06 Manus Sedlacek 1 Background Document to en 1991 Part 2 - Traffic Loads for Road Bridges

8/9/2019 06 Manus Sedlacek 1 Background Document to en 1991 Part 2 - Traffic Loads for Road Bridges

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Background document to EN 1991- Part 2- Traffic loads for road bridges - andconsequences for the designG. Sedlacek, G. Merzenich, M. Paschen, A. Bruls, L. Sanpaolesi, P. Croce, J.A. Calgaro, M. Pratt,Jacob, M. Leendertz, v. de Boer, A. Vrouwenfelder, G. Hanswille

Support to the implementation, harmonization and further development of the Eurocodes

First Edition, XXXXX 2008


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List of contents of the background document to EN 1991-2

Part A Background to load models for road bridges

Traffic and impact loadsWind loadsTemperature effectsSafety approach

Part B Consequences for design

Bridge structuresBridge decksBearingsTransition joints

Part C Aspects related to sustainability of load assumptions

New measurements and conclusionsTrends and requirements for transport policy


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6.4.1.3 Dynamic effects with several vehicles6.4.2 Impact factors for fatigue assessments6.4.3 Conclusions for loading codes

7. Dynamic simulations for justifying the load models in EN 1991-27.1 Procedure and assumptions

7.2 Results of the simulations for LM1 and LM27.3 Determination of representative values7.4 Dynamic simulations for fatigue assessments

7.4.1 General procedure7.4.2 Span factor a 7.4.3 Factors 432 , , 7.3.4 Results of simulations

8. Braking and acceleration forces8.1 Code provisions

8.2 Calculative model for braking forces on stiff bridges8.3 Dynamic effects from braking8.4 Determination of braking forces in dependence of the bridge length8.5 Specification of braking forces in EN 1991-2, 4.4.1 and conclusions


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1. Objectives

(1) This Part A of the background report deals with actions on road bridges andgives the results of studies that justify the rules and recommendations fortraffic loads on bridges, see EN 1991-Part 2.

(2) It also gives the background for the recommended choices of NationallyDetermined Parameters in the National Annexes.

(3) The background document to the vertical traffic loads is structured such that itgives in its section 2 to 5 the basic justification for the development of the loadpattern and of the amplitudes of the loads by evaluations of traffic with staticinfluence lines and in its sections 6 to 7 by using dynamic analysis of bridges.

2. Basic procedure

(1) The basis for the preparation of the traffic loads model in EN 1991-Part 2 hasbeen developed in parallel at various locations in Europe with studiesperformed at SETRA, LCPC, University of Pisa, University of Liege, RWTH

Aachen, TU Darmstadt, Flint & Neill, London that comprised the followingsteps:

Step 1: Evaluation of measured data related to the composition of traffic,traffic density, axle loads, vehicle loads, inter-axle distances andinter-vehicle distances.These evaluations gave the basis for the choice of thegeometrical pattern of the traffic load model being composed of agroup of single loads and uniformly distributed loads suitable todetermine the effects of traffic both for local assessments of thecomponents of the bridge deck and for global verifications of themain structure of the bridge.

Step 2: Calculative determination of the magnitudes of the loads in thetraffic load model by simulating traffic effects with a staticsimulation dynamic model on one side and a dynamic simulationmodel on the other side, that comprised a dynamic model for thebridge and their parts, models for the various vehicles as rigid

body kinematic systems and a model for the surface of the bridgeincluding the surface roughness and its effect on excitation of theaxles of the vehicles.

For consistency reasons the static simulation model was run withdynamic magnification factors obtained from the dynamicsimulation model.

Step 3: Comparison of the results of static and dynamic simulations toagree on conclusions for the load model in the Eurocode.

Step 4: Probabilistic limit state assessments to determine recommendedvalues for Nationally Determined Parameters as the partialfactors for the traffic load model.


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(2) In the following the works executed in these steps are reported.

3. Evaluation of measured traffic data to construct the traffic load model forULS assessments

3.1 History of measurements

(1) Until 1970 singular vehicles were taken from the flowing traffic and weighed onmovable weigh-bridges, so that the statistical relevance of data was doubtful.

(2) Since 1970 the LCPC used weigh-bridges for weigh in motion measurementsso that automatic records of axle loads with time gaps were possible.

(3) In 1975 to 1978 a measuring campaign was carried out supported by theECSC to determine axle loads and inter-axle distances for fatigueassessments of steel bridges.

(4) Since 1980 the LCPC has worked with piezo-electric measuring methods tosimplify the measurements.

(5) From 1988 on the data available from measurements in Europe were collectedby the Eurocode-working group to identify the development and distribution ofthe vehicle weights.

3.2 Result s of the measurements and evaluations3.2.1 Choice of representative traff ic

(1) The fig. 3-1 gives a survey on the measurements that were carried out adifferent European locations, which comprise different types of roads as

highways other roads urban roads

with different types of traffic as

long distance traffic short distance traffic special traffic, e.g. in the vicinity of gravel pits, quarries, etc.


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Figure 3-1: Measurements of traffic at various locations in Europe

(2) Significant are differences in the average daily traffic density (number ofvehicles/24 h), e.g. for Doxey (UK) with 34.500 vehicles/24 h. Important for theextreme traffic situations are however roads with an extreme density of heavyvehicles, for which fig. 3-2 gives an indication.

(3) Fig. 3-3 shows the accumulated distribution of the axle loads as measured. Inthis figure n10 is the number of axle loads with P A 10 kN. In fig. 3-4 also theaccumulated distribution of total vehicle loads is given, where n 30 is thenumber of vehicles with G 30 kN.

(4) It is apparent, that the traffic at Auxerre does not exhibit the largest axle loads

but the largest frequency of large axle loads; this is in particular caused by thelarge frequency of articulated vehicles in the traffic, see fig. 3-1.


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Figure 3-2: Percentage of types of vehicles

(5) In conclusion the data from the Auxerre traffic were selected as the basis forthe development of the Eurocode traffic load model.

Figure 3-3: Comparison of accumulated densities of axle loads


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Figure 3-4: Justification for choosing the Auxerre traffic data as basis for the

European traffic loading model

(6) In order to produce the pattern of the Eurocode traffic-load model moredetailed measurements were necessary to complete the informations on

- the frequency distribution of axle loads,- the frequency distribution of the distances between axles,- the frequency distributions of different types of vehicles,- the frequency distribution of distances between vehicles.

3.2.2 Measurements of axle-weight s and conclus ion for axle loads

(1) Fig. 3-5 gives as an example the frequency distribution of loads for axle no 2of an articulated vehicle in Rheden.

The frequency distribution is bimodal caused by the frequencies of unloadedand loaded vehicles.

(2) These frequency distributions can be approximated by Rayleigh-distributionswhich are close to normal distributions for large values.

Figure 3-5: Frequency distribution of weights of axle no 2 of articulatedvehicles in Rheden


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(3) From the extreme values P A 14 kN an extreme value distribution as in fig. 3-

6 for the Caronte-bridge can be derived that allows to extrapolaterepresentative values for the code.

Figure 3-6: Extreme value distribution and extrapolation

(4) From the half normal distribution y

2

2

2

1 Z e y

=

where

o x x z

= is the variable of distribution

Q x = is the variable of weight Q oo Q x = is the average value of weight Q

( ) f x x 2o= is the standard deviation f is the frequency of x

and the following input values from measurements


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kN 90Qo = kN 64 ,32=

the following representative values can be derived

daily extreme: 195 kN (measured 1 x /day)annual extreme: 236 kN1000 years extreme: 279 kN (characteristic value).

(5) Such extrapolations have been carried out for various locations wheremeasured data were available, see fig. 3-7.

Figure 3-7: Characteristic values for single axle loads and tandem, tridemand vehicle loads

(6) On the basis of fig. 3-7 the characteristic value of the axle load in Load-Model1 LM1 of EN 1991-2 was taken as

kN 300Q =

It is also the basis for the axle load in Load-Model LM2, kN 400Q = .

(7) The model for the axle load in LM1 includes a certain dynamic factor resultingfrom the roughness of the road surface where the measurements were made.

The magnitude of the dynamic factor has been determined according to fig. 3-8 from dynamic simulations of the flowing traffic at the points ofmeasurements.


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Figure 3-8: Determination of dynamic factor implicite in weight

measurements

(8) For Load-Model LM2 the amplification factor 14.1= for axle loads wasconsidered to be not sufficient. From dynamic simulations with a localirregulatory as given in fig. 3-9 on additional amplification factor of 1.3 wasobtained, that leads to the value

kN 4003003.1Q =

Figure 3-9: Impact factor from irregulatories on the road


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(9) The following conclusions can be drawn:

- the characteristic value of the axle load in LM1 is about 2,5 timeshigher than the legally permitted load. This is a result of systematic

overloading,- the characteristic value of the axle load in LM1 differs from location tolocation,

- the variation of the characteristic value of the axle load in LM1 withtime is small.

3.2.3 Measurements of dist ance of axles

(1) To choose an appropriate distance of axle loads in the traffic load modelfrequency distributions of distances were determined for articulated vehicles,see fig. 3-10.

Figure 3-10: Frequency distributions of distances between axles for

articulated vehicles for Rheden(2) According to fig. 3-10 the distances between axles 1 and 2 are in the range of

m20 ,1 x .

(3) Therefore for the double axle in Load-Model LM1 a distance of 1,20 m waschosen.

3.2.4 Measurements of vehicle weight s

(1) For determining the effects of traffic for lengths of influence areas greater than10 m the statistics of vehicle loads and of the inter-vehicle distance arenecessary.


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(2) As reference location for the measured data Auxerre was chosen, because ofthe following reasons, see fig. 3-1:

- the composition of the traffic corresponds to the estimation of futuretrends,

- the portion of lorries in the traffic composition is 32% in lane 1 and10% in lane 2 and in relation to other locations rather high,- the portion of loaded lorries from all lorries is 66% and hence mirrors

the trend for an improved transport management,- data were fully documented for a large time period for lane 1 and lane

2 in a 4-lane highway.

(3) In fig. 3-11 the most important types of vehicles from the full traffic are isolatedas given in fig. 3-2 for which this figure also shows the distribution of thesevehicle to lane 1 and lane 2 if the full lorry traffic is reduced to these 4 types ofvehicles.

Figure 3-11: Reduction of full lorry traffic to 4 important types of vehicles anddistributions of these types to lane 1 and lane 2 of the 4 lanehighway at Auxerre

(4) Fig. 3-12 gives the distributions of weights for those 4 vehicle types.


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Figure 3-12: Distribution of vehicle weights

(5) The most important vehicle type is type 3 (articulated vehicle), which isdominant both for ultimate limit state and fatigue verifications.

(6) The distributions in fig. 3-12 can be approximated by 2 normal distributions asillustrated for type 3 vehicle in fig. 3-13, one for light, one for heavy traffic.

Figure 3-13: Approximation of distributions as measured by normaldistributions

(7) In fig. 3-14 the statistical data of the distributions are given from the data asmeasured. In fig. 3-15 the statistical data are corrected after filtering out thedynamic effects. This filtering influences mainly the standard deviations.


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Figure 3-14: Statistical data for the lorry-traffic in lane 1 and lane 2 at Auxerre

from measured data

Figure 3-15: Statistical data of lorry-traffic in lane 1 and lane 2 at Auxerre afterfiltering out dynamic effects

(8) Furthermore the distributions of the vehicle loads on the various axles and thedistribution of the distances of axles can be described in statistical terms, seefig. 3-16.


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Figure 3-18: Simulation of traffic effects with influence lines

(14) The simulation was carried out with the following conditions:

Traffic on one lane: - flowing traffic as recorded for lane 1 in Auxerre (25% lorries, 75% cars)

- congested traffic: only 100 % lorries asrecorded in lane 1 in AuxerreTraffic on two lanes: - flowing traffic as recorded for lane 1 in

Auxerre (for each lane) (25% lorries, 75% cars)- congested traffic as recorded in lane 1 in

Auxerre (25% lorries, 75% cars)Traffic on three or four lanes: - flowing traffic artificially generated from

Auxerre (for each lane) data (10% lorries, 90% cars)- congested traffic artificially generated from

Auxerre data (10% lorries, 90% cars).

(15) These conditions are by 1 step more severe than the conditions measured in Auxerre. One could imagine that for bridges with 2, 3 or 4 lanes in each lanecongestions of only lorries could occur. This case has however not beenconsidered in the loading model because of the low probability of occurrence.


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3.2.5 Definit ion of flowing and congested traffic

(1) In congested traffic situations the inter-vehicle distance was assumed to be 1m (3 m between axles for jam).

(2) For defining flowing traffic the distances between vehicles depending on thespeed must be known.

(3) A possibility to define the density function for the inter-vehicle distance is bythe density function y of Davenport

x1k k

e x)k (

y

=

whered x = is the distance between the vehicles

oo d x = is the mean value of distancenn d x = is the modal values corresponding to the maximum frequency

no x x1

=

no

o

x x x

k

=

( )k is the Gamma-function of k

Hence the function is controlled by two parameters:

the mean value od andthe modal value nd .

(4) Fig. 3-19 gives distributions of measured distances and the calculativedistribution using od = 120 m and nd = 30 m.


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Figure 3-19: Distribution of distances between vehicles

(5) Another possibility for density distributions is given in fig. 3-20, where theconstant part between 20 m and 100 m covers the probability of developmentof convoy and a linear increase up to 20 m is due to the minimum distance.The exponentially decreasing part for distances greater than 100 m coversfree flowing traffic.

Figure 3-20: Comparison of measured and theoretical values for densityfunction of inter-vehicle distances

(6) The -value in fig. 3-20 gives the probability of occurrence for lorry distancesless than 100 m, and the -value has been obtained from traffic records of 24representative traffics in Germany.

(7) A simplified solution that could also be used for the inter-vehicle distance isthe minimum distance that results from the reaction time of a driver to avoid acollision with the front vehicle in case of braking. Assuming a minimum brakingreaction time sT = 1 s of the driver, the minimum distances is give by


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sT va =

where v is the mean speed of the vehicles. The distance is limited to aminimum value a = 1 m in case of jam situations.

3.2.6 Results of simulations with influence lines

(1) From the simulations that have been carried out to obtain-load effects, e.g. thebending moment M or the shear force V , the equivalent load Q effectingthese load effects can be determined by

k V ' Qor k L M ' Q ==

where

k is a factor resulting from the influence line considered.

(2) Fig. 3-21 and fig. 3-22 give these characteristic equivalent loads for traffic onone lane and the associated values from Load-Model LM1.

Figure 3-21: Characteristic values of equivalent loads ' Q determined fromtraffic simulations for the mid-span bending moment of a singlespan bridge for 1 lane traffic


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Figure 3-22: Characteristic values of equivalent loads ' Q determined fromtraffic simulations for the hogging moment of a two spancontinuous bridge for 1 lane traffic

(3) Fig. 3-23 gives the single loads Q [kN] and the uniformly distributed loads q[kN/m] that result from various influence lines.

Figure 3-23: Characteristic value of traffic loads for 1 lane traffic

(4) Fig. 3-24 demonstrates the effects of the Auxerre traffic as measured (25 %lorries, 75 % cars), that can be represented for flowing traffic by

L12800Q K +=

and for congested traffic


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L31800Q K +=

Figure 3-24: Global characteristic loads ' Q

(5) For traffic on more than 1 lane the studies have shown, that for L > 30 malways congested traffic is relevant.

(6) Fig. 3-25 and fig. 3-26 give the equivalent loadings for 2 lane traffic.

Figure 3-25: Characteristic values of equivalent loads ' Q determined fromtraffic simulations for the sagging moment of a single span bridgefor 2 lane traffic


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Figure 3-26: Characteristic values of equivalent loads ' Q determined fromtraffic simulations for the hogging moment of two span bridgesfor 2 lane traffic

(7) Fig. 3-27 and fig. 3-28 give the relevant equivalent loads for 4 lane traffic.


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Figure 3-27: Characteristic values of equivalent loads ' Q determined fromtraffic simulations for the sagging moment of single span bridgesfor 4-lane traffic

Figure 3-28: Characteristic values of equivalent loads ' Q determined fromtraffic simulations for the hogging moment of two spancontinuous bridges for 4-lane traffic

(8) For other influence curves the results are in between these extreme values.

(9) Fig. 3-29 gives the single loads Q [kN] and the uniformly distributed loads q [kN/m] that result from various influence lines and numbers of lane.

Figure 3-29: Characteristic values of traffic loads for 2, 3 and 4 lane traffic


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(10) The fig. 3-30 and fig. 3-31 demonstrate the effect of the characteristic vehiclein fig. 3-16 on the equivalent value ' Q . Such a vehicle could be used for theassessment of existing bridges.

Figure 3-30: Effects of the characteristic value in fig. 3-16 on equivalentvalues ' Q for sagging moments

Figure 3-31: Effects of characteristic vehicle in fig. 3-16 on equivalent values' Q for hogging moments


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3.2.7 Composi tion of load models LM1 and LM2

(1) The composition of Load-Model LM1 as concluded from the evaluations ofmeasurements is given fig. 3-32.

Figure 3-32: Load-Model LM1 in EN 1991-2

(2) Using a reference length of 11 m it can be interpreted as a simultaneousassembly of

- a 900 kN-vehicle in the first lane in a row of 450 kN vehicles in a jamsituation with 5 m inter-vehicle distance

- a 500 kN-vehicle in the second lane with 120 kN-vehicles in a row- a 300 kN vehicle in the third lane with 120 kN vehicles in a row

where the 120 kN vehicles may also be understood as a mixture of 450 kNlorries with personal cars.

(3) The Load-Model LM2 is simply a single axle representing the impact effect of acharacteristic axle load from irregularities on the road surface.

4. Representative values of traffi c loads for SLS assessments4.1 General

(1) Other representative values of traffic loads than characteristic values are nonfrequent values, frequent values and quasi permanent values.

(2) The non frequent or rare values are those having a mean return period of 1year from either flowing or congested traffic, e.g. considered for the limit stateof decompression in concrete.


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(3) The frequent values have a mean return period of 1 week. They are basedon frequent traffic situations as e.g. flowing traffic and good surface quality ofroad. The extreme situations applied for the determination of characteristicloads are not considered for frequent values.

(4) For quasi-permanent values of traffic (e.g. for creeping effects) usually thevalues are zero.

(5) To determine the magnitudes of representative values the half-normal densitydistribution yfor extreme action effects is used as for characteristic values:

2

2 z

e21 y

=

where

0 x x z

= is the reduced variable

x is the effect consideredo x is the particular average value of x used in fitting to the real

distribution curve is the standard deviation.

The values o x and are adjusted in such a way that the correlation coefficientis maximized.

(6) The value corresponding to a return period Ris

=+= R

00 R0 R z x

1 x z x x

where R z results from the following equation

( )

== R

2

z R

s

S

2 z

R T T

N 21dze

21 zY

where:

= iS n N is the number of vehicles corresponding to oi x x > and thesimulated time period

S T is the simulated time period RT is the required return period.

4.2 Result s of extrapolation

(1) A typical example for such a simulation is given in fig. 4-1.


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Figure 4-1: Example for the approximation of the distribution of bridge

response by a half-normal distribution and determination ofrepresentative values

(2) Fig. 4-2 gives in the scale of half-normal distributions straight lines withintersection points for the return periods required, where o x and hence o z areadjustment values for the effect considered (best fit parameters, see fig. 4-1).

Figure 4-2: Identification of ( ) zY depending on the return period required fordifferent curve fitting parameters

(3) Fig. 4-3 gives the results of many simulations of the Auxerre traffic. In thisfigure K x corresponds to a return period of 1000 years. The factor

L

s

n N 2


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where Ln is the number of lorries used in the simulation varies between 1 and4, but R z various only by 10 % for short return periods (1 day) and by 5 % forlong return periods (1 year).

Figure 4-3: Factorsk

R

x x to determine representative values from

characteristic values of traffic effects

4.3 Conclusions

(1) Taking account of the results in fig. 4-3 the following choices have been madein EN 1991-2:

- 1 (non frequent) = 0,80In this choice a reduction of about 10 % from 92 ,0 1 = has been madeto take account of advantages of good surface quality (better than forcharacteristic values) for the mayor part of service life and the fact thatcharacteristic values for large spans in EN 1991-2 are conservative.

- 1 (frequent) = 0,75 for axle loads k Q

= 0,40 for uniformly distributed loads k q In this choice also a reduction of about 10 % from 1 = 0,82 for flowingtraffic and 1 = 0,85 for congested traffic has been made due to goodsurface quality of for the mayor part of service life.

Also the frequent values from flowing traffic are not greater than 50 % ofthe frequent values for congested traffic, because the characteristicvalues differ by this amount, see fig. 3-23.

The values 1 for axle loads and uniformly distributed loads are alsoquite close to the values for fatigue-frequent loading model FLM1 forfatigue assessments.


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5. Fatigue loading models5.1 Basic princip les of fatigue

(1) Fatigue is the progressive, localised and permanent structural change

occurring in materials subjected to fluctuating stresses initiating andpropagating cracks through a structural part after a sufficient number of loadcycles.

(2) Fatigue is induced in bridges mainly by heavy vehicles. The development ofappropriate load models and verification concepts is a main topic of EN 1991-2and the bridge parts of the other Eurocodes.

(3) The same reasons that apply for the choice of the Auxerre traffic for thedevelopment of the load models LM1 and LM2 also apply for the preparationof the Load-Models for fatigue.

(4) In principle the way of determining the fatigue load models is as follows:

1. Choose typical bridges for simulating bridge responses to traffic floweither by static simulations using influence lines with dynamic factorsor by dynamic simulations with appropriate vibrating systemscomprising the mass and damping systems of the bridges, the vehiclesand the surface roughness of the road surface.

2. Evaluate the bridge-responses documented as stress-histories by anappropriate counting method to achieve a histogramme of the stressranges applicable as fatigue actions to the material.

3. Apply an appropriate damage accumulation rule (in general the Minerrule) to transfer the fatgiue actions expressed by a histogramme withvarying stress ranges into a histogramme (or spectrum) with constantstress ranges (damage equivalent stress ranges), to make the fatigueactions comparable with the fatigue resistances.

4. Select an appropriate fatigue resistance curve (Whler curve) for thematerial and structural detail chosen that originates from large scale

test specimen fabricated in the way as the real structure and thatconstitutes a characteristic value of fatigue resistance.

In general such fatigue resistance curves have been determined forconstant amplitude stress-ranges.

5- Apply a reliability concept to achieve sufficient reliability for the fatigueassessment.

6. Simplify the procedure by producing damage-equivalent fatigue loads,the effects of which can be more easily compared with the fatigue

resistances.(5) Fig. 5-1 gives the main steps of this procedure .


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Figure 5-1: Fatigue assessment: from influence lines to damage assessment

(6) For steel structures and for reinforcing and prestressing steel the fatiguestrength curves according to fig. 5-2 may be used.

Figure 5-2: Fatigue strength curves for structural steel and reinforcementand prestressing steel

(7) The fatigue strength curves for welded steel structures are defined by thefatigue strength category c (fatigue strength at 2 10

6 cycles) and theconstant amplitude fatigue limit D at 5 10

6 cycles.

For stress ranges above D the slope m of the curve in a double logarithmicscale is 3m = . Stress ranges below

D dont produce fatigue if the maximum

stress range Dmax .


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For stress spectra with stress range above D and below D the damageaccumulation may be performed with a modified fatigue resistance curve witha slope 5m = for stress ranges below D . This modification takes intoaccount that the constant amplitude fatigue limit D is reduced by damage

effects from stress ranges D > For stress ranges of the stress range spectrum which are below the cut-off-limit L at 10

8 cycles it may be assumed that these stress ranges do notcontribute to the calculated cumulated damage.

(8) Typical examples for fatigue strength categories in steel and compositebridges are shown in fig. 5-3. The stress ranges in this figure relate to nominalstresses.

Figure 5-4: Typical examples for fatigue strength categories in steel andcomposite bridges

(9) For reinforcement the fatigue strength curve is given in EN 1992-1-1 anddescribed by a two linear functions in the double logarithmic scale without anyconstant amplitude fatigue limit, see fig. 5-2.

(10) Whereas for steel structures normally a linear relation can be assumedbetween the fatigue loading and the stresses, for concrete structures the non-

linear behaviour due to cracking of concrete has to be taken into account forthe determination of the time history of the stresses. In this case in addition tothe fatigue load also the permanent loads and effects from climatetemperature actions have to be considered.

(11) In the following the effects of the counting method and of the damageaccumulation hypothesis (Miner Rule) are explained in more detail.

5.2 Counting method

(1) The following counting methods give the same results and can also identify themean stress level for each cycle

- the reservoir method,


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- the rainflow method

(2) The reservoir method, see fig. 5-5 and also fig. 5-1 defines a cycles by thedifference of water level in emptying a reservoir. Secondary stress rangesoriginate by water pockets left when starting with emptying the reservoir at the

lowest point. The method is easily understood but difficult to programme.

Figure 5-5: Reservoir method

(3) A method more suitable for programming is the rainflow method, see fig. 5-6.This method follows the flow of rain-drops from the top of a fictive roof asexplained by the example in fig. 5-6.

Figure 5-6: Rainflow counting method


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5.3 Damage accumulation hypothesis

(1) For a given stress-range histogramme (spectrum) the application of the Miner-Rule for a stress spectrum that has stress ranges above and below theconstant amplitude endurance limit D , is demonstrated in fig. 5-7.

Figure 5-7: Linear damage accumulation and equivalent amplification factor. fat

(2) The damage accumulation related to steel structures with 3m1 = and 5m2 = therefore reads:

> >

+=

Di

D

Li

5 D D

5ii

3 D D

32i

nn

nn D

(3) A damage accumulation can be neglected if the frequent values max donot exceed D .

5.4 The concept of damage equivalence

(1) The use of the Miner-rule together with the fatigue resistance curve allows to

transfer any stress range spectrum with variable stress ranges into a damageequivalent spectrum with constant stress-ranges e .

(2) For a fatigue resistance curve with an unlimited slope m and a spectrum with

)6 m

c

imi

102

n D

=

a damage equivalent constant amplitude spectrum with e and en resultsfrom:


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)

6 mc

eme

6 mc

imi

102

n

102

n D

=

=

and reads

( )m

e

imi

e nn =

(3) In taking into account a linear relationship between a histogramme of loadsii n ,S and the associated histogramme of stress-ranges ii n , , it is possible

to adopt the notion of damage-equivalent stress-ranges to damage equivalentloads.

Fig. 5-8 shows for the example of axle-loads measured for the traffic inRheden the distribution of axle loads ( )ii n ,S and the distribution of damages

i D calculated by

imii nS D =

that for the reference number en gives

m

e

ie n

DS

=

(4) Hence it is possible to calculate fatigue damage equivalent loads from loadhistogrammes on the basis of the shape of the fatigue resistance curve withoutknowing the relevant fatigue detail class.

(5) Fig. 5-8 shows that very high loads do not contribute to damage because oftheir small number, and small loads with large numbers do not contributebecause of their small amplitudes.


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Figure 5-8: Distribution of fatigue damage from a load histogramme anddetermination of fatigue resistance load

(6) There are various possibilities to define the damage equivalent load rangeseS depending on the choice of en .

Examples are:

- reference to the total number of cycles = ie nn which gives

m

i

ien n

DS

=

- reference to the maximum value of load ranges S maxS e = whichgives

mi

e S max

Dn

=

- reference to the most damaging load range egS which gives

( )

=i

iieg

D

S DS


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( )megi

1mi

meg

ieg

S D

D

S

Dn

+==

egS is equal to the mean values (gravity centre)of the damage distribution i D .

(6) Fig. 5-8 gives the results of a damage calculation for an axle loadhistogramme and shows the values egS and eS .

The definition egS gives larger values than enS resulting in less conservatismin particular where the values of eS are close to L , see fig. 5-9 and fig. 5-10.

(7) All definitions of eS are equivalent and based on the same fatigue damage

D DnS ieme ==

Figure 5-9: Errors in using an unlimited slope 3m = instead of trilinear fatigueresistance curve


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Figure 5-10: Errors in damage calculation when using an unlimited slope

3m = instead of trilinear fatigue resistance curve

5.5 Results of evaluation of load-distributions

(1) The evaluations were performed on the basis of load distributions as given infig. 5-8 to determine the following values:

Ln = number of lorries per day

321 n ,n ,n = number of single axles, tandem axles or tridem axles

30n = number of single axles exceeding kN 30P A =

d Q = loads with a return period of 1 day

f Q = loads, the exceedance of which would produce a damageof less than 1 % of the total damage. This load can beinterpreted as the frequent value for fatigue

gQ = load level that produces the maximum damage i D in the

i D - damage distribution. Together with the number gn itsignifies the damage equivalent load

gn = number of cycles that together with gQ produce the totalfatigue damage D on the basis of an unlimited slope of

the resistance curve 3m =

(2) The results of the evaluations are given in fig. 5-11 for single loads, fig. 5-12for tandem loads, fig. 5-13 for tridem loads and in fig. 5-14 for the full vehicles.


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Figure 5-11: Evaluation of distribution of single loads

Figure 5-12: Evaluation of distribution of tandem loads


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Figure 5-13: Evaluation of distribution of tridem loads

Figure 5-14: Evaluation of distribution of vehicles loads

(3) Results of the evaluations are the following

1. for frequent values for fatigue f Q :

single axle: 200 kNtandem axle: 300 kN = 2 x 150 kN


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tridem axle: 385 kN = 3 x 130 kNvehicle: 600 kN

2. for damage equivalent loads defined by the maximum damage ofdamage distribution gQ , gn :

single axle: 131 kN

5.043.0nn

;37 .1nn

30

g

L

g =

tandem axle: 252 kN = 2 x 125 kN

5.042.0nn

2

g =

tridem axle: 266 kN = 3 x 90 kN

75.0

n

n

3

g =

vehicle: 469 kN

5.043.0nn

L

g =

3. for damage equivalent loads defined by the number of vehicles Ln ,

calculated from 3 L

ref

ref

gge n

nnn

QQ =

single axle: 151,10 kNtandem axle: 120,00 kNtridem axle: 183,00 kNvehicle: 378,00 kN

(4) EN 1991-2 specifies as fatigue Load-Model LM3 a four axles vehicle, see fig.5-15-

Figure 5-15: Fatigue LM3 in EN 1991-2


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(5) This model has a total weight of 480 kN and therefore is slightly heavier thanthe most damaging equivalent fatigue load of 469 kN in Auxerre, see fig. 5-14.This model has 4 axles, as 4 axles represent a lorry in the mean, see

L

1

nn in

fig. 5-11.

The tandem load of 240 kN corresponds approximately to the most damagingequivalent load for tandem axles 252 kN, and the single axle load of 120 kN isapproximately the value of the most damaging single load of 131 kN

When compared with the damage equivalent loads defined by the number ofvehicles Load-Model 3 is conservative except for single axles.

(6) For the fatigue assessment of details the stress-situation of which is influencedby local loads as from single axles the fatigue loading model LM3 is normallynot precise enough, e.g. for bridge decks made of concrete or orthotropic steeldecks.

For that case EN 1991-2 has specified the fatigue load models LM2 and LM4,where LM2 gives the frequent values and LM3 gives the damage equivalentloads for 5 important vehicle silhouettes, see fig. 5-16.

The silhouettes give the mean value geometry, and the axle loads reflect moreprecise load-distributions for single axles, covering also the maximum value ofthe damage equivalent load eQ = 151 kN, determined for Le nn = .

(7) These load models LM2 and LM4 which are assembled in fig. 5-16 give alsomore precise contact surfaces of the wheels needed for local assessment, seefig. 5-17.


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Figure 5-16: Fatigue load models LM2 and LM4

Figure 5-17: Dimensions of pressure areas of wheels

(8) The damage equivalent load gQ representing the most damaging weight hasthe advantage, that it is less sensitive to a variation of the slope of the fatigueresistance curves than the value eQ related to = ie nn , see fig. 5-18.


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Figure 5-18: Histogramme of axle loads and associated distribution of

damage i D for 3m = and 4m=

5.6 Simulation of bridge responses for static fatigue actions5.6.1 Purpose of the simulation

(1) The simulation of bridge responses to traffic, that has been performed fordetermining characteristic values of load effect, as described in 3.26 has alsobeen used to determine the frequent Fatigue Load-Model (with 1 % fatiguedamage only) and to check the damage equivalent fatigue loading models asdescribed in 5-5 in view of the dependency on the shape and size of influencelines.

(2) To this end the following loading scenarios have been used:

AMT as recorded in Auxerre on lane 1 (25 % lorries and 75 % cars)T 1 AJ the same traffic as AMT , but with an inter-vehicle distance of

1 m


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Figure 5-20: Example for calculative service life of composite box girderbridges for different span length and deck widths for (1)continuous girders with reinforcements at supports, (2)continuous girders, (3) single span girders

5.6.2 Results for fatigue loads dependant on span L

(1) Fig. 5-21 gives the total frequent fatigue load f Q and the damage equivalentload gQ related so the length L of single span bridges for the flowing traffic and

the congested traffic chosen.

Figure 5-21: Total fatigue loads on span L


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(2) The frequent fatigue load is composed of a single load of 600 kN which is the

frequent load of a lorry and a uniformly distributed load that depends on thetype of traffic being either 9 kN/m for flowing traffic or 22 kN/m for congestedtraffic.

If frequent loads for fatigue are related to flowing traffic only, then thefrequent values can be defined by coefficients applied to the characteristictraffic loads:

Q1 to axle loads k Q : ( )75 ,0~8 ,01 = q1 to uniformly distributed loads k q ( )40 ,0~30 ,01 =

The reduction of q1 in relation to Q1 is caused by the increase of distances

between vehicles.

(3) The values Q1 and q1 for frequent loading for fatigue are very close to thefrequent values defined by a return period of 1 week in 4.3 (see values in ( )).

(4) Fig. 5-22 gives a comparison of frequent and damage equivalent fatigue loadmodels from fig. 5-19.

Figure 5-22: Comparison of frequent and damage equivalent fatigue loadmodels

(5) Figure 5-21 also gives the damage equivalent loads gQ , with the maximum

damaging effect together with L

g

n

n and the damage equivalent loads eQ related

to Ln for the Auxerre traffic.


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It is evident, that gQ for a span length of 10 m corresponds to the damageequivalent load determined from the Auxerre vehicle loads, see fig. 5-14, thatis constant for a length up to m50 L = , if flowing traffic is used as the basis.For congested traffic the maximum span length is about 20 m.

(6) Fig. 5-23 shows for traffic other than in Auxerre that for fatigue the critical inter-vehicle distance is not as for congested traffic, e.g. for a inter-vehicle distanceof 4,8 m, but in the range of 24 m.

Figure 5-23: Damage equivalent loads dependant on inter-vehicle distance

(7) The values L

e

nn represent more or less the ratio of the number en of loaded

lorries to the total number Ln of lorries.

For larger span lengths m50 L > resp. m20 L > the fatigue load grows slowly,see figure 5-23, so that an adjustment is necessary.

5.6.3 Comparison of the fatigue effects of the Auxerre traff ic with the fatigueload models in EN 1991-2

(1) To check the effects of the various Load-Models in EN 1991-2 a comparison ismade between the effects of these load models and the fatigue effects of the

Auxerre traffic.

(2) Fig. 5-24 gives the ratio of moment ranges due to Fatigue Load Model(LM1) 2 LM , f M which is the frequent fatigue model related to the

characteristic Load-Model (LM1), and the frequent fatigue value fA M fromthe Auxerre traffic, for different slopes m


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Figure 5-24: Comparison FLM1 and Auxerre-traffic

(3) For the influence lines 0 M , 1 M , 3 M the Eurocode-model is safe-sided, and for

m20 L also sufficient, whereas for influence line 2 M the model is not safe.

(4) Fig. 5-25 gives the ratio A , f

2 fLM

M M

for the fatigue load model LM2 for frequent

fatigue load. The greater precision of this model is reflected by a smallerscatter for the various influence lines in the range m20 L . It needs howeveran adjustment by an impact factor of about 1,10 to be safe sided.


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Figure 5-25: Comparison FLM2 and Auxerre traffic

(5) Fig. 5-26 gives the ratio A ,e

3 LM ,e

M M

for Le nn =

Figure 5-26: Comparison FLM3 and Auxerre traffic for Le nn =


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5.7 Conclusions

(1) EN 1991-2 gives fatigue loads, from which FLM1 and FLM2 are mainlyintended to perform a check for infinite durability:

D frequent

This check is always relevant for a large number of cycles as e.g. experiencedin short spans and structural components of the deck.

Note: If each lorry produces a stress range cycle this would lead to 1000 to8000 cycles per working day, e.g. 25 to 200 millions of cycles for 100 years.For small components loaded by axles or wheels this number is even higher.

(2) Fig- 5-30 shows the cut-off by the fatigue frequent Load-Models FLM1 andFLM2.

Figure 5-30: Cut-offs for fatigue load range spectra

(3) The damage-equivalent Load-Models FLM3 and FLM4 are intended to assessa specified limit of service life of the bridge.

To this end FLM3 is for main components of the bridge with influence lengthsm20 L whereas FLM4 is for deck-components with m20 L .

(4) The fatigue Load-Model FLM3 needs a supplementary vehicle to cope withlarger spans and with influence line 2 M applicable for hogging momentsabove supports.

(5) The supplementary vehicle to FLM3 and any preparation of time histories andcycle counting can be avoided, if FLM3 is used together with damageequivalent factors , directly obtained from the Auxerre traffic for differenttypes of influence lines.

(6) Any fatigue assessment both for infinite durability and limited service life needs

partial factors for reliability sake. These partial factors are specified in thematerial related fatigue codes or bridge design codes.


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(8) The afore described modelling of the structure allows the simulation fordifferent bridge structures as shown in fig.6-2. Simple and continuous beams,frames and arch bridges as well as truss bridges may he analysed includingalso secondary structural elements as e.g. cross-beams and stringers oforthotropic decks.

Figure 6-2: Bridge structures considered in the calculation-model


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Figure 6-3: Vehicle vibration model for a) a single-vehicle and b) an

articulated vehicle

6.2.3 The vehicle model

(1) The vehicles represent the exciters of the vibrating system bridge', and haveto be described as vibrating systems for themselves. Due to the roughness of

the roadway the vehicles are excited and produce dynamic wheel loads. Thefrequency spectrum of the wheel loads is controlled by:

- the speed of the vehicles;- the spectrum of the roadway roughness;- the characteristics of the vehicles.

(2) Therefore it is necessary that the modelling of the vehicle allows a goodapproximation of the actual behaviour of the vehicle for a wide range ofvarying conditions. For this modelling discrete rigid multibody systems withnonlinear behaviour are used [3].

(3) Two types of vehicles are discerned according to fig.6-3 : On one hand singlevehicles with a set of individual axles which are connected to a rigid body


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mass, on the other hand articulated vehicles where the trailor is connected tothe motor tractor by an elastic coupling.

(4) For each individual axle the idealized tire suspension system according to fig.6-4 is assumed. The mass of the wheels and the axle structure of each axle

are represented by a concentrated mass between the tire spring and thesuspension structure. The suspension structure is idealised as a parallelsystem consisting of a linear elastic spring for the suspension, a frictionelement and a viscous damper. The body mass is considered as a rigidelement with translatory and rotatory mass inertia.

Figure 6-4: Vibration model for a single-axle

(5) The degrees of freedom of the vibrating system of the vehicle are determinedby the vertical translatory displacements of the axle and the translatory androtatory movements of the rigid body mass. Using the equilibrium conditionsthe following coupled differential equation can be obtained

[ ] [ ] [ ] ( )t F zC z D z M F F F F =++ &&&

where

[ ]F M = mass matrix of the vehicle[ ]F D = damping matrix of the vehicle[ ]F C = stiffness matrix of the vehicle

z , z , z &&& = vector of the translatory displacements, speeds and accelerations( )t F F = vector of the exciting forces

(6) These differential equations formally are identical with those of the bridgesystem and therefore are resolved with the same numerical time step method.


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(7) Particular attention has to be paid to the suspension structure with inter-leaf

friction which is frequently used in commercial vehicles.

(8) The inter-leaf friction can result in vibrations where the leaf spring between the

axles and the rigid body may be blocked and only the tire springs may act.Due to the small damping effect of the tires then great dynamic effects may becaused also for even road surfaces. This particular behaviour of the leaf springis considered by a nonlinear load deflection characteristic according to fig. 6-5,which represents the "friction hysteresis".

(9) In this vehicle model all significant parameters of the vehicle which influencethe dynamic behaviour are considered. These parameters are:

- the vehicle structure (number of axles, inter-axle distance, distributionof masses);

- the suspension system of the axle (leaf spring suspension, pneumaticsuspension, hydraulic suspension);

- the damping system of the suspensions (pneumatic, hydraulic or byfriction);

- the properties of the tires.

Figure 6-5: Force-Displacement diagram of a suspension with inter-leaffriction

6.2.4 Model for the roadway roughness

(1) Three different kinds of roadway roughness may be found on road bridges:


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- local extreme unevennesses, e.g. transition joints, road holes orartificial obstacles (timber);

- regular unevennesses, e.g. surface waves due to fabrication (e.g.according to distance of cross-beams);

- irregular unevennesses.

(2) The two unevennesses mentioned first are deterministic and may bedescribed geometrically. The irregular unevenness however is stochastic andstatistic characteristics and functions have to be used to describe them.

For this the unevenness ( ) xu , see fig. 6-6, is presented by its complex fourierspectrum

( ) ( ) +

= d eU xu xi

Figure 6-6: Stochastic unevenness profile ( ) xu

(3) The fourier spectrum displays the amplitudes and phase relations of thedifferent harmonic components of the unevenness. In assuming a Gaussiandistribution of the irregular unevenness and the applicability of a stationary andergodic random process a relation between the square of the mean value ofthe unevenness profile and its power spectral density may be derived.

( ) ( ) +

+

== d dx xu

X 21limu

X

X

2

x

2

(4) Hence it follows that the power spectral density indicates how the squares of

the mean values of small frequency ranges are distributed in dependence ofthe frequencies. It represents a scale for the intensity of the unevenness indifferent ranges of wave lengths.

(5) From measurements on roadways the power spectral densities for theroughness of different roadway pavements are known [5] . In plotting thesevalues versus the cyclic frequency of path in a double logarithmic scaleaccording to fig. 6-7 it becomes obvious that all kinds of roadway pavementscan be characterized by similar functions. The densities of roughnessdecrease by an exponential function with the frequency of path and may beapproximated sufficiently by straight lines.


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Figure 6-7: Power spectral density of roadway roughness of different roadpavements (from [3])

Table 6-1: Classification of roadway roughness (ISO-TC 108)

Quality of pavement ( )[ ] [ ] 2w ,m1 for cm 1030 == lowerlimit

meanvalue

upperlimit

very goodgoodaveragepoorvery poor

0.52832128

141664256

< 2< 8< 32< 128< 512

(6) Therefore for the approximation of the power spectral density an exponentialfunction is used as mathematical model.

( ) ( )w

00

=

(7) The spectral roughness coefficient ( )0 and the spectral roughnessexponent w are used as characteristics for the description of the roadwayroughness. In using mean and extreme values for the parameter ( )0 a

classification of roadway roughnesses as given in table 6-1 may be used.


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(8) In order to use this stochastic description of the unevenness for a deterministiccalculation the unevenness has to be described deterministically as a discreteprofile. This is realized by approximating the profile by a fourier series. Theamplitudes of the harmonic elements of the fourier series can be determinedfrom a given power spectral density distribution by

( )iii 2 =

(9) As the power density distribution does not contain any information on thephase difference of the different components of the fourier series, the phaseangles for the different harmonic waves are determined by the Monte-Carlo-Method on the base of a Gaussian normal distribution. Roughness profileswhich have been generated that way showed a good approximation to theinput power spectral density distributions and allow to consider the stochasticroadway roughness in a realistic way.

6.3 Calibration of the models with test results

(1) The mechanical models of the bridge, the vehicles and the roadway surface asdescribed above and their interaction were used for the development of acomputer program. This computer program can simulate the crossing of oneor more vehicles on a bridge. The results of the simulation are time histories ofany displacements, forces or action effects of the vehicle or the bridge. Themain interest is in the bridge response, in particular in the ratio of the extremedynamic response in the bridge to the static response of the bridge which bydefinition is the dynamic increment or the impact factor .

(2) The program was used to simulate the behaviour of bridges wheremeasurements for crossing of test vehicles were carried out in order to provethat the calculations yielded to reliable results.

(3) In fig.6-8 a summary of the input parameters for dynamic loading tests on aroad bridge in Switzerland is given [5]. The bridge system is a continuousthree span prestressed concrete sway frame bridge with a hollow section. Thisbridge was crossed in several tests series by a 160 kN-vehicle with two axlesand leaf suspension with different crossing speeds in the range of 10 to 70km/h. The test series were carried out with two different pavements with

different roughnesses. Both roughness profiles were measured to be useddirectly and the power spectral density distribution was determined to generateroughness profiles indirectly. To get the bridge responses a set of timehistories for deflections and for strains which were measured by strain gaugeswere recorded. In the following only the results from the measurement ofdeflections in the third bridge span are considered.


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Figure 6-8: Input parameters of the calibration experiment

(4) For the numerical simulations of the tests the bridge was modelled as a frame[7]. The discretisation of the system allowed for 32 beam-elements with 72degrees of freedom. The vehicle was modelled as a two-axle single vehiclewith nonlinear behaviour the dynamic characteristics of which weredetermined in vibration tests. The roughness profiles used in the simulationswere those directly recorded.

(5) Fig. 6-9 shows a plot of the dynamic increments of the mid-span deflection inthe third span versus the vehicle speed. The figure presents the measuredand the calculated values. The strong variations of the dynamic incrementsreveal that there is no significant functional dependence between the dynamicincrement of a bridge and the crossing speed of the vehicles. There are great

values of dynamic increments for small speeds between 10 and 20 km/h. Thereason for this effect is the inter-leaf friction of the leaf suspension whichcauses frequent blockings of the leaves for small speeds by which the totalvehicle vibrates on the tires which exhibit only a small damping. The


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Figure 6-10: Comparison of measured and calculated time-histories ofdeflection for a vehicle speed of = 15 km/h

Figure 6-11: Comparison of measured and calculated time-histories ofdeflection for a vehicle speed of = 65 km/h

6.4 Parameter stud ies to determine physical impact factor s6.4.1 Impact factors for the ultimate limit st ate verification

(1) In order to clarify the mechanical causes for the observed dynamic responsesof road bridges and the main controlling parameters, parameter studies werecarried out.

(2) The dynamic effects of crossing vehicles may be best differentiated accordingto the influence length of the structural elements in question. The followingdistinctions can be made:

- local dynamic effects caused by the load of a wheel or of an axle;- dynamic effects caused by the loading of a complete vehicle;- dynamic effects caused by the simultaneous action of several vehicles.

(3) In each of these cases different mechanical processes, influencing parametersand dynamic magnification factors are relevant.

6.4.1.1 Local dynamic effects

(1) Local dynamic effects are investigated for the Stresses of stringers inorthotropic decks. For this investigation the stringer which is directly situatedunder the wheel track of a vehicle is modelled as a continuous beam on elasticsupports with masses according to fig. 6-12. The results of the study are thefollowing:


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Figure 6-12: Calculation model for the stringers of an orthotropic deck

(2) In general the stringers have short influence lengths which result in very higheigenfrequencies. Therefore these structural parts are not excited to significanteigenvibrations by the dynamic wheel loads of the crossing vehicles. Hencethere is no dynamic interaction between the vibrations of the stringers and thevibration of the vehicles. The dynamic stresses in the stringers are mainlydetermined by the dynamic wheel loads and therefore controlled by thedynamic properties of the vehicle, the speed and the surface roughness. Fig.6-13 demonstrates that the dynamic wheel loads increase with the square ofthe speed. The same functional dependence can be observed between thespeed and the dynamic increment of the load effects in the stringers.

Figure 6-13: Relation between dynamic wheel-loads and the dynamicincrement of direct loaded components


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(3) As the dynamic properties of the vehicles vary in a large range, the maximumvalues of the dynamic increments have a large scatter. Fig. 6-14 gives thescatter and the extreme values of the calculated dynamic increments versusthe type of vehicle and the roughness of the surface for a maximum speed of100 km/h. It seems not to be justifiable to differentiate according to structural

parameters, as e.g. cross beam distance or stringer stiffness.

Figure 6-14: Dynamic increments of the stringers depending on the vehicletype and quality of pavement

(4) A particular attention has to be paid to structural elements adjacent to thetransition joints, because the dynamic wheel loads in this region aresignificantly greater than in other areas of the bridge due to the localunevenness of the joints. From these local unevenness the dynamicincrements may attain values which may be up to 50 % greater than thosegiven in fig. 6-14. In concluding a constant dynamic magnification factor forlocal wheel effects depending only on the roadway roughness seems to beappropriate.

6.4.1.2 Dynamic effects of indi vidual vehicles

(1) For the dynamic action effects of the main girders of a bridge with mean orlong spans the frequency contents of the wheel load variation which influencesthe resonance behaviour has greater influence than the maximum values ofthe wheel loads.

(2) Therefore the dynamic behaviour of the total vehicle plays an important roleand the dynamic increment of the bridge is mainly controlled by the interactionof vehicle and bridge.

(3) These mechanical processes shall be first considered for the case of a singlevehicle crossing the bridge. The calculations have been carried out for three

span bridges with equal span lengths where the geometry and stiffness werechosen such that a wide spectrum of usual span lengths and eigenfrequencies


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were represented. Each dot in fig. 6-15 is referred to one of 35 bridges thatwere investigated in the study.

Figure 6-15: Relation between the fundamental frequency and span length of

investigate bridges

(4) As the regression curve, which from empirical studies gives a relation betweenthe fundamental frequency of a bridge and its span length [6], is independenton the type of material and way of construction, the following results areapplicable to steel bridges, reinforced concrete bridges and compositebridges.

- The dynamic response of a bridge is mainly determined by thefundamental frequency of the bridge and the spectrum of the dynamic

wheel loads. The bridges exhibit increased dynamic increments whentheir fundamental frequency is in the range of the frequencies of thevehicles. It is assumed however that the roughness of the surface issuch that the dynamic wheel loads in this frequency range are excited.

- The spectrum of the wheel loads is determined by the dynamicproperties of the vehicle and the roadway roughness. Hence in additionto the fundamental frequency of the bridge these two parameters arethe significant parameters for the dynamic reactions of a bridge undertraffic loads.

- Bridges with short spans have an increased sensitivity to resonancevibrations due to critical frequencies caused by the sequence of axles.

- The feedback effects of the vibrations of the bridge to the vibrations ofthe vehicles are rather small, i.e. the amplitudes and frequencies of thewheel loads are almost not influenced by the vibrations of the bridge.

- There is a great scatter of the maximum dynamic increments for a givenbridge even in case of constant surface quality and constant vehicles.The reason for this scatter is that a given power spectral densityfunction for the roadway roughness can be modelled by an infinitenumber of deterministic roughness profiles which yield to differentresults.


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- The dynamic increments for the deflections are greater than those forthe bending moments. The dynamic increments for the hoggingmoments are greater than those for the mid-span moments.

- The damping of the bridge has less importance than the other

influence-parameters.(5) The significance of the fundamental frequency of the bridge and of the

characteristics of the vehicles can be taken from the fig. 6-16 6-18 which aregiven as examples.

Figure 6-16: Relation between the fundamental frequency and span length ofinvestigate bridges

(6) Fig. 6-16 demonstrates the dynamic increments of the bending momentsversus the fundamental frequency of the bridge for a two axle vehicle withpneumatic suspension and a good quality roadway pavement. The envelopedemonstrates resonance effects in the ranges of 1.5 Hz to 2.5 Hz and beyond7.0 Hz. In the ranges between these resonance areas the dynamic incrementsare significantly reduced.

(7) The resonance effects are caused by the fundamental vibrations of the bridgeand the eigenfrequencies of the rigid body masses and the axle masses of thevehicles. The frequency ranges with low dynamic increments are outside theexcitement ranges of the vehicles. The results of the calculations are in goodagreement with the results of experimental tests [6].

(8) The influence of the variation of the dynamic characteristic of the vehicle onthe dynamic increments of the bridge can be taken from fig. 6-17. The vehicleconsidered had a conventional leaf suspension. The difference in the axle loadspectrum yields to a widening of the frequency ranges where the bridges gethigher dynamic increments and to greater values of the dynamic increments.


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Figure 6-17: Dynamic increments of the bending moment as a function of thefundamental frequency (vehicle B)

(9) Fig. 6-18 shows a reduction of the resonance effects in the range of lowerfrequencies due to articulated vehicles. Only for bridges with short Spans thedynamic increments are increased due to critical frequencies caused by thesequence of double and triple axles.

Figure 6-18: Dynamic increments of the bending moment as a function of thefundamental frequency (vehicle )

(10) The influence of the stochastically distributed roadway roughness on thedynamic increments can only be determined by statistics, because a givenpower spectral density distribution gives a set of equivalent deterministic

roughness profiles which cause maximum dynamic increments that scattervery much. In ordering the impact factors of different bridges versus meanvalues it can be shown, that the scatter increases with increasing mean valuesand a correlation between the means values and the standard deviation ispossible. The mean values have been obtained on the basis of ten differentroughness profiles generated for a given roughness quality. From this anyextreme values of the dynamic increment can be determined when theprobability of exceedance is specified. The study has yielded to the result thatthe maximum values of the impact factors of a bridge may be correlated to theroughness quality "very good", "good" and "average" by the factors 1 : 2 : 4.


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6.4.1.3 Dynamic effects with several vehicles

(1) A third item of the parameter study was devoted to determine the dynamiceffects caused by the simultaneous action of several vehicles of equaldynamic properties. The study was carried out for a convoy of 32t-vehicles

with four axles and an inter-vehicle distance of 5.0 with by a maximum speedof 40 km/h.

(2) The results of the study demonstrate that the magnitudes of the impact factorsdue to resonance with the rigid body eigenfrequency of the vehicles aresignificantly reduced and the dependences on the fundamental frequency ofthe bridge is less significant than for single vehicles, see fig. 6-19. The reasonfor this result is the wide band spectrum of excitation where the dynamiceffects of different vehicles are partly compensated. The influence of the spanlength of the bridge becomes more important. In fig. 6-20 the results aretherefore plotted versus the span length. The tendency of the dynamicincrements decreasing with increasing span length can be clearly identified.This is caused by the decreasing influence of the vehicle dynamics in relationto the static vehicle effects with increasing span length.

Figure 6-19: Dynamic increments of the bending moment as a function of thefundamental frequency for a convoy of heavy vehicles


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Figure 6-20: Dynamic increments of the bending moment as a function of thespan for a convoy of heavy vehicles

6.4.2 Impact factors for fatigue assessments

(1) Road bridges are loaded by varying traffic loads and have to be verified inview of sufficient fatigue safety. In the fatigue assessment the fatigue loadingeffects during the expected service life have to be accumulated and introducedinto a damage calculation. As the fatigue strength of steel parts is mainlydefined in terms of stress ranges, the traffic influences on fatigue have to bedescribed by loading ranges which include dynamic increments. The dynamicincrements to be used for the fatigue assessment do not represent extremevalues but have to be derived from damage calculations with actual timehistories where the dynamic response of the structure is considered. Due tolack of knowledge in this field the fatigue impact factors so far had beenestimated only. In the following a damage equivalent impact factor for thestringers of an orthotropic deck is determined.

(2) When a vehicle crosses a certain section of the stringer a set of stress rangesi

and cyclic numbers in is effected. The damage caused by this crossingcan be determined as

( ) =

i cmic

i

i N /

nd

(3) Here m is the slope of the S-N-curve and c is the reference fatigue strengthfor a certain detail for c N = 2 10

6 cycles. The consideration of the dynamicinfluence means that in relation to the spectrum of stress ranges i ; which isobtained for static loading only another spectrum of stress ranges i including dynamic effects is necessary. The fatigue impact factor can bederived from a comparison of the different damages caused by these spectra.

(4) This comparison of damage is carried out for single vehicles. The definition ofthe damage equivalent impact factor of a single vehicle is such, that thedamage yielding from a static calculation with a vehicle load multiplied with


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E , is the same as the damage calculated for dynamic stress ranges. Inassuming a linear relation between the stresses i of the stress rangespectrum, the action effects and the actions, the damage equivalent impactfactor may be determined on the level of action effects:

( )( )

=

mstat ,istat ,i

mdyn ,idyn ,i

E M n

M n

(5) The moment ranges which are used for the determination of E , have beenderived from the time histories for the bending moment by using the rainflowcounting method.

(6) The magnitude of the damage equivalent impact factor for local effects iscontrolled by the same parameters as the impact factor for ultimate limit state

verifications. These parameters are the type of vehicle, the vehicle speed andthe roadway roughness. The fatigue impact factor however representsapproximately an average dynamic increment and not an extreme value, dueto the effect of the damage calculation. This also applies to the scatter ofdynamic influences arising from vehicle characteristics. Fig. 6-21 shows themaximum values of the damage equivalent impact factor as well as the densitydistribution of the values depending on the type of vehicle. The distributionassumed is a Gaussian distribution.

Figure 6-21: Damage equivalent impact factor for the bending moment of thestringers of an orthotropic deck as a function of vehicle type andquality of pavement

(7) For a mean pavement quality and a speed of 100 km/h the mean value for thedamage equivalent impact factor for the fatigue assessment of a stringer of anorthotropic deck is in the range E = 1.28.

6.4.3 Conclus ions for loading codes

(1) The study on dynamic influences explains the physical phenomenon ofinteraction between moving vehicles, surface roughness of the road andvibration behaviour of bridges all focussed on the action effects needed.


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(2) The basis of the study are deterministic traffic scenarios that give an insight

into this interaction.

(3) Realistic traffic scenarios however are random in view of the composition of

the traffic, the vibration properties of vehicles depending on the loading rate,the intervehicle distances etc.

(4) Therefore the results of dynamic analysis need statistical evaluations that maylead to

- characteristic values and fatigue values of action effects from a staticanalysis,

- characteristic values and fatigue values of action effects from a dynamicanalysis.

(5) If dynamic impact factors are then defined as the ratios between e.g. thecharacteristic values of action effects from the dynamic analysis on one handand from the static analysis on the other hand they are in general no morerelated to the behaviour of the same vehicle but to the behaviour of differentvehicles and therefore may substantially differ from the dynamic impact factoras defined for deterministic loading situations as assumed in this section, seefig. 6-22.

Figure 6-22: Definitions of impact factors

(6) Statistically oriented dynamic analysis using the simulation method explainedin this section are given in section 7 of this report.


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7. Dynamic simulations for jus tifying the load models in EN 1991-27.1 Procedure and assumpt ions

(1) The simulation program explained in section 6 of this report has been used todetermine dynamic response histories of structures under moving traffic

- to check the results obtained with evaluations of response historiesbased on static influence lines

- to determine damage equivalent factors applicable together with thefatigue load models FLM3 and FLM4.

(2) The basis of such dynamic simulations were the statistical data of traffic asindicated in fig. 3-15 that were filtered from dynamic amplification factors by anumerical simulation of the measured data at the measurement point at

Auxerre, see fig. 3-8.

(3) This filtering was achieved by calculating the dynamic wheel loads and vehicleloads from the axle weights with dynamic effects as recorded with assuming agood surface roughness of the road at the point of measurement, so thatdynamic increments could be obtained, that then could be subtracted from thedynamic wheel loads and vehicle loads in order to get the static wheel loadsand vehicle loads.

(4) In this simulation the dynamic properties of the vehicles were described bystatistical parameters as given in table 7-1.

Table 7-1: Statistical parameters of dynamic properties of vehicles

(5) In order to obtain a large number of data necessary because of the definitionof characteristic values by

- a mean return period of 1000 years or equivalently- a return period of 50 years for 5% of the bridges or- a 10% fractile for a period of 100 years equal to the nominal service life

of a bridge

a sufficiently large number of simulations had to be performed.


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(6) Table 7-2 gives a survey on the bridges that fulfil the frequency conditions asgiven in fig. 6-15, that were used for the simulations for

- single span bridges with span L,- continuous bridges with 3 equal spans L.

Damping was assumed to be of Rayleigh-type with a critical value 1 = 0.6%for the first mode and 2 = 0.9% for the second mode.

Table 7-2: Characteristics of the bridges simulated

(7) Fig. 7-1 gives a survey on the action effects as bending-moments and shearforces that were determined by the simulation.


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Figure 7-1: Survey on the influence lines considered in the simulations(8) In the transverse direction a constant value of the influence line was assumed

which corresponds to box girder type bridges.

(9) The assumptions made for the surface quality ( = 2.0) were

- for characteristic values (related to ULS-verifications):

- good ( )o = 4.0 cm- average ( )o = 16.0 cm- poor ( )o = 64,0 cm

In addition a local irregularity as specified in fig. 3-9 waschecked.

- for representative values (related to SLS-verifications):

- only a good and average surface quality was assumed

- for fatigue equivalent values:

- only a good surface quality was assumed,- the effects of an irregularity was checked to model the effects of

transition joints.

(10) The assumption for the type of traffic were:

Traffic 1 (artificial): slow lane (lane 1) as in Auxerre with lorries only (byeliminating all cars)

Traffic 2 (measured): Auxerre traffic as measured in the slow lane(mixture of lorries and cars)

Traffic 3 (measured) Auxerre traffic as measured in lane 2


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Traffic 4 (artificial): traffic with cars only.

(11) Depending on the deck width the allocations of the various types of traffic to

the lanes are given in fig. 7-2.These assumptions correspond to the assumptions made for the evaluation ofstatic influence lines, see 3.2.4(14).

Figure 7-2: Deck widths and assumption for composition of traffic

(12) Inter-vehicle distances have been fixed as specified in 3.2.5(7) with aminimum value a = 5m.

(13) In total for each type of bridge, each width of bridge deck, each surface qualityand speed of vehicle 100 simulations were performed with about 25 vehiclesselected according to the Monte-Carlo method.

For each simulation the static and dynamic maximum values were determinedand plotted in a diagramme with an accumulated normal frequencydistribution.

Using the data from Auxerre the characteristic value was determined byextrapolating the distributions to the 8 105.11 fractiles which in the normaldistribution corresponds to the mean value plus 5 x standard deviations. Forthe extrapolation only the half-normal distribution fitted to the upper part of thereal distribution was used.

(14) Fig. 7-3 shows the example of a bridge response (static and dynamic) for 10vehicles with a speed h / km80v = and the selection of the maximum static anddynamic values.


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Figure 7-3: History of static and dynamic bridge responses and choice of

extreme values

(15) From 100 extreme values determined for the situation in fig. 7-3 theaccumulated frequency distributions as given in fig. 7-4 were determined thatcould be further evaluated to get characteristic values, see fig. 7-4.

Figure 7-4: Example for the accumulated frequency distributions for staticand dynamic bridge responses

(16) Various simulations have shown that flowing traffic (with a speed of vehicleh / km80h / km60v = ) is relevant for span lengths m20 L , whereas

congested traffic ( )h / km20h / km10 or jam situations ( )h / km0 are relevantfor span lengths m30 L , see fig. 7-5.


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Figure 7-5: Characteristic values of bridge responses for flowing and

congested traffic

(17) Fig. 7-6 shows the influence of different surface qualities and flowing andcongested traffic on the bridge responses, and fig. 7-7 gives the associateddynamic factors.

Figure 7-6: Characteristic values of bridge responses for different surfacequalities and flowing and congested traffic


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Figure 7-7: Dynamic factors resulting from the characteristic values in fig. 7-6

(18) In order to mirror traffic situations more realistically also the assumption ofdifferent speeds of vehicles in various lanes was made (mixed traffic):

- for 2 lane traffic: lane 1 h / km80v = lane 2 h / km10v =

- for 4 lane traffic: lane 1 h / km10v = lane 2 h / km80v = lane 3 h / km80v = lane 4 h / km10v =

however comparative studies showed that either flowing or congested trafficwith equal speeds are relevant depending on span lengths.

7.2 Resul ts of the simulations for LM1 and LM2

(1) Characteristic values of uniformly distributed loads determined from dynamicsimulations and from evaluations for static influence lines are compared forflowing traffic in fig. 7-8 and for congested traffic in fig. 7-9.


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Figure 7-8: Comparison of characteristic values of equivalent uniformlydistributed loads from dynamic simulations and static simulationsusing influence lines for flowing traffic

Figure 7-9: Comparison of characteristic values of equivalent uniformlydistributed loads from dynamic simulations and staticsimulations using influence lines for congested traffic

(2) The comparison demonstrates that the results obtained in different ways bydifferent authors are sufficiently accurate, differences result mainly fromdynamic effects.

06 Manus Sedlacek 1 Background Document to en 1991 Part 2 - Traffic Loads for Road Bridges

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