,5
2,0
2,5
0 20000 40000 60000 80000 100000
N
Incr
ease
in
Tem
pera
ture
[oC
]Centre oCEdge oC
Forced Convection
Heat Transfer Coefficient h = 40 W/m2/K
UPP Test ε = 172.5 μm/m
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
E13D 180-17
0200040006000
8000100001200014000
0 20000 40000 60000 80000 100000 120000
N
Stiff
ness
[Mpa
]
0,0
0,5
1,0
1,5
2,0
2,5
Tem
pera
ture
In
crea
se [o
C]
Decrease in Stiffness due to
Temperature
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
E 13D 180-17H eat T ran sfer h =4 kg /s3 /o K
0,0
1 ,0
2 ,0
3 ,0
4 ,0
5 ,0
6 ,0
7 ,0
0 50000 100000 150000
N
Tem
pera
ture
Incr
ease
[o
C]
Centre oC
Edge oC
Free Convection instead of Forced Convection
Heat Transfer Coefficient h = 4 W/m2/K
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
E 13D 180-17H eat T ransfer h =13 kg /s3/o K
0,0
0,5
1,0
1 ,5
2 ,0
2 ,5
3 ,0
3 ,5
4 ,0
0 50000 100000 150000
N
Tem
pera
ture
Incr
ease
[o
C]
Centre oC
Edge oC
“Bad” Forced Convection
Heat Transfer Coefficient h = 13 W/m2/K
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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4PB test, ε = 200 μm/m, T = 00C
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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Source: LCPC; Chantal de La Roche
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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Conclusion/Recommendation
Use Force Convection in the 4PB cabin
Check by temperature measurements the effectiveness of the cooling (using temperature
couples inside the specimen)
The heat ‘creation’ is related to: σ.ε or (ε 2), Temperature increases fast with higher strains
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
Ad Pronk ; Delft University of Technology
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2nd European 4PB Workshop, 24/25-09-2009, Univ. of Minho
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