Background to Fatigue Load Models

8/2/2019 Background to Fatigue Load Models

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Background to fatigue load models forEurocode 1: Part 2 Traffic LoadsPietro Croce

University of Pisa, Italy

SummaryThe paper deals with background studies on

definition and calibration of fatigue traffic load

models for road bridges in Eurocode 1: Part 2

Traffic Loads on Bridges. Starting from real traffic

data measurements, the derivation of the load

models is illustrated step by step, emphasizing

the peculiarities of the calibration methods

adopted in prenormative research. The basic

assumptions as well as the main features, the

accuracy and the philosophy of each load model

are discussed, and important questions pertaining

to vehicle interactions, not fully clarified in the

Eurocode, are tackled and solved in a general

way, in the framework of queueing theory.

Key words: Eurocode; bridges; fatigue; traffic loads; vehicle interactions; l method; load spectrum

Prog. Struct. Engng Mater. 2001; 3:335345 (DOI: 10.1002/pse.93)

Introduction

This paper is devoted to the critical illustration of

prenormative background studies which have beencarried out within the framework of EC1-2[1] to definefatigue load models for road traffic.

Fatigue is the progressive, localized and permanentstructural change occurring in a material subjected toconditions that produce fluctuating stresses and strains atsome point or points and that may culminate in cracks orcomplete fracture after a sufficient number offluctuations[2,3]. Fatigue is induced in engineeringstructures by actions and loads varying with timeand/or space and/or by random vibrations. Thus,fatigue can be originated by natural events, such as

waves or wind, or by loads deriving from the normalservice of the structure itself.

Among other structures, bridges are exposed tofatigue, under the action of lorries or trains crossingthem. The assignment of appropriate fatigue loadmodels is therefore a key topic in modern bridgecodes. In principle, modelling of fatigue loadsrequires complete knowledge of the so-called loadspectrum, expressing the load variation or the numberof recurrences of each load level during the design lifeof the structure. The load spectrum is generally givenin terms of an appropriate function, graph, histogram

or table. It is often deduced from recorded data,referring to relatively short time intervals. In this case,additional problems must be faced regarding thestatistical processing, the reliability over longerperiods and the future trends of available data.

Whenever the real load spectrum is so complicatedthat it cannot be directly employed for fatigue checks(the common situation for bridges), it is replaced by

some conventional load spectrum, aimed to reproducethe fatigue damage induced by the real one.The evaluation of conventional load spectra is

particularly thorny, because it requires considerationof the actions from the point of view of resistance. Infact, fatigue depends on the nature of the varyingactions and loads, and additionally on structuralmaterial details, through the shape and the propertiesof the relevant SN curves. Problems become eventougher when there is an endurance (fatigue) limit.Because the fatigue limit under constant amplituderepresents a threshold value for the damaging stress

range, one must distinguish between equivalent loadspectra, reproducing the actual fatigue damage, andfrequent load spectra, reproducing the maximum loadrange significant for fatigue, depending on whetherfatigue verifications require cumulative damagecomputations or boundless fatigue life assessments.

Moreover, the powerful methods of stochasticprocess theory, often used in defining fatigue loadspectra in other engineering structures, cannot beapplied to bridges, as road traffic loads induce broad-band stress histories. All that implies that the linkbetween action and effect cannot be expressed by

simple formulae, while further difficulties arise whenvehicle interactions, whether due to simultaneity ornot, become significant. Nevertheless, provided thatvehicle interaction problems can be solved in someway, as shown in the following, it is sufficient to

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Copyright & 2002 John Wiley & Sons, Ltd. Prog. Struct. Engng Mater. 2001; 3:335345


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consider that fatigue load spectra for bridges arecomposed of suitable sets of standardized lorries,where each lorry is associated with its own relevantproperties, i.e. frequency, number of axles, axle loads,inter-axle distances, as deduced by processing therelevant traffic measurements.

At this stage, it appears quite evident that thedefinition of load spectra for bridges requires carefulconsideration of fatigue assessment methodology, toassure that conventional spectra and real spectrainvolve the same fatigue resistance.

Fatigue verification methods

The preliminary explanation of fatigue assessmentmethodology based on conventional load spectra is acrucial question in studying fatigue load models. Itcan be easily recognized that fatigue verificationmethods follow a well-defined procedure,characterized by the following steps:

1. assignment of fatigue load spectra, discriminating,if necessary, equivalent ones from frequent ones;

2. detection and classification of structural detailsmost vulnerable to fatigue cracking and selection ofthe appropriate SN curves;

3. choice of the pertinent partial safety factors gMf;4. evaluation, for each detail, of the appropriate

influence surface.

At this stage, the methodology branches according

to whether fatigue verification is devoted tocomputation of fatigue damage or to assessment of boundless fatigue life.

DAMAGE COMPUTATION PROCEDURE

5a. calculation of the design stress history s stproduced in the detail by the equivalent loadspectrum traversing the influence surface;

6a. analysis of the stress history by means of asuitable cycle counting method, such as thereservoir method or the rainflow method[4], to

obtain the stress spectrum, giving the number ofoccurrences of each stress range in the referencetime interval;

7a. computation of the cumulative damage D usingthe PalmgrenMiner rule[5,6]: if D41, the fatiguecheck is satisfied, otherwise, it is necessary toraise the fatigue strength of the detail. Fatigueresistance can be enhanced both by reducing thestress range, i.e. enlarging the dimensions, or byincreasing the fatigue category, i.e. adoptingmore refined workmanship or details.

BOUNDLESS FATIGUE LIFE ASSESSMENT

5b. calculation of the design stress history s stproduced in the detail by the frequent loadspectrum traversing the influence surface;

6b. computation of the maximum stress rangeDsmax smax smin, where smax and smin are theabsolute maximum and the absolute minimumof the stress history;

7b. boundless fatigue life assessment. If theverification is not satisfied, it is possible toimprove fatigue resistance using the provisions

described in 7a, or to attempt a fatigue damagecomputation.

Evidently, in bridges exposed to high-densitytraffic, typical concrete slab and orthotropic steel deckdetails are subject to such huge number of stresscycles, that boundless fatigue life assessment usingfrequent load spectra becomes obligatory.

Reference traffic measurements

The fatigue load models of Eurocode 1 have beendefined and calibrated on the basis of a wide range ofEuropean traffic data measurements, resulting fromtwo large measurement campaigns carried out duringthe years 19771982 and 19841988, in severalEuropean countries. In particular, the statisticalanalyses of the recorded data allowed to conclude that

* daily flows of lorries range between 1000 and 8000in the slow lanes of motorways, between 600 and1500 in the slow lanes of main roads, reducingdrastically to 100200 lorries in both slow lanes of

secondary roads and fast lanes of motorways[79];* traffic measurements made in Auxerre (F) on the

motorway A6 Paris-Lyon are a good representationof heavy continental traffic in Europe[10].

Unlike static loads, which depend only on theupper tail, fatigue loads are influenced by the wholetraffic load distribution. Fatigue models have beenrefined accordingly, widening their field ofapplication, supplementing the main calibration,based on data from Auxerre, with secondarycalibration for other traffic measurements.

As motorways and main roads, serving longdistance itineraries, are affected by heavy commercialtraffic, characterized by a high percentage ofarticulated lorries, while secondary roads, servinglocal itineraries, are affected by lighter commercialtraffic, composed mostly by two axle lorries, thesecondary calibration considered motorwaytraffic}Auxerre (France), Brothal (Germany),Piacenza, Fiano Romano, Sasso Marconi (Italy)}aswell as local traffic on secondary roads}Epone(France). The studies also take into account theexpected traffic trend, that should cause, as confirmed

by new measurements[11,12],

* a marked increase in the percentage of articulatedlorries with a simultaneous reduction in thepercentage of lorries with trailers;

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* a reduction in the percentage of three-axle lorries infavour of two-axle lorries;

* an increase in the average load per lorry.

Fatigue load models of Eurocode 1

The calibration method, philosophy and mainfeatures of the fatigue models of Eurocode 1 aresummarized below, stressing the methodologicalapproach.

CALIBRATION METHODFatigue load models have been defined consideringreference influence surfaces relative to simplysupported and continuous bridges with spans of 3200m.

In agreement with the fatigue verification

procedure, calibration has been implementedaccording to the following scheme:

* choice of the most significant European traffic data;* selection of appropriate SN curves;* evaluation of the stress histories in reference

bridges;* cycle counting and stress spectra computation;* preliminary identification of fatigue models;* definition of standardized lorry geometries;* calibration of frequent load models, optimally

fitting the maximum stress range Dsmax induced byreal traffic;

* calibration of equivalent load models, optimallyfitting the fatigue damage D induced by real traffic.

REFERENCE S2N CURVESReference SN curves pertain to steel details,characterized by the endurance limit. In a logarithmicSN plot these curves are represented by a bilinearcurve, characterized by a branch of constant slope,m 3 (Fig. 1), or by a trilinear curve, characterized bytwo sloping branches, m 3 and m 5, (Fig. 2),according to whether boundless fatigue life or fatiguedamage is to be assessed[13,14]. As the fatigue limit DsDis taken into account, the maximum stress rangeDsmax of the real stress history can be above or belowthis limit. The conventional load models are then usedto reproduce the actual fatigue damage, ifDsmax > DsD, or Dsmax otherwise.

To be significant for fatigue, Dsmax must beexceeded several times during the bridge life and itsdefinition is by no means trivial. Two differentapproaches, leading to similar results, have beenproposed. In the former, Dsmax is defined as the stress

range such that 99% of the total fatigue damageresults from all stress ranges below Dsmax[14]. In thelatter, Dsmax is the stress range exceededapproximately 5 104 times during the bridge life.This last definition implies that the return period for

Dsmax is about half a day, this directly explaining whythe resulting load spectrum is said frequent.

To derive equivalent spectra independent of thefatigue classification, in EC1 studies cumulativedamage has generally been computed by referring tosimplified SNcurves with unique slope, either m 3(Fig. 3) or m 5 (Fig. 4), while SN curves with bothvalues of the slope (Fig. 5) have been used for someadditional calculations. Some comparisons show thatload spectra obtained from the simplified curve m 5are free from significant errors and reproduce theactual fatigue damage very well.

FATIGUE LOAD MODELSFrom the above-mentioned considerations, it followsthat at least two conventional fatigue load modelsmust be considered: one for boundless fatigue lifeassessments, the other for fatigue damage

S

N

m=3

O

Fig.1 Bilinear S^N curve

S

N510

610

m=3

m=5L

D

8O

Fig. 2 Trilinear S^N curve

S

N

m=3

O

Fig. 3 Simplied linear S^N curve (m 3)

FATIGUE LOAD MODELS FOR EUROCODE 1 PART 2 337



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calculations. Besides, since adequate fitting of theeffects induced by real traffic requires verysophisticated load models, whose application is oftendifficult, the introduction of simplified and

conservative models, to be used when sophisticatedchecks are unnecessary, seems appropriate.For this reason, in EC1, two fatigue load models are

foreseen for each kind of fatigue verification: theformer is essential, conservative and easy to use, thelatter is more refined and accurate, but a little morecomplicated. Finally, four conventional models aregiven:

* models 1 and 2 for boundless fatigue checks;* models 3 and 4 for damage computations.

Fatigue load model 1 is extremely simple andgenerally very conservative. It directly derives fromthe main load model used for assessing staticresistance, where the load values are simply reducedto the frequent values (Fig. 6a), multiplying thetandem axle loads Qik by 0.7 and the weight density ofthe uniformly distributed loads qik by 0.3. Obviously,

for local verifications, fatigue load model 1 applies toan isolated concentrated axle weighing Q 280 kN(Fig. 6b).

The verification consists of checking that themaximum stress range Dsmax induced by the model issmaller than the fatigue limit DsD. The applicationrules for load model 1 agree exactly with those givenfor the main load model[8], so that the absoluteminimum and maximum stresses correspond as a ruleto different load configurations. The model alsoallows one to make coarse verifications in multi-laneconfigurations, and generally tends towards the safe

side.The simplified fatigue model 3, conceived fordamage computation, applies to a symmetricalconventional four-axle vehicle, also termed thefatigue vehicle (Fig. 7). The equivalent load of eachaxle is 120 kN. This model is accurate enough forspans bigger than 10 m, while for smaller spans ittends towards the safe side.

The most refined fatigue models are load spectrafor five standardized vehicles, representative of themost common European lorries. Fatigue load model 2,

S

N

m=5

O

Fig. 4 Simplied linear S^N curve (m 5)

510

D

6

S

N

m=5

m=3

O

Fig. 5 Simplied bilinear S^N curve

Fig. 6 (a) Fatigue load model 1 (b) Fatigue load model 1 for local verications

40

160

40

200

40 80 40

120

40

160

40

200

40 80 40

120

traffic flow

direction

600

Fig. 7 Fatigue loadmodel 3}fatigue vehicle

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involving a set of lorries with frequent values of axleloads, and fatigue model 4, involving a set of lorrieswith equivalent values of the axle loads, are

illustrated in Tables 1 and 2, respectively. They allowone to perform very precise and sophisticatedverifications, provided that the interactions amongst

vehicles simultaneously crossing the bridge arenegligible, or taken appropriate account of.

In EC1 a further general purpose fatigue model isanticipated also, fatigue model 5. This model isconstituted by a sequence of consecutive axle loads,directly derived from traffic measurements, duly

supplemented to take into account vehicleinteractions, where relevant. Fatigue model 5 isdesigned to allow accurate fatigue verifications inparticular situations, such as suspended or cable-stayed bridges, important existing bridges or bridgescarrying unusual traffic, whose relevance justifies adhoc investigations[15].

Accuracy of fatigue load models

In the following, some significant results obtainedfrom the fatigue load models are compared with thosepertaining to reference traffic, allowing one to pointout the accuracy and the field of application of eachconventional model.

Essentially, the comparison concerns the influencelines illustrated in Fig. 8, for bridges of spanL 32100 m. The influence lines pertain to bendingmoment M0 at midspan of simply supported beams; bending moments M1 and M2 at midspan and on thesupport, respectively, of two-span continuous beams,and bending moment M3 at midspan of three-span

continuous beams. The comparison are summarizedin Figs. 913, depending on influence line andspan L.

.........................................................................

.........................................................................

Table 1 Fatigue load model 2: set of frequent lorries

Lorry silhouette Interaxles

(m)

Frequent axle

loads (kN)

4.50 90190

4.201.30

80140140

3.205.201.301.30

90180120120120

3.406.001.80

90190140140

4.803.604.401.30

90180120110110

..................................................................

......................................................................................................................................................

......................................................................................................................................................

Table 2 Fatigue load model 4: set of equivalent lorries

Traffic composition (%)

Lorry silhouette Interaxles

(m)

Equivalent axle

loads (kN)

Long distance Medium distance Local traffic

4.50 70130

20.0 50.0 80.0

4.20

1.30

70

120120

5.0 5.0 5.0

3.205.201.301.30

70150

909090

40.0 20.0 5.0

3.406.001.80

70140

9090

25.0 15.0 5.0

4.803.604.401.30

70130

908080

10.0 10.0 5.0




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Ratios DMmax;LM1=DMmax;real between themaximum stress range DMmax;LM1 due to fatiguemodel 1 and the maximum stress range DMmax;real dueto the Auxerre traffic are plotted against span L in Fig.9, while ratios DMmax;LM2=DMmax;real are similarlyplotted in Fig. 10, where DMmax;LM2 is the maximumstress range due to fatigue load model 2. Clearly,model 1 appears very conservative, especially forshort spans, while model 2 is much more reliable.Values slightly below the actual ones are estimated forM2 in the span range 2050 m, because of theparticular shape of the influence line.

Ratios DMeq;LM3=DMeq;real between the equivalentstress range DMeq;LM3 due to fatigue load model 3 andthe equivalent stress range DMeq;real due to theAuxerre traffic are plotted in Figs. 11 and 12,assuming m 3 and m 5, respectively, for the

slope of the linear SN curve. Similarly, ratiosDMeq;LM4=DMeq;real are plotted in Fig. 13 for m 3,where DMeq;LM4 is the equivalent stress range due tofatigue load model 4. As expected, model 4 fits theactual results for short influence lines very well.

Fatigue model 3 looks unsafe for M2 influence lineswhen spans are above 30 m, in particular for higher mvalues. To solve the problem it has been proposed[16]to modify model 3 by considering an additionalfatigue vehicle, running on the same lane 40 m behind

the first and having equivalent axle loads reduced to40 kN, each time that the influence surface exhibitstwo contiguous areas of the same sign. The inclusionof such an additional vehicle should mitigate the errorin computation ofDM2;eq, as it appears evident in

M0

M1

M2

L LL

L L

L L

L

M3

Fig. 8 Reference inuence lines

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

0 20 40 60 80 100

L [m]

M0

M1

M2

M3

realmax,

1LMmax,

M

M

Fig.9 Accuracy of fatigue loadmodel 1

0.8

0.9

1

1.1

1.21.3

1.4

0 5 10 15 20 25 30

L [m]

M0

M1

M2

realmax,

2LMmax,

M

M

Fig. 10 Accuracy of fatigue load model 2

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 20 40 60 80 100

L [m]

M0

M1

M2

M3

real,eq

3LM,eq

M

M

Fig.11 Accuracy of fatigue load model 3 (m 3)

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 20 40 60 80 100

L [m]

M0

M1

M2

M3

real,eq

3LM,eq

M

M

Fig.12 Accuracy of fatigue load model 3 (m 9)

0.8

0.9

1

1.1

1.2

0 5 10 15 20 25 30

L [m]

M0

M1

M2

M3

real,eq

4LM,eq

M

M

Fig.13 Accuracy of fatigue load model 4 (m

3)

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Fig. 14, where M2 calculations with the additionalfatigue vehicle are summarized for m 3 andm 9.

The k-coefficient method

Besides the usual damage computations based on thePalmgrenMiner rule, EC1 also foresees aconventional simplified fatigue assessment method,the l-coefficient method.

This method, derived from the methodologyemployed for railway bridges, is based on fatiguemodel 3 (fatigue vehicle) and is aimed at converting

fatigue verifications into conventional resistancechecks, comparing a conventional equivalent stressrange Dseq, depending on appropriate l-coefficients,with the detail category[16,17].

The equivalent stress range Dseq is given by

Dseq l1l2l3l4jfatDsp ljfatDsp (1)where Dsp sp;max sp;min is the maximum stressrange induced by fatigue model 3; l1 is a coefficientdepending on the shape and on the base length of theinfluence surface, i.e. on the number of secondarycycles in the stress history; l2 is a coefficient allowing

one to pass from reference traffic, used in fatiguemodel calibration, to expected traffic; l3 depends onthe design life of the bridge; l4 takes into accountvehicle interactions amongst lorries simultaneouslycrossing the bridge; and jfat is the equivalent dynamicmagnification factor for fatigue verifications.

The l1 values, given in graphical or tabular form,are derived in the calibration phase, comparing thedamage due to the fatigue vehicle with the damageproduced by a single stress cycle of maximum stressrange Dsp. If m is the slope of the SN curve

l1 P

i niDsmiDsmp

1m

(2)

l2 depends on the annual lorry flow and on trafficcomposition. In general, if N1 and Qm1 are the flow

and the equivalent weight of the actual traffic

Qm1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

i niQmiP

i ni

m

s(3)

and N0 and Q0 are the flow and the equivalent weightof the reference traffic

l2 k Qm1Q0

N1N0

1m

(4)

In eq. (4), k represents a conversion parameter, given by

k DefDv

Q0Qm1

(5)

where Dv is the damage produced by N0 fatiguevehicles and Def is the damage produced by N0 actuallorries. For the Auxerre traffic it is Q0 480 kN andN0 2 106 lorries=year.l3 is given by:

l3 ffiffiffiffiffiffi

TTR

mr

(6)

where TR is the reference design life (TR 100 yr) andT is the actual design life.l4, which accounts for vehicle interactions, can be

expressed as:

l4l; N1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN1N1

X

i

NiN1

ZiZ1

m !X

i

NcombN1

ZcombZ1

m !m

s

(7)

where N1 is the lorry flow on the main lane, Ni thelorry flow on the ith lane, Zi the maximum ordinate ofthe influence surface, corresponding to ith lane, Ni isthe lone, i.e. non-interacting, lorry flow on the ith lane,Ncomb the number of interacting lorries and Zcomb theoverall ordinate of the influence surface for theinteracting lanes, being the second summationextended to all relevant combinations of lorries onseveral lanes. A closed-form expression for l4 can bederived for two simultaneously loaded lanes, asshown in a subsequent section.

The equivalent impact factor jfat is the ratiobetween the damage due to the dynamic stress historyand the damage due to the corresponding static stresshistory:

jfat ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

ni;dynDsi;dynmPni;statDsi;statm

m

s(8)

In conclusion, ifDsc is the detail category, the fatigueassessment reduces to checking that

Dseq ljfatDsp4Dsc (9)

Partial safety factors for steel bridges

The partial safety factors gf, regarding action, and gm,regarding fatigue resistance, cover uncertainties in the

0.70.8

0.9

1

1.1

1.2

1.3

1.4

0 20 40 60 80 100

L [m]

M2 - 2 v - m=3

M2 - 2 v - m=9

real,eq

v2,3LM,eq

M

M

Fig. 14 Accuracy of fatigue load model 3 with additional fatiguevehicle




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evaluation of loads and stresses as well as fatiguestrength scattering. These partial safety factorsaffecting stress ranges are generally combined in aunique factor gMf gf gm. The numerical value ofgMfdepends on the possibility to detect and repair fatiguecracks and on the consequences of fatigue failure.Table 3 shows partial safety factors gMf values, as

suggested by EC3 for steel details[13,14].

Vehicle interaction modelling

As stated earlier, if vehicle interaction is relevant,stress histories cannot be determined by conventionalfatigue models or recorded traffic data, unlessappropriate additional information is available. Theachievement of general theoretical results inmodelling vehicle interactions could sensibly enlargethe field of application of the fatigue load models, and

it represents a main objective in the improvement ofEurocode 1-2.The probability that several vehicles are crossing

the bridge simultaneously, in the same lane or inseveral lanes, can be found theoretically[18] within theframework of queueing theory[1921], considering thebridge as a service system, with or without a waitingqueue, and the stochastic processes as Markovprocesses. This allows one to determine a suitablymodified load spectrum, composed of single vehiclesor vehicle convoys travelling on the bridge, so that thecomplete stress history results from a random

assembly of their individual stress histories.

BASIC ASSUMPTIONSLet the load spectrum consist of a set of q types oflorries and let Nij be the annual flow of the ith vehiclein the jth lane. The total flow in the jth lane is then:

Nj Xq

i1 Nij

Obviously, as the characteristic length L of theinfluence line increases, the probability that several

lorries are simultaneously crossing the bridgebecomes more and more relevant. The basichypotheses of the theory are that the vehicle arrivalsare distributed according to a Poisson law, and thatthe transit time Y on L is exponentially distributed.

INTERACTION BETWEEN LORRIES TRAVELLINGSIMULTANEOUSLY IN ONE LANE

The probability Pn that n lorries are travellingsimultaneously on L can be calculated by consideringthe bridge as a single-channel system with a waitingqueue, in which the waiting time, depending on the

number of requests in the queue, and that number ofrequests are limited. As there is a minimum value forthe time interval Ts between two consecutive lorries,the waiting time for the ith vehicle in the queue isgiven by Ti Y iTs and the number of requests inthe queue is limited to w IntYT1s 1.

Under the assumption that each Ti is distributedaccording to exponential law of parameter ji T1i ,the problem can be solved in closed form[18]. Theprobability Pn to have n vehicles in the lane, i.e. n 1requests in the queue, is then given by

Pnd

a

n1 d

aXw

i2 di a

Yi1s1 a

Xsj1 jj

1 !& '1

(10)

for n 0; 1, and by:

Pn dn

a

Yn1s1 a

Xsj1 jj

h i1& '

1 da

Xw

i2 di a

Yi1s1 a

Xsj1 jj

1 !& '1

(11)

for 24n4w, where d represents the lorry flow densityand a Y1. The annual number of interactionsbetween n vehicles i1; . . . ; in in the jth lane can then beobtained by substituting these formulae in the generalexpression

Ni1;i2;...;in;j Pn

1 P0Njn

Qnk1 NikjP

qnQn

s1 Nitsj (12)

where qn below the summation symbol indicates the

sum over all possible choices with repetitions of nelements among q.

In practice, the problem is reduced to consideringthe simultaneous presence of two lorries r and t only,so that:

P0 1 da

1 da j1

!1;

P2 d2

aa j11 d

a1 d

a j1

!1 (13)

and the annual number of interactions becomes:

Nr;t;j NrjNtjd

d a j1P

q2Q2

s1 Nitsj Nj

2(14)

........................................................................

........................................................................

Table 3 Partial safety factors cMf cfcm (EC3)Inspection and

accessibility

Fail-safe

components

cMf

Non-fail-safe

components

cMf

Periodic inspection andmaintenance; good accessibility

1.00 1.25

Periodic inspection andmaintenance; poor accessibility 1.15 1.35

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For a single vehicle model, eq. (14) simplifies furtherto:

N1;1;j Njd

2d a j1(15)

INTERACTION BETWEEN LORRIES TRAVELLINGSIMULTANEOUSLY IN SEVERAL LANES

Interactions between lorries travelling simultaneouslyin several lanes can be tackled by analogy, consideringthe bridge as a multiple-channel system without awaiting queue, where new requests are refused if allchannels are occupied. In this case the probability Pkto have vehicles simultaneously in k lanes, i.e. koccupied channels, can be deduced by solving anErlang-type system[1921]. Ifm is the density of the totalflow N, and recalling that a Y1,

Pk mk

akk!Xm

i0mi

aii! 1

(16)

for 04k4m. Substituting eq. (16) in the generalexpression:

Ni1h1;i2h2;...;ikhk Pk

1 P0Yk

j1NijhjNhj

N

k

Qkj1 NhjP

mk

Qks1 Nhts

(17)

where the sums taken over all the possible choices of kelements among m, it is possible to derive the annualnumber of interactions of k lorries, i1 on the h1th

lane,. . ., ik on the hkth lane:

Ni1h1;i2h2;...;ikhk mk

akk!Pmj1

mj

ajj!

Ykj1

NijhjNhj

N

k

Qk

j1 NhjPmk

Qks1 Nhts

(18)As stated earlier, often only the case in which twolorries r and t are simultaneously present on the hthand jth lanes is relevant, so that

P2 m2

2a2

X2i0

mi

aii!

1and

Nrh;tj NrhNtjNhNj

m2

2a2

X2i1

mi

aii!

1Nh Nj

2(19)

or, simply, when a single vehicle is considered:

Nh;j m2

2a2

X2i1

mi

aii!

1Nh Nj

2(20)

THE TIME-INDEPENDENT LOAD SPECTRUMThe procedures described above allow one to obtainthe previously mentioned lone vehicle spectrum (LVS),which is time-independent, being composed of

individual vehicles or of vehicle convoys travellingalone on the bridge.

Generally, the evaluation of the LVS requiresapplication of both procedures: simultaneous transitin the same lane is considered first, in order to obtaina new load spectrum for each lane, composed of

individual vehicles and of vehicle convoys travellingalone in the lane, which can be used to solve themulti-lane case.

TIME-INDEPENDENT INTERACTIONSOnce the LVS is determined, the complete stresshistory can be derived as a random assembly of theindividual stress histories. Unfortunately, the stressspectrum cannot be determined, in general, as asimple sum of the individual stress spectra. In factwhen maximum and minimum stresses are given bydifferent members of the spectrum, the individual

stress histories can combine, depending on the cyclecounting method adopted, giving rise to a time-independent interaction.

If cycles are identified by the reservoir method orthe rainflow method, the problem can be solved forthe general case[18,22]. The demonstration is beyondthe scope of the present paper, which shows only themain results.

Two individual stress histories sAi and sAj interact ifand only if

max sAi4max sAj and min sAi4min sAj (21)

or

max sAj4max sAi and min sAj4min sAi (22)

If the couples of interacting histories are sorted insuch a way that the corresponding Dsmax are indescending order, the number of the combined stresshistories as well as the residual numbers of eachindividual stress history can be computed in a verysimple recursive way.

In general, an individual stress history can interactwith several others; therefore, the number ofcombined stress histories Ncij, obtained as the hth

combination of the stress history sAi and the kthcombination of the stress history sAj is given by:

Ncij h1Nk1i Nj

h1Ni k1 Nj(23)

where h1Ni and k1Nj are the number of theindividual stress histories sAi and sAj not yetcombined, and 0Ni NAi and 0Nj NAj are thenumber of repetitions ofsAi and sAj in the LVS. Theactual number of individual stress histories sAi whichdo not combine with other stress histories is given by:

pNi 0 Ni Xk6i Nik Nki (24)where the sum is extended to all the stress historiessAk which combine with sAi itself. In conclusion, anew modified load spectrum is obtained, the




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members of which, represented by the lone individualvehicles and convoys and by their time-independentcombinations, are interaction free, so that it can bedefined as an interaction-free vehicle spectrum.

RELEVANT RESULTS

This method allows one to derive some importantgeneral results. It can be used to tackle relevantquestions concerning the calculation of the maximumlength of the influence line for which lorry interactionin the same lane can be disregarded or concerning thecalibration of the l4-factor accounting for multi-laneeffect in the l-coefficient method.

The analysis, briefly illustrated below, has beenperformed on the basis of the following assumptions:

* linear SN curve of slope m 5;* four different annual lorry flow rates,

N1 2:5 105; N2 5:0 105; N3 1:0 106;N4 2:0 106, distributed over 280 working days;

* constant lorry speed v 13:889 m/s.

Assuming an inter-vehicle interval Ts 1:5 s,application of eq. (15) allows, for example, thedetermination of how many vehicles per year aretravelling simultaneously in the same lane, dependingon the annual flow and on the length L, assummarized in Table 4.

These theoretical results, which are in goodagreement with numerical simulations, confirm thatthe simultaneous presence of several lorries in thesame lane is generally not relevant for spans below75 m. On the contrary, when the bending moment atthe support of two-span continuous beams isconsidered under high traffic flows, simultaneitybecomes significant, starting from 30 m span.

A closed-form expression for the l4-coefficient canbe obtained in a particularly relevant case, byapplying eq. (20). If two lanes carrying the same lorryflow are considered, the number of interactingvehicles per year is computed as shown in Table 5.Starting from this table, an equivalent stress rangeDseq, taking into account the interactions as well as allpossible relative positions of the two lorries, can beeasily evaluated, provided that the influencecoefficient of each lane is known. IfDs1 is theequivalent stress range induced by single-lane flowonly the required l4-coefficient is simply given byDseq=Ds1.

If the two lanes have the same weight, i.e. theinfluence surface is cylindrical, l4 values are inaccordance with Table 6, being 1:149 ffiffiffi25p the basicvalue for l4, corresponding to zero interactions. Theseresults demonstrate that l4 is a quasi-linear function

ofY

N, which can be expressed as:

l4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ1 Z2Z1

5

r 1:03 0:01 L N

v 106

(25)

where L is in m and v in m/s, with Z1 and Z2, Z15Z2,the influence coefficients of the two lanes,respectively.

Conclusions

The load models for fatigue verifications of road

bridges specified in EC1-2 are the result of extensivepre-normative research, which is essential for dealingwith these complicated problems. Following thecalibration procedures, the basic features of variousfatigue load models have been discussed, focusingparticularly on the philosophy, the accuracy and thefield of application of each of them, while somerelevant questions have been highlighted, especiallyconcerning vehicle interactions, which were thesubject of specific studies.

Besides a concise exposition of these furtherstudies, which allowed a general solution of the

problem of vehicle interactions, whether due tosimultaneity or not, noteworthy applications of themain results are illustrated, enabling one to expressthe l4-coefficient, accounting for vehicle interactions,in closed form.

......................................................................

......................................................................

Table 4 Number of interacting vehicles per year in one lane

L (m) N1 N2 N3 N4

40 1190 4729 18566 7160550 1690 6670 25987 9881360 2165 8515 32940 12361875 2858 11177 42796 157689

100 3978 15423 58110 208240

......................................................................

......................................................................

Table 5 Number of interacting vehicles per year in two lanescarrying equal flows

L (m) N1 N2 N3 N4

10 1846 7331 28901 11235820 3666 14450 56179 21276430 5458 21367 81966 30302850 8967 34626 129532 45871275 13213 50200 182480 617280

100 17312 64766 229356 746264150 25100 91240 308640 943390200 32383 114678 373132 1086953

......................................................................

......................................................................

Table 6 k4-factors for two lanes (cylindrical influence surface)

L (m) N1 N2 N3 N4

10 1.156 1.162 1.174 1.19720 1.162 1.174 1.197 1.23430 1.168 1.186 1.217 1.264

50 1.180 1.207 1.250 1.31075 1.194 1.230 1.283 1.351100 1.207 1.250 1.310 1.381150 1.230 1.283 1.351 1.423200 1.250 1.310 1.381 1.450

LOADING344



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1989.

Pietro Croce, PhD

Associate Professor,

Department of Structural Engineering,

via Diotisalvi 2, I-56126 Pisa, Italy,

E-mail: [email protected]


Background to Fatigue Load Models

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