1.1 Initiation period

Carbonation and chloride ingress are identified as the two main causes of rebar corrosion (^{Bazant, 1979}; ^{Tuutti, 1982}; ^{CONTECVET Manual, 2001}; ^{DURACRETE, 1998}; ^{MC 2010, 2012}). Carbonation is a process in which its rate is almost constant along the life, and then the typical square root law can be applied (see equation [1]) where V _CO2 is the rate of carbonation under reference conditions and x_c is the carbonation depth.

This V _CO2 depends on the chosen concrete mix (w/c-ratio, cement type and mineral additions) and on the influence of the execution or of the environmental conditions (mean relative humidity and CO2-concentration). In general, the carbonation progresses faster when the RH if the exterior of concrete is between 45 and 65%. The carbonation depth decreases with higher RH (^{Bakker, 1994}) (figure 1). The introduction of this effect into equation [1] is made by the available models in several manners. They are:

By using a root different (for instance cubic root) than the square. The result is in general an attenuation of the carbonation depth

By multiplying by a « climatic factor » (W_Rain) (^{Bakker, 1994}; ^{Galan et al. 2010}) usually smaller than the unity, for instance by taking into account « the wetness period » per year (days of rain per year with respect to 365).

In figure 1 it is given results of the comparison of rates of carbonation (^{Galan et al. 2010}) of tests of specimens in Madrid’s climate during 4 years of two kind of concretes (with w/c ratios of 0.45 and 0.6 and in accordance, 400 and 300 Kg of cement by m³ of concrete) exposed outdoor sheltered and exposed to rain. The resulting equation in these tests is given next. This relation may be different in other climates.

Figure 1 Relation between the velocity of carbonation in non-sheltered from rain and protected from rain conditions in the climate of Madrid. 

This equation leads to apply an exponent of 0.31 instead of 0.5 in [1’] and as indicated in the figure 1, means that the rate of carbonation exposed to rain is approximately the half of that when the same concrete is sheltered form rain (Madrid climate).

In the case of chloride penetration, the calculation is made through the expression named “equation of the error function” (equation [2]) where C(x, t) is the critical chloride concentration leading to depassivation inside the concrete at a distance x for a time t, C_s is the surface or external chloride concentration, and D_ap is the Apparent Diffusion Coefficient:

With respect to the chloride ingress three main limitations have been identified in the application of equation [2] (^{Andrade, 2014}):

The decrease with time of the chloride diffusion coefficient is empirically addressed (^{Mangat and Molloy, 1994}) by considering that D_ap decreases with time (figure 2) following equation [3] where n=“aging factor”, D₀ = the diffusion coefficient at the initial testing time t₀ (usually 28 days) and t_i is the period until when the coefficient is assumed to decrease:

This equation indicates that the initial value of the D_ap is changed for a smaller one for the calculations in equation [2]. Although generally accepted, it is not indicated how or when to test the decrease and the period of time of its application. In consequence, if the period of application is long time, it leads into a too low diffusion coefficient as shown in figure 2, where it can be deduced that depending on the aging factor (n) considered, the decrease may be of one or two orders of magnitude. Then, due to the uncertainties on the extrapolation from a short term test to long term, the decrease has to be limited. One possible “good practice” is to limit, depending on the type of cement, the decrease to 1year (Portland cement), 5 years (cement with mineral addition other than slags or limestone) or 10 years (cements with slag as mineral addition) or to a decrease of to limit to a maximum of one order of magnitude. Finally, a modification to this equation has been proposed by Gulikers (^{Tang and Gulikers, 2007}) (see equation [4])

By calculating D_ap through this expression, the same result is obtained by means of finite elements methods or by the analytical direct calculation of equation [2].

Figure 2 Decrease of Apparent Diffusion coefficient with time in function of aging factors ranging from 0.3 to 0.7. 

To overcome the problem of a variable surface concentration it is viable to use the surface concentration obtained in a natural diffusion test lasting 90 days. This time is long enough to allow biding to reach equilibrium with the external chloride concentration.

The other main limitation is related to the building up of a “pic” or maximum in the chloride concentration of a profile (figure 3-left) (^{Andrade, 2015}). This maximum has been attributed to “convection” by capillary absorption , although it seems more likely that it is due to the carbonation of the surface, but these reasons are not sufficient to explain how it is possible that the surface concentration could be lower than the external one, because the chloride concentration is almost nil at the concrete surface in figure 3. It has to be also remarked, that, in order to calculate the Apparent diffusion coefficient, Dap,, equation [2] can be fitted into this profile shape with the maximum, providing the “zero” depth (figure 3-right) is considered that of the maximum position (^{Andrade, 2015}) because the profile from the maximum evolves following Fick’s law. That is, the fitting of equation [2] should not be made by extrapolating the maximum back to the surface, because the surface concentration would be much higher than the real one. The value of x0 in the figure (distance from surface to the maximum position) should be added to the result of calculation of the depth of the chloride threshold (^{Andrade, 2015}) from equation [2]. What is unknown is how the chlorides deplete at the surface and how the maximum moves inwards, that is, the law of its evolution with time to long term.

Figure 3 Left: Typical maximum in chloride concentration in the interior of the concrete and not at its surface. Right: extrapolation to obtain the chloride surface concentration to be used in the calculations 

1.2 Depassivation step

It is assumed that depassivation means starting of corrosion, but it is not easy to identify the onset of corrosion in the real structures because the initial depassivated zone can be so small (figure 4) that it is not noticed. No external sign can be detected until concrete cover cracks or corrosion spots appear on the surface. No changes in the serviceability of the concrete are detected at depassivation onset. Then, unless electrochemical monitoring of the corrosion parameters is made, there is not the possibility to detect corrosion initiation. Even performing an electrochemical monitoring, the identification of depassivation in real conditions is not an easy task due to the changes in temperature and moisture. This difficulty of how to detect and measure the depassivation onset will not be addressed in present work; however some considerations on the consequences of this fact will be mentioned next.

Figure 4 Small spots of corrosion due to chloride attack 

Two main aspects will be analysed on the depassivation process: its duration and the variability of the chloride threshold. The duration because depassivation is not an instantaneous process, but it lasts a period of time due to the intrinsic heterogeneity of the concrete which make the threshold front to be irregular.

For carbonation the corrosion threshold is reached when the pH value drops below 8-9, which precisely coincide with the pnolphthalein colour-change from purple to colourless. Due to the presence of aggregates this carbonation front is not fully steep (^{Thiery, 2007}) but additionally is irregular and then the surface of the bar will be depassivated irregularly (figure 5), what is maintained until the front advances deeper. The variability of the front can be characterized by an averaged value and a standard deviation (for instance resulting in between the 15% and the 25% of the coefficient of variation).

Figure 5 Irregularities that may appear in the carbonation front which lead into a localized depassivated rebar surface. 

With respect to the chloride process, the corrosion starts when certain critical chloride concentration entries in contact to the steel surface. The normal value considered in numerous standards is of 0.4% by weight of cement, although values as higher as 2-3% may not developed corrosion in some circumstances. Then, the threshold is not a fixed quantity, but is a range of values (^{Izquierdo et al. 2004}; ^{Markeset, 2009}) that can be treated statistically. The distribution from Izquierdo et al. (^{Izquierdo et al. 2004}) is given in figure 6. It has an averaged value of Cx= 0.70 % (by concrete mass) with a standard deviation of ± 0.20%. The statistical distribution of Markeset (^{Markeset, 2009}) found in real structures is almost similar (bridges in Norway where cores were drilled until the bar and observation of incipient corrosion was identified as the chloride threshold).

Figure 6 Statistical distributions of the chloride threshold by Izquierdo et al (plain line) (^{Izquierdo et al. 2004}) made in laboratory conditions and by Markeset (discontinuous line) (^{Markeset, 2009}) measured in real structures. 

1.3 Propagation of corrosion

The progressive corrosion advance and loss in cross diameter of the bar and the oxides formed will induce several structural consequences which were described in ^{CONTECVET Manual (2001)} (figure 7):

(a) Decrease in cross section and bar ductility.

(b) Cover cracking

(c) Steel/concrete bond deterioration and

(d) Load-bearing capacity loss in regarding ULS.

Figure 7 Consequences of rebar corrosion (^{CONTECVET Manual, 2001}). 

In order to calculate the time to reach one of these “failure” conditions it is necessary to have a model for the evolution of the bar diameter loss. A model was proposed years ago by Andrade et al (^{Andrade, 1989}; ^{Andrade, 2004}) based in that of Tuutti (^{Tuutti, 1982}). The preliminary model has been updated more recently (^{Andrade, 2017}) which is shown in figure 8 as a bilinear propagation model expressed by the equation [5]:

Where tp is the propagation period, Pcorr is the loss in bar diameter or “accumulated corrosion”, Vcorr is the corrosion rate, (0 is the initial diameter, (t is the final one after t years of corrosion. The conversion factor between Vcorr and Icorr is: Vcorr (mm/year) = 0.0116 ( Icorr ((A/cm2). Figure 8 presents the complete service life model (^{Tuutti, 1982}; ^{Andrade, 2017}) having in mind the possibility that the corrosion rate is not only constant (line A in figure 8) but that it might be also presenting a two-step behaviour: line B would be the case when the corrosion decreases with time and line C that when it increases with time (for example when the cover cracks due to the corrosion in very wet environment). It also shows a “failure condition” of cracking (limit state of cover cracking parallel to the bars) depending on the degree of corrosion (^{Andrade, 2017}).

The corrosion rate depends on the concrete porosity, the degree of water saturation and the temperature of each local environment. Taking the exposure classes defined in the European standard EN 206, in Table 1 are given values of the Vcorr (considering linear its progress) and time of wetness (Wt) with the expected standard deviations (^{CONTECVET Manual, 2001}; ^{DURACRETE, 1998}). It is also given the pitting factor, (.

Figure 8 Service life model with the trends of the propagation period 

As the climate may be constant or repeated every year, the accumulated corrosion or corrosion depth, Pcorr, (see figure 9), which is the loss in diameter Pcorr, = ((= (0- (i, can be, as previously commented, linear with respect to the time (line A in figure 8) or can be considered a two-lineal trend where the corrosion decreases with time (Vcorr1>Vcorr2 -line B) or increases (Vcorr1<Vcorr2 - line C).

Figure 9 Evolution of the loss in diameter Pcorr with the advance of the aggressive front. 

If the corrosion is localized, there is a dependence of the maximum pit depth from the homogeneous corrosion by which it is possible to establish that (^{Gonzalez et al,1995}, ^{CONTECVET Manual, 2001}):

Where α is the “pitting factor” (see table 3) which accounts for the maximum pit depth. That is, the maximum pit depth is ( times deeper than if the corrosion would have been homogeneous. The average value is α =10, although it may range in the case of chloride attack from 3 to 15 times.

2.1 Example: Verification of cover cracking induced corrosion

For the sake of describing an example, cover cracking induced corrosion (parallel to the bars and not transversal to them due to the flexural loads) is proposed to be considered as a SLS, because when the cover in parallel to the bars a “condition state beyond that of the design” is reached in the same manner than when the cracks are transversal to the bars. The limit state of cracking induced corrosion can be calculated as suggested in MC2010 (^{MC 2010, 2012}) through the equation [8]:

where: P{} is probability that parallel cracks are produced, tSL is the design service life [years], tini is initiation period [years], tprop is propagation period [years], P0 is target failure probability. Figure 10 shows for the sake of illustration of the example, the probability of cover cracking (for bar diameter loss of Pcorr= 100 (m (^{Andrade et al. 2017}) associated to the cover cracking) for several chloride diffusion coefficients (and corrosion rates associated to the concrete quality) in a hypothetical element of 5 cm in cover and a service life of 100 years.

The most important deductions from these figures are that the cover cracking probability is not unique, but varies with the rate of chloride ingress and of the corrosion rate. Other input parameters, as the aging factor or the coefficient of variation also influence the probability of reaching Pcorr= 100 (m of diameter loss within the life time period. It is worth to notice that, logically, the probability of cover cracking when reaching 100 (m of diameter loss is smaller as the concrete is of higher quality, because the depassivation will be produced later. Reversely, lower quality concretes show that a higher probability is found at the end of the target service life. Then, it cannot be established a single value as a unique failure probability, but would depend on the concrete quality. The fact that the reliability is structure-dependent is general and that is why the semi-probabilistic format with general “safety factors” is the normal method in the existing Codes. In order to take a particular safety factor numerous cases and their economic consequences have to be analyzed, in order to select the optimum giving the highest safety at the minimum cost.

Figure 10 In the upper part is given the time to reach Pcorr = 100 (m of diameter loss for several Dap,Cl . In the bottom part is given the probability of failure which is different depending on the value of Dap,Cl and associated corrosion rate. 

2.2 Verification of ULS

The methodology of calculation of the ULS is similar to the one of calculation for the DLS or SLS through the determination of the accumulated corrosion, Pcorr, from the exterior part of the reinforcement (figure 9) and the consideration of the corresponding general or localized loss of section. Then, from the reduced section of the bar, concrete and from the decreased steel/concrete bond, the loss in load bearing capacity of the concrete section can be recalculated. It is out of present work a detailed explanation. This detailed explanation can be found in Contecvet Manual (^{CONTECVET Manual, 2001}) and the published Reports from Duracrete project (^{DURACRETE, 1998}).