FAILURE PROBABILITY OF DETERIORATING REINFORCEDCONCRETE BEAMSFailure probability of reinforced concrete beams

B. TEPL, D. NOVK, Z. KERNER and W. LAWANWISUTTechnical University of Brno, Faculty of Civil Engineering, Institute of StructuralMechanics, Brno, Czech Republic

Abstract

For a specific reinforced concrete beam the depassivation of reinforcement is assessedand a consequent corrosion process in time is evaluated using the available numericalmodels. The cross-section of a beam is analyzed using the layer approach. Divisioninto layers enables us the efficient modelling of the carbonation process and theconsequent corrosion of reinforcement. The influence of deterioration on failureprobability is assessed.

Keywords: Bayesian updating, carbonation of concrete, corrosion of reinforcement,deterioration, failure probability, importance sampling, reliability

1 Introduction

The deterioration caused by aggressive environment may have a significantimpact on safety and serviceability of reinforced concrete (RC) structures. Theimportant degradation processes are the carbonation of concrete, the ingress ofchloride ions (both these effects cause the depassivation of reinforcement) and thecorrosion of reinforcement. Such time-dependent processes are governed by manyuncertainties - random parameters. Consequently, by these stressors the strength andstiffness of RC members can be considerably decreased. Therefore it is important todevelop a suitable tool for the prediction of such processes and to utilize them whileinvestigating the reliability or the repair strategy of structural members.

The problem of deteriorating RC beams have gained recently a strong attentionof several researchers, Lee et al. (1996, 1998): A nonlinear finite element (NL-FE)

Durability of Building Materials and Components 8. (1999) Edited by M.A. Lacasseand D.J. Vanier. Institute for Research in Construction, Ottawa ON, K1A 0R6,Canada, pp. 1357-1366. National Research Council Canada 1999

-) is excluded in this case. The physical, chemical,technological and environmental characteristics are considered to be random variableswithin the framework of a fully probabilistic approach.

If the bending action prevails and the attention is paid to the ultimate limit stateof simply supported beams only, the limit state function may be represented bycomparing the ultimate bending moment (resistance) and load action in one cross-section. First, the carbonation depth as a main factor influencing the initiation ofcorrosion of reinforcement bars is treated more precisely using the so-called Bayesianupdating method based on short-term in-situ measurements (measured carbonationdepth statistics). Secondly the appropriate advanced simulation techniques(importance sampling) are used for the probabilistic prediction of deterioration in timemeasured by the theoretical probability of failure pf. Using pf -time curves andisoprobability curves the residual life-time of the structure can be predicted.

2 Deterioration processes of reinforced concrete

2.1 Carbonation of concreteThe model of time-dependent carbonation depth xc of ordinary Portland cement

(OPC) concrete by Papadakis et al. (1992) is used. This computational model is basedon the mass conservation of CO2, Ca(OH)2 and CSH (hydrated calcium silicate) in anycontrol volume of the concrete mass. The simplified carbonation depth formula forOPC concrete is expressed here as:

( ) (1)tcc

a

c

wRHf

c

wc

w

x COa

cc

c

cc6

210

44

4.22

10001

10001

3.035.0

++

+

-

=r

rr

rr

where xc is the carbonation depth (mm) for time t (years), r c, r a are the mass density(kg/m3) of cement and aggregates, resp., w/c, a/c are the water/cement,aggregate/cement ratio, resp., RH is the ambient relative humidity,

2COc is CO2 content

in atmosphere (mg/m3), a is the aggregate unit content. In the original formulation(Papadakis et al. 1992) the model for high values of RH does not provide satisfactoryresults. This has been overcome by implementing the step-wise linear relationshipf(RH) extracted from experiments reported by Matouek (1997), see also Novk et al.(1996).

A total of 11 variables (see details in Table 1) are involved in this model, and insome cases not all appropriate information is available. Therefore, a simpler modelmay be useful. For this purpose the model by Bob (1996) seems to be appropriate:

(2)tf

dkCx

c

c 150=

where xc is the average depth of carbonation (or chloride penetration) (mm), fc isconcrete compressive strength (MPa), t is time of CO2 or/and Cl

- action (years), C isthe coefficient of cement type, k introduces the influence of humidity (environmentalconditions) and d is coefficient for CO2 content. Both models were randomized andcompared - see Kerner et al. (1996). In these forms the models are utilized in thepresent paper. The purpose of modelling the carbonation depth is basically to calculatethe time at which depth xc reaches the concrete cover c (initiation time Ti).

2.2 Corrosion of reinforcementAn accurate and reasonably general computational model for corrosion of

depassivated steel bars embedded in concrete is still missing. Note that both theuniform and pitting type of corrosion should be modelled. Presently the models usede.g. by Andrade et al. (1996); Rodriguez et al. (1996) and Stewart and Rosowsky(1998) seems to be sufficient for the prediction of uniform corrosion. The formula forthe time related rebar diameter decrease is as follows:

( ) ( )( )

(3)

+>+

3 Computational model

3.1 Nonlinear analysisThe codes give approximate expressions for computing the ultimate flexural

resistance of reinforced concrete sections. In order to make the computation moreaccurate a simple method that satisfies equilibrium and compatibility is utilized. Thereinforced concrete section is divided into layers, each layer is considered to be in astate of uniaxial tension or compression. Maximum strains are considered initially,then stresses in layers (both concrete and reinforcement) are computed using theidealized stress-strain diagrams given in Fig. 1. By iterative algorithm equilibrium ischecked resulting in an ultimate bending moment Mu. This layered approach is alsosuitable for modelling the deterioration.

Fig. 1: Stress-strain diagrams: a) concrete; b) reinforcing steel.

3.2 Limit state functionFor a reliability analysis a function g(X) of basic random variables X = X1, X2,...,

Xm is defined comparing ultimate bending moment Mu (resistance) with load bendingmoment M (action of load). Limit state function is then given as:

g(X,t) = Mu(X,t) M (4)

and is a function of time t, model uncertainty factor and m random variables. Notethat the decrease of Mu(X,t) is strong when t > Ti. A beam is considered to be safe if:

g(X,t) > 0 (5)

For the purpose of the present study the load action M is taken as a deterministicvalue. This arrangement enables us to predict the structural life-time of deterioratingstructure for a prescribed load level.

4 Reliability analysis

The material of concrete and reinforcing steel deteriorate in time andconsequently the ultimate bending moment decreases. The goal is to quantify theinfluence of deterioration on the reliability. The failure probability is defined as:

b)a)

fy

ee

ss

e s

ss

eee c

pf = P(g(X,t) 0) (6)

and can be evaluated at several time points ti (i = 1, 2,..., n). The increase of failureprobability in time can be estimated here by advanced simulation techniques e.g.importance sampling (Bourgund and Bucher, 1986).

5 Numerical example case study

5.1 General remarksA particular storage building structure is composed of prefabricated RC units.

The study is focused on the roof beam where the carbonation depth was investigatedin-situ in several positions (the age being 11 years). The cross-section of the beam isshown in Fig. 2.

Fig. 2: Critical cross-section with reinforcing bars.

5.2 Random variablesIn-situ investigation also provided other values of several input variables and

their statistical parameters; some other parameters were gained by engineeringjudgement. All of them are summarized in Table 1.

5.3 Results of parametric studyThe approach described above was used and the results of the probabilistic

analysis are as follows:The probability of failure is obviously increasing in the course of time due to

corrosion of reinforcement. This effect can be observed in Fig. 3 for different levels ofload (where the pf - time curves are shown). Similar results are depicted in a differentway by the mean of the isoprobabilities (i.e. curves of constant probability) - seeFig. 4. The pf -time curves or the isoprobabilities can be used for the structural life-time (L) prediction. It could be observed in Fig. 4 e.g. for M =550 kNm: If theadmissible pf would be 10

-5 then L 75 years; for pf = 10-6 L 25 years only.The in-situ measurements of carbonation depth xc after 11 years of the utilization

of the building were performed. The results of the mean value of 6 mm and standard

H1=0.15 m

H=1.25 m

B1=0.30 m

B2=0.15 m

deviation of 1.2 mm were obtained, and the Bayesian updating (see e.g. Novk et al.1996) of the xc evaluation in time could be assessed. Based on this new xc-t functionalso an updated pf -t curve could be computed. The comparison of a prior and posteriorpf -time curves is shown in Fig. 5, the latter curve being more accurate as the utilizationof in-situ measurements takes into account the effect of local conditions. This cannotbe accounted for by a general analytical model itself. Such updated pf -time curvesmake it possible to gain a more reliable prediction of the structural life-time.

Table 1: Basic random variables.

Variable Symb. Unit Mean COV(%)

PDF

Uncertainty factor of model - 1 0.1 NAmbient CO2 content

2COc mg/m3 800 0.15 N

Relative humidity RH - 0.85 0.1 NUnit content of cement c kg 291 0.02 LNUnit content of water w kg 189 0.01 LNUnit content of sand r kg 853 0.015 LNUnit content of gravel 4-8 mm sizeg4-8 kg 382 0.015 LNUnit content of gravel 8-16 g8-16 kg 630 0.015 LNMass density of cement r c kg/m

3 3100 0.015 NMass density of sand r s kg/m

3 2590 0.02 NMass density of gravel 4-8 r g4-8 kg/m

3 2540 0.02 NMass density of gravel 8-16 r g8-16 kg/m

3 2660 0.02 NConcrete cover cc mm 20.2 0.2 NDiameter of steel Di mm 22 0.05 LNHeight of cross-section H m 1.25 0.02 NTop flange depth H1 m 0.15 0.02 NTop flange width B1 m 0.3 0.02 NWeb thickness B2 m 0.15 0.02 NStrength of concrete in compressionfc MPa 54.1 0.15 LNStrength of concrete in tensionRbt MPa 3.1 0.25 LNStrength of steel Rs MPa 410 0.05 LNModulus of elasticity for steel Es GPa 210 0.03 NModulus of elasticity for concreteEb GPa 23 0.05 NType of cement C - 1.4 0.247 U Influence of humidity k - 0.65 0.311 U Influence of CO2 content d - 1.5 0.192 U Current density icorr mA/cm2 1.5 0.33 U Remarks: N...Normal, LN...Lognormal and U...Uniform on interval

In Fig. 5 a) homogenous corrosion is treated and updated values of carbonationdepth (smaller values) decreased the failure probability to the level of the failureprobability for zero time. For pitting corrosion, this decrease to zero time level worksapproximately up to 40 years, then there is an increase of the failure probability (but oflower values), see Fig. 5 b). It has to be realized that we start from time zero where afailure probability exists (10-5) and this failure probability is the result of the random

variability of geometrical and material characteristics only. Also the influence of twodifferent models for the carbonation process was studied. The curves in Fig. 6 indicateonly negligible effects in this particular beam. A more serious degradation effect is thepitting corrosion. Using the same approach also the influence of a type of corrosion onthe probability of failure was studied - as it can be observed in Fig. 7. The results areonly indicative as the pitting process is strongly uncertain and is often started due tothe chlorides attack.

Fig. 3: Failure probability vs. time for different levels of bending moments(carbonation model by Papadakis and homogenous corrosion)

Fig. 4: Isoprobabilities (Papadakis, homogenous corrosion)

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

0 25 50 75 100

Time (years)

Failu

re p

robabili

ty (

-)

.

500 kNm 600 kNm 700 kNm

0 25 50 75 100

Time (years)

80

70

60

50

1E

1E

1E1E

1E

1E

Fig. 5: Bayesian updating of pf (carbonation model by Papadakis and M= 600kNm): a) homogenous corrosion; b) pitting corrosion

6 Conclusions

The influence of the degradation process on reliability is modelled and shown ina quantitative probabilistic way - failure probability vs. time.

The approach described above may conveniently serve for the assessment of theresidual life of a RC structure taking into account the degradation due tocarbonation of concrete and corrosion of steel reinforcement.

1.E-06

1.E-05

1.E-04

1.E-03

0 20 40 60 80 100

Time (years)

Failu

re p

robabili

ty (

-)--

Prior Posterior

a)

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

0 20 40 60 80 100

Time (years)

Failu

re p

robabili

ty (

-)..

Prior Posterior

b)

M= 600 kNm)

Fig. 7: Influence of type of corrosion - homogenous (aa = 2) and pitting (aa = 6)(carbonation model by Papadakis and M=600 kNm)

1.E-06

1.E-05

1.E-04

1.E-03

0 25 50 75 100

Time (years)

Papadakis Bob

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

0 25 50 75 100

Time (years)

Pitting Homogenous

Utilizing the Bayesian updating procedure and site-oriented measurements thepredictions of degradation processes and the evaluation of their impact on thereliability measure are more effective. The local conditions are accounted for.

The approach is rather general and different models and other degradationprocesses may be incorporated.

7 Acknowledgement

The research was funded under grant No. 103/97/K003 from the Grant Agency ofthe Czech Republic. The authors thank for this funding.

8 References

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Bob, C. (1996) Probabilistic Assessment of Reinforcement Corrosion in ExistingStructures. Concrete Repair, Rehabilitation and Protection, London, pp. 17-28.

Boer, A. and Veen, C. (1998) Simple Way of Finding Residual Strength of ActualStructures after Influences of Deterioration. Diana World, issue No.1, pp. 8-11.

Bourgund, U. and Bucher, C.G. (1986) Importance Sampling Procedure Using DesignPoints - A Users Manual. Internal Working Report, Institute of EngineeringMechanics, University of Innsbruck, Austria.

Kerner, Z., Tepl, B. and Novk, D. (1996) Uncertainty in Service Life PredictionBased on Carbonation of Concrete. Durability of Building Materials andComponents 7 (Volume One), London, pp. 13-20.

Lee, H.S., Tomosawa F. and Noguchi, T. (1996) Effects of Rebar Corrosion on theStructural Performance of Singly Reinforced Beams. Durability of BuildingMaterials and Components 7 (Volume One), London, pp. 571-580.

Lee, H.S., Noguchi, T. and Tomosawa F. (1998) FEM Anal. for Struct. Performance ofDeteriorated RC Struct. Due to Corrosion. CONSEC, Tromso, pp. 327-336.

Matouek, M. (1977) Effects of some Environmental Factors on Structures. Ph.D.thesis, Technical University of Brno, Czech Republic (in Czech).

Novk, D. Kerner, Z. and Tepl, B. (1996) Prediction of Structure DeteriorationBased on the Bayesian Updating. Proc. of the 4th int. symp. on natural draughtcooling towers, Kaiserslautern, pp. 417-421.

Papadakis, V.G., Fardis, M.N. and Vayennas, C.G. (1992) Effect of Composition,Environmental Factors and Cement-lime Mortar Coating on ConcreteCarbonation. Materials and Structures, Vol. 25, No. 149, pp. 293-304.

Rodriguez, J., Ortega, L.M., Casal, J. and Diez, J.M. (1996) Corrosion ofReinforcement and Service Life of Concrete Structures. Durability of BuildingMaterials and Components 7 (Volume One), London, pp. 117-126.

Stewart, M.G. and Rosowsky, D.V. (1998) Structural and Serviceability Reliabilitiesfor Chloride Diffusion, Cracking, Spalling and Corrosion of Concrete Bridges,Research Report No. 162.02.1998, Department of Civil, Surveying andEnvironmental Engineering, The University of Newcastle, Australia.