Reliability of RC beams under chloride-ingress

Construction and Building

MATERIALS

Construction and Building Materials 21 (2007) 1605–1616

www.elsevier.com/locate/conbuildmat

Reliability of RC beams under chloride-ingress Fre´de´ric Duprat

*

National Institute of Applied Sciences – INSA, Department of Civil Engineering, 135 Avenue de Rangueil, 31077 Toulouse Cedex 4, France Received 10 March 2006; accepted 1 August 2006 Available online 20 September 2006

Abstract An assessment of the reliability of reinforced concrete beams with corroded reinforcement was proposed based on the computation of the reliability index. The corrosion was induced by chlorides provided from de-icing salts or marine breeze. Models for chloride diffusion and corrosion kinetics were chosen depending on the availability of consistent statistical data, and as well as to represent the physical phenomena realistically. The statistical distributions were specified from a wide review of previous studies for environmental parameters (surface chloride concentration or flux, coefficient of diffusion, corrosion current density) and also for geometrical parameters and mechanical properties of materials. Various design specifications were accounted for in designing the beams corresponding to different expected degrees of aggressiveness of the environment and the possibility of concrete cracking. The time-dependant reliability was evaluated taking into account the previous survival period of the structure. The results show the effect on reliability of exposure conditions, quality of concrete and design option according to the french design rules.  2006 Elsevier Ltd. All rights reserved. Keywords: Reinforced concrete; Chloride ingress; Corrosion risk; Reliability; Design provisions

1. Introduction Chloride-induced corrosion of steel in reinforced concrete structures is one of the major causes of their deterioration over time. Chlorides from de-icing salts or marine breeze penetrate through the concrete cover and break down the natural protective oxide layer formed around the reinforcements by the strong alkalinity of pore solution. Once the protective layer has dissolved, corrosion is initiated if chloride concentration exceeds a threshold value. Pitting corrosion typically arises under chlorideingress, in contrast to general corrosion occurring for similar reasons (destruction of the protective oxide layer) under exposure to atmospheric carbon dioxide or other pollutants. General corrosion affects large areas of reinforcement with more or less uniform loss over the perimeter of the rebars. Pitting corrosion is localized on small areas but causes a substantial reduction of the cross-section

*

Corresponding author. Tel.: +35 561 559 930; fax: +35 561 559 900. E-mail address: [email protected].

0950-0618/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2006.08.002

of the reinforcement and makes it brittle, implying that failure of the structure might occur without warning. The prediction of the aging of concrete structures subjected to corrosion of reinforcement is complicated not only because the mechanical and chemical phenomena concerned are complex but also because of their random nature. There is indeed a high degree of uncertainty associated with the environmental parameters, physical properties of materials and loading. In order to protect structures against early deterioration, design considerations often include particular specifications for concrete cover, concrete mix (type of cement, aggregates, compacity) and reinforcement placement. Particular quality assurance during construction is also added to these design provisions. However the owners of structures frequently resort to maintenance strategies that limit the cost of possible repairs to or replacement of degraded existing structures. Such strategies require planning and in-site measurements which are also costly. The optimal decisions about the experimental plan and the maintenance methods can be made on the basis of a probabilistic assessment of the risk

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F. Duprat / Construction and Building Materials 21 (2007) 1605–1616

of failure. In addition, this provides a rational criterion for comparison of the various choices for design. In the present paper, a time-dependent reliability analysis of a reinforced concrete girder is developed. The girder is exposed to de-icing salt or to marine breeze. Probabilistic information about uncertainties related to the environment and physical properties of the materials have been estimated from a wide review of previous experimental or statistical studies. The probability of failure is given throughout the time for which the beam is subject to deterioration. Various design options are considered in order to emphasize their influence on reliability and their consequences on the maintenance strategy.

where C0 is the surface chloride concentration (time-invariant – kg/m3); x is the penetration depth (m); t is the time (s); Da is the apparent diffusion coefficient (m2/s) and erfc() is the complementary error function. Eq. (4) is clearly a simplified diffusion model. In fact, binding is not linear and other parameters interact with chloride-ingress in concrete: –the binding between chlorides and other ionic species in the pore solution, –the binding between these other species and the cement paste, –the porosity of concrete and the internal pore structure, –the external conditions (humidity and temperature).

2. Chloride penetration The flexural cracking of concrete structures under service loading allows direct ingress of chlorides if crack widths are sufficiently large. Nevertheless it can stated that cracking can only affect the initiation of corrosion and does not affect the subsequent propagation process [1,2]. Therefore, it is assumed here that chloride transport is a diffusion process through the saturated concrete cover. 2.1. Diffusion model According to Fick’s first law, the free chloride flux FL(x, t) in the pore solution (in only one direction of propagation) is given by F L ðx; tÞ ¼ DL

oC L ðx; tÞ ox

ð1Þ

where DL is the diffusion coefficient of free chlorides (m2/s) and CL(x, t) is the free chloride concentration (kg/m3). It is assumed that the diffusion coefficient DL does not depend on the concentration. The mass balance leads to an expression for Fick’s second law oC T ðx; tÞ oF L ðx; tÞ ¼ ot ox

ð2Þ

The total chloride concentration CT(x, t) is the sum of free chlorides and bound chlorides chemically reacting with the cement paste or physically adsorbed to the cement gel. Consequently expression (2) becomes oC T ðx; tÞ oC T ðx; tÞ oC L ðx; tÞ ¼ ot oC L ðx; tÞ ot   oC I ðx; tÞ oC L ðx; tÞ oF L ðx; tÞ ¼ ¼ 1þ oC L ðx; tÞ ot ox

More accurate and comprehensive models of chloride penetration have been developed; most of them based on numerical solving of the transport equations together with empirical expressions for the physical parameters (binding capacities derived from Freundlinch or Langmuir isotherms, flux or molar concentrations of ionic species in the pore solution, diffusion coefficients). Meijers et al. [3], Johannesson [4] and Kong et al. [5] have proposed such models. These models are suitable and efficient as long as the experimental data required for empirical expressions are available. These input data can be w/c ratio, porosity of concrete, weight of hydration products, weight and volume percentages of cement paste and aggregates, temperature and humidity of concrete, and intrinsic diffusion coefficients of ionic species in cement paste and aggregates. Although possible in the laboratory, obtaining such properties for existing concrete structures is very difficult. There is therefore a lack of statistical data even for the w/c ratio. In such conditions, the use of a simplified model (Eq. (4)) appears justified provided that the parameters C0 and Da are estimated from numerous measurements. Hence in Eq. (4) CL(x, t) will be replaced by CT(x, t) because nearly all values of C0 and Da given in the literature have been obtained by fitting Fick’s law to total concentration measured profiles (acid-soluble). 2.2. Surface concentration and diffusion coefficient In the case of structures exposed to de-icing salts, statistical distributions have been suggested (see Tables 1a and 1b) from wide experimental investigations. Most of them were conducted on bridge decks in the USA [6–8]. Wallbank also gathered numerous data on bridges in the UK [9].

ð3Þ

where CI(x, t) is the bound chloride concentration. The binding isotherm is necessary to solve Eq. (3). If this isotherm is considered linear, the free chloride concentration in a semi-infinite field can be estimated by   x C L ðx; tÞ ¼ C 0 erfc pffiffiffiffiffiffiffi ð4Þ 2 Da t

Table 1a Surface concentration (kg/m3) for structures exposed to de-icing salts Reference

Range

Mean

Coefficient of variation

Distribution

[9] [6] [7] [8]

0.25–15 1.2–8.2 2.45–9.8 0.15–5.25

6.5 3.5 5.63 2.51

0.7 0.5 0.1 0.68

Lognormal Lognormal Lognormal Gamma

F. Duprat / Construction and Building Materials 21 (2007) 1605–1616

Only a few in-site surveys have been undertaken for structures exposed to marine breezes. The data reported in Tables 2a and 2b relate to pontoon piles in Japan [10], bridge piles in Scotland [11] and bridge girders in Oregon [12]. The altitude of the specimens extracted for measurements lay between 2 m and 35 m above high water level and the distance from sea front did not exceed 250 m. Only data related to faces directly exposed to the sea wind have been retained. The authors did not draw conclusions about the statistical distributions. The mean strengths of concrete observed lie between 20 MPa and 40 MPa, with w/c ratios varying from 0.45 to 0.7, which correspond to those of ordinary concrete structures. A constant surface chloride content is a commonly accepted assumption in the case of structures exposed to de-icing salts and is also well supported by experience [6]. A lognormal distribution of the surface chloride content is taken in the present paper. The mean and coefficient of variation are 3.5 kg/m3 and 0.6, respectively. In a coastal zone, air-borne chlorides carried by the wind accumulate on the concrete surface. This has been appraised by several authors [10,13–15] and such a hypothesis is recommended in the Hetek Manual published by the Danish Road Directorate [16]. According to the latter, a significant increase occurs in the surface concentration between 1 year and 100 years of exposure to a marine atmosphere. For ordinary concrete (without blended materials) with w/c = 0.5 the expected values are C1 = 1.1% and C100 = 7.5% by mass of cement. Table 1b Apparent diffusion coefficient (·1012 m2/s) for structures exposed to deicing salts Reference

Range

Mean

Coefficient of variation

Distribution

[9] [6] [7] [8]

0.03–0.65 0.6–7.5 1–8.2 0.16–1.64

0.15 2 4.3 0.85

0.7 0.75 0.28 0.51

Lognormal Lognormal Lognormal Gamma

Uji [10] suggests resorting to a constant diffusion flux on the concrete surface provided that the environment remains nearly constant. If the surface concentration increases with the square root of time, the chloride content can be expressed as Cðx; tÞ ¼ 2F 0 rffiffiffiffiffiffiffiffi     rffiffiffiffiffiffiffi t x2 x p x  exp   erfc pffiffiffiffiffiffiffi pDa 2 Da t 4Da t 2 Da t ð5Þ where F0 is the diffusion flux on the concrete surface (kg/ m2 s). A mean value of F0 equal to 7.5 · 1011 kg/m2 s and a coefficient of variation of 0.6 have been proposed by Stewart [17] derived from data obtained by Uji [10] and Ohta [13]. A higher mean value of 3.5 · 1010 kg/m2 s is adopted here. Introducing this value in Eq. (5) (with Da = 1.3 · 1012 m2/s) gives a surface concentration of about 12 kg/m3, as has been reported by Cramer [12] after 40 years’ exposure time under very severe conditions. A lognormal distribution was chosen for the diffusion flux with a coefficient of variation of 0.6. Chloride contents obtained from Eqs. (4) and (5) are shown in Fig. 1 using data for a bridge studied by Cramer. Recent outcomes from the survey performed by Castro on concrete specimens exposed to the natural marine environment of Yucata´n [14] show that the chloride surface concentration (and hence the diffusion flux) decreases as the distance from the coastline increases. The mean values of the maximum surface concentration obtained after one year of exposure from a set of specimens made with ordinary concrete (twenty for each distance to sea front) were 8.3 kg/m3 at 50 m, 2.9 kg/m3 at 100 m and 2 kg/m3 at 780 m. An average decrease of 20% at 200 m and 50% at 500 m can be proposed. A significant variation of the diffusion coefficient over time has not been raised by the authors previously quoted for measurements made on existing structures. However, as concrete matures, additional hydration of cement and interaction between chlorides and hydration products

Table 2a Surface concentration (kg/m3) for structures exposed to marine breeze Reference

Range

Mean

Coefficient of variation

[10] [11] [12]

0.08–1.69 0.43–4.1 1.67–17

0.44 1.52 9.75

0.83 0.63 0.51

Table 2b Apparent diffusion coefficient (· 1012 m2/s) for structures exposed to marine breeze Reference

Range

Mean

Coefficient of variation

[11] [12]

1.1–6.81 0.41–1.71

4.75 0.98

0.38 0.57

1607

Fig. 1. Comparison of chloride concentrations.

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F. Duprat / Construction and Building Materials 21 (2007) 1605–1616

occur and consequently the diffusion coefficient is expected to decrease. This decrease is particularly noticeable for concrete containing blended materials like fly ash or silica fume [18,19]. Nevertheless for ordinary concrete without blended materials, it can be stated that the diffusion coefficient remains nearly constant [20]. It is not straightforward to state the mean value of the diffusion coefficient because of the scatter on the experimental data, which underlines the influence of many factors: cement content in concrete mix, weight and volume percentages of cement paste and aggregates, w/c ratio, and degree of cement hydration. Based on relationships obtained from least squares regression among these factors, simplified models for determining the diffusion coefficient have been proposed by Bentz [21] and Papadakis [22]. For an ordinary concrete mix, the outcomes of these models are 4.8 · 1012 m2/s and 4.3 · 1012 m2/s, respectively. Such values of Da are close to the mean values reported by Enright (see Table 1b) and by Wood (see Table 2b) and also close to the value 3.9 · 1012 m2/s provided by the Hetek Manual for the same concrete mix. Unfortunately statistical data on parameters involved in these models are very limited. Hence their utilization in the present probabilistic approach is not envisaged. In this work it was found convenient to propose three mean values of the diffusion coefficient depending on the quality of concrete and in compliance with available data: 1012 m2/s for good quality concrete, 4 · 1012 m2/s for ordinary quality concrete and 7 · 1012 m2/s for poor quality concrete. A lognormal distribution was chosen with a coefficient of variation of 0.7. 2.3. Critical chloride concentration The corrosion of reinforcements is initiated when the chloride content exceeds a threshold value that depassivates the steel embedded in the concrete provided that sufficient moisture and oxygen are present. In a recent study bringing together numerous data from different authors, Alonso et al. [23] pointed out that the critical concentration may lie within a large range, namely 0.2–3% by weight of cement, or approximately 0.5–10 kg/m3 with few values up to 3 kg/m3. This range agrees with the survey by Glass (0.17–2.5%) also combining numerous data [24]. The influence of several factors such as concrete mix proportions, cement type, C3A content of cement, materials incorporated, w/c ratio, relative humidity, and temperature is one reason for the lack of agreement among the measured values. Another reason is the definition of the threshold itself (stated from corrosion potential, visual inspection, corrosion or galvanic current, or anodic polarization) [23]. It is recognized that poor quality of concrete, i.e. low cement content and/or high w/c ratio in the concrete mix, tends to bring down the critical concentration. Although accurate statistical distributions cannot be derived from experimental data, some propositions are reported in Table 3.

Table 3 Statistical distribution of critical chloride concentration (kg/m3) Reference

Mean

Coefficient of variation

Distribution

[25] [26] [17] [7]

1.38 3.4 0.9 1.0

0.2 0.6 0.19 0.1

Normal Lognormal Uniform [0.6–1.2] Lognormal

Table 4 Distribution parameters of critical chloride concentration (in kg/m3) Concrete quality

Field

Mean

Coefficient of variation

Good Ordinary Poor

1.5–2.5 1.0–2.0 0.5–1.5

2 1.5 1

0.14 0.19 0.29

The mean values fall on either side of the value of 1.9 kg/m3 provided by the Hetek Manual for an ordinary concrete mix without blended materials. Because of the wide scatter of the observed values, and also as a consequence of the lack of consistent statistical data, the use of a uniform distribution for the critical chloride concentration seems relevant. The mean of the distribution was chosen so as to reflect three concrete qualities as described in Table 4. 3. Corrosion rate Once the protective oxide layer around the reinforcement bars has been broken down the corrosion is initiated and occurs at a rate depending on moisture and oxygen availability in the neighborhood of the rebars. The corrosion current has been recognized to be a suitable and relevant indicator of the corrosion rate because it relates directly to the corrosion penetration depth in the steel. In the case of uniform corrosion, a corrosion current of 1 lA/cm2 leads theoretically to a penetration depth of 11.6 lm/year [27]. From experimental investigations, Gonzalez indicates that values of 1–3 lA/cm2 are frequent in active corrosion and equivalent to a penetration depth of 11–33 lm/year [28]. Based on a survey by Yokozeki et al. [29], Vu proposed an empirical expression for a constant annual corrosion rate as a function of the depth of the reinforcements, which is related to the oxygen availability, and the w/c ratio, which is related to the moisture, and are more convenient to use [30] I corr ð1Þ ¼

3:78ð1  w=cÞ d

1:64

ð6Þ

where Icorr is the corrosion current (lA/cm2) and d is the depth of the reinforcement (cm). Eq. (6) has been found to be in agreement with experimental results. After an immersion of 6 years in sea water, the average corrosion rate for ordinary concretes specimens (Portland cement content = 400 kg/m3, w/c = 0.6) containing rebars with a cover of 2–5 cm is close to 87 lm/year, and corresponds to a corrosion current of 7.5 lA/cm2 (maxi-

F. Duprat / Construction and Building Materials 21 (2007) 1605–1616

mum values lying between 10 and 20 lA/cm2) [28]. Eq. (6) gives Icorr(1) = 7.5 lA/cm2 with w/c = 0.6 and d = 2.3 cm. For bridge decks exposed to de-icing salt, Enright indicates corrosion rates varying from 13 lm/year up to 127 lm/year for ordinary concrete with an average compressive strength of 20.7 MPa [7]. Enright suggests an average rate of 76 lm/ year for a concrete cover of 6.9 cm corresponding to a corrosion current of 6.5 lA/cm2. Eq. (6) gives Icorr(1) = 6.5 lA/cm2 with w/c = 0.76 and d = 6 cm. As pointed out by Liu and Weyers [31] and Yalc¸yn and Ergun [32], the corrosion rate decreases rapidly during the first few years after initiation and then tends to a nearly constant value. This results from the formation of rust products which slow down the diffusion of irons away from the steel surface. In the present paper Eq. (6) is used to estimate the corrosion current, weighted by the factor t0.3 where t is the corrosion time (in years), and with a w/c ratio corresponding to the quality of concrete (0.57 for ordinary, 0.65 for poor and 0.5 for good). 4. Loss of reinforcing steel cross-section After an effective corrosion period of T years, the penetration depth (in cm) into the reinforcement can be expressed as   Z T 4 0:3 pðT Þ ¼ 11:6  10 I corr ð1Þ 1 þ t dt 

1

T0:7  1 ¼ 11:6  104 I corr ð1Þ 1 þ 0:7

8 2 pD0 > > < 4  A1  A2 Ar ðT Þ ¼ A1  A2 > > : 0

1609

if pðT Þ 6 pD0ffiffi2 if

D0ffiffi p 2

6 pðT Þ 6 D0

if pðT Þ > D0

where

i h  2 Þ2 A1 ¼ 12 a1 D20  a D20  pðT D0 h i 2 Þ2 A2 ¼ 12 a2 pðT Þ  a pðT D0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ 2pðT Þ 1  ðpðT Þ=D0 Þ2

a1 ¼ 2 arcsinða=D0 Þ a2 ¼ 2 arcsinða=2pðT ÞÞ 5. Illustrative example A typical simple-span floor slab girder will be considered (Fig. 3). The web and the base slab are precast. The topping slab is cast-in-place on top of the precast units. The characteristic value of the industrial live load is Qk = 4.5 kN/m2. The design dimensions are hc = 20 cm, hw = 55 cm and

 ð7Þ

Eq. (7) applies in the case of uniform corrosion arising from concrete carbonation for example. For localized pitting corrosion, typically under chloride-ingress, the maximum penetration pitting pmax exceeds the average penetration given by Eq. (7). Gonzalez et al. found from their experimental results that the ratio R = pmax/p(T) varied from 2.8 to 8.9 [28]. A uniform distribution for R between 3 and 9 is taken here because of the lack of statistical data. If a hemispherical form is assumed for pits (Fig. 2), the net cross-sectional area of a corroded bar is calculated by [7]

Fig. 2. Pit configuration.

ð8Þ

Fig. 3. Floor slab girder.

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F. Duprat / Construction and Building Materials 21 (2007) 1605–1616

bw = 35 cm. Two stirrups are used as shear reinforcement and the spacing between stirrups near to the supports is s. Stirrups are not exactly centered in the web and a slight eccentricity ec of their position is expected to occur. The reinforcement is designed in compliance with the French BAEL rules [33]. Three design cracking conditions are considered, denoted slightly severe, severe and very severe conditions. In addition the concrete cover is specified according to the three exposure conditions applied in practice in connection with the earlier cracking conditions, denoted slightly aggressive, aggressive and very aggressive conditions. The characteristic values of the concrete compressive strength are fc28 = 22 MPa, fc28 = 25 MPa, fc28 = 28 MPa for a concrete of poor, ordinary or good quality. The characteristic value of the steel yield strength is fy = 500 MPa. These values correspond to the 5% percentile of the distributions. The concrete strength is involved in the calculation of the flexural reinforcement only under the assumption of the slightly severe cracking condition (Ultimate Limit State calculation). For the other cracking conditions the Serviceability Limit State criterion on steel stress applies. Hence the difference between the cross-sectional areas of reinforcing bars due to the quality of concrete (and occurring only for the ULS calculation) is not significant and will be neglected in what follows. Design specifications are reported in Table 5. For the SLS calculation of flexural reinforcement the tensile steel stress is limited according to BAEL provision A.4.5.3 by

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  rs ¼ min 2=3f y ; maxð0:5f y ; 110 1:6f t28 Þ ð9Þ where ft28 is the concrete tensile strength. This limit is recommended by the rules so as to avoid excessive cracking of concrete, which could facilitate the ingress of chlorides or other aggressive chemical agents and compromise the structural resistance. Nevertheless, as indicated earlier, the flexural cracking of concrete structures under service load does not affect the propagation process of corrosion and is not a determining factor in the structural behavior with time, even if reinforcement is calculated according to the SLS criterion [1]. As a consequence the present paper addresses the structural reliability of the beam towards ULS rather than towards SLS. The flexural strength of the beam corresponds to the top of the moment-curvature diagram (Fig. 4). For a fixed cur-

Fig. 4. Flexural strength definition.

vature the neutral axis position is calculated from the force equilibrium conditions. Sargin’s strain–stress relationship is used for calculating the concrete force. Its advantage resides in the fact that it can be analytically integrated over the compression zone. The connection between flange and web is only ensured by the tying transverse reinforcement (because of the construction system of the floor). Consequently the effective flange width taken into account in the calculation of bending moment is computed from the cross-sectional area of the tying rebar and the height of the compressive zone, in an iterative procedure. The shear strength of the beam is estimated in accordance with BAEL provision A.5.1.2 (with inclination of concrete struts equal to 45). The decrease of bond strength between the concrete and reinforcements occurring particularly for general corrosion affects the structural behaviour as far as the deflection under service load is concerned, because of the subsequent reduction of tension stiffening effect. Nevertheless, in the case of localized corrosion where loss of bond is located around pits, the ultimate flexure and shear strengths are very slightly modified due to the decrease of bond strength. Classical structural models assuming a perfect bond between concrete and steel have thus been found to be in fairly good agreement with experimental results [34]. This assumption is adopted here. Error factors are applied to the outcomes of structural models, respectively aM and aV to the flexure and shear strengths, in order to account for inaccuracies and approximations made in modelling. The statistical features of aM and aV are proposed on the basis of comparison between

Table 5 Design specifications Cracking condition

Flexural reinforcements (2 layers) Bottom

Upper

Slightly severe

4Ø25

Severe

2Ø32 2Ø25 4Ø32

Very severe

Stirrups

Transverse ties

Spacing s (mm)

Covers cl, cb (mm)

Cover cu (mm)

2Ø25 2Ø20 4Ø25

4 Ø8

1Ø10

170

25

15

4Ø8

1Ø10

170

32

30

4Ø32

4Ø8

1Ø10

110

50

50

F. Duprat / Construction and Building Materials 21 (2007) 1605–1616

experimental strengths and theoretical resistances obtained from models similar to those used in the present study. The mean value of aM and aV is expected to be equal to one. McGregor indicates a coefficient of variation of 0.11 for aM in the case of the ACI model for the flexure strength [35], and Nowak suggests a coefficient of variation of 0.12 in the case of the AASHTO model [35]. A coefficient of variation of 0.18 for aV was taken by Zararis in the case of the EC2 model for the shear strength. [36]. It is assumed that all the external sides of the structure are subjected to chloride-ingress. 6. Probabilistic approach Two ultimate limit states are considered here with time t. The flexure strength margin at mid-span can be expressed as EM ðtÞ ¼ M R ðtÞ  M S ðtÞ

ð10Þ

and the shear strength margin at a support can be expressed as EV ðtÞ ¼ V R ðtÞ  V S ðtÞ

ð11Þ

6.1. Random variables The variables previously described and related to exposure conditions and corrosion kinetics are reported in Table 6.

1611

The geometrical variables together with the mechanical properties of materials considered as random variables are reported in Table 7. They were defined from propositions and data available in the literature [37–39]. The coefficient of variation of the concrete strength depends on its quality and is lower as the quality increases. The coefficient of variation of the lateral cover cl is generally high and reaches 30% in certain cases [7]. This value is not surprising because the uncertainty on the cover results from the eccentricity of the stirrups ec together with the deviation of the dimensions bw and bs: cl = (bw  bs)/2 + ec. In the case of precast units the coefficient of variation of cl is reduced to about 24% (maximum value). For the covers cu and cb the coefficients of variation correspond to various exposure conditions. The characteristic value of the live load is defined according to structural life-cycle of 50 years and to a 90% percentile of the distribution. The Gumbel distribution was chosen here. 6.2. Probability of failure Because of the continuous degradation of the structure with time, a time-dependant reliability analysis has to be applied. Such an analysis requires the features of the stochastic process of loading to be established and particularly its correlation function for the whole structural life-cycle envisaged. Due to the lack of data and for the sake of simplicity, a non-stochastic discrete approach may also

Table 6 Environmental variables Variables

Mean or range

Coefficient of variance

Distribution

Surface concentration, C0 (kg/m3) Surface flux, F0 (kg/m2 s) R factor

3.5 3.5 · 1010 [3–9]

0.6 0.6 0.29

Lognormal Lognormal Uniform

Diffusion coefficient, Da (m2/s) Critical concentration, Ccr (kg/m3) Corrosion current, Icorr (lA/cm2)

Quality of concrete Poor 7 · 1012 [0.5–1.5] 21.1 t0.3/c

0.7 [0.14–0.29]

Lognormal Uniform

Ordinary 4 · 1012 [1–2] 15.1 t0.3/c

Good 1012 [1.5–2.5] 11.8 t0.3/c

Table 7 Geometrical and mechanical variables Variables

Mean

Coefficient of variance

Distribution

Steel strength, fs (MPa) Concrete Strength, fc28 (MPa) Web width, bw (cm) Web height, hw (cm) Flange thickness, hc (cm) Stirrup width, bs (cm)a Stirrup eccentricity, ec (cm) Stirrup spacing, s (cm)a Cover, cu or cb (cm)a Spacing between steel layers, dl (cm)a Flexure model uncertainty, aM Shear model uncertainty, aV

xk/(1–1.56 cvx) xk/(1–1.56 cvx) xd = 35 xd = 55 xd = 20 xd = (35; 28.6; 25) 0 xd = (17; 17; 11) xd = (2.5; 3.2; 5) xd = (2.5; 3.2; 3.2) 1.0 1.1

0.10 (0.20; 0.15; 0.10) rx = 0.5 rx = 1.0 0.05 rx = 0.5 rx = 0.5 0,10 (0.20; 0.15; 0.10) 0.15 0.10 0.15

Lognormal Lognormal Normal Normal Normal Normal Normal Normal Lognormal Lognormal Normal Normal

xd: design value, xk: characteristic value, cvx: coefficient of variation, rx: standard deviation. a Depends on cracking or exposure design conditions.

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F. Duprat / Construction and Building Materials 21 (2007) 1605–1616

be relevant if the discretized periods are long enough for the correlation between periods to be negligible. For industrial warehouse loading, the length of independent periods can be estimated at 2 years [37,40]. The elementary 2 year probability of failure for any ith period is expressed as:

is the frontier between the safe set and the failure set. In order to compute b, the random physical variables with any probability functions are transformed into normalized and uncorrelated variables. The reliability index was computed in this study using the ComRel software [41].

P f2 ðiÞ ¼ Prob½Ri 6 S i  ¼ Prob½ðM R 6 M S Þi [ ðV R 6 V S Þi 

7. Results

ð12Þ where MS and VS are flexure and shear actions. For each limit state the reliability index is computed, e.g. bM and bV. It follows from a FORM approach that: P f2 ðiÞ ¼ UðbM Þ þ UðbV Þ  U2 ðbM ; bV ; qMV Þ

ð13Þ

where U(), U2() are the standard normal and binormal distribution functions and qMV is the correlation between limit states. After T = 2n years the cumulative probability of failure of the series system of n independent events is expressed as P f ðT Þ ¼ 1  Prob½ðR1 > S 1 Þ \ ðR2 > S 2 Þ \    \ ðRn > S n Þ n Y ¼1 ð1  P f2 ðiÞÞ ¼ Prob½t 6 T  ð14Þ

The following results are expressed in terms of the reliability index instead of the probability of failure so that the outcomes can be easily compared to the minimum reliability index recommended in design codes for ultimate limit states. The relationship between reliability index and probability of failure is given by P f ¼ UðbÞ

and

b ¼ U1 ð1  P f Þ

ð16Þ

The reliability is estimated for the three cracking conditions and for the four exposure conditions. For each case three qualities of concrete are considered. The exposure time ranges between 1 and 100 years. 7.1. Basic results

i¼1

Eq. (14) is the cumulative distribution of the structural lifetime T. The conditional probability that the structure fails after 2(n + 1) years given that it has survived 2n earlier years can be deduced from Eq. (14) P f ðT Þ ¼ Prob½ðRnþ1 6 S nþ1 ÞjðRn > S n Þ Prob½ðRnþ1 6 S nþ1 Þ \ ðRn > S n Þ Prob½Rn > S n  P f ðT þ 2Þ  P f ðT Þ ¼ 1  P f ðT Þ ¼

ð15Þ

The reliability index used here is the Hasofer–Lind reliability index b. This is defined in the standard space as the minimum distance from the origin to the failure surface, which

The variation of the reliability index as a function of time is shown in Figs. 5 and 6, for all exposure conditions to chloride-ingress and for the design specifications corresponding to the slightly severe cracking condition. An ordinary quality of concrete is considered. It can be seen in Fig. 5 that large negative values are reached for T=100 years in the case of a non-conditional estimation of the reliability (cumulative probability without updating). For such values of the reliability index, the probability of failure exceeds 0.5 and proven cases of failure would be expected to have been numerous or at least more numerous than those recognized. Thus it seems more relevant to assess the reliability by using the conditional probability of failure, which integrates the survival

Fig. 5. Variations of the reliability index (Pf non-updated).

F. Duprat / Construction and Building Materials 21 (2007) 1605–1616

period of the structure previous to the time at which the reliability is estimated. 7.2. Influence of the exposure condition The corrosion-induced degradation of the beam leads to a broad increase of the risk of failure as shown in Fig. 6: the risk is multiplied by more than 550 after T = 50 years, and more than 14,800 after T = 100 years in the case of exposure to de-icing salts or at the sea front. This increase is more marked for exposure to de-icing salts in the first forty years. After this time, the risk level is similar for both types of exposure because the air-borne chloride surface concentration also reaches high values. Different choices for the design cracking condition for calculating the cross-sectional area of reinforcement and

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the associated design specifications for the concrete cover (see Table 5) yield radically different subsequent reliability levels as shown in Fig. 7. If the beam is designed according to the very severe cracking condition, the reliability index at the end of the design life-cycle (T = 50 years) remains greater than the threshold value of 3.8 recommended by design codes for ultimate limit states (dotted line on Fig. 7). Hence for exposure at the sea front, only a slight decrease of b* can be seen from 5.4 to 3.2 at T = 100 years. In contrast the reliability index is lower than 3.8 at the end of the design life-cycle if design specifications according to the slightly severe cracking condition are applied. Even for the exposure at a distance of 500 m from the coast b* falls from 4.3 to 1 at T = 100 years. A deep concrete cover depth allows the reinforcing bars to be protected from chlorides longer, thus lengthening the

Fig. 6. Variations of the reliability index (P f updated).

Fig. 7. Variations of the updated reliability index (exposure to marine breeze).

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time before corrosion initiation. Moreover it slows down the corrosion rate after corrosion is initiated because of a lower oxygen availability in the neighborhood of rebars. This well known favorable effect restricts the increase of the corrosion risk with time. 7.3. Influence of the quality of concrete The following figures show the variation of the reliability for three qualities of concrete associated with two cracking conditions. In Fig. 8 exposure at the sea front is considered while in Fig. 9 exposure to de-icing salts is considered. The influence of the quality of the concrete is clear in every case. Under the SLS criterion, the design of the beam is not modified according to the quality of concrete and, provided

that the characteristic value of the concrete strength is specified in compliance with design rules, the structural reliability remains constant as long as no deterioration occurs (as shown on the curves at the beginning of exposure). The duration of this first protective period depends on the quality of concrete and on the concrete cover depth (design cracking condition). It can be estimated from curves and is reported in Table 8 for all cases. Once corrosion starts to act, the rate of decrease of the reliability index with time is obviously more pronounced when the quality of concrete is lower, whatever design cracking condition is applied. The application of the most severe design specifications for reinforcement and concrete cover cannot therefore be deemed to guarantee a satisfactory reliability level during the life-cycle of the structure. Hence the reliability index obtained for these design specifications is lower than 3.8 for T = 50 years.

Fig. 8. Variations of the updated reliability index for various qualities of concrete (exposure at sea front).

Fig. 9. Variations of the updated reliability index for various qualities of concrete (exposure to de-icing salts).

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Table 8 Duration of the protective period

References

Exposure

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Design cracking condition

Quality of concrete Good

Ordinary Poor

Sea front

Slightly severe 15 years Very severe 45 years

10 years 30 years

<5 years 15 years

De-icing salts

Slightly severe 10 years Very severe 40 years

<5 years <5 years 15 years <5 years

From a probabilistic and quantitative point of view, the overall results of this study justify the necessity for good engineering practice in design as well as in construction to impose specific requirements for the quality of concrete in addition to the requirements for the cover of embedded steel bars and rules for determining their cross-sectional area. 8. Conclusion Because of the random uncertainties affecting the transport and mechanical properties of concrete, the environmental conditions, and the loading and geometrical features of structures, it is necessary to resort to a probabilistic approach to assess the reliability of reinforced concrete structures with corroded embedded steel bars. Such an approach has been carried out in the present study for concrete slab girders exposed to de-icing salts (or equivalent saline industrial solutions) or to a marine breeze at various distances from the coast (or equivalent saline atmosphere with higher or lower concentrations). Based on the computation of the Hasofer–Lind index for flexure and shear, a time-updated estimation of the reliability has been developed allowing the previous survival period of the structure to loading under degradation to be taken into account. Laws of probability have been chosen from various publications available today, bringing together numerous statistical data. The beam was designed in compliance with the French BAEL rules and according to more or less severe design specifications for concrete cover and calculation of the cross-sectional area of steel bars depending on the expected degree of aggressiveness of the environment and the possibility of concrete cracking. The favorable influence on reliability of the quality of concrete as well as the application of severe design specifications is clearly shown. Nevertheless the risk of failure increases continuously as the structure deteriorates even if the most severe design specifications tend to attenuate this effect. In order to obtain a satisfactory reliability level for the whole design life-cycle of concrete structures, it seems obvious from such quantitative outcomes that specific requirements for concrete properties should be combined with the requirements for concrete cover and the calculation of reinforcements. Efforts currently being made in that sense by numerous research laboratories and organizations, and by the committees drawing up design codes are significant and still necessary to give concrete structures better durability.

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