use of mathematical models for predicting the service life...
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Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Use of Mathematical Models for Predicting the Service Life of
Concrete Structures
• Service life modelling in general• Mathematics in modelling of chloride ingress• Analytical sensitivity of models • Mechanistic modelling• Comparison of different models• Conclusions
Contents:
Tang Luping, Chalmers University of Technology
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Deterioration of Concrete
• Corrosion induced by carbonation• Corrosion induced by chlorides• Freeze/thaw attack• Chemical attack (ground water, soil,…)• Abrasion (erosion, wearing)• Leaching, AAR/ASR,…
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Deterioration of Concrete
• Corrosion induced by carbonation• Corrosion induced by chlorides• Freeze/thaw attack• Chemical attack (ground water, soil,…)• Abrasion (erosion, wearing)• Leaching, AAR/ASR,…
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Service Life of Concrete Structures Exposed to Chloride Environment
Func
tiona
lity
t
Designed functionality
Structural failure
Corrosion propagation
Corrosion initiation
Actual functionality
Designed service life
Chloride ingress
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Definition of Service Life
There are some arguments on the definition of service life, especially in the considerationof propagation period.
From the viewpoint of structural design, service life is often defined as the time of damage initiation, especially in the case of chloride initiated pitting corrosion.
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Probabilistic Service Life Design
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Probabilistic Service Life Design
“Probabilistic service life design” is in factin a statistic way to handle
“Deterministic models”,─ Key to proper prediction of service life
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Empirical Modelling Based on phenomena, observations and
experimental data Much depending on the availability of
representative data for model building and validation
Apart from cause-and-effect between variables, not much requiring in terms of knowledge or understanding
A “trial and error” approach
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Questionable Empirical Modelling
y = 8.43Ln(x) + 10.65R2 = 0.94
y = 1.78x + 11.52R2 = 0.96
y = 12.62e0.08x
R2 = 0.95
x
y
y = 8.48Ln(x) + 9.26R2 = 0.79
y = 1.93x + 10.05R2 = 0.89
y = 11.77e0.09x
R2 = 0.90
xy
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Mechanistic Modelling Based physics and chemistry governing
the behaviour of the process Not requiring much data for model
development, and hence not subject to the hypersensitivities in data
Requiring a fundamental understanding of the physics and chemistry governing the process
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Modelling for Chloride Ingress
Simple Fick’s 2nd LawModified Fick’s 2nd Law – DuraCrete ModelMechanistic models (e.g. ClinConc Model) –
Based on physical and chemical processes
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Simple Model based on Fick’s 2nd Law
Error-functional solution to Fick’s 2nd Law2
2
xCD
tC
∂∂
=∂∂
=
−−
tDx
CCCC
2erf
is
s
−=
tDx
CC
2erf1
s
when Ci = 0
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Assumptions in the simple Fick’s lawmodel for choride transport
Non-charged particles (but Cl- is actually charged ones)
Constant diffusion coefficient D (but D may change with many factors, such as coexisting ions, concrete age, temperature, and in some cases with depth)
Constant binding capacity (but it may change with pH, temperature, etc.)
No effect of temperature
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Time-dependent Diffusion Coefficient
Instantaneous D (e.g. from rapid migration test)
t' – concrete age; t'0 – age when D0 is tested; t'ex – age when concrete is exposed; t – exposure duration
( )nn
tttD
tt
DtD−−
′′+
⋅=
′′
⋅=′0
ex0
00
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Mathematics of Time-Dependent D
−=
−=
tDx
Tx
CC
as 2erf1
2erf1 Da – apparent
( ) ( ) ( ) ( )nn
nnb
ttD
ttDttDtatD
′′
⋅=
′′
⋅=′⋅′⋅=′⋅=′−
−− 00
0000
( ) ( ) ( ) ( )[ ]nnntt
tttt
ntDtdtDT −−′+
′′−′+⋅
−′⋅
=′′= ∫ 1ex
1ex
0n0
1ex
ex
( ) ttDtDttt
tt
tt
nDT
nnn
⋅′≠⋅=⋅
′⋅
′
−
′+⋅
−=
−−
a0
1ex
1exn0 1
1
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Mathematics of Time-Dependent D
( )tDD ′≠a
Apparent Da is an average of instantaneous D(t) under the exposure duration from 0 to t
( ) ( ) tdtDtttt
TDtt
t′′
′−′+== ∫
′+
′
ex
exexexa
1
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Mathematical solution to Fick’s Law with Instantaneous D(t)
( )n
tttDtD
−
′′+
⋅=0
ex0
⋅
′⋅
′
−
′+⋅
−
−=
⋅−=
−=
−−
ttt
tt
tt
nD
x
tDx
Tx
CC
nnn0
1ex
1ex0
as
11
2
erf1
2erf1
2erf1
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Mathematical solution to Fick’s Law with Instantaneous D(t)
( )n
tttDtD
−
′′+
⋅=0
ex0
⋅
′⋅
′
−
′+⋅
−
−=
⋅−=
−=
−−
ttt
tt
tt
nD
x
tDx
Tx
CC
nnn0
1ex
1ex0
as
11
2
erf1
2erf1
2erf1
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Apparent Diffusion Coefficient
( )mmm
ttD
tttD
ttDtD
−−−
′
⋅≈
′′+
⋅=
⋅=
exaex
ex
exaex
1a1a
(curve-fitted from the field exposure data)
Impossible to obtain Daex without exposure (t = 0)!
Extrapolation from a number of Da(ti) is needed to obtain Daex
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Mathematical solution to Fick’s Law with Apparent Da(t)
⋅
′⋅
−=
⋅
⋅
−=
⋅−=
−=
tt
tD
x
tttD
x
tDx
Tx
CC
mmex
aex1
a1
as
2
erf1
2
erf1
2erf1
2erf1
( )mm
ttD
ttDtD
−−
′
⋅=
⋅=
exaex
1a1a
It is practically difficult to obtain Daex, since it needsextrapolation from field data after exposure for many years.
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Mathematical solution to Fick’s Law with Apparent Da(t)
⋅
′⋅
−=
⋅
⋅
−=
⋅−=
−=
tt
tD
x
tttD
x
tDx
Tx
CC
mmex
aex1
a1
as
2
erf1
2
erf1
2erf1
2erf1
( )mm
ttD
ttDtD
−−
′
⋅=
⋅=
exaex
1a1a
It is practically difficult to obtain Daex, since it needsextrapolation from field data after exposure for many years.
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
DuraCrete Model for Chloride Ingress
( )
⋅
⋅⋅⋅
−=
tttDkk
xCtxCncl
0cl0,clc,cle,
cls,cl
2
erf1,
D0,cl – Instantaneous D measured using RCM at age t0
ke,cl, kc,cl – Environmental and curing factor, respectively
No real mathematical relationship between D0,cl and Daex!
Not an analytical solution to Fick’s 2nd law!
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Relationship between Instantaneous D0and apparent Da(t)
( )
mm
nnn
ttD
ttD
tt
tt
tt
nDtD
′⋅=
⋅=
′⋅
′
−
′+⋅
−=
−−
exaex
1a1
01
ex1
ex0a 1
1
At age t' ex the exposure duration t = 0! No real mathematical relationship between D0 and Daex!
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Da – t from Field Exposures (Bamforth)
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Analytical Sensitivity of Models
′
−
′+
′⋅
′
−
′+⋅
′+
−−
+
′⋅
⋅π
=⋅∆∆
−−
−−
−
nn
nn
z
tt
tt
tt
tt
tt
tt
ntt
CCezn
Cn
nC
1ex
1ex
1exex
1exex
0
s1
ln11ln
11ln
2
′⋅
⋅π′
=′
⋅′∆
∆ −
tt
CCezn
Cn
nC z
0
s
ln2
s
0
0
22 11
CCezez
CC
CD
DC z
zs−
− ⋅⋅
π=⋅⋅⋅
π=⋅
∆∆1s
s
=⋅∆∆
CC
CC
'0
na'n
ttDD
′′
⋅=
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Most Sensitive Parameter n
-10
-8
-6
-4
-2
0
2
4
0.1 0.2 0.3 0.4 0.5
C /C s
Cs
Do or k
n = 0.2
n' = 0.2
n = 0.5
n' = 0.5
n = 0.8
n' = 0.8
t0 (n' = 0.2)
t0 (n' = 0.5)
t0 (n' = 0.8)
Cf
fC i
i
⋅∆∆
n as in Eq (6)n' as in Eq (9)
z = 1
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Da – t from Field Exposures (Bamforth)
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Mechanistic Models for Chloride TransportThe sophisticated models consider the interaction between the following processes: Free chlorides as driving force, Intrinsic diffusion coefficient, Non-linear chloride binding Effect of co-existing ions, Effect of pore structures, and Effect of temperature (Arrhenius law).
Numerical techniques have to be employed to simulate the above processes!
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
TWO Procedures:
Mechanistic Model ClinConc
1. Simulation of free chloride penetration through the pore solution in concrete using a genuine flux equation based on the principle of Fick’s law with the free chloride concentration as the driving potential
2. Calculation of the distribution of the total chloride content in concrete using the mass balance equation combined with non-linear chloride binding
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
ClinConc Model for Chloride IngressMass balance equation:
Rate of accumulation
of mass in the system
Rate ofmass
flow in=
Rate ofmass
flow out–
qCl qCl + dqClcf
dx
cbdcb
( ) ClClClClCl ddd qqqqxt
c−=+−=⋅
∂∂
xq
tc
∂∂
−=∂∂ ClCl
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
ClinConc Model for Chloride IngressTransport function:
∂∂
+∂∂
=∂∂
+∂∂
=∂∂
f
bfbftot 1cc
tc
tc
tc
tc
∂∂
∂∂
=∂∂
−=∂∂
xcD
xxq
tc fCltot
qCl qCl + dqClcf
dx
cbdcb
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Chloride Ingress FunctionsAs an engineering expression for a time-dependent diffusion process
⋅
′
−
′+⋅
′⋅
−ξ
−=−−
−−
tt
tt
tt
tn
D
xcccc
nnnD
i
i
1ex
1ex6m6m
s1
12
erf1
( )100
c
b ×+⋅ε
=B
ccC
Free chlorides
Total chlorides
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Chloride Ingress FunctionsAs an engineering expression for a time-dependent diffusion process
⋅
′
−
′+⋅
′⋅
−ξ
−=−−
−−
tt
tt
tt
tn
D
xcccc
nnnD
i
i
1ex
1ex6m6m
s1
12
erf1
( )100
c
b ×+⋅ε
=B
ccC
Free chlorides
Total chlorides Factor bridging the gap between lab test and actual field exposure taking into account effects of binding, temperature, etc.
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Chloride Ingress FunctionsAs an engineering expression for a time-dependent diffusion process
⋅
′
−
′+⋅
′⋅
−ξ
−=−−
−−
tt
tt
tt
tn
D
xcccc
nnnD
i
i
1ex
1ex6m6m
s1
12
erf1
( )100
c
b ×+⋅ε
=B
ccC
Free chlorides
Total chlorides Factor bridging the gap between lab test and actual field exposure taking into account effects of binding, temperature, etc.
Binder content
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Concrete and Input ParametersMix No.
Binder type Binder[kg/m3]
w/b DRCM[×10-12
m2/s]
Cs[%binder]
n
123
SRPC 500450420
0.300.350.40
2.523.612.2
3.754.384.12
0.3
4 10% SF 630 0.30 0.34 3.75 0.62
5 20% FA (fly ash) 630 0.30 1.49 3.24 0.69
6 5% SF + 10% FA 450 0.35 1.04 4.38 0.69
789
10
5% SF (silica fume)
550500450420
0.250.300.350.40
0.850.622.934.43
3.133.754.385.0
0.62
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Results from Modelling for Concrete with Portland Cement
0
2
4
6
0 20 40 60 80 100
Depth [mm]
Cl [
% b
y w
t of b
inde
r] [No. 3]: SRPC, w/b 0.40
Double sides ingress
0
2
4
6
0 10 20 30 40 50
Cl [
% b
y w
t of b
inde
r]
Field M1 M2
M2b M3
[No. 1]: SRPC, w/b 0.30
0
2
4
6
0 10 20 30 40 50
Cl [
% b
y w
t of b
inde
r] [No. 2]: SRPC, w/b 0.35
M1 – Simple Fick’s 2nd lawM2 – DuraCrete equationM2b – DuraCrete with analytical eq.M3 – ClinConc model
No. 1: SRPCw/b 0.30
No. 2: SRPCw/b 0.35
No. 3: SRPCw/b 0.40
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Results from Modelling for Concrete with Blended Cement
0
2
4
6
0 10 20 30 40 50
Cl [
% b
y w
t of b
inde
r]
Field M1 M2
M2b M3
[No. 4]: 10% SF, w/b 0.30
0
1
2
3
4
5
0 10 20 30 40 50
Cl [
% b
y w
t of b
inde
r]
[No. 5]: 20% FA, w/b 0.30
0
1
2
3
4
5
0 10 20 30 40 50
Depth [mm]
Cl [
% b
y w
t of b
inde
r]
[No. 6]: 5% SF + 10% FA, w/b 0.35
M1 – Simple Fick’s 2nd lawM2 – DuraCrete equationM2b – DuraCrete with analytical eq.M3 – ClinConc model
No. 4: 10%SFw/b 0.30
No. 5: 20%FAw/b 0.30
No. 6: 5%SF+10%FAw/b 0.35
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Results from Modelling for Concrete Blended with 5% Silica Fume
0
1
2
3
4
5
0 10 20 30 40 50
Cl [
% b
y w
t of b
inde
r]
[No. 7]: 5% SF, w/b 0.25
0
1
2
3
4
5
0 10 20 30 40 50
Depth [mm]
Cl [
% b
y w
t of b
inde
r] [No. 9]: 5% SF, w/b 0.35
0
1
2
3
4
5
0 10 20 30 40 50
Cl [
% b
y w
t of b
inde
r] Field
M1
M2
M2b
M3
[No. 8]: 5% SF, w/b 0.30
0
1
2
3
4
5
0 20 40 60 80 100
Depth [mm]
Cl [
% b
y w
t of b
inde
r]Field
M1
M2
M2b
M3
[No. 10]: 5% SF, w/b 0.40
No. 7: 5%SFw/b 0.25
No. 8: 5%SFw/b 0.30
No. 9: 5%SFw/b 0.35
No. 10: 5%SFw/b 0.40
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Modelling for 100 years Chloride Ingressin PC Concrete
0
1
2
3
4
5
6
0 20 40 60 80 100
Cl [
% b
y w
t of b
inde
r]
[No. 1]: SRPC, w /b 0.30
0
2
4
6
8
0 50 100 150 200
Depth [mm]
Cl [
% b
y w
t of b
inde
r]
Field 10y
M1-100y
M2-100y
M2b-100y
M3-100y
[No. 3]: SRPC, w/b 0.40
0
1
2
3
4
5
6
0 10 20 30 40 50
Depth [mm]
Cl [
% b
y w
t of b
inde
r]
[No. 4]: 10% SF, w /b 0.30
0
1
2
3
4
5
6
0 20 40 60 80 100
Depth [mm]
Cl [
% b
y w
t of b
inde
r] Field 10y
M1-100y
M2-100y
M2b-100y
M3-100y
[No. 5]: 20% FA, w/b 0.30
No. 1: SRPCw/b 0.30
No. 3: SRPCw/b 0.40
No. 4: 10%SFw/b 0.30
No. 5: 20%FAw/b 0.30
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Modelling for 100 years Chloride Ingress in Concrete Blended with Silica Fume
0
1
2
3
4
5
0 10 20 30 40 50
Cl [
% b
y w
t of b
inde
r]
[No. 7]: 5% SF, w/b 0.25
0
1
2
3
4
5
0 10 20 30 40 50
Depth [mm]
Cl [
% b
y w
t of b
inde
r] [No. 9]: 5% SF, w/b 0.35
0
1
2
3
4
5
0 10 20 30 40 50
Cl [
% b
y w
t of b
inde
r]
Field 10y
M1-100y
M2-100y
M2b-100y
M3-100y
[No. 8]: 5% SF, w/b 0.30
0
1
2
3
4
5
0 20 40 60 80 100
Depth [mm]
Cl [
% b
y w
t of b
inde
r]Field 10y
M1-100y
M2-100y
M2b-100y
M3-100y
[No. 10]: 5% SF, w/b 0.40
No. 7: 5%SFw/b 0.25
No. 8: 5%SFw/b 0.30
No. 9: 5%SF, w/b 0.35 No. 10: 5%SF, w/b 0.40
Chalmers University of Technology
Research group Building Materials Belfast Workshop, 19 June 2010, by Tang Luping
Conclusions
– Mathematics in a prediction model should be carefully examined in order to understand the physical meaning of input parameters
– The actual data from the field are very useful for verification of different prediction models
– Mechanistic model ClinConc reveals good prediction for all the 10 cases studied
– The model using simple error function to Fick’s 2nd law significantly overestimates chloride ingress
– DuraCrete model seems underestimating chloride ingress in concrete with pozzolanic additives