Emilio Bastidas-Arteaga

Mauricio Sánchez-Silva

Alaa Chateauneuf

Moema Ribas

Coupled reliability model of biodeterioration, chloride ingress and cracking for reinforced concrete structures

Keywords: Biodeterioration, chloride ingress, corrosion, crack initiation, crack propagation, reliability

Maintaining and operating civil infrastructure systems has been recognized as a critical issue worldwide. Among all possible causes of safety reduction during the structural lifetime, deterioration is particularly important. Structural deterioration is usually a slow time-dependent process controlled by safety and operation threshold specifications. This paper presents a model of RC deterioration by coupling biodeterioration (i.e., chemical, physical and mechanical action of live organisms), steel corrosion, and concrete cracking. The final purpose of the model is to compute the reduction of the concrete section and the area of steel reinforcement in order to assess the change of structural capacity with time. Given the uncertainties in both the parameters and the model, the probabilistic nature of loads, the material properties and the difussion process are taken into account to evaluate structural reliability. The model is illustrated with an example where the inelastic behavior of a pile subject to random loading is considered. The results of the analysis have shown that the effect of biodeterioration on the structural performance is significant and can cause an important reduction of its lifetime. On the whole, The paper states that modeling the effects of biodeterioration in RC structures should be included as part of infrastructure planning and design, especially, when they are located in aggressive environments.

Introduction

Civil infrastructure systems are critical assets for any country's socioeconomic development. Designing these systems for a particular service lifetime and maintaining them in operation has been recognized as a critical issue worldwide. The life-cycle of a structure can be thought of as the time during which the structure is able to meet specific technical requirements with an acceptable level of safety. In lifecycle analysis, the technical problem focuses on defining the interaction among three main processes:

(1) the structural deterioration (e.g., aging, obsolescence), (2) the occurrence of unexpected extreme events (e.g., earthquakes, floods); and (3) the maintenance/rehabilitation program [START_REF] Santander | Design and maintenance-program optimization for large infrastructure systems[END_REF] [START_REF] Santander | Design and maintenance-program optimization for large infrastructure systems[END_REF]). This paper will focus only on the first aspect. Nowadays, many deteriorated structures are evaluated for possible repair and continued service because they are in a situation where their replacement would be economically unfeasible. Thus, developing robust models for prediction and strategies for periodic inspection and maintenance play a significant role in enabling target reliabilities to be met over a period of continued service (Mori and Ellingwood, 1995 [6], Clifton, 1993 [7]).

Deterioration is common in structures located in aggressive environments and subjected to, for instance, sulfate attack or chloride penetration causing corrosion and biodeterioration. Numerous studies have addressed the problem of chloride ingress in concrete structures. However, in addition to chloride penetration, biological processes can accelerate the degradation process by modifying severely the structural durability and reliability. This aspect is particularly important in marine structures (e.g., ports and offshore platforms), sewage systems and waste water treatment plants (Piedrahita, 2004 [2]; Cho and Mori, 1995[3]; Morton et al., 1991[4]).

The cause of failure of reinforced concrete (RC) structures subject to deterioration is a loss of capacity resulting from a combination of steel corrosion and concrete cracking. Corrosion leads to a loss of the effective area of the reinforcement and has proven to be a common cause of failure. In a study conducted by CC Technologies Laboratories Inc. in 2001 [START_REF]Corrosion Costs and Preventive Strategies in the United States, Report by to Federal[END_REF], it was found that the total direct cost of corrosion in USA is close to US$137.9 billion/year. The industrial and infrastructure sectors more susceptible to corrosion damage are shown in Table 1. On the other hand, concrete cracking is the result of many factors among which loading, steel corrosion and the action of live organisms are of particular interest.

Structural deterioration is usually a slow time-dependent process controlled by a safety/operation threshold specification. In RC structures, cover reduction is caused by the joint action of live organisms and the accumulation of large amounts of chlorides and carbon dioxide on the structure surface. As concrete cover is the principal mean of protection from corrosion, live organisms facilitate the diffusion process by eroding the concrete surface and destroying the concrete microstructure. This accelerates the initiation of steel corrosion and increases the corrosion rate. Predicting the deterioration of RC structures due to chloride penetration when accelerated by live organisms depends on many factors. It is first necessary that the chloride and carbon dioxide concentration on the structural surface become high enough. Besides, it is required that the environment provides conditions for live organisms to survive and to carry out their vital processes, i.e., appropriate climate conditions, temperature, light intensity, oxygen availability, surface roughness and so forth. Biodeterioration of concrete structures is a subject that requires further analytical and experimental investigation.

This paper presents a work on modeling RC deterioration by coupling biodeterioration (i.e., action of live organisms), steel corrosion, and concrete cracking. The final purpose of the model is to compute the reduction of the concrete section and the area of steel reinforcement in order to assess the change of structural capacity with time. Given the uncertainties in both the parameters and the model, the probabilistic nature of loads, the material properties and the difussion process are taken into account to evaluate structural reliability. The RC deterioration is discussed in section 2, reinforcement corrosion induced by chloride ingress is treated in section 3 and the proposed model is presented in section 4. Finally, as an illustrative example, the model has been applied to evaluate the inelastic performance of RC piles located in aggressive environments.

Reinforced concrete deterioration 2.1 Cracking and corrosion

In optimal conditions, the durability of RC structures is high and the variation of the structural reliability with time is not significant. However, for special structures located in aggressive environments this might not be the case. Some examples of aggressive environments are those in which there is:

1. high relative humidity (i.e., between 60% and 98%); 2. cycles of humidification and drying, of freezing and defrosting; 3. high carbon dioxide concentrations (e.g., carbonation in urban atmospheres); 4. high concentration of chlorides or other salts (e.g., marine environments); or 5. high concentration of sulfates and small amounts of acids (e.g., sewer pipes or residual water treatment plants).

In any deterioration process, the structural reliability becomes a time-dependent process; for the particular case of chloride penetration leading to steel corrosion in RC structures, this process has been widely studied. The time-dependent reliability problem is described schematically in Figure 1. The structure life-cycle can be divided into three stages (Mori and Ellingwood, 1993 [8]; and Thoft-Christensen, 2002 [START_REF] Thoft-Christensen | Deterioration of concrete structures[END_REF]):

• Stage 1 : after construction structural reliability is maximum; then, deterioration initiates as a result of the environmental conditions such as: chloride ingress, carbonation, sulfate attack, biodeterioration, erosion, etc. Nevertheless, the consequences of these actions do not have a significant impact in the reliability until their individual or joint actions create the necessary conditions to depassivate the protective layer of the steel, e.g., low pH of concrete, optimal temperature, water and oxygen. As a result of this new state, the reinforcement starts to be corroded actively; this time is called time of corrosion initiation (τ ini ) (see also Figure 2a).

• Stage 2 : as soon as corrosion starts, corrosion products appear. Since these products have a lower density, they occupy the porous zone of concrete around the steel. When the total amount of corrosion products exceeds the amount needed to fill the porous zone, an expansive pressure is induced on the concrete initiating the cracking process (Liu, 1996 [10]; and Liu and Weyers, 1998 [11]). In this stage, two phenomena called crack initiation and crack propagation are present. The former is defined as the condition for which a hairline crack of 0.05mm width appears (Vu and Stewart, 2005 [12] and Vu et al., 2005 [13]). The time required to reach this point is called time to crack initiation (τ cr ) (Figure 2b). The last is measured in terms of the crack width, cw.

The time required to reach the threshold crack size, cw lim (e.g., cw lim = 0.5mm) is called time to severe cracking, (τ sp ). Once this stage starts, the reliability decay can play an important role in the structural safety.

• Stage 3 : when crack width reaches the threshold value, there is a significant increment of the corrosion rate as a result of an increase in water and oxygen availability and in concrete electrical conductivity (i.e. Schiessl and Raupach, 1997 [START_REF] Schiessl | Laboratory Studies and Calculations on the Influence of Crack Width on Chloride-Induced Corrosion of Steel in Concrete[END_REF]). This increment induces a reliability decay of an appreciable magnitude.

Biodeterioration

The process of deterioration of RC structures as a result of the action of live organisms (chemical and physical) is usually called concrete biodeterioration. Biodeterioration was defined by Hueck, 1965 [15] as: "...any undesirable change produced by normal activities of live organisms on materials". According to Gaylarde, 2003 [16], biodeterioration can be classified within three types:

• Aesthetic: it is concerned with the presence of microorganisms on the surface of the structure producing a disagreeable aspect.

• Chemical : it refers to the degradation of cement and aggregates as a result of substances emanated by live organisms.

• Physical : encompasses all physical actions of organisms leading to a physical deterioration of RC.

Biodeterioration is caused by live organisms that can be found in the ground, the water or the air. In many cases, they can be found on concrete surfaces that offer favorable conditions for them to survive; for example, places where the surface texture allows them to settle down, there is water available, the pH of concrete is low, the temperature conditions are appropriate and where there are nutrients for their survival. When these organisms settle on the concrete surface, they form a film called biofilm, from where they start to deteriorate the concrete.

Biodeterioration is caused mainly by: (1) bacteria; (2) fungi; (3) algae and lichens; and (4) organisms that erode and drill the concrete (Gaylarde et al., 2003 [17]).

Bacteria

They are unicellular microorganisms with the size of one micron or less. They can be classified as autotrophic or heterotrophic; also, they can be aerobe, anaerobe or aerotolerant anaerobes (see Appendix A for definitions). Among those that cause concrete deterioration some of the most important are: (1) cyanobacteria; (2) nitrobacteria; (3) sulfur-reducing bacteria; and (4) sulfur-oxidizing bacteria. Their main characteristics are presented in Table 2.

The chemical action of cyanobacterias does not affect RC considerably, however, its organic material secretion serves as nutrient for fungi and heterotrophic bacteria that can be very aggressive to RC. Regarding the physical degradation, the endolithic cells penetrate in concrete through micro-cracks absorbing water, which causes a growth of their molecular mass generating tensile stresses leading to an increment in the size of cracks (Gaylarde, 2003 [16]).

Nitrobacteria transform ammonia from the atmosphere into nitrites and nitrates; this process is called nitrification. Nitrites and nitrates attack the concrete and as a result of the reaction, "calcium nitrate" is produced. It affects the concrete since it results from the solubilization of one of the cement components (Biczok, 1972 [18]; Kumar and Kumar, 1999 [19]; Sand and Bock, 1999 [START_REF] Sand | Biodeterioration of mineral materials by microorganisms -biogenic sulfuric and nitric acid corrosion of concrete and natural stone[END_REF]; and Shirakawa, 1994 [START_REF] Shirakawa | Estudo da biodeterioração do concreto por Thiobacillus. São Paulo: Instituto de Pesquisas energéticas e nucleares[END_REF]) .

Sulfur-reducing bacteria oxidize the sulfate existing in the water and transform it into hydrogen sulfides and H 2 S in gaseous state. It does not affect the concrete directly; nevertheless, the H 2 S is used for the Sulfur-oxidizing bacteria to produce sulfuric acid that degrades concrete chemically. This process commonly called Concrete Corrosion (Olmstead and Hamlin, 1900 [22]; Nica et al., 2000 [23]; and Roberts et al., 2002 [24]). It can be found frequently in closed structures such as: sewage systems and waste water treatment plants.

Sulfur-oxidizing bacteria are divided in two groups: (1) Neutrophilic Sulfur Oxidizing Microorganisms (NSOM); and (2) the Acidophilic Sulfur Oxidizing Microorganisms (ASOM). The oxygen required for their existence causes that they oxidize the H 2 S (produced by the Sulfur-reducing bacteria), transforming it into sulfuric acid H 2 SO 4 , which reacts with the cement forming calcium sulfate (gypsum). Furthermore, when gypsum is combined with the hydrated aluminate of calcium, it forms sulfur-aluminate of calcium (ettringite). In addition, it can produce acetic acid, sulfates, sulfur, sulfites and polythionates, substances that affect the concrete chemically. Roberts et al., 2002 [24], reports five species of this bacteria called Thiobacillus; their main characteristics are shown in Table 3. The first four species are NSOM and the last is ASOM. The last species (i.e., T. thiooxidans) are the group of microorganisms that causes more damage to concrete. Table 4 shows some values of concrete biodeterioration rates as a result of the action of ASOM.

Fungi

Fungi can be found almost everywhere (even in adverse atmospheres with little water and without air) as long as there is organic matter available. Generally, they survive in temperatures between 25 and 30 o C. Usually, fungi found in RC surfaces are simple fungi, mold and yeast (Cho and Mori, 1995 [3]). They produce mechanical damage in the concrete micro-structure by the hyphae penetration. In addition, they deteriorate concrete chemically by the generation of organic and inorganic acids that precipitate salts.

Algae and lichens

The algae are a complex and varied vegetal group, for this reason their description and classification is not an easy task. They can survive in any habitat where there are both light and humidity; then, they can be found in the sea, humid soils and snow. Their action on concrete structures consists on taking calcium and magnesium for their metabolic processes from de cement paste generating small cavities in the concrete surface.

Lichens are the product of the symbiosis of an alga and a fungus. The fungus provides water to alga and the alga takes from the fungus inorganic substances; then, they complement each other to survive in very hostile environments. On concrete surfaces colonized by lichens there can be found many micro-perforations with diameters between 0,5 and 10 microns as well as perforations with diameters between 0,1 to 0,5 mm (Piedrahita, 2004 [2]).

Organisms that erode and drill the concrete

When concrete biodeterioration is not controlled, a series of organisms of appreciable size appears (Piedrahita, 2004 [2]). These types of organisms are found generally in tropical environments and on the concrete surface, where the biofilm has been formed. Between the organisms that erode concrete, the most representative are the mollusk (Cantera et al., 1996 [28]) whose action consists on continuously scrape the surface. Among the second class of organisms, there are the bivalves, the crustaceans and the sipunculids (Cantera et al., 1996 [28]). They make perforations of cavern type by means of chemical and physical processes.

Biodeterioration control measures

There are many preventive measures to control the action of live-organisms which depend basically on the characteristics of the species and the environmental conditions. Any procedure of countermeasures to diminish or mitigate damage must include the following actions: (1) to establish a regime of inspection or monitoring that detects immediately the appearance of the biofilm; (2) to determine the organisms that constitute the biofilm, including its density and location; (3) to elaborate a integrated plan against the involved organisms in order to solve the problem; and (4) to supervise the integrated plan and its results. Depending on the type of organism detected (i.e. step two) and the degree of damage of the structure, the integrated plans can include a combination of the following measures:

(1) surface cleaning using chemical agents (i.e. biocides), biological and physical means; (2) surface painting using admixtures with biocides; and (3) replacement of concrete cover. Specific information on these counter-measures is beyond the scope of this paper, but they can be found in the literature (see Keevil et al., 1990 [29]; De Graef et al., 2000 [START_REF] De Graef | Bacterial cleaning of concrete surfaces[END_REF]; Martinez Outeriño, 2003 [31] and Giúdice, 2003 [32]).

Corrosion of steel reinforcement

In addition to concrete biodeterioration, RC structural integrity may be seriously affected by steel corrosion. Steel corrosion caused by chloride ingress has been widely studied and some basic concepts are presented in this section. Tuutti, 1982 [33] proposed a model of the ingress of chlorides in concrete as function of time and depth using the Frick's second law of diffusion,

∂C ∂τ = D cl ∂ 2 C ∂x 2 (1)
where, C is the chloride ion concentration, D cl is the chloride diffusion coefficient in concrete, τ is the time and x is depth in the diffusion direction. Assuming that concrete is an homogenous and isotropic material with the initial conditions: (1) when time is zero, the concentration is zero and (2) the chloride surface concentration is constant, then, the chloride ion concentration C(x, τ ) at depth x after time τ is:

C(x, τ ) = C s 1 -erf x 2 √ D cl τ (2)
where C s is chloride surface concentration and erf is the error function.

The threshold concentration C th is defined as the chloride concentration for which the rust passive layer of steel is destroyed and the corrosion reaction begins, that instant is called initiation time τ ini .

From equation 2, when C(x, τ ) is equal to C th and x is equal to the bar cover c, τ ini is computed as:

τ ini = c 2 erf -1 1 -C th Cs -2 4D cl (3)
The depth of threshold concentration C th at time τ called "penetration depth of chloride threshold aggressive front" (Liang and Yang, 2005 [START_REF] Liang | Theoretical elucidation on the on-site measurements of corrosion rate of reinforcements[END_REF]) D th (τ ) is obtained if the concentration C(x, τ ) = C th and x = D th (τ ), then, equation 2 becomes:

D th (τ ) = 2 D cl τ erf -1 1 - C th C s (4)
At time τ ,the corrosion density i corr (τ ) is expressed according to the Faraday's law as:

i corr (τ ) = J c (τ )n O2 F (5)
where, n O2 is the obtained electric number of O 2 molecule participated in chemical reaction (n

O2 = 4), J c (τ ) is the diffusion flow of O 2 in concrete at time τ and F is Faraday's constant, F = 96500 [C/mol].
On the other hand, based on electrochemistry principles, the diffusion flow of O 2 in concrete is equal to (Liang et al., 2004 [35]):

J c (τ ) = D O2 ∂C(x) ∂x (6)
where D O2 is the diffusion coefficient of O 2 in the concrete [mm 2 /year] and x is the space coordinate in the diffusion direction. According to the Frick's diffusion law there is a linear variation of the oxygen concentration. Therefore, if the oxygen concentration on the surface is C O2 = 8.93 × 10 -10 [mol/mm 3 ] and at the depth D th (τ ) the concentration is C st = 0 (Liang et al., 2004 [35]),

∂C(x) ∂x = C O2 -C st D th (τ ) (7)
Replacing equations 6 and 7 into equation 5 yields to:

i corr (τ ) = n O2 F D O2 C O2 D th (τ ) (8)
On the other hand, according to Faraday's law, the product of steel corrosion during time dτ is:

dW τ = i corr (τ ) n a F M A a (τ )dτ (9)
where dW τ is the amount of corrosion products in grams at time dτ , M is the atomic weight of F e = 55.85 [g/mol], n a is the electric charge of iron ion (n a = 2) and A a (τ ) is the corrosion surface area of steel in concrete at time τ considering a 1 [mm] unit length bar (Figure 2), and is equal to:

A a (τ ) = d cos -1 d + 2c -2D th (τ ) d ( 10
)
where d is the initial bar diameter in [mm] and c is the cover thickness in [mm]. Niu et al., 1996 [36] proposed an equation for calculating D O2 as a function of the concrete compressive strength f ′ cu in [N/mm 2 ] (f ′ cu is obtained from the test over a concrete cube sample):

D O2 = 3.1536 × 10 5 32.15 f ′ cu -0.44 mm 2 year ( 11
)
If τ w is the time of full corrosion, this occurs when A a (τ ) is equal to total surface area of the steel bar (i.e., A a (τ ) = πd), then, for τ w ≥ τ , the amount of corrosion products, W (τ ), is obtained by replacing equation 8 into equation 9 and integrating with respect to time:

W (τ ) = 0 grams if τ ini ≥ τ kD O2 τ τ ini Aa(τ ) D th (τ ) dτ if τ w ≥ τ > τ ini (12)
where

k = n O2 C O2 M/n a = 7.125 × 10 -8 [g/mm 3 ]. Finally, if τ > τ w , A a (τ ) = πd and D th (τ ) = c + d, then, W (τ ) =    0 grams if τ ini ≥ τ kD O2 τ τ ini Aa(τ ) D th (τ ) dτ if τ w ≥ τ > τ ini kD O2 τ w τ ini Aa(τ ) D th (τ ) dτ + πd c+d τ τ w dτ if τ > τ w    (13)
Based on equation 13, it is then possible to compute the effective diameter of the bar as function of time as follows:

d bar (τ ) = d 0 1 - W (τ ) W 0 (14

Coupled deterioration model

The coupled model presented in this section takes into account three effects (Figure 3): (1) corrosion induced by chloride ingress; (2) cracking as a result of corrosion products; and (3) reduction of concrete cover caused by biodeterioration. In the first part of this section, the corrosion model presented in previous section will be modified to include the cracking. Afterwards, a time-dependent biodeterioration rate function is obtained by using a fuzzy logic inference system. Finally, all three effects are coupled into one single model.

RC cracking as a result of corrosion products

Once corrosion starts there are stresses resulting from the expansion produced by corrosion. Then, in order to take into account this effect, Liu and Weyers,1998 [11] proposed a model to find the time to crack initiation, τ cr , that is function of the amount of rust products. They defined the amount of critical rust products W crit [g] as the amount for which all free spaces between the steel bar and the concrete are filled and the cracking begins. It can be calculated as:

W crit = ρ steel ρ steel -0.57ρ rust (W porous + W Expand ) ( 15
)
where ρ steel is the density [g/mm 3 ] of reinforced steel, ρ rust is the density of corrosion products, W porous is the amount of corrosion products necessary to fill the porous zone around the steel/concrete interface and is defined as:

W porous = πρ rust t por d ( 16
)
where t por is the thickness of porous band around the steel concrete interface and d is the diameter of the steel bar. Besides, W Expand is the amount of corrosion products needed to fill in the space due to the expansion of the concrete around the reinforcement; this is:

W Expand = πρ rust (d + 2t por ) t crit ( 17
)
where t crit is the thickness of corrosion products needed to generate the tensile stresses (Liu and Weyers, 1998 [11]) and it is computed as:

t crit = cf ′ t E c k 2 + l 2 l 2 -k 2 + ν c (18)
where f ′ t is the tensile strength of concrete, E c is the elastic modulus of concrete, k = (d + 2t por ) /2, l = c + (d + 2t por ) /2 and ν c is Poisson's ratio of concrete.

The amount of corrosion products W (τ ) is zero if the age of the concrete τ is less than τ ini (Figure 4a). When τ ≤ τ sp , the value of D O2 is obtained by using equation 11. The time to severe cracking, τ sp , is equal to (Vu and Stewart, 2005 [12] and Vu et al., 2005 [13]):

τ sp = τ cr + τ pr ( 19
)
where τ pr is the time of crack propagation. Based on experimental data obtained from accelerated corrosion tests with a corrosion rate of i corr(acc) = 100 µA/cm 2 , Vu et al (2005) [13] found that the accelerated time to crack propagation in hours can be estimated as:

τ pr(acc) = ξ(c/wc) λ ( 20
)
where c is the cover in mm, wc is the water/cement ratio, and ξ and λ are parameters that depend on the limit crack width. For instance, when cw lim = 0.5 mm, ξ = 225 and λ = 0.29 (Vu et al., 2005 [13]). Since equation 20 was obtained from accelerated tests, a correction factor, k R , has been deduced by Vu et al. (2005) [13], based on the experimental values obtained by Alonso et al (1998) [START_REF] Alonso | Factors controlling cracking of concrete affected by reinforcement corrosion, materials and structures[END_REF] as:

k R ≈ 0.95 exp - 0.3i corr(acc) i corr(real) - i corr(acc) 2500i corr(real) + 0.3 (21)
where i corr(real) corresponds to the real corrosion rate in µA/cm 2 . Therefore, the time to crack propagation in years is:

τ pr = k R 0.0114 i corr(real) τ pr(acc) (22)
After cracking begins (i.e., τ > τ sp ) the diffusion coefficient D O2 becomes time-dependent and a linear behavior is assumed. Consider that at time τ n (with τ n ≫ τ sp ; e.g. τ n = 500 years) the oxygen concentration at the steel surface (i.e., C nO2 ) is closer to the oxygen concentration at the concrete surface C O2 (i.e., C nO2 ≈ C O2 ). Then substituting these values into Frick's second law (equation 2), it is possible to compute a new oxygen diffusion coefficient at time τ n , as:

D nO2 (τ n ) = c 2 4τ n erf -1 1 - C nO 2 C O 2 2 (23)
and the diffusion coefficient of oxygen in the concrete (DC O2 ) at time τ becomes (Figure 4b):

DC O2 (τ ) = D O2 if τ ≤τ sp D O2 + D O 2 -D nO 2 τ sp-τ n (τ -τ sp ) if τ > τ sp (24)
then, substituting equation 24 into 13, the amount of corrosion products when considering the RC cracked, W crack (τ ), in [g] by 1 mm of length at time τ is given by:

W crack (τ ) =    0 grams if τ ini ≥ τ k τ τ ini DC O 2 Aa(τ ) D th (τ ) dτ if τ w ≥ τ > τ ini k τ w τ ini DC O 2 (τ )Aa(τ ) D th (τ ) dτ + πd c+d τ τ w DC O2 (τ ) dτ if τ > τ w    (25)
The consequences of cracking on the corrosion rate is not easy to evaluate; Figure 5 presents several models. In the first model, the time-variant corrosion rate has been computed by using the values reported by Vu et al. (2005) [13]. In this model, corrosion rate remains constant during the first year, after which, the formation of rust products on the steel surface reduces this rate substantially. The second model considers a reduction of corrosion rate, from the corrosion initiation time, (τ ini ) until the time of full corrosion, (τ w ). After this point, the corrosion rate remains constant (see also Figure 4). In the third case, the model includes concrete cracking and assumes that the change of the diffusion coefficient of concrete (equation 20) begins only after the time to crack initiation, (τ cr ). After this point, diffusion grows due to the increase in the diffusion oxygen coefficient. The last case is similar to the previous one but the effect of cracking has only an important impact when the crack width reaches a limit value of cw lim = 0.5 mm, (τ sp ). In this case, the corrosion rates are lower than in the third case.

The first two models presented in Figure 5 do not take into account concrete cracking to estimate corrosion rate, while the last two do. In this paper the last model was used to compute the amount of corrosion products. It is important to note that the corrosion rates used in this paper are within the range suggested by EN 206 (Geocisa and Torroja Institute, 2002 [START_REF] Geocisa | Contecvet: A validated users manual for assessing the residual service life of concrete structures. Manuel for assessing corrosion-affected concrete structures[END_REF]) for corrosion induced by chloride ingress in tidal zones.

Time-dependent biodeterioration rate function

Biodeterioration is a very complex process where many classes of organisms, environmental variables and other deteriorating processes interact. Computing the RC time-dependent biodeterioration rate is a crucial aspect for the accuracy of the numerical model. However, to the authors knowledge, no much work have been done in this field and there is not yet a robust model of this process. As no sufficient data are available to perform appropriate statistical analysis, the application of probability theory is not possible. Thus, non precise data is considered based on existing experimental reports (Table 4). The proposed model is based on fuzzy inference, which is a valuable tool for managing situations where information about a process can only be described conceptually or there is not enough data to build a robust statistical model. Then, based on fuzzy logic, a model to obtain a time-dependent function of the biodeterioration rate is proposed.

Conceptually, the process can be divided into three stages (Figure 6):

1. RC immunity: in this stage, due the concrete high alkalinity (pH between 11 and 13), the organisms cannot survive nor adhere to it; therefore, the RC is immune to biological damage.

2. Biofilm formation: the presence of CO 2 in the atmosphere causes carbonation, which in turn leads to a reduction of the pH of the RC surface (until approximately 9). In addition, the erosive action of the water and/or the friction with other materials generate certain roughness on the concrete surface that allows microorganisms to adhere forming the biofilm. Experimental evidence has shown that microorganisms can also ingress into concrete by a diffusion process alterating the concrete microstructure (Ribas and Pinheiro, 2006).

3. Active biodeterioration of concrete: in this stage, the concrete pH continues decreasing, by the joint action of carbonation and organisms until it reaches a value of pH < 5. The RC surface is highly deteriorated and the cracks have an important size. All those conditions make possible that other organisms also stick to biofilm on the surface contributing to RC chemical deterioration. When the concrete is cracked, some organisms like the endolithic cells, algae and fungi ingress into cracks, generating tensile stresses that deteriorate the concrete by increasing the crack size.

Crack formation can also be promoted as a result of the weakened on the concrete microstructure caused by fungi and other microorganisms that might have already enter into concrete matrix by diffusion.

Based on the conceptual model the membership functions for the age of the structure (i.e., in time units [years], τ ) and the biodeterioration rate γ [mm/year] can be defined. Then, The structure life can be divided in two stages:

1. Initial Age: at this age, microorganisms can hardly live on the surface of the concrete because it is even and the pH of concrete is high.

2. Active Biodeterioration Age: at this stage, the biodeterioration is active because the surface offers optimal conditions for colonization of microorganisms.

These stages are represented by membership functions of sigmoidal form (see Figure 7a):

µ age (τ , a, b) = 1 1 + exp(-a(τ -b)) ( 26
)
where τ is the time in years and a, b are constants that define the shape of the function. In addition, it is necessary to define two membership functions for the corresponding biodeterioration rate γ. The first corresponds to the initiation of the biodeterioration process for which the maximum membership value assigned is γ ini = 0 [mm/year]. Then, in this case the modified Gaussian function that describes the initial biodeterioration rate membership is given by (see Figure 7b):

µ γ 1 (γ, σ) = exp -1 2 γ σ 2 if γ ≥ 0 0 otherwise ( 27
)
where σ is a shape parameter. The second function corresponds to the time within which biodeterioration is active; thus, based on the values presented in Table 4, the average active biodeterioration rate, γ av , is assigned to the maximum membership value. In this case, the membership function that describes the corrosion rate is:

µ γ 2 (γ, γ av , σ) = exp - 1 2 γ -γ av σ 2 (28)
Figure 7 illustrate these functions for the following parameters: a τ ini = -0.8 and b τ ini = 6.5 for the Initial Age; a τ act = 0.7 and b τ act = 12 for the Active Biodeterioration Age; σ γ ini = 0.1 for the Initial Biodeterioration Rate; and γ av = 2.5 mm/yr and σ γ act = 0.4 for the Active Biodeterioration Rate.

The rules used for the fuzzy inference system are:

1. If the structure is in the initial age, then the biodeterioration rate is the initial.

2. If the structure is in the active biodeterioration age, then the biodeterioration rate is active.

Based on these rules a response surface can be determined as follows:

S (τ , γ) = max µ age1 (τ , a τ ini , b τ ini )µ γ 1 (γ, σ γ ini ), µ age2 (τ , a τ act , b τ act )µ γ 2 (γ, γ av , σ γ act ) ( 29
)
where S (τ , γ) is the surface generated by the maximum of products composition, µ age1 (τ , a τ ini , b τ ini ) is the membership function corresponding to concrete initial age, µ age2 (τ , a τ act , b τ act ) to active biodeterioration age, µ γ 1 (γ, σ γ ini ) to initial biodeterioration rate and µ γ 2 (γ, γ av , σ γ act ) to active biodeterioration rate. The surface generated by equation 29 is shown in Figure 8a. Finally, by using the strategy of the center of gravity for the surface defuzzification, it is possible to obtain a time-dependent biodeterioration rate function, γ(τ ), as (Figure 8b):

γ(τ ) = γ • S (τ , γ) dγ S (τ , γ) dγ (30)
The advantage of this solution is that it provides an estimation of the time-dependent change of the biodeterioration rate, that depends on the expert's knowledge. Hence the expert has to define the conceptual model of the problem as presented above. Moreover, he/she determines the form of the membership functions, since, he/she has to assign values to the variables a, b, γ av and σ based on his/her expertise. For example, an expert can define the membership functions of the biodeterioration rate considering environmental factors and nutrients availability of determinate place, and the age membership functions taking into account the type of cement or the condition of the structure at a given time. It is important to stress on the fact that the biodeterioration rate function can be updated permanently by modifying the membership functions based on field measurements and experimental data.

Coupled model

Due to action of live organisms, the thickness of the concrete cover is a time-dependent function. Then, the thickness at time τ (i.e., cover (τ )) is calculated as:

cover (τ ) = c - τ 0 γ (τ ) dτ (31)
where c is the cover at τ = 0 and γ (τ ) corresponds to the biodeterioration rate function showed in Figure 8b. Then, the initiation time τ ini is calculated by replacing equation 31 into equation 3,

τ ini = cover (τ ) 2 erf -1 1 -C th Cs -2 4D cl (32)
For time τ > τ ini , the advance of the aggressive corrosion front, D th (τ ) (equation 4), and the reduction of steel area grow rapidly affecting considerably the properties of the structural member. Therefore, equation 10 is modified simply by making the cover thickness a function of time. That is:

A am (τ ) = d cos -1 d + 2cover (τ ) -2D th (τ ) d (33)
where A am (τ ) is corroded surface area of the reinforcement at time τ (taking 1 mm unit length) when the biodeterioration process is taken into account. Consequently, the amount of corrosion products W m (τ ), which is function of A am (τ ), can be expressed as:

W m (τ ) =    0 grams if τ ini ≥ τ k τ τ ini DC O 2 Aam(τ ) D th (τ ) dτ if τ w ≥ τ > τ ini k τ w τ ini DC O 2 (τ )Aam(τ ) D th (τ ) dτ + πd c+d τ τ w DC O2 (τ ) dτ if τ > τ w    (34)
On the whole, the proposed model couples three processes: (1) steel corrosion; (2) concrete cracking; and (3) biodeterioration. The flow diagram that summarizes the entire procedure is presented in Figure 9. The ultimate purpose of the model is to compute the reduction of steel reinforcement area of the RC element. As it will be shown in the numerical example, this approach to RC biodeterioration can be used to make better estimates of structural reliability and is an important tool for intervention and rehabilitation purposes.

Numerical Example

Basic considerations and assumptions

In this section, the coupled model described in the previous section is illustrated with a numerical example. The goal of this example is to compute the probability of failure of a seaport reinforced concrete pile in terms of its curvature ductility capacity. The pile showed in Figure 10 is part of a port structural system subjected to axial loading produced by the loading and unloading of containers on the platform (see Figure 10a). The port is located in a tropical region where the environmental conditions make the structure susceptible to biological deterioration. The geometric characteristics of the pile are presented in Figure 10b.

For structures immersed in the sea, there are three zones (see figure 10a), with different surface chloride concentration and availability of oxygen and water for which the cinematic of the corrosion process is different (Luping and Andersen, 2000 [40]). These are: (1) the Submerged zone, (2) the Splash and tidal zone, and (3) the Atmospheric zone. In the first zone, the surface chloride concentration is higher that in the other zones, but the risk of corrosion and biodeterioration are low because the oxygen and light are not available. In the splash and tidal zone, the RC surface is exposed to wetting and drying cycles that provide the optimal conditions (chlorides, water and oxygen) for corrosion to initiate. Furthermore, the existence of light, nutrients and water in sufficient quantities promotes the appearance of the biofilm. It is widely accepted that the highest risks of corrosion and biodeterioration occur in this zone. Finally, in the last region the wind is the main source of chlorides and the water available is low, thus, the risks of corrosion and biodeterioration are small. Therefore, the present example focuses on the splash and tidal zone.

The following assumptions about the problem have been made:

• The density of steel reinforcement is assumed deterministic (ρ steel = 8000 kg/m 3 ),

• The tensile strength of concrete is given by a percentage of its compressive strength (i.e. f ′ t = 0.10f ′ c ). Perfect correlation between tensile and compressive strengths is assumed.

• In order to estimate the time to severe cracking, a limit crack width of cw lim = 0.5 mm has been chosen.

• The water/cement ratio is calculated by using the Bolomey's formula.

• The corrosion of steel is assumed to be uniform.

• The deterioration process is continuous, i.e. there is not maintenance.

Reliability problem formulation

Ductility capacity is the structural ability to sustain, without failure, deformations well beyond the elastic limit. In this case, the curvature ductility capacity µ φ is equal to,

µ φ = φ u φ y (35)
where φ u and φ y are the ultimate and yield curvatures respectively. Stress-strain relationship for the steel follows an elasto-plastic model and for concrete it was assumed an inelastic behavior (Kent and Park, 1971 [44]). In this application, the failure probability is calculated with respect to the following limit state function:

g µ φ ini , µ (τ ) = θµ φ ini -µ (τ ) (36)
where θµ φ ini corresponds to a fraction (i.e., θ) of the initial ductility, µ φ ini , and µ (τ ) is the ductility of the pile at time τ . Three variations of the same limit state function were considered based on a fraction of the structural initial capacity, µ φ ini , θ: (1) θ = 0.75; (2) θ = 0.50; and (3) and θ = 0.25. The probabilistic models of the variables used are shown in Table 5. The reliability was calculated by using a "pointin-time" analysis by using using the first-order second-moment reliability method FORM, which is conservative (upper bound) (Stewart and Val, 1999 [45]).

Results of the model

The failure probability of the three limit state functions for various average active biodeterioration rates, (γ av ), is presented in Figure 11. It can be observed that the shape of the fragility curves, i.e., slope and time to failure failure (i.e., p f = 1) depend upon: (1) the average active biodeterioration rate and (2) the limit state function (i.e., defined as a fraction of the initial ductility θ). Comparing the cases with and without biodeterioration it is observed that the overall behavior is similar, but all failure times with biodeterioration occur earlier. In addition, as the average active biodeterioration rate grow, the time to failure decreases and the slope of the fragility curves increase. It can be observed also that the overall effect of γ av is larger when the fraction of the initial ductility θ decreases.

The relationship between the average active biodeterioration rate and the time required to reach failure are plotted in Figure 12. This figure shows that for all limit state functions, the time to failure grows as γ av decreases. This time period also grows as θ decreases. For instance, for θ = 0.75, the time to failure is 40 years for γ av = 2.5 mm/yr, and 80 years for γ av = 0 mm/yr respectively. An inverse analysis can also be made from the data presented in Figure 12; for a predefined life-cycle it is possible to estimate the average biodeterioration rate required to reach a given limit state. For example, if a time window of 60 years is selected, a γ av = 0.9 mm/yr is required to meet the failure limit condition as defined by a 75% of its initial curvature ductility (i.e., (p f ) θ=0.75 = 1). The same values for θ = 0.5 and θ = 0.25 are γ av = 1.3 mm/yr and γ av = 1.7 mm/yr, respectively (Figure 12).

Finally, it is important to notice the effect of deterioration in the reduction of the ductility capacity of the structural component. The change of ductility with time for a probability of failure equal to p f = 1 and p f = 0.5, and for various average biodeterioration rates is shown in Figure 13. As expected, the ductility capacity decreases with time, but if biodeterioration is not considered, the structural ductility capacity is larger than in the case where this effect is included. For example, if biodeterioration is not taken into account, the time required to reach failure, for θ = 0.75, will be 80 years. However, when considering biodeterioration this time becomes t(γ av = 0.5) θ=0.75 = 69 years (13%) and t(γ av = 2.5) θ=0.75 = 40 years (50%). It is evident from the results that preventive maintenance against biodeterioration is paramount to extend the structural life.

Conclusions

Modeling the effects of biodeterioration in RC structures should to be explicitly included as part of infrastructure planning and design, in particular, when structures are located in aggressive environments. Biodeterioration might affect structural performance significantly causing an important reduction in the lifetime of the structure. The proposed model couples three processes: (1) steel corrosion; (2) concrete cracking; and (3) biodeterioration. This coupling process is used to compute the reduction of steel reinforcement area and concrete cross section as a function of time. Furthermore, a fuzzy inference system based on experimental data and expert opinions was used to model the biodeterioration rate which is an essential part of the integrated model. The fuzzy model is a very convenient tool which combines data bases with expert knowledge about the local environmental variables. The proposed model is illustrated by studying the effects of biodeterioration in the ductility capacity of sea port piles. A difference between 13% and 50% in the structural life was observed. The high variability is related to the value of γ av , which should be a subject of further experimental research.

Appendix A -Glossary

Term Description

Aerobe: organisms that require oxygen for their vital processes. Aerotolerant anaerobes: organisms that not require oxygen, but are not affected by their exposure to air. Anaerobe:

organisms that are unable to use oxygen for their vital processes. Autotroph:

these are organisms that do not need organic carbon to survive because their carbon source is the inorganic CO 2 and some car-

  ) where d bar (τ ) is the diameter of the bar at time τ , d 0 is the bar initial diameter, W (τ ) is the amount of corrosion products in [g] by 1 [mm] of length at time τ calculated by equation 13 and W 0 is the initial weight in [g] by 1 [mm] length.

Figure 1 :

 1 Figure 1: Reliability profile for RC structures subject to corrosion.

Figure 2 :

 2 Figure 2: Steel corrosion process combined with concrete cracking and biodeterioration effects.

Figure 3 :

 3 Figure 3: Conceptual description of the coupled model for biodeterioration, chloride ingress and cracking of concrete structures.

Figure 4 :

 4 Figure 4: (a) Change of the amount of corrosion products and the diameter of the steel bar with time; (b) Variation of the diffusion oxygen coefficient with time.

Figure 5 : 22 Figure 6 :

 5226 Figure 5: Time-dependent corrosion rates models.

Figure 7 :

 7 Figure 7: (a) Age membership functions. (b) Biodeterioration rate membership functions.

Figure 8 :

 8 Figure 8: (a) Surface generated in FIS type Mamdani. (b) Time-dependent biodeterioration rate function.

Figure 9 :

 9 Figure 9: Flow diagram describing the coupled model.

Figure 11 :

 11 Figure 11: Fragility curves for various average active biodeterioration rates. (a) θ = 0.75; (b) θ = 0.5; and (c) θ = 0.25.

Figure 12 :

 12 Figure 12: Relationship between average active biodeterioration rate and structural lifetime for the three limit states considered. θ = 0.75; θ = 0.5; and θ = 0.25.

Table 1 :

 1 Cost of corrosion in several industrial and infrastructure sectors

	Industrial sector	Cost of corrosion 2001 (US$Billion/year)
	Gas & Liquid Transmission pipelines	7.0
	Hazardous Materials Storage	7.0
	Highway bridges	8.3
	Waterways & ports	0.3
	Drinking water and sewer systems	36.0
	Transportation	29.7
	Defense and nuclear waste storage	20.1
	Production and manufacturing	17.6
	Others	11.9
	Total	137.9

Table 2 :

 2 Effects of bacteria on RC structures

	Bacteria type	Lifestyle		Temperature	Consequences on concrete
					and	pH
					range
	1. Cyanobacteria	Autothophic,	-60 to 85	Generate tensile stresses leading
			aerobe	or	• C	(wide	to an increment in the size of
			anaerobe		range pH)	cracks
	2. Nitrobacteria	Heterothophic	18 to 25 o C	Nitrates and nitrates attack the
			and anaer-	pH < 7.5	concrete produce "calcium ni-
			obe			trate". this process affects the
						concrete since it results from the
						solubilisation of one of the ce-
						ment components.
	3. Sulfur-reducing	Heterothophic	25 to 44 o C	Produce H2S that is used for the
	bacteria		and anaer-	5.5 < pH <	Sulfur-oxidizing bacteria to pro-
			obe		9	duce sulfuric acid. This process is
						commonly called Concrete Corro-
						sion
	4. Sulfur-oxidizing	Heterothophic	25 to 44 o C	Produce sulfuric acid, acetic acid,
	bacteria		and aerobe	2 < pH < 9	sulfates, sulfur, sulfites and poly-
						thionates that affect the concrete
						chemically.
	4.1	Neutrophilic			pH ≈ 7
	Sulfur	Oxidizing
	(NSOM)
	4.2	Acidophilic			pH < 3.0
	Sulfur	Oxidizing
	(ASOM)

Table 3 :

 3 Five species of Thiobacillus and their characteristics

	Species	pH range	Lifestyle	Products
	T. thioparus	4.5-10	Autotrophic aerobe	Sulfur, Polythionic acids
	T. novellus	5-9.2	Mixotroph	Sulfur
	T. neapolitanus	4-9	Autotrophic	Polythionic acids, sulfuric acids
	T. intermedius	1.7-9	Mixotroph	Polythionic acids, sulfuric acids
	T. thiooxidans	0.5-4	Autotrophic	Sulfur, sulfuric acids

Table 4 :

 4 Biodeterioration rates caused by ASOM

	Source	Biodeterioration rate (mm/yr)
	Morton et al., 1991 [4]	2.7
	Mori et al., 1992 [25]	4.6 -4.7
	Ismail et al., 1993 [26]	2 -4
	Davis, 1998 [27]	3.1
		20

Table 5 :

 5 Probabilistic description of the variables used in the model.

	Variable Distribution	Mean	Est. Dev.	Source
	Load	Lognormal	400 kN	30 kN
	f 'c	Normal	28 MPa	2.8 MPa	Melchers, 1999 [41]
	fy	Normal	420 MPa	42 MPa	Melchers, 1999 [41]
	C th	Lognormal	0.37	0.04	Hong, 2000 [42]
	Cs	Lognormal	2	0.6	Luping and Andersen, 2000 [40]
	D cl	Lognormal	1.62×10 -8 cm 2 /s 0.5×10 -8 cm 2 /s	Hong, 2000 [42]
	ρ rust	Normal	3600 kg/m 3	360 kg/m 3	Thoft-Christensen, 2000 [43]
	tpor	Lognormal	12.5×10 -6 m	2.54×10 -6 m	Thoft-Christensen, 2000 [43]


Emilio Bastidas-Arteaga

Philippe Bressolette

Alaa Chateauneuf

Mauricio Sánchez-Silva

Probabilistic lifetime assessment of RC structures under coupled corrosion-fatigue deterioration processes

Keywords: corrosion-fatigue, chloride ingress, cyclic load, lifetime assessment, reinforced concrete, reliability

Structural deterioration is becoming a major problem when considering long term performance of infrastructures. The existence of a corrosive environment, cyclic loading and concrete cracking cause structural strength degradation. The interaction of these conditions can only be taken into account when modeling the coupled action. In this paper, a new model to assess lifetime of RC structures subject to corrosion-fatigue deterioration processes, is proposed. Separately, corrosion leads to cross section reduction while fatigue induces the nucleation and the propagation of cracks in steel bars. When considered together, pitting corrosion nucleates the crack while environmental factors affect the kinematics of crack propagation. The model is applied to the reliability analysis of bridge girders located in different chloride-contaminated environments. Overall results show that the coupled effect of corrosion-fatigue on RC structures affects strongly its performance leading to large reduction in the expected lifetime.

Introduction

Long-term performance of infrastructure systems is dominated by structural deterioration. Structural deterioration is loss of capacity due to physical, chemical, mechanical or biological actions. Since a corrosive environment and cyclic loading are two of the main causes of reinforced concrete (RC) deterioration, a significant amount of research has been devoted to these two specific damage mechanisms [START_REF] Schijve | Fatigue of structures and materials in the 20th Century and the state of the art[END_REF] [2] [START_REF] Jones | Principles and prevention of corrosion[END_REF]. Corrosion is the most common form of steel deterioration and consists in material disintegration as a result of chemical or electrochemical action. Most metals corrode on contact with water (or moisture in the air), acids, bases, salts, and other solid and liquid chemicals. Metals will also corrode when exposed to gaseous materials like acid vapors, formaldehyde gas, ammonia gas, and sulfur containing gases [START_REF] Jones | Principles and prevention of corrosion[END_REF]. Depending upon the case, corrosion can be concentrated locally to form a pit, or it can extend across a wide area to produce general wastage. On the other hand, fatigue is the damage of a material resulting from repeatedly stress applications (e.g. cyclic loading). Fatigue is conditioned by factors such as high temperature, i.e. creep-fatigue, and the presence of aggressive environments, i.e. corrosion-fatigue [START_REF] Schijve | Fatigue of structures and materials in the 20th Century and the state of the art[END_REF] [START_REF] Suresh | Fatigue of Materials[END_REF].

The damage to RC structures resulting from the corrosion of reinforcing is exhibited in the form of steel cross section reduction, loss of bond between concrete and steel, cracking, and spalling of concrete cover [4] [5]. The corrosion of steel reinforcement has been usually associated to chloride ingress and carbonation [START_REF] Clifton | Predicting the service life of concrete[END_REF]; however, recent studies have shown that other deterioration processes like biodeterioration might contribute significantly to this process [START_REF] Bastidas-Arteaga | Coupled reliability model of biodeterioration, chloride ingress and cracking for reinforced concrete structures[END_REF]. In RC structures, the coupled effect of corrosion and fatigue has not been studied in as much detail as their separated effects. Coupled corrosion-fatigue deterioration results from the combined action of cycling stresses in corrosive environments. Localized corrosion leading to pitting may provide sites for fatigue crack initiation. Corrosive agents (e.g., seawater) increase the fatigue crack growth rate [START_REF] Gangloff | Environmental cracking corrosion fatigue[END_REF], whereas the morphology of metals/alloys at micro-level governs the pit nucleation sites [START_REF] Rajasankar | A probability-based model for growth of corrosion pits in aluminium alloys[END_REF]. Under these conditions, the formation and growth of pits is influenced by both a corrosive environment and cyclic loads and become a coupled damage mechanism.

Examples of structures that experience this type of damage are bridges, chimneys, towers and offshore structures situated close to the sea or exposed to the application of de-icing salts. The effects of gradually accumulating corrosion on the low cycle fatigue of reinforcing steel have been recorded experimentally by Apostolopoulos et al. [START_REF] Apostolopoulos | Low cycle fatigue behavior of corroded reinforcing bars S500s tempcore[END_REF] showing that corrosion implies an appreciable reduction in the ductility, the strength and the number of cycles to failure.

Large research efforts have been made to predict the corrosion-fatigue life of structural members constituted by aluminum, titanium and steel alloys. Goswami and Hoeppner [START_REF] Goswami | Pitting corrosion fatigue of structural materials[END_REF] proposed a seven-stage conceptual model in which the electrochemical effects in pit formation and the role of pitting in fatigue crack nucleation were considered. Other research studies focused on particular stages of the process. For instance, a transition model from pit to crack based in two criteria: stress intensity factor and the competition between pit growth and crack growth, was proposed by Kondo [START_REF] Kondo | Prediction of fatigue crack initiation life based on pit growth[END_REF], and further discussed by Chen et al. [START_REF] Chen | Transition from pitting to fatigue crack growth modeling of corrosion fatigue crack nucleation in a 2024-T3. aluminum alloy[END_REF]. In order to take into consideration the entire progressive damage process and the uncertainties in each stage, Shi and Mahadevan [START_REF] Shi | Damage tolerance approach for probabilistic pitting corrosion fatigue life prediction[END_REF] proposed a mechanics-based probabilistic model for pitting corrosion fatigue life prediction of aluminum alloys.

The objective of this paper is to develop a probabilistic lifetime prediction model for RC structures under the coupled effect of corrosion and fatigue. The model assesses the total corrosion-fatigue life as the sum of three critical stages: (1) corrosion initiation and pit nucleation; (2) pit-to-crack transition, and (3) crack growth. The first considers the time from the end of construction until the generation of a pit. The length of this stage is estimated by considering Frick's diffusion law and electrochemical principles. The second stage includes the pit growth until crack nucleation. In this stage the interaction between electrochemical and mechanical processes is taken into account. The latter stage covers the time of crack growth until it reaches a critical crack size. The critical crack size is defined as the crack size at which the RC member reaches a limit state of resistance.

The model proposed is exposed in section 2. Section 3 presents a discussion about time-dependent reliability analysis. Finally, an application to bridge girders is exposed in section 4.

Coupled corrosion-fatigue model

The corrosion-fatigue damage process in RC structures is conceptually depicted in Figure 1. The process takes into account the interaction between: (1) chloride ingress, (2) RC cracking and (3) cyclic load. Chloride ingress leads to steel depassivation, and takes part in the kinematics of the corrosion process. Besides, corrosion as a result of chloride ingress induces a high localized corrosion (i.e., pitting corrosion) leading to reinforcing steel crack nucleation [START_REF] Jones | Principles and prevention of corrosion[END_REF]. Concrete cracking generated by the accumulation of corrosion products in the steel/concrete interface plays an important role in the steel corrosion rate. Its importance depends of both the width of the crack in the concrete and the aggressiveness of the environment. On the other hand, cyclic loading is important in the transition from pit to crack and crack growth.

The corrosion-fatigue deterioration process is basically divided in two stages: (1) pit formation and growth; and (2) fatigue crack growth. Pit formation and growth involves electrochemical processes depending predominantly on environmental factors. Crack growth is estimated in terms of Linear Elastic Fracture Mechanics (LEFM) and depends mostly upon both cyclic loads and material properties. Goswami and Hoeppner [START_REF] Goswami | Pitting corrosion fatigue of structural materials[END_REF] proposed to separate conceptually the corrosion-fatigue life into the following stages: (1) electrochemical stage and pit nucleation, (2) pit growth, (3) competitive mechanisms between pit growth and fatigue crack nucleation, (4) chemically, "short" crack growth, (5) transition from "short crack" to "long crack", (6) long crack growth, and (7) corrosion fatigue crack growth until instability. However, assuming initial immunity of RC structures and the fact that some of these stages proposed by Goswami and Hoeppner are transitional stages, the total corrosion-fatigue life, τ T , will be divided in the following stages (Figure 1):

(i) corrosion initiation and pit nucleation, τ cp , (ii) pit-to-crack transition, τ pt , and (iii) crack growth, τ cg .

Corrosion initiation and pit nucleation

This stage is divided into two sub-stages: (i) time to corrosion initiation, τ ini , and (ii) time to pit nucleation, τ pn . The first describes the time from the end of construction until the depassivation of the corrosion protective layer of reinforcing steel, and subsequently, corrosion initiation. For RC structures, the length of this stage depends mainly upon the concrete characteristics and the thickness of the cover. It is calculated by using the Frick's second law under the assumption that concrete is an homogenous and isotropic material [START_REF] Tuutti | Corrosion of steel in concrete[END_REF]:

∂C ∂τ = D cl ∂ 2 C ∂x 2 (1)
where C is the chloride ion concentration, D cl is the chloride diffusion coefficient in concrete, τ is the time and x is depth in the diffusion direction. If the following initial conditions are defined: (1) at time τ = 0, the chloride concentration is zero; (2) the chloride concentration on the surface is constant. The solution to eq. 1 leads to the chloride ion concentration C(x, τ ) at depth x after time τ :

C(x, τ )=C s  1 -erf ✓ x 2 √ D cl τ ◆ (2)
where C s is the surface chloride concentration and erf() is the error function. The threshold concentration C th is defined as the chloride concentration for which the rust passive layer of steel is destroyed and the corrosion begins. When C(x, τ ) is equal to C th and the depth x is equal to the bar cover c, i.e. x = c, the time to corrosion initiation, τ ini , becomes:

τ ini = c 2 4D cl  erf -1 ✓ 1 - C th C s ◆ -2 (3)
Pit nucleation is the result of an electrochemical processes induced by corrosion. Computing time to pit nucleation, τ pn , is a non-trivial task because its value depends on several environmental, material and loading factors whose interaction is not well understood yet. The pit depth at time τ , p(τ ) (Figure 2b), can be calculated as [START_REF] Jones | Principles and prevention of corrosion[END_REF]:

p(τ )=0.0116α Z i corr (τ )dτ (4)
where p(τ ) is given in mm, α is the ratio between pitting and uniform corrosion depths, and i corr (τ ) is the time-variant corrosion rate which is given in µA/cm 2 . Time to pit nucleation is estimated by defining a threshold, p 0 , for the pit depth, p(τ ). For example, according to Harlow and Wei [START_REF] Harlow | A probability model for the growth of corrosion pits in aluminium alloys induced by constituent particles[END_REF] this value is p(τ pn )= p 0 =1.98×10 -6 m. During a brief time period after steel depassivation, i.e. τ ini + 1yr≥ τ > τ ini , it is reasonable to assume that i corr (τ ) remains constant (i.e. i corr (τ )=i ini ) and is equal to [16]:

i ini = 37.8(1 -wc) -1.64 c ( 5
)
where wc is the water-cement ratio and c is given in mm. Consequently, by making p(τ pn )=p 0 in eq. 4, substituting eq. 5 in eq. 4 and by integrating, the time to pit nucleation, τ pn , can be written:

τ pn = 2.281cp 0 α (1 -wc) 1.64 (6)
Equation 6 gives a simple relationship to estimate τ pn based on electrochemical principles. Other proposals to obtain τ pn can be found in the literature [START_REF] Shi | Damage tolerance approach for probabilistic pitting corrosion fatigue life prediction[END_REF], however, this discussion is beyond the scope of this paper.

Pit-to-crack transition

The time of transition from pit to crack, τ pt , is defined as the time at which the maximum pit depth reaches a critical value leading to crack nucleation. Crack nucleation depends on the competition between the processes of pit and crack growth. To estimate this transition period, two criteria can be taken into account [START_REF] Kondo | Prediction of fatigue crack initiation life based on pit growth[END_REF] [START_REF] Chen | Transition from pitting to fatigue crack growth modeling of corrosion fatigue crack nucleation in a 2024-T3. aluminum alloy[END_REF]:

(i) Rate competition criterion: the transition takes part when the crack growth rate, da/dτ , exceeds the pit growth rate, dp/dτ , as illustrated in Figure 2a.

(ii) Fatigue threshold criterion: the transition occurs when the stress intensity factor of the equivalent surface crack growth for the pit, ∆K eq , reaches the threshold stress intensity factor for the fatigue crack growth, ∆K tr . In this study, the fatigue threshold criterion was not taken into consideration because experimental observations indicate that this criterion is not appropriate to estimate fatigue crack nucleation at low loading frequencies [START_REF] Chen | Transition from pitting to fatigue crack growth modeling of corrosion fatigue crack nucleation in a 2024-T3. aluminum alloy[END_REF]. Therefore, the rate competition criterion where pit growth rate is described by electrochemical mechanisms and fatigue crack growth rate is estimated in terms of LEFM will be discussed in the following sections. This discussion will focus on:

(i) Pit growth rate (ii) Fatigue crack growth rate (iii) Computation of pit-to-crack transition 2.2.1. Pit growth rate After pit nucleation and as a result of a localized galvanic corrosion, pit growth can be estimated in terms of a change in the volumetric rate by using the Faraday's law [11] [15]:

dV dτ = Mi corr nF ρ ( 7
)
where M is the molecular weight of iron, (i.e. M = 55.85 g/mol), i corr is the corrosion rate, n is the valence of iron, (i.e. n = 2), F is the Faraday's constant, (i.e. F = 96500 C/mol), and ρ is the density of iron, (i.e. ρ = 8000 kg/m 3 ). Corrosion rate is the effective galvanic current between the particles and the steel matrix.

The relationship between corrosion rate and temperature is taken into account by using the Arrhenius equation:

i corr = i p0 exp ✓ -∆H RT ◆ (8)
where i p0 is the pitting current coefficient, ∆H is the activation enthalpy, R is the universal gas constant (R =8 .314/mol K), and T is the absolute temperature. It is clear, from eq. 8, that the corrosion rate increases when temperature is raised. Nevertheless, given the complexity of the corrosion process in RC, the relationship between i corr and T is not simple [START_REF] López | Influence of temperature on the service life of rebars[END_REF]. For instance, it has been found that the corrosion kinetics of different RC structures under conditions of constant temperature can vary widely [START_REF] López | Influence of temperature on the service life of rebars[END_REF]. This variability results from the interaction between temperature and other factors such as conductivity and relative humidity. In addition, corrosion rate in RC is a function of environmental aggressiveness. Thus, the corrosion rate becomes a time-variant function depending mostly upon: concrete pH, oxygen, water and chlorides availability. In this context, Bastidas-Arteaga et al. [START_REF] Bastidas-Arteaga | Structural reliability of RC structures subject to biodeterioration, corrosion and concrete cracking[END_REF] proposed a time-variant corrosion rate function, i corr (τ ), which takes into account environmental hostility. This function is based on a fuzzy inference system under the following assumptions: (i) during the period corresponding to the initial corrosion age, the initial corrosion rate function, i ini (τ ) is estimated as [16]:

i ini (τ )= 2 6 4 37.8(1 -wc) -1.64 c if τ ini + 1yr ≥ τ > τ ini 32.13(1 -wc) -1.64 c (τ -τ ini ) -0.3 if τ > τ ini + 1yr 3 7 5 ; (9)
(ii) during the period corresponding to the maximum corrosion age, i corr leads to a maximum corrosion rate, i max . The value of i max depends principally on the availability of water, oxygen and salts at the corrosion cell as a result of severe concrete cracking. If the membership function for the initial corrosion age is µ ini (τ ) and for the maximum corrosion age is µ max (τ ), the time-variant corrosion rate considering concrete cracking and environmental aggressiveness takes the form:

i corr (τ )= µ ini (τ )i ini (τ )+µ max (τ )i max µ ini (τ )+µ max (τ ) (10)
Figure 2b depicts the time-variant corrosion rate functions. It is observed that the initial corrosion rate function (i.e. dotted line) decreases after τ ini + 1 yr. This is caused by the formation of rust products on the steel surface that reduce the diffusion of iron ions away from the steel surface. Moreover, the accumulation of corrosion products reduces the area ratio between the anode and the cathode [16]. The time-variant corrosion rate (i.e. continuous line) follows the initial corrosion rate function until the time for several concrete cracking. After the time of severe cracking, τ sc , there is a transition towards the maximum corrosion rate. The advantage of this fuzzy approach is that the corrosion rate can be estimated based on local conditions, experts' knowledge and empirical models or observations. For example, for a given site, an expert can define the shape of the membership functions or the value of the maximum corrosion rate taking into account its knowledge about concrete quality, material properties, environmental hostility, RC behavior in similar conditions etc. Finally, it is important to clarify that the function can be permanently updated based on field measurements and experimental data.

Pit formation and shape are random phenomena which depend mainly on material properties, fabrication processes and electrochemical factors [START_REF] Goswami | Pitting corrosion fatigue of structural materials[END_REF]. Currently this dependence is not well understood. Some authors assume that pit growth follows the shape of a hemisphere or a prolate spheroid [START_REF] Harlow | A probability model for the growth of corrosion pits in aluminium alloys induced by constituent particles[END_REF] [START_REF] Kondo | Prediction of fatigue crack initiation life based on pit growth[END_REF]. For simplicity, a spherical shape of the pit is assumed herein (Figure 2b) and its radius, p(τ ), can be estimated from eq. 4.

Then, the volumetric rate of pit growth (eq. 7) can be directly rewritten in terms of pit growth rate by deriving eq. 4 with respect to time:

dp dτ =0.0116αi corr (τ ) (11)

Fatigue crack growth rate

The fatigue crack growth rate is estimated by means of the Paris-Erdogan law:

da dN = C p (∆K) m ( 12
)
where a is the crack size, N is the number of cycles, ∆K is the alternating stress intensity factor, and C p and m are material constants. It is know that there is a strong correlation between C p and m, and their values are highly depending on the environmental aggressiveness. For RC structures, Salah el Din and Lovegrove [START_REF] El Din | Fatigue of cold worked ribbed reinforcing bar -A fracture mechanics approach[END_REF] reported experimental values for C p and m corresponding to medium and long crack growth stages, leading to:

da dN = 2 4 3.83 × 10 -29 (∆K)
and the stress intensity factor, ∆K, is computed as:

∆K(a)=∆σY (a/d 0 ) √ πa ( 14
)
where ∆σ is the stress range, i.e. ∆σ = σ max -σ min , d 0 is the initial diameter of the bar and Y (a/d 0 ) is a dimensionless geometry notched specimen function, which can be approximated by [START_REF] Murakami | Stress intensity factor for circumferentially cracked round bar in tension[END_REF]:

Y (a/d 0 )= 1.121 -3.08(a/d 0 )+7.344(a/d 0 ) 2 -10.244(a/d 0 ) 3 +5.85(a/d 0 ) 4 [1 -2(a/d 0 ) 2 ] 3/2 (15)
The dotted line in Figure 2a follows the shape of the fatigue crack growth rate as function of time, which starts growing after corrosion initiation and pit nucleation.

Computation of pit-to-crack transition

In order to find the time for pit-to-crack transition, τ pt , an equivalent stress intensity factor for the pit, ∆K pit must be estimated. ∆K pit is found by substituting a in eq. 14 by the pit depth p(τ ) (eq. 4):

∆K pit (τ )=∆σY (p(τ )/d 0 ) p πp(τ ) (16)
Then, the equivalent crack growth rate becomes:

da dτ = C p [∆K pit (τ )] m f ( 17
)
where f is the frequency of the cyclic load. Therefore, the time of pit-to-crack transition is obtained by equating the pit growth rate (eq. 11) with the equivalent crack growth rate (eq. 17), and solving for τ pt in:

0.0116αi corr (τ pt )=C p (∆K pit (τ pt )) m f (18)
In order to calculate τ pt , eq. 18 must be solved numerically. Figure 2a illustrates the graphical solution for τ pt . Pit-to-crack transition is found at the intersection between the continuous and dotted lines.

Crack growth

Crack growth modeling is concerned with the time from the crack initiation until the crack size has reached a value that defines the failure of the cross section. The size of the initial crack on the steel reinforcement, a 0 , is estimated as the pit depth when the transition from pit to crack is reached, i.e. a 0 = p(τ pt ) (eq. 4). The size of the critical crack, a c , is defined as the crack size at which the RC member reaches a limit state of resistance (e.g. bending capacity). This time is obtained as:

τ cg = 1 f 2
where a 1 is the crack size at which the crack growth rate changes from medium to long crack growth. The transition between medium and long crack growth occurs when the crack size reaches a threshold stress intensity factor -i.e. ∆K(a 1 ) = 9 MPa √ m [START_REF] El Din | Fatigue of cold worked ribbed reinforcing bar -A fracture mechanics approach[END_REF].

The flowchart that summarizes the entire procedure is presented in Figure 3. The corrosion initiation process depends upon both environmental features and structural configuration. After corrosion initiation, the accumulation of corrosion products in the concrete/steel interface increases the corrosion rate as a result of concrete cracking. By considering the interaction between pit growth and cyclic load, the rate competition criterion is evaluated, and consequently, the time of pit-to-crack transition is assessed. Finally, once crack growth has become the preponderating process, the crack growth induces failure in the cross section.

This model allows the evaluation of the total structural lifetime including: corrosion initiation, pit-tocrack transition and crack propagation. Its robustness lies in the implicit integration of various parameters and processes affecting the RC lifetime, which is very convenient for reliability analysis.

Time-dependent reliability analysis

The cumulative distribution function (CDF) of the total corrosion-fatigue life, F τT (τ ), is defined as [START_REF] Zhang | Reliability-based reassessment of corrosion fatigue life[END_REF]:

F τT (τ ) = Pr{τ T ≤ τ } = Z τcp+τpt+τcg≤τ f (x) dx ( 20
)
where x is the vector of the random variables taken into account and f (x) is the joint probability density function of x. If structural failure is achieved when the crack size reaches a critical value inducing failure of the cross section, the limit state function becomes:

g (x,τ )=a c (x) -a τ (x,τ ) (21)
where a τ (x,τ ) is the crack size at time τ and a c (x) is the critical crack size defined in terms of a limit state of resistance. For example, if the limit state of bending capacity is chosen, since the crack growth induces a reduction in the bending capacity of the structural member, the ultimate bending capacity is reached when a c is achieved. By taking into account eq. 21, the failure probability, p f , can be estimated as:

p f = Z g(x,τ )≤0 f (x) dx (22)
Corrosion-fatigue deterioration process is a time-dependent problem where conditional probabilities are involved. Given the complexity of the problem, closed form solutions for both the CDF of the total corrosionfatigue lifetime and the failure probability are difficult to obtain. To solve this problem, it is necessary to use Monte-Carlo simulations or analytical approximations such as FORM/SORM.

Application to bridge girder

This section presents an illustrative example describing the coupled effect of corrosion-fatigue of a simply supported RC bridge girder subject to cyclic loading. The span of the girder is 10 m with the geometrical characteristics of the cross section and the steel reinforcement shown in Figure 4. The girder has been designed according to EUROCODE 2 [START_REF]Eurocode 1 and 2: Basis of design and actions on structures and design of concrete structures[END_REF]. Besides the dead load, a truck wheel load is applied on the girder. The design load, P k , corresponds to a wheel load located in the center of the span. Table 1 presents the load and the material properties used in the design.

In order to study the implications of the traffic frequency, f , the values of 50, 500, 1000 and 2000 cycles/day are considered. It is important to stress that these frequencies are in the range defined by the EUROCODE 1 for heavy trucks [START_REF]Eurocode 1 and 2: Basis of design and actions on structures and design of concrete structures[END_REF]. The effect of environmental hostility on structural reliability, was considered by taken into consideration four levels of aggressiveness (Table 2). Each level is characterized by: (1) a chloride surface concentration, C s , (2) a maximum expected value of the corrosion rate, i max , and (3) a concrete cover, c. The chloride surface concentration is treated as a random variable for which the mean depends largely on proximity to sea and is computed based on the work of McGee [23]. The maximum expected values of the corrosion rate are based on the EN206 [START_REF] Geocisa | Contecvet: A validated users manual for assessing the residual service life of concrete structures. Manual for assessing corrosion-affected concrete structures[END_REF]. The cover is defined according to EUROCODE 2 [START_REF]Eurocode 1 and 2: Basis of design and actions on structures and design of concrete structures[END_REF]. Table 2 summarizes the characteristics of each level of environmental aggressiveness.

The basic considerations and assumptions of the study are: -Section confinement is taken into account by using the Kent and Park model [START_REF] Kent | Flexural members with confined concrete[END_REF] and stress-strain relationship for steel follows an elasto-plastic model. -The elasticity modulus and the tensile strength of concrete are given in terms of the compression strength according to EUROCODE 2 [START_REF]Eurocode 1 and 2: Basis of design and actions on structures and design of concrete structures[END_REF]. -A limit crack width of 0.5 mm is used as a threshold for severe concrete cracking, τ sc .

-The water/cement ratio is calculated by using the Bolomey's formula [16].

-A threshold pit depth, p 0 , of 1.98×10 -6 m is utilized to estimate the time to pit nucleation.

-The deterioration process is continuous, -i.e. there is no maintenance.

The reliability analysis was carried out using Monte-Carlo simulations. The probabilistic models for the variables are given in Table 3. It is worthy to notice the high variability of surface chloride concentration and the chloride diffusion coefficient. In the case of C s , data reported by McGee [23] were obtained from a field-based study of 1158 bridges in the Australian state of Tasmania. This work appears to be the most comprehensive study for bridges in different environmental conditions. The chloride diffusion coefficient is influenced by many factors such as mix proportions (i.e. water-cement ratio, cement type), curing, compaction and environment (i.e. relative humidity and temperature), among others. Therefore, the high variability of the chloride diffusion coefficient results from the variability of the factors influencing D cl .

The stress range induced by the cyclic load, ∆σ, was computed by evaluating the equilibrium conditions between the internal forces and external actions. Tension at bottom reinforcement was defined as critical since this reinforcement is subject to both pitting corrosion and cyclic tension forces. Tension force inducing fatigue varies between T min corresponding to the dead load, and T max resulting from the combined action of to dead and live loads. Based on these considerations, the stress range, ∆σ, is computed as:

∆σ (P )= T max (P ) -T min A s ( 23
)
where A s is the initial cross section area of the reinforcement in tension (i.e. at the bottom).

Distributions of life stages

Corrosion initiation and pit nucleation

For all levels of aggressiveness, the goodness-of-fit test indicates that both τ ini and τ pn follow a lognormal distribution. The Kolmogorov-Smirnov test (KS-test) with a level or significance of 5% was chosen as selection criterion. As expected, the mean, µ τini , and standard deviation, σ τini , decrease when the environmental aggressiveness is increased (Figure 5). The high standard deviations result from the high COVs of C s and D cl (Table 3). The impact of the high variability of C s and D cl on the PDF of τ ini , is studied by varying their COVs, as shown in Figure 6. For all the considered COVs, µ τini diminishes when µ Cs increases, and tends to larger values when the COVs of C s and D cl are augmented. It is also observed that the relationship between µ Cs and µ τini is not linear and there is a limit (e.g. µ Cs = 2 kg/m 3 ) from which an increase of µ Cs does not produce significant variation of µ τini . In addition, it should be noted that the COV(C s ) has a higher impact on µ τini than the COV(D cl ), particularly, for smaller values of µ Cs (Figure 6b).

For the time to pit nucleation, the mean of τ pn varies between 3.9 -5.4 days with a COV(τ pn ) of 0.25 for all levels of aggressiveness. Therefore, given both the higher magnitude and variability of the time to corrosion initiation, it can be concluded that the time to pit nucleation is negligible.

Pit-to-crack transition

For all levels of aggressiveness and traffic frequencies, the KS-test has shown that the time to transition from pit to crack follows a lognormal distribution. Figure 7 presents the mean and 90% confidence interval of τ pt , for several traffic frequencies. Values between 8.3 -51.4 years and 3.2 -42.8 years are obtained for µ τpt and σ τpt , respectively. In all cases, the mean and confidence interval of τ pt reduce for higher values of traffic frequency. This decrease is expected because the crack growth rate is function of f , and therefore, the intersection between dp/dτ and da/dτ occurs early when f is increased. It can also be observed that the mean and the dispersion of τ pt are smaller when environmental aggressiveness is increased. This reduction is explained by fact that high levels of environmental aggressiveness are related with high values of i max . Thus, when the i max leads to high values, the fatigue crack growth rate is increased producing an early crack nucleation.

Since a crack is nucleated at the end of this stage, and the initial crack size is an input data to compute the length of the following stage, Figure 8 depicts the PDFs of a 0 , for both various levels of aggressiveness and several traffic frequencies. In all cases, the goodness-of-fit test shows that a 0 follows a lognormal distribution with µ a0 varying between 1.6-9 mm and σ a0 of 0.98-5.99 mm. The overall behavior indicates that both the mean and standard deviation tend to higher values when the level of aggressiveness is increased (Figure 8a). This increase is expected because given the preponderance of pitting corrosion on fatigue for higher levels of aggressiveness, the transition occurs for higher initial crack sizes. From the relationship between the mean and standard deviation of a 0 and traffic frequency is possible to notice that both µ a0 and σ a0 decrease when f is increased (Figure 8b). This reduction is due to the fact that the augmentation of the traffic frequency turns the fatigue into the preponderating process, and subsequently, accelerates the pit-to-crack transition.

Crack growth

In all cases considered, the KS-test gives as a result that τ cg follows a lognormal distribution with µ τcg varying between 1.6 -48.6 years and σ τcg between 0.31 -11.14 years. Figure 9 depicts the relationship between the mean and the 90% confidence interval of τ cg and the traffic frequencies. The overall behavior indicates that the mean and dispersion of τ cg diminishes when traffic frequency is increased. This behavior is explained by the fact that failure, i.e. eq. 21, is reached in a shorter period when the frequency tends to higher values (i.e. eq. [START_REF] El Din | Fatigue of cold worked ribbed reinforcing bar -A fracture mechanics approach[END_REF]. Furthermore, it can be noted that the mean and dispersion decrease when the level of aggressiveness is increased. This decrement is related to the fact that the size of the initial crack leads to larger values for higher levels of environmental aggressiveness, and therefore, the time to reach failure is lower.

For all traffic frequencies and levels of environmental aggressiveness, the KS-test shows that the size of the critical crack at which the RC member reaches the bending limit state, a c , follows a Weibull distribution with mean and standard deviation of 11.86 and 0.97 mm, respectively. This non dependency of f and environmental aggressiveness is expected because the bending limit state depends mainly on the effective cross section of the bar.

Total corrosion-fatigue lifetime

The goodness-of-fit test demonstrates that τ T follows a lognormal distribution with mean varying between 35.8 -448.8 years and standard deviation between 49 -264.7 years. The PDFs of τ T for various levels of aggressiveness and f = 500 cycles/day are depicted in Figure 10a. As expected, µ τT and σ τT decrease when the level of aggressiveness is increased. Figure 10b presents the mean of τ T for both, various levels of aggressiveness and traffic frequencies. In general, the critical time, τ T , occurs earlier when the traffic frequency is augmented.

The relationship between the mean of τ T and the level of aggressiveness is plotted in Figure 11a. It should be noted that the mean of τ T without considering fatigue damage (i.e. f = 0 cycles/day), is also included. The increase of traffic frequency induces an appreciable reduction of µ τT , in particular, in environments with low aggressiveness. By taking as a reference the case without considering fatigue damage, it is possible to estimate the additional lifetime reduction induced by the coupled action of corrosion and fatigue (Figure 11b). The additional reduction of τ T is high for smaller levels of aggressiveness and bigger traffic frequencies. It can be also noticed that considering the coupled effect of corrosion and fatigue can represent an additional lifetime reduction from 4 -43%. Considering the effect of corrosion and fatigue does not produce an appreciable additional lifetime reduction for both smaller traffic frequencies and very aggressive environments. However, for traffic frequencies between 500-2000 cycles/day, the coupled effect of corrosion-fatigue induces additional lifetime reductions between 21-43% of τ T . These results strengthen the importance of including the coupled effect of corrosion and fatigue in the lifetime assessment, in particular, when the structure is placed in corrosive environments under cyclic loading.

The percentage of participation of each stage in the total corrosion-fatigue life is estimated by the ratios of the mean values (Figure 12). In all cases the participation is meanly due to τ cp , followed by τ pt and τ cg . Whereas for τ cp , the participation decreases when the level of aggressiveness is increased, for τ pt and τ cg , it grows when the level of environmental aggressiveness is augmented. This behavior seems logical because an increment in the level of environmental aggressiveness accelerates the time to corrosion initiation and, subsequently, augments the participation of τ pt and τ cg . By considering the relationship between τ cp and f , the percentage of participation of τ cp increases when the traffic frequency grows. Contrary, for τ pt and τ cg , their percentages decrease when f leads to higher values. This behavior is explained by the fact that larger traffic frequencies reduce the length of the crack nucleation and propagation stages. To conclude, given the larger impact of the time to corrosion initiation and pit nucleation on the total corrosion-fatigue life (i.e. 47 to 92% of τ T ), the maintenance efforts must be directed towards controlling this stage of the process.

The failure probability of the girder, p f is plotted in Figure 13. It can be observed that the importance of traffic frequency on the failure probability for higher frequencies is minor. For instance, for a time window of 100 years, a moderate level of aggressiveness, without considering fatigue damage, a failure probability of 2.7×10 -2 is obtained; for the same conditions but different frequencies, failure probability becomes 1.6×10 -1 , 3.4×10 -1 , 3.5×10 -1 and 3.7×10 -1 for traffic frequencies of 50, 500, 1000 and 2000 cycles/day, respectively. That means, the failure probability is increased by 6, 12.5, 13.1 and 13.5 times for the traffic frequencies of 50, 500, 1000 and 2000 cycles/day, respectively. Whereas this difference varies considerably between 50 and 500 cycles/day, for values between 500 and 2000 cycles/day the increment is not significant. From these results it is possible to affirm that, although the coupled effect of corrosion and fatigue causes a significant increase of p f , there is a limit after which the increase of f does not produce an appreciable change in p f . The influence of reinforcement, A s , on failure probability for a traffic frequency of 1000 cycles/day is plotted in Figure 14. If p ft represents the target failure probability -i.e. p ft = 10 -4 , and a lifetime of 50 yr is chosen, it can be observed from Figure 14 that the reinforcement resulting from the design does not assures p ft . For the case of low aggressiveness (Figure 14a), increasing A s can guarantee this condition. Nevertheless, although increasing A s tends to amplify the time to reach p ft , for the other levels the failure probability is lower than p ft . Therefore, to ensure p ft during the lifetime of the structure, it is necessary to implement other countermeasures such as inspection and maintenance programs for corrosion control.

Conclusions

The combined action of corrosion and fatigue influences strongly the structural performance of RC structures and reduces substantially their lifetime. In this paper, a new model that couples these two phenomena is developed and the consequences on the life-cycle of RC structures are assessed. The model takes into account the interaction between the following processes: (1) corrosion induced by chloride ingress, (2) concrete cracking resulting from the accumulation of corrosion products and (3) reinforcement fatigue produced by cyclic loading. The corrosion-fatigue deterioration process is divided within the following three stages: (1) corrosion initiation and pit nucleation, (2) pit-to-crack transition and (3) crack growth. Corrosion initiation is induced by chloride ingress causing the nucleation of a pit. Pit-to-crack transition is achieved when the crack growth produced by cyclic loading becomes the preponderating process. A crack is nucleated at the end of this stage. Finally, crack growth ends when the crack size causes the structural failure.

In order to illustrate the model, a bridge girder located in several chloride-contaminated environments was studied. The time-dependant reliability analysis of the girder included the random nature of the material parameters, the loading scheme and the environmental conditions; the solution was found by using Monte-Carlo simulation. After the analysis, it was noted that the failure probability depends highly on the maximum corrosion rates, surface chloride concentration and traffic frequency.

The results also show that the time to corrosion initiation is highly influenced by the coefficient of variation of the chloride surface concentration. The mean of the time to pit nucleation had a smaller value in comparison to the mean of the time to corrosion initiation; therefore, it is possible to affirm that the length of this sub-stage can be neglected. It was also observed that pit-to-crack transition and crack growth occur early when both the level of aggressiveness and traffic frequency are increased. Regarding external loading, it was remarked that for traffic frequencies between 500 -2000 cycles/day, the coupled effect of corrosionfatigue becomes paramount by leading to additional lifetime reductions between 21 -43%. However, there is a limit after which the increase of traffic frequency does not produce an appreciable change in the failure probability.

Finally, it was observed that if the target failure probability is set as 10 -4 , for a structure with a lifetime of 50 years, the structural configuration selected does not reach this value for all environmental conditions considered. Therefore, to guarantee the operation above a predefined target failure probability it is necessary to implement countermeasures such as inspection and maintenance programs for corrosion control.

List of Tables

-Table 1. Design load and material constants. -Table 2. Description of the studied environments. -Table 3. Probabilistic models of the variables used in the model.

List of Figures

  × 10 -29 (∆K) 20.863 + Z ac a1 da 3.16 × 10 -12 (∆K) 3.143 if a 0 <a 1 Z ac a0 da 3.16 × 10 -12 (∆K)

-

  Fig.

1 .

 1 Scheme of corrosion-fatigue deterioration process in RC structures. -Fig. 2. (a) Rate competition criterion. (b) Time-variant corrosion rate. -Fig. 3. Flowchart of the proposed model. -Fig. 4. Configuration of the bridge girder. -Fig. 5. PDFs of τ ini for various levels of aggressiveness. -Fig. 6. Relationship between µ τini and µ Cs in terms of: (a) COV(D cl ), and (b) COV(C s ). -Fig. 7. Mean and 90% confidence interval of τ pt for several traffic frequencies. -Fig. 8. PDFs of a 0 for: (a) various levels of aggressiveness and f = 50 cycles/day; (b) low level of aggressiveness and various traffic frequencies. -Fig. 9. Mean and 90% confidence interval of τ cg for several traffic frequencies. -Fig. 10. (a) PDFs of τ T for various levels of aggressiveness and f = 500 cycles/day; (b) mean of τ T for both, various levels of aggressiveness and traffic frequencies. -Fig. 11. (a) Mean of τ T for various traffic frequencies. (b) Additional lifetime reduction induced by the coupled action of corrosion and fatigue. -Fig. 12. Percentage of participation of each phase in the total corrosion-fatigue life. (a) Low aggressiveness. (b) Extreme aggressiveness. -Fig. 13. Time-dependent structural reliability of the considered bridge girder. -Fig. 14. Influence of positive steel reinforcement area on failure probability for several levels of aggressiveness: (a) low, (b) moderate.

2. Pit-to-crack transition, v pt 3 .Fig. 1 .Fig. 2 .Fig. 3 .

 3123 Fig. 1. Scheme of corrosion-fatigue deterioration process in RC structures.

Fig. 4 .

 4 Fig. 4. Configuration of the bridge girder.

Fig. 5 .Fig. 6 .Fig. 7 .

 567 Fig. 5. PDFs of τ ini for various levels of aggressiveness.

Fig. 8 .

 8 Fig. 8. PDFs of a 0 for: (a) various levels of aggressiveness and f = 50 cycles/day; (b) low level of aggressiveness and various traffic frequencies.

Fig. 13 .Fig. 14 .

 1314 Fig. 13. Time-dependent structural reliability of the considered bridge girder.

High structures situated to 0.1 km or less from the coast, but without direct contact to seawater. RC structures subject to deicing salts can also be classified in this level.

5 50

Extreme structures subject to wetting and drying cycles; the processes of surface chloride accumulation are wetting with seawater, evaporation and salt crystallization.

10 55

Table 3 Probabilistic models of the variables used in the model.

Variable