International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Calibration of partial safety factors for fatigue design of steel bridges D’Angelo, Luca; Faber, Michael Havbro; Nussbaumer, Alain Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Calibration Of Partial Safety Factors For Fatigue Design Of SteelBridgesLuca D’AngeloPhD Candidate,Civil Engineering Institute, ICOM, EPFL, Lausanne, SwitzerlandMichael Havbro FaberProfessor, Dept. of Civil Engineering, DTU, Lyngby, DenmarkAlain NussbaumerProfessor,Civil Engineering Institute, ICOM, EPFL, Lausanne, SwitzerlandABSTRACT: The aim of this paper is to propose a new framework for assessment of fatigue partial safetyfactors with focus on steel bridges welded joints. Fatigue resistance S-N curves for constant amplitude(CA) and variable amplitude (VA) loadings are defined using a novel probabilistic approach based onMaximum Likelihood (ML) and Monte-Carlo Simulations (MCS) methods. The proposed frameworkincludes ML-MCS-based S-N curves of different welded detail categories and it is applicable for the caseof deterministic and probabilistic CA loadings as well as for VA loadings. One example is developed,that of a typical bridge fatigue sensitive welded joint. The results are compared to Eurocode standardsboth in terms of partial safety factor values and of characteristic S-N curves.1. INTRODUCTIONOver the last thirty years significant attention hasbeen paid to the probabilistic methods for the as-sessment of fatigue reliability. By the early 90’sconsiderable effort has been made to implementthese methods for application in steel bridges withfocus on fatigue sensitive details under the long-term effect of traffic loading.The common approach to the formulation of thefatigue limit state is based on S-N curves in com-bination with Miner’s linear damage accumulationrule (Miner (1945)). Fatigue life assessment of steelbridges asks for consideration of welded joints; acommon assumption in fatigue analysis of weldedjoints is that the crack initiation phase is almost nonexistent and that all the fatigue life is taken by thecrack propagation phase (Gurney (1979)). Withinthis hypothesis the linear elastic fracture mechanicsshows that the S-N curve is a straight line havingslope of −m1 in the log(S-N) plane, where S is thestress range and N is the number of cycles to fail-ure; the line is assumed to become horizontal at thecrack growth propagation threshold (constant am-plitude fatigue limit (CAFL)). Traditional fatigueanalysis of welded joints is based on the nominalstress approach wherein characteristic S-N curvesare based on constant amplitude (CA) fatigue tests;due to inherent randomness in fatigue life, a statisti-cal evaluation of fatigue test data is required. Char-acteristic S-N curves in the Eurocode standards aredetermined by fitting a linear regression to the fail-ure data, disregarding all run-outs and somewhat ar-bitrarily fixing the CAFL at 5 ·106 cycles (EuropeanCommittee for Standardization (1989) and Eulerand Kuhlmann (2014)). Since most structures expe-rience variable amplitude (VA) loading during theirlife the fatigue life assessment of welded details un-der VA loading is needed. It is generally acceptedthat even infrequent CAFL-exceeding cycles are re-sponsible for lowering the stress range threshold forcrack propagation, thus enabling stress ranges be-low CAFL to also contribute to damage as crackdevelops. To take into account this behavior, amodified S-N curve having a slope of −m2 belowthe CAFL can be considered for damage accumula-tion (modified Miner’s rule, Haibach (1970)). The112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015target value of the accumulated damage indicat-ing fatigue failure is subject to considerable uncer-tainty and it is still under debate. For welded jointsthe common assumption is to consider the criticaldamage, D, as a Log-Normal random variable withmean equal to 1 and standard deviation equal to 0.3(Joint Committee Structural Safety (2013)).The fatigue verification using the concept of par-tial safety factors, as recommended in EN 1993-1-9(European Committee for Standardization (2005))is based on characteristic CA S-N curves and onmodified Miner’s rule .The aim of this paper is to suggest a new frame-work for assessment of fatigue resistance partialsafety factor for the case of CA deterministic andprobabilistic loadings and VA loadings. Fatigue re-sistance S-N curves for CA and VA loadings are de-fined using a novel probabilistic approach based onthe Maximum Likelihood (ML) and Monte-CarloSimulations (MCS) methods. Both failure and run-out points of experimental CA and VA fatigue data-sets are used. The ML-MCS based approach allowsto overcome the following limitations of Eurocodeapproach: 1) Run-out data points are not consid-ered; 2) the CAFL is arbitrarily fixed at 5 · 106 cy-cles; and 3) the choice of second slope, −m2, andof critical damage, D, for welded joints, has notbeen rigorously validated by experimental VA fa-tigue test results.The paper is organized as follows:• The ML-MCS based probabilistic approachfor assessment of fatigue resistance modelsunder CA and VA loadings is shortly recalled;• The framework for partial safety factor cal-ibration using ML-MCS based fatigue resis-tance S-N curves is presented;• An application of the framework to a typicalbridge fatigue sensitive welded joint is consid-ered;• Results are discussed and comparisons withpartial safety factor values recommended inEN 1993-1-9 are made.2. RESISTANCE MODELSIn this section the ML-MCS based probabilistic ap-proach for assessment of fatigue resistance modelsunder CA and VA loadings is shortly recalled. De-tailed description of the approach can be found in(D’Angelo and Nussbaumer (2014)).2.1. CA S-N curveThe relationship between the fatigue log-life, Y ,and the nominal applied log-stress range, X , ismodeled as:Y =m0 +m1Xstep(X−V )+ ε(0,exp(σ)) (1)where X is the natural logarithm of the stress rangeS, Y is the natural logarithm of number of cycles,N, and V is the natural logarithm of the CAFL. Theerror term, ε , is modeled as a Normal random vari-able with location parameter, 0, and scale parameterexp(σ).The CA stochastic model is characterized usingmodel parameter vector θ = (m0,m1,σ ,µV ,σV ),where m0 is the intercept of the S-N curve in log-logplane, m1 is the slope of the S-N curve, σ is the nat-ural logarithm of the scale parameter of the Normalrandom variable Y and µV , σV are respectively thelocation and the scale parameter of the Log-NormalCAFL distribution. The expected value of modelparameter vector, θˆ , and the Variance-Covariancematrix, ρ , are estimated by applying ML approachto experimental CA fatigue data points (xi,yi). Bothfailure and run-out data points are taken into con-sideration, the latter in terms of censored data.2.2. VA S-N curveThe relationship between the fatigue log-life, Y ,and the nominal applied log-stress range, X , ismodeled as:Y =m0 +m1X + ε, for X > Vm0 +V∆m+(m1−∆m)X + ε, for X ≤V(2)where (m1−∆m) is the slope of the S-N curve be-low the CAFL.Fatigue resistance under VA loadings is ex-pressed using Miner’s linear accumulation rule.Critical damage, D, is modeled as a Log-Normal212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015random variable with location parameter, µD, andscale parameter, σD. The parameters of D Log-Normal distribution are assessed by applying theML approach to observed values of cumulateddamage, di; di values are generated from VA ex-perimental test results using MCS approach. Bothfailure and run-out data points are considered.The complete stochastic model for CA and VAloadings is defined by the model parameter vec-tor Θ = (θ ,µD,σD), the parameter ∆m and theVariance-Covariance matrix Σ.3. FRAMEWORK FOR PARTIAL SAFETY FACTORCALIBRATIONThe typical design equation for verification of astructural component (Faber and Sorensen (2003))is:G =zRcγM− (γF1Fc1 + . . .+ γFnFcn) = 0 (3)where Rc is the characteristic value for the resis-tance, z is a design variable, Fci is the characteristicvalue of the ith action effect, γM is the partial safetyfactor for the resistance and γFi is the partial safetyfactor for the ith action effect.According to the design equation (3) a reliabilityanalysis can be made with the following limit statefunction:g = zR− (F1 + . . .+Fn) = 0 (4)The general limit state equation (4) and the gen-eral design equation (3) will be adapted to the con-sidered fatigue case (CA deterministic loadings,CA probabilistic loadings, VA loadings) in the fol-lowing three subsections. The partial safety fac-tor for fatigue loadings γF can be settled to 1.0 byproper choice of load model.3.1. CA deterministic loadings3.1.1. Definition of Load CasesIn order to compare findings for VA and CA load-ings on a common basis, load case points are de-fined using the concept of CA equivalent stressrange. Three different Rayleigh VA loading spec-tra are considered in order to take in account theinfluence of the loading spectrum position on theLorry Perc. Nv Nax Ncyc40% 800 ·103 2 1.6 ·10610% 200 ·103 3 0.6 ·10630% 600 ·103 5 3.0 ·10615% 300 ·103 4 1.2 ·1065% 100 ·103 5 0.5 ·1066.9 ·106Table 1: Yearly loading cycles according to Tab 4.7 ofEN 1991-2-2003, short influence line bridge.assessment of the partial safety factor for fatigue re-sistance, γM. The scale parameters σR of the threeconsidered loading spectra are defined in orderto have ς = 50%,5%,0.01% (σR = 5.4,9.4,19.5MPa), where ς is the percentage of cycles in thespectrum with stress ranges exceeding the ML esti-mate of the 0.05 quantile of the CAFL.The total number of cycles, Ntot , is definedaccording to Tab 4.7 of EN 1991-2-2003 (Euro-pean Committee for Standardization (2002)) forthe case of roads with high percentage of trucks(Nobs = 2 · 106 per year and slow lane). The caseof road bridges with short influence line (one axlegives one stress cycle) is considered in this study.The relationship between the number of cycles andthe stress range of the loading spectrum is:n(s) = Ntot · fS(s;σR)ds = Ntotsσ2Rexp(−s22σ2R)ds(5)The load case l is represented by the pair(Seq,Neq|σR)l; a total number of 30 load cases areconsidered: Seq = (5,15, . . . ,95)T MPa and σR =(5.4,9.4,19.5)T MPa.For each load case l, the equivalent number ofcycles NEq,l is then computed as follows:NEq,l =Smˆ1Eq,lexp(µD)∫ ∞0fS(s;σR)s−mˆ1 ds (6)For simplicity, median S-N curve having singleslope equal to m1, is used for the definition of theequivalent number of cycles, NEq,l .312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 20 40 60 80 10000.0050.010.0150.020.0250.030.0350.04Stress range, S [MPa]pdf←σR,.05←σR,.95←σR,.50Figure 1: Rayleigh loading spectrum, σR = 19.5 MPa.3.1.2. Definition of objective functionThe design equation is defined as:G =yc(z ·XEq)γM−YEq = 0 (7)where z is a design variable, yc is the character-istic value of the fatigue resistance (expressed inlog-number of cycles), which depends on the de-terministic log-stress range XEq = ln(SEq), and YEqis the equivalent log-number of cycles (determinis-tic). The characteristic value of the fatigue resis-tance, yc, corresponds to 5% probability of failure.The limit state equation is defined as follows:g = Y (z · ln(SEq))−YEq = 0 (8)The probability of failure is:Pf =∫g≤0fX(x)dx (9)where fX(x) is the joint probability density functionof the model random variables.The objective function is:T (γM) =Ndet∑d=1Nlc∑l=1(Ptf −Pf )2 (10)where Nlc is the number of load cases, Ndet is thenumber of the considered details and Ptf = Φ(−β t)is the annual target failure probability. Accordingto JCSS PMC Part 1 (Joint Committee StructuralSafety (2013)), the tentative annual target reliabil-ity index β t = 4.2 should be considered as the mostcommon fatigue design situation (safe life, highconsequences); this target level can be then conser-vatively used for existing structures.The value of the partial safety factor γˆM, whichcorresponds to the annual target failure probabilityPtf , is computed by minimizing the objective func-tion T (γM) with a Newton Raphson search algo-rithm. At each step of the minimum search algo-rithm, the probability of failure Pf is computed bythe MCS method.3.2. CA probabilistic loadings3.2.1. Definition of load casesIn order to have probabilistic definition of loadcases, the scale parameter of the Rayleigh spectrumis modeled as a Normal random variable, havingmean equal to the values used for CA deterministiccase and coefficient of variation equal to 0.1. Byusing Equations (5) and (6) it is possible to relatethe uncertainty on NEq,l to the uncertainty on σR:NEq,l =Ntot Smˆ1Eq,lexp(µD)∫ ∞0sσ2Rexp(−s22σ2R)s−mˆ1 ds(11)The uncertainty on Neq,l is quantified by sam-pling random values of σR and fitting genericprobability distribution to sampled values of NEq,l .Since the transformation given in Equation (11)is not linear, non-Normal probability distributioncould be chosen to fit sampled values of NEq,l .3.2.2. Definition of objective functionThe design equation and the limit state equation aredefined by considering NEq,l as a random variableinstead of a deterministic value:G =yc(z ·XEq))γM− yEq,c = 0 (12)where yEq,c is the characteristic value of the equiv-alent number of cycles, corresponding to the 0.5quantile of YEq distribution. The limit state equa-tion, the formulation of the probability of failureand the objective function are the same as for thecase of CA deterministic loadings (see Equations(8), (9) and (10)).412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20153.3. VA loadings3.3.1. Definition of load casesVA load cases are defined using a Rayleigh loadingspectrum, having scale parameter σR∗ . Eightdifferent values of σR∗ are considered, giving ς =(50%,75%,5%,1%,0.5%,0.1%,0.05%,0.01%),where ς is the percentage of cycles in the spectrumwith stress ranges exceeding the ML estimate ofthe 0.05 quantile of the CAFL.3.3.2. Definition of objective functionThe design equation is defined as:G = dc−Ntot ·(∫ ∞exp(vc)fS(s;σ∗R)exp(yc(z · s)/γM)ds ++∫ exp(vc)0fS(s;σ∗R)exp(yc(z · s)/γM)ds)(13)where dc is the characteristic value of the criti-cal damage (corresponding to the median value ofLog-Normal distribution) and yc is the character-istic value of fatigue resistance in terms of log-number of cycles (see Equation (2)).The limit state function is defined as follows:g = D−Ntot ·(∫ ∞exp(V )fS(s;σ∗R)exp(Y (z · ln(s)))ds ++∫ exp(V )0fS(s;σ∗R)exp(Y (z · ln(s)))ds)(14)The probability of failure is defined similarly tothe case of CA loadings (see Equation 9).The objective function is:T (γM) =Ndet∑d=1Nlc∑j=1(Ptf −Pf )2 (15)where where Nlc is the number of load cases, Ndetis the number of the considered details and Ptf =Φ(−β t) is the annual target failure probability.The value of the partial safety factor γˆM, whichminimizes the objective function T (γM), is com-puted again by using a Newton Raphson search al-gorithm in combination with MCS method.4. STUDY CASE: WELDED COVER PLATEThe framework built in Section 3 was applied toa typical fatigue-sensitive bridge detail: a weldedcover plate. The parameters of CA S-N curvewere estimated using 26 experimental CA test re-sults from NCHRP Project 12-7 and from PennDotProject 72-3 (Fisher et al. (1982)); in tested beams,the cover-plate thickness tc, is lower than the beamflange thickness tb, which is lower than 20 mm:the tested specimen is classified as FAT50 accord-ing to EN 1993-1-9. The parameters of VA S-Ncurve were estimated using 32 experimental VAtest results from NCHRP Report 354 (Fisher et al.(1993)); in tested beams, tc = tb = 25 mm: thetested specimen is classified as FAT45 according toEN 1993-1-9. Considered cover-plate componentcan be then classified conservatively as FAT45 de-tail according to EN 1993-1-9 for generic CA andVA loadings.4.1. S-N curvesThe ML estimate of model parameter vector is:ΘˆT =(29.02,−3.42,−0.54,3.55,−1.62,0.78,1.39).The parameter ∆m is equal to 17.The approximate Variance-Covariance matrix ofmodel parameter vector is:Σ =∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣8.61 −2.30 0.04 −0.01 0.07 0.00 0.00−2.30 0.61 −0.01 0.00 −0.02 0.00 −0.000.04 −0.01 0.04 −0.00 0.00 0.00 0.00−0.01 0.00 −0.00 0.00 −0.00 −0.00 −0.000.07 −0.02 0.00 −0.00 0.17 0.00 −0.000.00 0.00 0.00 −0.00 0.00 0.01 0.010.00 −0.00 −0.00 −0.00 −0.00 0.01 0.01∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣4.2. CA loadingsThe partial safety factor γM for CA deterministicand CA probabilistic loadings were computed byconsidering the sum of 30 load cases, as describedin Section 3.1.1; the three loading spectra used todefine CA load cases are represented in Figure 2.For the case of CA deterministic loadings the pa-rameter σR is deterministic. For the case of CAprobabilistic loadings the parameter σR is a nor-mal random variable having coefficient of variationequal to 0.1. The uncertainty on NEq is related tothe uncertainty on σR as described in Section 3.2.1.Figure 3 shows that γˆM is equal to 1.36 for the case512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015105 106 107100101102ς=0.01%ς=5%ς=50%Characteristic S−N curveStress range, S [MPa]Number of cycles, NFigure 2: VA loading spectra with ς = 50%,5%,0.01%.1.3 1.35 1.4 1.4501234 x 10−4γM∂(Pf−Pft )∂Pf  γM=1.36γM=1.37CA det.CA prob.Figure 3: γM search, CA loadings.of CA deterministic loadings and to 1.37 for thecase of CA probabilistic loadings.4.3. VA loadingsThe partial safety factor γˆM for VA loadings wascomputed by considering the sum of 8 defined loadspectra, as described in section 3.3.1. Figure 4shows that γˆM is equal to 1.56.5. RECOMMENDATIONS FOR FATIGUE DESIGNIn this section two verification schemes are pro-posed for fatigue design under CA loadings and VAloadings.The verification scheme for fatigue design underCA loadings is shown in Figure 5: γˆM = 1.37 (CA1.52 1.54 1.56 1.58 1.6 1.62 1.640246810x 10−4γM∂(Pf−Pft )∂PfγM=1.56Figure 4: γM search, VA loadings.105 106 107 108 109101102Number of cycles, NStress range, S [MPa]Verification:Seq ≤ exp(vc/1.37) = 10 MPaorNeq(Seq) ≤ exp(yc(Seq)1.37)Seqexp (yc(Seq))exp(v.05) 23Figure 5: Fatigue verification, CA loadings.probabilistic loadings) has been chosen; this verifi-cation is based on one-slope CA fatigue resistancecharacteristic S-N curve, assuming that the equiv-alent stress range, SEq exceeds the design value ofCAFL, exp(v.05/γˆM).The verification scheme for fatigue design un-der VA loadings is shown in Figure 6; this verifi-cation is based on critical damage accumulation onthe assumption that at least one cycle of the load-ing spectrum exceeds the design value of CAFL,exp(v.05/γˆM).The verifications schemes proposed above for fa-tigue design under CA and VA loadings are com-pared to EN 1993-1-9 format for fatigue reliabil-ity verification. The cover plate detail is classi-612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015105 106 107 108 109101102Siniexp (yc(Si))Verification:∑iniexp (yc(Si)/1.56) ≤ 0.54exp(v.05)Number of cycles, NStress range, S [MPa]Figure 6: Fatigue verification, VA loadings.fied as FAT45 as explained in Section 4. Sincecrack formation in the cover-plate could rapidlylead to structural failure, the case -Safe life, Highconsequences- is chosen in Table 3.1 of EN 1993-1-9.Under these assumptions, the EN 1993-1-9format for fatigue verification under CA loadingsis the following:γF f ·SE,2 ≤ScγM f⇐⇒ SE,2 ≤Sc1.35(16)where SE,2 is the equivalent CA stress range re-lated to 2 million cycles, Sc is the characteristicvalue of the fatigue strength at 2 million cycles, γF fis the partial factor for fatigue loadings (which isset to 1.0), and γM f is the partial factor for fatiguestrength (which is set to 1.35). The verification for-mat at 2 million cycles, given in Equation (16), isextended to the generic case of N∗ number of cy-cles:γF f ·SE,N∗ ≤(2N∗) 13 ScγM f(17)The definition of partial resistance factor γM f inEN 1993-1-9 (Section 1.4, pp. 9, European Com-mittee for Standardization (2005)) is ambiguousbecause γM f is strictly defined for fatigue strengthat 2 million cycles and it is generically applied forfatigue strength at all number of cycles .The EN 1993-1-9 format for fatigue verificationunder VA loadings uses the damage sum Dd:Dd =Ntot∑iniNi≤ 1.0 (18)where Ni is the endurance obtained from the fac-tored scγM f −N curve.The ML-MCS approach-based partial resistancefactor, γˆM, and the EN 1993-1-9 format-based par-tial resistance factor, γM f , cannot be directly com-pared since γM f is only valid for verification at 2million cycles while γˆM is valid for all numbers ofcycles. The comparison between EN 1993-1-9 for-mat and ML-MCS format for fatigue reliability ver-ification has to be done in terms of design value offatigue strength at 2 million cycles, design value ofCAFL and critical value of cumulated damage Dtd .Param. EN 1993-1-9 ML-MCS CA ML-MCS VAγM f = 1.36 γˆM = 1.37 γˆM = 1.56m1 -3 -3.4 -3.4m2 -5 - -20.4Sc,2·106 45 MPa 51 MPa 51 MPaSd,2·106 33 MPa 18 MPa 12 MPaexp(vc) 33 MPa 23 MPa 23 MPaexp(vd) 24 MPa 10 MPa 8 MPaDtd 1.00 - 0.54Table 2: Comparison between EN 1993-1-9 format andML-MCS format for fatigue reliability verification.6. CONCLUSIONSThe suggested approach for calibration of fatigueresistance partial safety factor improves the fatiguereliability verification format by: 1) A re-definitionof the verification format in the log(S-N) plane, in-stead of (S-N) plane, allowing for the related par-tial factor, γˆM, to be valid for all numbers of cycles(and not only at 2 million cycles); and 2) A more re-alistic consideration of the CAFL position and theS-N curve in the high-cycle fatigue (HCF) region(N > 106cycles).Both of points above improve consistency inachieving target levels of safety.In order to improve the accuracy in estimation ofthe CAFL position and its variability, we need to712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015consider experimental fatigue data-sets having sig-nificant number of data points in the HCF region.The study case considered in this paper showedthat:• The Eurocode-based characteristic S-N curveand the ML-MCS-based characteristic S-Ncurve are slightly different for N ≤ 5 · 106 cy-cles;• The Eurocode format gives an under conserva-tive estimate of the 0.05 quantile of the CAFLwith respect to the ML-MCS approach;• The Eurocode format gives an under conserva-tive estimate of the design fatigue strength at2 million cycles with respect to the ML-MCSapproach;• The Eurocode format gives an under conserva-tive estimate of the design value of the criticaldamage, Dtd , with respect to the ML-MCS ap-proach (1.0 vs 0.54);• The uncertainty associated with the loadingdoes not affect the partial resistance factor forCA loadings (it goes from 1.36 for CA deter-ministic loadings to 1.37 for CA probabilisticloadings);• The uncertainty associated with the loading af-fects the partial resistance factor for VA load-ings (it goes from 1.37 for CA deterministicloading to 1.56 for VA probabilistic loadings);It is recalled here that in order to set the partialloading factor to 1.0 the characteristic load modelhas to represent an extreme fatigue loading condi-tion.The reliability framework for calibration of fa-tigue resistance partial safety factor, set up in thispaper and applied to welded cover plate detail, con-stitutes a powerful tool that can be used to revisethe Eurocode basis for fatigue design of structures.The EN 1993-1-9 format for fatigue reliability ver-ification and associated partial safety factors can berevised by considering different fatigue details andby further differentiating between 1) CA verifica-tion; 2) VA verification using lambda factors; and3) VA verification using damage sum. The frame-work presented in this work can be easily adaptedfor the differentiation above.7. REFERENCESD’Angelo, L. and Nussbaumer, A. (2014). “Evalua-tion of S-N-P curves under variable amplitude load-ings using novel probabilistic approach.” Report No.204233, EPFL, Lausanne.Euler, M. and Kuhlmann, U. (2014). “Statistical inter-vals for evaluation of test data according to Eurocode3 part 1-9.” Report No. TC6-WG3, ECCS, Bruxelles.European Committee for Standardization (1989). “Eu-rocode EN 1993 - Part 1 - Background Documenta-tion.European Committee for Standardization (2002). “Eu-rocode 1 : Actions on structures. Part 2: Traffic loadson bridges.European Committee for Standardization (2005). “Eu-rocode 3: Design of steel structures - Part 1-9: Fa-tigue.Faber, M. H. and Sorensen, J. D. (2003). “Applicationsof Statistics and Probability in Civil Engineering.”Proceedings of the 9th International Conference onApplications of Statistics and Probability in Civil En-gineering, A. Der Kiureghian, S. Madanat, and J. M.Pestana, eds., San Francisco, California, Millpress,927–935.Fisher, J. W., Bellenoit, J. R., and Yen, B. T. (1982).“High cycle fatigue behavior of steel bridges–a finalreport.” Report No. 386-13(82), Lehigh University,Bethlehem, Pennsylvania.Fisher, J. W., Nussbaumer, A., Keating, P. B., and Yen,B. T. (1993). “Resistance of Welded Details Un-der Variable Amplitude Long-Life Fatigue Loading.”Report No. 354, National Cooperative Highway Re-search Program, Bethlehem, Pennsylvania.Gurney, T. (1979). Fatigue of welded structures. Cam-bridge University Press, 2nd edition.Haibach, E. (1970). Modifizierte lineareSchadensakkumulations-Hypothese zur Berücksichti-gung des Dauerfestigkeitsabfalls mit fortschreitenderSchädigung. Technische Mitteilungen: Labora-torium für Betriebsfestigkeit. Laboratorium fürBetriebsfestigkeit.Joint Committee Structural Safety (2013). “JCSS Prob-abilistic Model Code: Resistance Models.Miner, M. A. (1945). “Cumulative damage in fatigue.”Journal of Applied Mechanics, 12, 159–164.8

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