# Fatigue Damage Estimation in Existing Railway Steel Bridges by Detailed Loading History Analysis

Authors: Alessio Pipinato; Carlo Pellegrino; Claudio Modena
Journal: ISRN Civil Engineering
 (2012)

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## Abstract

Fatigue life estimation of metal historical bridges is a key issue for managing cost-effective decisions regarding rehabilitation or replacement of existing infrastructure. Because of increasing service loads and speeds, this type of assessment method is becoming relevant. Hence there is a need to estimate how long these structures could remain in service. In this paper a method to estimate fatigue damage in existing steel railway bridges by detailed loading history analysis is presented. The procedure is based on the assumption that failure probability is a function of the number of predicted future trains and the probability of failure is related to the probability of reaching the critical crack length.


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## Body

## 1. Introduction

A relevant amount of the bridges in the European railway networks are metal made and have been built during the last 100 years. The increasing volume of traffic and axle weight of trains means that the current loads are much higher than those envisaged when the bridge was designed. In this context, issues as maintenance, assessment, rehabilitation, and strengthening of existing bridges assume a significant importance [1, 2]. The authors have developed some works concerning assessment and fatigue behavior of metal railway bridges by means of full-scale experimental testing. In particular in Pipinato et al. [3, 4] full-scale tests on dismantled steel bridges have been developed, whereas assessment of existing bridges and estimation of their remaining fatigue life are shown in Pipinato and Modena [5] and Pipinato et al. [6]. Moreover, a comprehensive method to assess the reliability of existing bridges taking fatigue into account has been recently published [7]. Among historical metal bridges, riveted structures are the most common; the role of riveted connections in the fatigue assessment is documented by several researches, such as, in Bruhwiler et al. [8], Kulak [9], Akesson and Edlund [10], Di Battista et al. [11], Bursi et al. [12], Matar and Greiner [13], Boulent et al. [14], Albrecht and Lenwari [15], Kühn et al. [16], Albrecht and Lenwari [17], and Brühwiler et al. [18]. Fatigue is one of the most common causes of failure in riveted bridges, as highlighted by the ASCE Committee on Fatigue and Fracture Reliability [19] and confirmed by Byers et al. [20]. Increasing loads on existing riveted bridges and the fact that these bridges were not explicitly designed against fatigue-raised questions regarding their remaining fatigue life. As a consequence, a better knowledge of the loading history is needed, having a relevant role in the fatigue damage assessment. The fatigue damage mainly depends on the following three main parameters: the stress range amplitude due to traffic load; the geometry of the construction details; the number of stress cycles due to the past traffic which directly influences the remaining fatigue life of a structure. In the context of structural reliability assessment, a comprehensive examination of fatigue safety and remaining service life of railway bridges is based on these three main parameters. The main objective of this paper is to estimate the fatigue damage in existing railway metal bridges and at the same time to determine the remaining fatigue life according to a step-by-step procedure referring to the LEFM (Linear Elastic Fracture Mechanics) theory by means of detailed loading history analysis. The method is then applied to a real case study, the Meschio railway bridge briefly described in the following paragraph.


## 2. Case Study

The Meschio bridge, a short span riveted flanged railway bridge built in 1918, was taken out of service in 2005 (Figure1). It has been used in the line Mestre-Cormons, which is located in the North-eastern part of Italy. The net span of the bridge was 12.40 m. The main horizontal structure was made of two couples of twinned riveted composite flange beams. Wooden beams were located between the coupled beams, with a net distance of 565 mm from web to web of the beams, while the beam height was 838 mm. In this open-deck riveted railway bridge, transversal short shear diaphragms are riveted with double angles to both webs carried the rails. Each twinned beam supported the wooden elements of a single rail. The thickness of the main beam plates was 11 mm. The web was reinforced by 1 m spaced shear stiffeners, whereas the flanges were reinforced with 10 mm thick plates. The plate thickness increased from the abutment to the midspan. Each pair of twinned beams was linked to the corresponding pair with transverse bracing frames. The characteristics of the materials and more details on the geometry are described in Pipinato et al. [3, 4], while a typical cross section has been reported in Figure 2. Because the examined bridge was characterized by a simple structure, that is, a statically determinate bridge, it was rather easy to evaluate the nominal stress on members and connections of the bridge. In order to check these results, a stress analysis has been performed with a simplified FEM model, calibrated with observed strains derived from real scale testing results [3, 4].Figure 1
Dimensional overviews of the Meschio bridge.Figure 2
Cross section of the of the Meschio bridge girder at midspan.


### 2.1. Load History Assumption

Railway traffic estimation has been based on International Union of Railways, UIC [21] and on real data observed in the railway network from 1900 to 1990. A quadratic polynomial regression based on these data was performed and used for future traffic estimation (Figure 3). The maximum traffic capacity of the railway line in which the bridge is included has been assumed equal to 235 trains/day, following to the maximum capacity of the main national railway lines. According to these assumptions, the adopted traffic model are shown in Figures 3 and 4: the past real traffic and its growth tendency by adopting the UIC regression is shown in Figure 3, while the capacity of the line is developed in Figure 4 according to different traffic trend evolution, until a maximum of 235 trains/day. Figure 4 shows the traffic estimation from 1900 to 2020, with increasings of 1%, 2%, 3% and the UIC regression tendency (percentage increasing versus time) to estimate the number of trains passed on the bridge. UIC traffic data have been considered for the past, whereas some assumptions have been proposed for the future. The method described in the following is based on the aforementioned traffic estimation, and this scenario is also in accordance with CER [23]: in fact for the period 1995–2004, CER traffic increase is of 16%, just like UIC traffic increase. The past traffic has been assumed with a rate of 50% of passengers and 50% of freight trains [24]. Traffic assumption includes train type and convoy numbers: from 1900 to 1990 these data are presented in Table 1, while from 1991–2000 traffic was estimated according to Instruction 44/F [25], as described in Table 2. As a matter of fact, load models have been implemented as they were coded in the different historical periods analyzed:(i)
the historical loadings have been assumed according to UIC 779-1 [21];(ii)
from 1991 to 2000, the traffic spectrum has been based on the Instruction 44/F [25];(iii)
from 2001 up to now, loads refers to LM71 (Figure5; [22]) load model (method A) or to Instruction 44/F (method B) [25].Table 1
Traffic type for the historical train (1900–1990).
PeriodTrain/dayTypeUIC ref.Train type [%]Train/day for type1900–190834,3Tot.17,2PassengerA03100%17,217,2FreightA04100%17,21909–192347,2Tot.23,6PassengerA0560%14,1A0640%9,423,6FreightA07100%23,61924–193860,0Tot.30,0PassengerA0860%18,0A0940%12,030,0FreightA0760%18,0A1040%12,01939–195355,7Tot.27,8PassengerA1150%13,9A1250%13,927,8FreightA1060%16,7A1340%11,11954–196872,8Tot.36,4PassengerA1425%9,1A1517%6,2A1733%12,0A1825%9,136,4FreightA1640%14,6A2060%21,91969–1983102,8Tot.51,4PassengerA1428%14,4A1836%18,5A1936%18,551,4FreightA2040%20,6A2160%30,91984–1990120,0Tot.60,0PassengerS0156%33,6S0244%26,460,0FreightS0337%22,2S0437%22,2S0513%7,8S0613%7,8Table 2
Traffic type for the period 1991–2000.
Train typeNameTrain/dayAxle/dayLocomotive (L) and carriages (Ci)t/axleWagon numberWagon typeAxle spacing [m]1IC20960L20,2512.6-6.4-2.6IntercityC11552.56-16.44-2.56C212,7562.56-16.44-2.562EC10340L2012.85-2.35-2.85-2.35-2.85EurocityC114,2522.56-16.44-2.56C21252.56-16.44-2.563EXPR15990L2012.85-2.35-2.85-2.35-2.85ExpressC114,25102.56-16.44-2.56C21252.56-16.44-2.564DIR301380L18,612.85-2.35-2.85-2.35-2.85DirectC110,675102.4-16.6-2.45ETR10480L2023-9-3EurostarC111,6103-17.3-36TEC15990L18,712.85-2.35-2.85-2.35-2.85Container freightC120151.8-12.8-1.87Merci acciaio10720L18,722.85-2.35-2.85-2.35-2.85Steel freightC120151.8-13.06-1.88Trenomerci tipo D45380L2022.85-2.35-2.85-2.35-2.85D4 freightC122,5161.8-4.65-1.89Trenomerci misto5270L18,712.85-2.35-2.85-2.35-2.85MixedfreightC116249Figure 3
Traffic increment regression based on UIC [21] real train (percentage increment versus time).Figure 4
Traffic estimation from 1900 to 2020, with increments of 1%, 2%, 3% and UIC regression tendency (percentage increment versus time).Figure 5
Load Model 71 according to EN 1991-2 [22].For historical trains and for Instruction 44/F trains, dynamic amplification factorφ is calculated according to EN 1991-2 [22]:(1)φ=1+12(φ′+1/2φ′′),
where:(2)φ′=K1-K+K4,K=ν160forL<20m,K=ν47.16⋅L0.408forL>20m,φ′′=0.56⋅e-L2/100,
where ν is the train speed (m/s) and L is the determinant length LF (m). For LM71 load model (method A), the dynamic amplification factor has been calculated according to EN 1991-2 [22](3)ϕ3=2,16Lϕ-0,2+0,73
assuming 1,00≤Φ3≤2,00, and LF = determinant length. All the dynamic amplification factors applied have been reported in Table 3.Table 3
Dynamic amplification factor (detail 1).
PeriodTrain typeTrain nameSpeed [km/h]D.A.F. [dynamic amplification factor]φ,ϕ3Code ref.1900–1983Passenger/freightA03/A211001,135UIC1984–1990PassengerS011251,168UICPassengerS021401,190UICFreightS03801,111UICFreightS04801,111UICFreightS05801,111UICFreightS06801,111UIC1991–2000PassengerIC2001,290Instr.44/fPassengerEC1601,221Instr.44/fPassengerEXPR1501,205Instr.44/fPassengerDIR1401,190Instr.44/fPassengerETR2501,391Instr.44/fFreightTEC1201,161Instr.44/fFreightMerci1001,135Instr.44/fFreightTreno merci1001,135Instr.44/fFreightTreno merci1201,161Instr.44/f2001–:Fatigue loadLM 71—1,380EN1991-2


### 2.2. Assessment Procedure

The assessment method is based on a probabilistic evaluation of the reliability marginG defined in the form:(4)G=R-E≥0,
where R is the structural resistance and E represents action effects. The probability of failure is defined as:(5)Pf=P(G<0)=P(R-E<0).
Assuming statistical independence of R and E, the probability of failure can be defined as:(6)Pf=∫-∞∞fE(x)ϕR(x)dx,
where ϕR(x) is the cumulative function distribution of structural resistance R,     (7)P(R<x)=ϕR(x).fE(x) expresses probability occurrence of action effects E in the near of the point x(8)P(x-dx2≤E≤x+dx2)=fE(x)dx.
As a consequence, the probability of failure could be expressed as:(9)Pf=∫-∞0fG(G)dG,
and the reliability index could be expressed as [26]:(10)β=mGsG=mR-mEsR2+sE2.
The aforementioned procedure could be specified to the bridge case study, where the calculated probability of fatigue fracture to obtain the probability of failure could be estimated as:(11)Pf=Pfat(1-Pdet⁡),
where Pf is failure probability; Pfat is probability of fatigue fracture; Pdet is probability of crack detection that is considered zero (Pdet=0) since structural health monitoring system has been used on the bridge [27, 28]. The probability of failure can also be expressed with the reliability index according to the normal standard distribution. Finally the reliability of a structural element is compared to the target value:(12)βfail≥βtarget,
where βfail is reliability index with respect to failure; βtarget is target reliability index. This model implies the use of the fatigue action effect (the required nominal fatigue strength) as “required operational load factor αreq” which is obtained by dividing the required nominal fatigue strength by the action effect of the fatigue load:(13)αreq=ΔσC,reqΔσ(ΦQfat),
where ΔσC,req is the required nominal fatigue strength; αreq is the required operational load factor; Δσ(ΦQfat)-stress range due to the load model adopted at worst position (Qfat), considering the dynamic amplification (Φ,e.g., maximizing the fatigue stress amplitude). For a simplified probabilistic approach, a relation between mean value of required operational load factor m(logαreq) and number of future train passages Nfut has been introduced [28]. The mean of the required operational load factor m(logαreq) is then read for fatigue category chosen (expressed as ND, number of load cycles corresponding to the constant-amplitude fatigue limit) starting from a number of future trains Nfut (as from 2005); this relation could be used for any influence lengths [28], commissioning time, and freight traffic fraction. According to the same model, a value of 0.04 may be taken as standard deviation of the required operational load factors, resulting from the assumed fuzziness of the traffic model [28]. Adopting the following notation and assumption:(14)β(Nfut)=mR-mE(Nfut)sR2+sE2,
where sE is the standard deviation of the required fatigue strength; β(Nfut) is the reliability index; mr=logΔσc+2sr is the mean of the fatigue strength (logΔσ relating to 2 × 106 cycles); mE(Nfut)=m(logαreq)+logΔσ(Φ·Qfat) is the mean of the required fatigue strength as a function of the number of future trains Nfut; sR=slog(N)/m is the standard deviation of the fatigue strength where m is the slope of the S-N curve, and s(logN) is the standard deviation of test results; sE is the standard deviation of the fatigue strength. It should be observed that the variability in S-N curve is only on life and not on stress range and that the variability of stress for a given life is not statistically related to variability of life for a given stress. Specific values of β are recommended for a determinate remaining service life, according to ISO/CD 13822 [29]: for the assessment of existing structures and the fatigue limit state, reference indexes should be βmax=3.1 for not visible detail and βmin=2.3 for visible detail. The reliability model herein presented has been related to the aforementioned traffic spectra and loadings.


### 2.3. Damage Accumulation

Due to the inherent disadvantage of theS-N curve approach, which cannot incorporate information on crack size, an alternative approach based on LEFM concepts [30, 31] is considered in this study. The LEFM approach is based on crack propagation theory [32–35]. The Paris law [36], the most common LEFM-based crack growth model, is used since it retains the simplicity of the fatigue evaluation process. This can be described as:(15)dadN=C⋅ΔKm,
where a is the crack size, N is the number of stress cycles, C and m are material constants and ΔK is the stress intensity range. According to LEFM theory [31], ΔK can be estimated as:(16)ΔK=Y(a)⋅Δσπ⋅a,
where Δσ is the tensile stress range, Y(a) is the geometry function to take into account stress concentrations [37], such as, the stress concentration coefficient and the dimensions of the specimen under consideration. It is not completely clear how the stress cycles below a constant amplitude fatigue limit affect the fatigue life [38]. Stress cycles due to live loads could be lower than the fatigue limit [39] and in these particular cases, according to Miner’s rule [40] and to damage verifications based on Eurocodes (e.g., [41]), they should not produce any damage. The damage model developed in this paper considers the damage due to stress cycles below the cut-off limit which causes a damage in term of crack propagation according to LEFM principles [42]. The adopted damage model implies that stress ranges are damage effective only if Δσth is exceeded, where Δσth is the damage limit and ΔσD is the fatigue limit for constant amplitude stress ranges at the number of cycles N=5*106, defined by [30] and taking a0 as the initial crack size, the Δσth could be expressed as:(17)Δσth=ΔσDY(a0)⋅π⋅a0Y(a)⋅π⋅a.
That could be written as(18)Δσth=ΔσDf(D)
being(19)f(D)=Y(a0)⋅π⋅a0Y(a)⋅π⋅a.
Combining (15) and (16), with Y = constant,(20)dadN=C⋅Ym⋅Δσm⋅(π⋅a)m/2,
and taking(21)B=C⋅Ym⋅Δσm⋅πm/2,(22)dadN=B⋅am/2.
Equation (22) can be written as(23)∫a0aiam/2da=∫0NiBdN,
in which ai is the depth of the crack at a number of cycles equal to Ni. According to Kunz [30] the initial size of the crack a0=0.1 mm. According to Bremen [43], (15) could be also written as:(24)dadN=C⋅(ΔKm-ΔKthm).
And according to (16),(25)ΔKth=Y(a)⋅Δσthπ⋅a.
It follows that(26)dadN=C⋅Y(a)m⋅(π⋅a)m/2⋅(Δσm-Δσthm)∫a0acritY(a)-m⋅a-m/2da=∫0NC⋅πm/2⋅(Δσm-Δσthm)dN=C⋅πm/2⋅(Δσm-Δσthm)⋅N.
And being constant(27)∫a0acritY(a)-m⋅a-m/2daC⋅πm/2.
It follows that(28)(Δσm-Δσthm)⋅N=constant(Δσim-Δσthm)⋅Ni=(Δσkm-Δσthm)⋅Nkdi=1Ni.
The single damage increases, taking Δσth as the cut-off limit and Δσk as the category detail at the number of cycles N=2*106, according to Kunz [30] is represented by:(29)di=Δσim-ΔσthmΔσkm-Δσthm⋅1Nk,
where Δσi = applied stress range, m=S-N curve slope, D = total damage.Failure will occur when the accumulated damageD=Σdi=1, according to Miner [40]. In the case study analyzed, the bridge has been built in 1918 and dismantled in 2005. Table 4 shows that damage accumulation starts with A05 UIC train (1909–1923), and all the following trains contribute to the damage as reported. For every train type contributing to damage, the number of cycles of the detail category (Nc), the damage increasing (di), the axles number passed at the end of the period (Ri) are reported. Category C=112 for bending detail and C=100 for shear detail has been assumed, as suggested by EN 1993-1-9 [41]: the category detail reference has been made according to Eurocode indications. Load models are described by concentrated characteristic axle load that implies cycle fluctuation in the structural components: stress variations (Δσ,Δτ) have been counted as per ASTM [44]. Basing on real scale structural tests [45] a linear correlation between axle load (P) and stress variations, without dynamic amplification factor, has been reported in Figures 6 and 7. Moreover, Table 5 reports the cycle counting model adopted, while in Table 6 a damage calculation example making reference to a single train, UIC A03 from 1900–1908 is reported.Table 4
Damage accumulation from 1918.
D=ΣdiNcdiRidi×RiTrain A03   (1900–1908)0,000E + 002,0E+ 060,00000000970,0E +000,0000,000E + 002,0E +060,00000000970,0E +000,0000,000E + 002,0E +060,00000000970,0E +000,000Train A04   (1900–1908)0,000E + 002,0E +060,00000018230,0E +000,0000,000E + 002,0E +060,00000018230,0E +000,0000,000E + 002,0E +060,00000018230,0E +000,000Train A05    (1909–1923)0,000E + 002,0E +060,00000031466,2E +040,0191,950E − 022,0E +060,00000031936,2E +040,0203,928E − 022,0E +060,00000032351,2E +050,0407,938E − 022,0E +060,00000033051,9E +050,0611,408E − 012,0E +060,00000033852,8E +050,094Train A06   (1909–1923)2,352E − 012,0E +060,00000048954,1E +040,0202,554E − 012,0E +060,00000048964,1E +040,0202,757E − 012,0E +060,00000048962,1E +040,0102,858E − 012,0E +060,00000048972,1E +040,0102,959E − 012,0E +060,00000048974,1E +040,0203,161E − 012,0E +060,00000048975,0E +050,243Table 5
Model adopted for the cycle counting (e.g., UIC A03 train, from 1900–1908).
PeriodTrainTrain/dayTractor (L) carriage (Ci)kN/axlen° axlesn° carriageAxles/dayΣF [kN]n° Axles/year17,2L140,003151,54201,9E +0417,2L120,002134,32401,3E +041900–1908A0317,2L100,001117,21006,3E +0317,2C150,0036308,99001,1E +05Tot.411,816601,5E +05Table 6
Model adopted for the damage calculation (e.g., UIC A03 train, from 1900–1908).
ΔσΦϕ*ΔσmΔσthΔσc,reqD=ΣdiNcdiRidi×Ri62,01,24577,19373,701000,00E +002,0E +063,62E − 081,7E +050,00660,01,24574,70373,701006,12E − 032,0E +061,38E − 081,1E +050,00222,01,24527,45373,551007,67E − 032,0E +060,00E +005,6E +040,00011,01,24513,72373,511007,67E − 032,0E +060,00E +001,0E +060,000Figure 6
Load-stress relation for detail 1.Figure 7
Load-stress relation for detail 2.


### 2.4. Calculation of Reliability Index

The reliability analysis of detail 1 and 2 (Figures8, 9, and 10) has been performed assuming the methodology described above and by adopting the aforementioned load models, applied to the hot-spot details: the midspan bottom flange and the riveted connection of the short shear diaphragm transverse connecting the principle beams. Table 7 reports the analysis from 2005-future for detail 1 and load method A, while Table 8 for load method B. Table 9 reports the analysis from 2005-future for detail 2 and load method A, Table 10 for load method B: as could be observed, these tables illustrate the increasing number of future trains (Nfut), the value of βmax and βmin, the decreasing value of β(Nfut), and the bridge status according to this analysis; moreover, the precise value of the out of service year is reported at the end of the bridge damage lifecycle. According to these values, the reliability index trends are plotted (Figures 11, 12, 13, and 14). Moreover, the hypothesis to repair the detail 2 has been analyzed: the precise time interval has been identified in the average value β=2,70=(βmax-βmin)/2 (Figure 15). Results have highlighted that the detail 1, assuming the load model A, will reach the out of service in 2043, while the same detail loaded with the model B terminates in 2049; the detail 2, the hot spot of this structure, based on the load model A will be the out of service in 2021, while the same detail loaded with the model B goes out of service in 2027: by repairing the more damaged detail 2, it is possible to increase the service life of the bridge of about 12 years.Table 7
Reliability analysis 2005-future, detail1, load model A.
YearNfutβmaxβminβ(Nfut)Status20052,09E +053,12,32,71In service20105,36E +053,12,32,65In service20158,76E +053,12,32,59In service20201,23E +063,12,32,54In service20251,59E +063,12,32,48In service20301,96E +063,12,32,42In service20352,35E +063,12,32,37In service20402,74E +063,12,32,32In service20453,14E +063,12,32,26Out of service in 2043Table 8
Reliability analysis 2005-future, detail1, load model B.
YearNfutβmaxβminβ(Nfut)Status20052,09E +053,12,32,57In service20105,36E +053,12,32,48In service20158,76E +053,12,32,41In service20201,23E +063,12,32,36In service20251,59E +063,12,32,34In service20301,96E +063,12,32,34In service20352,35E +063,12,32,34In service20402,74E +063,12,32,34In service20453,14E +063,12,32,32In service20503,55E +063,12,32,29Out of service in 2049Table 9
Reliability analysis 2005-future, detail 2, load model A.
YearNfutβmaxβminβ(Nfut)Status20052,09E +053,12,32,80In service20105,36E +053,12,32,63In service20158,76E +053,12,32,46In service20201,23E +063,12,32,31In service20251,59E +063,12,32,17Out of service in 2021Table 10
Reliability analysis 2005-future, detail 2, load model B.
YearNfutβmaxβminβ(Nfut)Status20052,09E +053,12,32,65In service20105,36E +053,12,32,52In service20158,76E +053,12,32,43In service20201,23E +063,12,32,36In service20251,59E +063,12,32,32In service20301,96E +063,12,32,28Out of service in 2027Figure 8
Details considered: midspan inferior flange (1) and short shear diaphragm (2).Midspan bottom flange (a) and typical failure (b).
(a)(b)Riveted connection of the short-shear diaphragm (a) and typical failure (b).
(a)(b)Figure 11
Reliability trend, detail 1, load model A.Figure 12
Reliability trend, detail 1, load model B.Figure 13
Reliability trend, detail 2, load model A.Figure 14
Reliability trend, detail 2, load model B.Figure 15
Reliability trend, detail 2, load model B, repair hypothesis.


## 2.1. Load History Assumption

Railway traffic estimation has been based on International Union of Railways, UIC [21] and on real data observed in the railway network from 1900 to 1990. A quadratic polynomial regression based on these data was performed and used for future traffic estimation (Figure 3). The maximum traffic capacity of the railway line in which the bridge is included has been assumed equal to 235 trains/day, following to the maximum capacity of the main national railway lines. According to these assumptions, the adopted traffic model are shown in Figures 3 and 4: the past real traffic and its growth tendency by adopting the UIC regression is shown in Figure 3, while the capacity of the line is developed in Figure 4 according to different traffic trend evolution, until a maximum of 235 trains/day. Figure 4 shows the traffic estimation from 1900 to 2020, with increasings of 1%, 2%, 3% and the UIC regression tendency (percentage increasing versus time) to estimate the number of trains passed on the bridge. UIC traffic data have been considered for the past, whereas some assumptions have been proposed for the future. The method described in the following is based on the aforementioned traffic estimation, and this scenario is also in accordance with CER [23]: in fact for the period 1995–2004, CER traffic increase is of 16%, just like UIC traffic increase. The past traffic has been assumed with a rate of 50% of passengers and 50% of freight trains [24]. Traffic assumption includes train type and convoy numbers: from 1900 to 1990 these data are presented in Table 1, while from 1991–2000 traffic was estimated according to Instruction 44/F [25], as described in Table 2. As a matter of fact, load models have been implemented as they were coded in the different historical periods analyzed:(i)
the historical loadings have been assumed according to UIC 779-1 [21];(ii)
from 1991 to 2000, the traffic spectrum has been based on the Instruction 44/F [25];(iii)
from 2001 up to now, loads refers to LM71 (Figure5; [22]) load model (method A) or to Instruction 44/F (method B) [25].Table 1
Traffic type for the historical train (1900–1990).
PeriodTrain/dayTypeUIC ref.Train type [%]Train/day for type1900–190834,3Tot.17,2PassengerA03100%17,217,2FreightA04100%17,21909–192347,2Tot.23,6PassengerA0560%14,1A0640%9,423,6FreightA07100%23,61924–193860,0Tot.30,0PassengerA0860%18,0A0940%12,030,0FreightA0760%18,0A1040%12,01939–195355,7Tot.27,8PassengerA1150%13,9A1250%13,927,8FreightA1060%16,7A1340%11,11954–196872,8Tot.36,4PassengerA1425%9,1A1517%6,2A1733%12,0A1825%9,136,4FreightA1640%14,6A2060%21,91969–1983102,8Tot.51,4PassengerA1428%14,4A1836%18,5A1936%18,551,4FreightA2040%20,6A2160%30,91984–1990120,0Tot.60,0PassengerS0156%33,6S0244%26,460,0FreightS0337%22,2S0437%22,2S0513%7,8S0613%7,8Table 2
Traffic type for the period 1991–2000.
Train typeNameTrain/dayAxle/dayLocomotive (L) and carriages (Ci)t/axleWagon numberWagon typeAxle spacing [m]1IC20960L20,2512.6-6.4-2.6IntercityC11552.56-16.44-2.56C212,7562.56-16.44-2.562EC10340L2012.85-2.35-2.85-2.35-2.85EurocityC114,2522.56-16.44-2.56C21252.56-16.44-2.563EXPR15990L2012.85-2.35-2.85-2.35-2.85ExpressC114,25102.56-16.44-2.56C21252.56-16.44-2.564DIR301380L18,612.85-2.35-2.85-2.35-2.85DirectC110,675102.4-16.6-2.45ETR10480L2023-9-3EurostarC111,6103-17.3-36TEC15990L18,712.85-2.35-2.85-2.35-2.85Container freightC120151.8-12.8-1.87Merci acciaio10720L18,722.85-2.35-2.85-2.35-2.85Steel freightC120151.8-13.06-1.88Trenomerci tipo D45380L2022.85-2.35-2.85-2.35-2.85D4 freightC122,5161.8-4.65-1.89Trenomerci misto5270L18,712.85-2.35-2.85-2.35-2.85MixedfreightC116249Figure 3
Traffic increment regression based on UIC [21] real train (percentage increment versus time).Figure 4
Traffic estimation from 1900 to 2020, with increments of 1%, 2%, 3% and UIC regression tendency (percentage increment versus time).Figure 5
Load Model 71 according to EN 1991-2 [22].For historical trains and for Instruction 44/F trains, dynamic amplification factorφ is calculated according to EN 1991-2 [22]:(1)φ=1+12(φ′+1/2φ′′),
where:(2)φ′=K1-K+K4,K=ν160forL<20m,K=ν47.16⋅L0.408forL>20m,φ′′=0.56⋅e-L2/100,
where ν is the train speed (m/s) and L is the determinant length LF (m). For LM71 load model (method A), the dynamic amplification factor has been calculated according to EN 1991-2 [22](3)ϕ3=2,16Lϕ-0,2+0,73
assuming 1,00≤Φ3≤2,00, and LF = determinant length. All the dynamic amplification factors applied have been reported in Table 3.Table 3
Dynamic amplification factor (detail 1).
PeriodTrain typeTrain nameSpeed [km/h]D.A.F. [dynamic amplification factor]φ,ϕ3Code ref.1900–1983Passenger/freightA03/A211001,135UIC1984–1990PassengerS011251,168UICPassengerS021401,190UICFreightS03801,111UICFreightS04801,111UICFreightS05801,111UICFreightS06801,111UIC1991–2000PassengerIC2001,290Instr.44/fPassengerEC1601,221Instr.44/fPassengerEXPR1501,205Instr.44/fPassengerDIR1401,190Instr.44/fPassengerETR2501,391Instr.44/fFreightTEC1201,161Instr.44/fFreightMerci1001,135Instr.44/fFreightTreno merci1001,135Instr.44/fFreightTreno merci1201,161Instr.44/f2001–:Fatigue loadLM 71—1,380EN1991-2


## 2.2. Assessment Procedure

The assessment method is based on a probabilistic evaluation of the reliability marginG defined in the form:(4)G=R-E≥0,
where R is the structural resistance and E represents action effects. The probability of failure is defined as:(5)Pf=P(G<0)=P(R-E<0).
Assuming statistical independence of R and E, the probability of failure can be defined as:(6)Pf=∫-∞∞fE(x)ϕR(x)dx,
where ϕR(x) is the cumulative function distribution of structural resistance R,     (7)P(R<x)=ϕR(x).fE(x) expresses probability occurrence of action effects E in the near of the point x(8)P(x-dx2≤E≤x+dx2)=fE(x)dx.
As a consequence, the probability of failure could be expressed as:(9)Pf=∫-∞0fG(G)dG,
and the reliability index could be expressed as [26]:(10)β=mGsG=mR-mEsR2+sE2.
The aforementioned procedure could be specified to the bridge case study, where the calculated probability of fatigue fracture to obtain the probability of failure could be estimated as:(11)Pf=Pfat(1-Pdet⁡),
where Pf is failure probability; Pfat is probability of fatigue fracture; Pdet is probability of crack detection that is considered zero (Pdet=0) since structural health monitoring system has been used on the bridge [27, 28]. The probability of failure can also be expressed with the reliability index according to the normal standard distribution. Finally the reliability of a structural element is compared to the target value:(12)βfail≥βtarget,
where βfail is reliability index with respect to failure; βtarget is target reliability index. This model implies the use of the fatigue action effect (the required nominal fatigue strength) as “required operational load factor αreq” which is obtained by dividing the required nominal fatigue strength by the action effect of the fatigue load:(13)αreq=ΔσC,reqΔσ(ΦQfat),
where ΔσC,req is the required nominal fatigue strength; αreq is the required operational load factor; Δσ(ΦQfat)-stress range due to the load model adopted at worst position (Qfat), considering the dynamic amplification (Φ,e.g., maximizing the fatigue stress amplitude). For a simplified probabilistic approach, a relation between mean value of required operational load factor m(logαreq) and number of future train passages Nfut has been introduced [28]. The mean of the required operational load factor m(logαreq) is then read for fatigue category chosen (expressed as ND, number of load cycles corresponding to the constant-amplitude fatigue limit) starting from a number of future trains Nfut (as from 2005); this relation could be used for any influence lengths [28], commissioning time, and freight traffic fraction. According to the same model, a value of 0.04 may be taken as standard deviation of the required operational load factors, resulting from the assumed fuzziness of the traffic model [28]. Adopting the following notation and assumption:(14)β(Nfut)=mR-mE(Nfut)sR2+sE2,
where sE is the standard deviation of the required fatigue strength; β(Nfut) is the reliability index; mr=logΔσc+2sr is the mean of the fatigue strength (logΔσ relating to 2 × 106 cycles); mE(Nfut)=m(logαreq)+logΔσ(Φ·Qfat) is the mean of the required fatigue strength as a function of the number of future trains Nfut; sR=slog(N)/m is the standard deviation of the fatigue strength where m is the slope of the S-N curve, and s(logN) is the standard deviation of test results; sE is the standard deviation of the fatigue strength. It should be observed that the variability in S-N curve is only on life and not on stress range and that the variability of stress for a given life is not statistically related to variability of life for a given stress. Specific values of β are recommended for a determinate remaining service life, according to ISO/CD 13822 [29]: for the assessment of existing structures and the fatigue limit state, reference indexes should be βmax=3.1 for not visible detail and βmin=2.3 for visible detail. The reliability model herein presented has been related to the aforementioned traffic spectra and loadings.


## 2.3. Damage Accumulation

Due to the inherent disadvantage of theS-N curve approach, which cannot incorporate information on crack size, an alternative approach based on LEFM concepts [30, 31] is considered in this study. The LEFM approach is based on crack propagation theory [32–35]. The Paris law [36], the most common LEFM-based crack growth model, is used since it retains the simplicity of the fatigue evaluation process. This can be described as:(15)dadN=C⋅ΔKm,
where a is the crack size, N is the number of stress cycles, C and m are material constants and ΔK is the stress intensity range. According to LEFM theory [31], ΔK can be estimated as:(16)ΔK=Y(a)⋅Δσπ⋅a,
where Δσ is the tensile stress range, Y(a) is the geometry function to take into account stress concentrations [37], such as, the stress concentration coefficient and the dimensions of the specimen under consideration. It is not completely clear how the stress cycles below a constant amplitude fatigue limit affect the fatigue life [38]. Stress cycles due to live loads could be lower than the fatigue limit [39] and in these particular cases, according to Miner’s rule [40] and to damage verifications based on Eurocodes (e.g., [41]), they should not produce any damage. The damage model developed in this paper considers the damage due to stress cycles below the cut-off limit which causes a damage in term of crack propagation according to LEFM principles [42]. The adopted damage model implies that stress ranges are damage effective only if Δσth is exceeded, where Δσth is the damage limit and ΔσD is the fatigue limit for constant amplitude stress ranges at the number of cycles N=5*106, defined by [30] and taking a0 as the initial crack size, the Δσth could be expressed as:(17)Δσth=ΔσDY(a0)⋅π⋅a0Y(a)⋅π⋅a.
That could be written as(18)Δσth=ΔσDf(D)
being(19)f(D)=Y(a0)⋅π⋅a0Y(a)⋅π⋅a.
Combining (15) and (16), with Y = constant,(20)dadN=C⋅Ym⋅Δσm⋅(π⋅a)m/2,
and taking(21)B=C⋅Ym⋅Δσm⋅πm/2,(22)dadN=B⋅am/2.
Equation (22) can be written as(23)∫a0aiam/2da=∫0NiBdN,
in which ai is the depth of the crack at a number of cycles equal to Ni. According to Kunz [30] the initial size of the crack a0=0.1 mm. According to Bremen [43], (15) could be also written as:(24)dadN=C⋅(ΔKm-ΔKthm).
And according to (16),(25)ΔKth=Y(a)⋅Δσthπ⋅a.
It follows that(26)dadN=C⋅Y(a)m⋅(π⋅a)m/2⋅(Δσm-Δσthm)∫a0acritY(a)-m⋅a-m/2da=∫0NC⋅πm/2⋅(Δσm-Δσthm)dN=C⋅πm/2⋅(Δσm-Δσthm)⋅N.
And being constant(27)∫a0acritY(a)-m⋅a-m/2daC⋅πm/2.
It follows that(28)(Δσm-Δσthm)⋅N=constant(Δσim-Δσthm)⋅Ni=(Δσkm-Δσthm)⋅Nkdi=1Ni.
The single damage increases, taking Δσth as the cut-off limit and Δσk as the category detail at the number of cycles N=2*106, according to Kunz [30] is represented by:(29)di=Δσim-ΔσthmΔσkm-Δσthm⋅1Nk,
where Δσi = applied stress range, m=S-N curve slope, D = total damage.Failure will occur when the accumulated damageD=Σdi=1, according to Miner [40]. In the case study analyzed, the bridge has been built in 1918 and dismantled in 2005. Table 4 shows that damage accumulation starts with A05 UIC train (1909–1923), and all the following trains contribute to the damage as reported. For every train type contributing to damage, the number of cycles of the detail category (Nc), the damage increasing (di), the axles number passed at the end of the period (Ri) are reported. Category C=112 for bending detail and C=100 for shear detail has been assumed, as suggested by EN 1993-1-9 [41]: the category detail reference has been made according to Eurocode indications. Load models are described by concentrated characteristic axle load that implies cycle fluctuation in the structural components: stress variations (Δσ,Δτ) have been counted as per ASTM [44]. Basing on real scale structural tests [45] a linear correlation between axle load (P) and stress variations, without dynamic amplification factor, has been reported in Figures 6 and 7. Moreover, Table 5 reports the cycle counting model adopted, while in Table 6 a damage calculation example making reference to a single train, UIC A03 from 1900–1908 is reported.Table 4
Damage accumulation from 1918.
D=ΣdiNcdiRidi×RiTrain A03   (1900–1908)0,000E + 002,0E+ 060,00000000970,0E +000,0000,000E + 002,0E +060,00000000970,0E +000,0000,000E + 002,0E +060,00000000970,0E +000,000Train A04   (1900–1908)0,000E + 002,0E +060,00000018230,0E +000,0000,000E + 002,0E +060,00000018230,0E +000,0000,000E + 002,0E +060,00000018230,0E +000,000Train A05    (1909–1923)0,000E + 002,0E +060,00000031466,2E +040,0191,950E − 022,0E +060,00000031936,2E +040,0203,928E − 022,0E +060,00000032351,2E +050,0407,938E − 022,0E +060,00000033051,9E +050,0611,408E − 012,0E +060,00000033852,8E +050,094Train A06   (1909–1923)2,352E − 012,0E +060,00000048954,1E +040,0202,554E − 012,0E +060,00000048964,1E +040,0202,757E − 012,0E +060,00000048962,1E +040,0102,858E − 012,0E +060,00000048972,1E +040,0102,959E − 012,0E +060,00000048974,1E +040,0203,161E − 012,0E +060,00000048975,0E +050,243Table 5
Model adopted for the cycle counting (e.g., UIC A03 train, from 1900–1908).
PeriodTrainTrain/dayTractor (L) carriage (Ci)kN/axlen° axlesn° carriageAxles/dayΣF [kN]n° Axles/year17,2L140,003151,54201,9E +0417,2L120,002134,32401,3E +041900–1908A0317,2L100,001117,21006,3E +0317,2C150,0036308,99001,1E +05Tot.411,816601,5E +05Table 6
Model adopted for the damage calculation (e.g., UIC A03 train, from 1900–1908).
ΔσΦϕ*ΔσmΔσthΔσc,reqD=ΣdiNcdiRidi×Ri62,01,24577,19373,701000,00E +002,0E +063,62E − 081,7E +050,00660,01,24574,70373,701006,12E − 032,0E +061,38E − 081,1E +050,00222,01,24527,45373,551007,67E − 032,0E +060,00E +005,6E +040,00011,01,24513,72373,511007,67E − 032,0E +060,00E +001,0E +060,000Figure 6
Load-stress relation for detail 1.Figure 7
Load-stress relation for detail 2.


## 2.4. Calculation of Reliability Index

The reliability analysis of detail 1 and 2 (Figures8, 9, and 10) has been performed assuming the methodology described above and by adopting the aforementioned load models, applied to the hot-spot details: the midspan bottom flange and the riveted connection of the short shear diaphragm transverse connecting the principle beams. Table 7 reports the analysis from 2005-future for detail 1 and load method A, while Table 8 for load method B. Table 9 reports the analysis from 2005-future for detail 2 and load method A, Table 10 for load method B: as could be observed, these tables illustrate the increasing number of future trains (Nfut), the value of βmax and βmin, the decreasing value of β(Nfut), and the bridge status according to this analysis; moreover, the precise value of the out of service year is reported at the end of the bridge damage lifecycle. According to these values, the reliability index trends are plotted (Figures 11, 12, 13, and 14). Moreover, the hypothesis to repair the detail 2 has been analyzed: the precise time interval has been identified in the average value β=2,70=(βmax-βmin)/2 (Figure 15). Results have highlighted that the detail 1, assuming the load model A, will reach the out of service in 2043, while the same detail loaded with the model B terminates in 2049; the detail 2, the hot spot of this structure, based on the load model A will be the out of service in 2021, while the same detail loaded with the model B goes out of service in 2027: by repairing the more damaged detail 2, it is possible to increase the service life of the bridge of about 12 years.Table 7
Reliability analysis 2005-future, detail1, load model A.
YearNfutβmaxβminβ(Nfut)Status20052,09E +053,12,32,71In service20105,36E +053,12,32,65In service20158,76E +053,12,32,59In service20201,23E +063,12,32,54In service20251,59E +063,12,32,48In service20301,96E +063,12,32,42In service20352,35E +063,12,32,37In service20402,74E +063,12,32,32In service20453,14E +063,12,32,26Out of service in 2043Table 8
Reliability analysis 2005-future, detail1, load model B.
YearNfutβmaxβminβ(Nfut)Status20052,09E +053,12,32,57In service20105,36E +053,12,32,48In service20158,76E +053,12,32,41In service20201,23E +063,12,32,36In service20251,59E +063,12,32,34In service20301,96E +063,12,32,34In service20352,35E +063,12,32,34In service20402,74E +063,12,32,34In service20453,14E +063,12,32,32In service20503,55E +063,12,32,29Out of service in 2049Table 9
Reliability analysis 2005-future, detail 2, load model A.
YearNfutβmaxβminβ(Nfut)Status20052,09E +053,12,32,80In service20105,36E +053,12,32,63In service20158,76E +053,12,32,46In service20201,23E +063,12,32,31In service20251,59E +063,12,32,17Out of service in 2021Table 10
Reliability analysis 2005-future, detail 2, load model B.
YearNfutβmaxβminβ(Nfut)Status20052,09E +053,12,32,65In service20105,36E +053,12,32,52In service20158,76E +053,12,32,43In service20201,23E +063,12,32,36In service20251,59E +063,12,32,32In service20301,96E +063,12,32,28Out of service in 2027Figure 8
Details considered: midspan inferior flange (1) and short shear diaphragm (2).Midspan bottom flange (a) and typical failure (b).
(a)(b)Riveted connection of the short-shear diaphragm (a) and typical failure (b).
(a)(b)Figure 11
Reliability trend, detail 1, load model A.Figure 12
Reliability trend, detail 1, load model B.Figure 13
Reliability trend, detail 2, load model A.Figure 14
Reliability trend, detail 2, load model B.Figure 15
Reliability trend, detail 2, load model B, repair hypothesis.


## 3. Conclusions

This work deals with the estimation of the fatigue damage in existing railway metal bridges and the remaining fatigue life according to a detailed loading history analysis. The method is then applied to a real case study. In terms of loadings, a detailed loading history analysis has been performed by adopting two different load methods, A and B: the historical loadings have been assumed according to UIC 779-1 [21]; from 1991 to 2000, the traffic spectrum has been based on the Instruction 44/F [25]; from 2001 up to now, loads refer to LM71 load model (method A) or to Instruction 44/F (method B). It was confirmed that the critical fatigue detail of the bridge is located in the riveted connection of the short-shear diaphragm connecting principle beams as shown in previous mentioned studies. Moreover, different results according to different load methods related to the future traffic (i.e., load A and load B) should influence the expected fatigue life; in particular, a more realistic fatigue load model (load A) could lead to an extension of the fatigue life of the investigated bridge; on the contrary, an assessment based on conservative and approximate code load model could lead to a shorter lifetime prediction. The method described in this work enables a better understanding of the damage level in steel bridges, and could help to maintain in service existing bridges, adopting detailed loading history analysis. As a matter of fact, managing authorities should be aware of the possibility to correctly estimate residual life of existing infrastructure, for example, by implementing maintenance program based on advanced analytical assessment.



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*Source: 231674-2012-07-24.xml*