On Mathematical Models of Degradation Processes According to ISO16204 and fib Model Code

On Mathematical Models of Degradation Processes According to ISO 16204 and fib Model Code Martina Šomodíková1 | Jiří Doležel1 | David Lehký1 1 Introduction In the fib Model Code 2010 [1], service life is defined as “the period in which the required performance of a structure or structural element is achieved, when it is used for its intended purpose and under the expected conditions of use”. It is a temporal (quantitative) value that is not a material property but is related to the ability of materials, components and systems to maintain specific utility and other properties at the required level under normal maintenance, over a certain time period and under given operating and environmental conditions. The methods for determining the residual service life of new and existing structures may be as follows: Based on knowledge of the service life of a similar structure (with similar characteristics) located in similar conditions; On the basis of accelerated tests; Using mathematical models; Using stochastic method – a method using a reliability model or a method using a combination of statistical and deterministic models may be used. In the case of the determination of service life based on the knowledge of the service life of a similar structure, it is rather an estimate due to the large variability, especially in the properties of the materials used and the influence of the various environments. For the determination based on accelerated tests, it is necessary that the degradation mechanisms of the material in accelerated conditions are the same as in the real environment. Even if this condition is met, the lack of data on the degradation rate under normal conditions, based on long-term monitoring of the condition of structures and long-term testing, is a major problem in the determination of service life. Therefore, mathematical modelling and stochastic methods, which are the focus of this paper, appear to be appropriate methods for the service life assessment. Related to this issue the utilization of a standardized methodology implemented in the industry worldwide plays an important role. At the turn of the millennium, efforts were initiated to develop a platform for durability design of concrete structures that contained the same elements and philosophy as that of modern structural design [2]. In following few years the fib Bulletin No. 34 [3] was endorsed. The close cooperation between the fib and ISO commitees was established and based on the principles of ISO 2394 [4] the fib Model Code 2010 [1] and the ISO 16204 [5] documents were finalized. These two documents are today close to being identical when assessing the service life and durability of conrete structures. 2 Mathematical modelling of degradation processes In most cases, the service life of a structure, tS, related to ORIGINAL ARTICLE Abstract The main factors affecting the service life of concrete structures are the presence of chloride ions and carbon dioxide, alternating frost action and mechanical stresses on the structure. In practice, these effects can be taken into account by using mathematical modelling of these phenomena. In this contribution, the authors focus on the processes of chloride ions and carbon dioxide (carbonation process) diffusion through concrete and the associated subsequent corrosion of reinforcement. Mathematical models recommended by the fib Model Code and ISO 16204 are described and the effect of simultaneous environmental and mechanical load on the service life of the structure is discussed. The models are used to study the residual service life of a simple reinforced concrete frame bridge based on the percentage loss of reinforcement area analysis over time. Keywords fib Model Code, ISO 16204, Concrete Carbonation, Chloride Ions Ingress, Corrosion of Reinforcement Correspondence Ing. Martina Šomodíková, Ph.D. Brno University of Technology Faculty of Civil Engineering, Institute of Structural Mechanics Veveří 331/95 60200 Brno Email: somodikova.[email protected].cz 1 Brno University of Technology, Faculty of Civil Engineering, Brno, Czech Republic Proceedings in civil engineering This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. https://doi.org/10.1002/cepa.2135 wileyonlinelibrary.com/journal/cepa ce/papers 6 (2023), No. 5 © 2023 Ernst & Sohn GmbH. 1221 the durability limit state, is defined as the time that elapses from the time the structure is put into service to the time when the reinforcement corrosion process is initiated, i.e., tS = ti. In the case of penetration of chloride ions and air CO2 through the concrete, the corrosion process is initiated by diffusion of chloride ions in the concrete to the depth at which the steel reinforcement is located, or by carbonation, which lowers the pH of the concrete at its contact with the steel reinforcement. Taking a less conservative view, the service life can be determined as the sum of two time periods, i.e., the initiation phase, ti, and the propagation phase, tp, as tS = ti + tp. Since the corrosion of the reinforcement reduces its effective area, the safety criterion can be defined, for example, on the basis of reaching a limit value of percentage loss of area of the reinforcement bars. Note here that while the area of the reinforcement decreases, the corrosion products formed have a twofold to sixfold increase in volume, which leads to an increase in tensile stress in the surrounding concrete, and thus to the formation of longitudinal continuous cracks and consequent spalling of the concrete cover. 2.1 Corrosion of reinforcement The course of reinforcement corrosion, which takes place during the propagation stage of the assumed service life of the structure, i.e., after the initiation time of reinforcement depassivation is reached, can be modelled according to [6]. For the case of uniform corrosion, the following equations are used to estimate the reinforcement diameter d [mm] over time: 𝑑(𝑡)={ 𝑑i 𝑑i−0.0116𝑖corr𝑅corr(𝑡−𝑡i) 0 (1a) for the prescribed conditions of: { 𝑡≤𝑡i 𝑡i<𝑡≤𝑡i+𝑑i 0.0116𝑖corr𝑅corr 𝑡>𝑡i+𝑑i 0.0116𝑖corr𝑅corr (1b) where di [mm] is the initial diameter of the reinforcement, icorr [μA/cm2] takes the corrosion current density (corrosion rate) into account and Rcorr [-] reflects the type of corrosion (uniform or pitting). Similarly, the pitting depth p [mm] can be determined over time for pitting corrosion as [7]: 𝑝(𝑡)={ 0 for 𝑡≤𝑡i 0.0116𝑖corr𝑅corr(𝑡−𝑡i)for 𝑡>𝑡i (2) Assuming a hemispherical shape of the pit the residual (net) cross-sectional area of the corroded bar (Ar [mm2], grey area in Fig. 1 left) at time t > ti can be calculated as [8]: 𝐴r(𝑡)= { π𝑑i2 4−𝐴1−𝐴2for 𝑝(𝑡)≤√2 2𝑑i 𝐴1−𝐴2for √2 2𝑑i<𝑝(𝑡)≤𝑑i 0 for 𝑝(𝑡)>𝑑i (3a) with 𝐴1=12[𝜃1(𝑑i 2)2−𝑎p|𝑑i 2−𝑝(𝑡)2 𝑑i|] 𝐴2=12[𝜃2𝑝(𝑡)2−𝑎p𝑝(𝑡)2 𝑑i] 𝑎p=2𝑝(𝑡)√1−(𝑝(𝑡) 𝑑i)2 𝜃1=2arcsin(𝑎p 𝑑i) 𝜃2=2arcsin(𝑎p 2𝑝(𝑡)) (3b) The cross-sectional area can be calculated without significant loss of accuracy using a simpler equation that assumes a circular pit shape and a reinforcement area calculated based on the overlap of two circles of radius di (see Fig. 1 right), as in [9]: 𝐴r(𝑡)={π𝑑i2 4−𝐴pfor 𝑝(𝑡)≤𝑑i 0pro 𝑝(𝑡)>𝑑i (4a) with 𝐴p=𝑑i2 4(𝛿−sin 𝛿) 𝛿=2arccos(1−𝑝(𝑡) 𝑑i) (4b) Figure 1 Area of reinforcing bar with pitting corrosion – hemispherical pit (left), circular pit (right) 2.2 Concrete carbonation and chloride ions ingress As already mentioned above, the depassivation of reinforcement can be caused mainly by carbonation process and/or by the chloride ions ingress, where the initiation time is determined on the basis of the assumption that the carbonation front reaches the depth at which the steel members are located or the chloride ion concentration at the depth of the steel reinforcement reaches a critical value. The mathematical models of carbonation and chloride penetration are based on Fick's second law of diffusion: 𝜕𝐶 𝜕𝑡=𝐷𝜕2𝐶 𝜕𝑥2 (5) where C is the concentration of the permeant, x is the distance from the surface, t is the diffusion time and D is generally the diffusion coefficient, which depends on the properties of the concrete and the environment. The solution to this differential equation is found by the Crank procedure using the Gaussian error function “erf(.)”. According to ISO 16204 [5], the carbonation depth xc [mm] at time t [years] can be calculated as: 𝑥c(𝑡)=𝑊𝑘√𝑡 (6) 1222 25097075, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cepa.2135 by Technical University In Brno, Wiley Online Library on [23/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License where the parameter W [-] takes changing climatic conditions (such as humidity and temperature) into account, k [-] is a factor describing the basic resistance of the concrete mixture to the carbonation process. When designing new structures, the parameters W and k can be determined based on data measured on similar structures in similar climatic conditions; when verifying existing structures, they must be determined from measurements. For modelling, the sophisticated model recommended in the fib Bulletin No. 34 [3] can be used, where the carbonation depth is determined based on the formula: 𝑥𝑐(𝑡)=√2𝑘e𝑘c(𝑘t𝑅ACC,0 −1 +𝜀t)𝐶CO2∙𝑊(𝑡)∙√𝑡 (7) The environmental function parameter, ke, includes the influence of the moisture content of the concrete surface (influence of the relative humidity RH [%]) on the value of the diffusion coefficient, the parameter kc takes the influence of the curing time of the fresh mix (parameters tc [days] and bc [-]) on the effective resistance of the concrete to carbonation into account. The parameter 𝑅ACC,0 −1 [(m2/s)/(kg/m3)] is the inverse of the effective resistance of the dry mix to carbonation, the value of which is determined by testing concrete specimens exposed to conditions in which the carbonation process is accelerated (the ACC test). The parameters kt [-] and εt [(m2/s)/(kg/m3)] cover the differences between samples tested under accelerated conditions and the structure tested under natural conditions. The CCO2 parameter includes the effect of the CO2 concentration and the time-dependent weather function W(t) [-] accounts for the weather conditions due to rain (parameters pSR [-], tw [days] and bw [-]). We refer the reader to [3] for detailed recommendations on the individual model parameters. In the case of chloride ions ingress, the chloride concentration, C(x,t) [wt. %/cement] (relative to the weight of cement), at depth x [mm] (most often at the depth of the concrete cover of the steel reinforcement) and time t [years] can be expressed for 1D cases by a simple analytical formula: 𝐶(𝑥,𝑡)=𝐶0+(𝐶S−𝐶0)[1−erf(𝑥 2√𝐷c𝑡)] (8) where CS [wt. %/cement] is the chloride concentration on the concrete surface, Dc [mm2/year] is the diffusion coefficient of chloride penetration through the concrete and C0 [wt. %/cement] is the initial chloride concentration in the concrete. The solution of Eq. (8) is valid assuming homogeneous material fully saturated with water and constant CS and Dc values over time. Due to its simplicity, this solution has been used for several decades and is also recommended in various modifications by a number of national and international normative documents. Critical points of the use of models of chloride penetration through concrete are summarized in e.g. [10], where, among other things, the high variability of modelling results caused by often considerable uncertainty in the values of model parameters is mentioned. Another widely discussed topic is the simultaneous action of multiple degradation processes and the time dependence of diffusion coefficient and surface chloride concentration The time dependence of the diffusion coefficient Dc can be modelled according to the ISO 16204 [5] using the formula: 𝐷c(𝑡)=𝐷app(𝑡)=𝐷app(𝑡0)(𝑡0 𝑡)𝛼 (9) where Dapp(t0) [mm2/year] is the actual value of the diffusion coefficient determined at the reference time, t0 [years] (t0 = 28 days = 0.0767 years) and α [-] is the ageing factor of the concrete, which takes the increase in the resistance of the concrete to the penetration of aggressive substances due to its ageing into account, i.e., it takes into account the decrease in the Dapp value over time due to the hydration of the cement components (changes in the pore structure). It should be noted that, both in the design of new structures and in the assessment of the residual life of existing structures, the ageing factor α should be obtained from observations of structures in situ, where the concrete composition, performance and conditions of exposure to aggressive substances are similar to those of the actual structure. To calculate the ageing factor, observations during at least two periods of exposure (with a sufficient interval between observations) are necessary. The model according to the fib Bulletin No. 34 [3] accounts for the time dependence of the diffusion coefficient as follows: 𝐷c(𝑡)=𝐷app(𝑡)=𝑘env∙𝐷RCM(𝑡0)∙𝑘u∙(𝑡0 𝑡)𝛼 (10) where kenv [-] is the environmental parameter that takes the effect of temperature on the diffusion coefficient value (parameters T [°C] and be [°C]) into account, DRCM(t0) [m2/s] is the chloride migration coefficient, the value of which can be determined from the Rapid Chloride Migration test (RCM), and kt [-] is the conversion factor of the diffusion coefficient unit (ku = 3.1536∙106; 1 m2/s = 3.1536∙106 mm2/year). The concentration of chloride ions is then determined based on Eq. (8). At the surface of concrete exposed to external chloride, a layer where diffusion is not the main process and the chloride ions penetration process differs from the Fick's second law of diffusion due to exposure to frequent wetting and subsequent evaporation, the so-called convection zone, Δx, usually forms. Taking into account this fact, the Eq. (8) can be further modified to the form of [11]: 𝐶(𝑥,𝑡)=𝐶0+(𝐶S,∆𝑥−𝐶0)[1−erf(𝑥−∆𝑥 2√𝐷app(𝑡)∙𝑡)] (11) The maximum concentration of chloride ions here is not at the outer surface (x = 0), but arises progressively at the position x = ∆x, with this thickness given in the range of 6–11 mm. In the convection zone, chloride concentrations can deviate considerably from normal values, so the model according to Eq. (11) neglects these values and works with the so-called surrogate chloride concentration CS,∆x [wt. %/cement], which is applied only from a depth greater than the depth of the convection zone. This modification in determination of the chloride concentration in time is only meaningful for long exposure times when the convection zone has formed and remained more or less constant for a long time. 1223 25097075, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cepa.2135 by Technical University In Brno, Wiley Online Library on [23/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License To express the time dependence of the chloride concentration on the surface of concrete, CS, a linear or rootmean-square dependence on time with constant k according to [12] can be used: 𝐶S(𝑡)=𝐶0+𝑘𝑡 or 𝐶S(𝑡)=𝐶0+𝑘√𝑡 (12) Note here that the functions in Eq. (12) are not clearly suitable for describing the chloride ions ingress from the spreading salts, which is related to the change of seasons, the action of rain (washing the salts from the concrete surface) and possibly other climatic or other influences. The time dependence of CS value is also not included in ISO 16204 due to the complexity of the whole issue and therefore the use of these models should always be carefully considered in view of possible errors in the determination of the service life of structures. As an example, the effect of seasonal application of thawing salts has been addressed for 2D cases by the cellular automata technique, see e.g. [13]. 2.3 Simultaneous effects of environmental actions and mechanical load Intuitively, it can be assumed that degradation processes will be faster in structures with cracks. However, neither the fib nor the ISO committees have been able to come up with any general model that takes this effect into account. Therefore, it was agreed to use a simplified approach that assumes that reinforcement corrosion is not affected up to a certain crack width. Depending on the severity of the environmental influence and the sensitivity of the structure, the limiting crack width is usually given by a characteristic value (i.e., 5% upper quantile) in the range of 0.2 to 0.4 mm [2]. According to studies reported in e.g. [14], the influence of stress on the rate of carbonation and chloride diffusion can be easily taken into account by means of correction factors. The depth of carbonation can be predicted with respect to the stress state according to the formula: 𝑥c(𝑡)=𝑘𝜎𝐴√𝑡 (13) where the constant A can be calculated using any suitable carbonation model depending on the composition and curing of concrete, type of cement, humidity of the environment, CO2 content, or other parameters (see e.g. the above mentioned model according to [3]). The correction factor kσ [-] is defined separately for elements under tension (stress σt) and compression (stress σc) as: 𝑘𝜎(𝜎t/𝜎u,t)=1+1.41(𝜎t/𝜎u,t)+0.82(𝜎t/𝜎u,t)2 𝑘𝜎(𝜎c/𝜎u,c)=1−2.27(𝜎c 𝜎u,c)+4.86(𝜎c/𝜎u,c)2 (14) where σu,t and σu,c represent the ultimate tensile and compressive stress of the concrete. The dependence of kσ on the tensile/compressive stress ratio and its limiting value is shown in Fig. 2. It is evident that a modest compressive load, i.e., values of (σc/σu,c) in the range of 0 to approximately 0.5, decreases the rate of concrete carbonation. This effect is due to partial closing of micro-cracks under an applied compressive stress. However, if the load is increased above approximately 50 % of the ultimate compressive strength, the on-going closing micro-cracks is overcompensated by formation of new cracks through which air CO2 can penetrate the concrete. This accelerates the carbonation process. As the tensile load increases, the carbonation process is only accelerated, since even a modest tensile stress leads to the formation and opening of micro-cracks, which serve as new pathways for the penetration and migration of CO2 into the concrete. Figure 2 Dependence of the kσ factor on the ratio of tensile/compressive stress and its limit values As in the case of carbonation, the models of chloride ions ingress can be extended by the effect of mechanical load on the structure. With respect to cracks in the concrete, formatted due to stress from loading, the diffusion of chloride ions is accelerated. When the effect of cracks is considered, the value of the diffusion coefficient (generally denoted by D) changes. This can be divided into two parts, D0 [m2/s] and Dt [m2/s], as shown in Fig. 3. The diffusion coefficient D [m2/s] is determined with respect to the width of the cracks formed, w [mm], and their maximum distance sr,max [mm] as [15]: 𝐷=(1− 𝑤 𝑠r,max)𝐷0+𝑤 𝑠r,max𝐷t (15) where D0 is the value of the diffusion coefficient for intact concrete and Dt is the value of the diffusion coefficient inside the crack, the parameters w and sr,max can be obtained e.g. by measurement on a real structure or by calculation according to e.g. [16]. The values of Dt can be determined with respect to the crack width as follows [17]: 𝐷t={0 m2/s (0,16𝑤−3)∙10−10 m2/s 13∙10−10 m2/s (16a) for the crack width of: {𝑤<30 μm 30 μm ≤ 𝑤≤100 μm 𝑤>100 μm (16b) Figure 3 Diffusion coefficient D for chloride penetration through cracked concrete 1224 25097075, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cepa.2135 by Technical University In Brno, Wiley Online Library on [23/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License The application of these relations for modelling the simultaneous effect of mechanical and environmental load and the verification of the ability of these models to achieve results closer to reality is presented in [18]. 3 Study on mathematical models with or without the effect of mechanical load A simple reinforced concrete frame bridge was selected for the residual service life study. The bridge was put into operation in 2019, and after two years of use (2021), a continuous crack greater than 0.1 mm wide was observed in the middle of the span of the frame girder (see Fig. 4). Figure 4 Continuous crack greater than 0.1 mm wide in the middle of the frame span, recorded after two years of bridge use – upstream side (left), substructure bottom view (middle) and downstream side (right) The service life of the bridge was analysed using mathematical modelling of degradation processes. The input material parameters considered were as follows: C30/37 concrete class (exposure class XD1 – medium wet, humid environment and concrete surface exposed to chlorides dispersed in air; water to cement ratio w/c = 0.55), B500B,A reinforcement profile of 32 mm diameter, concrete cover 50 mm, stirrups of 12 mm diameter (i.e., main reinforcement cover a = 62 mm). For modelling of degradation processes, the aforementioned models according to the fib Bulletin No. 34 [3] and ISO 16204 [5], respectively, were used. The input parameters were defined according to Tab. 1. Note here that a combination of deterministic and stochastic models was used to determine the residual service life. Deterministic values of carbonation depth, chloride concentration and loss of reinforcement area over time were calculated based on the mean values of the input parameters (column “Mean” in Tab. 1). To take the variability in material properties and environmental characteristics into account, the same outputs were calculated using stochastic modelling based on a suitably chosen statistical model of the input variables (columns “COV” and “PDF” in Tab. 1, meaning the coefficient of variation and probability density function, respectively). The results are summarized in Fig. 5, which shows the mean values (deterministic model) ± standard deviations (stochastic model) of carbonation depth and chloride concentration over time. FReET-D software was used for stochastic modelling (see e.g. [19]), statistical characteristics were calculated based on one hundred of random simulations. Nor the deterministic nor the stochastic calculation does not assume that the reinforcement will be depassivated due to carbonation or chloride attack over the design life of the structure (solid lines in Fig. 5). The reinforcement depassivation is not assumed even if the effect of cracks in concrete is taken into account (dashed lines in Fig. 5). The results of the stochastic modelling then show some variability in the modelling results. However, using the fib and ISO model, the depth of concrete cover and the critical value of chloride ions concentration at the depth of the concrete cover is not reached even when considering the variability of the input variable, and it is therefore clear that the design of the structure with respect to the limit state of durability is perfectly fine with respect to a sufficient concrete cover. Table 1 Definition of input parameters Input parameter [Unit] Mean COV [-] PDF* Note CCO2 [mg/m3] 820 0.12 N RH [%] 70 0.07 B Limits: 0–100 tc [days] 7 - Det. bc [-] –0.567 0.04 N 𝑅ACC,0 −1 [(m2/s)/(kg/m3)] 9.8×10–11 0.42 N Limits: 1×10–12–1×10–9 kt [-] 1.25 0.28 N εt [(m2/s)/(kg/m3)] 1×10–11 0.15 N pSR [-] 1 - Det. tw [days] 60 - Det. bw [-] 0.446 0.37 N x [mm] 62 0.12 LN σt/σu,t [-] 1 - Det. Assuming reaching the tensile strength of concrete CS [wt. %/cement] 0.465 0.75 N Lower limit : 0 C0 [wt. %/cement] 0.0 - Det. DRCM [m2/s] 1.97×10–11 0.20 N be [-] 4526.85 0.15 N T [°C] 10 0.05 N α [-] 0.3 0.40 B Limits : 0–1 Ccr [wt. %/cement] 0.6 0.15 B Limits : 0.2–2.0 w [mm] 0.3 - Det. Based on measurements (effect of temperature changes and concrete shrinkage) sr,max [mm] 600 - Det Calculated according to [16] di [mm] 32 0.02 N icorr [-] 3 0.5 R Rcorr [-] 6 - Det Mean = 2 for uniform corrosion Note: *Det. = deterministic value; B = beta, LN = lognormal, N = normal, R = rectangular probability density function Although the mathematical modelling does not assume depassivation of the reinforcement and subsequent corrosion of the reinforcement bars, a theoretical analysis of the loss of reinforcement area was performed; see Fig. 6. Here, 1225 25097075, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cepa.2135 by Technical University In Brno, Wiley Online Library on [23/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License different formulas were used for reinforcement area calculations in order to compare the potential percentage loss of reinforcement area over corrosion propagation period. Uniform as well as pitting corrosion was compared for this purpose. Figure 5 Results of a study of the residual service life assessment – modelling of concrete carbonation and chloride ions ingress Figure 6 Results of a study of the residual service life assessment – modelling of reinforcement corrosion The safety criterion was determined according to the national standard ČSN 73 6221 [19] for very poor (VI) to emergency condition (VII) as reaching the limiting value of the corroded reinforcement area at the level of 15 %. This value would be achieved after approximately 40 years of propagating corrosion (Fig. 6). Based on the theoretical mathematical models a corrosion loss of reinforcement area at the level of 15 % was reached after 40 years of corrosion propagation with a probability of 47.6 to 56.4 % (according to the formula used to calculate the residual area of corroded reinforcement, Ar). A normal distribution was assumed in this case. Note that the progress of the reinforcement area is very similar for uniform and pitting corrosion during the first 50 years of propagation. In the following years, pitting corrosion progresses much faster. If no additional measures are taken to take into account the identified defects in the form of a localized crack, the structure will be classified as being in structural condition VI to VII after 40 years of use with this probability. Thereafter, restrictive measures will have to be taken within the framework of its use, in the extreme case resulting in its closure. It should be noted here that if the loss of the reinforcement area due to corrosion up to 1 % is detected, it is recommended to recalculate the load-bearing capacity in relation to the classification of the structure in structural condition IV (satisfactory). Based on mathematical modelling, such a loss of reinforcement area is achieved with a 50% probability in 2 years to 11 years after the initiation of corrosion process. 4 Conclusions When the bridge owner/manager or structural engineer is faced with the need to assess the residual service life of concrete structures in relation to ongoing degradation processes, reinforcement corrosion and the current structural condition, the use of appropriate mathematical models in combination with stochastic methods appears to be a very fast and efficient tool compared to residual service life assessment procedures based on knowledge of the service life of a similar structure located in similar conditions or based on accelerated testing. The use of advanced methods of stochastic service life analysis allows a greater insight into the progress of the degradation processes taking place, with the ability to make predictions of the damage over time. The values of the theoretical probability of failure when limiting values are exceeded, such as loss of reinforcement area due to corrosion, can also be calculated. Finally, we note that mathematical and stochastic models of concrete degradation and reinforcement corrosion require knowledge of the input parameters and their probabilistic models, which in some cases may appear to be a disadvantage. However, the mathematical models can be calibrated based on the results of real in-situ measurements and diagnostic surveys carried out. Suitable probability distribution functions and statistical characteristics can also be obtained from the available literature and some normative documents. 5 Acknowledgements The authors would like to gratefully acknowledge the financial support of the Czech Science Foundation project No. 22-00774S. References [1] fib Bulletins Nos. 65 and 66 (2012) Model Code 2010 1226 25097075, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/cepa.2135 by Technical University In Brno, Wiley Online Library on [23/10/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License – Final draft, Volume 1 and 2. International Federation for Structural Concrete (fib), Lausanne, Switzerland. [2] Helland, S. (2013) Design for service life: implementation of fib Model Code 2010 rules in the operational code ISO 16204. 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