use of mathematical models for predicting the service life...

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



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,…



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



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.



Probabilistic Service Life Design



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



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



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



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



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



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



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



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



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



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



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



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



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.



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!



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!



Da – t from Field Exposures (Bamforth)



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

′′

⋅=



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



Da – t from Field Exposures (Bamforth)



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!



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



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



ClinConc Model for Chloride IngressTransport function:

∂∂

+∂∂

=∂∂

+∂∂

=∂∂

f

bfbftot 1cc

tc

tc

tc

tc

∂∂

∂∂

=∂∂

−=∂∂

xcD

xxq

tc fCltot

qCl qCl + dqClcf

dx

cbdcb



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




⋅

′

−

′+⋅

′⋅

−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.




⋅

′

−

′+⋅

′⋅

−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



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



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



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



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



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



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



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

use of mathematical models for predicting the service life...

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