Fatigue reliability assessment of steel bridges considering spatial correlation in system evaluation

Abstract Fatigue is among the most critical forms of deterioration damage that occurs to steel bridges. The performance of steel bridges, which may be seriously affected due to fatigue, should be assessed and predicted. However, the code-based reliability assessment only provides a clear criterion for single details. The whole bridge or a splitting system from the bridge may contain many details which brings system impact on the fatigue assessment. In this paper the system reliability of steel bridges is addressed with focus on the influence of spatial correlations between similar fatigue sensitive details. The study is conducted on a typical riveted railway bridge, from which a subsystem is selected and modelled as a weakest link system. It is assumed that all details in the system are equicorrelated. An efficient modeling approach has been developed to quantify the correlation and help estimate the system reliability analytically. A parametric study was performed to determine the system effects on the fatigue reliability assessment and identify the decisive variables.


Introduction
Engineering structures are generally subjected to deterioration processes such as fatigue and corrosion, and their structural reliability may thus reduce over time. Components of the structure might be built with the same materials, and possibly subjected to similar environment conditions and load fluctuations. Consequently, a spatial correlation is formed between components. This leads to a dependence among deterioration failures of different components, which will affect the system safety and, therefore it is necessary to consider the structure as a system with the spatial correlation taken into account to analyze the structural reliability.
Several recent contributions have considered spatial correlation among components deterioration and investigated its influence on the whole system. De Leon and Mac ıas (2005) investigated how the spatial correlation between corroded pipeline segments affects the system reliability. They also showed that the correlation degree between the failure modes at two pipeline segments increases with the degree of correlation between the initial defects at these segments. Hajializadeh et al. (2016) predicted the probability of failure for chloride-induced corroding reinforced concrete (RC) structures accounting for the effect of the spatial correlation associated with the loading and resistance. Titi et al. (2018) investigated on the corrosion process of RC frames considering the spatial correlation of the random variables and showed that the role of correlation is emphasized under severe exposures. Geyer et al. (2020) considered the spatial variability of material parameters in the reliability assessment of hydraulic structures through a detailed random field modelling. Abyani and Bahaari (2020) investigated the effects of correlation of random variables between the adjacent components on system reliability of corroded pipelines for four different failure modes.
Most of the previous studies considered one component at a time when considering spatial correlation in reliability assessment. Schneider, Th€ ons, and Straub (2017) enabled an integral assessment by proposing two coupled sub-models: a probabilistic system deterioration model for considering stochastic dependence among component deterioration, and a probabilistic structural model for calculating the failure probability of the system. Kang and Song (2010) proposed an efficient method named Sequential Compounding Method (SCM) in which the reliabilities of components are firstly computed, and the components are subsequently combined by an iterative procedure. Roscoe, Diermanse, and Vrouwenvelder (2015) developed the Equivalent Planes Method (EPM) for the Dutch flood defense system. The components are combined two at a time, similar as SCM. These two methods differ notably that SCM only requires correlation between components while EPM requires information about the auto-correlation of the underlying random variables contributing to the failure.
For bridges, it is common to have similar details appear several times within the structure. Using a system approach to estimate reliability has distinct advantage since it considers both the reliability of components and their relationship and importance to the entire system. Hendawi and Frangopol (1994) aimed to improve current design and evaluation specifications by including system reliability. A two-member parallel system and a n-member parallel system were studied theoretically. Estes and Frangopol (1999) proposed a system reliability approach for optimizing the lifetime repair strategy for highway bridges. The approach was demonstrated by an existing Colorado State highway bridge, which was modeled as a series-parallel combination of failure modes. Nowak and Cho (2007) proposed an innovative prediction method for the combination of failure modes of a selected arch bridge in Korea. The method was demonstrated by a series-parallel system and results showed that the proposed method could significantly reduce calculation time compared to the conventional system reliability analysis method. Imam, Chryssanthopoulos, and Frangopol (2012) presented a system-based model of a stringer-to-cross-girder connection, the probabilities of fatigue failure of individual hot-spots on various connection components were estimated first, then combined using a system reliability approach to estimate the overall reliability of the connection. Maljaars and Vrouwenvelder (2014) provided a method to determine the reliability of a structure considering fatigue failure of a detail repeated multiple times in that structure, where spatial correlations are accounted for. The method was demonstrated for the case of an orthotropic bridge deck.
Most of the previous studies are case specific which means the systems were defined and modelled following selected bridges. As those work shows, the actual probability of failure of a bridge can be very complex. It will lead to very specific conclusions depending on how the systems are combined on the selected bridge. An idealized series system is studied in this paper as a base case, which enables a parametric study on how spatial correlation influences the system reliability. "Failure" in this study is defined as the first presence of damage, which is based on the strategy adhered by the bridge managers in Sweden, that no presence of fatigue cracks is allowed in the load bearing structure. An efficient modelling approach is proposed for system reliability assessment of fatigue deterioration. It is based on the assumption that all details within the series system are equicorrelated and the correlation between two details can be used to estimate the reliability of the whole system analytically. Consequently, a comparable accuracy is gained with less computational time than with common simulation techniques. Even a rearrangement of the system or a parametric study about the number of details will not add extra calculation time. Besides, this approach provides a more accurate estimation compared to the traditional probability bounds method.
In Section 2, a brief introduction of system reliability methods is provided and based on them an efficient modelling approach is illustrated for a series system as supplement of common simulation techniques. In Section 3, the probabilistic model of the fatigue deterioration problem on steel structures is illustrated. In Section 4, the proposed modelling approach was applied on the Rautasjokk Bridge as a case study, quantifying the equicorrelation between details and evaluating the system reliability. Both Monte Carlo Simulation and Subset Simulation were applied for prior and posterior reliability assessment. Then a parametric study was performed, showing how the correlations influence the system reliability and identifying the decisive variables. All the results are discussed extensively in the same chapter. In Section 5, limitations of the current study have been discussed and in Section 6, conclusions are drawn from the case study.

System reliability
Structural systems or their subsystems may be idealized into two simple categories: series and parallel. In a series system, typified by a chain, and also called a "weakest link" system, attainment of any one component limit state constitutes failure of the structure (Melchers & Beck, 2018, p.139). If the failures of all components are needed to obtain the failure of the system, the system is said to be a "parallel" system (Sørensen, 2004, p.30).

System safety considering bridges
The Swedish Transport Administration (Trafikverket), has developed their own management system BaTMan to manage their stock of bridges (and tunnels). They collect and store various data from inspections and condition assessments about the bridges required to make decision about their maintenance at operational, tactical and strategic levels (Hallberg & Racutanu, 2007, p.628). Based on the bridge inspection, the physical and functional condition of both the structural members and the entire bridge is determined and a condition class (CC) is assigned by the bridge inspector. The CC spans from 0 to 3 as listed in Table 1 and describes to what extent structural members fulfill the functional requirements at the time of inspection. If a fatigue crack in one of the main load carrying members is detected, the bridge is counted as CC 3 and immediate action is required. The actions can for example consist of closure of the bridge, traffic limitations, extra maintenance, or repeated inspections. From that perspective, the appearance of a crack stipulates a failure of the required functionality of the bridge, even though it is not seen as a structural failure in reliability theory. Therefore a model without redundancy, a series system model, is motivated. Table 1. Condition classes (CC) system used in BaTMan (Trafikverket (2015)).

CC
Assessment Follow-up 3 Defective function Immediate action is needed 2 Defective function expected within 3yrs Action is needed within 3yrs 1 Defective function expected within 10yrs Action is needed within 10yrs 0 Defective function expected beyond 10yrs No action is needed within 10yrs In reality the actual probability of failure of a bridge is very complex and depends on how different types of systems are combined. It can be very specific when it comes to different bridges. This paper presents results on how spatial correlation influences the system reliability and an idealised system model has been applied as simplification. The definition of failure in the current study is limited as the first presence of damage, which is based on the requirement from the Swedish Transport Administration.

Bounds for system reliability
In a series system, two extreme conditions are considered as the bounds of the system failure probability. One is to consider that all details are fully correlated, which means that one detail's failure implies a failure of all details simultaneously. The other case is to consider that all details are independent. All details should function to keep the system function.
In real applications, those two extreme cases are relatively rare. Two similar details at different locations are usually partially correlated. Literature as Ditlevsen (1979) and Ditlevsen and Madsen (1996), considering the correlation coefficient q ij between failure events of different details i and j, suggested the Ditlevsen bounds method and estimated an upper and lower bounds on the possible probability of failure of a multi-details system P sys . Many studies have applied Ditlevsen bounds to investigate system failure (Al-Harthy and Frangopol (1994), Okasha and Frangopol (2010), Rezaei et al. (2017)).

Exact solution for equicorrelated details
Assuming the details being equicorrelated with the correlation coefficient q ij ¼ q and having the same reliability level b will lead to an exact solution of the system reliability. Ditlevsen (1983) considers the safety margin of the selected system to be the intersection of the m linear safety margins, corresponding to m details. Each individual safety margin is M i expressed as where X, X 1 , :::, X m are mutually independent standardized normally distributed random variables. Then the system safety margin is expressed as (Ditlevsen (1983)): Therefore, the probability of the system failure is expressed as: The reliability of this system could be measured by the parameter b, often called the "Hasofer-Lind reliability index", as follows: where U À1 ðÞ is the inverse of the standardized normal distribution function.

Equicorrelation-based modelling approach
For the simplified series system model, an efficient modelling approach is proposed. This modelling approach is based on the assumption that similar details at different locations could be grouped and those details within the same system have the equivalent correlation among each other. The value of the correlation is dependent on the correlations of underlying random variables contributing to the failure. This modelling approach facilitates the common simulation techniques to have a comparable accuracy with low time-consumption. To distinguish the difference, in the rest of the paper q v is applied to represent the correlation between underlying stochastic variables, while q eq represents the equivalent correlation between similar details within the same system. This method is mainly based on the derived equation Equation (3). The main procedure is as follows: Perform a prior reliability assessment of the known detail and a two-details series system considering the correlated stochastic variables. Then the prior reliability of the known details b and the probability of failure for the two-details series system P sys are obtained respectively from simulations. Apply Equation (3) inversely to derive the equicorrelation q eq between these two details. Since all details are assumed to be equicorrelated, q eq represents the correlation of any two details in the system. Use Equation (3) with the known values of b and q eq , the probability of failure for the m-details series system can be computed analytically instead of performing simulations.
The equicorrelation-based modelling approach can be flexibly applied to large systems but not necessarily increase the computation time. Besides the necessary simulations of the single detail and two-details system, the proposed approach estimates the reliability of the whole system analytically. It avoids to assess all details one by one, which reduces the simulation time dramatically. Even changing the number of details consisted in the system, the new estimation can be obtained without extra effort. And this feature also allows the proposed method deal with large systems with better accuracy than using bounds methods. The word choice "efficient" means that accurate estimations can be gained with little computational effort. The accuracy, however, depends on the model assumption.
An application of the proposed approach is presented in Section 4.2. To be clear and concise, the proposed equicorrelation-based modeling approach is abbreviated as ECM in the rest of the paper.

Probabilistic model considering fatigue
Many studies have been accomplished in fatigue reliability assessment of steel bridges. Wang, Chen, Chen, and Xu (2006) developed a single-angle probabilistic fracture failure model based on fracture mechanics principles investigating the behaviour of a riveted built-up girder. Imam, Righiniotis, and Chryssanthopoulos (2008) presented a probabilistic fatigue assessment methodology for riveted railway bridges. The S-N curves and the cumulative damage model were treated probabilistically for resistance estimation. Imam, Chryssanthopoulos, and Frangopol (2009) presented a system-based model for the fatigue assessment of old, deteriorating riveted railway bridge connections. Fatigue damage calculations were based on the theory of critical distances which considered the entire stress distribution ahead of any given stress concentration. Kwon and Frangopol (2010) evaluated the fatigue reliability assessment of steel bridges by using probability density functions of equivalent stress range based on field monitoring data combined with linear S-N curve. An overview of the development of fatigue design provisions and detailing for steel bridge structures was presented by Fisher and Roy (2011). Guo and Chen (2013) performed the fatigue reliability analysis of welded details of a 40-year old steel box-girder bridge, based on the linear elastic fracture mechanics (LEFM) and the long-term stress monitoring.
In this paper, a probabilistic fatigue model was established based on LEFM and Paris law for crack growth. Paris (1961) proposed a crack-growth model, which describes the relationship between cyclic crack growth and the stress intensity factor range: where a is the crack size, N is the number of stress cycles, A and n are empirical, material related parameters. The variable K r is the stress intensity factor range, depending on the stress range S, the crack size and the geometry of the considered detail. K r is expressed as: where YðaÞ is a geometry correction factor considering the geometry of the unwelded component, and M k ðaÞ is a stress magnification factor due to the weld geometry and evolving in crack shape (Hobbacher, 1992, pp.899-900).
For the analyses presented in this paper, the linear crack growth rate relation by Paris and Erdogan (1963) was considered as the basis. However, King, Stacey, and Sharp (1996) presented a review of existing data on fatigue crack growth rates in structural and engineering steel and proposed bi-linear constants that offers improved accuracy. The bi-linear crack growth rate relation has also been suggested in BSI (2013) and JCSS (2001) for its good performance and has been applied in many fatigue analysis of bridges (Righiniotis and Chryssanthopoulos (2004), Kwon, Frangopol, and Soliman (2012), Maljaars, Steenbergen, and Vrouwenvelder (2012), Maljaars and Vrouwenvelder (2014), Leander and Al-Emrani (2016)). It is expressed as: where K th is the crack growth threshold below which K r causes no crack growth. The transition point K ab is calculated as: The material parameters as well as the crack growth threshold are regarded as stochastic variables with uncertainty. Therefore the bi-linear crack growth rate in Equation (7) is presented with mean value and 95% confidence interval, as shown in Figure 1. The characteristic values of each stochastic variable considered can be found in Table 2.
Stress cycles accumulate over time since vehicles/passengers pass the bridge frequently. When the number of cycles increases, it is possible to induce the crack initiation and/or propagation phenomenon. An incremental numerical procedure is carried out to evaluate the equations above.
The procedure starts at an initial defect, expressed as the initial crack size a 0 , and ends at the critical crack size,  expressed as a cr : Therefore the number of cycles to reach a critical crack size is obtained. The vector x contains the stochastic variables considered in the probabilistic model. Leander and Al-Emrani (2016) has shown that the load sequence has a minor influence on the crack growth for typical bridge loading. This allows the simplification of the analysis by using an approximate solution suggested in Equation (10) based on the expectation of the power of the stress ranges.
where E ½ denotes the expected value and S ab corresponds to the value reaching the transition point of the stress intensity factor K ab :

Reliability assessment
A general limit state function (LSF) considering fatigue for one single detail is formulated on the number of cycles as: where N cr ðXÞ is described by Equation (9) and NðtÞ is the number of cycles at the studied point in time. For a series system, the LSF is modified as all details within the system do not fail: where i counts the number of details in this system. The probability of failure is defined as: As stated in Equation (4), the corresponding system reliability index b sys can be obtained.
For fatigue assessment of existing structures, a target value of the failure probability P f ¼ 10 À3 is suggested in the standard ISO 13822 (ISO, 2010) for single uninspectable details; while in combination with inspections, a value of P f ¼ 10 À2 is suggested. The corresponding values expressed in reliability index are b ¼ 3:1 and b ¼ 2:3 respectively. These values are stated for a reference period equal to the intended remaining service life. However, there is no such clear criteria in system reliability. Therefore, a simulationbased assessment is performed. In this paper, b ¼ 3:1 is selected as the target reliability level for single details and has been applied in the Section 4.2.1.

Detection event
If the theoretical assessment is based on a measured indication of damage, the prior estimation of the reliability can be updated considering results from inspection. The detection event can be established as: where aðx, N ri Þ is the estimated crack size at cycle N ri and a d is the lower bound limit of detectable crack size described by a probability of detection (PoD) curve. Then the estimated probability of failure can be updated assuming no crack is detected (H D 0) as:

PoD curve
The accuracy of the inspection is dependent on the PoD curve. Here the recommended curve issued by DNV GL (2015) was adopted due to its connection to specific nondestructive testing (NDT) techniques. It has been widely used in Offshore Technology, and proved to be robust under application. The selected PoD model facilitates the study to consider various inspection methods with the original reliability assessment. It also allows a consideration of different environment conditions for the structure. The curve is expressed as: where x indicates the lower bound of detectable crack length for visual inspection and crack depth for other methods. Parameters X 0 and X 1 are depending on the detection method and prevailing conditions of the detail. In this paper, visual inspection is selected due to its simplicity and low-cost in practise, though with low accuracy. Values for X 0 and X 1 for visual inspection are 15.78 and 1.079 respectively (DNV GL (2015)). Those numbers are valid for good conditions above water.

Description of the system
The Rautasjokk Bridge is located along the iron ore railway line in Northern Sweden. A photo of the bridge is shown in Figure 2. Its original abutments were built in 1902 and the present superstructure was built in 1962. The load carrying structure consists of two simply supported parallel steel trusses, with a span length of 33 meters. The single track is carried by stringer beams, which in turn spans between crossbeams. The main truss of the bridge is assembled using rivets but the secondary structure including the stringers and crossbeams have welded connections. The material in the stringer beams is denominated SS 1311 in the Swedish system with yield strength 240 MPa and ultimate strength 360 MPa. The Rautasjokk Bridge is selected as a case study because it is representative for many similar railway bridges in Sweden and globally. And it also showed strong indications of an exhausted fatigue life due to high load level from iron ore trains (H€ aggstr€ om (2015)).
The monitoring campaign of the Rautasjokk Bridge consists of six uni-axial strain gauges, shown in Figure 3(a). The monitoring started 25 October 2017 and the data evaluated in the paper comprised passages until 30 April 2018. A total of 4,029 train passages were registered during this period. A deterministic codes based assessment for single details has been performed by H€ aggstr€ om (2015), showing that the locations of gauges 101 and 102, located close to the gusset plate on the stringer beam, to be the most critical sections. Figure 3(b) shows the dimension of the gusset plate, which is prone to failure under the repeated load due to the passing trains. A crack is assumed to initiate at the end of the butt weld connecting the plate to the flange of the stringer beam, then propagate into the flange. This detail is categorized as detail 5 in Table 8.5 in Eurocode 3 Part 1-9 (CEN (2005)). For this detail, YðaÞ in Equation (6) is defined by Newman and Raju (1981) and M k ðaÞ is given by Leander, Ayg€ ul, and Norlin (2013).
There are eight similar details on the stringer beams selected and grouped since they have the same design and properties as shown in Figure 4. They are separated from the load bearing structure on the bridge and regarded as a subsystem. According to H€ aggstr€ om (2015), the stringers can be adequately modelled as simply supported between the crossbeams, which entails a similar response to passing trains for all eight joints. With a refined Finite Element model of the bridge, Lundman and Parn eus (2018) showed that the results from modelling the stringers with joints at the crossbeams matches the measured response better than modelling stringers as continuous. Considering these previous investigations, these eight details have been modelled as equicorrelated.
The reliability analyses were performed considering the probabilistic model in Section 3. The stochastic variables considered are listed in Table 2. The stress range spectrum is generally based on the traffic load measurement. The uncertainty in stress spectrum is considered by including variables C S and C SIF , which represent model uncertainties for the stress range and the stress intensity factor range respectively. C S represents the uncertainty related with how the stresses are measured. The monitoring campaign measures the strain which are further converted into stresses. It is the global uncertainty involved in the procedure. C SIF represents the uncertainty involved with how to calculate the stress intensity factor from the nominal stresses. It also includes the uncertainty in parameters YðaÞ and M k ðaÞ in Equation (6). Compared to C S , C SIF is the local uncertainty of the selected detail. JCSS (2002) recommends log-normal distributed uncertainty factors with an expected value of 1. The distributions of C S and C SIF are suggested in Leander et al. (2013). The crack propagation life is determined using distribution functions for the material related constants A and n. Since the scatter in n is very small for the same material, BS 7910:2013 suggests using fixed values for n 1 and n 2 and all scatter in crack growth is incorporated in the distributions of A 1 and A 2 in the bi-linear crack growth relation (BSI (2013)).  The crack dimension a covers propagation both in depth and in width. The crack propagation curve, in which crack size is plotted against the number of cycles, usually presents an exponential shape. It means that the number of cycles to reach a critical crack depth N cr is insensitive to the critical crack size a cr if it is sufficiently large. The crack will grow from half of the plate width through the whole width within a few cycles. Therefore, an uncertainty of the critical crack size does not influence the estimated service life, which has also been shown by Maljaars et al. (2012). Also Luki c and Cremona (2001) modelled the final crack size as deterministic and showed small differences in service life for different critical sizes. For the gusset plate detail, the critical crack size a cr is assumed to be deterministic as half of the plate width.
The subsystem is modelled as a series system with noredundancy and the motivation has been discussed in Section 2.1. One detail has been monitored and the reliability status of the monitored detail is expected to provide information about similar details at other locations. Therefore the spatial correlation of loading, model uncertainties and material parameters between the different locations were considered. The conditional correlated samples were generated in the normal space by applying Cholesky factorization firstly, then transformed to the log-normal space. The prior reliability of the system was evaluated by both Monte Carlo Simulation (MCS) and Subset Simulation (SS).

Equivalent correlation by Monte Carlo simulation
Monte Carlo Simulation (MCS) is an easy and direct simulation method to estimate the prior reliability of single details. However, its calculation time will dramatically increases while more details are included as a system. Therefore, the equicorrelation-based modelling approach (ECM) is applied supplementing MCS.
Here a particular example was performed with the correlation of model uncertainties is set to be 0.2, while material parameters are regarded fully correlated (q v ¼ 1) and initial crack sizes are regarded independent (q v ¼ 0). The sample size was selected preliminarily to be 10 5 to save analysis time.
Reliability assessment was firstly performed for the monitored detail and a two-detail series system and results are shown in Figure 5(a). It is observed that the prior reliability of the system will decrease when one extra correlated detail is considered. The equivalent correlation between these two details was reversely derived by Equation (3) and plotted as shown in Figure 5(b). However, it has noticeable variations along time with a relatively slow convergence speed.
The explanation is that the sample size is not large enough to provide an accurate result of q eq , especially when the cycle numbers are small and the probability of failure is low. Several sample sizes were tested for the same case and the corresponding equivalent correlation curves were plotted in Figure 6. In Figure 6(a) it is noticed that all cases ended up with a consistent convergence value and the case with sample size 10 6 provides a faster convergence in q eq compared to other cases. In Figure 6(b), the equivalent correlation is plotted against the number of samples in the failure region. It is unanimous for all cases that results came to convergence after having at least 5000 failure samples. In order to keep a good accuracy, sample size 10 6 was selected and applied for the subsequent simulations.
After the equivalent correlation q eq between two details was obtained, the multi-details system reliability can be estimated at low time-consumption. Three sets of estimations have been made respectively based on the purple curve with sample size 10 6 in Figure 6(a), by using: 1) the instant value of q eq corresponding to each cycle number, 2) the median value and 3) the mean value of q eq . The total number of details in the system is increased by one at each time, from three to eight, and the estimation results were compared with an explicit MCS. In this paper only the results of three and eight details system are plotted as Figure 7. When the number of details increases, correspondingly the system reliability becomes lower overall, implying that the series system become weaker when more details are included. All three analytical estimations show reassuring consistence with the simulation results, except the one using the instant value of q eq shows a small deviation at the beginning. It is due to that the q eq has not converged at small number of cycles. A more extreme case is considered by extending the number of details to 100. It is not the real case for the Raustasjokk Bridge, but it is possible to have such large system on other bridges, e.g. the S€ oderstr€ om Bridge studied by Leander et al. (2013). The proposed ECM is expected to be applicable in general. Here the Ditlevsen bounds method is applied under the assumption that every detail has the same reliability level. The Ditlevsen bounds can prove if the estimations by using ECM are within the reasonable ranges. As the Ditlevsen bounds formulated, the bounds are determined by the reliability of the monitored detail b and the correlation between two details q (replaced by q eq in the rest of paper). In this case, b is a time-variant parameter, while q eq is an unknown variable. Two different ways of applying the reliability bounds are presented.
In Figure 8(a), the reliability of the monitored detail is set as the target reliability b ¼ 3:1 and the equivalent correlation q eq varies within (0,1). The solid blue and red line show the bounds determined by the Ditlevsen bounds. The grey area is computed by exact integral (Ditlevsen and Madsen (1996)), which makes the bounds narrower. The exact solution is obtained by using Equation (3). After obtaining the converged correlation q eq ¼ 0:8 between details, its corresponding value on the "exact solution" curve is the system reliability value for 100 details system, which is b sys ¼ 2:16.
In Figure 8(b), the reliability of the monitored detail is timevariant and obtained from the prior reliability assessment. The correlation between two details was fixed to the converged value q eq ¼ 0:8 obtained in Figure 6. Similarly as in Figure  8(a), the bounds were determined by the Ditlevsen bounds. Three estimations were plotted by using the ECM, and they show reassuring consistency and are within the reliability bounds. When the target reliability b ¼ 3:1 for a single detail is fulfilled, the system reliability is obtained as b sys ¼ 2:15. The Ditlevsen bounds provide an estimation on the range of fatigue lives corresponding to the same reliability level. Table 3 summarizes the results presented in Figure 8. The results based on Figure 8(a) implies that when the single detail fulfills the target reliability, under the condition that q eq ¼ 0:8, the system reliability is estimated to be 2.16. However, the value of the fatigue life (the critical number of cycles) cannot be estimated through the procedure. For a comparable system reliability level, the Ditlevsen bounds provides a lower bound 0.42. But the upper bound is undetermined. The results based on Figure 8   more comprehensive. By using the instant value of the reliability level of the single detail, the corresponding system reliability is estimated. When the single detail's reliability falls to 3.1, the system reliability is 2.15, which corresponding to a fatigue life of 11:01 Ã 10 5 cycles. For a comparable reliability level, Ditlevsen bounds provide an estimation of fatigue life, ranging from 8:38 Ã 10 5 cycles to 17:23 Ã 10 5 cycles. 11:01 Ã 10 5 is indeed within the range. Both estimations of system reliability are consistent. Those two ways of using the Ditlevsen bounds complement each other and bring out much useful information about the system reliability. Comparing the bound methods and the equicorrelation-based approach, the latter one is recommended due to its time-efficiency and accurate results while the bounds provide a relatively wide range. Based on the example performed above, it is confirmed that ECM is able to estimate the system reliability when more similar details are considered and it provides an accurate result at low time-consumption.

Equivalent correlation by subset simulation
The Subset Simulation (SS) is an adaptive stochastic simulation procedure which calculates the product of conditional probabilities of several chosen intermediate failure events (Au and Beck (2001)). In Wang, Leander, and Karoumi (2019), MCS and SS were applied respectively for reliability assessment of a single detail. SS showed promising feasibility and high efficiency compared to MCS. Therefore, the SS has been applied for system reliability assessment and expected to have good performance. According to Wang et al. (2019), the sample number N s for each subset iteration can provide   Table 3. System reliability and bounds. a good accuracy when it is around one thousand or higher, and an empirical value of the conditional probability p 0 ¼ 0:1 is recommended. Here N s ¼ 1000 and p 0 ¼ 0:1 were applied. In Figure 9(a), the system reliability curves from SS are plotted with the MCS results. A good consistence is observed, however, with certain variation in the high reliability region. It unfortunately includes the target reliability for a single detail b ¼ 3:1, which is not desired in seeking for an accurate estimate. Three more simulations of system reliability assessment were performed as presented in Figure  9(b) to examine the robustness of SS. The deviations still exist, as Wang et al. (2019) claimed that the deviation exists due to inevitable randomness.
The system reliabilities from SS are estimated and summarized in Table 4. They are estimated at the critical cycle numbers when the target reliability of the single detail is fulfilled. The MCS result is regarded as an accurate result, and the deviations of SS are all within an acceptable range.
SS is feasible in reliability assessment of single details, however, it is not the best choice for the system reliability assessment. The main shortages are: Time-consuming When more details, i.e. m details, considered in the system, SS needs m Ã N s calls of the main calculation script in each subset iteration. Since SS solves the LSF with several iterations by generating seeds based on the previous iteration and gradually approaching the accurate result, it is not possible to shorten the calculation procedure. The calculation time for a system would be at least m times the calculation time for a single detail. Though MCS uses a large sample size as 10 6 , it is able to run the simulation in parallel. Therefore the calculation time for a system doesn't have much difference from solving only one detail.

Low accuracy
Due to the inevitable randomness, the SS curves have certain variations along the time. It becomes more difficult to find a converged equivalent correlation between two details. Therefore the further estimations with more details has relatively low accuracy compared to MCS and the proposed ECM cannot provide the best performance as supplement. It implies that the SS routine has to assess the whole system to maintain an acceptable accuracy, no matter how many details are included. In this aspect, it becomes less efficient.

Parametric study
In the previous sections of researching on equivalent correlation, a partial correlation q v ¼ 0:2 was considered for the model uncertainties. In this section, a parametric study is performed by varying the correlation value between different variables by MCS, investigating their individual impact. Five cases are listed in Table 5.
The model uncertainties for stress and stress intensity factor are considered as one joint uncertainty variable by multiplying them with each other. The joint uncertainty variable still follows a log-normal distribution, only changing the characteristic value in mean and standard deviation. For material parameters, A a and A b are categorized within one group, related with the bi-linear crack growth law; while K th is a different topic as near-threshold crack growth.
The equivalent correlation between two details are plotted case by case in Figure 10. The curve of the equivalent correlation is influenced by the correlations of different variables contributing to the failure. Case 1 mainly investigates the influence of the model uncertainties, while Case 2, 3 and 5 investigate how the correlation of initial crack size would influence the system, since the crack size is the only detectable and measurable variable among others. Case 4 investigates the influence of material parameters by setting A a and A b as partially correlated, however, the K th is regarded as fully correlated.
As shown in Figure 10(a), when the correlation between model uncertainties q v increases from 0 to 1, the equivalent correlation also increases correspondingly and steadily, with a converged value of q eq ranging from 0.77 to 0.95. In Case 2 as shown in Figure 10(b), when the correlation of model uncertainties are assumed independent and the correlation of initial crack size varies, the converged value of q eq is more concentrated, and a modest increase is observed from 0.77 to 0.81. Both cases have relatively big numbers in equicorrelation which implies that the two details have a quite high correlation. It is due to the fully correlated material parameters. In Case 3 as shown in Figure 10(c), the model uncertainties are assumed fully correlated, with only the correlation of initial crack size varying. The curve of "fully correlated" case was omitted in the figure. Because when every variable is fully correlated, the two details are fully correlated, which means q eq ¼ 1. The converged values of q eq in Case 3 are as concentrated as Case 2, with a higher convergence value starting from 0.975. Different from Case 1, which presents that increasing the correlation of model uncertainties induces spreading and steadily increasing curves, Case 2 and Case 3 both show concentrated curve groups, which implies that the variation in the correlation of initial crack size doesn't influence q eq as much as model uncertainties. The difference between Case 2 and Case 3 is the starting value of the q eq . Case 3 has an overall higher value in q eq because both model uncertainties and material parameters are regarded as fully correlated, which raises up the values of q eq : The curves in Case 4 show a similar trend as Case 1, spreading and increasing; however with a lower start value of q eq and a bigger increase gap in q eq from each increase in the correlation of material parameters. It is clear that material parameters are the most dominant variables among others.
Case 5 is abnormal compared to others, which didn't show any sign of correlation. The possible explanation is: when other variables are independent, their influence on the system is dominating and the influence of the initial crack size is too tiny to be noticed. To prove this explanation, the system reliability curve of Case 5 with a selected correlation value q v ¼ 0:2 is plotted against Case 2 as well as the independent case as shown in Figure 11. It is observed that the system reliability curve of Case 5 is lower than Case 2, and it overlaps with the totally independent case. Therefore it is proved that in Case 5 the correlation of initial crack size doesn't have much influence on the system and could be simply regarded as independent.
In Figure 12, the system reliability is estimated for the Case 1-4 at the critical number of cycles when the single detail target reliability b ¼ 3:1 is fulfilled. For Case 1, the increase in correlation of model uncertainties also induces a steady increase in system reliability, while for Case 2 the increase in correlation of initial crack size doesn't have much impact on the system reliability. This phenomena corresponds to the sensitivity test from Leander and Al-Emrani (2016), showing that model uncertainties of stress (C S ) and stress intensity factor (C SIF ) have a significant influence on the reliability, while the uncertainty of initial crack size (a 0 ) has only a modest influence on the reliability. In Case 3, the system reliabilities have an overall higher value than Case 2, which is due to the fully correlated model uncertainties. The curve gradually approaches the value b sys ¼ 3:1, which corresponds to the target reliability of a single detail, which is also the system reliability when all details are fully correlated. The influence of the initial crack size could be observed only when other variables are fully correlated. In Case 4, the b sys shows a relatively low value compared to other cases at the beginning, because the q eq is the lowest as showed in Figure 10(d). Later the b sys has a sharper raise if compared to Case 1, since the correlation of material parameters increases and it has a larger impact on the system reliability.
It is concluded that material parameters have the most dominating influence on the system, followed by the model uncertainties. The variations in the correlation of material parameters and model uncertainties give an obvious change in the equivalent correlation curve, as well as the system reliability curve. However, the influence from the correlation of initial crack size is very little.

Updated reliability
Correlation is especially relevant when inspections are considered in the reliability assessment (Vrouwenvelder (2004)). Moan and Song (2000) investigated the effect of inspection on the reliability of both inspected and uninspected joints dependent upon the correlation of properties of different joints. Maljaars and Vrouwenvelder (2014) presented the influence of various inspection strategies on the reliability of the bridge deck structure when spatial correlation is considered. When an inspection is performed of the monitored detail, with the assumption of no crack is detected, a significant increase in the fatigue life is expected. If two similar details are correlated with each other, the updating of the monitored detail is expected to provide some information about the second detail, and probably also increases the fatigue life of the second detail.
Here only Case 1 and Case 2 were researched. An inspection was performed on the monitored detail first. Then the   correlation varying from 0 to 1 with an increment of 0.1 was considered for the selected samples both satisfying gðx, tÞ 0 and H D ðxÞ 0. As shown in Figure 13, the updated reliability curves of the second detail are plotted as a group of dashed grey lines. The second detail has an increase in its fatigue life in both cases. It confirms that with the correlation between details considered, the monitored detail will provide information about other similar details and an increase in their reliabilities is expected as well.
In Case 1, the grey lines show a trend as moving upwards gradually. It implies that the higher the correlation of model uncertainties is, the more increase the reliability level gets. In Case 2, varying the correlation of the initial crack size doesn't give much influence on the updating. The grey lines are more concentrated and couldn't be distinguished from each other. Similar with the results in Section 4.3, the correlation of model uncertainties has more dominating impact on the reliability updating, compared to the correlation of the initial crack size. Even when the model uncertainties and the initial crack size are both assumed to be independent, the reliability of the second detail gets updated (referred to the lowest grey line). The result is a consequence of the fully correlated material parameters.

Discussions
In this paper an idealised series model has been studied and all details in the system are assumed to be equicorrelated. If other studies use this simplified model for a preliminary assessment, or as a basic component of a more complex system, the conclusions drawn from the current study could be applicable and helpful. There are two directions of further work that the simplified model could be improved. One is to increase the complexity of the system while the other one is to consider different correlations among details.
It is of great interest to explore how complicated systems can be expanded considering spatial correlation. The model can be gradually improved from a series system to a basic series-parallel system, or even a more realistic mixed system according to the selected structures. Besides, for those different system settings, it is also important to figure out to which extent the spatial correlation will have non-negligible influence on the system. All the details are assumed to have equal correlation among each other in this paper. As explained in Section 4.1 it is an acceptable assumption for the selected system based on previous investigations about loading. Besides the Rautasjokk Bridge hasn't been maintained or repaired, which implies that the material parameters should have the same correlation. However, the equicorrelation condition is not necessarily applicable for other bridges, especially the structures which have been subjected to repairs and other maintenance actions. After these actions, the equicorrelation condition will normally be broken. To consider different correlations among details requires advanced simulation techniques. Methods which allow details to be combined one at a time offer the possibility for various correlations, however, the difficulty to suggest realistic correlations still exists.

Conclusions
Spatial correlation between several fatigue prone details has been considered in a reliability assessment of an isolated part of the load bearing system of a steel bridge. The study is carried out by idealising the isolated part into a series system. Besides, the comprehensive case study is valid under the assumptions that details within the system are equicorrelated with each other and have the same individual reliability level.
Within the scope of the idealised series system model and the assumption that all details are equicorrelated, the following conclusions can be drawn: By considering spatial correlation among similar details at different locations, it is possible to estimate the prior system reliability based on the information of the single monitored detail. An efficient modelling approach ECM was outlined and it shows high efficiency and accuracy in system reliability estimation as supplement to common simulation techniques. It is able to deal with large structural systems without inducing extra calculation time. The approach has been validated by using crude Monte Carlo simulations and results are consistent. Compared to SS, MCS shows better feasibility when integrating with ECM. In the parametric study, the correlations of material parameters have the most dominating influence on the system reliability, followed by the model uncertainties. Increasing the correlations of material parameters or model uncertainties will induce a higher value of the equivalent correlation between two details, as well as an increase in system reliability. The influence of the initial crack size could be noticed only when all other variables are fully correlated. Crack size is the only measurable parameters among all stochastic variables. However, no significant gain can be expected from the actual crack size at other locations even with high spatial correlations. Combined with inspection, the inspected detail is able to provide information about other similar details depending on the level of correlation. Others will obtain an increase in the reliability level.
As illustrated in Section 5, there are possibilities to further expand the study about system reliability of steel bridges, both in system modelling and in values of correlations. Besides, up to now the research on the Rautasjokk Bridge is limited to the reliability level. A risk-based assessment of the bridge is the next step by considering possible failure scenarios. A framework will be further developed from those outcomes.