http://www-gci.insa-tlse.fr/lmdc/
Probabilistic Assessment of Corrosion Riskdue to Concrete Carbonation
Frédéric DupratAlain Sellier
Materials and Durability of Constructions LaboratoryINSA / UPS - Toulouse - France
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
CO2
CO2
CO2
CO2 pressure on external edges for most of concrete structures
CO2 ingress:carbonation
CO2
CO2
CO2
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
CO2 pressure on external edges for most of concrete structures
CO2 ingress:carbonation
Precipitation of calcite
CO2
CO2
CO2
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
CO2 pressure on external edges for most of concrete structures
CO2 ingress:carbonation
Precipitation of calcite
Dissolution of calciumfixed by cement hydrates
OHCaCOOH
COCa(OH) 232
22
CO2
CO2
CO2
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
CO2 pressure on external edges for most of concrete structures
CO2 ingress:carbonation
Precipitation of calcite
Dissolution of calciumfixed by cement hydrates
Decrease of pHin pore solution
CO2 pressure on external edges for most of concrete structures
CO2 ingress:carbonation
Precipitation of calcite
Dissolution of calciumfixed by cement hydrates
Decrease of pHin pore solution
Favourable conditions to initiation and development of corrosion
CO2
CO2
CO2
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
c
g2CO
D
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Physical parameters: - diffusion coefficient - concrete cover thickness
Predictingmodel
Mean values
Given date:depassivation
no depassivation
g2CO
D
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Physical parameters: - diffusion coefficient - concrete cover thickness
Predictingmodel
Given date:depassivation
no depassivation
Mean values
Random incertainties
c
c
(c)
Laws of probability
Physical parameters: - diffusion coefficient - concrete cover thickness
g2CO
D
Predictingmodel
Given date:depassivation
no depassivation
Mean values
Random incertainties
Laws of probability
Probabilistic approach
Given date:probability of depassivation
e
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Diffusion term Capacity term
��������������g g2 2
g g2 2CO CO
CO φ 1-Sr div D grad CO μt
Concentration
Porosity
Saturation
Diffusion
Sink term : precipitation of calcite
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
gCO2 Strongly non-linear term
Numerical instability around the carbonation
front
Change of variable
Agressivespecies
CO2g
Dissolved species
CaS
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005
Diffusion term Capacity term
��������������g g2 2
g g2 2CO CO
CO φ 1-Sr div D grad CO μt
Introduction
Carbonation modeling
Probabilistic approach
Results
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0 500 1000 1500 2000 2500
CO2g (mol/m3)
CaS (mol/m3)
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005
Strongly non-linear term
Numerical instability around the carbonation
front
Change of variable
Diffusion term Capacity term
��������������g g2 2
g g2 2CO CO
CO φ 1-Sr div D grad CO μt
Introduction
Carbonation modeling
Probabilistic approach
Results
Agressivespecies
CO2g
Dissolved species
CaS
( ) ( )
1
��������������
g2s
eq
g g2 2s g
2 sCOCas s s
a D
CO COφ 1-SrCaφ 1-Sr CO div D grad Ca
t Ca Ca Ca
All consumed CO2 reacts with CaS in hydrates
10-3< (a) <10-2
negligible (Deq) non-linear
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005
Diffusion term Capacity term
��������������g g2 2
g g2 2CO CO
CO φ 1-Sr div D grad CO μt
Introduction
Carbonation modeling
Probabilistic approach
Results
S
S
g2
COS Cagrad
Ca
CODdiv
t
Cag2
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005
Diffusion term Capacity term
Introduction
Carbonation modeling
Probabilistic approach
Results
G
CaSm
CaSM
Deqm
DeqM
Deq CaS
SCa
Deq(CaS)
L
Conservation of flowSCa
*eq
L
0 eq D
L
xD
dx
SM
Sm
Ca
Ca Seq
S
SmSM*eq
CaD
dCa
CaCaD
CaS(G)
Deq*
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005
S
S
g2
COS Cagrad
Ca
CODdiv
t
Cag2
Diffusion term
Introduction
Carbonation modeling
Probabilistic approach
Results
airCO0COCO g2
g2
g2
DtrDD
Influence of cracking
Reference diffusion
Tortuousity, connectivity of cracks
Tension volumic strain
Gazeous diffusion
Magnifying the diffusion coefficient
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Equivalent CaS diffusion coefficient field
Mechanical strain field
Loading and mechanical properties
Magnified CO2 diffusion coefficient field
Physical properties: CO2 diffusion coefficients
tortuousity, saturation degree
Initial condition:CaS=2500 mol/m3
Initial equivalent CaS diffusion coefficient field
Boundary condition:CaS=0 along the edges
t=t0
Solid calcium field: CaS
Convergencefor CaS field ?
no t=tf ?
yes
t=t +t
no
Start
Endyes
Practical application: reinforced concrete beam
6 m 25 cm
5.2 kN/m
55 cm
Eb 35000 MPa
0COg2
D
airCOg2
D10-8 m2/s
1.39.10-5 m2/s 0.5 0.15Sr 0.3
Carbonation profiles
1month 5 years 20 years 35 years 50 years
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Carbonation depth
Zone saineCas=2500mol/m3
(b) Zone carbonatée Cas <<2500mol/m3(a) ½ section centrale de la poutre
A
A
B
B
(c) Evolution du profil en calcium noncarbonaté entre les points A et B enfonction du temps
Position de l’armature
Carbonated zoneCaS<< 2500 mol/m3
Non-carbonated zoneCaS= 2500 mol/m3
A
BA B
Non-carbonated CaS profiles between
A and B
Eb 35000 MPa
0COg2
D
airCOg2
D10-8 m2/s
1.39.10-5 m2/s 0.5 0.15Sr 0.3
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005
Practical application: reinforced concrete beam
0.0
0.5
1.0
1.5
2.0
2.5
0 10 20 30 40 50
Time (year)
Car
bona
ted
dept
h (c
m)
Introduction
Carbonation modeling
Probabilistic approach
Results
6 m
5.2 kN/m
25 cm
55 cm
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
• Diffusion coefficient• Tortuousity / Connectivity• Concrete Young's modulus• Loading• Cover thickness
G(U) = 0
u1
u2
Concrete cover cAB
P*
O
Carbonation depth dAB
Finite elementanalysis
FailureG(U) < 0[ cAB < dAB ] A
B
A
BPerformanceG(U) > 0[ cAB > dAB ]
(k)(k)
(k)(k)(k)(k)T1)(k
UGgrad
UG UU
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
ResultsReliability index = min(UTU)1/2 with G(U)=0
Rackwitz-Fiessler's algorithm
• Significant computational cost
• Very much time consuming
• Non-guaranteed convergence
Non-linear FEM
1 G(U) computation at T=60 years 12 minutes CPU time
Gradient not accurately estimated
Direct approach
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
N
1i
1N
1i
N
1ijjiij
2iii
N
1iii0 UUaUaUaaUQ
Response surface approach
Reliability index = min(UTU)1/2 with Q(U)=0
(k)(k)
(k)(k)(k)(k)T1)(k
UQgrad
UQ UU
Quadratic response surfacewith mixed terms
• a0, ai, aii, aij determined by least square method
• (N+1)(N+2)/2 numerical observations
1 "center point"2N axial points
1N
1iiN out-of-axes points
star shape experimental design
• Successive experimental designs are necessary
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
N
1i
1N
1i
N
1ijjiij
2iii
N
1iii0 UUaUaUaaUQ
Reliability index = min(UTU)1/2 with Q(U)=0
Response surface approach
(k)(k)
(k)(k)(k)(k)T1)(k
UQgrad
UQ UU
u1
u2
ED(1)
Q(U)(1)=0P*
G(U)=0
P*(1)
P*
G(U)=0
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
N
1i
1N
1i
N
1ijjiij
2iii
N
1iii0 UUaUaUaaUQ
Reliability index = min(UTU)1/2 with Q(U)=0
Response surface approach
(k)(k)
(k)(k)(k)(k)T1)(k
UQgrad
UQ UU
u1
u2
ED(2)Q(U)(2)=0
P*(2)
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
P*(m)
#1 Previous P*(m) outside the ED(m)
ED(m+1) "recentered" on P*(m)
ED(m)
P0
u1
u2
Building the experimental design
P*
G(U)=0
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
u1
u2
Building the experimental design
ED(m+1)
|1||2|
ED(m+1) "recentered" on P*(m)
#1 Previous P*(m) outside the ED(m)
+
i
m*m
m*m
0i U
UQ
UQgrad
1ΔΔ
i| 0.25
0 = N½
+
U*(m)
P*
G(U)=0
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
u1
u2
Building the experimental design
Q(U)(m)=0
Q(U)(m)<0
Q(U)(m)>0
ED(m+1) "recentered" on P*(m)
#1 Previous P*(m) outside the ED(m)
+
i
m m
m m
0i U
UQ
UQgrad
1ΔΔ
i| 0.25
0 = N½
+
ED(m+1)
2
U(m)
P*
G(U)=0
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
u1
u2
Building the experimental design
#2 Previous P*(m) inside the ED(m)
ED(m)
P*(m)
P0
P1
P2P0
P*
G(U)=0
Retained points:
cos(P0Pi,P0P*(m)) > 0
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
u1
u2
Building the experimental design
Retained points:
cos(P0Pi,P0P*(m)) > 0
#2 Previous P*(m) inside the ED(m)
ED(m+1)
P*(m)
P2
Complementary points:
symmetrical transformed / P*(m)P*
G(U)=0
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
u1
u2
Building the experimental design
Retained points:
cos(P0Pi,P0P*(m)) > 0
#2 Previous P*(m) inside the ED(m)
ED(m+1)
P*(m)
Complementary points:
symmetrical transformed / P*(m)P*
G(U)=0Bringing the transformed points
closer to P*(m)
P2
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
u1
u2
Building the experimental design
Retained points:
cos(P0Pi,P0P*(m)) > 0
#2 Previous P*(m) inside the ED(m)
ED(m+1)
P*(m)
Complementary points:
symmetrical transformed / P*(m)P*
G(U)=0Bringing the transformed points
closer to P*(m)
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
u1
u2
Building the experimental design
Retained points:
cos(P0Pi,P0P*(m)) > 0
#2 Previous P*(m) inside the ED(m)
ED(m+1)
Complementary points:
symmetrical transformed / P*(m)P*
G(U)=0Bringing the transformed points
closer to P*(m)
P0
ED(m+1) "recentered" on P*(m)
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Start
First experimental design ED(0)
RF algorithm: P*(0)
ED(m)ED(0) ; P*(m)P*(0)
P*(m) inside
the ED(m) ?
Finite elementanlysis
RF algorithm: P*(m+1)
| P*(m+1) P*(m) | < 0.15
End
Building the ED(m+1)
with procedure #2
yes
Building the ED(m+1)
with procedure #1
no
Reliability index yes
ED(m)ED(m+1)
P*(m)P*(m+1)
no
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Practical application: reinforced concrete beam
Material properties:• Concrete strength• Reference diffusion coefficient• Turtuousity factor
Concrete probes of low scale
Variance reduction
Concrete probes of similar scale
No change
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Practical application: reinforced concrete beam
Material properties:• Concrete strength• Reference diffusion coefficient• Turtuousity factor
Distribution Mean CoV
Lognormal 35 MPa 0.1
Concrete probes of similar scale
No change
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Practical application: reinforced concrete beam
Material properties:• Concrete strength• Reference diffusion coefficient• Turtuousity factor
Distribution Mean CoV
Lognormal 35 MPa 0.1Lognormal 10-8 m2/s 0.8 Uniform 0.5 0.46[0.1 to 0.9]
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Practical application: reinforced concrete beam
Material properties:• Concrete strength• Reference diffusion coefficient• Turtuousity factor
Distribution Mean CoV
Lognormal 35 MPa 0.1Lognormal 10-8 m2/s 0.8 Uniform 0.5 0.46[0.1 to 0.9]Loading parameter:
• Live load E1max 1.04 kN/m2 0.38
Geometrical parameter:• Concrete cover thickness Lognormal 2 cm 0.2
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Practical application: reinforced concrete beam
Efficiency of the adaptative RSM
Features Iteration 1 Iteration 2 Iteration 3New experimental points 21 21 1
Designpoint
U1 (diffusion of CO2)U2 (tortuousity)U3 (concrete strength)U4 (concrete cover)U5 (live load)
5.51795.306 10-2
-1.366 10-4
-2.95753.595 10-2
4.84167.734 10-2
-4.478 10-4
-2.08927.531 10-2
4.80968.518 10-2
-4.978 10-4
-2.01298.372 10-2
Reliability index 6.2608 5.2737 5.2145Iterations in -search algorithm 6 4 3
T=2 years
Features Iteration 1 Iteration 2New experimental points 21 21
Designpoint
U1 (diffusion of CO2)U2 (tortuousity)U3 (concrete strength)U4 (concrete cover)U5 (live load)
1.19540.2712
-2.094 10-3
-0.79053.533 10-2
1.22910.2552
-1.790 10-3
-0.71442.995 10-2
Reliability index 1.4590 1.4447Iterations in -search algorithm 4 3
T=30 years
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Practical application: reinforced concrete beam
Variation of the reliability with time
0
1
2
3
4
5
6
0 10 20 30 40 50 60Time (years)
Re
liab
ility
ind
ex
SLS
• Significant decrease of the reliability index
• Reliability index lower than threshold value recommended by Eurocodes after T=30 years
Probabilistic Assessment of Corrosion Risk due to Carbonation Baltimore January 5-7 2005Introduction
Carbonation modeling
Probabilistic approach
Results
Practical application: reinforced concrete beam
Variation of the sensitivity factors with time
• Diffusion coefficient and cover thickness for T < 35 years
• Tortuousity factor and loading play a role for T > 35 years
0.00.10.2
0.30.40.50.60.7
0.80.91.0
0 10 20 30 40 50 60
Diffusion Cover thickness Tortuousity Load
Time (years)
Se
nsi
tivity
fa
cto
r