Traffic load modeling based on structural health monitoring data

Chengming Lan & Hui LiSchool of Civil Engineering, Harbin Institute of Technology, Harbin, China

Jinping OuSchool of Civil Engineering, Harbin Institute of Technology, Harbin, China; School of Civil and HydraulicEngineering, Dalian University of Technology, Dalian, China

ABSTRACT: Live load models are foundation for life-cycle design of highway bridges. Many highway bridgesare now equipped with structural health monitoring systems, which provide valuable data to establish loadmodels. In this paper, traffic load models of the Binzhou Yellow River Highway Bridge are developed based onthe field measurement of vehicles by an already installed structural health monitoring system. The probabilisticdistribution model and extreme value distribution of gross vehicle weight are statistically analyzed using themonitoring data and the results indicate that they follow the Bimodal-Lognormal Distribution and GumbelDistribution, respectively.

1 INTRODUCTION

Traffic load is one of the most critical factors thatinfluence a bridge design, analysis and maintenance.Important traffic load information includes the mostpossible maximum gross vehicle weight during thedesign return period. The extrapolation and statisticalanalysis are frequently employed to obtain the mostpossible maximum gross vehicle weight within thedesign return period. The extrapolation approach wasproposed by Nowak (1994). This approach provides aneasy and effective way to obtain the maximum valueof related parameters. It can avoid complicated sim-ulation. Nowak (1994) used this approach to obtainthe traffic load model of the Ontario Highway BridgeDesign Code. However, the approach is subjectiveand the accuracy depends on the experience of theresearchers. The statistical analysis approach is aneffective method to obtain the cumulative distribu-tion function (CDF) of gross vehicle weight and axleweight. Miao and Chan (2002) used this approach toobtain the traffic load model for Hong Kong. Thedisadvantage of the statistical analysis approach is itscomplexity.

In this paper, traffic load models are establishedbased on the measurement of vehicles by a structuralhealth monitoring system installed on the Binzhou Yel-low River Highway Bridge (BYRHB). The probabilis-tic distribution model and extreme value distributionof gross vehicle weight are statistically analyzed usingthe monitoring data.

2 DESCRIPTION OF THE BRIDGE

The Shandong Binzhou Yellow River Highway Bridgeis a three-tower cable-stayed bridge carrying a dualthree-lane carriageway over the Yellow Rive, as shownin Figure 1. This bridge is located in ShandongProvince, China, providing an important connectionbetween Eastern and Northern China. The construc-tion of the bridge began in August, 2001, and wascompleted in November, 2003, and opened to traf-fic in July, 2004. The entire length of the bridge is1698.4 m with a 13.102 km highway approach andconnection line. The main bridge has a total length of768 m, consisting of two 300 m spans and two 84 mside-spans.

A sophisticated structural health monitoring systemwas installed on this bridge and became operationalsince 2004 (Ou, 2003; Li et al, 2006). The measure-ment of vehicles passing through this bridge providesthe valuable raw data to model the traffic loads.

3 EXTREME VALUE DISTRIBUTIONOF VEHICLE LOAD

3.1 Probability distribution of vehicle load

The data of gross vehicle weight have been collectedby the SHM system since 2004. One-day data is takento be an observation unit for analyzing the probabilitydistribution and extreme value distribution of vehicle

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84m 300m 300m 84m

Figure 1. General view of the Shandong Binzhou Yellow River Highway Bridge.

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Histogram of gross vehicle weight

Veh

icle

num

ber

Gross vehicle weight kN

Figure 2. The histogram of gross vehicle weight.

gross weight for this bridge. Statistic operation isrepeatedly conducted on every day data and the resultsindicate that the statistic characteristics of gross weightfor all days are almost the same. Therefore, a sampledrawn randomly from the data in 2005 is employed tomodeling the traffic load. The histogram of gross vehi-cle weight is shown in Figure 2. It can be seen fromFigure 2 that vehicle with weights less than 100 kN(light vehicles) are dominant, however, heavy truckswith weight of more than 700 kN are also observed.The mean, standard deviation and coefficient of varia-tion of the sample are 104.94 kN, 172.83 kN and 1.647,respectively.

Generally, the Lognormal Normal Distribution(LND) and Inverse Gaussian Distribution (IGD) whichare the unimodal distribution are frequently used to

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CDF of LND

CDF of IGD

CD

F F

(x)

Gross vehicle weight kN

Figure 3. Cumulative distribution functions of LND andIGD of the gross vehicle weight.

fit the cumulative distribution function (CDF) of thesamples and the results are shown in Figure 3. TheKolmogorov-Smirnov test (KS-test) (Hahn, 1994) isadopted herein to determine if a sample comes from apopulation with a specific distribution, which can offera critical value following a given reliable parameter.The advantage of this approach is that all deviationscan be obtained between every observed distributionpoint and theoretical distribution point. The LND andIGD cannot be accepted by the KS-test at the 5% sig-nificance level. Therefore, the LND and IGD cannotbe used to model the probability distribution of thegross vehicle weight. Since the regularity of the loga-rithm of gross load is clearly observed from Figure 4,

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CDF of sample

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CD

F F

(y)

PDF

f(y)

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Histogram of PDF

Figure 4. Probability density function and cumulative dis-tribution function of variable Y .

the logarithmic distribution of gross vehicle weightcan be obtained by univariate finite mixtures of Nor-mal Distribution. X denotes the gross vehicle weightwith a unit of kN and Y is the natural logarithm of X ,i.e. Y = ln(X ).F̂(y) is the CDF of Y and expressed by

F̂(y) = p1�

(y − μY 1

σY 1

)+ p2�

(y − μY 2

σY 2

)(1)

where p1 and p2 respectively represent the weightor mixing coefficients for the first and second term,and p1 + p2 = 1; �(·) is the CDF of standard normaldistribution; μY 1, σY 1, μY 2 and σY 2 are the statisticalparameters of the distribution in Eq. (1). The Maxi-mum Likelihood Estimation (MLE) method is used toobtain the statistical parameters from the samples inthis paper. Let fX (x; θ) be the density function of vari-able X , where, for simplicity, θ is the only parameter tobe estimated from a set of sample valuesx1, x2, . . ., xn.The likelihood function of θ is defined as,

L(θ) =n∏

i=1

fX (xi; θ) (2)

When the sample values are given, the likelihood func-tion L becomes a function of a single variable θ . Theestimation procedure for θ based on the method ofmaximum likelihood consists of choosing, as an esti-mate of θ , the particular value of θ that maximizesL. The maximum of L(θ) occurs in most cases at thevalue of θ , where dL/dθ is zero. Hence, the maxi-mum likelihood estimate θ̂ of θ based on sample valuesx1, x2, . . ., xn can be determined from,

dL(x1, x2, . . ., xn; θ̂ )

d θ̂= 0 (3)

L is always nonnegative and attains its maximum forthe same value of θ̂ as ln L. Since ln L is in the formof a sum rather than a product, it is generally easier toobtain θ̂ by solving

d ln L(x1, x2, . . ., xn; θ̂ )

d θ̂(4)

The single parameter estimation procedure can bedirectly extended to multi-parameter estimation. In thecase of m parameters, the likelihood function becomes

L(θ) =n∏

i=1

fX (xi; θ1, θ2, . . ., θm) (5)

and the MLEs of θj , j = 1, 2, . . ., m, are obtainedby simultaneously solving the system of likelihoodequations

∂lnL

∂θ̂j

= 0, j = 1, 2, . . ., m (6)

In this way the greatest probability is given to theobserved set of events, provided that the true formof probability density distribution is known.

A 180-day vehicle weight data are used as samplesto estimate the statistical parameters in Eq. (1). Theestimated values of the statistical parameters in Eq.(1) are 0.543, 0.457, 2.542, 0.342, 4.901 and 1.024 forp1, p2, μY 1, σY 1, μY 2 and σY 2. The PDF and CDF of Yare shown in Figure 4. Based on the estimated results,the CDF of X can be written as,

FX (x) = P(X ≤ x) = P(Y ≤ ln x) = FY (ln x)

= p1�

(ln x − μY 1

σY 1

)+ p2�

(ln x − μY 2

σY 2

)(7)

It is clear that the CDF of X are the mixturesof two LND (called Bimodal-Lognormal Distribu-tion, BLD) with different parameters. For the grossweight of light cars, it follows the LND (LN (13.472,4.7462)) with a probability of occurrence p1 = 0.543,whereas, for the gross weight of heavy trucks, itfollows LND (LN (227.125, 309.3962)) with a prob-ability of occurrence p2 = 0.457. The CDF of Xobtained from Eq. (7) are shown in Figure 5. It isobserved that the curve of CDF of X shown in Figure 5can better trace the measured one than that shownin Figure 3. The CDF of BLD with the estimatedparameters is accepted by KS-test at the significancelevel of 1%. Therefore, the Bimodal-Lognormal Dis-tribution can be used to model the CDF of the grossvehicle weight.

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CD

F F

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Figure 5. Cumulative distribution function of BLD of thegross vehicle weight.

3.2 Extreme value distribution of traffic load

Since the load duration of a vehicle on a bridge is veryshort, the gross vehicle load stochastic process on abridge can be described approximately by a filteredPoisson process (Lin, 1990). Hence, a filtered Pois-son process is employed to simulate the gross vehicleload stochastic process {s(t), t ∈ [0, T ]} in this paper.The diagrammatic sketch of filtered Poisson processis shown in Figure 6.

The filtered Poisson process which is used tosimulate gross vehicle load stochastic process is

s(t) =N (t)∑n=0

ξn · I (t, τn) (8)

where{N (t), t ∈ [0, T ]} is a Poisson process withparameterλ, and ξn(n = 1, 2, . . . ) are variables fol-lowing F(x), independent of each other and ξ0 = 0.The responding function is expressed by

I (t, τn) ={

1, t ∈ τn;0, t /∈ τn, (9)

where τn is the load duration of the nth vehicle andτ0 = 0.

The extreme value probability distribution of afiltered Poisson processis expressed by

FM (x) ={

exp [−λT (1 − F(x))] , x ≥ 0;0, x < 0, (10)

where F(x) is the CDF of gross vehicle weight men-tioned above; λ is the parameter of Poisson processand can be calculated by the MLE approach based onthe survey data; and T is the period of requirement.

Figure 6. Diagrammatic sketch of filtered Poisson process.

For the purpose of obtaining the extreme value prob-ability distribution of gross vehicle weight, the firstpeak associated with light vehicles is not of interest.For this reason, the splitting of Poisson process is dis-cussed next. Let {N (t), t ≥ 0} be a Poisson processwith rate λ′. Suppose that each arrival of the processis classified as being either type 1 arrivals or type 2arrivals with respectively independent probabilities p′

1and p′

2 of all other arrivals. Let Ni(t) be the number oftype i (i = 1, 2) arrivals up to time t. Then {N1(t)} and{N2(t)} are two independent Poisson processes withrespective rates λp′

1 and λp′2 (Tijms, 2003). Therefore,

stochastic process of the gross vehicle weight can besplit into two independent Poisson processes with rateλp1 and λp2, the former for modeling light vehicles,while the latter for heavy trucks. The extreme valuedistribution of heavy truck in one day is expressed as,

FM (x) = exp {−λp2T1 [1 − FZ (x)]} , x ≥ 0 (11)

where FM (x) is the CDF of extreme value in one day,the time intervals can be approximately described asthe Exponential Distribution, FZ (x) is the CDF ofheavy trucks followed LND (LN (227.125, 309.3962)),obviously, T1 = 86400 s. For the Shandong BinzhouYellow River Highway Bridge, the estimation valueofλobtained using measurement data is 0.02385Based on the Unified Standard for Reliability Designof Highway Engineering Structures (China, GB/T50283-1999), the extreme value distribution of grossvehicle weight in one day is used to represent theirextreme value distribution in one year, and then theextreme value distribution of vehicles in service-lifecan be written as

FT (x) = exp {−λp2T2 [1 − FM (x)]} , x ≥ 0 (12)

where FT (x) is extreme value distribution of grossvehicle weight in service-life; T2 is service-life orremaining service life with units of year.

Solving Eq. (12) directly is more difficult due to thecomplicated function FM (x)in this equation. There-fore, the Monte Carlo simulation method (Kottegoda,1998), which is one of the most commonly effectiveapproaches to simulate complicated random variables

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and stochastic processes, is adopted to solve Eq. (12).It has been proven that the CDF is always uniformlydistributed on [0, 1] (Eckhardt et al. 1987). Since therandom variable x and the CDF F(x) are 1-to-1, onecan sample x by first sampling y = F(x) and thensolving for x by inverting F(x), or x = F−1(y). There-fore, a random numbers sample ξ from U[0, 1] aregenerated and the value of x is determined by inver-sion, x = F−1(ξ). This method sometimes calledthe ‘‘Golden Rule for Sampling’’. According to thismethod, many counterfeit functions could be producedby means of the Monte Carlo approach. Supposed thatthe service-life is 100 years, 100 counterfeit randomnumbers of extreme value distribution are obtainedby Monte Carlo approach and checked by KS-testwhether the Normal, the Lognormal, the Weibull, theGamma, the Inverse Gauss, and the Gumbel Distribu-tion (Extreme-Value Type-I Distribution) can describethe extreme value distribution of the gross vehicleweight or not. Based on the simulated results, only theGumbel distribution can be accepted by KS-test at thesignificance level of 1%.. Therefore, the extreme valuedistribution of the gross vehicle weight is as follows

FT (x) = exp(− exp(−(x − b)/a)), x ≥ 0 (13)

where a = 38.79and b = 1676.51 which are obtainedby MLE and the units of gross vehicle weight is kN.

4 CONCLUSION

Traffic load models of the Shandong Binzhou YellowRiver Highway Bridge are established in this casestudy based on the field measurement by a struc-tural health monitoring system installed on this bridge.The following conclusions are obtained from this casestudy:

i. The probability distribution of the gross vehicleweight follows the Bimodal-Lognormal Distribu-tion, one peak for light cars and another for heavytrucks. In this case study, for the gross weightof light cars, it follows the LND (LN (13.472,4.7462)) with a probability of occurrence p1 =0.543; whereas, for the gross weight of heavytrucks, it follows LND (LN (227.125, 309.3962))with a probability of occurrence p2 = 0.457.

ii. The gross vehicle load stochastic process on abridge can be described approximately by a fil-tered Poisson process. Further study indicates thatthe extreme value distribution of the gross vehicleweight follows the Gumbel Distribution.

ACKNOWLEDGEMENT

This study is financially supported by the NationalNatural Science Foundation of China under the fol-lowing grants: 50525823, 50538020 and 50278029.

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