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A condition-based maintenance policy based on a probabilistic meta-model in the case of chloride-induced… Boutros, El Hajj; Castanier, Bruno; Schoefs, Frank; Bastidas-Arteaga, Emilio; Yeung, Thomas Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 A Condition-Based Maintenance Policy Based On A Probabilistic Meta-Model In The Case Of Chloride-Induced Corrosion Boutros El Hajj PhD student, University of Nantes, Nantes, France Bruno Castanier Professor, University of Angers, Angers, France Frank Schoefs Professor, University of Nantes, Nantes, France Emilio Bastidas-Arteaga Associate Professor, University of Nantes, Nantes, France Thomas Yeung Assistant Professor, Ecole de Mines de Nantes, Nantes, France ABSTRACT: Maintenance and management policies are usually focused on minimizing the life-cycle cost only. Therefore the optimal solution in this context does not necessarily result in a satisfactory long-term structural performance. In this paper, we will present an approach for modeling the degradation of structures and infrastructures for maintenance purposes. The degradation is modeled using probabilistic data-driven state dependent stochastic processes, hereafter called meta-model. This work implements this degradation model into a maintenance framework and carries out two numerical examples in order to show the applicability of our meta-model in a maintenance and management optimization context. This paves the road for future work on meta-model updating and maintenance optimization by consider-ing a multi-objective optimization policies.1. INTRODUCTION Growth and achievement in society are ensured by the quality of operation and performance of the infrastructure: roads, power lines, ports, dams, bridges, etc... Previous studies have always searched for methods to measure and monitor the performance of structures in order to avoid excessive degrada-tion leading to unacceptable risky situations. It is highly important to keep track of the evolution of degradation in materials and predict its level to avoid failure by maintaining the structure in the “safe” zone. In civil engineering, inspections can be classed into two categories: destructive and non- destructive techniques (DT and NDT respec-tively). As the name indicates, in order to carry out a DT inspection the structure is harmed. It’s up to the inspection consultant to decide whether the inspection will alter the global performance of the structure or just give a bad visual sensation (not less important than the performance in some cases). On the other hand, NDT do not affect the performance of the structure. From a structural and managerial point of view, it is clearly prefer-able to use NDT over DT, but NDT measurements could be costly and in some cases, results are less reliable than the obtained by DT. Corrosion of reinforcement steel is known to be one of the major causes of deterioration of re-inforced concrete (RC) structures (Bastidas-Arteaga et al., 2012, Bastidas-Arteaga et al., 2011). In this paper, we focus on the case of chlo-ride-induced corrosion in submerged reinforced concrete. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 The purpose of this paper is to propose a maintenance and management model for this pa-thology. Most existing maintenance systems fo-cus on life-cycle cost minimization only. There-fore, the obtained solution does not necessarily re-sult in satisfactory long-term structural perfor-mance (Frangopol & Liu, 2007). Moreover, com-plex multi-parametric degradation models were developed for representing the main trends, but not for (i) uncertainty propagation, and (ii) updat-ing from NDT results that are generally not di-rectly linked to the model. Meta-models have been shown to be efficient in the case where the degradation can be modeled with a single Markov matrix (O’Connor et al., 2013, Bastidas-Arteaga et al., 2012). This paper addresses the case of a complete three-step problem. The aim of the proposed approach is to esti-mate the life-cycle cost and the condition index for different proposed solutions. Therefore, on one hand we calculate a “condition index” through which overall structural condition is rep-resented. On the other hand, we calculate life-cy-cle cost from a set of possible maintenance strate-gies, inspections, and design costs. Section 2, introduces the degradation pro-cess. Section 3, shows the development of the meta-model. Section 4 describes the maintenance actions and performance indexes that can be con-sidered. Section 5 illustrates the proposed ap-proach with numerical examples of different maintenances policies. And finally, the conclu-sions and perspectives of this work are drawn in section 6. 2. PROBLEM STATEMENT Chloride ingress into RC structures leads to ser-viceability and safety losses. Deterioration mod-elling allows estimating the effects of chloride in-gress with regard to serviceability or ultimate limit states. Ultimate limit states are highly de-pendent on both, geometrical characteristics (cross-sectional dimensions, span length, etc.) and loading (dead, live, seismic, etc.). Therefore, to generalize the results, this work focuses on a ser-viceability limit state related to the time to corro-sion damage of the concrete cover (severe crack-ing or spalling). Corrosion-induced cover crack-ing and damage occurs on the concrete surface above and parallel to the rebars. The time to cor-rosion damage, (severe cracking or spalling), is thus obtained as the sum of three stages (Figure 1): (i) corrosion initiation; (ii) crack initiation (time to first cracking - hairline crack of 0.05 mm width), and; (iii) crack propagation (time for crack to develop from crack initiation to a limit crack width, wlim). The corrosion initiation phase is controlled by the diffusion of chlorides into concrete. When the chloride concentration at the surface of the steel (cover depth) exceeds a threshold concentra-tion, there is steel depassivation followed by cor-rosion initiation.  The crack initiation phase is dominated by the chemical reaction of corrosion generating cor-rosion products. These corrosion products or rust slowly fill the pores surrounding the reinforce-ment steel creating an internal pressure. When the internal pressure exceeds the concrete resistance, hairline cracks appear. Finally, the propagation phase lies in the con-tinuous accumulation of rust generating more in-ternal tensile stress, resulting in an excessive cracking of the concrete cover.   Figure 1. Degradation by corrosion of reinforced concrete (50 mm cover) by (Youping Liu, 1996) The proposed probabilistic degradation meta-model is based on a small number of “phys-ical” indicators (two per phase) chosen in a way 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 to be accessible through NDT inspections and to provide a truthful degradation level. For each phase, we selected the following physical indica-tors.   1st phase: Chloride concentration at the sur-face of the steel [Cl-] and the concrete pH.  2nd phase: Internal tensile stress and corrosion current density.  3rd phase: Crack width and corrosion current density (Li et al.,  2006). 3. META-MODELING This section shows the formulation of the degra-dation meta-model. Since this is a three-phased model, we will start by laying down the common ideas behind the model. Then we proceed to ex-plicit the mathematical equations. For each phase, we propose to define a biva-riate process written  (𝜌𝑡, 𝜃𝑡)∀𝑡≥0  as a state de-pendent stochastic process similar to the one in-troduced in Zouch et al. (2011): 𝜌𝑡  describing a condition state and 𝜃𝑡 a potential of its evolution (observed), both being dependent. The evolution of degradation over a period of time is given by positive increments for the degradation processes respectively (Δρ, Δθ) which are continuous ran-dom variables. We assume that the degradation increments in a given time interval τ are random variables which are a function of the present deg-radation state(𝜌𝑡, 𝜃𝑡). The degradation process is therefore assumed to be a Markov process. A suit-able candidate for the distribution laws of each in-crement (Δρ, Δθ) is the gamma distribution (Van Noortwijk, 2009) defined by two parameters (α and β where: α is the shape parameter and β is the scale parameter). In the described bivariate state-dependent model, we will consider that only the shape parameter is a function of the current state (𝜌𝑡, 𝜃𝑡)and τ, but independent of time. The construction of the dependence of the two sub processes is motivated by mechanical ex-pert judgments; there is a cause-effect relationship between the two processes. For instance, for the second and third phases, the corrosion current density is the cause, and the width of the crack and internal stress are the effect. When corrosion cur-rent density increases, the tensile stresses on con-crete also increase accelerating concrete cracking initiation and propagation. At the same time, the presence of cracks induces more oxygen and hu-midity near the corrosion cell resulting in an in-creasing of the corrosion current density (mutual dependencies). The correlation is modeled in terms of mutual acceleration effects directly in each of the shape parameters of the gamma distributions. This model is sequential in the sense that for each phase, we first seek to characterize the evolution in terms of the causal process then doing so for the respective effect process. To simplify the identification step, we con-sider that the state dependence is exclusively gov-erned by the shape functions; the scale functions 𝛽𝜃 and 𝛽𝜌 are considered constant throughout the life cycle. Therefore, we have to model and iden-tify the shape functions  𝛼𝜃  and 𝛼𝜌 which are re-spectively, in the case of the gamma distribution, proportional to the expected value of the incre-ments for 𝜃 and 𝜌 on the time interval τ. The choices for each shape function is moti-vated by the evolution of its respective physical parameter over time. Figure 1 shows simulations of the whole process.  Figure 2. Degradation simulations  A detailed example of the construction and performance analysis of a meta- model applied to the third phase can be found in El Hajj et al. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 (2014, a). Further studies on the estimation pro-cess in the case of censored data can be found in El Hajj et al. (2014, b). 3.1. First phase The corrosion initiation phase is then character-ized by two parameters:  (𝜌1,𝑡)∀𝑡≥0  represents the concentration of chloride at the surface of the steel [Cl-].  (𝜃1,𝑡)∀𝑡≥0 models the basicity of the concrete pH. Inspired from the trends appropriate to these physical indicators, we propose: ∀(𝜌1, 𝜃1) > 0, ∆𝜌1(𝜏1;  𝜌1, 𝜃1)  ~ 𝑔𝑎𝑚𝑚𝑎(𝛼𝜌1( 𝜌1, 𝜃1). 𝜏1, 𝛽𝜌1) (1)  ∆𝜃1(𝜏1; 𝜌1, 𝜃1, ∆𝜌1)              ~ 𝑔𝑎𝑚𝑚𝑎(𝛼𝜃1(𝜌1, 𝜃1, ∆𝜌1). 𝜏1, 𝛽𝜃1)  (2)  with the suitable shape functions: 𝛼𝜌1(𝜌1, 𝜃1) = (𝑎3. 𝜃1 + 𝑎4). 𝑒−(𝜌1−𝑎1)2𝑎2  (3)  𝛼𝜃1(𝜌1, 𝜃1, ∆𝜌1) = (𝑎6. (𝜌1 +∆𝜌12) + 𝑎7) . 𝑒−𝑎5.𝜃1 (4)  For this study we consider the following pa-rameters: 𝑎1 = 2.8 , 𝑎2  = 4.2 , 𝑎3 = 0.15 , 𝑎4 = 0.15 , 𝑎5 = 0.2 , 𝑎6 = 0.1 , 𝑎7 = 0.15 , 𝛽𝜌1 = 0.2 and 𝛽𝜃1 =  0.2. 3.2. Second phase The crack initiation phase is characterized by:  (𝜌2,𝑡)∀𝑡≥0  represents the internal tensile stress (MPa).  (𝜃2,𝑡)∀𝑡≥0models the corrosion current den-sity « icorr » (μA/cm2). From the trends appropriate to these physical indicators, we propose: ∀(𝜌2, 𝜃2) > 0, ∆𝜃2(𝜏2 ;  𝜌2, 𝜃2)             ~𝑔𝑎𝑚𝑚𝑎(𝛼𝜃2( 𝜌2, 𝜃2). 𝜏2 , 𝛽𝜃2) (5)  ∆𝜌2(𝜏2 ;  𝜌2, 𝜃2, ∆𝜃2)             ~ 𝑔𝑎𝑚𝑚𝑎(𝛼𝜌2(𝜌2, 𝜃2, ∆𝜃2). 𝜏2 , 𝛽𝜌2) (6)  with the following shape functions: 𝛼𝜃2(𝜌2, 𝜃2) = (𝑏3. 𝜌2 + 𝑏4). 𝑒−(𝜃2−𝑏1)2𝑏2  (7)  𝛼𝜃2(𝜌2, 𝜃2, ∆𝜃2) = (𝑏6. (𝜃2 +∆𝜃22) + 𝑏7) . 𝑒−𝑏5.𝜌2  (8)  For this study we consider the following pa-rameters: 𝑏1 = 3.1 , 𝑏2  = 3.2 , 𝑏3 = 1 , 𝑏4 = 0.15, 𝑏5 = 0.25, 𝑎6 = 0.05, 𝑎7 = 1, 𝛽𝜌1 =  0.2 and 𝛽𝜃1 =  0.2. 3.3. Third phase The crack propagation phase is characterized by:  (𝜌3,𝑡)∀𝑡≥0  represents the width of the crack a (mm).  (𝜃3,𝑡)∀𝑡≥0models the corrosion current den-sity icorr  (μA/cm2). From the trends appropriate to these physical indicators, we propose: ∀(𝜌3, 𝜃3) > 0, ∆𝜃3(𝜏3 ;  𝜌3, 𝜃3)             ~𝑔𝑎𝑚𝑚𝑎(𝛼𝜃3( 𝜌3, 𝜃3). 𝜏3 , 𝛽𝜃3) (9)  ∆𝜌3(𝜏3 ;  𝜌3, 𝜃3, ∆𝜃3)              ~ 𝑔𝑎𝑚𝑚𝑎(𝛼𝜌3(𝜌3, 𝜃3, ∆𝜃3). 𝜏3 , 𝛽𝜌3) (10)  with the suitable shape functions: 𝛼𝜃3(𝜌3, 𝜃3) = (𝑐3. 𝜌3 + 𝑐4). 𝑒−(𝜃3−𝑐1)2𝑐2  (11)  𝛼𝜃3(𝜌3, 𝜃3, ∆𝜃3) = (𝑐6. (𝜃3 +∆𝜃32) + 𝑐7) . 𝑒−𝑐5.𝜌3  (12)  For this study we consider the following pa-rameters: 𝑐1 = 2.5 , 𝑐2  = 4 , 𝑐3 = 1 , 𝑐4 =  1.2 , 𝑐5 = 0.4, 𝑐6 = 0.9, 𝑐7 = 1, 𝛽𝜌3 =  0.2 and 𝛽𝜃3 = 0.2. 4. MAINTENANCE ACTIONS This section illustrates how maintenance actions can be represented by the meta-model. First, we will describe three maintenance actions adopted in the European EN 1504 applied to the case of chloride-induced corrosion. Then we explain the mathematical modeling of these maintenance ac-tions. 4.1. Maintenance catalogue Three maintenance techniques are chosen for this study: Cathodic protection (applied for the three phases), Chloride extraction (used in the diffusion phase) and Concrete replacement (available for 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 the three phases). A technical guide on mainte-nance techniques can be found on http://du-rati.lnec.pt. 4.1.1. Cathodic protection Cathodic protection (CP) is a technique used to control the corrosion by making it the cathode of an electrochemical cell. CP is an electrochemical technique installed permanently. 4.1.2. Chloride extraction Chloride extraction (CE), sometimes called desal-ination, is an electrochemical process to remove chloride ions from a chloride contaminated con-crete through ion migration. An anode embedded in an electrolyte medium is temporarily applied on the surface of the concrete. 4.1.3. Concrete replacement Concrete replacement is used for restoring the original load-carrying capacity of damaged con-crete or replacing a highly contaminated concrete. In the case of chloride-induced corrosion, the concrete replacement can be applied at three dif-ferent levels:  CR1: it is a preventive repair strategy in which the structure is repaired before corro-sion initiation. Chloride-contaminated con-crete cover is repaired by removing few cen-timeters of material for slabs and beams (be-fore concrete cover depth). Corroded bars are not replaced.  CR2: it is a corrective repair strategy in which repair takes place after corrosion initiation but the loss of cross-sectional area of rebars is not significant. Cracked/chloride-contami-nated concrete cover is repaired by removing about 6 cm of material for slabs and beams. Corroded bars are not replaced.  CR3: it is a corrective repair strategy in which repair takes place after severe concrete 
  cracking where the loss of cross-sectional area of rebars is significant. Cracked/chlo-ride- contaminated concrete cover is repaired by removing about 6 cm of material for slabs and beams. Corroded bars are replaced.  4.2. Effect of a maintenance action on the meta-model Maintenance actions can have different ef-fects on the parameters: they can modify its speed by decelerating (i.e. CP) or accelerating the pro-cess (i.e. after CR, a small chloride content re-mains in the unremoved concrete inducing a chlo-ride diffusion from the old material that can accel-erate corrosion initiation). They can also change the level of degradation (i.e. CE removes given quantity of chlorides inside the concrete). Table 1 summarizes the possible effect of each mainte-nance action on the meta-model.  Table 1: Maintenance action effect on the meta-model Maintenance actions 1st Phase 2nd Phase 3rd Phase 𝜌1 𝜃1 𝜌2 𝜃2 𝜌3 𝜃3 CP d d d d d d CE a* a*     CR1 a* a*     CR2 a* a* a* a*   CR3 a* a* a* a* a* a* d: decelerates, a: accelerates, a*: changes the degradation level to a lower one and a. We propose to model the effect on mainte-nance action on the processes directly in the shape functions by introducing two new parameters 𝑚1and𝑚2which can be defined respectively as the degradation acceleration factor and the effect of unremoved concrete after maintenance factor. Fi-nally, for the average degradation rate, we multi-ply the shape function by a constant 𝑚1 and for translation we introduce 𝑚2 (Eq. 13, 14 and Fig-ure 2). On the other hand, a maintenance action can also modify the level of degradation. Having a state-dependent meta-model, with a markovian property, modifying the level does not require the modification of any of the model’s functions. All it takes is to put the degradation at the appropriate level. As a result, the shape function for an “S-shaped” trend is a “bell-shaped” function (Eq. 3, 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 7 and 11), after maintenance the shape function would be (Figure 2): 𝛼𝑆(𝜌, 𝜃) = 𝒎𝟏 × (𝑎3. 𝜌+ 𝑎4). 𝑒−((𝜃−𝒎2)−𝑎1)2𝑎2  (13)  The shape function for an “L-shaped” trend is an “L-shaped” function (Eq. 4, 8 and 12), after 7 maintenance the shape function would be: 𝛼𝐿(𝜌, 𝜃) = 𝒎𝟏 × (𝑎3. 𝜌+ 𝑎4). 𝑒−𝑎1.(𝜃−𝒎𝟐) (14)  Maintenance techniques have been widely used and their effect on the physical process are rigorously studied. So the harder part in modelling the maintenance actions is to quan-tify 𝑚1 and 𝑚2. The estimation process is beyond the scope of this paper, but Maximum Likelihood Estimation procedure can be used towards this aim by using experimental data.  Figure 3. Mathematical model of a maintenance ac-tion on the shape functions It is advantageous to be able to model a maintenance action using only two parameters per process instead of updating 9 parameters. 4.3. Performance indicators The maintenance decision is chosen accordingly with the condition of the structure. In order to quantify the quality of a structure we need to choose an appropriate performance indicators.  Performance indicators can be classified un-der three categories (Frangopol et al., 2007):  Condition indexes: based on inspection and then classified in discrete states.  Safety indexes: the structure can be either safe or unsafe.  Reliability indexes. In this study we consider a condition index (CI) based on the inspection values of the param-eter of interest ρ. It is worth to be mentioned that a performance indicator can be a combination these categories. We classify the structure into 10 discrete in-spection-based states. Each phase is divided into 3 states starting from a CI=9 and going down CI=0. A CI=9 is associated with a low concentra-tion of chloride (early phase of chloride diffusion) and ending in a CI=0 for a severely cracked struc-ture (crack width of 3mm). The range of each CI is defined by dividing the region between the horizontal axis and the threshold line into three un-even regions using square root intervals. The closer to the threshold a region gets, the narrowest it becomes (Figure 3).  Figure 4. Condition states setup 5. APPLICATION OF THE MODEL IN A MAINTENANCE CONTEXT This section illustrates the use of the meta-model for the assessment of life-cycle costs and condi-tion indexes for different maintenance policies. Preventive and corrective maintenance poli-cies are considered. We will assume three life-0 0.2 0.4 0.6 0.8 100.10.20.30.40.50.60.70.80.91Condition index setup1/3 1/3 1/3sqrt (x)CI = 3CI = 2CI = 112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7 times of 50, 75 and 100 years. The inter-inspec-tion time is fixed at a 5 years. Inspections are con-sidered as perfect. This example aims at assessing the mainte-nance actions and costs for a target CI. We evalu-ate maintenance costs and condition index by car-rying out Monte Carlo stochastic simulations un-der the under the Matlab® environment. The out-puts of simulations are simulated inspection data and the history of the structures (e.g., loss of steel). The decision of a repair after inspection is based on these outputs and depends on both maintenance policy and extent of damage. For this example we consider only the con-crete replacement repair methods: CR1, CR2 and CR3. This choice was made since we dispose of real costs for the repair of a marine harbor (Srifi, 2012). For each policy, 1000 simulations are car-ried out to determine the costs (inspection, maintenance and total) and the CI of the structure. Total cost for each policy are assessed at present time without considering a discounting factor. Since we focus on existing structures, construc-tion and salvage cost are not included in the anal-ysis. Table 2: Costs of maintenance and inspections Phase CR1  CR2 CR3 Maintenance cost (€/m²) 263.2 323.0 353.4 Inspection (€/m²) 25 25 10 5.1. Preventive Maintenance The preventive maintenance (PM) policy aims at repairing the structure for a target CI = 7 (end of the 1st phase). The objective of this policy is to prevent the initiation of corrosion. However, having a 5 years inter-inspection interval, it is possible to miss an inspection for a CI = 7. In this case, we have to inspect for (𝜌2, 𝜃2) to make sure that the structure is in the 2nd phase, and the corrosion have already started. In that case, a CR2 is required instead of the origi-nally planned CR1 generating more costs, hereaf-ter called over cost.  From a practical point of view, if corrosion starts many times during the structural lifetime, the loss of steel might be significant enough to consider a CR3 (concrete replacement with re-placement of reinforcement steel) adding more over-cost. Table 3 gives the costs and the CI for three different life times of a structure maintained through a preventive policy. The Condition Index in the tables is the mean of all the inspections’ condition indexes. The over costs represents about the 11% of the total cost. Clearly the inter-inspection time have an impact on this value. Also, the relatively small size of the CI=7 zone compared to the two zones of the 1st phase (9 and 8) is another factor. If the zone 7 was bigger, it is more certain to per-form a maintenance for the aimed CI rather than skipping to the 2nd phase, but also we can trigger an early maintenance. Table 3: Costs and Condition index for a preventive maintenance policy Lifetime (years) 50 75 100 Inspections (€/m²) 318.2 465.4 619.9 Maintenance (€/m²) 864.3 1299.5 1804.2 Total cost (€/m²) 1182.5 1764.9 2424.1 Annual cost (€/m²) 23.6 23.5 24.2 Over cost (€/m²) 146.6 221.8 321.9 Condition Index 8.22 8.2 8.15 5.2. Corrective Maintenance The corrective maintenance policy aims at repair-ing the structure after sever cracking (CI=0). Ta-ble 4 provides the have the costs and the CI for three lifetimes. Table 4: Costs and Condition indexes for a corrective maintenance policy Lifetime (years) 50 75 100 Inspections (€/m²) 322 468.7 620.6 Maintenance (€/m²) 352 628.3 783.1 Total cost (€/m²) 674 1097 1403.8 Annual cost (€/m²) 13.5 14.6 14 Condition Index 6.38 5.85 5.86 5.3. Comparison  From Table 3 and Table 4 we can see that the cor-rective maintenance compromises the CI (~6) but 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8 reduces the costs (almost by half). In contrast, the preventive maintenance policy maintains the structure at a higher level of performance (CI>8) with a larger maintenance cost.  It is essential to point out that the ultimate ob-jective of this study is to show the applicability of the meta-model in a maintenance optimization context, which was done here. 6. CONCLUSIONS AND PERSPECTIVES Degradation models are an essential tool to pre-dict the evolution in time of structural perfor-mance. These models are then paramount for planning and optimizing maintenance and man-agement policies. This work proposed a new degradation meta-model that accounts for the above mentioned pur-poses. It can be seen as an intermediate between physical models (that can be complex to apply in a reliability context), and pure probabilistic mod-els (that fail to reflect the physical degradation and therefore suffer from lack of acceptability by the Civil engineering community). Moreover, the construction and calibration of the model are done via NDT data. This paper also illustrated the applicability of the meta-model in a maintenance context. Future works will consider real data for the calibration and will focus on maintenance optimi-zation. Concerning maintenance optimization, the next step would be to set up and define the multi-objective optimization problem. 7. ACKNOWLEDGEMENT This work is a part of the SI3M project (2012-2016 Identification of Meta-Model for Mainte-nance Strategies) funded by Region Pays de la Loire (France). 8. REFERENCES Duratinet project, Durable Transport Infrastructure in the Atlantic Area-Network, http://durati.lnec.pt. 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A survey of the application of gamma processes in maintenance. Reliability Engineering & System Safety, 94(1), 2–21.  Zouch, M., Yeung, T., & Castanier, B. (2011). "Optimizing Road Milling and Resurfacing Actions", Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability. 

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