The practical application of reliability assessments for large infrastruc- ture, such as steel railway truss bridges, is impeded by many factors. These factors include the high dimensionality of the problem, the difficulty of inte- gratingnon-linearfiniteelementanalysis, andthegeneralneedformoreinfor- mation related to the probabilistic assessment of steel railway truss bridges. Additionally, most studies on steel railway truss bridges generally overlook performance at the ultimate limit state under combined actions and focus on fatigue assessment. Therefore, in this study, we propose a scalable and gener- alized framework for the probabilistic assessment of steel railway bridges us- ingadjustedpartialandcombinationfactors. Theframeworkaddressesmany practical challenges associated with the probabilistic modelling for steel rail- way truss bridges, such as load and resistance uncertainties, horizontal forces such as braking and nosing, wind loads and railway line operator decisions, ∗Corresponding author Email addresses: shihab.khan@monash.edu (Mohammad Shihabuddin Khan), colin.caprani@monash.edu (Colin Caprani), roman.lenner@fsv.cvut.cz (Roman Lenner), miroslav.sykora@cvut.cz (Miroslav Sýkora) 1 Preprint submitted to Reliability Engineering & System Safety October 31, 2023 load effect combinations and correlations, and factor adjustment. The frame- work is demonstrated for assessing two longitudinal girders of a single-span 60 m steel railway truss bridge. Relevant input data and results are provided as a dataset for complete ease of reproducibility of the results. Notably, it is found that nosing loads are critical for the girders due to the induced large minor axis bending moments. Furthermore, some potential ways of reducing the dimensionality of the resulting factors are explored, which significantly impacts the resulting design’s efficiency. Keywords: reliability assessment, code calibration, LRFD calibration, railway bridge, steel truss bridge, partial safety factors 1. Introduction 1 The performance inadequacy of degrading infrastructure is a widely rec- 2 ognized problem that affects infrastructure globally. ASCE [1] noted that 3 approximately 46 000 (or 7.5%) of all American bridges are structurally defi- 4 cient,withabridgerepairbacklogestimatesexceeding$125billion. Similarly, 5 Bień et al. [2] report that close to 35% of all European railway bridge stock is 6 over 100 years old. Furthermore, around 36% of all European railway bridges 7 are composed of steel [2]. These degrading steel railway bridges are subject 8 to higher traffic volumes, travelling more frequently and at higher speeds. 9 Therefore, an accurate safety assessment of these deficient railway bridges is 10 urgently required. 11 Typically,engineersadoptsemi-probabilisticassessmentmethodsforbridges 12 [3, 4]. However, these methods fail to capture the full range of underlying un- 13 certainties within the load and resistance variables. Therefore, probabilistic 14 2 approaches are increasingly favoured, as they can help uncover the reserve 15 structure capacity of built infrastructure stemming from a conservative de- 16 sign process [5, 6, 7]. 17 Various studies have noted the benefits of reliability assessment for as- 18 sessing old railway bridges [8]. Casas et al. [9] recommends an “enhanced 19 assessment” including non-linear finite element analysis (FEA) coupled with 20 reliability assessment as the ideal assessment strategy. Practically, “enhanced 21 assessments” for steel railway truss bridges are impeded by three major fac- 22 tors. 23 Firstly, most studies overlook the performance of the bridges at the ulti- 24 mate limit state (ULS) and focus on fatigue performance [10, 11, 12]. A few 25 studies have considered ULS though: O’Connor et al. [13] conducted a sim- 26 plified reliability of riveted connections of a steel bridge with a single variable 27 load using a generic probabilistic model; Nadolski et al. [8] presented a sim- 28 plified reliability analysis of the bridge with just a single variable load, and; 29 Žitný et al. [14] analyzed the reliability of the steel truss bridge at the ULS 30 of equilibrium. However, the findings of these works are not general, and 31 cannot be considered where uncertainties such as wind-on-train load, resis- 32 tance characteristics, resistance model uncertainty, dynamic and horizontal 33 effects of traffic load and thermal load effects are significant. Therefore, the 34 performance of steel girders under combined actions, such as train, braking 35 and nosing forces, requires more investigation. 36 Secondly, the integration of non-linear FEA and reliability assessment 37 models is impeded by the scale of the assessment problem in three aspects: 38 (i) the large number of structural elements constituting the truss bridge; (ii) 39 3 multiple simultaneously acting load cases and consequent effects, and; (iii) 40 veryhighcomputationalcostsofFEAandreliabilityassessment. Whilesome 41 studies have explored metamodelling, and simplified methods for assessment 42 [15], their applicability and accuracy are limited by the numerous simplifying 43 modelling assumptions. 44 Finally, while Bień et al. [2] and Casas et al. [9] provide guidelines for the 45 probabilistic assessment of railway bridges, many essential aspects specific to 46 steel railway truss bridges lack detailed exposition. In particular, the model 47 errors, uncertainties within horizontal loads, and load effect correlations and 48 combinations have been little considered. 49 Toaddresstheaforementionedproblems,inthisstudy,theauthorspresent 50 a scalable and generalized framework for the probabilistic assessment of steel 51 railway bridges using adjusted partial and combination factors. This frame- 52 work leverages the strengths of linear-elastic analysis methods and reliabil- 53 ity assessments to develop adjusted safety factors for application in semi- 54 probabilistic non-linear structural analysis assessments. Relevant input data 55 and results are provided as a dataset for complete ease of reproducibility of 56 the results [16]. The study presents the framework in Section 2. Section 3 57 discusses some modelling challenges, complexities, and potential solutions. 58 The study applies the framework on an existing steel railway bridge in Sec- 59 tion 4. Section 5 concludes with some remarks and recommendations for 60 further research. 61 4 2. Framework for reliability assessment of steel railway truss bridges 62 2.1. Overview 63 The overall flowchart of the proposed framework is illustrated in Figure 1. 64 The methodology is divided into four stages with six mutually cooperating 65 analysis ‘modules’, as explained below. 66 2.2. Stage I: initial analysis and database generation 67 This stage is used to reduce the scale of the assessment problem while 68 maintaining the accuracy of the overall assessment framework. The objec- 69 tives of this stage are to: (i) identify a set of critical members for probabilis- 70 tic assessment; (ii) generate a database of load effects acting on the critical 71 members, and; (iii) generate a database of relevant structural and material 72 properties for all critical members. A linear elastic analysis model helps 73 achieve all three objectives of this stage, although in some cases, a dedicated 74 structural properties model may be required. 75 Linear elastic 3D finite element (FE) frame models can analyze truss 76 bridges more quickly than non-linear FE shell models, to estimate the load 77 effect. The relevant design code provisions specify the magnitude and ap- 78 plication of time-invariant and time-varying loads, for example, in Australia 79 thisis[17]. Themaximumvaryingloadeffect, suchasthatforwindontrains, 80 horizontal nosing, and vertical train loads, can be estimated using influence 81 lines generated from the analysis model. Furthermore, such frame mod- 82 els facilitate investigating alternative load paths stemming from structural 83 degradation. Based on the resulting load effects, the most critical structural 84 members are identified for a detailed probabilistic assessment. 85 5 Figure 1: A conceptual illustration of the probabilistic assessment framework. 6 Furthermore, the critical elements’ section properties, such as plastic sec- 86 tion modulus, can also be generated. These section properties are required 87 for estimating the resistance of the structural member quickly as per a struc- 88 tural design standard. They should also account for any reported structural 89 degradation, such as corrosion. 90 2.3. Stage II: uncertainty quantification 91 The objectives of this stage are to: (i) establish the statistical prop- 92 erties of the resistance and load parameters; (ii) incorporate measurement 93 data/expert opinion for updating the statistical properties, and; (iii) estab- 94 lish the relevant limit states and load combinations for reliability analysis. 95 An essential source for uncertainty quantification is the structural design 96 code. It typically specifies the following relevant information: (i) the return 97 period of load actions for the nominal value [18]; (ii) the performance re- 98 quirements on the sections for identifying the relevant limit state(s), and; 99 (iii) the interrelationship between various load effects for load combinations 100 and correlations. 101 2.3.1. Statistical properties 102 Generally, the parameters of a statistical distribution can be established 103 by specifying the Xkis the nominal or characteristic value (e.g. the 95- 104 percentile or loading for a return period specified by the design code), bias 105 (λ), and Coefficient of Variation ( CoV), as: 106 λ=µX Xk, (1) 7 and 107 CoV =σX µX, (2) where µis the mean, and σis the standard deviation of the distribution. 108 The design code, site-specific information, and other technical guidance 109 can be used to establish these parameters [19, 20, 21]. The maximum load 110 effects from the analysis model can be taken as the nominal value. Simi- 111 larly, the section properties database can be used with design code and other 112 specifications to establish the nominal value of the resistance variables. 113 2.3.2. Site-specific information 114 Bayesian statistics provides a formal mathematical framework to update 115 statistical distributions of prior information by utilizing site-specific infor- 116 mation [22]. Such information can be available through limited in-situ mea- 117 surements (such as historical traffic load data, material properties, wind drag 118 coefficient, and structural degradation) and expert opinion. This information 119 acquisition can incur high costs and, therefore, it should be justified by using 120 Bayesian preposterior analysis to quantify the potential monetary benefits of 121 such information [23]. 122 2.3.3. Load combinations for reliability analyses 123 The probability of simultaneous occurrence of multiple extreme loads is 124 very low, thereby necessitating proper probaility modelling of load combina- 125 tions. Keeping the focus on stationary situations and ultimate limit states 126 only, a few studies and documents [21, 24, 25, 26, 27] applied: 127 1. the Ferry Borges-Castanheta (FBC) processes — sequences of indepen- 128 8 dent, identically distributed random variables, each acting over a given 129 (deterministic) time interval; 130 2. various types of intermittent or non-intermittent renewal process, and; 131 3. outcrossing approach for random continuous process. 132 An approximate method is Turkstra’s rule [28], which considers a dom- 133 inant action at its maximum at time tand non-dominant action(s) at their 134 expected point-in-time value. When the chosen reference basis is annual and 135 daily maximum, it can be generalized for any number of load effects Sias: 136 Smax= max i" max yearSi+X j̸=imax daySj# (3) In comparison to the aforementioned higher-level approaches, Turkstra’s rule 137 has been found to provide reasonably accurate in many situations [29] and 138 has been thus adopted as the basis for load combination rules in codes of 139 practice. Its application requires fewer input data, such as information about 140 distributions for short reference periods, seasonal variability of climatic and 141 traffic loads, and dependencies between loads. Nevertheless, more detailed 142 load combination analysis may be needed in case of several equally critical 143 variable load effects, large number of loads, large failure probabilities (e.g. 144 for serviceability), and strongly non-stationary cases. 145 2.3.4. Maximum and parent Distribution Statistics 146 In many cases, explicit statistical data for both, parent and maximum 147 distributions, maynotbeavailable. Undersuchcircumstances, extremevalue 148 theory can be utilized for inferring the statistics of either the parent or the 149 9 maximum distribution. For some distribution families, such as the Gumbel, 150 the stability postulate can be used to determine the parameters of the parent 151 distribution analytically from its ( n-maximum) using linear transformation 152 [30]. 153 2.4. Stage III: adjustment and calibration of factors 154 The primary objectives of this stage are: (i) adjustment of the partial 155 safety and combination factors relative to their design code stipulated values 156 based on a chosen target reliability index; (ii) calibration or optimization 157 of the factors for a family or population of structural members, and; (iii) 158 check of calibrated factors using a forward reliability analysis of calibrated 159 factor-based design. These calibrated factors are then employed in a final 160 performance check of the section. 161 2.4.1. Factor adjustment 162 Factors can be adjusted using expert judgement, design value method, or 163 reliability optimization [31]. Factor adjustment using optimization involves 164 adjusting the design parameter zin the limit state function (LSF) such that 165 the probability of failure corresponds to that of the target reliability index 166 (βT) and identifying the most probable failure point, i.e. the design point 167 [32, 33]. For example, consider the LSF G(X): 168 G(X) =zR−Q1−Q2, (4) where zis a scalar design parameter, Ris the resistance variable, and Qiare 169 the transient loads. 170 10 This problem considers two load cases. In first, Q1has the maximum 171 distribution for a chosen reference period (e.g. annual maximum), and Q2172 has a point-in-time distribution for a chosen basis (e.g. daily maximum). In 173 second, the converse situation exists. Therefore, once calibrated, z=z∗such 174 that: 175 −Φ−1[P[G(z∗)≤0]] = βT (5) The design point at βTfor the two load cases can be represented as 176 X∗ βT= z1R∗ 1S∗ 11S∗ 12 z2R∗ 2S∗ 21S∗ 22  (6) The objective of a typical semi-probabilistic design is to ensure that the 177 design resistance Rdis greater than the design loads Sd,i, i.e. 178 Rd≥X iSd,i (7) where the design load and resistance variables are given by, 179 Sd,i=ψiγiSk,i Rd=z ϕ R k(8) where Sk,iis the characteristic value of the ith load, Rkis the characteristic 180 value of the resistance, ϕis the resistance factor, ψiare the combination fac- 181 tors, and γiare the load factors. Therefore, by comparing the corresponding 182 terms in Equations (6) to (8), the estimates of ϕ,ψ, and γadjusted to βT183 can be obtained. 184 11 2.4.2. Factor calibration 185 The aforementioned method of estimating factors yields unique values 186 of partial load factors for the time-varying load effects, and estimates of 187 partial load factors for time-invariant load effects, partial resistance factors, 188 and combination factors for time-varying load effects differing per load case. 189 In cases where the number of critical sections under consideration is large, 190 this will result in a very high volume of data generation, thus complicating 191 assessment for the asset manager. Therefore, these adjusted partial factors 192 can be further calibrated (or optimized) for all or a family of sections to 193 obtain an optimized set of partial factors [8]. 194 2.4.3. Design check 195 Such calibration of partial factors and the design check are conducted 196 iteratively, as shown in Figure 1. The design check uses the adjusted (or cal- 197 ibrated) partial and combination factors to estimate the corresponding scalar 198 design parameter zusing Equations (7) and (8). Then, based on the esti- 199 mated z, a forward reliability analysis is conducted to estimate the achieved 200 (design) reliability. This process mimics the design process and, therefore, 201 is termed a design check. For any valid set of partial and combination fac- 202 tors for an assessment problem, for the assessment of a single structure, the 203 obtained reliability must be greater than βTin all load combinations. Any 204 conservative enveloping of adjusted factors (e.g., taking the maximum factor 205 for a specific load effect type across a population of sections) would increase 206 the design reliabilities greater than βT. 207 12 2.5. Stage IV: final performance check 208 Using the adjusted and calibrated partial factors, the calibrated load ef- 209 fects acting at a section can be estimated. These calibrated load effects can 210 be applied to a non-linear analysis model of the section, and correspond- 211 ing resistance at failure can be estimated. Therefore, utilizing Equation (7), 212 the final performance check of the section using the adjusted and calibrated 213 partial and combination factors at βTis obtained. Such a targeted prob- 214 abilistic assessment helps the asset owner avoid unnecessary conservatism 215 stemming from the design code stipulated partial and combination factors, 216 while retaining the simplicity of a rating factor check. 217 3. Modelling and assessment challenges and considerations for steel 218 railway truss bridges 219 The framework proposed in Section 2 and Figure 1 presents numerous 220 modelling considerations, complexities, and challenges for the asset owner 221 which are discussed below. 222 3.1. Load effect and resistance uncertainties 223 3.1.1. Model uncertainties 224 In scientific models, some uncertainties are reduced by repeated obser- 225 vations, i.e. epistemic uncertainty, and others remain due to the stochastic 226 nature of the phenomenon, i.e. aleatory, [34]. Repeated observations indi- 227 cate less uncertainty and, hence, a narrower probability distribution. These 228 uncertainties cannot be quantified separately in many practical structural en- 229 gineering scenarios. Under such circumstances, uncertainties characterizing 230 13 the FE model ( ωSf) and the analytical model of structural “strength” ( ωSm) 231 can be considered. 232 Consider a steel section under the combined action of major axis bending 233 moment Mz, minor axis bending moment My, and axial force N. The ULS 234 considering structural uncertainties can be written as, 235 G= 1−ωSmωSfMz ωRMzR+ωSfMy ωRMyR+ωSfN ωRNR (9) where MzRis the nominal section major bending capacity, MyRis the nom- 236 inal section minor bending capacity, and NRis the axial strength of the 237 section. ωSmaccounts for the structural strength model uncertainty under 238 the combined actions, i.e. the failure surface. The phenomenon’s complexity 239 results in various approaches in engineering practice. For example, EN 1993 240 1-1 [35] provides two alternative methods for evaluating the performance 241 under combined actions. Similarly, AS 4100 [36], AS 5100.6 [37] specify a 242 power function with reduced section bending capacities for compact doubly 243 symmetric sections. Therefore, ωSmrepresents a significant uncertainty that 244 must be incorporated within the LSF. 245 The uncertainty within the FEA model, ωSf, stems from various sources. 246 Firstly, any simplifications of the numerical model describing the develop- 247 mentofinternalforces, suchastheidealizationofsupports, spatialvariability 248 considerations, and composite actions of structural members, add uncertain- 249 ties. Secondly, there exist uncertainties particularly relevant for steel railway 250 bridges, such as the behaviour of joints, supports, initial imperfections, be- 251 haviour of material, section and members under different load levels and 252 14 related changes in stiffness, and load redistribution. Material and physical 253 non-linearity can significantly affect load redistribution and resulting load 254 effects on steel structures. The stiffness of joints or the stiffness of secondary 255 elements (such as bracing) in the actual structure can deviate from that as- 256 sumedinthestructuralanalysismodel. Thirdly, thenumericalalgorithmsfor 257 FEA solutions can impart errors. Finally, the uncertainty can vary per load 258 effect magnitude. For example, bolted connections work linearly at low load 259 levels while at higher load levels slips and deformations increase uncertainty 260 [8]. 261 Incaseswheretheseuncertaintiescannotbedefinedexplicitly, acombined 262 uncertainty model can be considered that accounts for both ωSfandωSmas, 263 G=ωR−ωSMz MzR+My MyR+N NR (10) 3.1.2. Statistical models for load uncertainties 264 There exist few studies that guide modelling load uncertainties. In par- 265 ticular, explicit statistical models for ωSmare not readily available. Cajot 266 et al. [38] indicates a mean ranging between 1.28−1.58and a CoVof0.10, 267 approximately, for the EN 1993 strength model. 268 Some existing studies offer guidance on modelling ωSf. JCSS [39] offers 269 models for the load effect calculation and resistance models for steel and con- 270 crete. These models are enhanced by Braml et al. [40] considering work done 271 by Faber [41] and Hansen [42] leading to adjusted CoVfor the resistance. 272 Scholten et al. [43] lists case-specific uncertainties for variable loading based 273 on the confidence level in modelling of a structural system, geometric proper- 274 15 Table 1: Statistical models for load uncertainty, ωS. Type Distribution Mean CoV Ref ωSmfor EN 1993 1-1 [35] — 1.28-1.58 0.10 [38] ωSffor axial forces Lognormal 1.0 0.05 [39] ωSffor beam bending Lognormal 1.0 0.10 [39] ωSffor plate bending Lognormal 1.0 0.20 [39] ωSf — 1.00.10- 0.20[43] ωSffor permanent loads — 1.0 0.07 [45] ωSffor traffic loads — 1.0 0.10 [45] ties and crossing mode. James [44] discusses model uncertainties for various 275 limit states of railway concrete bridges, and notes that for the train loading, 276 model uncertainty is already included in the uncertainty of the static load; 277 hence omitted in the reliability calculations. 278 Consideringastandardnumericalfiniteelementmodel,modeluncertainty 279 statistics are summarized in Table 1. JCSS [39] specifies models for load 280 effects on beams and plates. Danish practice [43] specifies models based 281 on low/high load confidence level. These values partially align with the 282 recommendation of Steenbergen and Vrouwenvelder [45]. 283 3.1.3. Statistical models for resistance uncertainties 284 For members in compression, the resistance models address the complex 285 situation where [46]: 286 1. Depending on the shape of a cross-section, steel members are suscepti- 287 ble to torsional deformations; 288 2. One buckling mode dominates – commonly, spatial buckling deforma- 289 16 Table 2: Statistical models for resistance uncertainty, ωR. Type Distribution Mean Bias CoV Ref Bending and shear Lognormal 1.0 — 0.05 [39] Axial compression — — 1.15 0.10 [47] Bending — — 1.10-1.15 0.08-0.10 [47] Bending — 1.1-1.3 — 0.04-0.07 [38] Axial — 1.1 — 0.045 [38] tion of such members is divided into the two buckling modes about the 290 strong and weak axis, and; 291 3. End and intermediate lateral restraints affect buckling lengths. 292 Design code stipulations suggest that member compactness has a crucial 293 bearing on estimated structural capacity [36, 35]. The differing theoretical 294 models suggest difficulty of capturing the various mechanisms contributing 295 to the overall resistance of steel members. Further, statistical information 296 on related resistance model uncertainties, ωR, is scarce. These uncertainties 297 will depend upon the dominating failure mode. 298 Table 2 summarizes some statistical models for steel resistance uncer- 299 tainties from different sources. Note that applying a non-linear FE method 300 will reduce resistance model uncertainty in cases when a dominating failure 301 mechanism is difficult to identify. Therefore, reliability-adjusted and cali- 302 brated safety and combination factors will significantly benefit a non-linear 303 FE analysis. 304 3.2. Horizontal force modelling and uncertainties 305 Braking and nosing forces, acting horizontally at the top of the rails, are 306 often important in reliability verifications of railway bridges. Their effects 307 17 depend on the speed and type of the train, track curves, and geometry of 308 the bridge deck. Generally, no dynamic amplification is considered. Braking 309 forces act in the longitudinal direction along the track. They can be con- 310 sidered as uniformly distributed over the corresponding influence length. In 311 contrast, nosing forces are concentrated, acting perpendicular to the centre 312 line of a straight or curved track. 313 The major considerations associated with these horizontal forces for re- 314 liability assessment of railway bridges are: (i) scarce statistical models for 315 horizontal force uncertainties; (ii) low frequency of accidental braking events; 316 (iii) potential high conservatism in load models for assessment, and; (iv) their 317 interrelationships and correlations. 318 Detailed statistical models for braking and nosing loads are scarce. Mea- 319 surements covering the effects of representative types of trains can substan- 320 tially decrease design horizontal forces for purposes of assessment. 321 3.2.1. Braking force 322 A comparison of the Eurocode, UIC, and several national braking force 323 models (Austrian, Czech, German, Swedish, and Swiss) with in-situ mea- 324 surements indicates general overestimation of load effects by the code models 325 [48]. Furthermore, braking at high speeds (“accidental braking” in some un- 326 expected extreme situations) on bridges is rare. Czech railway operator data 327 analysis reveals that about one accident in 10 years can occur in one kilome- 328 tre of a track [49]. Additionally, the frequency of such events on bridges is 329 much lower than that in open terrain because accidental braking is usually 330 caused by some obstacles on the track like fallen trees. Therefore, the low 331 18 measured forces and low accidental braking frequency on a bridge suggest a 332 low mean braking force relative to the design code load effect. 333 There needs to be more information on the uncertainties within the brak- 334 ing load models of various design codes such as EN 1991-2 [50] and AS 5100.2 335 [17]. Without detailed statistical models, the uncertainties within braking 336 force estimates can be considered the same as those for vertical train loads 337 due to their close phenomenological relationship. Note that traction forces 338 are generally ignored in reliability analyses as they commonly lead to lower 339 load effects than braking forces [49]. 340 3.2.2. Nosing force 341 While the general ambiguity behind the exact phenomenological basis of 342 nosing forces indicates its consideration as a notional load representative of 343 various phenomena, these forces are usually associated with the rail hunting 344 or lateral track shifting forces exerted by each train wheelset on the rails 345 and ballast. AS 7508 [51] suggests that they result from the unbalanced 346 lateral acceleration and can be especially significant on curves with high 347 cant deficiency at high speeds. 348 UIC Code: 776-1 [52], which forms the basis of most EN 1991-2 [50] 349 provisions for railway bridges, recommends a characteristic value of 100 kN 350 for nosing loads. On the other hand, AS 5100.2 [17] also recommends 100 kN 351 but with provisions to scale the characteristic value in proportion with the 352 specified design traffic load. AS 7508 [51] recommends a maximum value 353 of lateral track shifting forces ( SLTSF, max) as a function of the vertical axle 354 19 loads as: 355 SLTSF, max = 0.85 10 +A 3 (11) where Ais the vertical axle load in kN. 356 These design values seem conservative since the railway operators do not 357 experience many related failures. Georgiev et al. [53] indicates that the Eu- 358 rocode nosing forces are overestimated by a factor of 4 times for freight trains 359 at speeds up to 80 km/h. Additionally, passenger trains induce even lower 360 nosing forces. Furthermore, AS 7508.1 [54] suggests an inversely correlated 361 relationship between vertical axle loads and horizontal nosing loads due to 362 increased frictional resistance for heavier axles. Therefore, measurements of 363 the effects of representative trains can substantiate decreasing design nosing 364 forces, particularly for railway bridge assessment. Detailed statistical models 365 for nosing loads are unavailable at present. Without any established statis- 366 tical model for nosing loads, they can be considered the same as that for 367 vertical train loads. 368 3.3. Wind load effect and operator decisions 369 3.3.1. Operator decisions 370 The probability of simultaneous strong wind and heavy trains crossing 371 the bridge can be estimated from the databases for wind speeds and traffic 372 flows. However, railway-line operators usually interrupt the railway traffic 373 when forecasted wind speed exceeds a specified threshold to ensure safety 374 and minimize rail derailment. For instance, in Australia many railway op- 375 erators restrict train passage at v≥28m/s; this threshold is adopted in the 376 case study in Section 4. In Germany, Deutsche Bahn AG [55] introduces a 377 20 threshold of 30 m/s. Fujii et al. [56] addressed traffic operation control under 378 high wind speeds in Japan. 379 Wind speed operating thresholds reduce the probability of extreme wind- 380 on-train load effects. The consequent impact on reliability is significant but 381 difficult to capture [14]. Probabilistic approaches, such as Bayesian Net- 382 works, can be adopted to develop the marginal distribution of the wind-on- 383 train load effect. Such a strategy would integrate the uncertainties from the 384 independent variables, such as the wind speed, forecasting model, and oper- 385 ator decisions. Moreover, such an imperfect decision-making process entails 386 many uncertainties that must be captured within these probabilistic models. 387 3.3.2. Wind load direction reversal 388 While modelling wind loads, the structural members are analyzed for one 389 ormorepossibledirectionsofthewindsbasedontheanticipatedmostadverse 390 outcomes. The assessment results for all wind load directions considered can 391 be then enveloped to establish the most critical wind direction case specific 392 to each member. However, within each assessment scenario, the direction of 393 the wind loads is considered to be fixed and unchanging. 394 In cases when the annual maximum wind load distribution has a real- 395 valued support, such as the Gumbel distribution, it is possible that some 396 probability content of the inferred parent distribution can spill over to the 397 opposite, negative or positive, domain. In physical terms, such a change of 398 sign indicates a change of wind load direction within an assessment scenario 399 and, therefore, it must be restricted. 400 Such a restriction on the wind load direction can be done in the following 401 21 ways. Firstly, if suitable, a strictly positive probability distribution (such as 402 exponential distribution) can be adopted. When the load effect is negative 403 as per the chosen sign convention (e.g. sagging is positive and hogging is 404 negative), the nature of the load effect can be captured by encoding a −1 405 multiplierwiththeloadeffectwithintheLSF.Alternatively, actualstatistical 406 data regarding the wind load effects can be developed instead of adopting 407 extreme value theory for inferring the distribution parameters. In any case, 408 such a wind load direction reversal must be restricted, but it must be done 409 so carefully so as not to introduce articifical truncated distributions into 410 the reliability assessment. Otherwise, the reliability results will erroneously 411 magnify (or diminish) the impact of wind loads on the structural member. 412 3.4. Load effect combinations and correlations 413 3.4.1. Load combinations specification 414 For ULS, most structural design codes specify load cases a combination 415 between the permanent effects (including prestressing) and transient load 416 effects. In contrast with BS EN 1990 [57], AS 5100.2 [17] specifies the load 417 combination factor for the non-dominating transient load effects as zero but 418 accounts for the rare probability of joint simultaneous occurrence of various 419 transient loads by adjusting the characteristic value of some accompanying 420 loads in various subclauses. 421 Permanent load effects, such as Dead Load (DL), Superimposed Dead 422 Load (SDL), and Track Load (TL), retain their distributions throughout 423 the reliability assessment. For steel railway bridge assessment at ULS, it is 424 important to consider load combinations: 425 22 Table 3: Load combinations considered in the study. Label T & BF NO WB & WT C1: T max Max Parent Parent C2: N max Parent Max Parent C3: W max Parent Parent Max •C1: T max : annual maximum of Train Load (T) and Braking Force 426 (BF) is combined with daily values of Nosing Load (NO), Wind on 427 Bridge(WB),andWindonTrain(WT);thelatterbeingtruncatedwith 428 the upper bound at the threshold of wind speed for traffic interruption 429 (Section 3.3.1). 430 •C2: N max : annual maxima of NO are combined with daily values of 431 T, BF, WB, and WT. 432 •C3: W max : annual maximum of wind pressure is applied as follows: 433 1. WB and WT is combined with daily values of T, BF, and T for 434 wind speeds below the threshold, 435 2. for wind speeds exceeding the threshold, WB is considered only. 436 For this study, additional numerical analyses revealed a good match be- 437 tween reliability indices based on Turkstra’s rule and FBC processes. Fur- 438 ther, when point-in-time (parent) distributions of variable loads are based 439 on weekly maxima, slightly conservative estimates are obtained. On the 440 other hand, consideration of monthly maxima for accompanying variable 441 loads leads to conservative estimates. Therefore, Turkstra’s rule is deemed 442 appropriate for this study. 443 23 3.4.2. Load correlations specification 444 Correlations are particularly important for: (i) the correlations between 445 loadeffectsforaparticularloadtype(e.g. major-axisbendingandminor-axis 446 bending of a given load type), and; (ii) correlations between various loads 447 themselves (e.g. vertical train loads and nosing). Based on the phenomeno- 448 logical inter-relationships between various loads, Tables 4 and 5 present a 449 qualitative recommendation on the degree of correlation for steel railway 450 bridge assessment. These qualitative correlation intensities range from very 451 low to perfect correlation. 452 Table 4: Qualitative correlation between bending moment and axial forces load effects for a given load case. MzS MyS NS MzSFull Very High High MySVery High Full High NSHigh High Full Table 5: Qualitative correlation between bending moment and axial forces load effects for a given load case. T BF NO WB WT T Full High Low-High Very Low Very Low BF High Full Low-High Very Low Very Low NO Low-High Low-High Full Very Low Very Low WB Very Low Very Low Very Low Full Very High WT Very Low Very Low Very Low Very High Full The correlations between the various load effects for a given load, as 453 specified in Table 4, are self-explanatory. However, for those between various 454 24 loads, it is considered that T, BF, and NO have higher mutual correlations 455 relativetothatwithWBandWTduetotheirphenomenologicalbasisintrain 456 and track geometry. The correlation of T and BF with NO is more complex 457 and, therefore, considered to be varying between medium to high ranges. 458 Nibloe et al. [58] noted that the proportion of lateral forces to vertical forces 459 changes significantly at higher vehicle speeds and is significantly larger for 460 empty trains. Therefore, the correlations between train and nosing loads can 461 besuitablyadjustedbetweenlowtohighrangestominimizethesimultaneous 462 occurrencesofhighverticaltrainandlateralnosingloadsbasedontrainspeed 463 and occupancy. 464 3.4.3. Isoprobabilistic transformation considerations 465 Due to the high reliance on engineering expert judgement for the specifi- 466 cation of correlation values, the constructed correlation matrix can be sym- 467 metric but not necessarily positive semi-definite as is mathematically re- 468 quired. Physically, it indicates that at least one load’s correlations can be 469 expressedasalinearcombination oftheothers. Thissituationinvalidatesthe 470 use of Cholesky decomposition in the standard first-order reliability method 471 (FORM) algorithms for isoprobabilistic transformation of the random vari- 472 ables, i.e. the transformation of the random variables from the physical space 473 to the reduced space. There can be two solutions to this problem which can 474 work together. 475 Firstly, instead of Cholesky decomposition, the singular value decompo- 476 sition (SVD) of the correlation matrix in physical space, CX, can be used to 477 determine the isoprobabilistic transformation matrix based on the orthonor- 478 25 mal eigenvectors corresponding to the eigenvalues of CX. The main advan- 479 tage of utilizing SVD is that it imposes a symmetric positive semi-definite 480 requirement on CXas opposed to Cholesky decomposition, which imposes a 481 positive definite requirement on CX. 482 Finally, when engineering expert judgement results in a symmetric but 483 non-positive definite correlation matrix, its nearest positive semi-definite ma- 484 trixcanbedeterminedusingHigham’sNearestCorrelationMatrixalgorithim 485 [59]. For example, for a case with three loads with correlations C12= 0.9, 486 C13= 0.1, and C23= 0.9, then Higham’s algorithm gives modified correla- 487 tion values of ˜C12= 0.77,˜C13= 0.185, and ˜C23= 0.77. This strategy has 488 many applications in finance but is also appropriate for structural reliability 489 assessments. Thus, a significant constraint associated with load correlations 490 can be remedied. 491 3.5. Factor adjustment challenges 492 3.5.1. Model uncertainty considerations 493 While the LSF usually includes variables other than load and resistance, 494 such as model errors as shown in Equations (9) and (10), the design equation 495 (Equation (7)) typically includes only load and resistance variables. There- 496 fore, the effects of such accompanying variables must be incorporated in the 497 consequent estimates of adjusted factors. This is done by scaling the design 498 values of the load and resistance variables with those of load and resistance 499 model errors before estimating the adjusted partial factors. 500 26 3.5.2. Calibrating adjusted factors 501 Adjusting partial factors can lead to many estimates for high-dimensional 502 problems. The generally adopted method for adjusting factors, as discussed 503 in Section 2.4.1, yields estimates of combination factors that vary per load 504 case. When a structural member is assessed at ncross sections for jload 505 caseswith lloads, thetotalnumberofestimatedadjustedfactorsis n×j×l. 506 Ideally, there should only be lpartial factors and jcombination factors. This 507 problem will be further compounded when there are be multiple structural 508 members of the same type (e.g. two longitudinal girders for the same span 509 with the same geometry). Therefore, obtaining an optimized set of adjusted 510 factors isa significant problemthat compounds higher dimensional problems. 511 A potentially quicker approach to calibrate the adjusted factors is to 512 envelope the factor estimates based on their maximum. Such an enveloping 513 for a structural member can be across all cross sections, load effects, and load 514 casesunderconsideration, anditsimpactcanbeassessedusingadesigncheck 515 (Section 2.4.3). Any such conservative enveloping will result in a structural 516 design greater than the initially targeted reliability index βT. Therefore, 517 the primary advantage of a probabilistic assessment can be reduced with 518 enveloped factors. However, a more targeted and thematic enveloping of 519 factors (e.g. by enveloping factors for a particular structural member type) 520 can minimize the efficiency lost. 521 27 4. Case study: probabilistic assessment of old steel railway truss 522 bridge 523 4.1. Bridge overview 524 The probabilistic framework presented in Section 2 with the modelling 525 approaches presented in Section 3 is applied and demonstrated on a case 526 study inspired by an existing steel truss railway bridge in Australia as il- 527 lustrated in Figure 2. The two longitudinal girders of the bridge, spanning 528 60min length over eight bays, are examined. The cross sections of these 529 girders are presented in Figure 3. The cross sections at intermediate lengths 530 between intersections with transverse girders have additional plates at the 531 top and bottom. Each girder is analyzed at cross sections at increments of 532 2.5m. Therefore, the total number of sections for reliability assessment is 533 2×60/2.5 = 48sections. Considering the widespread literature on non-linear 534 FEA modelling of steel structures [35], this case study focusses on calibrating 535 the factors for such an assessment. 536 Longitudinal Girders60 m 7.5 mELEVATION PLAN Figure 2: Illustration of the truss bridge in the case study. 28 (a) Intersection (b) Intermediate Figure 3: Cross-sectional geometry of longitudinal girders. 4.2. Uncertainty quantification and probabilistic modelling 537 Forultimatelimitstate, thecombinedactionofvariousloadsisconsidered 538 for the assessment: Dead Load (DL), Superimposed Dead Load (SDL), Track 539 Load (TL), Train Load (T), Braking Force (BF), Nosing Load (NO), Wind 540 on Bridge (WB), and Wind on Train (WT), i.e. 541 S={DL, SDL, TL, T, BF, NO, WB, WT } (12) . The statistics of the random variables are summarized in Table 6. Some es- 542 sential