TRANSLATION OF MEASURED VEHICULAR WEIGHTS INTO DESIGN LOADS TO BE USED FOR BRIDGE ENGINEERING

Transcription

1 7th International Symposium on Heavy Vehile Weights & Dimensions Delft. The Netherlands. June TRANSLATION OF MEASURED VEHICULAR WEIGHTS INTO DESIGN LOADS TO BE USED FOR BRIDGE ENGINEERING ABSTRACT Measurements performed in 1978, 1994, 1996, and 1999 at three loations in The Netherlands were used to derive a statistial model for the axle loads and vehiular loads. The paper disusses the proedure that was used to fit statistial distributions to the measured data and to derive design loads for both axle and vehile loads. Moreover, a trend analysis of the design loads is presented. Finally, the influene of speial permit vehiles on the design load was evaluated. INTRODUCTION During the last ten years, TNO Building and Constrution Researh has arried out an investigation into the modelling aspets of traffi loads on bridges, see for instane Vrouwenvelder & Waarts (1993) and Vrouwenvelder et al (1999). The Duth ministry of Transport, Publi Works and Water Management ommissioned the investigations. The aim of the investigations was to find a design proedure, whih guarantees a struture with a presribed level of safety with respet to the various limit states for the intended lifetime, and to replae the old ode VOSB (1963). Aording to present day standards, the required level of safety is expressed as an aeptable probability of failure or a target reliability index P in ombination with a referene period of time for whih the probability should be applied. For the Ultimate Limit State (ULS) the antiipated life time of the struture has been set equal to 1 years. For this period a reliability index P = 3.6 has been hosen aording to the urrent Duth building regulations (NEN67). This value orresponds to a probability of 1-4. For sake of omparison: the informative annex of the Euroode requires a reliability index P = 3.8. For the servie-ability limit states with respet to stati defletions and vibrations an ourrene rate of one per year on the average was thought to be appropriate. DERIVATION OF DESIGN LOADS For the design of bridges with respet to strength or servieability a designer needs axle and vehiular loads for the details and distributed lane loads for the design of prinipal onstrution parts like the main girders or the able stayes. Derivation of these loads involves two important unertainties: Extrapolation of (short time) measurement results to design loads; Future trends in traffi loads. A set of measurements done in 1978 (one loation: Rheden) and in 1994, 1996 and 1998 at three loations (M oerdijk, Arnhem and Eindhoven) in The Netherlands were used to derive a statistial model for the axle loads and vehiular loads. The measurements lasted some 2 to 7 days eah, registrating about 1, to 8, heavy vehiles per measurement. This large set of data enabled an extensive statistial analysis. Furthermore it was expeted that the trend in vehiular load ould be analysed in view of the long period between the Rheden (1978) and later measurements. This paper will only onsider the ultimate limit state. Furthermore only part of the study is presented, i.e. onerning the axle and vehiular loads. Given the vehiular load, the distributed lane loads an be derived. The proedure for this is already explained in Vrouwenvelder & Waarts (1993) and will not be further disussed in this paper. 513

2 PROBABILISTIC DESIGN PHILOSOPHY For the fundamental ase, where there is one load effet parameter S and one resistane parameter R, the basi design requirement in present-day odes is given by: (1) The index d indiates "design value". The value of Sd should follow from the requirement that the probability of "S > St" in the referene period is equal to: P{Sd> S} = <1>(-as) (2) Where <1> = standard normal distribution; as = influene oeffiient for the load; = target reliability index. In the ode one may use any ombination of harateristi value and partial safety fator that leads to Sd. To find as in equation (2) one should in priniple perform a omplete reliability analysis. For reason of simplifiation, aording to the appendix of ISO 2394, a standardised value of a s =.7 is adopted. This means that for the ultimate limit state the design load has a probability of exeedane of <p(-a s ) = <1>(-.7*3.6) = <1>(-2.5) =.62 in 1 years. If all unertainties in the load model are of an ergodi (not time-invariant) nature, the value Sd has a return peri od equal to: (3) Non-ergodiity may be aused, for instane, by statistial systemati unertainties, by unertainties in the traffi or vehile models, and so on. So, if the load proess is ergodi, this orresponds to a load effet with a return period Td = = 16 year. For the servieability limit state, the value equals. and so the value of as does not matter. The design value simply equals the maximum load effet to be expeted in one year. AXLE LOAD Distribution type A statistial analysis was arried out to fit probability distributions to the measured axle loads. On the basis of these probability distributions design values for the axle loads ould be derived. The statistial analysis was arried out in two parts. The first part, whih is disussed in this setion, onerns the derivation of the distribution type. The seond part, i.e. the assessment of the distribution parameters, is addressed in the next setion. The largest dataset (Moerdijk '98) was used to derive the distribution type. Beause of physial irumstanes it is supposed that the distribution type is equal for all measurements (loations and dates). Figure 1 shows the frequeny distribution of the measured Moerdijk '98 data; Figure 2 shows the umulative frequeny on logarithmi sale. Both figures are based on 284,881 measured axle loads over a period of 7 days. The shape of the distributions are harateristi for all data sets: at least one mode at about 4 kn, a main mode around 6 kn and a "shoulder' that starts above about 9 kn. These frequeny distributions annot be desribed by one single standard distribution type. As we are interested in the extreme values, we aim to desribe only the tail of the distribution (above a suitable threshold value Yo) with a standard distribution. There is no other information available exept the measured data. Figure 2 shows relative frequenies of exeedane ranging down to about The probability of exeedane of the design load is 1-11 (the target probability level of.62 mentioned in the previous paragraph divided by the number of vehiles per referene period of 1 years). So, the umulative frequeny distribution needs to be extrapolated over about five orders of magnitude. As illustrated in Figure 3 the result of the extrapolation may highly depend on the hoie of the standard distribution type. The four distributions shown Figure 3 result in a wide satter of design loads. It is assumed that only analysing these four distributions desribes the various extrapolation possibilities suffiiently. 514

3 The determination of the distribution type is based on the Bayesian approah. For eah of the four distribution types the probability is alulated based on the measured data (posterior probability P{iI.:d ): P{il y}= 4 P {YI i}p{i} i 1",,,4 LP{I i}p{i} ;=1 (4) where y = the distribution type (see indies in Figure 3); = measured data; P{ il y} = the posterior probability of the data being distributed aording to distribution type i; P{yl i} = likelihood of data y in distribution i; P{i} = the prior probability of distribution i. Distribution parameters and design loads The design load is derived from extrapolating the tail of the frequeny distribution. The result depends on the threshold load Yo. Two onsiderations lead to the best hoie of the threshold load Yo. The higher Yo the better the statistial distribution desribes the tail of the frequeny distribution. On the other hand: a higher Yo leads to less data and therefore a larger unertainty in the fitted distribution. The optimal threshold load Yo balanes the two onsiderations. The umulative distribution in Figure 6 shows a small kink above 1 kn. A threshold load lower than 1 kn is therefore supposed to be inappropriate. The design load is presented against the threshold load Yo in Figure 5 to dedue the most suitable value for Yo. The figure also shows the unertainty margins belonging to the design load for the various threshold loads. The inrease of the unertainty margins with inreasing Yo is based on the derease of used data. To alulate the unertainty margins, 2 samples of axle loads were randomly drawn from the Weibull distribution shown in Figure 6. Eah sample had the same size as the measured data set (284,881 axle loads). Subsequently on eah of the samples a Weibull distribution was fitted for various threshold loads. This resulted for eah threshold load in 2 fitted Weibull distributions from whih 2 design loads were alulated. The unertainty margin per threshold load was estimated as +/- two times the standard deviation of the assoiated 2 design loads. Figure 5 shows that a threshold load of 125 kn is the optimum hoie for the Moerdijk '98 axle loads. Higher threshold loads lead to some variation in design values, but this variation an be interpreted as statistial t1utuation as the values remain within the unertainty bands. Apparently, the distribution found at a threshold load of 125 kn also gives an aeptable fit to the data above higher thresholds. If a lower threshold value had been seleted, the drawn horizontal line in Figure 5 would have moved up, together with the unertainty bands, to math the design value (irle) assoiated with that threshold. Due to this shift, the unertainty bands would no longer have overed some of the design values for higher thresholds. This proedure is repeated for the nine other measurements. Figure 7 shows the design loads. It is remarkable that all ' 94 to '99 measurements result in a higher design load than the old "Rheden '78" measurement. It is also remarkable that the '94 to '99 measurements show no trend in axle load. The analysed data give no basis to onlude whether there is a trend in axle load (based on ' 78 data ompared to '94-'99 data) or that the differene an be explained by the differene in measurement loation. VEHICULAR LOADS Distribution type, parameters and design loads The frequeny distribution for the vehile loads resulting from the Moerdijk ' 98 dataset is shown in Figure 8. The distribution has a similar shape as the axle load distribution, exept for one thing: there seems to be a pronouned heavy tail starting above 65 kn. About 3 vehiles form this tail. In a first analysis, the data points in this heavy tail were onsidered as outliers produed by measurement errors. The onsiderations were: these data points formed only a very small fration of the data (3 in 8 ) 515

4 the axle onfigurations of the vehiles were not reognised by the vehile lassifiation system Moreover, a possible explanation for the extremely heavy loads ould be that two vehiles at lose distane were interpreted as one single vehile by the measurement system. On the basis of this assumption, distributions were fitted to the data in a similar way as desribed in setion for the axle loads. For the Moerdijk '98 data this resulted in a Weibull distributions as shown in Figure 9 with a orresponding vehile design load of 11 kn. Figure 9 shows an overview of the design loads obtained for other loations and measurement periods. As for the axle loads all reent measurements ('94-'98) result in a higher design load than the older "Rheden '78" measurement. The reent measurements on its own do not show a trend in vehiular load. Whether the differene in design load between the '78 data and '94-99' data an be explained as trend in load or as a result of different measurement loations an not be onluded from this data. Speial permit vehiles At the time of a seond analysis, data had beome available on permit appliations for heavy vehiles. In The Netherlands, it is ompulsory to apply for a permit for vehiles with a weight in exess of 1 kn. Figure 1 shows the frequeny distribution of the weights of vehiles, whih were granted a permit over the period The distribution is saled to enable a omparison with the measured data. In the light of this information, the assumption that the heavy tail of the measured frequeny distribution results from measurement errors needed to be reonsidered. Indeed, this tail onnets smoothly to the distribution of the permit data. The axle onfigurations of the vehiles responsible for this tail were srutinised one by one and it appeared that although these are not standard vehiles, most of them have realisti axle onfigurations. Hene, it was onluded that the earlier assumption of measurement errors ould possibly be unjustified. To explore the onsequenes of the heavy tail being real, a distribution was also fitted to the distribution inluding this tail. The proedure differed from the proedure for the axle loads in two respets. First, it was onsidered that the fit and the resulting design load should be based on unontrolled traffi only, i.e. vehiles without a permit. A frequeny distribution for the vehiles without permit was obtained by subtrating the frequeny distribution of the permits from the frequeny distribution of all measured vehile loads. The result is shown in Figure 1]. On this frequeny distribution the fit was arried out. Seond, two separate populations of vehiles were distinguished, eah with its own probability distribution. The probability distribution of the first population, the 'standard' vehiles, is the fitted distribution as obtained in the first analysis. The probability distribution of the seond population, the 'speial' vehiles, was alulated with a similar fitting proedure, but applied to the vehiles in the heavy tail only. The probability distributions for both populations of vehiles are shown in Figure 11. This figure also shows that if the heavy tail is real, a vehile design load of ] 6 kn should be onsidered in stead of the 11 kn that was derived in the first analysis. To enable better founded onlusions about the tail of the frequeny distributions for vehile loads, additional measurements are planned, in whih traffi information inluding axle and vehile loads wil be reorded during a period of one year on various loations in The Netherlands. CONCLUSIONS The distribution type and the threshold value for the tail of the frequeny distribution heavily influene the extrapolation from measured data to design load. Both items should be given extensive attention when deriving design loads for bridges. A trend in axle or vehiular load an not be observed from the '78 to '99 measurements. The various loations do not show a onsistent view. Loads in the dataset that are extremely high should not be regarded as outliers from the distribution too easily. Data in the Netherlands leads to a first impression that this data are real and should not be exluded from the statistial analysis. Bruls, P., Jaob, M. and Sedlaek, P., "Traffi data of the European ountries", 1st draft, Euroode on ations No. 9, Part12-W.G. 2, Feb Jaob, B., "Methods for the predition of extreme vehiular loads and load effets on bridges", Revised draft, Euroode on ations "Bridge loading", WG8, august Vrouwenvelder, A.C.W.M., P,.H. Waarts, "Traffi loads on bridges", Strutural engineering International, 3/1993 VOSB, NEN 18, "Diretions for the designing of steel bridges", Aug Vrouwenvelder, A.C.W.M., de Wit, P.H. Waarts, TNO report CON-DYN-R1813, Rijswijk,

5 TABLES & FIGURES.1.9 Moerdijk '98 data >-.8 u >.7 :::J -.6 >.5 >.4 Q5.3.2 P.1 ol eb o axle load (kn) Figure 1 - Frequeny distribution of axle loads > 1 v u Moerdijk '98 data as 1O-1 > U X > 1. 2 ' >. g1-3 > :::J - 1O-4 > > axle load (kn) Figure 2 - umulative frequeny of axle loads Frehet (4) data Gumbel (3) Gamma (2) Weibull (1) extrapolation o axle load (kn) Figure 3 - Typial extrapolations depending on the distribution type 517

6 Moerdijk '98, threshold load <12 kn Moerdijk '98, threshold load 13 kn g m m.8 e. Cl. (;.6 - Cl..6. (; 17i i Cl Cl distribution type Moerdijk '98, threshold load 15 kn 1. g m m e Cl. (;.6 (; i.4 17i.4 Cl.. Cl e. e Cl. 1.2 f-- I I distribution type Moerdijk '98, threshold load 17 kn distribution type Figure 2.. Posterior distributions of distributions distribution type " m.2 2 :: Ol 'u; 15 Cl> ( r ro.- '" u U T- T -r -' ( o O'"" G ",... 'u; 15 Cl> " " (f) (f) -l 1 -l 1 ::J ::J ( D '"'./' 25. =--=1'-. :,:r..-t' r. " -= J,I,I u m.2 2 :: Ol threshold load (kn) threshold load (kn) Figure 5 - Left figure: The irles show the alulated design loads as a funtion of the threshold value Yo' The horizontal line is the hypothetial 'true' design load, alulated from the data at the seleted threshold load of 15 kn. The grey area shows whih deviations from the 'true value' may our if the design load would be estimated using a higher threshold value. Apparently, the threshold value at 15 kn yields a biased design load: many of the irles fall outside the grey area, i.e. their deviation from the horizontal line annot be explained from statistial unertainty. Right figure: The The horizontal line is the hypothetial 'true' design load, alulated from the data at the seleted threshold load of 125 kn. Now all irles fall inside the grey area, whih indiates that the design load, alulated at a threshold of 125 kn is unbiased. 518

7 axle load (kn) Figure 6 - The irles show the umulative distribution and the fitted Weibull distribution at a threshold load of 125 kn Z ::.. 2 Cl) ::::J iii 15 en "iii Cl) " 1 5 o +---"-"""'""-----,----- Rheden Arnhem Eindhoven Moerdijk Figure 7 - Design axle loads for the various loations and positions in time 1 ID ts -2 " ID 1 ID X ID -4 ' 1 >- C -6 Moerdijk '98 data measured vehile loads ID 1 :::::l - -8 '-fitted Weibull-distributio ID 1 > vehile load O<N) Figure 8 - Frequeny distribution of vehile loads from Moerdijk '98 data and extrapolation in the first analysis 519

8 14 12 Z 1 "' tl.2 8 (ij "S E 6 Q) > 4 > w hc=l.---- r-- > r- < > r-- > 11 > O>-r- N > f--- r-....!:"... r-- > -r- >-> > o > < > r '- f- I'- I O> -r- f- > I-- > -r- - f--- - i- - I-- - i- Q) "' f- f Rheden Eindhoven Arnhem Moerdijk Figure 9 - Design vehiular loads (return period 16 years) for all measurement loations tl - :R 1 x ' > a year.g 8 > 1 measured vehile loads... permit data ", ---""\ \, o vehile load (kn) 2 25 Figure 1 - Comparison of the measured data with permit data. o 1 1IiIIIIi---r-----FMerdijk'98=data... tl - X ' > a5 6 :J 1 CT.g 8 > 1 unontrolled vehile loads standard vehiles o vehile load (kn) Figure 11 - Fitted distributions on both the population of 'standard' vehiles and the population of 'speial' vehiles 52