Structure and Infrastructure Engineering
Maintenance, Management, Life-Cycle Design and Performance
ISSN: 1573-2479 (Print) 1744-8980 (Online) Journal homepage: https://www.tandfonline.com/loi/nsie20
Improved Bayesian network configurations for probabilistic identification of degradation mechanisms: application to chloride ingress
Thanh-Binh Tran, Emilio Bastidas-Arteaga & Franck Schoefs
To cite this article: Thanh-Binh Tran, Emilio Bastidas-Arteaga & Franck Schoefs (2016) Improved Bayesian network configurations for probabilistic identification of degradation mechanisms: application to chloride ingress, Structure and Infrastructure Engineering, 12:9, 1162-1176, DOI: 10.1080/15732479.2015.1086387
To link to this article: https://doi.org/10.1080/15732479.2015.1086387
Published online: 02 Oct 2015.
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Structure and InfraStructure engIneerIng, 2016 VOL. 12, nO. 9, 1162–1176 http://dx.doi.org/10.1080/15732479.2015.1086387
Improved Bayesian network configurations for probabilistic identification of degradation mechanisms: application to chloride ingress
Thanh-Binh Tran, Emilio Bastidas-Arteaga and Franck Schoefs
department of Physics, L’unaM université, université de nantes-ecole centrale nantes, geM, Institute for research in civil and Mechanical engineering/ Sea and Littoral research Institute, nantes, france
ABSTRACT
Probabilistic modelling of deterioration processes is an important task to plan and quantify maintenance operations of structures. Relevant material and environmental model parameters could be determined from inspection data; but in practice, the number of measures required for uncertainty quantification is conditioned by time-consuming and expensive tests. The main objective of this study was to propose a method based on Bayesian networks for improving the identification of uncertainties related to material and environmental parameters of deterioration models when there is limited available information. The outputs of the study are inspection configurations (in space and time) that could provide an optimal balance between accuracy and cost. The proposed methodology was applied to the identification of random variables for a chloride ingress model. It was found that there is an optimal discretisation for identifying each model parameter and that the combination of these configurations minimises identification errors. An illustration to the assessment of the probability of corrosion initiation showed that the approach is useful even if inspection data are limited.
1. Introduction
Chloride-induced corrosion is frequently considered as the main cause of deterioration of various types of reinforced concrete (RC) structures (bridges, wharves, offshore platforms, marine structures, etc.) that are located close to the seashore or in contact with de-icing salts. Corrosion in reinforcing bar will affect load-carrying capacity through: reduction of reinforce- ment cross section, loss of bond between steel and concrete, concrete cracking and delamination (Lounis & Amleh, 2003). These consequences will lead to the reduction of serviceability and safety levels as well as the shortening of service life in RC structures. Most of RC structures are designed for a lifetime of 50–100 years.
However, under chloride attack, important damages are reported after 10–20 years (Kumar Mehta, 2004; Poupard, L’Hostis, Catinaud, & Petre-Lazar, 2006). For these reasons, many owners of large-scale structures proposed a routine schedule for maintenance in which structures are inspected periodically (every ∆t years) to ensure optimal levels of serviceability and safety during the structural life. In infrastructure management, maintenance costs have represented a significant portion of the corrosion management budgets. Koch, Brongers, Thompson, Virmani, & Payer (2002) reports that the annual direct cost of corrosion for highway bridges in US was about $8.3 billion, where $4.5 billion was spent on maintenance and $3.8 billion was used to replace structurally deficient bridges over 10 years.
CONTACT emilio Bastidas-arteaga emilio.bastidas@univ-nantes.fr © 2015 Informa uK Limited, trading as taylor & francis group
ARTICLE HISTORY
received 14 april 2015 revised 10 July 2015 accepted 11 august 2015 Published Online 2 October 2015
KEYWORDS
reliability and risk analysis; identification; Bayesian network; deterioration; chlorides; corrosion
Maintenance strategies are divided into two stages: inspection and repair; in which inspection results are necessary for the diag- nosis: detecting corrosion in early stages, evaluating the extent of damage as well as implementing repair measures. Nowadays, a large number of effective condition assessment techniques consisting of destructive and non-destructive methods have been developed to facilitate the assessment of corrosion con- sequences in RC structures. Among non-destructive methods, visual inspection is a technique usually used for evaluating the condition of RC structures and for providing qualitative infor- mation about the condition state (Roelfstra, Hajdin, Adey, & Brühwiler, 2004), but it is currently combined with image to pro- vide also quantitative assessment (Ghosh, Pakrashi, & Schoefs, 2011). However, its results are highly affected by environmental conditions and human errors reducing its accuracy.
Semi-destructive inspection techniques (coring), on the other hand, could provide accurate inspection results; however, they are more expensive and time-consuming. Non-destructive techniques are less expensive but require still technical develop- ments (Torres-Luque, Bastidas-Arteaga, Schoefs, Sánchez-Silva, & Osma, 2014) and specific post-treatment methods for assess- ing the chloride content from multi-technique measurements (Lecieux, Schoefs, Bonnet, Lecieux, & Lopes, 2015; Ploix, Gar- nier, Breysse, & Moysan, 2011). The study will focus on quanti- tative methods, semi- and non-destructive ones, whose common objective is to assess the chloride content profile in depth or at
least near the rebar. The outputs after such inspection campaigns are discrete chloride content measurements in depth and time; the continuous monitoring is beyond the scope of this study. Note that the information in depth gives a significant informa- tion about the chloride concentration level inside the concrete and thus on the history and it is also useful for prediction from updating of time-variant models.
In order to obtain an accurate and reliable assessment of the extent of corrosion damage, engineers and inspectors combine information from different corrosion evaluation techniques. Data collected after inspection campaigns are often used to determine parameters for chloride ingress or corrosion propagation models. This information can be after used for lifetime assessment and optimisation of maintenance strategies. This study will focus on the corrosion initiation stage that is estimated from chloride penetra- tion models. Under natural exposure conditions, chloride ingress involves important uncertainties related mainly to material prop- erties and exposure conditions (Bastidas-Arteaga, Chateauneuf, Sánchez-Silva, Bressolette, & Schoefs, 2011; Bastidas-Arteaga, Schoefs, Stewart, & Wang, 2013; Deby, Carcasses, & Sellier, 2008; Lounis & Amleh, 2003; Saassouh & Lounis, 2012).
These uncertainties are also affected by temporal and spa- tial variability of associated deterioration processes and their characterisation requires larger amount of inspection data (O’Connor & Kenshel, 2013; Peng & Stewart, 2014; Tran, Schoefs, & Bastidas-Arteaga, 2013). Nevertheless, in real practice, the number of inspections is limited by the difficulties to imple- ment tests that increase inspection times and costs. Therefore, it is necessary to use the available information in the best way for uncertainty quantification using statistic and/or probabilistic methods. Within this context, the Bayesian method is a reason- able choice to deal with this problem.
The Bayesian approach has been applied to problems related to chloride ingress into RC in some previous studies (De-León-Escobedo, Delgado-Hernández, Martinez-Martinez, Rangel-Ramírez, & Arteaga-Arcos, 2013; Engelund & Sorensen, 1998; Enright & Frangopol, 1999; Keßler, Fischer, Straub, & Gehlen, 2013; Ma, Zhang, Wang, & Liu, 2013; Suo & Stewart, 2009; Wang & Liu, 2010). This approach provides a comprehen- sive and rational framework to update the estimations based on incorporating new information from inspection data for ser- vice-life prediction of deteriorating RC structures (Keßler et al., 2013; Ma et al., 2013; Wang & Liu, 2010) or parameter estima- tions (Engelund & Sorensen, 1998; Enright & Frangopol, 1999; Suo & Stewart, 2009).
The application of Bayesian methods for probabilistic mod- elling of chloride-induced corrosion is becoming popular and it has also been applied to multiple events in the form of Bayesian Network (BN). Deby et al. (2008) and Deby, Carcasses, and Sellier (2012) have used BN as a probabilistic approach for the assessment of probabilistic distribution of random variables in the chloride diffusion problem. The approach seems to be robust when allowing the possibility to update the statistical distributions with new information from experimental results. Bastidas-Arteaga, Schoefs, and Bonnet (2012) proposed a procedure to identify random parameters in chloride ingress models from experimental data. In this approach, the a priori information about statistical parameters such as type of distri- bution, mean and standard deviation are assumed unknown.
This assumption provided a generalised approach that could be applied for real structures.
Hackl (2013) proposed a framework that combines struc- tural analysis and BN for reliability assessment. This combina- tion allows Bayesian updating of the model with measurements, monitoring and inspection results. Recently, Ma, Wang, Zhang, Xiang, and Liu (2014) developed a BN combining in situ load testing to predict the strength degradation of bridge structures subjected to chloride attack. Nevertheless, the above-mentioned studies have not optimised the utilisation of inspection data, especially when the information is limited.
The main objective of this study is to propose a methodology for defining appropriate BN configurations for parameter iden- tification. Data from numerical simulations, representing dif- ferent measurement sets, will be used to propose an appropriate configuration for the identification of each model parameter. A brief description of the BN and its application for chloride ingress are presented in Section 2. Section 3 introduces the problem definition and the BN implementation. Section 4 analyses the influences of different BN configurations on the identification of parameters. For each BN configuration, we also discuss about numerical issues to minimise the error in the updating process. Next, the identified parameters are employed in Monte Carlo simulation for the assessment of the probability of corrosion initiation (Section 5). An improved procedure is also suggested in this study to optimise the evaluation when available data for updating are limited.
2. Bayesian identification and its application to chloride ingress
2.1. Introduction to BN
Generally, a BN is a specific type of graphical model that is rep- resented as a directed acyclic graph (DAG). Nodes in DAG are graphical representation of objects and events that exist in real world, and they are used to represent variables or deterministic states. Causal relations between nodes are represented by draw- ing an arc (edge) between them. If there is a causal relationship between the variables (nodes), there will be a directional edge, leading from the cause variable to the effect variable. Each vari- able in the DAG has a probability density function (PDF), which dimension and definition depend on the edges leading into the variables. For a set of m random variables X = [X1, X2, …, Xm], a BN represents the joint probability mass function. The BN allows an efficient probabilistic modelling of complex problems by factoring the joint probability distribution into conditional probability distribution for each variable.
Figure 1 describes a simple BN that consists of three nodes corresponding to three random variables X1, X2 and X3 in which X2 and X3 are children of the parent node X1. The children nodes have conditional probability distributions that depend on their parent node. The parent node has a marginal probability dis- tribution. The Bayes’ rule allows for computing the a posteri- ori probability p(X1|X2), given the a priori and the conditional probabilities p(X1) and p(X2|X1):
STRucTuRE And InFRASTRucTuRE EngInEERIng 1163
( ) p X1|X2
=
p(X2 |X1 )p(X1 )
p(X ) 2
(1)
1164 T.-B. TRAn ET Al.
X 1
XX 23
Figure 1. a simple Bayesian network.
This a posteriori probability is the key of model identification
from inspection data.
2.2. Application to chloride ingress
2.2.1. Chloride ingress and modelling
In saturated concrete, the Fick’s diffusion equation (Tuutti, 1982) is usually used to predict the unidirectional diffusion (in x-direction):
2.2.2. BN modelling of chloride ingress
Chloride ingress could be modelled by the BN described in Fig- ure 2 where Cs and D are the two parent nodes (random vari- ables to identify). There are n child nodes C(xi,tj) representing the discrete chloride concentration measurement in time and space, i.e. at depth xi and at inspection time tj. The number of child nodes is computed as:
n = nxnt (4)
where nx is the total number of points in depth and nt is the total number of inspection times. Assuming that Cs and D are two independent random variables, the values of C(xi,tj) could be eas- ily estimated from Equation (3). Most of parameters in chloride ingress models are defined in the continuous space. However, in order to avoid using approximate inference algorithms which will be a disadvantage when working with continuous variables, continuous variables must be replaced by discrete random var- iables (Straub, 2009). Each node is defined over a specific range (upper and lower bounds) and its probability distribution is dis- cretised into a given number of states per node, Ns. Figure 2 also illustrates the discretisation considered for each node. For this example, the node C(x1,tj) was divided into Ns = 6 states over a predefined range.
In this BN, if all nodes are discrete, the probability of chlo-
ride concentration p(C(xi,tj)) can be calculated as follows
(Bastidas-Arteaga et al., 2012):
(( )) ∑(( ) )( )
p C xi,tj = p C xi,tj |D,Cs p D,Cs D,Cs ( ) ()
with p D, Cs = p(D)p Cs
To estimate p(C(xi,tj)), the conditional probability p(C(xi,tj)|D,Cs) must be already known in Equation (5). This conditional prob- ability accounts for the dependence between the chloride con-
2 𝜕Cfc =D𝜕Cfc
(2)
𝜕t 𝜕x2
where C (kg/m3) is the concentration of chloride dissolved
fc
in pore solution, t (year) is the time and D (m/s2) is the effec-
tive chloride diffusion coefficient. Assuming that concrete is a homogeneous and isotropic material with the following initial conditions: (i) the chloride concentration is zero at time t = 0 and (ii) the chloride surface concentration is constant during the exposure time, the free chloride ion concentration C(x,t) at depth x after time t for a semi-infinite medium is
(5)
� � x ��
C(x,t)=Cs 1−erf √ (3) tent C(x ,t ) with the two model parameters (D and C ) and it is
where C (kg/m3) is the chloride surface concentration and erf(.) s
computed based on the conditional probability table (CPT) of the BN; Monte Carlo simulations of the model (from Equation (3)) are required for the estimation of CPT.
The BN allows entering evidences and then updating the probabilities in the network. In this study, the evidences corre- spond to measures of chloride concentration at given points and times (chloride profiles). Then, the term p(C(xi,tj)|o) represents the probability distribution of C(xi,tj) given evidence o and a posteriori distributions of D and Cs can be computed by applying
the Bayes’ theorem:
( ( ))(( ))
p(D|o)=p D|C xi,tj p C xi,tj |o
is the error function.
Equation (3) remains valid when RC structures are satu-
rated and subjected to constant concentration of chlorides on the exposure surfaces. In real structures, these conditions are rarely present because concrete is a heterogeneous material and the chloride concentration in the exposed surfaces could be time-variant. Besides, this solution does not consider chlo- ride binding capacity, concrete ageing and other environmental factors such as the influence of surrounding temperature and humidity in chloride ingress process Bastidas-Arteaga & Stewart, (2015).
Although this solution neglects some important physical phenomena, this model will be used herein to illustrate the proposed methodology for the identification of random vari- ables using BN because its complexity is sufficient to account for non-linear effects in x-direction and in time and to perform sensitivity analysis: two variables are involved. The method- ology can be after extended to more realistic chloride ingress models. The computational effort for building the BN will increase for chloride ingress models involving a large number of variables.
p(C(x , t )|D)p(D) (6) ij
2D⋅t ij s
))
( ( ))(( ))
( ( withp D|C xi,tj
=
( ( ))
and:
p(Cs|o) = p
withp Cs|C xi,tj
)) p(C(xi, tj)|Cs)p(Cs) (7) = ( ( ))
p C xi,tj Cs|C xi,tj p C xi,tj |o
( (
p C xi,tj
123456 Ns Range
Figure 2. general Bn configuration for modelling chloride ingress.
The determination of these conditional probabilities is carried out by a BN Tool Box which is built on the MATLAB® Software. Note that it is assumed that the measurements are not affected by errors. Measurement errors can be modelled by adding addi- tional nodes to represent its PDF and its dependence on the magnitude of the measured values (Schoefs, Boéro, Clément, & Capra, 2012).
3. Problem definition and BN implementation
3.1. Generation of numerical evidences
This study aims at determining improved BN configurations for the identification of parameters of chloride ingress models. It this case, the BN will be used to update probabilistic models for the parameters to identify. The evaluation of the effectiveness of a given configuration should be based on a given criterion. Preferably, it should include a larger amount of experimental data (chloride profiles) that can be used to estimate ‘real’ prob- abilistic models of model parameters and consequently to test and compare various BN configurations. However, such a data- base is in practice very hard to obtain because chloride profiles are computed from semi-destructive tests that are expensive and time-consuming. Therefore, in order to assess the error associated with each BN configuration and to provide gen- eral recommendations that minimise the identification errors, we consider a large number of numerical evidences (chloride profiles) generated from Monte Carlo simulations.
The numerical chloride profiles are generated from theoretical probabilistic models of the random variables to identify. Table 1 presents the considered probabilistic models for Cs and D. The mean values for each parameter were taken from (Bastidas- Arteaga, Bressolette, Chateauneuf, & Sánchez-Silva, 2009). Cs corresponds to a structure placed in an atmospheric zone, close to the seashore but without direct contact with sea water. D is a typical diffusion coefficient for ordinary Portland concrete. The COV for each parameter were reduced to 20 and 15% for Cs and D, respectively. This is due to the fact that within one type of concrete, the variation is narrowed (Duracrete, 2000; Tang & Nilsson, 1993; Vu & Stewart, 2000). The assumption that Cs and
… …
C(x , t ) C(x , t ) C(x , t ) C(x , t ) 1m2mimnm
D follow lognormal distributions is also in agreement with other studies (Duracrete, 2000; Vu & Stewart, 2000).
The theoretical probabilistic models presented in Table 1 were used to generate 9000 random values for Cs and D. Afterwards, each set of values of Cs and D was used to compute the 9000 independent chloride profiles from Equation (3). The evidences to be introduced into the BN are then computed from these chloride profiles. The same simulated values will be used for all configurations to ensure that we update the BN with the same information.
Different configurations of the BN corresponding to differ- ent inspection schemes will be analysed for selecting inspec- tion schemes that provide the best identification of parameters. Each configuration will be evaluated in terms of the error of the identified parameter Zidentified with respect to the theoretical value Ztheory as:
… …
C(x , t ) C(x , t ) C(x , t ) C(x , t ) 1121i1n1
Cs D
… …
C(x , t ) C(x , t ) C(x , t ) C(x , t ) 1j2jijnj
STRucTuRE And InFRASTRucTuRE EngInEERIng 1165
Prob. density
Error(Z) =
|Zidentified − Ztheory|
Z 100%
theory
(8)
where Z represents the mean or the standard deviation of the parameter to identify – e.g. mean or standard deviation of Cs and D, Zidentified is determined from the a posteriori histograms of parent nodes (Cs and D), and Ztheory is the value of the mean or standard deviation used to generate numerical evidences (Table 1).
In practice, it is unrealistic (almost impossible) to collect 9000 chloride profiles. However, this larger database is necessary for obtaining a convergence on the error assessment. Error assess- ment results are after used for studying how the configuration of the BN can be improved for identification processes. The final part of the study(Section 5.2.3) will consider the improved BN configurations to study the case in which the information (number of profiles) is limited.
Table 1. theoretical values of parameters to identify.
Parameters
C
D
distribution
Lognormal Lognormal
Mean cOV
2.95 kg/m3 0.20 1.33 × 10−12 m2/s 0.15
s
1166 T.-B. TRAn ET Al.
3.2. BN definition and identification procedure
We aim at identifying the parameters Cs and D using chloride profiles as evidences. Section 4 details the configuration of BNs considered in this study that are basically based on the general case described by Figure 2. Table 2 describes the discretisation of each node as well as the considered a priori distributions. As detailed in Figure 2, each node is divided into Ns states over a given range. Ns will vary to determine a value that diminishes identification errors. The range (upper and lower bounds) for each parameter should in theory contain all the possible values of each parameter. These ranges can be defined on the basis of existing databases, similar study cases or expert knowledge. Here, the ranges for Cs and D were defined enough large to contain values representative of the variability of environmental exposure and material properties when the a priori information about these parameters is very poor. The theoretical distributions pre- sented in Table 2 can be used in this case to estimate upper and lower bounds for a given confidence interval. The adopted values cover a confidence interval larger than 99% by ensuring that the parameters to identify belong to this wide a priori range.
A priori characteristics (type of distribution, mean, standard deviation, etc.) are commonly considered to define the configu- ration of the parent nodes Cs and D. To avoid hypothesis about a priori information, we suppose that Cs and D follow uniform distributions defined over given upper and lower bounds. How- ever, if more rich a priori information is available, the consider- ation of this information will decrease the identification errors. The assumption of uniform distribution for unknown parameter could avoid making any assumption about the distribution shape (Bastidas-Arteaga et al., 2012; Cao & Wang, 2014; Robinson & Hartemink, 2010). A priori distributions of parent nodes Cs and D are used to generate a sufficient random number of chloride concentrations at depth x and time t using Equation (3) for each child node. These a priori data are used to compute the CPT for each child node in the BN.
Data from numerical simulations will be introduced to the BN
as evidences (Section 3.1). The probability that C(xi,tj) belongs to
a given state for different depths is then computed for the identifi-
cation of the term p(C(xi,tj)|o). These probabilities are then added
to the BN as soft evidences for inference. We considered an exact
inference algorithm for static BN. In this process, the BN will
characteristic of BNT as compared with other tools because users can add, modify or make complements of functions in order to fit with the different using purposes. BNT supports many types of conditional probability distributions (nodes), many inference algorithms (exact and approximate) for both static BNs and dynamic BNs parameters and structure learning (Murphy, 2001).
4. Strategies for improvement of BN configurations for parameter identification
This section investigates the influence of the BN configuration on the identification error with respect to the theoretical parameters. The following configurations are covered:
(1) only one child node corresponding to one inspection point in depth and one inspection time (Section 4.1),
(2) various child nodes corresponding to several inspection points in depth and one inspection time (Section 4.2),
(3) many child nodes corresponding to several inspection points in depth and varying the inspection time (Section 4.3), and
(4) various child nodes corresponding to several inspection points in depth and the combination of various inspec- tion times (Section 4.4).
Each section includes a discussion about numerical aspects that can be considered to minimise identification errors and/or decrease computational effort.
4.1. Identification using one inspection point in depth
4.1.1. Problem statement
In this part, the estimation of the chloride surface concentration
(Cs) and chloride diffusion coefficient (D) will be analysed from
evidences obtained at one depth point (Figure 3). This study
case could be also applied to the identification of data coming
from other inspection techniques that focus on one-point meas-
urements: resistivity-based probe (Wenner probe) or resistivity
basic sensors with only two points of injection (Du Plooy, Palma
Lopes, Villain, & Derobert, 2013). The BN now consists of three
calculate the a posteriori distributions: p(C |o) and p(D|o) using s
node C(xi,tj) representing the chloride concentration at depth xi. The inspection time is tins = t1 = 10 years (which is compatible with actual practices) and the total inspection depth is 12 cm. Consequently, we have 13 different BNs corresponding to 13 points in depth varying from 0 to 12 cm.
Equations (6) and (7). The a posteriori distributions are after used to identify the parameter Zidentified and therefore to evaluate the error of the configuration using Equation (8).
The BN configurations were implemented in the Bayesian Network Toolbox (BNT) that is an open-source MATLAB pack- age for directed graphical models. This can be seen as a robust
4.1.2. Numerical issue: discretisation of child nodes
As previously mentioned in Section 3.2, continuous variables need to be discretised into several states. The number of states per node could be adjusted to obtain a balance between accuracy
Cs D
Table 2. discretisation of nodes and a priori distributions.
nodes (Figure 3): two parent nodes are C and D, and one child s
Parameters
number of states per node (Ns)
Variablea Variablea
A priori distribution
uniform uniform
Range
[0.5; 8]
[6 × 10−13;
3 × 10−12]
C (kg/m3) s
D (m2/s)
C(x ,t ) (kg/m3)
C(x ,t ) ij i1
Variablea
–b
[0; 8]
aN varies to determine a value that diminishes the identification error. s
bcomputed from a priori information of parent nodes. Figure 3. Bn configuration using one inspection point in depth.
STRucTuRE And InFRASTRucTuRE EngInEERIng
1167
50 40 30 20 10
200 states per node 15 states per node 30 states per node 60 states per node
4.5 4 3.5 3 2.5 2 1.5 1 0.5
00
Mean 5% 95%
15
00 2 4 6 8 10 12 Depth (cm)
Figure 4. Identification error of the mean value of Cs for child nodes discretised into different number of states.
of results and computational time. When an accurate result is
expected, a high number of states are often chosen. Figure 4
with different discretisations and inspection positions in depth. It is clear that fluctuations are less important when each node C(xi,t1) in the BN is divided into Ns = 200 states. This means that a high number of states could reduce the fluctuations of the values identified by the BN. Consequently, we will use a discretisation of Ns = 200 states per node C(xi,t1) for all BNs in this subsection.
4.1.3. Influence of measure depth on the identification error
5 10 Depth (cm)
20
describes the estimations of the error of the mean value of C s
Figure 6. Mean chloride profile and 5 and 95% percentiles at tins = 10 years.
set x ≈ 0, C(xi,tj) ≈ Cs. Consequently, the chloride concentration at the surface is most valuable in the identification of Cs.
It is also observed in Figure 5(a) that the error in the identifi- cation of D decreases until a depth x < 9 cm and it increases after this point. This behaviour corresponds to the fact that chloride content at deeper parts is more useful for predicting the diffu- sion coefficient. However, the error increases at deeper points where the chloride contents are close to zero for tins = 10 years (x > 9 cm). The errors in the identification of the standard devi- ation of D follow a similar trend; however, their values are very far from the theoretical values with important errors (more than 200%). Therefore, it can be concluded that it is very difficult to perform a good identification of D using evidences obtained from only one point in depth.
4.2. Identification using full inspection depth
4.2.1. Problem statement
In this section, we keep the same inspection time (tins = 10 years), but the BN configuration will use data from a total inspection depth Li. The total inspection depth (Li = 12 cm) is divided into several inspection points of the same discretisation size Δx. The total inspection depth is selected to cover all the potential chlo- ride presence (see Figure 6). Figure 7 illustrates the discretisa- tion of the total inspection length on elements of size Δx. In the experimental procedure to determine chloride profiles, concrete cores are grinded over predefined discretisation sizes (equivalent to the parameter Δx) at various depths. Subsequently, an average
(b) 300 250 200 150 100
00 2 4 6 8 10 12 Depth (cm)
Figure 5 shows the errors in the identification of the mean and standard deviation of C and D. For C , the evolution of the error
ss
of both mean and standard deviation increases with the depth.
These estimations are related to the evolution of the chloride profiles in depth. Figure 6 presents the mean chloride profile and 5 and 95% percentiles estimated by Monte Carlo simulations from the theoretical values given in Table 1 at tins = 10 years. It is noted that chloride content is lower when depth increases. This means that data from chloride profiles near to the surface provide more useful information for the identification of Cs, whereas less information in the deeper parts increases the error. When the chloride content is close to zero, these errors increase and they can even reach 40% for the mean. In contrast, with the evidences near to the surface (x ≈ 0), we can obtain the best estimation for the mean and standard deviation of Cs, with errors of 1 and 3%, respectively. This is due to the fact that in Equation (3), when we
(a) 50 40 30 20 10
Mean Cs Mean D
00 2 4 6 8 10 12 Depth (cm)
Figure 5. Identification error using one depth point: (a) mean – (b) standard deviation.
Std Cs 50 Std D
Error (%)
Error (%)
Error (%)
Chloride content (kg/m3)
1168 T.-B. TRAn ET Al.
Figure 7. Bn configuration with ∆x = 2 cm.
Table 3. discretisation cases and number of points in depth.
error decreases from more than 200% with Δx = 3 cm to about 20% with Δx = 0.3 cm. This behaviour is expected because when the inspection depth is divided into small ∆x, we could obtain more rich spatial information describing the level of chloride ingress that is more useful for characterising the diffusion coef- ficient. Hence, we can conclude that data from full inspection depth are more valuable in the identification of D.
4.2.3. Numerical issue: ranges for discretising child nodes
As discussed in Section 4.1.2, when the nodes are discretised by considering a small number of states, the errors in the estimation of chloride diffusion coefficient increase significantly. In such a case, the information used to update the BN becomes poor, par- ticularly for deeper points where chloride content is low. For exam- ple, Figure 9(a) shows the evidence at depth x = 6 cm, tins = 10 years and a discretisation within Ns = 15 states. It is noted that in such a case that all the information coming from the evidences is concen- trated in one state, and then, this evidence cannot provide good information for updating the BN. Increasing the number of states could solve this problem. Figure 9(b) indicates that Ns = 200 states are more convenient to represent the variability of the chloride concentration at this depth. Nevertheless, this solution increases the size of the CPTs and, therefore, the computational time.
In this section, a procedure to improve the discretisation of child nodes is proposed. The number of states in the discretisa- tion remains constant but we use different ranges (upper and lower boundaries) for the discretisation of all child nodes C(xi,tj). Figure 10(a) shows the boundaries for each node which represent for chloride content at depth x for tins = 10 years. From a priori information of Cs and D (Table 2), Monte Carlo simulation was employed to generate a sufficient number of chloride concen- tration at depth x and time t from Equation (3). From these simulations, it is possible to determine maximum and minimum values of chloride content at depth x and time t which are used as the boundaries of each child node. By considering these bound- aries, we take advantage of the information of deeper points in
case
1 2 3 4 5
Δx (cm) 0.3
0.5 1
discretisation (cm)
0:0.3:12 0:0.5:12 0:1:12
number of points in depth, nx
41 25 13
2 0:2:12 7 3 0:3:12 5
chloride content is determined for the powder extracted at each depth. This average concentration is frequently represented at the middle of each interval.
The discretisation size Δx should not be smaller than 0.3 cm due to the accuracy of the semi-destructive equipment for deter- mining chloride profiles. The BNs will now have a number of child nodes depending on the total inspection depth (Li) and the adopted discretisation size (Δx). For example, if Li = 12 cm and ∆x = 2 cm, the BN configuration consists of 2 parent nodes and 7 child nodes as described in Figure 7. Table 3 presents the 5 cases considered in this section where Δx varies between 0.3 and 3 cm.
4.2.2. Influence of the discretisation size ∆x on the identification error
Figure 8 presents the error in the identification using full inspec- tion depth and the same ranges (boundaries) for the child nodes. These results were obtained by discretising the child nodes into 15 states. For all Δx values (BN configurations), it is noted that there is no remarkable change in the identification of the mean of Cs (Figure 8(a)) because the errors in 5 surveyed cases are close to 5%. In addition, it seems that increasing the number inspection points might produce more errors for the standard deviation of Cs (Figure 8(b)).
Interestingly, the gap between identified values and theoretical values for D is reduced significantly when the size of the discreti- sation size ∆x is smaller. The errors in the estimation of the mean of D are less than 5% when the ∆x is smaller than 0.5 cm. The standard deviation of D also reveals a better evolution when the
(a) 25 20 15 10 5
Mean Cs Mean D
(b) 250 200 150 100 50
00 0.5 1 1.5 2 2.5 3
Figure 8. Identification error using full inspection depth: (a) mean – (b) standard deviation.
Std Cs Std D
00 0.5 1 1.5 2 2.5 3
(a)
(b)
States per node
States per node
Figure 9. evidence at x = 6 cm with different discretisations of child nodes: (a) 15 states – (b) 200 states.
(a) 9 8 7 6 5 4 3 2 1
(b)
00 5 10 15 Depth x (cm)
States per node
Figure 10. (a) ranges for each child node at depth x – (b) evidence at x = 6 cm with 15 states for each child node and different ranges.
the updating process. Figure 10(b) depicts the evidence at depth x = 6 cm with Ns = 15 states. It is clear that the evidence at depth x = 6 cm could provide now more valuable information for updating the BN in this case.
4.2.4. Analysis of results for improved discretisation of child nodes
Figure 11 compares the error in the identification of the mean and standard deviation of D before and after improvement
(rebuild case) of the discretisation of child nodes. It is clear that the consideration of large number of states and/or different ranges per node reduces drastically the errors even when the total inspection depth was divided into large Δx (Δx ∈ [2, 3 cm]). This proposed approach could be very useful in practice when the number of measured points is limited or when the level of chloride content inside concrete is low.
Moreover, it is worth noticing that there are optimal discre- tisation sizes Δxopt: for the standard deviation (Figure 11(b)),
Min values Max values
STRucTuRE And InFRASTRucTuRE EngInEERIng 1169
Chloride content (kg/m3) Probability density (%)
Error (%)
Probability density (%)
Error (%)
Probability density (%)
1170 T.-B. TRAn ET Al.
Δxopt ≈ 2 cm (for Ns = 200 states) and Δxopt ≈ 1 cm (for Ns = 15 states rebuild). This means that the common idea that increasing the number of inspection points over the total inspection depth is better and is not always true for Bayesian updating. In fact, by doing so, the large error obtained by considering some inspection depths can affect the updating: that is the case for depths with low chloride content (around x = 10 cm). Moreover, the difference between chloride content in adjacent inspection points is too small when the discretisation size is smaller (∆x = 0.3 cm for instance). Therefore, the probability densities of the evidences used for updating the BN are very similar for two neighbouring inspection points.
Figure 12(a) illustrates this point by showing the almost iden- tical probabilities at x = 0 and 0.3 cm. This increases the errors in estimating the mean and standard deviation when using close points. In contrast, when ∆x increases (∆x = 2 cm), the proba- bility densities of the evidences differ by reducing the identifica- tion errors (Figure 12(b)). That is why, even for the mean value (Figure 11(a)), the protocol with Δx = 3 cm is better. On the opposite, when ∆x is larger than the optimal value, the errors for both mean and standard deviation increase because the informa- tion becomes poor for describing the chloride ingress process.
4.3. Using evidences from different inspection times
4.3.1. Problem statement
In this section, evidences obtained from various inspection times and depths will be introduced in the BN for the identification
(a) 25 20 15 10 5
00 0.5 1 1.5 2 2.5 3
process. According to previous section, different ranges were used for each child node in the BN with a sufficient number of states per node (Ns > 15 states) to minimise the fluctuation effects/errors in the results. This analysis considers then that Ns = 15 states. Since the BN configuration considers evidences coming from several inspection depths, the results of the follow- ing subsections will be illustrated in terms of the discretisation size (Δx) in order to study the effects of considering more or less information in depth. For example, for a total inspection depth of 12 cm, the smallest value of Δx (Δx = 0.3 cm) implies that there are nx = 41 inspection points in depth. On the opposite case, Δx = 12 cm considers that there are nx = 2 inspection points in depth located at x = 0 and 12 cm.
4.3.2. Influence of the inspection time on the identification error
Figure 13 shows the identification error for D by considering evidences taken and different inspection times tins for various discretisation sizes Δx. It can be seen that the inspection time tins influences the estimation of both the mean and standard devia- tion of D. The identification is improved when tins increases until arriving at an optimal inspection value tins,opt that varies between 40 and 50 years for the identification of the mean and stand- ard deviation. This phenomenon can be explained by the fact that when tins ≈ 45 years, the chloride concentration in the total inspection length is sufficient for describing the chloride ingress process, i.e. there is sufficient chloride content at each point in the space to improve the identification. When tins > 50 years, the
(b) 250 200 150 100 50
15 states
200 states
15 states rebuild
00 0.5 1 1.5 2 2.5 3
Figure 11. Identification error of D for three discretisations of child nodes: (a) mean – (b) standard deviation.
(a) 6 (b) 6
55
44 3 x=0cm 3 2 x=0.3cm 2
11
x=0cm x=2cm
00 2 4 6 8 10 12 00 2 4 6 8 10 12 Chloride content (kg/m3) Chloride content (kg/m3)
Figure 12. effect of discretisation size ∆x on the distribution of chloride content: (a) ∆x = 0.3 cm – (b) ∆x = 2 cm.
Probability density (%) Error (%)
Probability density (%) Error (%)
(a) 20 18 16 14 12 10 8 6 4 2
00 2 4
(b) 80 70 60 50 40 30 20 10
STRucTuRE And InFRASTRucTuRE EngInEERIng 1171
tins
10 years 20 years 30 years 40 years 50 years 60 years
6 8 10 12 00 2 4 6 8 10 12
Figure 13. Identification error for D with evidences from different inspection times: (a) mean – (b) standard deviation.
(a) 7 6 5 4 3 2 1
(b) 80 70 60 50 40 30 20 10
tins
10 years 20 years 30 years 40 years 50 years 60 years
00 2 4 6 8 10 12 00 2 4 6 8 10 12
Figure 14. Identification error for Cs with evidences from different inspection times: (a) mean – (b) standard deviation.
chloride content at x = 12 cm is larger than zero and therefore the identification errors increase because the inspection length is not large enough to describe the problem.
It is also worth noticing to complete the analysis at the end of previous section that there is an optimal value of Δx for each inspection time that minimises the error. The optimum value decreases when tins increases. This is related to the fact that for larger tins, the chloride content inside the total inspection depth is larger. Consequently, it is necessary to add more information to improve the representation of the chloride profile. It is also noted that the error is larger for smaller values of Δx in comparison with the Δxopt. There is no remarkable change in the estimation of the mean value of D when ∆x varies from 0.3 cm to the optimal value. However, the variation is more important for the identi- fication of the standard deviation of D. This is mainly related to the discretisation effect described in the previous section.
For Cs, the results presented in Figure 14 reveal that it is bet- ter to use the evidences at early inspection times to obtain a better estimation of Cs. At the beginning of the exposure (e.g. tins = 1 year), the chloride concentration at x ≈ 0 cm is close to Cs and it decreases drastically until zero for the neighbouring points. In such a case, the larger differences between neigh- bouring points (chloride concentration gradient) reduce the identification errors as indicated in Figure 12. However, when inspection times increase, chlorides penetrate into concrete and the chloride concentration gradient decreases for neighbouring points by introducing inaccuracies in the assessment. The results also show that the consideration of fewer points in depth (larger
Δx) diminishes significantly the errors in the assessment of the standard deviation as previously discussed in Section 4.2.2.
4.4. Combining simultaneously evidences from different inspection times
4.4.1. Problem statement
Inspection campaigns on RC structures can be carried out at different times. In this section, the BN will be used to evaluate the efficiency of combining simultaneously evidences obtained after different inspection times. For comparative purposes, we consider the same number of chloride profiles (9000 chloride profiles) for different schemes of inspection, thus the same quan- tity of information for all configurations (Table 4). Thus, the evidences from each scheme may come from one inspection time or from the combination of several inspection times.
4.4.2. Identification errors for various inspection schemes
Figure 15 depicts the identification error for Cs with evidences from different schemes of inspection. As concluded in Section
Table 4. Schemes for combining evidences for different inspection times.
Inspection scheme
1 2 3 4 5
Inspection times (tins) (years)
10
20
30
10 + 20 10 + 20 + 30
number of chloride profiles
9000
9000
9000
4500 + 4500 3000 + 3000 + 3000
Error (%)
Error (%)
Error (%) Error (%)
1172 T.-B. TRAn ET Al.
(a) 7
10 years 6 20 years 30 years
5 (10 + 20)years
(b) 90 80 70 60 50 40 30 20 10
10 years
20 years
30 years
(10 + 20)years
(10 + 20 +30)years
4 3 2 1
(10 + 20 +30)years
00 2 4 6 8 10 12 00 2 4 6 8 10 12
Figure 15. Identification error for Cs with evidences from different schemes of inspection: (a) mean – (b) standard deviation.
(a) 25 20 15 10 5
(b) 90
80 20 years
70 30 years
60 (10 + 20)years
50 (10 + 20 +30)years
40 30 20 10
10 years
20 years
30 years
(10 + 20)years
(10 + 20 +30)years
10 years
00 2 4 6 8 10 12 00 2 4 6 8 10 12
Figure 16. Identification error for D with evidences from different schemes of inspection: (a) mean – (b) standard deviation.
4.3, since the deterioration model considers that the chloride surface concentration does not depend on time, it will be better to use the evidences obtained at early inspection times. Indeed, the identification of Cs from inspection data at 10 years will be more accurate than those at 20 or 30 years. The results in Figure 15 have pointed out that combining data from different tins could not improve the identification. For example, in the inspection scheme 4 (10 + 20 years), the identification errors are larger than inspection scheme 1 which uses data from tins = 10 years. This is because, for the inspection scheme 4, the 4500 chloride profiles obtained at 20 years induce errors in the identification process.
Figure 16 presents the identification errors for D with evi- dences for different schemes of inspection. As previously men- tioned in Section 4.3, evidences at tins = 30 years provide smaller identification errors than those at 10 or 20 years. Results in Fig- ure 16 show that from inspection scheme 3 with 9000 chloride profiles at tins = 30 years, the estimation of D is more close to the theoretical values than others inspection schemes that use data from early inspection times or combine the evidences from dif- ferent tins. It can be also concluded that there are optimal inspec- tion years for minimising the errors for the identification of each parameter. In this case, the consideration of several inspection times introduced errors in the identification process. However, inspection schemes that consider several inspection times could be more appropriated to identify time-dependent parameters.
For example, information of several inspection times could be useful in the identification of an ageing factor for the chloride diffusion coefficient that is considered for other models (Tran, Bastidas-Arteaga, Bonnet, & Schoefs, 2015). Although this point is important for chloride ingress modelling, the consideration of other models is beyond the scope of this study.
4.5. Summary
Parameter identification using BN is a complex problem that requires an improved configuration to reduce identification errors. Due to physical and model characteristics, it was found that there is a best BN configuration for the identification of each parameter. For Cs, it was the BN configuration with one child node close to the concrete surface. For D, it is necessary to use the information from total inspection length to provide a better characterisation of the kinetics of the chloride ingress process. For a certain inspection depth, an optimal inspection time exists and could describe adequately the chloride ingress process and minimise the errors in the identification of D. In particular, there is an optimal discretisation size for each inspection time. This improved configuration could be considered for recommending inspection strategies. The results also revealed that it is better to use the information at early inspection times for estimating the parameter Cs; for D, the data at later and specific inspection times are more useful.
Error (%) Error (%)
Error (%) Error (%)
STRucTuRE And InFRASTRucTuRE EngInEERIng 1173
(a) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
(b) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Theory
Theory
Theory x=0cm x =3cm x=6cm x=9cm
x=12cm
Theory
0 20 40 60 80 0 20 40 60 80 Time (years) Time (years)
Figure 17. Probability of corrosion initiation with data obtained: (a) from a single point inspection depth – (b) from full inspection depth.
5. Assessment of the probability of corrosion initiation
5.1. Probability of corrosion initiation
This section examines the influence of the data identified from
BN on the evaluation of the probability of corrosion initiation.
In fact, previous sections have shown that there are different
improved configurations for the identification of Cs and D: a
multi-criteria analysis will not be efficient because the sensitiv-
ity of the response to these variables vary with time (Bastidas-
Arteaga et al., 2011). Thus, the quantity of interest ‘probability
of corrosion initiation’, useful for decision makers, is considered
initiation, pini, is obtained by integrating the joint probability function over the failure domain, i.e. Equation (10). pini is estimated herein using Monte Carlo simulations.
5.2. Numerical example
5.2.1. Problem description
now. The time to corrosion initiation, t , is defined as the time
ini ()
= U 𝜇 = 0.9 kg/m3;𝜎 = 0.2 kg/m3 (Vu & Stewart, surface reaches a threshold value, Cth. 2000) and a concrete cover depth of 6 cm.
at which the chloride concentration at the steel reinforcement eters: C
th
This threshold concentration represents the chloride concen- tration for which the rust passive layer of steel is destroyed and the corrosion reaction begins. Cth depends on many parameters (Bastidas-Arteaga & Schoefs, 2012): type and content of cement, exposure conditions, time and type of exposure, distance to the sea, oxygen availability at the bar depth, type of steel, electrical potential of the bar surface, presence of air voids, definition of corrosion initiation, methods and techniques for measuring Cth, etc. Then, the determination of an appropriate Cth becomes a major challenge for the owner/operator and it will be therefore assumed herein that Cth is a random variable.
The time to corrosion initiation is calculated by evaluating the time-dependent variation of the chloride concentration at the reinforcing steel depth that is computed from Equation (3). The cumulative distribution function of the time to corrosion
initiation, Ft
ini
(t), is defined as:
Ftini(t)=p(tini ≤t)= � f(x)dx tini ≤t
where f(x) is the joint PDF of the vector of random variables X. The limit state function that defines corrosion initiation can be written as:
g(𝐗, t) = Cth(𝐗) − Ctc(𝐗, t) (10)
where Ctc(X,t) is the total concentration of chlorides at the concrete cover depth, ct, at time t. The probability of corrosion
C arelowforthisinspectionstrategy,theerrorsoftheidentification s
Let us consider a RC component placed in a chloride- contaminated environment. The parameters describing the expo- sure (Cs) and material properties (D) are described in Table 1. The probability of corrosion initiation is computed by considering that the threshold of chloride concentration for initiation of cor- rosion follows an uniform distribution with following param-
Various inspection schemes are examined considering one point or various points in depth for a single inspection time tins = 10 years. The study considers different ranges for the discre- tisation of each child node. We will also study the case in which the number of chloride profiles is limited and we will propose a strategy to improve the assessment in such a case. The histograms obtained after updating the BN for each parameter will be used directly in Monte Carlo simulations to estimate the probability of corrosion initiation to avoid any assumption about analytical distribution laws.
5.2.2. Assessment of pini for larger inspection data
Figure 17(a) presents the probability of corrosion initiation with data
obtained from a single point inspection depth (Section 4.1). These
results were obtained by identifying each parameter from 9000
numerical chloride profiles. The results indicate that data identified
from one depth point do not provide an acceptable prediction of the
probability of corrosion initiation in comparison with the theoretical
(9)
assessment (Section 4.1). Although the errors in the identification of
of D introduce larger differences in the assessment of pini. However, from Figure 17(b), it is noted that the prediction of the probability of corrosion initiation is more close to the theoretical values when the parameters are identified from sev- eral inspection points (Section 4.2). In this case, the considera- tion of more inspection points reduces identification errors by improving the assessment of pini. The error depends on the ‘time
Probability of corrosion initiation
Probability of corrosion initiation
1174 T.-B. TRAn ET Al.
(a) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
(b) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Theory
Probability of corrosion initiation
Probability of corrosion initiation
Theory
00 20 40 60 80 00 20 40 60 80
Time (years)
Figure 18. Probability of corrosion initiation with limited data: (a) before improvement – (b) after improvement.
Time (years)
D
. . .
Step 1
Cs
x=0
Step 2
Figure 19. two-step procedure for improving identification with limited data.
of interest’ or ‘target probability of corrosion initiation’. The latter is more suitable because it is in link with actual recommendations schemes. By considering a target probability of corrosion lower than 0.5, larger values of Δx provide a better assessment. If a higher probability of corrosion is acceptable (between 0.6 and 0.9), the opposite trend is observed.
5.2.3. Assessment of pini for limited inspection data
The results presented above considered a large number of sim- ulations for the generation of numerical evidences. However, in real practice, the number of profiles collected after an inspection campaign is very limited. Figure 18(a) presents the probability of corrosion initiation estimated with limited data obtained by considering the full inspection length. In this case, the numer- ical evidences were generated from 15 profiles of chloride con- tent obtained from Monte Carlo simulations. It is noted that the assessment of the probability of corrosion initiation is far from the theoretical values. Consequently, it is necessary to improve the use of the information utilised for updating the BN. To improve the identification, it is proposed to combine results of BN configura- tions with one and several inspection points (Figure 19). Therefore, the results of the identification of Cs obtained with one depth point configuration (Step 1) are used as a priori data for the estimation of D by considering the full depth configuration (Step 2).
The assessments of probability of corrosion initiation that compare these two cases (before and after improvement) are shown in Figure 18. It is clear that this strategy improves the identification when data are limited. It could be concluded that this approach would be very useful for limited data. The
assessment could be improved if data of other inspection times are considered, as mentioned in Section 4.3.
6. Conclusions
Chloride ingress is one of the main causes inducing corrosion of RC structures. The identification of parameters in chloride ingress modelling is crucial in predicting chloride ingress into concrete that will help to reduce maintenance costs of structures exposed to chloride-contaminated environments. Inspection data used for the identification are very limited due to time- consuming and expensive tests. Therefore, it is necessary to use these data in an optimal scheme. Within this framework, the BN could provide a possibility to identify model parameters with different information.
In this work, results based on numerical evidences revealed that there are optimal configurations of BN for the identification of each parameter (Cs or D). For Cs, an early inspection with one point close to the surface could provide a good identifica- tion. For D, the identification should use the evidences from full inspection depth. At a specific inspection time, there is an optimal discretisation size that could provide the best estimation for D. These configurations could be combined to improve the identification of the model parameters. An application to the assessment of the probability of corrosion initiation showed that the approach is useful even if information is limited. When the number of chloride profiles is limited, it is required to improve: (i) the BN configuration and (ii) the updating strategy (a two- step procedure is proposed).
C(x, t)
. . .
A priori D distribution
Cs
C(x , t )
1j 2j 3j ij nj
C(x , t )
C(x , t )
C(x , t )
C(x , t )
Theory
STRucTuRE And InFRASTRucTuRE EngInEERIng 1175
This study focused on the identification of model parame-
ters for a simple analytical chloride ingress model. Tran et al.
(2015) found that the main findings of this study for improving
models that account for instance for the time-dependency of D as the proposed by Nilsson and Carcasses (2004). However, the identification of parameters for more complex models requires developing other BN configurations, larger computational efforts and additional inspection data. In addition, further work in this area will focus on:
• the use of real data (real chloride profiles and/or study cases),
• the consideration of other deterioration models numeri- cal or analytical that account for more realistic conditions: unsaturated zones, binding, time-dependency of model parameters, etc.,
• the consideration of measurement errors and model un- certainty,
• the extension of the identification of random variables to the identification of random fields parameters, and
• the consideration of costs for recommending inspection procedures (inspection times, number of cores, number and positions of measures in each core, etc.) that provide a balance between error and cost.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This work was supported by the ERDF funding of the European Union, Interreg program, for the DuratiNet project (2009–2012 Durable Transport Infrastructure in the Atlantic Area-Network); the ‘Pays de la Loire’ region (France) for the SI3 M project (2012–2016 Identification of Meta-model for Maintenance Strategies).
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