Modelling the interactions between defect mechanisms on metal bridges

: Bridge asset managers have ﬁnite resources at their disposal to minimise the risk of structural failure and ensure bridges are maintained to a suitable safety threshold. Any maintenance strategy must be efﬁcient in its use of resources and deliver an optimal Whole Life Cycle Cost (WLCC). The calculation of an accurate WLCC is contingent on having an accurate deterioration model to predict future asset condition and sufﬁcient performance indicators to appraise targeted maintenance strategies in a decision model. Typically predictive bridge deterioration models output a probability distribution for a single condition indicator over time. However, bridge deterioration is a diverse physical process with different degradation mechanisms. For example, metallic bridge elements may undergo corrosion and loss of coating or paintwork, as well as suffer from structural component failure modes such as buckling, permanent distortion, tearing and fracture. This paper presents a multi-defect approach to modelling bridge asset management. A multi-defect deterioration model is implemented using a Dynamic Bayesian Network (DBN). The model can predict the simultaneous progression of multiple bridge defects. The decision model can utilise the multiple condition indicators to apply the most appropriate maintenance intervention to provide an uplift in condition. The industrial data used in this research takes the format of a longitudinal study, which is common for many transportation agencies. The use of such data restricts the deterioration model to use a memoryless distribution which assumes a constant failure rate. However, bridge deterioration has been empirically shown to be a non-constant process. By modelling bridge deterioration as a combination of interacting defects, non-constant behaviour can be modelled, even when the model itself is parametrised using an exponential distribution. The paper presents a case study of the model calibrated using data from 13,569 metallic railway bridge girders in the United Kingdom.


INTRODUCTION
The railway is a critical component of the transportation network in the United Kingdom, which delivers economic and social benefits to the country.Bridge assets are a requisite for the safe, capacious, and high performance operation of the transportation network.Network Rail (NR) own and are responsible for the management of over 28,000 bridges in Great Britain and have a well established asset management strategy to minimise the risk of structural failures occurring on any of their bridges and to maximise the impact of any invested resources (Network Rail 2017).
As part of the development of the bridge asset management strategy a Whole Life Cycle Cost (WLCC) analysis must be performed to determine the impact of particular strategies on WLCC alongside additional factors such as safety risk and service disruption.Such analysis is critical to an infrastructure manager's strategic planning and forecasting abilities.An accurate WLCC analysis is contingent on accurate bridge deterioration profiles.
Typically, bridge condition is denoted using a single condition index on a one dimensional scale.This approach assumes that bridge deterioration is a single homogeneous process.However, the physical phenomena of bridge deterioration for many bridge types and materials, is an aggregation of several simultaneous processes which all result in the reduction of the structural integrity of the bridge.In this research, a model is proposed to model deterioration which accounts for the interactions between different bridge deterioration mechanisms.Moreover, a maintenance intervention on one defect mechanism may slow down the progression of a different mechanism.A case study is considered for the interactions of paintwork condition, corrosion and structural component failure on metallic railway bridge girders in the United Kingdom.

BRIDGE DETERIORATION MODELLING
Stochastic approaches are favoured in analysing deteriorated structures for their ability to incorporate the intrinsic uncertainty of the physical bridge deterioration process (Frangopol, Kallen, & van Noortwijk 2004).The stochastic models are commonly calibrated using expert judgement and/or historic records.The historic records that commonly exist for bridges include: • Condition records from bridge examinations.
• Maintenance records from servicing and maintenance interventions.
Maintenance records outline the occurrence and types of intervention that a bridge has undergone, and can be used to perform lifetime analysis without the subjectivity of condition indices.However, maintenance records for many transportation agencies are sparse and of poor quality, which limits the applicability of their use on large, mature bridge portfolios.Empirical bridge measurements may include records detailing the geometry and material characterisations of bridge assets, although these records are commonly for a minority of bridges and would not be appropriate for decision making at the portfolio level.Condition records document the condition of a bridge when inspected by an examiner.The change in condition between multiple inspections can be used to estimate a deterioration profile.Many transportation records are mandated by their jurisdiction to maintain an inspection regime which results in sizeable condition record datasets.The use of condition records to estimate transition rates for deterioration models is noted to be the most common (Frangopol, Dong, & Sabatino 2017).It is crucial to note that these condition records are not necessarily indicative of the structural capability of a load bearing element (Neves & Frangopol 2005, Liu & Frangopol 2006, Saydam, Frangopol, & Dong 2013) but due to the prevalence of the data, are commonly used to estimate deterioration profiles (Frangopol & Liu 2007).
Commonly, deterioration models use a single condition scale index to quantify the structural integrity of bridges or bridge elements.This is a consequence of bridge exam condition records expressing the structural integrity of bridges as some sort of scale of 'overall' or 'worst' condition.Bridge deterioration is not a single physical process but is rather a composition of several simultaneous process which reduce the capability and structural integrity of a bridge.Any consolidation of these mechanisms into one score profile will introduce a level of subjectivity and arbitrariness to any predictive model.
Modelling deterioration with a memoryless distribution requires the assumption that the process has a constant failure rate, which is considered to be a limitation of the methodology.However, many studies adopt this limitation as a consequence of the available data being in the format of a longitudinal study.Sobanjo (2011) has shown empirically that bridge deterioration is a non-constant process.For the considered defect mechanisms of corrosion, damaged paintwork and structural component failure, engineering experience suggests that this finding would be true.In this study, bridge deterioration is modelled as a non-homogeneous process.By modelling the several simultaneous deterioration mechanisms distinctly, contextualised deterioration profiles can be estimated for particular bridge element histories, which exhibit non-constant deterioration behaviour despite being characterised by exponential distributions.

Bridge Deterioration Bayesian Networks
Bayesian Belief Networks (BBN) are a method that can be applied to reliability engineering problems to incorporate expert engineering knowledges into a model (Langseth & Portinale 2007).Attoh-Okine & Bowers (2006) developed a BBN that modelled the interactions between different bridge elements to compute probabilities of bridge condition at the levels of deck, sub-structure, super-structure and overall deterioration of bridge performance.The root variables that represented the condition of the bridge elements possessed multiple condition states, however the study did not model how these defects would progress or interact with each other but rather just reported the condition of the bridge at different hierarchical levels for a specific time.
Dynamic Bayesian Networks (DBN) are an extension of BBNs which can be used to model phenomena in the temporal domain (Murphy 2002) and have found applications in reliability and deterioration studies (Weber & Jouffe 2003).Rafiq, Chryssanthopoulos, & Sathananthan (2015) presented a similar BBN model for bridge condition as Attoh- Okine & Bowers (2006) and extended it to a DBN to consider the bridge deterioration process temporally and analyse 'what-if' scenarios.This study considered a UK railway masonry arch bridge, however it used a single condition scale of Poor, Fair and Good.Zhang & Marsh (2018) proposed an additional bridge deterioration model, which used BBNs based on existing statistical models and expert knowledge on factors that could alter the deterioration profiles of assets.The model considers visual examinations and detailed examinations and exploits the asset hierarchy similar to the model proposed by, Rafiq, Chryssan-thopoulos, & Sathananthan (2015), to provide a prediction of bridge strength.Additionally, the proposed BBN assigned prior probabilities to hyper-parameters which facilitates the incorporation of uncertainty in statistical variables.However, this model also used a single condition scale, and was not validated against a real dataset.A model of the deterioration of sewer pipelines, proposed by Elmasry, Hawari, & Zayed (2017) is an example of a defect-based BBN.
It can be concluded that the use of BBN based frameworks are popular for applications in reliability engineering and there are several examples of their application to the study of bridge degradation.In this research, a model is developed that enables the prediction of defect progression for several defects and defect progression being influenced by other interacting defects.The model parameters are estimated by analysing the historic bridge condition records from the NR bridge examination regime.The interactions between defect mechanisms are modelled using a DBN, with multiple predictive outputs for a condition score for each of the defects considered for any given instance.

Bayesian Belief Networks
Bayesian Belief Networks (BBN) or Bayesian Networks (BN) are probabilistic graphical models that are composed of a Directed Acyclic Graph (DAG), and a conditional probability distribution for each of the variables in the DAG (Pearl 1988, Jensen 2001).
A DAG denotes the random variables, X i , as nodes and nodes can be connected to form pairs with directed links.The nodes that are at the start of a link are known as parent nodes and nodes at the end of a link are known as child nodes.Each node, X i has a conditional probability distribution P(X i |pa(X i )) that quantifies the causal influence of the parent nodes on the child node.The random variable in each node may be discrete or continuous, however in this study only discrete variables will be considered.Conditional Probability Tables (CPTs) are used to tabulate the conditional probability distribution for each random variable and each of the different conditional permutations it may assume.The joint probability distribution can be calculated using recursive factorisation, where: pa(X j ) denotes the set of all variables X i , such that there is an arc from node i to node j in the graph (Pearl 1988).
A Dynamic Bayesian Network (DBN) is a temporal implementation of BBN, which considers the evolution of a BBN over time (Murphy 2002).Time is discretised, with a BBN model defined for each discrete time step, known as a 'time slice'.The multiple time slices are connected through temporal links to form the complete model.If the time slices and temporal links remain consistent, then the model is a DBN.

MULTI-DEFECT BRIDGE DETERIORATION MODEL
Bridges are heterogeneous assets with each having its own unique composition of elements.A bridge inspected by NR, like many transportation agencies, will be described by a defined hierarchical decomposition, with major elements and minor elements.
Major elements include: inner supports, end supports and decks, and each major element will comprise of a set of minor elements.Moreover, each minor element type may be attributed the status of being a 'principal load bearing element'.At each detailed examination of a bridge, a condition will be recorded for each minor element on the bridge asset.NR use an alpha-numeric condition scale known as Severity Extent (SevEx) to record the condition of the elements of bridges at inspection.An additional scale known as Coating-Metal (CM) is used to record the intactness of the coating/paintwork (Network Rail 2017).The definition for metallic severity scores can be found in Table 1, metallic CM scores in Table 2 and the extent score which is common to SevEx and CM in Table 3.
From analysing the SevEx and CM scale definitions there are three distinct defect mechanisms that could be identified and monitored in the NR metal bridge stock: • Loss of coating or paintwork; • Corrosion; • Structural Component Failure (SCF) -Includes: buckling permanent distortion/displacement and tearing/fracture.
To model the occurrence of each of these defects, their extensiveness and the interactions between them, a DBN was developed.The structure of the causal influences between the defects is shown in Figure 1 as a BBN and the DBN deterioration model in Figure 2.

Metallic Bridge Element Condition Scale
The SevEx and CM condition scales were transformed to a defect specific condition scale using internal NR weightings for the SevEx/CM states (Network Rail 2017).For the paintwork node there are four states for this study, where P 1 denotes no visible defects and P 4 denotes extensive paintwork damage, with P 2 and P 3 as intermediate states of paintwork damage.Corrosion states, C1, C2, C3 and C4 are defined in a similar manner with C4 corresponding as the poor condition state that would trigger major maintenance interventions.For SCF, there are two states: F 1 for when a SCF mode is absent and F 2 denotes its presence.Two states were used for SCF as NR policy suggests maintenance intervention is required once a SCF mode is identified.Moreover, there is less prevalence of SCF being present in the records and two states were used to reduce the likelihood of a model being over fitted to rare event data.4 Defect occupies between 5% to 10% of the surface of the element.5 Defect occupies between 10% to 50% of the surface of the element.6 Defect occupies more than 50% of the surface of the element. of the temporal links and these are tabulated in the CPT.The probability values used in this study were parametrised using a λ rate for the exponential distribution.This parametrisation was used for numerical stability and to aid the optimisation process.All the required λ values that describe the model's CPTs are denoted by θ.A transition rate matrix, Q(θ), was defined with a structure such that could be used to calculate,

Model
where t is the size of the interval between the time slices in the DBN, expm is the matrix exponential and P is a probability matrix which contains all the probabilities required to populate the model's CPTs.The matrix exponential can be calculated using numerical methods (Moler & Van Loan 2003).
To determine the appropriate set of values for θ, a method of maximum likelihood applied to panel data was applied.It was deemed to be the most appropriate for the NR bridge condition records which form a longitudinal study.A rate is required to characterise the deterioration process as to calculate probabilities from observed transitions would require the inspection interval to be of a fixed size.At NR, the interval between each inspection varies depending on the condition of the bridge at the previous inspection and/or the design specification of the bridge.The method is based on Kalbfleisch & Lawless (1985), Kallen & Noortwijk (2006) and Ferreira, Neves, Silva, & de Brito (2018) and does not implicitly assume that a bridge element remains in its most recently revealed condition state until a further inspection reveals it to be otherwise.This is deemed to be more reflective of the continuous progression of each defect mechanism.The likelihood of the observed condition transitions is given by, where N denotes the number of observed condition transition records and i is the joint condition score at the first inspection in record r, j is the joint condition score at the second inspection in record r, t is the length of the inspection interval between the first and second inspections and N is the number of exam pair records.The loglikelihood function should be used for numerical stability, To compute the appropriate value for p r using θ, the conditions of paintwork, corrosion and SCF at the first inspection of record r, were used as a belief state on the initial time slice.Using exact inference on the DBN populated with θ, the joint probability of all the defects being observed in a particular state at time t were calculated.A time step size of one week was used between time slices.
To determine the MLE θ values can be found by taking derivatives of the log-likelihood function.In this research the optimal θ values were determined using a derivative-free approach, using the loglikelihood function as an objective function for a Genetic Algorithm (GA) (Goldberg 1989).

CASE STUDY
NR are responsible for ensuring that the bridge portfolio of the British railway adheres to rigorous guidelines on the structural integrity of bridge assets.Part of the overall asset management strategy of the bridge portfolio is the execution of industry inspection policy.NR have a data set dating from 1999 that contains the recorded bridge conditions at inspection for their entire bridge portfolio.
As a case study of the metallic multi-defect DBN model, the records for all the exposed metallic girders in the database were used to train the model.This amounts to 13,569 unique elements and 20,397 pairs of exam records.To ascertain the impact of modelling the interactions between different defect mechanisms, an independent model for each defect was also calibrated using the same data.Each model had its optimal θ determined using the function shown in (5) in a GA optimisation.

Defect DBN Condition Profiles
The parameter values for the independent multidefect deterioration model are shown in Table 4, and the values for the DBN variant in Table 5.The probability profile of each defects condition state for 25 years from the DBN model are shown in Figures 3,   4 and 5.Note that the bridge element has an initial condition state of P 1, C1 and F 1, representing a new bridge element.
It can be observed from the condition profiles that corrosion and paintwork damage are much more rapid in progression than SCF.This model output aligns with the engineering expectation that SCF would develop over a much longer time frame.To compare the fit between the different models, a test statistic such as a Pearson's chi-squared test could be used.However, due to the variability in inspection intervals, there would be a considerable number of bins which have low frequencies and the test was deemed inappropriate.Instead, an analysis of observed final inspections compared to the predicted final inspections was performed.The process for this comparison is: • Calculate the total number of records observed in each condition state at final inspection.
• Compute the probabilities for each condition state at the final inspection for each observed record using the DBN model, taking the condition at first inspection as the belief state.
• Sum the probabilities for each predicted condition state for all predicted final conditions.
To compare the goodness of fit between the two different models the Mean Squared Error (MSE) was used.The MSE is given by, where n is the total number of predictions, generated from the n observations, across all variables.Y is a vector of the observations across all variables and Ŷ is a vector of the predictions across all variables.The MSE can only assume non-negative values and values closer to zero are deemed to represent the model generating a better fit.The MSE values for the independent defect model and the DBN model are shown in Table 6, with the DBN providing a better goodness of fit than the independent model.Additionally, a log-likelihood ratio test statistic can be used to show that the improved fit of the model is statistically significant given the increase in parameters for the DBN model.The log-likelihood ratio test is given by where F Ind is the log-likelihood of the independent model with its optimal parameter θ values and F DBN is similarly defined for the DBN model.The null hypothesis for the likelihood ratio test is true when LR is small and rejected if the LR values have a significant difference.The LR statistic approximately follows a Chi-square distribution, with the degrees of freedom equal to the difference between the number of parameters used in each model.For the independent and DBN models the difference between log-likelihood scores is 621.9.Using a significance level of 5%, the null hypothesis can be rejected and thus conclude that the models are significantly different.

Bridge condition under different maintenance strategies
The comparison of different asset management strategies is a significant task for transportation infrastructure asset managers, with industry stakeholders requiring prudent decisions.Such comparisons require an evaluation of the outcomes on bridge condition and capability, safety risk, service risk etc. for each strategy and the associated impact on WLCC.
To evaluate the contextualised deterioration rates and the effects of targeted maintenance interventions a simple maintenance case study was performed by applying maintenance interventions to uplift the paintwork score to perfect condition, i.e.P1 at predefined condition states.The three maintenance strategies considered are: • Strategy 1 -Instant repair of paintwork upon entering state P 2; • Strategy 2 -Instant repair of paintwork upon entering state P 3; • Strategy 3 -Instant repair of paintwork upon entering state P 4.
The instant repair strategy assumes a continuously revealed condition state, no budgetary limitations and allocates no delay for scheduling and performing the intervention.These assumptions do not relate to the real-world scenario of managing a bridge portfolio, however the example serves to show the impact of considering defect interactions and modelling using contextualised rates.The average condition of each defect predicted over 25 years with each maintenance strategy and a 'do-nothing' approach are shown in Figures 6, 7 and 8.It can be observed from Figure 6 that if a maintenance strategy is applied to service the paintwork, the average paintwork condition will ultimately reach an equilibrium value which is greater than the 'do nothing' approach, which continually degrades.Moreover, the earlier the paintwork intervention is scheduled for, the lower the equilibrium value obtained for average condition score.Recall that a lower average condition score denotes a better condition.
The multi-defect DBN deterioration model incorporates the influence of paintwork condition on the rate of corrosion.Whilst the maintenance strategies considered in this study only intervene on paintwork, that targeted intervention still results in an uplift of the average condition of corrosion for all three scenarios.Nonetheless, it can be observed from Figure 7 that the earlier paintwork is maintained, the greater the reduction in corrosion prevalence.For Strategy 3, the paintwork intervention seems to be so delayed that the difference in the average condition of corrosion for Strategy 3 and the 'do nothing' approach is minimal.Theoretically one may anticipate that if the paintwork is consistently in P 1, then corrosion should not occur.However, in reality there is no such thing as perfect paintwork repair and consequently corrosion would still occur.Additionally, bridge examinations are subjective and some examiners may incorrectly identify the paintwork as being in P 1.Thus, when the model parameters are estimated, the rate of corrosion remains as non-zero, see Table 5. Expert judgement could be used to set the rates for corrosion for the P 1 scenario, however the authors deemed that the current model should only be calibrated using the available data.
The model accounts for an influence of corrosion on the occurrence of SCF.The reduced prevalence of corrosion due to the paintwork interventions, does seem to elicit a slight reduction in the occurrence of SCF.However, without the uncertainty being quantified it is unreasonable to declare this definitively.
The reduced evolution of defects is a critical trend that represents a desired modelling capability for NR asset managers, as the absorbing states for each of these defects represent the bridge element being in poor condition and posing a risk to the structural integrity of the element and the overall bridge.Having a model that can isolate these physical phenomena and enables the study of the effects of maintenance strategy is a hugely desirable tool for asset managers to develop and present strategies to stakeholders and more efficiently allocate resources.

CONCLUSIONS
An accurate WLCC analysis is required to be able to distinguish the cost effects of competing asset management strategies.This is a requisite component to the decision making process for many infrastructure asset managers.Commonly, bridge deterioration is re-ported using a single condition index, however it is a heterogeneous process composed of several distinct processes acting simultaneously.This research proposed a modelling approach where distinct deterioration modes are modelled using a DBN, allowing the consideration of any interactions between the different deterioration mechanisms.
The presented model is for metallic bridge components.The evolution of damaged paintwork was modelled as an independent process that influenced the rate of corrosion.Moreover, the model incorporated the relationship between corrosion and structural component failure.By modelling the defects as distinct processes that interact with each other, the model enabled the calculation of non-constant deterioration profiles for defect mechanisms despite being limited to using an exponential distribution due to data constraints.
The approach of modelling multiple defect simultaneously enables an improvement in predictive accuracy of deterioration as well as outputting additional bridge condition indicators for informing decision modelling.The additional indicators specific for each defect can then be employed to develop specific maintenance strategies opposed to the traditional qualitative maintenance actions.A simple example was shown that exhibited the contextualised benefits of applying specific maintenance interventions on paintwork and the resultant reduction in the progression of corrosion and structural component failure.The example was limited by several assumptions and future work is planned to consider the inclusion of resources constraints as well as additional repair actions.

Figure 2 :
Figure 1: Bayesian Belief Network representing causal influences between metallic defect modes.

Figure 6 :
Figure 6: Average condition of paintwork under different paintwork maintenance strategies.

Figure 7 :
Figure 7: Average condition of corrosion under different paintwork maintenance strategies.

Figure 8 :
Figure 8: Average condition of SCF under different paintwork maintenance strategies.

Table 2 :
CM severity definitions for metallic bridge elements (Network Rail 2017).

Table 4 :
List of parameters for independent multidefect deterioration model.

Table 5 :
List of required parameters for multi-defect DBN deterioration model.

Table 6 :
Mean Squared Error for each model based on the predictions of final condition.