Research ArticleStudy Based on Bridge Health Monitoring System onMultihazard Load Combinations of Earthquake and Truck Loadsfor Bridge Design in the Southeast Coastal Areas of China

Dezhang Sun1 Bin Chen2 and Baitao Sun1

1 Institute of Engineering Mechanics China Earthquake Administration Harbin 150080 China2 College of Civil Engineering and Architecture Zhejiang University Hangzhou 310058 China

Correspondence should be addressed to Bin Chen jeetchen 123hotmailcom

Received 8 August 2014 Accepted 1 October 2014

Academic Editor Bo Chen

Copyright copy 2015 Dezhang Sun et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Similar to American LRFD Bridge Design Specifications the current Chinese bridge design code is fully calibrated against gravityload and live load Earthquake load is generally considered alone and has its own methodology however which is not coveredin the code in a consistent probability-based fashion Earthquake load and truck load are the main loads considered in the basisof bridge design in more than 70 of seismic areas in China They are random processes and their combination is the mainsubject of this paper Seismic characteristics of southeast coastal areas of China are discussed and an earthquake probability curve iscalculated through seismic risk analysis Using measured truck load data from a Bridge HealthMonitoring System the multimodalcharacteristics of truck load are analyzed and a probability model for a time interval t is obtained by fitting results and reliabilitytheory Then a methodology is presented to combine earthquake load and truck load on a probabilistic basis To illustrate thismethod truck load and earthquake load combinations are used Results conceptually illustrate that truck load and earthquakeload are not dominant in southeast coastal areas of China but the effect of their combination is This methodology quantitativelydemonstrates that the design is controlled by truck load in most ranges that is truck load is more important to bridge design inthe region

1 Introduction

In the current bridge design specifications of China [1] atypical bridge is designed for 100 yearsrsquo service life and thedesign limit states are only fully calibrated for dead and liveloads Consideration of earthquake load has its own uniqueapproach principally because data and statistics are rareTruck load data are old andmay not suit the current situationand they therefore need to be updated This fact makes itdifficult to properly consider both truck load and earthquakeload in a consistent fashion A research project is currentlybeing carried out to establish amethodology to systematicallycombine truck load and earthquake load Principle emphasisis given to establishing the proper ldquodemandrdquo In order topursue the demand side of bridge design specifications thereare numerical challenges that must be overcome in additionto the fact that very limited historical data are available Truck

load and earthquake load are time-variant random processesTruck load occurs once in a typical time span of minutes orseconds on a common bridge while earthquake load occursonce in a typical time span of years or decades Furthermorethere is no evidence that the occurrence of earthquake loadsfollows normal distributions Fortunatelymany bridge healthmonitoring systems (BHMS) have been established whichprovide a convenient way to get useful data for this research[2]

To overcome the challenges many efforts [3ndash9] have beenmade in the past decades However because numbers ofassumptions have to be made in each model no generalconclusions can be drawn about satisfactory approach to dealwith load combination of earthquake load and truck load Inmore recent papers a methodology is proposed by Liang andLee [10 11] however its accuracy is yet to be substantiated

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 829380 12 pageshttpdxdoiorg1011552015829380

2 Shock and Vibration

One objective of this paper is to describe a methodologyto handle truck load and earthquake load combinationsEarthquake load ismodeled using seismic risk analysis Truckload is modeled using Stationary Poisson processes based onthe BHMS and statistical analysis Two numerical examplesof truck load and earthquake load combinations are used toillustrate the methodology

2 Earthquake Load

A number of variables describe the effects that earthquakeshave on bridges such as the intensity of acceleration therate of earthquake occurrences the natural period of thebridge the seismic response coefficient and the responsemodification factor In order to explain the methodology ofload combinations only the intensity of acceleration and therate of occurrence are chosen as the main variables

Based on the Poisson process assumption the probabilityof exceedance (119875

119890) in a given exposure time (119879

119890) is related to

the annual probability of exceedance (120582) by [12]

119875119890= 1 minus 119890

minus120582sdot119879119890 (1)

Because the number of earthquakes varies widely from site tosite they are converted to Peak ground acceleration (PGA)and the return period curve (119879

119877= 1120582 119879

119877is return period)

The cumulative probability of an earthquake in time 119879 can bewritten as

119875 = 119890minus119879119879119877 (2)

The PGA and frequency of exceedance curve can be obtainedfrom US Geological Survey (USGS) mapping in the UnitedStates but cannot be obtained in China Therefore seismicrisk analysis is used to calculate earthquake probability curveThe procedures are presented just as follows

Formore than one potential seismic source zone supposethe parameters of the earthquake are random distributionsand the probability over 1 year is a stable Poisson processBased on the total probability theorem the probability ofexceeding a given earthquake intensity 119878

0in one site can

be expressed by (3) by considering the uncertainties ofoccurrence and the upper limit magnitude

119875119905(119878 gt 119878

0) = 1 minus

119870

prod

119896=1

119868

sum

119894=1

119882(119894)

119903119896exp[

[

minus

119869

sum

119895=1

120592(119894)

119896119895sdot 119875(119895)

119896119882(119895)

119906119896sdot 119905]

]

(3)

where 119875(119894)119896

is the probability of the 119895th upper limit magnitudeexceeding a given earthquake intensity 119878

0in potential seismic

source 119896 ](119894)119896119895

is the 119894th year occurrence probability of the 119895thupper limit magnitude in potential seismic source 119896119882(119894)

119903119896is

the weight of the 119894th year occurrence probability in potentialseismic source 119896 and119882(119895)

119906119896is the weight of the 119895th upper limit

magnitude in potential seismic source 119896The earthquake intensity could be acceleration velocity

or displacement For acceleration 119878 = ln 119886 (119878 is earthquakeintensity 119886 is acceleration)

Because of the uncertainties of direction impact of poten-tial seismic source zones for 119905 = 1 year the probability ofexceedance is

1198751(119878 gt 119878

0) =

119870

sum

119896=1

119868

sum

119894=1

119869

sum

119895=1

120592(119894)

119896119875(119895)

119896119882(119894)

119903119896119882(119895)

119906119896 (4)

where119875(119895)119896

is the conditional probability of the 119895th upper limitmagnitude

For disperse potential seismic source areas the probabil-ity of the 119895th upper limit magnitude can be expressed as

119875(119895)

119896(119878 gt 119878

0|119872119906119896

) =

sum119870

119896=1119875(119895)

120585(119878 gt 119878

0|119877) Δ119860120585

119860119896

(5)

where 119860119896is the area of the potential seismic source 119896 Δ119860

120585

is the area of zone x If the occurrence probability of 1 yearis divided by the weights in each upper limit magnitude ofpotential seismic source zone the exceedance probability ofthe 119895th upper limit magnitude of potential seismic source 119896can be given as

119875(119895)

120585=

sum119899120585119899119870119903[Φ10158401015840

119899+1minus Φ10158401015840

119899] + [119890

minus120573119872(119895)

119899 Φ1015840

119899minus 119890minus120573119872(119895)

119899+1Φ1015840

119899+1]

sum 120585119898[119890minus120573119872

(119895)

119899 minus 119890minus120573119872(119895)

119899+1]

+ Φ0

(6)

According to the seismic belt materials and reports thesoutheast coastal area of China has two I degree seismicareas namely the South China seismic area and the SouthChina Sea seismic area The seismic belt of southeast coastalareas of China is located south of the middle Yangtze Riverseismic belt bordering on the seismic region of the TibetanPlateau on the west and includes Kwangtung provinceHainan province most of Fujian and Guangxi provincesand part of Yunnan Guizhou and Jiangxi provinces Crustalthickness ranges between 28 and 40 km gradually increasingfrom the southeast coastal area of China to the northwestmountains An internal secondary elliptical gravity anomalyis relatively developed in the earthquake zones There areno obvious banded anomalies except the gravity gradientzones of southeast coastal areas and Wuling Mountain Inthe zones magnetic anomalies change gently and thereare no larger banded anomalies Because the southeastcoastal areas of China are in the same seismic belt andmost of the areas in the zone have a PGA seismic forti-fication level of 01 g Shenzhen city is then used for thebasic earthquake probability calculation and comparisonin this paper The South China belt is shown in Figure 1From Figure 1 it can be seen that there are many higherthan Ms 60 earthquakes in the southeast coastal areas ofChina present in the seismic analysis Earthquake load isstill the main load considered for bridge design in theseareas

Based on (3) to (6) the annual exceedance probability ofShenzhen is shown in Figure 2

Shock and Vibration 3

IIIminus2

Taiwan

Hainan

Yunnan

Guizhou

Hunan Jiangxi

Fujian

Kwangtung

Guangxi Taiwan

The South China Sea

105deg

ZunyiBijie

Liupanshui AnshunGuiyang

DuxunKaili

QujingXingyi

Wenshan

Baise

Henei

Liangshan

Nanning

Qinzhou

Fanggang

Sanya

Haikou

Zhanjing

MaomingYangjing

Yulin

TongrenHuaihua

JishouLoudi

ShaoyangHengyang

Yongzhou

Chenzhou

ZhuzhouXiangtan

ChangshaYiyang

YichunPingxiang

Xinyu LinchuanYingtan

Ningde

Fuzhou

Putian

Sanming

Nanping

Wenzhou

QuanzhouLongyanZhangzhou

XiamenMeizhouChaozhou

ShantouJieyangHuizhou

ShenzhenShanweiDongguan

Guangzhou

JiangmenZhongshan

Zhaoqing

QingyuanWuzhou

Shaoguan

Ganzhou

Jian

GuilinHechi

Liuzhou

Beihai Shangxiachuandao

Gaoxiong

Tainan

Taizhong

Zhejiang

Heyuan

The

Strait

110∘

115∘

120∘

25∘

20∘

1

2

3

4

5

6

7

8

9

13

14

15

16

17

IIIII

III

IV

IV

V

V

Iminus1Iminus2

Iminus2

IIIminus1

IIIminus2

IIIminus1

Iminus1

I

I

I

10

11

12

Figure 1The South China belt (1)The seismic area of South China (2)The seismic area of South China Sea (3)The seismic area of Taiwan(4)The earthquake subregion of middle of Qinghai-Tibet Plateau (5)The earthquake subregion of south of Qinghai-Tibet Plateau (6)Thesoutheast costal seismic belt (South China coastal seismic belt) (7)The seismic zone in the middle reach of Yangtze River (8) Seismic area ofthe west of Taiwan (9) Seismic area of the east of Taiwan (10) Boundaries of seismic zone (11) Boundaries of seismic belt (12) South Chinacoastal seismic rupture (13) Epicenter of Ms 7 and above (14) Epicenter of Ms 60ndash69 (15) Epicenter of Ms 50ndash59 (16) Epicenter of Ms47ndash49 (17) Calculation site

Assume that the probability density of earthquake loadintensity in time 119879 follows a distribution defined as 119891

119879(119910)

where 119910 is a variable of PGA intensity Based on the Poissonprocess assumptions the cumulative probability function119865119905(119910) over interval 119905 can be obtained using the following

119865119905(119910) = [119865

119879(119910)]119905119879

(7)

The probability density function can then be derived as

119891119905(119910) =

120597119865119905(119910)

120597119910=119905

119879sdot [119865119879(119910)](119905119879minus1)

sdot 119891119879(119910) (8)

Note that 119905 and 119879 should have the same dimension

4 Shock and Vibration

Intensity (PGA)

Prob

abili

ty

100

100

10minus2

10minus4

10minus6

10minus8

10minus12

10minus10

10minus14

10minus16

10minus18

10minus20

10minus3

10minus2

10minus1

Figure 2 Annual exceedance probability of PGA

3 Truck Load

Studies on truck load have been difficult historically princi-pally because weighing equipment was lacking and the dataare correspondingly rare [13 14] Fortunately the installationof BHMS is required on newly built long-span bridgesincluding the weighing-in-motion (WIM) system [15ndash17]Time gross weight axle weight wheel base velocity and soforth are measured and collected The probability model oftruck load can be obtained through statistical analysis

Nowak [18] indicated that at a specific site heavy trucksmay have an average number of 1000 which is also discussedby GhosnMoses [19] suggested heavy trucks follow a normaldistribution with a mean of 300 kN and a standard deviationof 80 kN (coefficient of variable COV = 265) Zhao andTabatabai [20] discussed the local standard vehicle modelusing data from about six million vehicles in Washingtonwhich can be used as a reference for a truck load model Inthis paper the truck load model is obtained through datamining from measured WIM data from three bridge sitesin Hangzhou Xiamen and Shenzhen (Figure 3) Truck loaddata of 5 axles and more in the three sites are filtered andselected Through WIM data analysis truck load probabilitycharacteristics of Hangzhou Xiamen and Shenzhen aresimilar even the shape of truck load probability curvesConsidering the three sites have very similar traffic flowalmost equal to 10 truck load probability curve of ShenzhenCity is used for analysis and validity of subsequent casestudies Truck load probability curve is shown in Figures 4and 5The fitted curve is obtained using normal distributionwhose mean value is 2949 kN and the coefficient of varianceis 374

For a typical bridge the truck loadwill consist of a varyingnumber of trucks on the bridge The probability functionfor such a bridge can be obtained using following analysisAssume 119860 is a set consisting of the elements 119860

1 1198602 119860

119898

which present 119898 events and the probability of 119860 is 119875(119860)while 119861 is a set consisting of the elements 119861

1 1198612 119861

119899

which present 119899 events and the probability of 119861 is 119875(119861) sothe probability 119875(119860 + 119861) presents the probability of intensity(119860 + 119861) Then 119875(119860 + 119861) can be calculated by

119875119860+119861

(119896) = sum

119894

119875119860(119894) sdot 119875119861(119896 + 1 minus 119894) (9)

Note that the length of 119875(119860) is119898 and the length of 119875(119861) is 119899The sum is over all the values of 119894which lead to legal subscriptsfor 119860(119894) and 119861(119896 + 1 minus 119894) where 119896 is the 119896th (119860 + 119861)(119896)119894 = max(1 119896 + 1 minus 119899) min(119896119898) Equation (9) reflects theprobability of combining two sets and when it comes to aseries of setsΦ = Φ

1+ Φ2+ sdot sdot sdot + Φ

119873 (9) can be extended to

119873 dimensions

119875Φ(1206011+ 1206012+ sdot sdot sdot + 120601

119873)

= sum(sdot sdot sdot (sum(sum119875Φ1

(1206011) sdot 119875Φ2

(1206012)) sdot 119875Φ3

(1206013)) sdot sdot sdot )

sdot 119875Φ119873

(120601119873)

(10)

where Φ119894in (10) is the 119894th set of event and 119875

Φ119894

(120601119894) is the

probability of set Φ119894

Based on total probability theory and Poisson processesthe truck load intensity function for an interval 119905 can becalculated using the following

119865Φ(120601)119905= 119875Φ0

+sum

119873

119875Φ1+Φ2+sdotsdotsdotΦ119873

sdot 119901 (119873) (11)

where 119875Φ0

is the probability with no truck passing on thebridge119873 = 1 2 maximumnumber of trucks119875

Φ1+Φ2+sdotsdotsdotΦ119873

is the probability of varying number of trucks passing thebridge 119901(119873) is the probability of occurrence of119873 trucks onthe bridge

4 Model of Combination

The intensity of dead load is usually defined as a time inde-pendent variable and that of truck load is a time dependentvariable both of whom follow normal distributions [5 818] In this paper a normal distribution is used for deadload which is considered to maintain more or less the samemagnitude such that it can be treated as a random timeindependent variable

As mentioned above earthquake load and time-variabletruck load are assumed to be Poisson processes each withsame distribution and time duration Based on these assump-tions as mentioned earlier load processes can be convertedto a small 119905 interval in which loads can be combinedThe objective of time-variable combination is to find themaximum value of different random variables namely thecombined value 119883 = 119883

1+ 1198832+ sdot sdot sdot + 119883

119899 with an interval

119905 and maximum value 119883max119879 in a time 119879 119899 is the numberof loads 119883 is the intensity of combined load and 119883

119894is the

load intensity of the 119894th loadThemaximum value of119883 in thelifetime of the bridge can be expressed as

119883max119879 = max119879

[1198831+ 1198832+ sdot sdot sdot + 119883

119899]119905 (12)

To simplify the discussion the number of loads is taken astwo the problem being reduced to the prediction of 119883

12=

1198831+ 1198832 where 119883

12is the intensity of the union of 119883

1and

1198832 Assume that 119891

1(119909) and 119891

2(119909) are the probability density

functions of 1198831and 119883

2in the interval 119905 and 119865

1(119909) and 119865

2(119909)

are the cumulative probability functions of 1198831and 119883

2in the

Shock and Vibration 5

HubeiAnhui Hangzhou Bay

Bridge

Zhejiang

Hunan

Jiangxi

Fujian

GuangdongGuangxi

Jimei Bridge

Shenzhen Bay

Bridge

Main city

Xiangyang Xinyang

Zhumadian

LiuanHefei

Nanjing

SuzhouShanghai

Hangzhou

Ningbo

Taizhou

Lishui

Wenzhou

Fuzhou

Nanping

Sanming

Putian

QuanzhouLongyan

Zhangzhou

JieyangShantou

HuizhouGuangzhou

Hong Kong

Qingyuan

Shaoguan

Ganzhou

Jian

Xinyu

YingtanShangrao

QuzhouJingdezhen

Jiujiang

Nanchang

Xianning

Wuhan

Jingzhou

Jingmen

Yichang

Zhangjiajie

Yueyang

Changde

Changsha

XiangtanLoudi

Shaoyang Hengyang

Yongzhou

Chenzhou

Guilin

Wuzhou

Yangjiang

Maoming

Yulin

HighwayRail way State road

Taiwan

Figure 3 Layout of monitoring sites

interval 119905 11989111988312

(119909) and 11986511988312

(119909) are the probability densityfunction and cumulative probability function of 119883

12 The

maximum value of combination in the entire bridge servicelife is defined as119883

12max119879 which is defined as

11988312max119879 = max

119879

[1198831+ 1198832]119905 (13)

The probability density of the combined loads 119883 can beobtained using the convolution integral

11989111988312

(119909) = 1198911lowast 1198912= int

+infin

minusinfin

1198911(120591) sdot 119891

2(119909 minus 120591) 119889120591 (14)

where ldquolowastrdquo is used as convolution symbol

6 Shock and Vibration

0 500 1000 1500

0

05

1

15

2

25

3

35

4

Intensity (kN)

Prob

abili

ty d

ensit

y

times10minus3

Figure 4 Histogram of the monitoring sites

0 200 400 600 800 1000 1200 1400

Density fittedActual density

0

05

1

15

2

25

3

35

4

Prob

abili

ty d

ensit

y

times10minus3

Intensity (kN)

Figure 5 Curve fitting of truck load density

Dirac Delta function is introduced to deal with thecharacteristics of119883 and 119884 in small 119905 interval

120575 (119909) = +infin 119909 = 0

0 119909 = 0(15)

Therefore the probability density can be illustrated as

119891119894(119909) = 119875

1198831198940

120575 (119909) + 1198911015840

119894(119909) 119894 = 1 2 (16)

Note that 1198751198831198940

is the probabilities of 119883119894(119894 = 1 2) in its ldquozero

pointsrdquo namely the events do not happen (eg themaximumtrucks on the bridge are 8 119875

1198831198940

is the probability of no trucks

on the bridge)1198911015840119894(119909) is its probability density functions with-

out ldquozero pointsrdquo Then the cumulative probability functionsof1198831and119883

2can be calculated through

119865119894(119909) = 119875

1198831198940

int

119909

minusinfin

120575 (120591) 119889120591 + int

119909

minusinfin

1198911015840

119894(119909) 119889120591

= 1198751198831198940

+ int

119909

minusinfin

1198911015840

119894(119909) 119889120591

(17)

where 119894 = 1 2Based on (16) and (17) the probability density of 119883 then

can be grouped as

11989111988312

(119909)

= int

+infin

minusinfin

[11987511988310

120575 (120591) + 1198911015840

1(120591)]

times [11987511988320

120575 (119909 minus 120591) + 1198911015840

2(119909 minus 120591)] 119889120591

= int

+infin

minusinfin

[11987511988310

11987511988320

120575 (120591) 120575 (119909 minus 120591)]

+ [11987511988310

120575 (119909) 1198911015840

2(119909 minus 120591)] + [119875

11988320

120575 (119909 minus 120591) 1198911015840

1(120591)]

+ [1198911015840

1(120591) 1198911015840

2(119909 minus 120591)] 119889120591

= 11987511988310

11987511988320

120575 (119909) + 11987511988310

1198911015840

2(119909) + 119875

11988320

1198911015840

1(119909)

+ int

+infin

minusinfin

1198911015840

1(120591) 1198911015840

2(119909 minus 120591) 119889120591

(18)

where based on the characteristic of Dirac Delta functionwhich isint+infin

minusinfin

120575(119909)119889119909 = 1 (18) can be converted to cumulativeprobability function Assume the cumulative probabilityfunction of 119865

11988312

(119909) then

11986511988312

(119909) = int

119909

minusinfin

11989111988312

(119909) 119889119909

= 11987511988310

11987511988320

+ 11987511988310

1198651015840

2(119909) + 119875

11988320

1198651015840

1(119909)

+ int

+infin

minusinfin

1198651015840

1(120591) 1198911015840

2(119909 minus 120591) 119889120591

(19)

From (19) it is clear that the combined load probabilityconsists of four parts events 119883

1and 119883

2are not happening

event 1198831is happening while 119883

2is not happening event 119883

1

is not happening while1198832is happening and119883

11198832are both

happeningTo further simplify the discussion without losing gener-

ality the probability of two loads occurring simultaneously isneglected Thus (19) is simplifying to

11986511988312

(119909) = int

119909

minusinfin

11989111988312

(119909) 119889119909

asymp 11987511988310

11987511988320

+ 11987511988310

1198651015840

2(119909) + 119875

11988320

1198651015840

1(119909)

(20)

Shock and Vibration 7

Then the cumulative probability function 11986511988312max119879

(119909) ofmaximum value of load combinations 119883

12max119879(119909) in time119879 can be obtained

11986511988312max119879

(119909) = [11986511988312

(119909)119905]119879119905

(21)

When the number of loads is more than two and theseloads apply to bridge directly while satisfying Poissonprocess (14) can be extended to 119899 dimensions Assume1198911 1198912 119891

119899are the probability densities functions of

1198831 1198832 119883

119899 1198651 1198652 119865

119899are the cumulative probability

functions of1198831 1198832 119883

119899 11989110158401 1198911015840

2 119891

1015840

119899are the probability

density functions of 1198831 1198832 119883

119899without ldquozerordquo points

119891119883(119909) and 119865

119883(119909) are the probability density function and

cumulative probability functionThen the probability densityfunction is deduced as

119891119883(119909) = 119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) (22)

where the cumulative probability 119865119883max(119909) similar to that

given by (19) is

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

= int

119909

minusinfin

1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) 119889119909

(23)

Equation (23) can account for load combinations of allloads which satisfy the first three assumptions If more thantwo events occurring simultaneously can be neglected (23)reduces to

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

asymp 11987511988310

11987511988320

sdot sdot sdot 1198751198831198990

+ 1198651015840

111987511988320

sdot sdot sdot 1198751198831198990

+ sdot sdot sdot + 11987511988310

11987511988320

sdot sdot sdot 119875119883119899minus10

1198651015840

119899

(24)

Then the maximum value of119883 in time 119879 can be obtained by

119865119883max119879

(119909) = [119865119883(119909)119905]119879119905

(25)

where 119865119883(119909)119905is 119865119883(119909) in 119905 interval 119865

119883max119879(119909) is the cumula-

tive probability function of maximum value of119883 in time 119879Although in our study emphasis is given to formulate

the ldquodemandrdquo to establish load combinations all events mustaddress a capacity issue of the bridge For example theearthquake load and truck load combination on a bridgecolumn can either consider the vertical load or the columnbase shear load Theoretically (23) can deal with most loadcombinations but as more loads are considered a more con-servative design will be adopted Based on the methodologyand assumptions described above the maximum load can becombined and the procedures are summarized as follows

(1) determine truck load and earthquake load distribu-tions over a particular period

(2) using Poisson processes convert earthquake load andtruck load distributions over a particular period to asufficiently small interval 119905

203 cm

60m

19m

442m442m442m

Figure 6 Longitudinal profile of the typical bridge

203 cm

190m

50m

152m

215m143m

215m

600m

Figure 7 Transverse profile of the typical bridge

(3) using (23) earthquake and truck load combinationsover an interval 119905 can be obtained

(4) using (25) the load combinations over an interval 119905can be converted to the bridge service life interval 119879

5 Numerical Examples

Example 1 Using the method of load combination describedin the preceding section a simple example of horizontal loadcombination is presented here Profiles of the typical bridgeare shown in Figures 6 and 7Theweight of the superstructureat each column is 538 tons the eccentricity of truck load is 50meters and the effects of soil and secondary effects of gravityare ignored Furthermore it is assumed that the maximumnumber of trucks on one lane is two The results are given inFigures 8 9 10 11 12 13 and 14 The interval 119905 is 10 secondsand the average daily truck traffic (ADTT) is about 1947

8 Shock and Vibration

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

ytimes10

minus4

Intensity (kN-M)

(a)

0 2000 4000 6000 8000 10000

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty

Intensity (kN-M)

(b)

Figure 8 Probability curve of each truck load effect

1 truck passing2 trucks passing

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M)

Figure 9 Truck load probability density curve for varied number oftrucks

Truck and earthquake load effects are the base momentcaused by trucks and earthquakes respectively Figure 8shows the probability curves of each truck load effectwhich has a similar shape to truck load Figure 9 shows thetruck load probability density curve for varied numbers oftrucks From Figure 10 we can see that over the interval 119905the probability of no truck on the bridge is much largerthan the other number of trucks passing Figure 11 showsthe combined probability curves for truck load over anearthquake load duration which indicate that the truck loadover an earthquake load duration is larger than each truckload

0 1 2

0

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 10 Probability of passing truck number simultaneously in 119905interval on the bridge

From the results shown in Figure 13 it can be seen thatcurve C and curve A have a similar shape and curve C isdisplaced to the right This means that mode and mean valueof the truck and earthquake load combination in this exampleis larger than those of truck load and also illustrates that truckload is more important to bridge design in this area FromFigure 14 we can seemore clearly that overmost ranges truckload is larger than earthquake load Because the truck loadhas a smaller tail their combination in the tail is close to theearthquake load Curve D of loads combined directly over100 years is further away from curve C which means thecombinations of truck load and earthquake load directly over100 years give a much larger value and it is not suggested thatthe bridge design uses the curve D method

Shock and Vibration 9

0

05

1

15

2

25

3

35

Prob

abili

ty d

ensit

y

times10minus4

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(a)

05

055

06

065

07

075

08

085

09

095

1

Cum

ulat

ive p

roba

bilit

y

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(b)

Figure 11 Combined probability curves for truck load in earthquake load duration

05 1 15 2 25

0

1

2

3

4

5

6

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M) times104

(a)

2000 4000 6000 8000 10000 12000 14000

0

01

02

03

04

05

06

07

08

09

1

Cum

ulat

ive p

roba

bilit

y

Intensity (kN-M)

(b)

Figure 12 Probability curves for earthquake load in 100 years

Example 2 Example 2 illustrates vertical load combinationsMost of the configurations are the same as used in Example 1The difference is in the maximum number of trucks namelythe maximum number of trucks on one lane in this exampleis four The results are shown in Figures 15 16 17 18 19 and20 Figures 13 and 14 are the results of truck load Note that inthis example the 119905 interval is 10 seconds and 119879 is taken as 100years

Figures 15 to 16 show vertical truck load probabilitycurves for varied numbers of trucks and probabilities ofpassing truck numbers on the bridge over the interval 119905From Figure 16 it can be seen that the probability is verylow when the maximum number of trucks on one lane isfour Figures 17 and 18 show similar curve shapes with those

in Figures 13 and 14 which indicate that there is the samerule in load combinations in both the horizontal and verticaldirections Comparing Figures 17 to 20 though themaximumnumber of trucks is four because dead load is combinedwith truck and earthquake load in the gravity direction thedead load contributes a substantial portion in vertical loadcombinations

6 Conclusions

This paper describes a method to combine earthquake loadand truck load in the service life of bridges The followingconclusions can be drawn

(1) Given the more than 70 seismic areas in Chinaearthquake load is a main consideration for bridge

10 Shock and Vibration

16000

05

1

15

2

2000 4000 6000 8000 10000 12000 14000

Prob

abili

ty d

ensit

y

times10minus3

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN-M)

Figure 13 Probability density of load combination in 100 years

05 1 15 2

05

055

06

065

07

075

08

085

09

095

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Cum

ulat

ive p

roba

bilit

y

times104Intensity (kN-M)

Figure 14 Cumulative probability of load combination in 100 years

design in the southeast coastal areas of China Theearthquake load probability curve is obtained usingseismic risk analysis

(2) Using measured truck load data from BHMS multi-modal characteristics of truck load are analyzed Thetruck load density of each truck is obtained by curvefitting Considering that truck load may consist ofvarying numbers of trucks truck load is calculatedthrough traffic analysis

(3) In thismethod themaximumvalue of combined loadis defined as 119883max119879 = max

119879[1198831+ 1198832+ sdot sdot sdot 119883

119899]119905

which means (1198831+ 1198832+ sdot sdot sdot 119883

119899) over the interval 119905

is first combined and then the maximum value in the

0 2000 4000 6000 8000 100000

05

1

15

2

25

3

35

4

1 truck passing2 trucks passing

3 trucks passing4 trucks passing

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 15 Vertical truck load probability density curves for variednumber of trucks

0 1 2 3 40

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 16 Probability of passing truck number simultaneously in 119905interval on the bridge with maximum number 4

bridge service life is determined 119883max119879 is based onprobability which covers all the probability combi-nations of the combined situations In this method aDiracDelta function is introduced to deal with119883 overa small interval 119905 To demonstrate the methodologyintuitively examples of load combinations in horizon-tal and vertical directions are provided

(4) The shape of the earthquake and truck load com-bination is similar to that of truck load alone butthe curve is displaced to the right which means themode and mean value of truck and earthquake loadcombination in this example is larger than that oftruck load aloneThis also illustrates that truck load is

Shock and Vibration 11

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 17 Probability density of vertical load combination in 100years

0 1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Figure 18 Cumulative probability of vertical load combination in100 years

more sensitive to bridge design and over most rangestruck load is larger than earthquake load in this area

(5) The curve from direct load combined over 100 yearsis further away from the curve obtained using themethod in the paper with the direct combinationof truck load and earthquake load over 100 yearsgiving much larger values It is not suggested that thismethod be used in bridge design considerations

(6) Because dead load is combined with truck and earth-quake load along the direction of gravity the dead

0 2000 4000 6000 8000 10000 12000 140000

1

2

3

4

5

6

7

8

Dead loadTruck and earthquakeTruck and earthquake and dead load

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 19 Probability density of dead truck and earthquake loadcombination

0 2000 4000 6000 8000 10000 12000 140000

02

04

06

08

1

12

14

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Dead loadTruck and earthquakeTruck and earthquake and dead load

Figure 20 Probability curves of dead truck and earthquake loadcombination

load contributes a substantial portion in vertical loadcombinations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Shock and Vibration

Acknowledgments

This study is jointly funded by Basic Institute ScientificResearch Fund (Grant no 2012A02) the National NaturalScience Fund of China (NSFC) (Grant no 51308510) andOpen Fund of State Key Laboratory Breeding Base of Moun-tainBridge andTunnel Engineering (Grant no CQSLBF-Y14-15) The results and conclusions presented in the paper areof the authors and do not necessarily reflect the view of thesponsors

References

[1] P R China Ministry of Communications ldquoGeneral code fordesign of highway bridges and culvertsrdquo Tech Rep JTG D60-200 China Communications Press Beijing China 2004

[2] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[3] C J Turkstra and H O Madsen ldquoLoad combinations incodified structural designrdquo Journal of Structural Engineeringvol 106 no 12 pp 2527ndash2543 1980

[4] ACI Publication SP-31 Probabilistic Design of Reinforced Con-crete Buildings American Concrete Institute Detroit MichUSA 1971

[5] B Ellingwood Development of a Probability Based Load Cri-terion for American National Standard A58-Building CodeRequirements forMinimumDesign Loads in Buildings andOtherStructures US Department of CommerceNational Bureau ofStandards 1980

[6] Y K Wen Development of Reliability-Based Design Criteria forBuildings Under Seismic Load National Center for EarthquakeEngineering Research Buffalo NY USA 1994

[7] Y-K Wen ldquoStatistical combination of extreme loadsrdquo Journalof Structural Engineering vol 103 no 5 pp 1079ndash1093 1977

[8] M Ghosn F Moses and J Wang ldquoDesign of highway bridgesfor extreme eventsrdquo NCHRP 489 Transportation ResearchBoard National Research Council Washington D C USA2003

[9] S E Hida ldquoStatistical significance of less common load combi-nationsrdquo Journal of Bridge Engineering vol 12 no 3 pp 389ndash393 2007

[10] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loadsmdashpart I comprehensive reliability and par-tial failure probabilitiesrdquo Journal of Earthquake Engineering andEngineering Vibration vol 11 no 3 pp 293ndash301 2012

[11] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loads Part II Examples for live and earthquakeload effectsrdquo Journal of Earthquake Engineering and EngineeringVibration vol 11 no 3 pp 303ndash311 2012

[12] E EMatheu D E Yule and R V Kala ldquoDetermination of stan-dard response spectra and effective peak ground accelerationsfor seismic design and evaluationrdquo Tech Rep no ERDCCHLCHETN-VI-41 2005

[13] HN Li TH YiMGu and LHuo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009

[14] B Chen Z W Chen Y Z Sun and S L Zhao ldquoConditionassessment on thermal effects of a suspension bridge basedon SHM oriented model and datardquo Mathematical Problems inEngineering vol 2013 Article ID 256816 18 pages 2013

[15] T-H Yi H-N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013

[16] Y Xia B Chen X Q Zhou and Y L Xu ldquoFieldmonitoring andnumerical analysis of Tsing Ma suspension bridge temperaturebehaviorrdquo Structural Control and HealthMonitoring vol 20 no4 pp 560ndash575 2013

[17] G Fu and J You ldquoExtrapolation for future maximum loadstatisticsrdquo Journal of Bridge Engineering vol 16 no 4 pp 527ndash535 2011

[18] A S Nowak ldquoCalibration of LRFD bridge design coderdquoNCHRPReport TransportationResearchBoard of theNationalAcademies Washington DC USA 1999

[19] F Moses ldquoCalibration of load factors for LRFD bridge evalua-tionsrdquo NCHRP Report 454 Transportation Research Board ofthe National Academies Washington DC USA 2001

[20] J Zhao and H Tabatabai ldquoEvaluation of a permit vehiclemodel using weigh-in-motion truck recordsrdquo Journal of BridgeEngineering vol 17 no 2 pp 389ndash392 2012

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2 Shock and Vibration

One objective of this paper is to describe a methodologyto handle truck load and earthquake load combinationsEarthquake load ismodeled using seismic risk analysis Truckload is modeled using Stationary Poisson processes based onthe BHMS and statistical analysis Two numerical examplesof truck load and earthquake load combinations are used toillustrate the methodology

2 Earthquake Load

A number of variables describe the effects that earthquakeshave on bridges such as the intensity of acceleration therate of earthquake occurrences the natural period of thebridge the seismic response coefficient and the responsemodification factor In order to explain the methodology ofload combinations only the intensity of acceleration and therate of occurrence are chosen as the main variables

Based on the Poisson process assumption the probabilityof exceedance (119875

119890) in a given exposure time (119879

119890) is related to

the annual probability of exceedance (120582) by [12]

119875119890= 1 minus 119890

minus120582sdot119879119890 (1)

Because the number of earthquakes varies widely from site tosite they are converted to Peak ground acceleration (PGA)and the return period curve (119879

119877= 1120582 119879

119877is return period)

The cumulative probability of an earthquake in time 119879 can bewritten as

119875 = 119890minus119879119879119877 (2)

The PGA and frequency of exceedance curve can be obtainedfrom US Geological Survey (USGS) mapping in the UnitedStates but cannot be obtained in China Therefore seismicrisk analysis is used to calculate earthquake probability curveThe procedures are presented just as follows

Formore than one potential seismic source zone supposethe parameters of the earthquake are random distributionsand the probability over 1 year is a stable Poisson processBased on the total probability theorem the probability ofexceeding a given earthquake intensity 119878

0in one site can

be expressed by (3) by considering the uncertainties ofoccurrence and the upper limit magnitude

119875119905(119878 gt 119878

0) = 1 minus

119870

prod

119896=1

119868

sum

119894=1

119882(119894)

119903119896exp[

[

minus

119869

sum

119895=1

120592(119894)

119896119895sdot 119875(119895)

119896119882(119895)

119906119896sdot 119905]

]

(3)

where 119875(119894)119896

is the probability of the 119895th upper limit magnitudeexceeding a given earthquake intensity 119878

0in potential seismic

source 119896 ](119894)119896119895

is the 119894th year occurrence probability of the 119895thupper limit magnitude in potential seismic source 119896119882(119894)

119903119896is

the weight of the 119894th year occurrence probability in potentialseismic source 119896 and119882(119895)

119906119896is the weight of the 119895th upper limit

magnitude in potential seismic source 119896The earthquake intensity could be acceleration velocity

or displacement For acceleration 119878 = ln 119886 (119878 is earthquakeintensity 119886 is acceleration)

Because of the uncertainties of direction impact of poten-tial seismic source zones for 119905 = 1 year the probability ofexceedance is

1198751(119878 gt 119878

0) =

119870

sum

119896=1

119868

sum

119894=1

119869

sum

119895=1

120592(119894)

119896119875(119895)

119896119882(119894)

119903119896119882(119895)

119906119896 (4)

where119875(119895)119896

is the conditional probability of the 119895th upper limitmagnitude

For disperse potential seismic source areas the probabil-ity of the 119895th upper limit magnitude can be expressed as

119875(119895)

119896(119878 gt 119878

0|119872119906119896

) =

sum119870

119896=1119875(119895)

120585(119878 gt 119878

0|119877) Δ119860120585

119860119896

(5)

where 119860119896is the area of the potential seismic source 119896 Δ119860

120585

is the area of zone x If the occurrence probability of 1 yearis divided by the weights in each upper limit magnitude ofpotential seismic source zone the exceedance probability ofthe 119895th upper limit magnitude of potential seismic source 119896can be given as

119875(119895)

120585=

sum119899120585119899119870119903[Φ10158401015840

119899+1minus Φ10158401015840

119899] + [119890

minus120573119872(119895)

119899 Φ1015840

119899minus 119890minus120573119872(119895)

119899+1Φ1015840

119899+1]

sum 120585119898[119890minus120573119872

(119895)

119899 minus 119890minus120573119872(119895)

119899+1]

+ Φ0

(6)

According to the seismic belt materials and reports thesoutheast coastal area of China has two I degree seismicareas namely the South China seismic area and the SouthChina Sea seismic area The seismic belt of southeast coastalareas of China is located south of the middle Yangtze Riverseismic belt bordering on the seismic region of the TibetanPlateau on the west and includes Kwangtung provinceHainan province most of Fujian and Guangxi provincesand part of Yunnan Guizhou and Jiangxi provinces Crustalthickness ranges between 28 and 40 km gradually increasingfrom the southeast coastal area of China to the northwestmountains An internal secondary elliptical gravity anomalyis relatively developed in the earthquake zones There areno obvious banded anomalies except the gravity gradientzones of southeast coastal areas and Wuling Mountain Inthe zones magnetic anomalies change gently and thereare no larger banded anomalies Because the southeastcoastal areas of China are in the same seismic belt andmost of the areas in the zone have a PGA seismic forti-fication level of 01 g Shenzhen city is then used for thebasic earthquake probability calculation and comparisonin this paper The South China belt is shown in Figure 1From Figure 1 it can be seen that there are many higherthan Ms 60 earthquakes in the southeast coastal areas ofChina present in the seismic analysis Earthquake load isstill the main load considered for bridge design in theseareas

Based on (3) to (6) the annual exceedance probability ofShenzhen is shown in Figure 2

Shock and Vibration 3

IIIminus2

Taiwan

Hainan

Yunnan

Guizhou

Hunan Jiangxi

Fujian

Kwangtung

Guangxi Taiwan

The South China Sea

105deg

ZunyiBijie

Liupanshui AnshunGuiyang

DuxunKaili

QujingXingyi

Wenshan

Baise

Henei

Liangshan

Nanning

Qinzhou

Fanggang

Sanya

Haikou

Zhanjing

MaomingYangjing

Yulin

TongrenHuaihua

JishouLoudi

ShaoyangHengyang

Yongzhou

Chenzhou

ZhuzhouXiangtan

ChangshaYiyang

YichunPingxiang

Xinyu LinchuanYingtan

Ningde

Fuzhou

Putian

Sanming

Nanping

Wenzhou

QuanzhouLongyanZhangzhou

XiamenMeizhouChaozhou

ShantouJieyangHuizhou

ShenzhenShanweiDongguan

Guangzhou

JiangmenZhongshan

Zhaoqing

QingyuanWuzhou

Shaoguan

Ganzhou

Jian

GuilinHechi

Liuzhou

Beihai Shangxiachuandao

Gaoxiong

Tainan

Taizhong

Zhejiang

Heyuan

The

Strait

110∘

115∘

120∘

25∘

20∘

1

2

3

4

5

6

7

8

9

13

14

15

16

17

IIIII

III

IV

IV

V

V

Iminus1Iminus2

Iminus2

IIIminus1

IIIminus2

IIIminus1

Iminus1

I

I

I

10

11

12

Figure 1The South China belt (1)The seismic area of South China (2)The seismic area of South China Sea (3)The seismic area of Taiwan(4)The earthquake subregion of middle of Qinghai-Tibet Plateau (5)The earthquake subregion of south of Qinghai-Tibet Plateau (6)Thesoutheast costal seismic belt (South China coastal seismic belt) (7)The seismic zone in the middle reach of Yangtze River (8) Seismic area ofthe west of Taiwan (9) Seismic area of the east of Taiwan (10) Boundaries of seismic zone (11) Boundaries of seismic belt (12) South Chinacoastal seismic rupture (13) Epicenter of Ms 7 and above (14) Epicenter of Ms 60ndash69 (15) Epicenter of Ms 50ndash59 (16) Epicenter of Ms47ndash49 (17) Calculation site

Assume that the probability density of earthquake loadintensity in time 119879 follows a distribution defined as 119891

119879(119910)

where 119910 is a variable of PGA intensity Based on the Poissonprocess assumptions the cumulative probability function119865119905(119910) over interval 119905 can be obtained using the following

119865119905(119910) = [119865

119879(119910)]119905119879

(7)

The probability density function can then be derived as

119891119905(119910) =

120597119865119905(119910)

120597119910=119905

119879sdot [119865119879(119910)](119905119879minus1)

sdot 119891119879(119910) (8)

Note that 119905 and 119879 should have the same dimension

4 Shock and Vibration

Intensity (PGA)

Prob

abili

ty

100

100

10minus2

10minus4

10minus6

10minus8

10minus12

10minus10

10minus14

10minus16

10minus18

10minus20

10minus3

10minus2

10minus1

Figure 2 Annual exceedance probability of PGA

3 Truck Load

Studies on truck load have been difficult historically princi-pally because weighing equipment was lacking and the dataare correspondingly rare [13 14] Fortunately the installationof BHMS is required on newly built long-span bridgesincluding the weighing-in-motion (WIM) system [15ndash17]Time gross weight axle weight wheel base velocity and soforth are measured and collected The probability model oftruck load can be obtained through statistical analysis

Nowak [18] indicated that at a specific site heavy trucksmay have an average number of 1000 which is also discussedby GhosnMoses [19] suggested heavy trucks follow a normaldistribution with a mean of 300 kN and a standard deviationof 80 kN (coefficient of variable COV = 265) Zhao andTabatabai [20] discussed the local standard vehicle modelusing data from about six million vehicles in Washingtonwhich can be used as a reference for a truck load model Inthis paper the truck load model is obtained through datamining from measured WIM data from three bridge sitesin Hangzhou Xiamen and Shenzhen (Figure 3) Truck loaddata of 5 axles and more in the three sites are filtered andselected Through WIM data analysis truck load probabilitycharacteristics of Hangzhou Xiamen and Shenzhen aresimilar even the shape of truck load probability curvesConsidering the three sites have very similar traffic flowalmost equal to 10 truck load probability curve of ShenzhenCity is used for analysis and validity of subsequent casestudies Truck load probability curve is shown in Figures 4and 5The fitted curve is obtained using normal distributionwhose mean value is 2949 kN and the coefficient of varianceis 374

For a typical bridge the truck loadwill consist of a varyingnumber of trucks on the bridge The probability functionfor such a bridge can be obtained using following analysisAssume 119860 is a set consisting of the elements 119860

1 1198602 119860

119898

which present 119898 events and the probability of 119860 is 119875(119860)while 119861 is a set consisting of the elements 119861

1 1198612 119861

119899

which present 119899 events and the probability of 119861 is 119875(119861) sothe probability 119875(119860 + 119861) presents the probability of intensity(119860 + 119861) Then 119875(119860 + 119861) can be calculated by

119875119860+119861

(119896) = sum

119894

119875119860(119894) sdot 119875119861(119896 + 1 minus 119894) (9)

Note that the length of 119875(119860) is119898 and the length of 119875(119861) is 119899The sum is over all the values of 119894which lead to legal subscriptsfor 119860(119894) and 119861(119896 + 1 minus 119894) where 119896 is the 119896th (119860 + 119861)(119896)119894 = max(1 119896 + 1 minus 119899) min(119896119898) Equation (9) reflects theprobability of combining two sets and when it comes to aseries of setsΦ = Φ

1+ Φ2+ sdot sdot sdot + Φ

119873 (9) can be extended to

119873 dimensions

119875Φ(1206011+ 1206012+ sdot sdot sdot + 120601

119873)

= sum(sdot sdot sdot (sum(sum119875Φ1

(1206011) sdot 119875Φ2

(1206012)) sdot 119875Φ3

(1206013)) sdot sdot sdot )

sdot 119875Φ119873

(120601119873)

(10)

where Φ119894in (10) is the 119894th set of event and 119875

Φ119894

(120601119894) is the

probability of set Φ119894

Based on total probability theory and Poisson processesthe truck load intensity function for an interval 119905 can becalculated using the following

119865Φ(120601)119905= 119875Φ0

+sum

119873

119875Φ1+Φ2+sdotsdotsdotΦ119873

sdot 119901 (119873) (11)

where 119875Φ0

is the probability with no truck passing on thebridge119873 = 1 2 maximumnumber of trucks119875

Φ1+Φ2+sdotsdotsdotΦ119873

is the probability of varying number of trucks passing thebridge 119901(119873) is the probability of occurrence of119873 trucks onthe bridge

4 Model of Combination

The intensity of dead load is usually defined as a time inde-pendent variable and that of truck load is a time dependentvariable both of whom follow normal distributions [5 818] In this paper a normal distribution is used for deadload which is considered to maintain more or less the samemagnitude such that it can be treated as a random timeindependent variable

As mentioned above earthquake load and time-variabletruck load are assumed to be Poisson processes each withsame distribution and time duration Based on these assump-tions as mentioned earlier load processes can be convertedto a small 119905 interval in which loads can be combinedThe objective of time-variable combination is to find themaximum value of different random variables namely thecombined value 119883 = 119883

1+ 1198832+ sdot sdot sdot + 119883

119899 with an interval

119905 and maximum value 119883max119879 in a time 119879 119899 is the numberof loads 119883 is the intensity of combined load and 119883

119894is the

load intensity of the 119894th loadThemaximum value of119883 in thelifetime of the bridge can be expressed as

119883max119879 = max119879

[1198831+ 1198832+ sdot sdot sdot + 119883

119899]119905 (12)

To simplify the discussion the number of loads is taken astwo the problem being reduced to the prediction of 119883

12=

1198831+ 1198832 where 119883

12is the intensity of the union of 119883

1and

1198832 Assume that 119891

1(119909) and 119891

2(119909) are the probability density

functions of 1198831and 119883

2in the interval 119905 and 119865

1(119909) and 119865

2(119909)

are the cumulative probability functions of 1198831and 119883

2in the

Shock and Vibration 5

HubeiAnhui Hangzhou Bay

Bridge

Zhejiang

Hunan

Jiangxi

Fujian

GuangdongGuangxi

Jimei Bridge

Shenzhen Bay

Bridge

Main city

Xiangyang Xinyang

Zhumadian

LiuanHefei

Nanjing

SuzhouShanghai

Hangzhou

Ningbo

Taizhou

Lishui

Wenzhou

Fuzhou

Nanping

Sanming

Putian

QuanzhouLongyan

Zhangzhou

JieyangShantou

HuizhouGuangzhou

Hong Kong

Qingyuan

Shaoguan

Ganzhou

Jian

Xinyu

YingtanShangrao

QuzhouJingdezhen

Jiujiang

Nanchang

Xianning

Wuhan

Jingzhou

Jingmen

Yichang

Zhangjiajie

Yueyang

Changde

Changsha

XiangtanLoudi

Shaoyang Hengyang

Yongzhou

Chenzhou

Guilin

Wuzhou

Yangjiang

Maoming

Yulin

HighwayRail way State road

Taiwan

Figure 3 Layout of monitoring sites

interval 119905 11989111988312

(119909) and 11986511988312

(119909) are the probability densityfunction and cumulative probability function of 119883

12 The

maximum value of combination in the entire bridge servicelife is defined as119883

12max119879 which is defined as

11988312max119879 = max

119879

[1198831+ 1198832]119905 (13)

The probability density of the combined loads 119883 can beobtained using the convolution integral

11989111988312

(119909) = 1198911lowast 1198912= int

+infin

minusinfin

1198911(120591) sdot 119891

2(119909 minus 120591) 119889120591 (14)

where ldquolowastrdquo is used as convolution symbol

6 Shock and Vibration

0 500 1000 1500

0

05

1

15

2

25

3

35

4

Intensity (kN)

Prob

abili

ty d

ensit

y

times10minus3

Figure 4 Histogram of the monitoring sites

0 200 400 600 800 1000 1200 1400

Density fittedActual density

0

05

1

15

2

25

3

35

4

Prob

abili

ty d

ensit

y

times10minus3

Intensity (kN)

Figure 5 Curve fitting of truck load density

Dirac Delta function is introduced to deal with thecharacteristics of119883 and 119884 in small 119905 interval

120575 (119909) = +infin 119909 = 0

0 119909 = 0(15)

Therefore the probability density can be illustrated as

119891119894(119909) = 119875

1198831198940

120575 (119909) + 1198911015840

119894(119909) 119894 = 1 2 (16)

Note that 1198751198831198940

is the probabilities of 119883119894(119894 = 1 2) in its ldquozero

pointsrdquo namely the events do not happen (eg themaximumtrucks on the bridge are 8 119875

1198831198940

is the probability of no trucks

on the bridge)1198911015840119894(119909) is its probability density functions with-

out ldquozero pointsrdquo Then the cumulative probability functionsof1198831and119883

2can be calculated through

119865119894(119909) = 119875

1198831198940

int

119909

minusinfin

120575 (120591) 119889120591 + int

119909

minusinfin

1198911015840

119894(119909) 119889120591

= 1198751198831198940

+ int

119909

minusinfin

1198911015840

119894(119909) 119889120591

(17)

where 119894 = 1 2Based on (16) and (17) the probability density of 119883 then

can be grouped as

11989111988312

(119909)

= int

+infin

minusinfin

[11987511988310

120575 (120591) + 1198911015840

1(120591)]

times [11987511988320

120575 (119909 minus 120591) + 1198911015840

2(119909 minus 120591)] 119889120591

= int

+infin

minusinfin

[11987511988310

11987511988320

120575 (120591) 120575 (119909 minus 120591)]

+ [11987511988310

120575 (119909) 1198911015840

2(119909 minus 120591)] + [119875

11988320

120575 (119909 minus 120591) 1198911015840

1(120591)]

+ [1198911015840

1(120591) 1198911015840

2(119909 minus 120591)] 119889120591

= 11987511988310

11987511988320

120575 (119909) + 11987511988310

1198911015840

2(119909) + 119875

11988320

1198911015840

1(119909)

+ int

+infin

minusinfin

1198911015840

1(120591) 1198911015840

2(119909 minus 120591) 119889120591

(18)

where based on the characteristic of Dirac Delta functionwhich isint+infin

minusinfin

120575(119909)119889119909 = 1 (18) can be converted to cumulativeprobability function Assume the cumulative probabilityfunction of 119865

11988312

(119909) then

11986511988312

(119909) = int

119909

minusinfin

11989111988312

(119909) 119889119909

= 11987511988310

11987511988320

+ 11987511988310

1198651015840

2(119909) + 119875

11988320

1198651015840

1(119909)

+ int

+infin

minusinfin

1198651015840

1(120591) 1198911015840

2(119909 minus 120591) 119889120591

(19)

From (19) it is clear that the combined load probabilityconsists of four parts events 119883

1and 119883

2are not happening

event 1198831is happening while 119883

2is not happening event 119883

1

is not happening while1198832is happening and119883

11198832are both

happeningTo further simplify the discussion without losing gener-

ality the probability of two loads occurring simultaneously isneglected Thus (19) is simplifying to

11986511988312

(119909) = int

119909

minusinfin

11989111988312

(119909) 119889119909

asymp 11987511988310

11987511988320

+ 11987511988310

1198651015840

2(119909) + 119875

11988320

1198651015840

1(119909)

(20)

Shock and Vibration 7

Then the cumulative probability function 11986511988312max119879

(119909) ofmaximum value of load combinations 119883

12max119879(119909) in time119879 can be obtained

11986511988312max119879

(119909) = [11986511988312

(119909)119905]119879119905

(21)

When the number of loads is more than two and theseloads apply to bridge directly while satisfying Poissonprocess (14) can be extended to 119899 dimensions Assume1198911 1198912 119891

119899are the probability densities functions of

1198831 1198832 119883

119899 1198651 1198652 119865

119899are the cumulative probability

functions of1198831 1198832 119883

119899 11989110158401 1198911015840

2 119891

1015840

119899are the probability

density functions of 1198831 1198832 119883

119899without ldquozerordquo points

119891119883(119909) and 119865

119883(119909) are the probability density function and

cumulative probability functionThen the probability densityfunction is deduced as

119891119883(119909) = 119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) (22)

where the cumulative probability 119865119883max(119909) similar to that

given by (19) is

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

= int

119909

minusinfin

1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) 119889119909

(23)

Equation (23) can account for load combinations of allloads which satisfy the first three assumptions If more thantwo events occurring simultaneously can be neglected (23)reduces to

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

asymp 11987511988310

11987511988320

sdot sdot sdot 1198751198831198990

+ 1198651015840

111987511988320

sdot sdot sdot 1198751198831198990

+ sdot sdot sdot + 11987511988310

11987511988320

sdot sdot sdot 119875119883119899minus10

1198651015840

119899

(24)

Then the maximum value of119883 in time 119879 can be obtained by

119865119883max119879

(119909) = [119865119883(119909)119905]119879119905

(25)

where 119865119883(119909)119905is 119865119883(119909) in 119905 interval 119865

119883max119879(119909) is the cumula-

tive probability function of maximum value of119883 in time 119879Although in our study emphasis is given to formulate

the ldquodemandrdquo to establish load combinations all events mustaddress a capacity issue of the bridge For example theearthquake load and truck load combination on a bridgecolumn can either consider the vertical load or the columnbase shear load Theoretically (23) can deal with most loadcombinations but as more loads are considered a more con-servative design will be adopted Based on the methodologyand assumptions described above the maximum load can becombined and the procedures are summarized as follows

(1) determine truck load and earthquake load distribu-tions over a particular period

(2) using Poisson processes convert earthquake load andtruck load distributions over a particular period to asufficiently small interval 119905

203 cm

60m

19m

442m442m442m

Figure 6 Longitudinal profile of the typical bridge

203 cm

190m

50m

152m

215m143m

215m

600m

Figure 7 Transverse profile of the typical bridge

(3) using (23) earthquake and truck load combinationsover an interval 119905 can be obtained

(4) using (25) the load combinations over an interval 119905can be converted to the bridge service life interval 119879

5 Numerical Examples

Example 1 Using the method of load combination describedin the preceding section a simple example of horizontal loadcombination is presented here Profiles of the typical bridgeare shown in Figures 6 and 7Theweight of the superstructureat each column is 538 tons the eccentricity of truck load is 50meters and the effects of soil and secondary effects of gravityare ignored Furthermore it is assumed that the maximumnumber of trucks on one lane is two The results are given inFigures 8 9 10 11 12 13 and 14 The interval 119905 is 10 secondsand the average daily truck traffic (ADTT) is about 1947

8 Shock and Vibration

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

ytimes10

minus4

Intensity (kN-M)

(a)

0 2000 4000 6000 8000 10000

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty

Intensity (kN-M)

(b)

Figure 8 Probability curve of each truck load effect

1 truck passing2 trucks passing

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M)

Figure 9 Truck load probability density curve for varied number oftrucks

Truck and earthquake load effects are the base momentcaused by trucks and earthquakes respectively Figure 8shows the probability curves of each truck load effectwhich has a similar shape to truck load Figure 9 shows thetruck load probability density curve for varied numbers oftrucks From Figure 10 we can see that over the interval 119905the probability of no truck on the bridge is much largerthan the other number of trucks passing Figure 11 showsthe combined probability curves for truck load over anearthquake load duration which indicate that the truck loadover an earthquake load duration is larger than each truckload

0 1 2

0

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 10 Probability of passing truck number simultaneously in 119905interval on the bridge

From the results shown in Figure 13 it can be seen thatcurve C and curve A have a similar shape and curve C isdisplaced to the right This means that mode and mean valueof the truck and earthquake load combination in this exampleis larger than those of truck load and also illustrates that truckload is more important to bridge design in this area FromFigure 14 we can seemore clearly that overmost ranges truckload is larger than earthquake load Because the truck loadhas a smaller tail their combination in the tail is close to theearthquake load Curve D of loads combined directly over100 years is further away from curve C which means thecombinations of truck load and earthquake load directly over100 years give a much larger value and it is not suggested thatthe bridge design uses the curve D method

Shock and Vibration 9

0

05

1

15

2

25

3

35

Prob

abili

ty d

ensit

y

times10minus4

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(a)

05

055

06

065

07

075

08

085

09

095

1

Cum

ulat

ive p

roba

bilit

y

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(b)

Figure 11 Combined probability curves for truck load in earthquake load duration

05 1 15 2 25

0

1

2

3

4

5

6

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M) times104

(a)

2000 4000 6000 8000 10000 12000 14000

0

01

02

03

04

05

06

07

08

09

1

Cum

ulat

ive p

roba

bilit

y

Intensity (kN-M)

(b)

Figure 12 Probability curves for earthquake load in 100 years

Example 2 Example 2 illustrates vertical load combinationsMost of the configurations are the same as used in Example 1The difference is in the maximum number of trucks namelythe maximum number of trucks on one lane in this exampleis four The results are shown in Figures 15 16 17 18 19 and20 Figures 13 and 14 are the results of truck load Note that inthis example the 119905 interval is 10 seconds and 119879 is taken as 100years

Figures 15 to 16 show vertical truck load probabilitycurves for varied numbers of trucks and probabilities ofpassing truck numbers on the bridge over the interval 119905From Figure 16 it can be seen that the probability is verylow when the maximum number of trucks on one lane isfour Figures 17 and 18 show similar curve shapes with those

in Figures 13 and 14 which indicate that there is the samerule in load combinations in both the horizontal and verticaldirections Comparing Figures 17 to 20 though themaximumnumber of trucks is four because dead load is combinedwith truck and earthquake load in the gravity direction thedead load contributes a substantial portion in vertical loadcombinations

6 Conclusions

This paper describes a method to combine earthquake loadand truck load in the service life of bridges The followingconclusions can be drawn

(1) Given the more than 70 seismic areas in Chinaearthquake load is a main consideration for bridge

10 Shock and Vibration

16000

05

1

15

2

2000 4000 6000 8000 10000 12000 14000

Prob

abili

ty d

ensit

y

times10minus3

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN-M)

Figure 13 Probability density of load combination in 100 years

05 1 15 2

05

055

06

065

07

075

08

085

09

095

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Cum

ulat

ive p

roba

bilit

y

times104Intensity (kN-M)

Figure 14 Cumulative probability of load combination in 100 years

design in the southeast coastal areas of China Theearthquake load probability curve is obtained usingseismic risk analysis

(2) Using measured truck load data from BHMS multi-modal characteristics of truck load are analyzed Thetruck load density of each truck is obtained by curvefitting Considering that truck load may consist ofvarying numbers of trucks truck load is calculatedthrough traffic analysis

(3) In thismethod themaximumvalue of combined loadis defined as 119883max119879 = max

119879[1198831+ 1198832+ sdot sdot sdot 119883

119899]119905

which means (1198831+ 1198832+ sdot sdot sdot 119883

119899) over the interval 119905

is first combined and then the maximum value in the

0 2000 4000 6000 8000 100000

05

1

15

2

25

3

35

4

1 truck passing2 trucks passing

3 trucks passing4 trucks passing

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 15 Vertical truck load probability density curves for variednumber of trucks

0 1 2 3 40

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 16 Probability of passing truck number simultaneously in 119905interval on the bridge with maximum number 4

bridge service life is determined 119883max119879 is based onprobability which covers all the probability combi-nations of the combined situations In this method aDiracDelta function is introduced to deal with119883 overa small interval 119905 To demonstrate the methodologyintuitively examples of load combinations in horizon-tal and vertical directions are provided

(4) The shape of the earthquake and truck load com-bination is similar to that of truck load alone butthe curve is displaced to the right which means themode and mean value of truck and earthquake loadcombination in this example is larger than that oftruck load aloneThis also illustrates that truck load is

Shock and Vibration 11

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 17 Probability density of vertical load combination in 100years

0 1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Figure 18 Cumulative probability of vertical load combination in100 years

more sensitive to bridge design and over most rangestruck load is larger than earthquake load in this area

(5) The curve from direct load combined over 100 yearsis further away from the curve obtained using themethod in the paper with the direct combinationof truck load and earthquake load over 100 yearsgiving much larger values It is not suggested that thismethod be used in bridge design considerations

(6) Because dead load is combined with truck and earth-quake load along the direction of gravity the dead

0 2000 4000 6000 8000 10000 12000 140000

1

2

3

4

5

6

7

8

Dead loadTruck and earthquakeTruck and earthquake and dead load

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 19 Probability density of dead truck and earthquake loadcombination

0 2000 4000 6000 8000 10000 12000 140000

02

04

06

08

1

12

14

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Dead loadTruck and earthquakeTruck and earthquake and dead load

Figure 20 Probability curves of dead truck and earthquake loadcombination

load contributes a substantial portion in vertical loadcombinations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Shock and Vibration

Acknowledgments

This study is jointly funded by Basic Institute ScientificResearch Fund (Grant no 2012A02) the National NaturalScience Fund of China (NSFC) (Grant no 51308510) andOpen Fund of State Key Laboratory Breeding Base of Moun-tainBridge andTunnel Engineering (Grant no CQSLBF-Y14-15) The results and conclusions presented in the paper areof the authors and do not necessarily reflect the view of thesponsors

References

[1] P R China Ministry of Communications ldquoGeneral code fordesign of highway bridges and culvertsrdquo Tech Rep JTG D60-200 China Communications Press Beijing China 2004

[2] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[3] C J Turkstra and H O Madsen ldquoLoad combinations incodified structural designrdquo Journal of Structural Engineeringvol 106 no 12 pp 2527ndash2543 1980

[4] ACI Publication SP-31 Probabilistic Design of Reinforced Con-crete Buildings American Concrete Institute Detroit MichUSA 1971

[5] B Ellingwood Development of a Probability Based Load Cri-terion for American National Standard A58-Building CodeRequirements forMinimumDesign Loads in Buildings andOtherStructures US Department of CommerceNational Bureau ofStandards 1980

[6] Y K Wen Development of Reliability-Based Design Criteria forBuildings Under Seismic Load National Center for EarthquakeEngineering Research Buffalo NY USA 1994

[7] Y-K Wen ldquoStatistical combination of extreme loadsrdquo Journalof Structural Engineering vol 103 no 5 pp 1079ndash1093 1977

[8] M Ghosn F Moses and J Wang ldquoDesign of highway bridgesfor extreme eventsrdquo NCHRP 489 Transportation ResearchBoard National Research Council Washington D C USA2003

[9] S E Hida ldquoStatistical significance of less common load combi-nationsrdquo Journal of Bridge Engineering vol 12 no 3 pp 389ndash393 2007

[10] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loadsmdashpart I comprehensive reliability and par-tial failure probabilitiesrdquo Journal of Earthquake Engineering andEngineering Vibration vol 11 no 3 pp 293ndash301 2012

[11] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loads Part II Examples for live and earthquakeload effectsrdquo Journal of Earthquake Engineering and EngineeringVibration vol 11 no 3 pp 303ndash311 2012

[12] E EMatheu D E Yule and R V Kala ldquoDetermination of stan-dard response spectra and effective peak ground accelerationsfor seismic design and evaluationrdquo Tech Rep no ERDCCHLCHETN-VI-41 2005

[13] HN Li TH YiMGu and LHuo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009

[14] B Chen Z W Chen Y Z Sun and S L Zhao ldquoConditionassessment on thermal effects of a suspension bridge basedon SHM oriented model and datardquo Mathematical Problems inEngineering vol 2013 Article ID 256816 18 pages 2013

[15] T-H Yi H-N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013

[16] Y Xia B Chen X Q Zhou and Y L Xu ldquoFieldmonitoring andnumerical analysis of Tsing Ma suspension bridge temperaturebehaviorrdquo Structural Control and HealthMonitoring vol 20 no4 pp 560ndash575 2013

[17] G Fu and J You ldquoExtrapolation for future maximum loadstatisticsrdquo Journal of Bridge Engineering vol 16 no 4 pp 527ndash535 2011

[18] A S Nowak ldquoCalibration of LRFD bridge design coderdquoNCHRPReport TransportationResearchBoard of theNationalAcademies Washington DC USA 1999

[19] F Moses ldquoCalibration of load factors for LRFD bridge evalua-tionsrdquo NCHRP Report 454 Transportation Research Board ofthe National Academies Washington DC USA 2001

[20] J Zhao and H Tabatabai ldquoEvaluation of a permit vehiclemodel using weigh-in-motion truck recordsrdquo Journal of BridgeEngineering vol 17 no 2 pp 389ndash392 2012

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Shock and Vibration 3

IIIminus2

Taiwan

Hainan

Yunnan

Guizhou

Hunan Jiangxi

Fujian

Kwangtung

Guangxi Taiwan

The South China Sea

105deg

ZunyiBijie

Liupanshui AnshunGuiyang

DuxunKaili

QujingXingyi

Wenshan

Baise

Henei

Liangshan

Nanning

Qinzhou

Fanggang

Sanya

Haikou

Zhanjing

MaomingYangjing

Yulin

TongrenHuaihua

JishouLoudi

ShaoyangHengyang

Yongzhou

Chenzhou

ZhuzhouXiangtan

ChangshaYiyang

YichunPingxiang

Xinyu LinchuanYingtan

Ningde

Fuzhou

Putian

Sanming

Nanping

Wenzhou

QuanzhouLongyanZhangzhou

XiamenMeizhouChaozhou

ShantouJieyangHuizhou

ShenzhenShanweiDongguan

Guangzhou

JiangmenZhongshan

Zhaoqing

QingyuanWuzhou

Shaoguan

Ganzhou

Jian

GuilinHechi

Liuzhou

Beihai Shangxiachuandao

Gaoxiong

Tainan

Taizhong

Zhejiang

Heyuan

The

Strait

110∘

115∘

120∘

25∘

20∘

1

2

3

4

5

6

7

8

9

13

14

15

16

17

IIIII

III

IV

IV

V

V

Iminus1Iminus2

Iminus2

IIIminus1

IIIminus2

IIIminus1

Iminus1

I

I

I

10

11

12

Figure 1The South China belt (1)The seismic area of South China (2)The seismic area of South China Sea (3)The seismic area of Taiwan(4)The earthquake subregion of middle of Qinghai-Tibet Plateau (5)The earthquake subregion of south of Qinghai-Tibet Plateau (6)Thesoutheast costal seismic belt (South China coastal seismic belt) (7)The seismic zone in the middle reach of Yangtze River (8) Seismic area ofthe west of Taiwan (9) Seismic area of the east of Taiwan (10) Boundaries of seismic zone (11) Boundaries of seismic belt (12) South Chinacoastal seismic rupture (13) Epicenter of Ms 7 and above (14) Epicenter of Ms 60ndash69 (15) Epicenter of Ms 50ndash59 (16) Epicenter of Ms47ndash49 (17) Calculation site

Assume that the probability density of earthquake loadintensity in time 119879 follows a distribution defined as 119891

119879(119910)

where 119910 is a variable of PGA intensity Based on the Poissonprocess assumptions the cumulative probability function119865119905(119910) over interval 119905 can be obtained using the following

119865119905(119910) = [119865

119879(119910)]119905119879

(7)

The probability density function can then be derived as

119891119905(119910) =

120597119865119905(119910)

120597119910=119905

119879sdot [119865119879(119910)](119905119879minus1)

sdot 119891119879(119910) (8)

Note that 119905 and 119879 should have the same dimension

4 Shock and Vibration

Intensity (PGA)

Prob

abili

ty

100

100

10minus2

10minus4

10minus6

10minus8

10minus12

10minus10

10minus14

10minus16

10minus18

10minus20

10minus3

10minus2

10minus1

Figure 2 Annual exceedance probability of PGA

3 Truck Load

Studies on truck load have been difficult historically princi-pally because weighing equipment was lacking and the dataare correspondingly rare [13 14] Fortunately the installationof BHMS is required on newly built long-span bridgesincluding the weighing-in-motion (WIM) system [15ndash17]Time gross weight axle weight wheel base velocity and soforth are measured and collected The probability model oftruck load can be obtained through statistical analysis

Nowak [18] indicated that at a specific site heavy trucksmay have an average number of 1000 which is also discussedby GhosnMoses [19] suggested heavy trucks follow a normaldistribution with a mean of 300 kN and a standard deviationof 80 kN (coefficient of variable COV = 265) Zhao andTabatabai [20] discussed the local standard vehicle modelusing data from about six million vehicles in Washingtonwhich can be used as a reference for a truck load model Inthis paper the truck load model is obtained through datamining from measured WIM data from three bridge sitesin Hangzhou Xiamen and Shenzhen (Figure 3) Truck loaddata of 5 axles and more in the three sites are filtered andselected Through WIM data analysis truck load probabilitycharacteristics of Hangzhou Xiamen and Shenzhen aresimilar even the shape of truck load probability curvesConsidering the three sites have very similar traffic flowalmost equal to 10 truck load probability curve of ShenzhenCity is used for analysis and validity of subsequent casestudies Truck load probability curve is shown in Figures 4and 5The fitted curve is obtained using normal distributionwhose mean value is 2949 kN and the coefficient of varianceis 374

For a typical bridge the truck loadwill consist of a varyingnumber of trucks on the bridge The probability functionfor such a bridge can be obtained using following analysisAssume 119860 is a set consisting of the elements 119860

1 1198602 119860

119898

which present 119898 events and the probability of 119860 is 119875(119860)while 119861 is a set consisting of the elements 119861

1 1198612 119861

119899

which present 119899 events and the probability of 119861 is 119875(119861) sothe probability 119875(119860 + 119861) presents the probability of intensity(119860 + 119861) Then 119875(119860 + 119861) can be calculated by

119875119860+119861

(119896) = sum

119894

119875119860(119894) sdot 119875119861(119896 + 1 minus 119894) (9)

Note that the length of 119875(119860) is119898 and the length of 119875(119861) is 119899The sum is over all the values of 119894which lead to legal subscriptsfor 119860(119894) and 119861(119896 + 1 minus 119894) where 119896 is the 119896th (119860 + 119861)(119896)119894 = max(1 119896 + 1 minus 119899) min(119896119898) Equation (9) reflects theprobability of combining two sets and when it comes to aseries of setsΦ = Φ

1+ Φ2+ sdot sdot sdot + Φ

119873 (9) can be extended to

119873 dimensions

119875Φ(1206011+ 1206012+ sdot sdot sdot + 120601

119873)

= sum(sdot sdot sdot (sum(sum119875Φ1

(1206011) sdot 119875Φ2

(1206012)) sdot 119875Φ3

(1206013)) sdot sdot sdot )

sdot 119875Φ119873

(120601119873)

(10)

where Φ119894in (10) is the 119894th set of event and 119875

Φ119894

(120601119894) is the

probability of set Φ119894

Based on total probability theory and Poisson processesthe truck load intensity function for an interval 119905 can becalculated using the following

119865Φ(120601)119905= 119875Φ0

+sum

119873

119875Φ1+Φ2+sdotsdotsdotΦ119873

sdot 119901 (119873) (11)

where 119875Φ0

is the probability with no truck passing on thebridge119873 = 1 2 maximumnumber of trucks119875

Φ1+Φ2+sdotsdotsdotΦ119873

is the probability of varying number of trucks passing thebridge 119901(119873) is the probability of occurrence of119873 trucks onthe bridge

4 Model of Combination

The intensity of dead load is usually defined as a time inde-pendent variable and that of truck load is a time dependentvariable both of whom follow normal distributions [5 818] In this paper a normal distribution is used for deadload which is considered to maintain more or less the samemagnitude such that it can be treated as a random timeindependent variable

As mentioned above earthquake load and time-variabletruck load are assumed to be Poisson processes each withsame distribution and time duration Based on these assump-tions as mentioned earlier load processes can be convertedto a small 119905 interval in which loads can be combinedThe objective of time-variable combination is to find themaximum value of different random variables namely thecombined value 119883 = 119883

1+ 1198832+ sdot sdot sdot + 119883

119899 with an interval

119905 and maximum value 119883max119879 in a time 119879 119899 is the numberof loads 119883 is the intensity of combined load and 119883

119894is the

load intensity of the 119894th loadThemaximum value of119883 in thelifetime of the bridge can be expressed as

119883max119879 = max119879

[1198831+ 1198832+ sdot sdot sdot + 119883

119899]119905 (12)

To simplify the discussion the number of loads is taken astwo the problem being reduced to the prediction of 119883

12=

1198831+ 1198832 where 119883

12is the intensity of the union of 119883

1and

1198832 Assume that 119891

1(119909) and 119891

2(119909) are the probability density

functions of 1198831and 119883

2in the interval 119905 and 119865

1(119909) and 119865

2(119909)

are the cumulative probability functions of 1198831and 119883

2in the

Shock and Vibration 5

HubeiAnhui Hangzhou Bay

Bridge

Zhejiang

Hunan

Jiangxi

Fujian

GuangdongGuangxi

Jimei Bridge

Shenzhen Bay

Bridge

Main city

Xiangyang Xinyang

Zhumadian

LiuanHefei

Nanjing

SuzhouShanghai

Hangzhou

Ningbo

Taizhou

Lishui

Wenzhou

Fuzhou

Nanping

Sanming

Putian

QuanzhouLongyan

Zhangzhou

JieyangShantou

HuizhouGuangzhou

Hong Kong

Qingyuan

Shaoguan

Ganzhou

Jian

Xinyu

YingtanShangrao

QuzhouJingdezhen

Jiujiang

Nanchang

Xianning

Wuhan

Jingzhou

Jingmen

Yichang

Zhangjiajie

Yueyang

Changde

Changsha

XiangtanLoudi

Shaoyang Hengyang

Yongzhou

Chenzhou

Guilin

Wuzhou

Yangjiang

Maoming

Yulin

HighwayRail way State road

Taiwan

Figure 3 Layout of monitoring sites

interval 119905 11989111988312

(119909) and 11986511988312

(119909) are the probability densityfunction and cumulative probability function of 119883

12 The

maximum value of combination in the entire bridge servicelife is defined as119883

12max119879 which is defined as

11988312max119879 = max

119879

[1198831+ 1198832]119905 (13)

The probability density of the combined loads 119883 can beobtained using the convolution integral

11989111988312

(119909) = 1198911lowast 1198912= int

+infin

minusinfin

1198911(120591) sdot 119891

2(119909 minus 120591) 119889120591 (14)

where ldquolowastrdquo is used as convolution symbol

6 Shock and Vibration

0 500 1000 1500

0

05

1

15

2

25

3

35

4

Intensity (kN)

Prob

abili

ty d

ensit

y

times10minus3

Figure 4 Histogram of the monitoring sites

0 200 400 600 800 1000 1200 1400

Density fittedActual density

0

05

1

15

2

25

3

35

4

Prob

abili

ty d

ensit

y

times10minus3

Intensity (kN)

Figure 5 Curve fitting of truck load density

Dirac Delta function is introduced to deal with thecharacteristics of119883 and 119884 in small 119905 interval

120575 (119909) = +infin 119909 = 0

0 119909 = 0(15)

Therefore the probability density can be illustrated as

119891119894(119909) = 119875

1198831198940

120575 (119909) + 1198911015840

119894(119909) 119894 = 1 2 (16)

Note that 1198751198831198940

is the probabilities of 119883119894(119894 = 1 2) in its ldquozero

pointsrdquo namely the events do not happen (eg themaximumtrucks on the bridge are 8 119875

1198831198940

is the probability of no trucks

on the bridge)1198911015840119894(119909) is its probability density functions with-

out ldquozero pointsrdquo Then the cumulative probability functionsof1198831and119883

2can be calculated through

119865119894(119909) = 119875

1198831198940

int

119909

minusinfin

120575 (120591) 119889120591 + int

119909

minusinfin

1198911015840

119894(119909) 119889120591

= 1198751198831198940

+ int

119909

minusinfin

1198911015840

119894(119909) 119889120591

(17)

where 119894 = 1 2Based on (16) and (17) the probability density of 119883 then

can be grouped as

11989111988312

(119909)

= int

+infin

minusinfin

[11987511988310

120575 (120591) + 1198911015840

1(120591)]

times [11987511988320

120575 (119909 minus 120591) + 1198911015840

2(119909 minus 120591)] 119889120591

= int

+infin

minusinfin

[11987511988310

11987511988320

120575 (120591) 120575 (119909 minus 120591)]

+ [11987511988310

120575 (119909) 1198911015840

2(119909 minus 120591)] + [119875

11988320

120575 (119909 minus 120591) 1198911015840

1(120591)]

+ [1198911015840

1(120591) 1198911015840

2(119909 minus 120591)] 119889120591

= 11987511988310

11987511988320

120575 (119909) + 11987511988310

1198911015840

2(119909) + 119875

11988320

1198911015840

1(119909)

+ int

+infin

minusinfin

1198911015840

1(120591) 1198911015840

2(119909 minus 120591) 119889120591

(18)

where based on the characteristic of Dirac Delta functionwhich isint+infin

minusinfin

120575(119909)119889119909 = 1 (18) can be converted to cumulativeprobability function Assume the cumulative probabilityfunction of 119865

11988312

(119909) then

11986511988312

(119909) = int

119909

minusinfin

11989111988312

(119909) 119889119909

= 11987511988310

11987511988320

+ 11987511988310

1198651015840

2(119909) + 119875

11988320

1198651015840

1(119909)

+ int

+infin

minusinfin

1198651015840

1(120591) 1198911015840

2(119909 minus 120591) 119889120591

(19)

From (19) it is clear that the combined load probabilityconsists of four parts events 119883

1and 119883

2are not happening

event 1198831is happening while 119883

2is not happening event 119883

1

is not happening while1198832is happening and119883

11198832are both

happeningTo further simplify the discussion without losing gener-

ality the probability of two loads occurring simultaneously isneglected Thus (19) is simplifying to

11986511988312

(119909) = int

119909

minusinfin

11989111988312

(119909) 119889119909

asymp 11987511988310

11987511988320

+ 11987511988310

1198651015840

2(119909) + 119875

11988320

1198651015840

1(119909)

(20)

Shock and Vibration 7

Then the cumulative probability function 11986511988312max119879

(119909) ofmaximum value of load combinations 119883

12max119879(119909) in time119879 can be obtained

11986511988312max119879

(119909) = [11986511988312

(119909)119905]119879119905

(21)

When the number of loads is more than two and theseloads apply to bridge directly while satisfying Poissonprocess (14) can be extended to 119899 dimensions Assume1198911 1198912 119891

119899are the probability densities functions of

1198831 1198832 119883

119899 1198651 1198652 119865

119899are the cumulative probability

functions of1198831 1198832 119883

119899 11989110158401 1198911015840

2 119891

1015840

119899are the probability

density functions of 1198831 1198832 119883

119899without ldquozerordquo points

119891119883(119909) and 119865

119883(119909) are the probability density function and

cumulative probability functionThen the probability densityfunction is deduced as

119891119883(119909) = 119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) (22)

where the cumulative probability 119865119883max(119909) similar to that

given by (19) is

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

= int

119909

minusinfin

1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) 119889119909

(23)

Equation (23) can account for load combinations of allloads which satisfy the first three assumptions If more thantwo events occurring simultaneously can be neglected (23)reduces to

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

asymp 11987511988310

11987511988320

sdot sdot sdot 1198751198831198990

+ 1198651015840

111987511988320

sdot sdot sdot 1198751198831198990

+ sdot sdot sdot + 11987511988310

11987511988320

sdot sdot sdot 119875119883119899minus10

1198651015840

119899

(24)

Then the maximum value of119883 in time 119879 can be obtained by

119865119883max119879

(119909) = [119865119883(119909)119905]119879119905

(25)

where 119865119883(119909)119905is 119865119883(119909) in 119905 interval 119865

119883max119879(119909) is the cumula-

tive probability function of maximum value of119883 in time 119879Although in our study emphasis is given to formulate

the ldquodemandrdquo to establish load combinations all events mustaddress a capacity issue of the bridge For example theearthquake load and truck load combination on a bridgecolumn can either consider the vertical load or the columnbase shear load Theoretically (23) can deal with most loadcombinations but as more loads are considered a more con-servative design will be adopted Based on the methodologyand assumptions described above the maximum load can becombined and the procedures are summarized as follows

(1) determine truck load and earthquake load distribu-tions over a particular period

(2) using Poisson processes convert earthquake load andtruck load distributions over a particular period to asufficiently small interval 119905

203 cm

60m

19m

442m442m442m

Figure 6 Longitudinal profile of the typical bridge

203 cm

190m

50m

152m

215m143m

215m

600m

Figure 7 Transverse profile of the typical bridge

(3) using (23) earthquake and truck load combinationsover an interval 119905 can be obtained

(4) using (25) the load combinations over an interval 119905can be converted to the bridge service life interval 119879

5 Numerical Examples

Example 1 Using the method of load combination describedin the preceding section a simple example of horizontal loadcombination is presented here Profiles of the typical bridgeare shown in Figures 6 and 7Theweight of the superstructureat each column is 538 tons the eccentricity of truck load is 50meters and the effects of soil and secondary effects of gravityare ignored Furthermore it is assumed that the maximumnumber of trucks on one lane is two The results are given inFigures 8 9 10 11 12 13 and 14 The interval 119905 is 10 secondsand the average daily truck traffic (ADTT) is about 1947

8 Shock and Vibration

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

ytimes10

minus4

Intensity (kN-M)

(a)

0 2000 4000 6000 8000 10000

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty

Intensity (kN-M)

(b)

Figure 8 Probability curve of each truck load effect

1 truck passing2 trucks passing

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M)

Figure 9 Truck load probability density curve for varied number oftrucks

Truck and earthquake load effects are the base momentcaused by trucks and earthquakes respectively Figure 8shows the probability curves of each truck load effectwhich has a similar shape to truck load Figure 9 shows thetruck load probability density curve for varied numbers oftrucks From Figure 10 we can see that over the interval 119905the probability of no truck on the bridge is much largerthan the other number of trucks passing Figure 11 showsthe combined probability curves for truck load over anearthquake load duration which indicate that the truck loadover an earthquake load duration is larger than each truckload

0 1 2

0

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 10 Probability of passing truck number simultaneously in 119905interval on the bridge

From the results shown in Figure 13 it can be seen thatcurve C and curve A have a similar shape and curve C isdisplaced to the right This means that mode and mean valueof the truck and earthquake load combination in this exampleis larger than those of truck load and also illustrates that truckload is more important to bridge design in this area FromFigure 14 we can seemore clearly that overmost ranges truckload is larger than earthquake load Because the truck loadhas a smaller tail their combination in the tail is close to theearthquake load Curve D of loads combined directly over100 years is further away from curve C which means thecombinations of truck load and earthquake load directly over100 years give a much larger value and it is not suggested thatthe bridge design uses the curve D method

Shock and Vibration 9

0

05

1

15

2

25

3

35

Prob

abili

ty d

ensit

y

times10minus4

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(a)

05

055

06

065

07

075

08

085

09

095

1

Cum

ulat

ive p

roba

bilit

y

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(b)

Figure 11 Combined probability curves for truck load in earthquake load duration

05 1 15 2 25

0

1

2

3

4

5

6

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M) times104

(a)

2000 4000 6000 8000 10000 12000 14000

0

01

02

03

04

05

06

07

08

09

1

Cum

ulat

ive p

roba

bilit

y

Intensity (kN-M)

(b)

Figure 12 Probability curves for earthquake load in 100 years

Example 2 Example 2 illustrates vertical load combinationsMost of the configurations are the same as used in Example 1The difference is in the maximum number of trucks namelythe maximum number of trucks on one lane in this exampleis four The results are shown in Figures 15 16 17 18 19 and20 Figures 13 and 14 are the results of truck load Note that inthis example the 119905 interval is 10 seconds and 119879 is taken as 100years

Figures 15 to 16 show vertical truck load probabilitycurves for varied numbers of trucks and probabilities ofpassing truck numbers on the bridge over the interval 119905From Figure 16 it can be seen that the probability is verylow when the maximum number of trucks on one lane isfour Figures 17 and 18 show similar curve shapes with those

in Figures 13 and 14 which indicate that there is the samerule in load combinations in both the horizontal and verticaldirections Comparing Figures 17 to 20 though themaximumnumber of trucks is four because dead load is combinedwith truck and earthquake load in the gravity direction thedead load contributes a substantial portion in vertical loadcombinations

6 Conclusions

This paper describes a method to combine earthquake loadand truck load in the service life of bridges The followingconclusions can be drawn

(1) Given the more than 70 seismic areas in Chinaearthquake load is a main consideration for bridge

10 Shock and Vibration

16000

05

1

15

2

2000 4000 6000 8000 10000 12000 14000

Prob

abili

ty d

ensit

y

times10minus3

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN-M)

Figure 13 Probability density of load combination in 100 years

05 1 15 2

05

055

06

065

07

075

08

085

09

095

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Cum

ulat

ive p

roba

bilit

y

times104Intensity (kN-M)

Figure 14 Cumulative probability of load combination in 100 years

design in the southeast coastal areas of China Theearthquake load probability curve is obtained usingseismic risk analysis

(2) Using measured truck load data from BHMS multi-modal characteristics of truck load are analyzed Thetruck load density of each truck is obtained by curvefitting Considering that truck load may consist ofvarying numbers of trucks truck load is calculatedthrough traffic analysis

(3) In thismethod themaximumvalue of combined loadis defined as 119883max119879 = max

119879[1198831+ 1198832+ sdot sdot sdot 119883

119899]119905

which means (1198831+ 1198832+ sdot sdot sdot 119883

119899) over the interval 119905

is first combined and then the maximum value in the

0 2000 4000 6000 8000 100000

05

1

15

2

25

3

35

4

1 truck passing2 trucks passing

3 trucks passing4 trucks passing

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 15 Vertical truck load probability density curves for variednumber of trucks

0 1 2 3 40

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 16 Probability of passing truck number simultaneously in 119905interval on the bridge with maximum number 4

bridge service life is determined 119883max119879 is based onprobability which covers all the probability combi-nations of the combined situations In this method aDiracDelta function is introduced to deal with119883 overa small interval 119905 To demonstrate the methodologyintuitively examples of load combinations in horizon-tal and vertical directions are provided

(4) The shape of the earthquake and truck load com-bination is similar to that of truck load alone butthe curve is displaced to the right which means themode and mean value of truck and earthquake loadcombination in this example is larger than that oftruck load aloneThis also illustrates that truck load is

Shock and Vibration 11

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 17 Probability density of vertical load combination in 100years

0 1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Figure 18 Cumulative probability of vertical load combination in100 years

more sensitive to bridge design and over most rangestruck load is larger than earthquake load in this area

(5) The curve from direct load combined over 100 yearsis further away from the curve obtained using themethod in the paper with the direct combinationof truck load and earthquake load over 100 yearsgiving much larger values It is not suggested that thismethod be used in bridge design considerations

(6) Because dead load is combined with truck and earth-quake load along the direction of gravity the dead

0 2000 4000 6000 8000 10000 12000 140000

1

2

3

4

5

6

7

8

Dead loadTruck and earthquakeTruck and earthquake and dead load

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 19 Probability density of dead truck and earthquake loadcombination

0 2000 4000 6000 8000 10000 12000 140000

02

04

06

08

1

12

14

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Dead loadTruck and earthquakeTruck and earthquake and dead load

Figure 20 Probability curves of dead truck and earthquake loadcombination

load contributes a substantial portion in vertical loadcombinations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Shock and Vibration

Acknowledgments

This study is jointly funded by Basic Institute ScientificResearch Fund (Grant no 2012A02) the National NaturalScience Fund of China (NSFC) (Grant no 51308510) andOpen Fund of State Key Laboratory Breeding Base of Moun-tainBridge andTunnel Engineering (Grant no CQSLBF-Y14-15) The results and conclusions presented in the paper areof the authors and do not necessarily reflect the view of thesponsors

References

[1] P R China Ministry of Communications ldquoGeneral code fordesign of highway bridges and culvertsrdquo Tech Rep JTG D60-200 China Communications Press Beijing China 2004

[2] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[3] C J Turkstra and H O Madsen ldquoLoad combinations incodified structural designrdquo Journal of Structural Engineeringvol 106 no 12 pp 2527ndash2543 1980

[4] ACI Publication SP-31 Probabilistic Design of Reinforced Con-crete Buildings American Concrete Institute Detroit MichUSA 1971

[5] B Ellingwood Development of a Probability Based Load Cri-terion for American National Standard A58-Building CodeRequirements forMinimumDesign Loads in Buildings andOtherStructures US Department of CommerceNational Bureau ofStandards 1980

[6] Y K Wen Development of Reliability-Based Design Criteria forBuildings Under Seismic Load National Center for EarthquakeEngineering Research Buffalo NY USA 1994

[7] Y-K Wen ldquoStatistical combination of extreme loadsrdquo Journalof Structural Engineering vol 103 no 5 pp 1079ndash1093 1977

[8] M Ghosn F Moses and J Wang ldquoDesign of highway bridgesfor extreme eventsrdquo NCHRP 489 Transportation ResearchBoard National Research Council Washington D C USA2003

[9] S E Hida ldquoStatistical significance of less common load combi-nationsrdquo Journal of Bridge Engineering vol 12 no 3 pp 389ndash393 2007

[10] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loadsmdashpart I comprehensive reliability and par-tial failure probabilitiesrdquo Journal of Earthquake Engineering andEngineering Vibration vol 11 no 3 pp 293ndash301 2012

[11] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loads Part II Examples for live and earthquakeload effectsrdquo Journal of Earthquake Engineering and EngineeringVibration vol 11 no 3 pp 303ndash311 2012

[12] E EMatheu D E Yule and R V Kala ldquoDetermination of stan-dard response spectra and effective peak ground accelerationsfor seismic design and evaluationrdquo Tech Rep no ERDCCHLCHETN-VI-41 2005

[13] HN Li TH YiMGu and LHuo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009

[14] B Chen Z W Chen Y Z Sun and S L Zhao ldquoConditionassessment on thermal effects of a suspension bridge basedon SHM oriented model and datardquo Mathematical Problems inEngineering vol 2013 Article ID 256816 18 pages 2013

[15] T-H Yi H-N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013

[16] Y Xia B Chen X Q Zhou and Y L Xu ldquoFieldmonitoring andnumerical analysis of Tsing Ma suspension bridge temperaturebehaviorrdquo Structural Control and HealthMonitoring vol 20 no4 pp 560ndash575 2013

[17] G Fu and J You ldquoExtrapolation for future maximum loadstatisticsrdquo Journal of Bridge Engineering vol 16 no 4 pp 527ndash535 2011

[18] A S Nowak ldquoCalibration of LRFD bridge design coderdquoNCHRPReport TransportationResearchBoard of theNationalAcademies Washington DC USA 1999

[19] F Moses ldquoCalibration of load factors for LRFD bridge evalua-tionsrdquo NCHRP Report 454 Transportation Research Board ofthe National Academies Washington DC USA 2001

[20] J Zhao and H Tabatabai ldquoEvaluation of a permit vehiclemodel using weigh-in-motion truck recordsrdquo Journal of BridgeEngineering vol 17 no 2 pp 389ndash392 2012

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Shock and Vibration

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International Journal of

4 Shock and Vibration

Intensity (PGA)

Prob

abili

ty

100

100

10minus2

10minus4

10minus6

10minus8

10minus12

10minus10

10minus14

10minus16

10minus18

10minus20

10minus3

10minus2

10minus1

Figure 2 Annual exceedance probability of PGA

3 Truck Load

Studies on truck load have been difficult historically princi-pally because weighing equipment was lacking and the dataare correspondingly rare [13 14] Fortunately the installationof BHMS is required on newly built long-span bridgesincluding the weighing-in-motion (WIM) system [15ndash17]Time gross weight axle weight wheel base velocity and soforth are measured and collected The probability model oftruck load can be obtained through statistical analysis

Nowak [18] indicated that at a specific site heavy trucksmay have an average number of 1000 which is also discussedby GhosnMoses [19] suggested heavy trucks follow a normaldistribution with a mean of 300 kN and a standard deviationof 80 kN (coefficient of variable COV = 265) Zhao andTabatabai [20] discussed the local standard vehicle modelusing data from about six million vehicles in Washingtonwhich can be used as a reference for a truck load model Inthis paper the truck load model is obtained through datamining from measured WIM data from three bridge sitesin Hangzhou Xiamen and Shenzhen (Figure 3) Truck loaddata of 5 axles and more in the three sites are filtered andselected Through WIM data analysis truck load probabilitycharacteristics of Hangzhou Xiamen and Shenzhen aresimilar even the shape of truck load probability curvesConsidering the three sites have very similar traffic flowalmost equal to 10 truck load probability curve of ShenzhenCity is used for analysis and validity of subsequent casestudies Truck load probability curve is shown in Figures 4and 5The fitted curve is obtained using normal distributionwhose mean value is 2949 kN and the coefficient of varianceis 374

For a typical bridge the truck loadwill consist of a varyingnumber of trucks on the bridge The probability functionfor such a bridge can be obtained using following analysisAssume 119860 is a set consisting of the elements 119860

1 1198602 119860

119898

which present 119898 events and the probability of 119860 is 119875(119860)while 119861 is a set consisting of the elements 119861

1 1198612 119861

119899

which present 119899 events and the probability of 119861 is 119875(119861) sothe probability 119875(119860 + 119861) presents the probability of intensity(119860 + 119861) Then 119875(119860 + 119861) can be calculated by

119875119860+119861

(119896) = sum

119894

119875119860(119894) sdot 119875119861(119896 + 1 minus 119894) (9)

Note that the length of 119875(119860) is119898 and the length of 119875(119861) is 119899The sum is over all the values of 119894which lead to legal subscriptsfor 119860(119894) and 119861(119896 + 1 minus 119894) where 119896 is the 119896th (119860 + 119861)(119896)119894 = max(1 119896 + 1 minus 119899) min(119896119898) Equation (9) reflects theprobability of combining two sets and when it comes to aseries of setsΦ = Φ

1+ Φ2+ sdot sdot sdot + Φ

119873 (9) can be extended to

119873 dimensions

119875Φ(1206011+ 1206012+ sdot sdot sdot + 120601

119873)

= sum(sdot sdot sdot (sum(sum119875Φ1

(1206011) sdot 119875Φ2

(1206012)) sdot 119875Φ3

(1206013)) sdot sdot sdot )

sdot 119875Φ119873

(120601119873)

(10)

where Φ119894in (10) is the 119894th set of event and 119875

Φ119894

(120601119894) is the

probability of set Φ119894

Based on total probability theory and Poisson processesthe truck load intensity function for an interval 119905 can becalculated using the following

119865Φ(120601)119905= 119875Φ0

+sum

119873

119875Φ1+Φ2+sdotsdotsdotΦ119873

sdot 119901 (119873) (11)

where 119875Φ0

is the probability with no truck passing on thebridge119873 = 1 2 maximumnumber of trucks119875

Φ1+Φ2+sdotsdotsdotΦ119873

is the probability of varying number of trucks passing thebridge 119901(119873) is the probability of occurrence of119873 trucks onthe bridge

4 Model of Combination

The intensity of dead load is usually defined as a time inde-pendent variable and that of truck load is a time dependentvariable both of whom follow normal distributions [5 818] In this paper a normal distribution is used for deadload which is considered to maintain more or less the samemagnitude such that it can be treated as a random timeindependent variable

As mentioned above earthquake load and time-variabletruck load are assumed to be Poisson processes each withsame distribution and time duration Based on these assump-tions as mentioned earlier load processes can be convertedto a small 119905 interval in which loads can be combinedThe objective of time-variable combination is to find themaximum value of different random variables namely thecombined value 119883 = 119883

1+ 1198832+ sdot sdot sdot + 119883

119899 with an interval

119905 and maximum value 119883max119879 in a time 119879 119899 is the numberof loads 119883 is the intensity of combined load and 119883

119894is the

load intensity of the 119894th loadThemaximum value of119883 in thelifetime of the bridge can be expressed as

119883max119879 = max119879

[1198831+ 1198832+ sdot sdot sdot + 119883

119899]119905 (12)

To simplify the discussion the number of loads is taken astwo the problem being reduced to the prediction of 119883

12=

1198831+ 1198832 where 119883

12is the intensity of the union of 119883

1and

1198832 Assume that 119891

1(119909) and 119891

2(119909) are the probability density

functions of 1198831and 119883

2in the interval 119905 and 119865

1(119909) and 119865

2(119909)

are the cumulative probability functions of 1198831and 119883

2in the

Shock and Vibration 5

HubeiAnhui Hangzhou Bay

Bridge

Zhejiang

Hunan

Jiangxi

Fujian

GuangdongGuangxi

Jimei Bridge

Shenzhen Bay

Bridge

Main city

Xiangyang Xinyang

Zhumadian

LiuanHefei

Nanjing

SuzhouShanghai

Hangzhou

Ningbo

Taizhou

Lishui

Wenzhou

Fuzhou

Nanping

Sanming

Putian

QuanzhouLongyan

Zhangzhou

JieyangShantou

HuizhouGuangzhou

Hong Kong

Qingyuan

Shaoguan

Ganzhou

Jian

Xinyu

YingtanShangrao

QuzhouJingdezhen

Jiujiang

Nanchang

Xianning

Wuhan

Jingzhou

Jingmen

Yichang

Zhangjiajie

Yueyang

Changde

Changsha

XiangtanLoudi

Shaoyang Hengyang

Yongzhou

Chenzhou

Guilin

Wuzhou

Yangjiang

Maoming

Yulin

HighwayRail way State road

Taiwan

Figure 3 Layout of monitoring sites

interval 119905 11989111988312

(119909) and 11986511988312

(119909) are the probability densityfunction and cumulative probability function of 119883

12 The

maximum value of combination in the entire bridge servicelife is defined as119883

12max119879 which is defined as

11988312max119879 = max

119879

[1198831+ 1198832]119905 (13)

The probability density of the combined loads 119883 can beobtained using the convolution integral

11989111988312

(119909) = 1198911lowast 1198912= int

+infin

minusinfin

1198911(120591) sdot 119891

2(119909 minus 120591) 119889120591 (14)

where ldquolowastrdquo is used as convolution symbol

6 Shock and Vibration

0 500 1000 1500

0

05

1

15

2

25

3

35

4

Intensity (kN)

Prob

abili

ty d

ensit

y

times10minus3

Figure 4 Histogram of the monitoring sites

0 200 400 600 800 1000 1200 1400

Density fittedActual density

0

05

1

15

2

25

3

35

4

Prob

abili

ty d

ensit

y

times10minus3

Intensity (kN)

Figure 5 Curve fitting of truck load density

Dirac Delta function is introduced to deal with thecharacteristics of119883 and 119884 in small 119905 interval

120575 (119909) = +infin 119909 = 0

0 119909 = 0(15)

Therefore the probability density can be illustrated as

119891119894(119909) = 119875

1198831198940

120575 (119909) + 1198911015840

119894(119909) 119894 = 1 2 (16)

Note that 1198751198831198940

is the probabilities of 119883119894(119894 = 1 2) in its ldquozero

pointsrdquo namely the events do not happen (eg themaximumtrucks on the bridge are 8 119875

1198831198940

is the probability of no trucks

on the bridge)1198911015840119894(119909) is its probability density functions with-

out ldquozero pointsrdquo Then the cumulative probability functionsof1198831and119883

2can be calculated through

119865119894(119909) = 119875

1198831198940

int

119909

minusinfin

120575 (120591) 119889120591 + int

119909

minusinfin

1198911015840

119894(119909) 119889120591

= 1198751198831198940

+ int

119909

minusinfin

1198911015840

119894(119909) 119889120591

(17)

where 119894 = 1 2Based on (16) and (17) the probability density of 119883 then

can be grouped as

11989111988312

(119909)

= int

+infin

minusinfin

[11987511988310

120575 (120591) + 1198911015840

1(120591)]

times [11987511988320

120575 (119909 minus 120591) + 1198911015840

2(119909 minus 120591)] 119889120591

= int

+infin

minusinfin

[11987511988310

11987511988320

120575 (120591) 120575 (119909 minus 120591)]

+ [11987511988310

120575 (119909) 1198911015840

2(119909 minus 120591)] + [119875

11988320

120575 (119909 minus 120591) 1198911015840

1(120591)]

+ [1198911015840

1(120591) 1198911015840

2(119909 minus 120591)] 119889120591

= 11987511988310

11987511988320

120575 (119909) + 11987511988310

1198911015840

2(119909) + 119875

11988320

1198911015840

1(119909)

+ int

+infin

minusinfin

1198911015840

1(120591) 1198911015840

2(119909 minus 120591) 119889120591

(18)

where based on the characteristic of Dirac Delta functionwhich isint+infin

minusinfin

120575(119909)119889119909 = 1 (18) can be converted to cumulativeprobability function Assume the cumulative probabilityfunction of 119865

11988312

(119909) then

11986511988312

(119909) = int

119909

minusinfin

11989111988312

(119909) 119889119909

= 11987511988310

11987511988320

+ 11987511988310

1198651015840

2(119909) + 119875

11988320

1198651015840

1(119909)

+ int

+infin

minusinfin

1198651015840

1(120591) 1198911015840

2(119909 minus 120591) 119889120591

(19)

From (19) it is clear that the combined load probabilityconsists of four parts events 119883

1and 119883

2are not happening

event 1198831is happening while 119883

2is not happening event 119883

1

is not happening while1198832is happening and119883

11198832are both

happeningTo further simplify the discussion without losing gener-

ality the probability of two loads occurring simultaneously isneglected Thus (19) is simplifying to

11986511988312

(119909) = int

119909

minusinfin

11989111988312

(119909) 119889119909

asymp 11987511988310

11987511988320

+ 11987511988310

1198651015840

2(119909) + 119875

11988320

1198651015840

1(119909)

(20)

Shock and Vibration 7

Then the cumulative probability function 11986511988312max119879

(119909) ofmaximum value of load combinations 119883

12max119879(119909) in time119879 can be obtained

11986511988312max119879

(119909) = [11986511988312

(119909)119905]119879119905

(21)

When the number of loads is more than two and theseloads apply to bridge directly while satisfying Poissonprocess (14) can be extended to 119899 dimensions Assume1198911 1198912 119891

119899are the probability densities functions of

1198831 1198832 119883

119899 1198651 1198652 119865

119899are the cumulative probability

functions of1198831 1198832 119883

119899 11989110158401 1198911015840

2 119891

1015840

119899are the probability

density functions of 1198831 1198832 119883

119899without ldquozerordquo points

119891119883(119909) and 119865

119883(119909) are the probability density function and

cumulative probability functionThen the probability densityfunction is deduced as

119891119883(119909) = 119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) (22)

where the cumulative probability 119865119883max(119909) similar to that

given by (19) is

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

= int

119909

minusinfin

1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) 119889119909

(23)

Equation (23) can account for load combinations of allloads which satisfy the first three assumptions If more thantwo events occurring simultaneously can be neglected (23)reduces to

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

asymp 11987511988310

11987511988320

sdot sdot sdot 1198751198831198990

+ 1198651015840

111987511988320

sdot sdot sdot 1198751198831198990

+ sdot sdot sdot + 11987511988310

11987511988320

sdot sdot sdot 119875119883119899minus10

1198651015840

119899

(24)

Then the maximum value of119883 in time 119879 can be obtained by

119865119883max119879

(119909) = [119865119883(119909)119905]119879119905

(25)

where 119865119883(119909)119905is 119865119883(119909) in 119905 interval 119865

119883max119879(119909) is the cumula-

tive probability function of maximum value of119883 in time 119879Although in our study emphasis is given to formulate

the ldquodemandrdquo to establish load combinations all events mustaddress a capacity issue of the bridge For example theearthquake load and truck load combination on a bridgecolumn can either consider the vertical load or the columnbase shear load Theoretically (23) can deal with most loadcombinations but as more loads are considered a more con-servative design will be adopted Based on the methodologyand assumptions described above the maximum load can becombined and the procedures are summarized as follows

(1) determine truck load and earthquake load distribu-tions over a particular period

(2) using Poisson processes convert earthquake load andtruck load distributions over a particular period to asufficiently small interval 119905

203 cm

60m

19m

442m442m442m

Figure 6 Longitudinal profile of the typical bridge

203 cm

190m

50m

152m

215m143m

215m

600m

Figure 7 Transverse profile of the typical bridge

(3) using (23) earthquake and truck load combinationsover an interval 119905 can be obtained

(4) using (25) the load combinations over an interval 119905can be converted to the bridge service life interval 119879

5 Numerical Examples

Example 1 Using the method of load combination describedin the preceding section a simple example of horizontal loadcombination is presented here Profiles of the typical bridgeare shown in Figures 6 and 7Theweight of the superstructureat each column is 538 tons the eccentricity of truck load is 50meters and the effects of soil and secondary effects of gravityare ignored Furthermore it is assumed that the maximumnumber of trucks on one lane is two The results are given inFigures 8 9 10 11 12 13 and 14 The interval 119905 is 10 secondsand the average daily truck traffic (ADTT) is about 1947

8 Shock and Vibration

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

ytimes10

minus4

Intensity (kN-M)

(a)

0 2000 4000 6000 8000 10000

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty

Intensity (kN-M)

(b)

Figure 8 Probability curve of each truck load effect

1 truck passing2 trucks passing

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M)

Figure 9 Truck load probability density curve for varied number oftrucks

Truck and earthquake load effects are the base momentcaused by trucks and earthquakes respectively Figure 8shows the probability curves of each truck load effectwhich has a similar shape to truck load Figure 9 shows thetruck load probability density curve for varied numbers oftrucks From Figure 10 we can see that over the interval 119905the probability of no truck on the bridge is much largerthan the other number of trucks passing Figure 11 showsthe combined probability curves for truck load over anearthquake load duration which indicate that the truck loadover an earthquake load duration is larger than each truckload

0 1 2

0

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 10 Probability of passing truck number simultaneously in 119905interval on the bridge

From the results shown in Figure 13 it can be seen thatcurve C and curve A have a similar shape and curve C isdisplaced to the right This means that mode and mean valueof the truck and earthquake load combination in this exampleis larger than those of truck load and also illustrates that truckload is more important to bridge design in this area FromFigure 14 we can seemore clearly that overmost ranges truckload is larger than earthquake load Because the truck loadhas a smaller tail their combination in the tail is close to theearthquake load Curve D of loads combined directly over100 years is further away from curve C which means thecombinations of truck load and earthquake load directly over100 years give a much larger value and it is not suggested thatthe bridge design uses the curve D method

Shock and Vibration 9

0

05

1

15

2

25

3

35

Prob

abili

ty d

ensit

y

times10minus4

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(a)

05

055

06

065

07

075

08

085

09

095

1

Cum

ulat

ive p

roba

bilit

y

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(b)

Figure 11 Combined probability curves for truck load in earthquake load duration

05 1 15 2 25

0

1

2

3

4

5

6

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M) times104

(a)

2000 4000 6000 8000 10000 12000 14000

0

01

02

03

04

05

06

07

08

09

1

Cum

ulat

ive p

roba

bilit

y

Intensity (kN-M)

(b)

Figure 12 Probability curves for earthquake load in 100 years

Example 2 Example 2 illustrates vertical load combinationsMost of the configurations are the same as used in Example 1The difference is in the maximum number of trucks namelythe maximum number of trucks on one lane in this exampleis four The results are shown in Figures 15 16 17 18 19 and20 Figures 13 and 14 are the results of truck load Note that inthis example the 119905 interval is 10 seconds and 119879 is taken as 100years

Figures 15 to 16 show vertical truck load probabilitycurves for varied numbers of trucks and probabilities ofpassing truck numbers on the bridge over the interval 119905From Figure 16 it can be seen that the probability is verylow when the maximum number of trucks on one lane isfour Figures 17 and 18 show similar curve shapes with those

in Figures 13 and 14 which indicate that there is the samerule in load combinations in both the horizontal and verticaldirections Comparing Figures 17 to 20 though themaximumnumber of trucks is four because dead load is combinedwith truck and earthquake load in the gravity direction thedead load contributes a substantial portion in vertical loadcombinations

6 Conclusions

This paper describes a method to combine earthquake loadand truck load in the service life of bridges The followingconclusions can be drawn

(1) Given the more than 70 seismic areas in Chinaearthquake load is a main consideration for bridge

10 Shock and Vibration

16000

05

1

15

2

2000 4000 6000 8000 10000 12000 14000

Prob

abili

ty d

ensit

y

times10minus3

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN-M)

Figure 13 Probability density of load combination in 100 years

05 1 15 2

05

055

06

065

07

075

08

085

09

095

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Cum

ulat

ive p

roba

bilit

y

times104Intensity (kN-M)

Figure 14 Cumulative probability of load combination in 100 years

design in the southeast coastal areas of China Theearthquake load probability curve is obtained usingseismic risk analysis

(2) Using measured truck load data from BHMS multi-modal characteristics of truck load are analyzed Thetruck load density of each truck is obtained by curvefitting Considering that truck load may consist ofvarying numbers of trucks truck load is calculatedthrough traffic analysis

(3) In thismethod themaximumvalue of combined loadis defined as 119883max119879 = max

119879[1198831+ 1198832+ sdot sdot sdot 119883

119899]119905

which means (1198831+ 1198832+ sdot sdot sdot 119883

119899) over the interval 119905

is first combined and then the maximum value in the

0 2000 4000 6000 8000 100000

05

1

15

2

25

3

35

4

1 truck passing2 trucks passing

3 trucks passing4 trucks passing

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 15 Vertical truck load probability density curves for variednumber of trucks

0 1 2 3 40

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 16 Probability of passing truck number simultaneously in 119905interval on the bridge with maximum number 4

bridge service life is determined 119883max119879 is based onprobability which covers all the probability combi-nations of the combined situations In this method aDiracDelta function is introduced to deal with119883 overa small interval 119905 To demonstrate the methodologyintuitively examples of load combinations in horizon-tal and vertical directions are provided

(4) The shape of the earthquake and truck load com-bination is similar to that of truck load alone butthe curve is displaced to the right which means themode and mean value of truck and earthquake loadcombination in this example is larger than that oftruck load aloneThis also illustrates that truck load is

Shock and Vibration 11

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 17 Probability density of vertical load combination in 100years

0 1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Figure 18 Cumulative probability of vertical load combination in100 years

more sensitive to bridge design and over most rangestruck load is larger than earthquake load in this area

(5) The curve from direct load combined over 100 yearsis further away from the curve obtained using themethod in the paper with the direct combinationof truck load and earthquake load over 100 yearsgiving much larger values It is not suggested that thismethod be used in bridge design considerations

(6) Because dead load is combined with truck and earth-quake load along the direction of gravity the dead

0 2000 4000 6000 8000 10000 12000 140000

1

2

3

4

5

6

7

8

Dead loadTruck and earthquakeTruck and earthquake and dead load

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 19 Probability density of dead truck and earthquake loadcombination

0 2000 4000 6000 8000 10000 12000 140000

02

04

06

08

1

12

14

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Dead loadTruck and earthquakeTruck and earthquake and dead load

Figure 20 Probability curves of dead truck and earthquake loadcombination

load contributes a substantial portion in vertical loadcombinations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Shock and Vibration

Acknowledgments

This study is jointly funded by Basic Institute ScientificResearch Fund (Grant no 2012A02) the National NaturalScience Fund of China (NSFC) (Grant no 51308510) andOpen Fund of State Key Laboratory Breeding Base of Moun-tainBridge andTunnel Engineering (Grant no CQSLBF-Y14-15) The results and conclusions presented in the paper areof the authors and do not necessarily reflect the view of thesponsors

References

[1] P R China Ministry of Communications ldquoGeneral code fordesign of highway bridges and culvertsrdquo Tech Rep JTG D60-200 China Communications Press Beijing China 2004

[2] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[3] C J Turkstra and H O Madsen ldquoLoad combinations incodified structural designrdquo Journal of Structural Engineeringvol 106 no 12 pp 2527ndash2543 1980

[4] ACI Publication SP-31 Probabilistic Design of Reinforced Con-crete Buildings American Concrete Institute Detroit MichUSA 1971

[5] B Ellingwood Development of a Probability Based Load Cri-terion for American National Standard A58-Building CodeRequirements forMinimumDesign Loads in Buildings andOtherStructures US Department of CommerceNational Bureau ofStandards 1980

[6] Y K Wen Development of Reliability-Based Design Criteria forBuildings Under Seismic Load National Center for EarthquakeEngineering Research Buffalo NY USA 1994

[7] Y-K Wen ldquoStatistical combination of extreme loadsrdquo Journalof Structural Engineering vol 103 no 5 pp 1079ndash1093 1977

[8] M Ghosn F Moses and J Wang ldquoDesign of highway bridgesfor extreme eventsrdquo NCHRP 489 Transportation ResearchBoard National Research Council Washington D C USA2003

[9] S E Hida ldquoStatistical significance of less common load combi-nationsrdquo Journal of Bridge Engineering vol 12 no 3 pp 389ndash393 2007

[10] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loadsmdashpart I comprehensive reliability and par-tial failure probabilitiesrdquo Journal of Earthquake Engineering andEngineering Vibration vol 11 no 3 pp 293ndash301 2012

[11] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loads Part II Examples for live and earthquakeload effectsrdquo Journal of Earthquake Engineering and EngineeringVibration vol 11 no 3 pp 303ndash311 2012

[12] E EMatheu D E Yule and R V Kala ldquoDetermination of stan-dard response spectra and effective peak ground accelerationsfor seismic design and evaluationrdquo Tech Rep no ERDCCHLCHETN-VI-41 2005

[13] HN Li TH YiMGu and LHuo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009

[14] B Chen Z W Chen Y Z Sun and S L Zhao ldquoConditionassessment on thermal effects of a suspension bridge basedon SHM oriented model and datardquo Mathematical Problems inEngineering vol 2013 Article ID 256816 18 pages 2013

[15] T-H Yi H-N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013

[16] Y Xia B Chen X Q Zhou and Y L Xu ldquoFieldmonitoring andnumerical analysis of Tsing Ma suspension bridge temperaturebehaviorrdquo Structural Control and HealthMonitoring vol 20 no4 pp 560ndash575 2013

[17] G Fu and J You ldquoExtrapolation for future maximum loadstatisticsrdquo Journal of Bridge Engineering vol 16 no 4 pp 527ndash535 2011

[18] A S Nowak ldquoCalibration of LRFD bridge design coderdquoNCHRPReport TransportationResearchBoard of theNationalAcademies Washington DC USA 1999

[19] F Moses ldquoCalibration of load factors for LRFD bridge evalua-tionsrdquo NCHRP Report 454 Transportation Research Board ofthe National Academies Washington DC USA 2001

[20] J Zhao and H Tabatabai ldquoEvaluation of a permit vehiclemodel using weigh-in-motion truck recordsrdquo Journal of BridgeEngineering vol 17 no 2 pp 389ndash392 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

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RotatingMachinery

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 5

HubeiAnhui Hangzhou Bay

Bridge

Zhejiang

Hunan

Jiangxi

Fujian

GuangdongGuangxi

Jimei Bridge

Shenzhen Bay

Bridge

Main city

Xiangyang Xinyang

Zhumadian

LiuanHefei

Nanjing

SuzhouShanghai

Hangzhou

Ningbo

Taizhou

Lishui

Wenzhou

Fuzhou

Nanping

Sanming

Putian

QuanzhouLongyan

Zhangzhou

JieyangShantou

HuizhouGuangzhou

Hong Kong

Qingyuan

Shaoguan

Ganzhou

Jian

Xinyu

YingtanShangrao

QuzhouJingdezhen

Jiujiang

Nanchang

Xianning

Wuhan

Jingzhou

Jingmen

Yichang

Zhangjiajie

Yueyang

Changde

Changsha

XiangtanLoudi

Shaoyang Hengyang

Yongzhou

Chenzhou

Guilin

Wuzhou

Yangjiang

Maoming

Yulin

HighwayRail way State road

Taiwan

Figure 3 Layout of monitoring sites

interval 119905 11989111988312

(119909) and 11986511988312

(119909) are the probability densityfunction and cumulative probability function of 119883

12 The

maximum value of combination in the entire bridge servicelife is defined as119883

12max119879 which is defined as

11988312max119879 = max

119879

[1198831+ 1198832]119905 (13)

The probability density of the combined loads 119883 can beobtained using the convolution integral

11989111988312

(119909) = 1198911lowast 1198912= int

+infin

minusinfin

1198911(120591) sdot 119891

2(119909 minus 120591) 119889120591 (14)

where ldquolowastrdquo is used as convolution symbol

6 Shock and Vibration

0 500 1000 1500

0

05

1

15

2

25

3

35

4

Intensity (kN)

Prob

abili

ty d

ensit

y

times10minus3

Figure 4 Histogram of the monitoring sites

0 200 400 600 800 1000 1200 1400

Density fittedActual density

0

05

1

15

2

25

3

35

4

Prob

abili

ty d

ensit

y

times10minus3

Intensity (kN)

Figure 5 Curve fitting of truck load density

Dirac Delta function is introduced to deal with thecharacteristics of119883 and 119884 in small 119905 interval

120575 (119909) = +infin 119909 = 0

0 119909 = 0(15)

Therefore the probability density can be illustrated as

119891119894(119909) = 119875

1198831198940

120575 (119909) + 1198911015840

119894(119909) 119894 = 1 2 (16)

Note that 1198751198831198940

is the probabilities of 119883119894(119894 = 1 2) in its ldquozero

pointsrdquo namely the events do not happen (eg themaximumtrucks on the bridge are 8 119875

1198831198940

is the probability of no trucks

on the bridge)1198911015840119894(119909) is its probability density functions with-

out ldquozero pointsrdquo Then the cumulative probability functionsof1198831and119883

2can be calculated through

119865119894(119909) = 119875

1198831198940

int

119909

minusinfin

120575 (120591) 119889120591 + int

119909

minusinfin

1198911015840

119894(119909) 119889120591

= 1198751198831198940

+ int

119909

minusinfin

1198911015840

119894(119909) 119889120591

(17)

where 119894 = 1 2Based on (16) and (17) the probability density of 119883 then

can be grouped as

11989111988312

(119909)

= int

+infin

minusinfin

[11987511988310

120575 (120591) + 1198911015840

1(120591)]

times [11987511988320

120575 (119909 minus 120591) + 1198911015840

2(119909 minus 120591)] 119889120591

= int

+infin

minusinfin

[11987511988310

11987511988320

120575 (120591) 120575 (119909 minus 120591)]

+ [11987511988310

120575 (119909) 1198911015840

2(119909 minus 120591)] + [119875

11988320

120575 (119909 minus 120591) 1198911015840

1(120591)]

+ [1198911015840

1(120591) 1198911015840

2(119909 minus 120591)] 119889120591

= 11987511988310

11987511988320

120575 (119909) + 11987511988310

1198911015840

2(119909) + 119875

11988320

1198911015840

1(119909)

+ int

+infin

minusinfin

1198911015840

1(120591) 1198911015840

2(119909 minus 120591) 119889120591

(18)

where based on the characteristic of Dirac Delta functionwhich isint+infin

minusinfin

120575(119909)119889119909 = 1 (18) can be converted to cumulativeprobability function Assume the cumulative probabilityfunction of 119865

11988312

(119909) then

11986511988312

(119909) = int

119909

minusinfin

11989111988312

(119909) 119889119909

= 11987511988310

11987511988320

+ 11987511988310

1198651015840

2(119909) + 119875

11988320

1198651015840

1(119909)

+ int

+infin

minusinfin

1198651015840

1(120591) 1198911015840

2(119909 minus 120591) 119889120591

(19)

From (19) it is clear that the combined load probabilityconsists of four parts events 119883

1and 119883

2are not happening

event 1198831is happening while 119883

2is not happening event 119883

1

is not happening while1198832is happening and119883

11198832are both

happeningTo further simplify the discussion without losing gener-

ality the probability of two loads occurring simultaneously isneglected Thus (19) is simplifying to

11986511988312

(119909) = int

119909

minusinfin

11989111988312

(119909) 119889119909

asymp 11987511988310

11987511988320

+ 11987511988310

1198651015840

2(119909) + 119875

11988320

1198651015840

1(119909)

(20)

Shock and Vibration 7

Then the cumulative probability function 11986511988312max119879

(119909) ofmaximum value of load combinations 119883

12max119879(119909) in time119879 can be obtained

11986511988312max119879

(119909) = [11986511988312

(119909)119905]119879119905

(21)

When the number of loads is more than two and theseloads apply to bridge directly while satisfying Poissonprocess (14) can be extended to 119899 dimensions Assume1198911 1198912 119891

119899are the probability densities functions of

1198831 1198832 119883

119899 1198651 1198652 119865

119899are the cumulative probability

functions of1198831 1198832 119883

119899 11989110158401 1198911015840

2 119891

1015840

119899are the probability

density functions of 1198831 1198832 119883

119899without ldquozerordquo points

119891119883(119909) and 119865

119883(119909) are the probability density function and

cumulative probability functionThen the probability densityfunction is deduced as

119891119883(119909) = 119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) (22)

where the cumulative probability 119865119883max(119909) similar to that

given by (19) is

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

= int

119909

minusinfin

1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) 119889119909

(23)

Equation (23) can account for load combinations of allloads which satisfy the first three assumptions If more thantwo events occurring simultaneously can be neglected (23)reduces to

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

asymp 11987511988310

11987511988320

sdot sdot sdot 1198751198831198990

+ 1198651015840

111987511988320

sdot sdot sdot 1198751198831198990

+ sdot sdot sdot + 11987511988310

11987511988320

sdot sdot sdot 119875119883119899minus10

1198651015840

119899

(24)

Then the maximum value of119883 in time 119879 can be obtained by

119865119883max119879

(119909) = [119865119883(119909)119905]119879119905

(25)

where 119865119883(119909)119905is 119865119883(119909) in 119905 interval 119865

119883max119879(119909) is the cumula-

tive probability function of maximum value of119883 in time 119879Although in our study emphasis is given to formulate

the ldquodemandrdquo to establish load combinations all events mustaddress a capacity issue of the bridge For example theearthquake load and truck load combination on a bridgecolumn can either consider the vertical load or the columnbase shear load Theoretically (23) can deal with most loadcombinations but as more loads are considered a more con-servative design will be adopted Based on the methodologyand assumptions described above the maximum load can becombined and the procedures are summarized as follows

(1) determine truck load and earthquake load distribu-tions over a particular period

(2) using Poisson processes convert earthquake load andtruck load distributions over a particular period to asufficiently small interval 119905

203 cm

60m

19m

442m442m442m

Figure 6 Longitudinal profile of the typical bridge

203 cm

190m

50m

152m

215m143m

215m

600m

Figure 7 Transverse profile of the typical bridge

(3) using (23) earthquake and truck load combinationsover an interval 119905 can be obtained

(4) using (25) the load combinations over an interval 119905can be converted to the bridge service life interval 119879

5 Numerical Examples

Example 1 Using the method of load combination describedin the preceding section a simple example of horizontal loadcombination is presented here Profiles of the typical bridgeare shown in Figures 6 and 7Theweight of the superstructureat each column is 538 tons the eccentricity of truck load is 50meters and the effects of soil and secondary effects of gravityare ignored Furthermore it is assumed that the maximumnumber of trucks on one lane is two The results are given inFigures 8 9 10 11 12 13 and 14 The interval 119905 is 10 secondsand the average daily truck traffic (ADTT) is about 1947

8 Shock and Vibration

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

ytimes10

minus4

Intensity (kN-M)

(a)

0 2000 4000 6000 8000 10000

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty

Intensity (kN-M)

(b)

Figure 8 Probability curve of each truck load effect

1 truck passing2 trucks passing

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M)

Figure 9 Truck load probability density curve for varied number oftrucks

Truck and earthquake load effects are the base momentcaused by trucks and earthquakes respectively Figure 8shows the probability curves of each truck load effectwhich has a similar shape to truck load Figure 9 shows thetruck load probability density curve for varied numbers oftrucks From Figure 10 we can see that over the interval 119905the probability of no truck on the bridge is much largerthan the other number of trucks passing Figure 11 showsthe combined probability curves for truck load over anearthquake load duration which indicate that the truck loadover an earthquake load duration is larger than each truckload

0 1 2

0

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 10 Probability of passing truck number simultaneously in 119905interval on the bridge

From the results shown in Figure 13 it can be seen thatcurve C and curve A have a similar shape and curve C isdisplaced to the right This means that mode and mean valueof the truck and earthquake load combination in this exampleis larger than those of truck load and also illustrates that truckload is more important to bridge design in this area FromFigure 14 we can seemore clearly that overmost ranges truckload is larger than earthquake load Because the truck loadhas a smaller tail their combination in the tail is close to theearthquake load Curve D of loads combined directly over100 years is further away from curve C which means thecombinations of truck load and earthquake load directly over100 years give a much larger value and it is not suggested thatthe bridge design uses the curve D method

Shock and Vibration 9

0

05

1

15

2

25

3

35

Prob

abili

ty d

ensit

y

times10minus4

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(a)

05

055

06

065

07

075

08

085

09

095

1

Cum

ulat

ive p

roba

bilit

y

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(b)

Figure 11 Combined probability curves for truck load in earthquake load duration

05 1 15 2 25

0

1

2

3

4

5

6

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M) times104

(a)

2000 4000 6000 8000 10000 12000 14000

0

01

02

03

04

05

06

07

08

09

1

Cum

ulat

ive p

roba

bilit

y

Intensity (kN-M)

(b)

Figure 12 Probability curves for earthquake load in 100 years

Example 2 Example 2 illustrates vertical load combinationsMost of the configurations are the same as used in Example 1The difference is in the maximum number of trucks namelythe maximum number of trucks on one lane in this exampleis four The results are shown in Figures 15 16 17 18 19 and20 Figures 13 and 14 are the results of truck load Note that inthis example the 119905 interval is 10 seconds and 119879 is taken as 100years

Figures 15 to 16 show vertical truck load probabilitycurves for varied numbers of trucks and probabilities ofpassing truck numbers on the bridge over the interval 119905From Figure 16 it can be seen that the probability is verylow when the maximum number of trucks on one lane isfour Figures 17 and 18 show similar curve shapes with those

in Figures 13 and 14 which indicate that there is the samerule in load combinations in both the horizontal and verticaldirections Comparing Figures 17 to 20 though themaximumnumber of trucks is four because dead load is combinedwith truck and earthquake load in the gravity direction thedead load contributes a substantial portion in vertical loadcombinations

6 Conclusions

This paper describes a method to combine earthquake loadand truck load in the service life of bridges The followingconclusions can be drawn

(1) Given the more than 70 seismic areas in Chinaearthquake load is a main consideration for bridge

10 Shock and Vibration

16000

05

1

15

2

2000 4000 6000 8000 10000 12000 14000

Prob

abili

ty d

ensit

y

times10minus3

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN-M)

Figure 13 Probability density of load combination in 100 years

05 1 15 2

05

055

06

065

07

075

08

085

09

095

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Cum

ulat

ive p

roba

bilit

y

times104Intensity (kN-M)

Figure 14 Cumulative probability of load combination in 100 years

design in the southeast coastal areas of China Theearthquake load probability curve is obtained usingseismic risk analysis

(2) Using measured truck load data from BHMS multi-modal characteristics of truck load are analyzed Thetruck load density of each truck is obtained by curvefitting Considering that truck load may consist ofvarying numbers of trucks truck load is calculatedthrough traffic analysis

(3) In thismethod themaximumvalue of combined loadis defined as 119883max119879 = max

119879[1198831+ 1198832+ sdot sdot sdot 119883

119899]119905

which means (1198831+ 1198832+ sdot sdot sdot 119883

119899) over the interval 119905

is first combined and then the maximum value in the

0 2000 4000 6000 8000 100000

05

1

15

2

25

3

35

4

1 truck passing2 trucks passing

3 trucks passing4 trucks passing

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 15 Vertical truck load probability density curves for variednumber of trucks

0 1 2 3 40

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 16 Probability of passing truck number simultaneously in 119905interval on the bridge with maximum number 4

bridge service life is determined 119883max119879 is based onprobability which covers all the probability combi-nations of the combined situations In this method aDiracDelta function is introduced to deal with119883 overa small interval 119905 To demonstrate the methodologyintuitively examples of load combinations in horizon-tal and vertical directions are provided

(4) The shape of the earthquake and truck load com-bination is similar to that of truck load alone butthe curve is displaced to the right which means themode and mean value of truck and earthquake loadcombination in this example is larger than that oftruck load aloneThis also illustrates that truck load is

Shock and Vibration 11

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 17 Probability density of vertical load combination in 100years

0 1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Figure 18 Cumulative probability of vertical load combination in100 years

more sensitive to bridge design and over most rangestruck load is larger than earthquake load in this area

(5) The curve from direct load combined over 100 yearsis further away from the curve obtained using themethod in the paper with the direct combinationof truck load and earthquake load over 100 yearsgiving much larger values It is not suggested that thismethod be used in bridge design considerations

(6) Because dead load is combined with truck and earth-quake load along the direction of gravity the dead

0 2000 4000 6000 8000 10000 12000 140000

1

2

3

4

5

6

7

8

Dead loadTruck and earthquakeTruck and earthquake and dead load

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 19 Probability density of dead truck and earthquake loadcombination

0 2000 4000 6000 8000 10000 12000 140000

02

04

06

08

1

12

14

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Dead loadTruck and earthquakeTruck and earthquake and dead load

Figure 20 Probability curves of dead truck and earthquake loadcombination

load contributes a substantial portion in vertical loadcombinations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Shock and Vibration

Acknowledgments

This study is jointly funded by Basic Institute ScientificResearch Fund (Grant no 2012A02) the National NaturalScience Fund of China (NSFC) (Grant no 51308510) andOpen Fund of State Key Laboratory Breeding Base of Moun-tainBridge andTunnel Engineering (Grant no CQSLBF-Y14-15) The results and conclusions presented in the paper areof the authors and do not necessarily reflect the view of thesponsors

References

[1] P R China Ministry of Communications ldquoGeneral code fordesign of highway bridges and culvertsrdquo Tech Rep JTG D60-200 China Communications Press Beijing China 2004

[2] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[3] C J Turkstra and H O Madsen ldquoLoad combinations incodified structural designrdquo Journal of Structural Engineeringvol 106 no 12 pp 2527ndash2543 1980

[4] ACI Publication SP-31 Probabilistic Design of Reinforced Con-crete Buildings American Concrete Institute Detroit MichUSA 1971

[5] B Ellingwood Development of a Probability Based Load Cri-terion for American National Standard A58-Building CodeRequirements forMinimumDesign Loads in Buildings andOtherStructures US Department of CommerceNational Bureau ofStandards 1980

[6] Y K Wen Development of Reliability-Based Design Criteria forBuildings Under Seismic Load National Center for EarthquakeEngineering Research Buffalo NY USA 1994

[7] Y-K Wen ldquoStatistical combination of extreme loadsrdquo Journalof Structural Engineering vol 103 no 5 pp 1079ndash1093 1977

[8] M Ghosn F Moses and J Wang ldquoDesign of highway bridgesfor extreme eventsrdquo NCHRP 489 Transportation ResearchBoard National Research Council Washington D C USA2003

[9] S E Hida ldquoStatistical significance of less common load combi-nationsrdquo Journal of Bridge Engineering vol 12 no 3 pp 389ndash393 2007

[10] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loadsmdashpart I comprehensive reliability and par-tial failure probabilitiesrdquo Journal of Earthquake Engineering andEngineering Vibration vol 11 no 3 pp 293ndash301 2012

[11] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loads Part II Examples for live and earthquakeload effectsrdquo Journal of Earthquake Engineering and EngineeringVibration vol 11 no 3 pp 303ndash311 2012

[12] E EMatheu D E Yule and R V Kala ldquoDetermination of stan-dard response spectra and effective peak ground accelerationsfor seismic design and evaluationrdquo Tech Rep no ERDCCHLCHETN-VI-41 2005

[13] HN Li TH YiMGu and LHuo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009

[14] B Chen Z W Chen Y Z Sun and S L Zhao ldquoConditionassessment on thermal effects of a suspension bridge basedon SHM oriented model and datardquo Mathematical Problems inEngineering vol 2013 Article ID 256816 18 pages 2013

[15] T-H Yi H-N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013

[16] Y Xia B Chen X Q Zhou and Y L Xu ldquoFieldmonitoring andnumerical analysis of Tsing Ma suspension bridge temperaturebehaviorrdquo Structural Control and HealthMonitoring vol 20 no4 pp 560ndash575 2013

[17] G Fu and J You ldquoExtrapolation for future maximum loadstatisticsrdquo Journal of Bridge Engineering vol 16 no 4 pp 527ndash535 2011

[18] A S Nowak ldquoCalibration of LRFD bridge design coderdquoNCHRPReport TransportationResearchBoard of theNationalAcademies Washington DC USA 1999

[19] F Moses ldquoCalibration of load factors for LRFD bridge evalua-tionsrdquo NCHRP Report 454 Transportation Research Board ofthe National Academies Washington DC USA 2001

[20] J Zhao and H Tabatabai ldquoEvaluation of a permit vehiclemodel using weigh-in-motion truck recordsrdquo Journal of BridgeEngineering vol 17 no 2 pp 389ndash392 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

0 500 1000 1500

0

05

1

15

2

25

3

35

4

Intensity (kN)

Prob

abili

ty d

ensit

y

times10minus3

Figure 4 Histogram of the monitoring sites

0 200 400 600 800 1000 1200 1400

Density fittedActual density

0

05

1

15

2

25

3

35

4

Prob

abili

ty d

ensit

y

times10minus3

Intensity (kN)

Figure 5 Curve fitting of truck load density

Dirac Delta function is introduced to deal with thecharacteristics of119883 and 119884 in small 119905 interval

120575 (119909) = +infin 119909 = 0

0 119909 = 0(15)

Therefore the probability density can be illustrated as

119891119894(119909) = 119875

1198831198940

120575 (119909) + 1198911015840

119894(119909) 119894 = 1 2 (16)

Note that 1198751198831198940

is the probabilities of 119883119894(119894 = 1 2) in its ldquozero

pointsrdquo namely the events do not happen (eg themaximumtrucks on the bridge are 8 119875

1198831198940

is the probability of no trucks

on the bridge)1198911015840119894(119909) is its probability density functions with-

out ldquozero pointsrdquo Then the cumulative probability functionsof1198831and119883

2can be calculated through

119865119894(119909) = 119875

1198831198940

int

119909

minusinfin

120575 (120591) 119889120591 + int

119909

minusinfin

1198911015840

119894(119909) 119889120591

= 1198751198831198940

+ int

119909

minusinfin

1198911015840

119894(119909) 119889120591

(17)

where 119894 = 1 2Based on (16) and (17) the probability density of 119883 then

can be grouped as

11989111988312

(119909)

= int

+infin

minusinfin

[11987511988310

120575 (120591) + 1198911015840

1(120591)]

times [11987511988320

120575 (119909 minus 120591) + 1198911015840

2(119909 minus 120591)] 119889120591

= int

+infin

minusinfin

[11987511988310

11987511988320

120575 (120591) 120575 (119909 minus 120591)]

+ [11987511988310

120575 (119909) 1198911015840

2(119909 minus 120591)] + [119875

11988320

120575 (119909 minus 120591) 1198911015840

1(120591)]

+ [1198911015840

1(120591) 1198911015840

2(119909 minus 120591)] 119889120591

= 11987511988310

11987511988320

120575 (119909) + 11987511988310

1198911015840

2(119909) + 119875

11988320

1198911015840

1(119909)

+ int

+infin

minusinfin

1198911015840

1(120591) 1198911015840

2(119909 minus 120591) 119889120591

(18)

where based on the characteristic of Dirac Delta functionwhich isint+infin

minusinfin

120575(119909)119889119909 = 1 (18) can be converted to cumulativeprobability function Assume the cumulative probabilityfunction of 119865

11988312

(119909) then

11986511988312

(119909) = int

119909

minusinfin

11989111988312

(119909) 119889119909

= 11987511988310

11987511988320

+ 11987511988310

1198651015840

2(119909) + 119875

11988320

1198651015840

1(119909)

+ int

+infin

minusinfin

1198651015840

1(120591) 1198911015840

2(119909 minus 120591) 119889120591

(19)

From (19) it is clear that the combined load probabilityconsists of four parts events 119883

1and 119883

2are not happening

event 1198831is happening while 119883

2is not happening event 119883

1

is not happening while1198832is happening and119883

11198832are both

happeningTo further simplify the discussion without losing gener-

ality the probability of two loads occurring simultaneously isneglected Thus (19) is simplifying to

11986511988312

(119909) = int

119909

minusinfin

11989111988312

(119909) 119889119909

asymp 11987511988310

11987511988320

+ 11987511988310

1198651015840

2(119909) + 119875

11988320

1198651015840

1(119909)

(20)

Shock and Vibration 7

Then the cumulative probability function 11986511988312max119879

(119909) ofmaximum value of load combinations 119883

12max119879(119909) in time119879 can be obtained

11986511988312max119879

(119909) = [11986511988312

(119909)119905]119879119905

(21)

When the number of loads is more than two and theseloads apply to bridge directly while satisfying Poissonprocess (14) can be extended to 119899 dimensions Assume1198911 1198912 119891

119899are the probability densities functions of

1198831 1198832 119883

119899 1198651 1198652 119865

119899are the cumulative probability

functions of1198831 1198832 119883

119899 11989110158401 1198911015840

2 119891

1015840

119899are the probability

density functions of 1198831 1198832 119883

119899without ldquozerordquo points

119891119883(119909) and 119865

119883(119909) are the probability density function and

cumulative probability functionThen the probability densityfunction is deduced as

119891119883(119909) = 119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) (22)

where the cumulative probability 119865119883max(119909) similar to that

given by (19) is

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

= int

119909

minusinfin

1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) 119889119909

(23)

Equation (23) can account for load combinations of allloads which satisfy the first three assumptions If more thantwo events occurring simultaneously can be neglected (23)reduces to

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

asymp 11987511988310

11987511988320

sdot sdot sdot 1198751198831198990

+ 1198651015840

111987511988320

sdot sdot sdot 1198751198831198990

+ sdot sdot sdot + 11987511988310

11987511988320

sdot sdot sdot 119875119883119899minus10

1198651015840

119899

(24)

Then the maximum value of119883 in time 119879 can be obtained by

119865119883max119879

(119909) = [119865119883(119909)119905]119879119905

(25)

where 119865119883(119909)119905is 119865119883(119909) in 119905 interval 119865

119883max119879(119909) is the cumula-

tive probability function of maximum value of119883 in time 119879Although in our study emphasis is given to formulate

the ldquodemandrdquo to establish load combinations all events mustaddress a capacity issue of the bridge For example theearthquake load and truck load combination on a bridgecolumn can either consider the vertical load or the columnbase shear load Theoretically (23) can deal with most loadcombinations but as more loads are considered a more con-servative design will be adopted Based on the methodologyand assumptions described above the maximum load can becombined and the procedures are summarized as follows

(1) determine truck load and earthquake load distribu-tions over a particular period

(2) using Poisson processes convert earthquake load andtruck load distributions over a particular period to asufficiently small interval 119905

203 cm

60m

19m

442m442m442m

Figure 6 Longitudinal profile of the typical bridge

203 cm

190m

50m

152m

215m143m

215m

600m

Figure 7 Transverse profile of the typical bridge

(3) using (23) earthquake and truck load combinationsover an interval 119905 can be obtained

(4) using (25) the load combinations over an interval 119905can be converted to the bridge service life interval 119879

5 Numerical Examples

Example 1 Using the method of load combination describedin the preceding section a simple example of horizontal loadcombination is presented here Profiles of the typical bridgeare shown in Figures 6 and 7Theweight of the superstructureat each column is 538 tons the eccentricity of truck load is 50meters and the effects of soil and secondary effects of gravityare ignored Furthermore it is assumed that the maximumnumber of trucks on one lane is two The results are given inFigures 8 9 10 11 12 13 and 14 The interval 119905 is 10 secondsand the average daily truck traffic (ADTT) is about 1947

8 Shock and Vibration

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

ytimes10

minus4

Intensity (kN-M)

(a)

0 2000 4000 6000 8000 10000

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty

Intensity (kN-M)

(b)

Figure 8 Probability curve of each truck load effect

1 truck passing2 trucks passing

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M)

Figure 9 Truck load probability density curve for varied number oftrucks

Truck and earthquake load effects are the base momentcaused by trucks and earthquakes respectively Figure 8shows the probability curves of each truck load effectwhich has a similar shape to truck load Figure 9 shows thetruck load probability density curve for varied numbers oftrucks From Figure 10 we can see that over the interval 119905the probability of no truck on the bridge is much largerthan the other number of trucks passing Figure 11 showsthe combined probability curves for truck load over anearthquake load duration which indicate that the truck loadover an earthquake load duration is larger than each truckload

0 1 2

0

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 10 Probability of passing truck number simultaneously in 119905interval on the bridge

From the results shown in Figure 13 it can be seen thatcurve C and curve A have a similar shape and curve C isdisplaced to the right This means that mode and mean valueof the truck and earthquake load combination in this exampleis larger than those of truck load and also illustrates that truckload is more important to bridge design in this area FromFigure 14 we can seemore clearly that overmost ranges truckload is larger than earthquake load Because the truck loadhas a smaller tail their combination in the tail is close to theearthquake load Curve D of loads combined directly over100 years is further away from curve C which means thecombinations of truck load and earthquake load directly over100 years give a much larger value and it is not suggested thatthe bridge design uses the curve D method

Shock and Vibration 9

0

05

1

15

2

25

3

35

Prob

abili

ty d

ensit

y

times10minus4

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(a)

05

055

06

065

07

075

08

085

09

095

1

Cum

ulat

ive p

roba

bilit

y

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(b)

Figure 11 Combined probability curves for truck load in earthquake load duration

05 1 15 2 25

0

1

2

3

4

5

6

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M) times104

(a)

2000 4000 6000 8000 10000 12000 14000

0

01

02

03

04

05

06

07

08

09

1

Cum

ulat

ive p

roba

bilit

y

Intensity (kN-M)

(b)

Figure 12 Probability curves for earthquake load in 100 years

Example 2 Example 2 illustrates vertical load combinationsMost of the configurations are the same as used in Example 1The difference is in the maximum number of trucks namelythe maximum number of trucks on one lane in this exampleis four The results are shown in Figures 15 16 17 18 19 and20 Figures 13 and 14 are the results of truck load Note that inthis example the 119905 interval is 10 seconds and 119879 is taken as 100years

Figures 15 to 16 show vertical truck load probabilitycurves for varied numbers of trucks and probabilities ofpassing truck numbers on the bridge over the interval 119905From Figure 16 it can be seen that the probability is verylow when the maximum number of trucks on one lane isfour Figures 17 and 18 show similar curve shapes with those

in Figures 13 and 14 which indicate that there is the samerule in load combinations in both the horizontal and verticaldirections Comparing Figures 17 to 20 though themaximumnumber of trucks is four because dead load is combinedwith truck and earthquake load in the gravity direction thedead load contributes a substantial portion in vertical loadcombinations

6 Conclusions

This paper describes a method to combine earthquake loadand truck load in the service life of bridges The followingconclusions can be drawn

(1) Given the more than 70 seismic areas in Chinaearthquake load is a main consideration for bridge

10 Shock and Vibration

16000

05

1

15

2

2000 4000 6000 8000 10000 12000 14000

Prob

abili

ty d

ensit

y

times10minus3

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN-M)

Figure 13 Probability density of load combination in 100 years

05 1 15 2

05

055

06

065

07

075

08

085

09

095

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Cum

ulat

ive p

roba

bilit

y

times104Intensity (kN-M)

Figure 14 Cumulative probability of load combination in 100 years

design in the southeast coastal areas of China Theearthquake load probability curve is obtained usingseismic risk analysis

(2) Using measured truck load data from BHMS multi-modal characteristics of truck load are analyzed Thetruck load density of each truck is obtained by curvefitting Considering that truck load may consist ofvarying numbers of trucks truck load is calculatedthrough traffic analysis

(3) In thismethod themaximumvalue of combined loadis defined as 119883max119879 = max

119879[1198831+ 1198832+ sdot sdot sdot 119883

119899]119905

which means (1198831+ 1198832+ sdot sdot sdot 119883

119899) over the interval 119905

is first combined and then the maximum value in the

0 2000 4000 6000 8000 100000

05

1

15

2

25

3

35

4

1 truck passing2 trucks passing

3 trucks passing4 trucks passing

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 15 Vertical truck load probability density curves for variednumber of trucks

0 1 2 3 40

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 16 Probability of passing truck number simultaneously in 119905interval on the bridge with maximum number 4

bridge service life is determined 119883max119879 is based onprobability which covers all the probability combi-nations of the combined situations In this method aDiracDelta function is introduced to deal with119883 overa small interval 119905 To demonstrate the methodologyintuitively examples of load combinations in horizon-tal and vertical directions are provided

(4) The shape of the earthquake and truck load com-bination is similar to that of truck load alone butthe curve is displaced to the right which means themode and mean value of truck and earthquake loadcombination in this example is larger than that oftruck load aloneThis also illustrates that truck load is

Shock and Vibration 11

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 17 Probability density of vertical load combination in 100years

0 1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Figure 18 Cumulative probability of vertical load combination in100 years

more sensitive to bridge design and over most rangestruck load is larger than earthquake load in this area

(5) The curve from direct load combined over 100 yearsis further away from the curve obtained using themethod in the paper with the direct combinationof truck load and earthquake load over 100 yearsgiving much larger values It is not suggested that thismethod be used in bridge design considerations

(6) Because dead load is combined with truck and earth-quake load along the direction of gravity the dead

0 2000 4000 6000 8000 10000 12000 140000

1

2

3

4

5

6

7

8

Dead loadTruck and earthquakeTruck and earthquake and dead load

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 19 Probability density of dead truck and earthquake loadcombination

0 2000 4000 6000 8000 10000 12000 140000

02

04

06

08

1

12

14

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Dead loadTruck and earthquakeTruck and earthquake and dead load

Figure 20 Probability curves of dead truck and earthquake loadcombination

load contributes a substantial portion in vertical loadcombinations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Shock and Vibration

Acknowledgments

This study is jointly funded by Basic Institute ScientificResearch Fund (Grant no 2012A02) the National NaturalScience Fund of China (NSFC) (Grant no 51308510) andOpen Fund of State Key Laboratory Breeding Base of Moun-tainBridge andTunnel Engineering (Grant no CQSLBF-Y14-15) The results and conclusions presented in the paper areof the authors and do not necessarily reflect the view of thesponsors

References

[1] P R China Ministry of Communications ldquoGeneral code fordesign of highway bridges and culvertsrdquo Tech Rep JTG D60-200 China Communications Press Beijing China 2004

[2] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[3] C J Turkstra and H O Madsen ldquoLoad combinations incodified structural designrdquo Journal of Structural Engineeringvol 106 no 12 pp 2527ndash2543 1980

[4] ACI Publication SP-31 Probabilistic Design of Reinforced Con-crete Buildings American Concrete Institute Detroit MichUSA 1971

[5] B Ellingwood Development of a Probability Based Load Cri-terion for American National Standard A58-Building CodeRequirements forMinimumDesign Loads in Buildings andOtherStructures US Department of CommerceNational Bureau ofStandards 1980

[6] Y K Wen Development of Reliability-Based Design Criteria forBuildings Under Seismic Load National Center for EarthquakeEngineering Research Buffalo NY USA 1994

[7] Y-K Wen ldquoStatistical combination of extreme loadsrdquo Journalof Structural Engineering vol 103 no 5 pp 1079ndash1093 1977

[8] M Ghosn F Moses and J Wang ldquoDesign of highway bridgesfor extreme eventsrdquo NCHRP 489 Transportation ResearchBoard National Research Council Washington D C USA2003

[9] S E Hida ldquoStatistical significance of less common load combi-nationsrdquo Journal of Bridge Engineering vol 12 no 3 pp 389ndash393 2007

[10] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loadsmdashpart I comprehensive reliability and par-tial failure probabilitiesrdquo Journal of Earthquake Engineering andEngineering Vibration vol 11 no 3 pp 293ndash301 2012

[11] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loads Part II Examples for live and earthquakeload effectsrdquo Journal of Earthquake Engineering and EngineeringVibration vol 11 no 3 pp 303ndash311 2012

[12] E EMatheu D E Yule and R V Kala ldquoDetermination of stan-dard response spectra and effective peak ground accelerationsfor seismic design and evaluationrdquo Tech Rep no ERDCCHLCHETN-VI-41 2005

[13] HN Li TH YiMGu and LHuo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009

[14] B Chen Z W Chen Y Z Sun and S L Zhao ldquoConditionassessment on thermal effects of a suspension bridge basedon SHM oriented model and datardquo Mathematical Problems inEngineering vol 2013 Article ID 256816 18 pages 2013

[15] T-H Yi H-N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013

[16] Y Xia B Chen X Q Zhou and Y L Xu ldquoFieldmonitoring andnumerical analysis of Tsing Ma suspension bridge temperaturebehaviorrdquo Structural Control and HealthMonitoring vol 20 no4 pp 560ndash575 2013

[17] G Fu and J You ldquoExtrapolation for future maximum loadstatisticsrdquo Journal of Bridge Engineering vol 16 no 4 pp 527ndash535 2011

[18] A S Nowak ldquoCalibration of LRFD bridge design coderdquoNCHRPReport TransportationResearchBoard of theNationalAcademies Washington DC USA 1999

[19] F Moses ldquoCalibration of load factors for LRFD bridge evalua-tionsrdquo NCHRP Report 454 Transportation Research Board ofthe National Academies Washington DC USA 2001

[20] J Zhao and H Tabatabai ldquoEvaluation of a permit vehiclemodel using weigh-in-motion truck recordsrdquo Journal of BridgeEngineering vol 17 no 2 pp 389ndash392 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

Then the cumulative probability function 11986511988312max119879

(119909) ofmaximum value of load combinations 119883

12max119879(119909) in time119879 can be obtained

11986511988312max119879

(119909) = [11986511988312

(119909)119905]119879119905

(21)

When the number of loads is more than two and theseloads apply to bridge directly while satisfying Poissonprocess (14) can be extended to 119899 dimensions Assume1198911 1198912 119891

119899are the probability densities functions of

1198831 1198832 119883

119899 1198651 1198652 119865

119899are the cumulative probability

functions of1198831 1198832 119883

119899 11989110158401 1198911015840

2 119891

1015840

119899are the probability

density functions of 1198831 1198832 119883

119899without ldquozerordquo points

119891119883(119909) and 119865

119883(119909) are the probability density function and

cumulative probability functionThen the probability densityfunction is deduced as

119891119883(119909) = 119891

1(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) (22)

where the cumulative probability 119865119883max(119909) similar to that

given by (19) is

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

= int

119909

minusinfin

1198911(119909) lowast 119891

2(119909) lowast sdot sdot sdot lowast 119891

119899(119909) 119889119909

(23)

Equation (23) can account for load combinations of allloads which satisfy the first three assumptions If more thantwo events occurring simultaneously can be neglected (23)reduces to

119865119883(119909) = int

119909

minusinfin

119891119883(119909) 119889119909

asymp 11987511988310

11987511988320

sdot sdot sdot 1198751198831198990

+ 1198651015840

111987511988320

sdot sdot sdot 1198751198831198990

+ sdot sdot sdot + 11987511988310

11987511988320

sdot sdot sdot 119875119883119899minus10

1198651015840

119899

(24)

Then the maximum value of119883 in time 119879 can be obtained by

119865119883max119879

(119909) = [119865119883(119909)119905]119879119905

(25)

where 119865119883(119909)119905is 119865119883(119909) in 119905 interval 119865

119883max119879(119909) is the cumula-

tive probability function of maximum value of119883 in time 119879Although in our study emphasis is given to formulate

the ldquodemandrdquo to establish load combinations all events mustaddress a capacity issue of the bridge For example theearthquake load and truck load combination on a bridgecolumn can either consider the vertical load or the columnbase shear load Theoretically (23) can deal with most loadcombinations but as more loads are considered a more con-servative design will be adopted Based on the methodologyand assumptions described above the maximum load can becombined and the procedures are summarized as follows

(1) determine truck load and earthquake load distribu-tions over a particular period

(2) using Poisson processes convert earthquake load andtruck load distributions over a particular period to asufficiently small interval 119905

203 cm

60m

19m

442m442m442m

Figure 6 Longitudinal profile of the typical bridge

203 cm

190m

50m

152m

215m143m

215m

600m

Figure 7 Transverse profile of the typical bridge

(3) using (23) earthquake and truck load combinationsover an interval 119905 can be obtained

(4) using (25) the load combinations over an interval 119905can be converted to the bridge service life interval 119879

5 Numerical Examples

Example 1 Using the method of load combination describedin the preceding section a simple example of horizontal loadcombination is presented here Profiles of the typical bridgeare shown in Figures 6 and 7Theweight of the superstructureat each column is 538 tons the eccentricity of truck load is 50meters and the effects of soil and secondary effects of gravityare ignored Furthermore it is assumed that the maximumnumber of trucks on one lane is two The results are given inFigures 8 9 10 11 12 13 and 14 The interval 119905 is 10 secondsand the average daily truck traffic (ADTT) is about 1947

8 Shock and Vibration

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

ytimes10

minus4

Intensity (kN-M)

(a)

0 2000 4000 6000 8000 10000

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty

Intensity (kN-M)

(b)

Figure 8 Probability curve of each truck load effect

1 truck passing2 trucks passing

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M)

Figure 9 Truck load probability density curve for varied number oftrucks

Truck and earthquake load effects are the base momentcaused by trucks and earthquakes respectively Figure 8shows the probability curves of each truck load effectwhich has a similar shape to truck load Figure 9 shows thetruck load probability density curve for varied numbers oftrucks From Figure 10 we can see that over the interval 119905the probability of no truck on the bridge is much largerthan the other number of trucks passing Figure 11 showsthe combined probability curves for truck load over anearthquake load duration which indicate that the truck loadover an earthquake load duration is larger than each truckload

0 1 2

0

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 10 Probability of passing truck number simultaneously in 119905interval on the bridge

From the results shown in Figure 13 it can be seen thatcurve C and curve A have a similar shape and curve C isdisplaced to the right This means that mode and mean valueof the truck and earthquake load combination in this exampleis larger than those of truck load and also illustrates that truckload is more important to bridge design in this area FromFigure 14 we can seemore clearly that overmost ranges truckload is larger than earthquake load Because the truck loadhas a smaller tail their combination in the tail is close to theearthquake load Curve D of loads combined directly over100 years is further away from curve C which means thecombinations of truck load and earthquake load directly over100 years give a much larger value and it is not suggested thatthe bridge design uses the curve D method

Shock and Vibration 9

0

05

1

15

2

25

3

35

Prob

abili

ty d

ensit

y

times10minus4

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(a)

05

055

06

065

07

075

08

085

09

095

1

Cum

ulat

ive p

roba

bilit

y

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(b)

Figure 11 Combined probability curves for truck load in earthquake load duration

05 1 15 2 25

0

1

2

3

4

5

6

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M) times104

(a)

2000 4000 6000 8000 10000 12000 14000

0

01

02

03

04

05

06

07

08

09

1

Cum

ulat

ive p

roba

bilit

y

Intensity (kN-M)

(b)

Figure 12 Probability curves for earthquake load in 100 years

Example 2 Example 2 illustrates vertical load combinationsMost of the configurations are the same as used in Example 1The difference is in the maximum number of trucks namelythe maximum number of trucks on one lane in this exampleis four The results are shown in Figures 15 16 17 18 19 and20 Figures 13 and 14 are the results of truck load Note that inthis example the 119905 interval is 10 seconds and 119879 is taken as 100years

Figures 15 to 16 show vertical truck load probabilitycurves for varied numbers of trucks and probabilities ofpassing truck numbers on the bridge over the interval 119905From Figure 16 it can be seen that the probability is verylow when the maximum number of trucks on one lane isfour Figures 17 and 18 show similar curve shapes with those

in Figures 13 and 14 which indicate that there is the samerule in load combinations in both the horizontal and verticaldirections Comparing Figures 17 to 20 though themaximumnumber of trucks is four because dead load is combinedwith truck and earthquake load in the gravity direction thedead load contributes a substantial portion in vertical loadcombinations

6 Conclusions

This paper describes a method to combine earthquake loadand truck load in the service life of bridges The followingconclusions can be drawn

(1) Given the more than 70 seismic areas in Chinaearthquake load is a main consideration for bridge

10 Shock and Vibration

16000

05

1

15

2

2000 4000 6000 8000 10000 12000 14000

Prob

abili

ty d

ensit

y

times10minus3

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN-M)

Figure 13 Probability density of load combination in 100 years

05 1 15 2

05

055

06

065

07

075

08

085

09

095

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Cum

ulat

ive p

roba

bilit

y

times104Intensity (kN-M)

Figure 14 Cumulative probability of load combination in 100 years

design in the southeast coastal areas of China Theearthquake load probability curve is obtained usingseismic risk analysis

(2) Using measured truck load data from BHMS multi-modal characteristics of truck load are analyzed Thetruck load density of each truck is obtained by curvefitting Considering that truck load may consist ofvarying numbers of trucks truck load is calculatedthrough traffic analysis

(3) In thismethod themaximumvalue of combined loadis defined as 119883max119879 = max

119879[1198831+ 1198832+ sdot sdot sdot 119883

119899]119905

which means (1198831+ 1198832+ sdot sdot sdot 119883

119899) over the interval 119905

is first combined and then the maximum value in the

0 2000 4000 6000 8000 100000

05

1

15

2

25

3

35

4

1 truck passing2 trucks passing

3 trucks passing4 trucks passing

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 15 Vertical truck load probability density curves for variednumber of trucks

0 1 2 3 40

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 16 Probability of passing truck number simultaneously in 119905interval on the bridge with maximum number 4

bridge service life is determined 119883max119879 is based onprobability which covers all the probability combi-nations of the combined situations In this method aDiracDelta function is introduced to deal with119883 overa small interval 119905 To demonstrate the methodologyintuitively examples of load combinations in horizon-tal and vertical directions are provided

(4) The shape of the earthquake and truck load com-bination is similar to that of truck load alone butthe curve is displaced to the right which means themode and mean value of truck and earthquake loadcombination in this example is larger than that oftruck load aloneThis also illustrates that truck load is

Shock and Vibration 11

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 17 Probability density of vertical load combination in 100years

0 1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Figure 18 Cumulative probability of vertical load combination in100 years

more sensitive to bridge design and over most rangestruck load is larger than earthquake load in this area

(5) The curve from direct load combined over 100 yearsis further away from the curve obtained using themethod in the paper with the direct combinationof truck load and earthquake load over 100 yearsgiving much larger values It is not suggested that thismethod be used in bridge design considerations

(6) Because dead load is combined with truck and earth-quake load along the direction of gravity the dead

0 2000 4000 6000 8000 10000 12000 140000

1

2

3

4

5

6

7

8

Dead loadTruck and earthquakeTruck and earthquake and dead load

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 19 Probability density of dead truck and earthquake loadcombination

0 2000 4000 6000 8000 10000 12000 140000

02

04

06

08

1

12

14

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Dead loadTruck and earthquakeTruck and earthquake and dead load

Figure 20 Probability curves of dead truck and earthquake loadcombination

load contributes a substantial portion in vertical loadcombinations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Shock and Vibration

Acknowledgments

This study is jointly funded by Basic Institute ScientificResearch Fund (Grant no 2012A02) the National NaturalScience Fund of China (NSFC) (Grant no 51308510) andOpen Fund of State Key Laboratory Breeding Base of Moun-tainBridge andTunnel Engineering (Grant no CQSLBF-Y14-15) The results and conclusions presented in the paper areof the authors and do not necessarily reflect the view of thesponsors

References

[1] P R China Ministry of Communications ldquoGeneral code fordesign of highway bridges and culvertsrdquo Tech Rep JTG D60-200 China Communications Press Beijing China 2004

[2] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[3] C J Turkstra and H O Madsen ldquoLoad combinations incodified structural designrdquo Journal of Structural Engineeringvol 106 no 12 pp 2527ndash2543 1980

[4] ACI Publication SP-31 Probabilistic Design of Reinforced Con-crete Buildings American Concrete Institute Detroit MichUSA 1971

[5] B Ellingwood Development of a Probability Based Load Cri-terion for American National Standard A58-Building CodeRequirements forMinimumDesign Loads in Buildings andOtherStructures US Department of CommerceNational Bureau ofStandards 1980

[6] Y K Wen Development of Reliability-Based Design Criteria forBuildings Under Seismic Load National Center for EarthquakeEngineering Research Buffalo NY USA 1994

[7] Y-K Wen ldquoStatistical combination of extreme loadsrdquo Journalof Structural Engineering vol 103 no 5 pp 1079ndash1093 1977

[8] M Ghosn F Moses and J Wang ldquoDesign of highway bridgesfor extreme eventsrdquo NCHRP 489 Transportation ResearchBoard National Research Council Washington D C USA2003

[9] S E Hida ldquoStatistical significance of less common load combi-nationsrdquo Journal of Bridge Engineering vol 12 no 3 pp 389ndash393 2007

[10] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loadsmdashpart I comprehensive reliability and par-tial failure probabilitiesrdquo Journal of Earthquake Engineering andEngineering Vibration vol 11 no 3 pp 293ndash301 2012

[11] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loads Part II Examples for live and earthquakeload effectsrdquo Journal of Earthquake Engineering and EngineeringVibration vol 11 no 3 pp 303ndash311 2012

[12] E EMatheu D E Yule and R V Kala ldquoDetermination of stan-dard response spectra and effective peak ground accelerationsfor seismic design and evaluationrdquo Tech Rep no ERDCCHLCHETN-VI-41 2005

[13] HN Li TH YiMGu and LHuo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009

[14] B Chen Z W Chen Y Z Sun and S L Zhao ldquoConditionassessment on thermal effects of a suspension bridge basedon SHM oriented model and datardquo Mathematical Problems inEngineering vol 2013 Article ID 256816 18 pages 2013

[15] T-H Yi H-N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013

[16] Y Xia B Chen X Q Zhou and Y L Xu ldquoFieldmonitoring andnumerical analysis of Tsing Ma suspension bridge temperaturebehaviorrdquo Structural Control and HealthMonitoring vol 20 no4 pp 560ndash575 2013

[17] G Fu and J You ldquoExtrapolation for future maximum loadstatisticsrdquo Journal of Bridge Engineering vol 16 no 4 pp 527ndash535 2011

[18] A S Nowak ldquoCalibration of LRFD bridge design coderdquoNCHRPReport TransportationResearchBoard of theNationalAcademies Washington DC USA 1999

[19] F Moses ldquoCalibration of load factors for LRFD bridge evalua-tionsrdquo NCHRP Report 454 Transportation Research Board ofthe National Academies Washington DC USA 2001

[20] J Zhao and H Tabatabai ldquoEvaluation of a permit vehiclemodel using weigh-in-motion truck recordsrdquo Journal of BridgeEngineering vol 17 no 2 pp 389ndash392 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

ytimes10

minus4

Intensity (kN-M)

(a)

0 2000 4000 6000 8000 10000

0

01

02

03

04

05

06

07

08

09

1

Prob

abili

ty

Intensity (kN-M)

(b)

Figure 8 Probability curve of each truck load effect

1 truck passing2 trucks passing

0 2000 4000 6000 8000 10000

0

1

2

3

4

5

6

7

8

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M)

Figure 9 Truck load probability density curve for varied number oftrucks

Truck and earthquake load effects are the base momentcaused by trucks and earthquakes respectively Figure 8shows the probability curves of each truck load effectwhich has a similar shape to truck load Figure 9 shows thetruck load probability density curve for varied numbers oftrucks From Figure 10 we can see that over the interval 119905the probability of no truck on the bridge is much largerthan the other number of trucks passing Figure 11 showsthe combined probability curves for truck load over anearthquake load duration which indicate that the truck loadover an earthquake load duration is larger than each truckload

0 1 2

0

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 10 Probability of passing truck number simultaneously in 119905interval on the bridge

From the results shown in Figure 13 it can be seen thatcurve C and curve A have a similar shape and curve C isdisplaced to the right This means that mode and mean valueof the truck and earthquake load combination in this exampleis larger than those of truck load and also illustrates that truckload is more important to bridge design in this area FromFigure 14 we can seemore clearly that overmost ranges truckload is larger than earthquake load Because the truck loadhas a smaller tail their combination in the tail is close to theearthquake load Curve D of loads combined directly over100 years is further away from curve C which means thecombinations of truck load and earthquake load directly over100 years give a much larger value and it is not suggested thatthe bridge design uses the curve D method

Shock and Vibration 9

0

05

1

15

2

25

3

35

Prob

abili

ty d

ensit

y

times10minus4

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(a)

05

055

06

065

07

075

08

085

09

095

1

Cum

ulat

ive p

roba

bilit

y

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(b)

Figure 11 Combined probability curves for truck load in earthquake load duration

05 1 15 2 25

0

1

2

3

4

5

6

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M) times104

(a)

2000 4000 6000 8000 10000 12000 14000

0

01

02

03

04

05

06

07

08

09

1

Cum

ulat

ive p

roba

bilit

y

Intensity (kN-M)

(b)

Figure 12 Probability curves for earthquake load in 100 years

Example 2 Example 2 illustrates vertical load combinationsMost of the configurations are the same as used in Example 1The difference is in the maximum number of trucks namelythe maximum number of trucks on one lane in this exampleis four The results are shown in Figures 15 16 17 18 19 and20 Figures 13 and 14 are the results of truck load Note that inthis example the 119905 interval is 10 seconds and 119879 is taken as 100years

Figures 15 to 16 show vertical truck load probabilitycurves for varied numbers of trucks and probabilities ofpassing truck numbers on the bridge over the interval 119905From Figure 16 it can be seen that the probability is verylow when the maximum number of trucks on one lane isfour Figures 17 and 18 show similar curve shapes with those

in Figures 13 and 14 which indicate that there is the samerule in load combinations in both the horizontal and verticaldirections Comparing Figures 17 to 20 though themaximumnumber of trucks is four because dead load is combinedwith truck and earthquake load in the gravity direction thedead load contributes a substantial portion in vertical loadcombinations

6 Conclusions

This paper describes a method to combine earthquake loadand truck load in the service life of bridges The followingconclusions can be drawn

(1) Given the more than 70 seismic areas in Chinaearthquake load is a main consideration for bridge

10 Shock and Vibration

16000

05

1

15

2

2000 4000 6000 8000 10000 12000 14000

Prob

abili

ty d

ensit

y

times10minus3

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN-M)

Figure 13 Probability density of load combination in 100 years

05 1 15 2

05

055

06

065

07

075

08

085

09

095

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Cum

ulat

ive p

roba

bilit

y

times104Intensity (kN-M)

Figure 14 Cumulative probability of load combination in 100 years

design in the southeast coastal areas of China Theearthquake load probability curve is obtained usingseismic risk analysis

(2) Using measured truck load data from BHMS multi-modal characteristics of truck load are analyzed Thetruck load density of each truck is obtained by curvefitting Considering that truck load may consist ofvarying numbers of trucks truck load is calculatedthrough traffic analysis

(3) In thismethod themaximumvalue of combined loadis defined as 119883max119879 = max

119879[1198831+ 1198832+ sdot sdot sdot 119883

119899]119905

which means (1198831+ 1198832+ sdot sdot sdot 119883

119899) over the interval 119905

is first combined and then the maximum value in the

0 2000 4000 6000 8000 100000

05

1

15

2

25

3

35

4

1 truck passing2 trucks passing

3 trucks passing4 trucks passing

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 15 Vertical truck load probability density curves for variednumber of trucks

0 1 2 3 40

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 16 Probability of passing truck number simultaneously in 119905interval on the bridge with maximum number 4

bridge service life is determined 119883max119879 is based onprobability which covers all the probability combi-nations of the combined situations In this method aDiracDelta function is introduced to deal with119883 overa small interval 119905 To demonstrate the methodologyintuitively examples of load combinations in horizon-tal and vertical directions are provided

(4) The shape of the earthquake and truck load com-bination is similar to that of truck load alone butthe curve is displaced to the right which means themode and mean value of truck and earthquake loadcombination in this example is larger than that oftruck load aloneThis also illustrates that truck load is

Shock and Vibration 11

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 17 Probability density of vertical load combination in 100years

0 1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Figure 18 Cumulative probability of vertical load combination in100 years

more sensitive to bridge design and over most rangestruck load is larger than earthquake load in this area

(5) The curve from direct load combined over 100 yearsis further away from the curve obtained using themethod in the paper with the direct combinationof truck load and earthquake load over 100 yearsgiving much larger values It is not suggested that thismethod be used in bridge design considerations

(6) Because dead load is combined with truck and earth-quake load along the direction of gravity the dead

0 2000 4000 6000 8000 10000 12000 140000

1

2

3

4

5

6

7

8

Dead loadTruck and earthquakeTruck and earthquake and dead load

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 19 Probability density of dead truck and earthquake loadcombination

0 2000 4000 6000 8000 10000 12000 140000

02

04

06

08

1

12

14

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Dead loadTruck and earthquakeTruck and earthquake and dead load

Figure 20 Probability curves of dead truck and earthquake loadcombination

load contributes a substantial portion in vertical loadcombinations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Shock and Vibration

Acknowledgments

This study is jointly funded by Basic Institute ScientificResearch Fund (Grant no 2012A02) the National NaturalScience Fund of China (NSFC) (Grant no 51308510) andOpen Fund of State Key Laboratory Breeding Base of Moun-tainBridge andTunnel Engineering (Grant no CQSLBF-Y14-15) The results and conclusions presented in the paper areof the authors and do not necessarily reflect the view of thesponsors

References

[1] P R China Ministry of Communications ldquoGeneral code fordesign of highway bridges and culvertsrdquo Tech Rep JTG D60-200 China Communications Press Beijing China 2004

[2] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[3] C J Turkstra and H O Madsen ldquoLoad combinations incodified structural designrdquo Journal of Structural Engineeringvol 106 no 12 pp 2527ndash2543 1980

[4] ACI Publication SP-31 Probabilistic Design of Reinforced Con-crete Buildings American Concrete Institute Detroit MichUSA 1971

[5] B Ellingwood Development of a Probability Based Load Cri-terion for American National Standard A58-Building CodeRequirements forMinimumDesign Loads in Buildings andOtherStructures US Department of CommerceNational Bureau ofStandards 1980

[6] Y K Wen Development of Reliability-Based Design Criteria forBuildings Under Seismic Load National Center for EarthquakeEngineering Research Buffalo NY USA 1994

[7] Y-K Wen ldquoStatistical combination of extreme loadsrdquo Journalof Structural Engineering vol 103 no 5 pp 1079ndash1093 1977

[8] M Ghosn F Moses and J Wang ldquoDesign of highway bridgesfor extreme eventsrdquo NCHRP 489 Transportation ResearchBoard National Research Council Washington D C USA2003

[9] S E Hida ldquoStatistical significance of less common load combi-nationsrdquo Journal of Bridge Engineering vol 12 no 3 pp 389ndash393 2007

[10] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loadsmdashpart I comprehensive reliability and par-tial failure probabilitiesrdquo Journal of Earthquake Engineering andEngineering Vibration vol 11 no 3 pp 293ndash301 2012

[11] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loads Part II Examples for live and earthquakeload effectsrdquo Journal of Earthquake Engineering and EngineeringVibration vol 11 no 3 pp 303ndash311 2012

[12] E EMatheu D E Yule and R V Kala ldquoDetermination of stan-dard response spectra and effective peak ground accelerationsfor seismic design and evaluationrdquo Tech Rep no ERDCCHLCHETN-VI-41 2005

[13] HN Li TH YiMGu and LHuo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009

[14] B Chen Z W Chen Y Z Sun and S L Zhao ldquoConditionassessment on thermal effects of a suspension bridge basedon SHM oriented model and datardquo Mathematical Problems inEngineering vol 2013 Article ID 256816 18 pages 2013

[15] T-H Yi H-N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013

[16] Y Xia B Chen X Q Zhou and Y L Xu ldquoFieldmonitoring andnumerical analysis of Tsing Ma suspension bridge temperaturebehaviorrdquo Structural Control and HealthMonitoring vol 20 no4 pp 560ndash575 2013

[17] G Fu and J You ldquoExtrapolation for future maximum loadstatisticsrdquo Journal of Bridge Engineering vol 16 no 4 pp 527ndash535 2011

[18] A S Nowak ldquoCalibration of LRFD bridge design coderdquoNCHRPReport TransportationResearchBoard of theNationalAcademies Washington DC USA 1999

[19] F Moses ldquoCalibration of load factors for LRFD bridge evalua-tionsrdquo NCHRP Report 454 Transportation Research Board ofthe National Academies Washington DC USA 2001

[20] J Zhao and H Tabatabai ldquoEvaluation of a permit vehiclemodel using weigh-in-motion truck recordsrdquo Journal of BridgeEngineering vol 17 no 2 pp 389ndash392 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

0

05

1

15

2

25

3

35

Prob

abili

ty d

ensit

y

times10minus4

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(a)

05

055

06

065

07

075

08

085

09

095

1

Cum

ulat

ive p

roba

bilit

y

0 2000 4000 6000 8000 10000

Intensity (kN-M)

(b)

Figure 11 Combined probability curves for truck load in earthquake load duration

05 1 15 2 25

0

1

2

3

4

5

6

Prob

abili

ty d

ensit

y

times10minus4

Intensity (kN-M) times104

(a)

2000 4000 6000 8000 10000 12000 14000

0

01

02

03

04

05

06

07

08

09

1

Cum

ulat

ive p

roba

bilit

y

Intensity (kN-M)

(b)

Figure 12 Probability curves for earthquake load in 100 years

Example 2 Example 2 illustrates vertical load combinationsMost of the configurations are the same as used in Example 1The difference is in the maximum number of trucks namelythe maximum number of trucks on one lane in this exampleis four The results are shown in Figures 15 16 17 18 19 and20 Figures 13 and 14 are the results of truck load Note that inthis example the 119905 interval is 10 seconds and 119879 is taken as 100years

Figures 15 to 16 show vertical truck load probabilitycurves for varied numbers of trucks and probabilities ofpassing truck numbers on the bridge over the interval 119905From Figure 16 it can be seen that the probability is verylow when the maximum number of trucks on one lane isfour Figures 17 and 18 show similar curve shapes with those

in Figures 13 and 14 which indicate that there is the samerule in load combinations in both the horizontal and verticaldirections Comparing Figures 17 to 20 though themaximumnumber of trucks is four because dead load is combinedwith truck and earthquake load in the gravity direction thedead load contributes a substantial portion in vertical loadcombinations

6 Conclusions

This paper describes a method to combine earthquake loadand truck load in the service life of bridges The followingconclusions can be drawn

(1) Given the more than 70 seismic areas in Chinaearthquake load is a main consideration for bridge

10 Shock and Vibration

16000

05

1

15

2

2000 4000 6000 8000 10000 12000 14000

Prob

abili

ty d

ensit

y

times10minus3

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN-M)

Figure 13 Probability density of load combination in 100 years

05 1 15 2

05

055

06

065

07

075

08

085

09

095

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Cum

ulat

ive p

roba

bilit

y

times104Intensity (kN-M)

Figure 14 Cumulative probability of load combination in 100 years

design in the southeast coastal areas of China Theearthquake load probability curve is obtained usingseismic risk analysis

(2) Using measured truck load data from BHMS multi-modal characteristics of truck load are analyzed Thetruck load density of each truck is obtained by curvefitting Considering that truck load may consist ofvarying numbers of trucks truck load is calculatedthrough traffic analysis

(3) In thismethod themaximumvalue of combined loadis defined as 119883max119879 = max

119879[1198831+ 1198832+ sdot sdot sdot 119883

119899]119905

which means (1198831+ 1198832+ sdot sdot sdot 119883

119899) over the interval 119905

is first combined and then the maximum value in the

0 2000 4000 6000 8000 100000

05

1

15

2

25

3

35

4

1 truck passing2 trucks passing

3 trucks passing4 trucks passing

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 15 Vertical truck load probability density curves for variednumber of trucks

0 1 2 3 40

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 16 Probability of passing truck number simultaneously in 119905interval on the bridge with maximum number 4

bridge service life is determined 119883max119879 is based onprobability which covers all the probability combi-nations of the combined situations In this method aDiracDelta function is introduced to deal with119883 overa small interval 119905 To demonstrate the methodologyintuitively examples of load combinations in horizon-tal and vertical directions are provided

(4) The shape of the earthquake and truck load com-bination is similar to that of truck load alone butthe curve is displaced to the right which means themode and mean value of truck and earthquake loadcombination in this example is larger than that oftruck load aloneThis also illustrates that truck load is

Shock and Vibration 11

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 17 Probability density of vertical load combination in 100years

0 1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Figure 18 Cumulative probability of vertical load combination in100 years

more sensitive to bridge design and over most rangestruck load is larger than earthquake load in this area

(5) The curve from direct load combined over 100 yearsis further away from the curve obtained using themethod in the paper with the direct combinationof truck load and earthquake load over 100 yearsgiving much larger values It is not suggested that thismethod be used in bridge design considerations

(6) Because dead load is combined with truck and earth-quake load along the direction of gravity the dead

0 2000 4000 6000 8000 10000 12000 140000

1

2

3

4

5

6

7

8

Dead loadTruck and earthquakeTruck and earthquake and dead load

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 19 Probability density of dead truck and earthquake loadcombination

0 2000 4000 6000 8000 10000 12000 140000

02

04

06

08

1

12

14

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Dead loadTruck and earthquakeTruck and earthquake and dead load

Figure 20 Probability curves of dead truck and earthquake loadcombination

load contributes a substantial portion in vertical loadcombinations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Shock and Vibration

Acknowledgments

This study is jointly funded by Basic Institute ScientificResearch Fund (Grant no 2012A02) the National NaturalScience Fund of China (NSFC) (Grant no 51308510) andOpen Fund of State Key Laboratory Breeding Base of Moun-tainBridge andTunnel Engineering (Grant no CQSLBF-Y14-15) The results and conclusions presented in the paper areof the authors and do not necessarily reflect the view of thesponsors

References

[1] P R China Ministry of Communications ldquoGeneral code fordesign of highway bridges and culvertsrdquo Tech Rep JTG D60-200 China Communications Press Beijing China 2004

[2] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[3] C J Turkstra and H O Madsen ldquoLoad combinations incodified structural designrdquo Journal of Structural Engineeringvol 106 no 12 pp 2527ndash2543 1980

[4] ACI Publication SP-31 Probabilistic Design of Reinforced Con-crete Buildings American Concrete Institute Detroit MichUSA 1971

[5] B Ellingwood Development of a Probability Based Load Cri-terion for American National Standard A58-Building CodeRequirements forMinimumDesign Loads in Buildings andOtherStructures US Department of CommerceNational Bureau ofStandards 1980

[6] Y K Wen Development of Reliability-Based Design Criteria forBuildings Under Seismic Load National Center for EarthquakeEngineering Research Buffalo NY USA 1994

[7] Y-K Wen ldquoStatistical combination of extreme loadsrdquo Journalof Structural Engineering vol 103 no 5 pp 1079ndash1093 1977

[8] M Ghosn F Moses and J Wang ldquoDesign of highway bridgesfor extreme eventsrdquo NCHRP 489 Transportation ResearchBoard National Research Council Washington D C USA2003

[9] S E Hida ldquoStatistical significance of less common load combi-nationsrdquo Journal of Bridge Engineering vol 12 no 3 pp 389ndash393 2007

[10] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loadsmdashpart I comprehensive reliability and par-tial failure probabilitiesrdquo Journal of Earthquake Engineering andEngineering Vibration vol 11 no 3 pp 293ndash301 2012

[11] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loads Part II Examples for live and earthquakeload effectsrdquo Journal of Earthquake Engineering and EngineeringVibration vol 11 no 3 pp 303ndash311 2012

[12] E EMatheu D E Yule and R V Kala ldquoDetermination of stan-dard response spectra and effective peak ground accelerationsfor seismic design and evaluationrdquo Tech Rep no ERDCCHLCHETN-VI-41 2005

[13] HN Li TH YiMGu and LHuo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009

[14] B Chen Z W Chen Y Z Sun and S L Zhao ldquoConditionassessment on thermal effects of a suspension bridge basedon SHM oriented model and datardquo Mathematical Problems inEngineering vol 2013 Article ID 256816 18 pages 2013

[15] T-H Yi H-N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013

[16] Y Xia B Chen X Q Zhou and Y L Xu ldquoFieldmonitoring andnumerical analysis of Tsing Ma suspension bridge temperaturebehaviorrdquo Structural Control and HealthMonitoring vol 20 no4 pp 560ndash575 2013

[17] G Fu and J You ldquoExtrapolation for future maximum loadstatisticsrdquo Journal of Bridge Engineering vol 16 no 4 pp 527ndash535 2011

[18] A S Nowak ldquoCalibration of LRFD bridge design coderdquoNCHRPReport TransportationResearchBoard of theNationalAcademies Washington DC USA 1999

[19] F Moses ldquoCalibration of load factors for LRFD bridge evalua-tionsrdquo NCHRP Report 454 Transportation Research Board ofthe National Academies Washington DC USA 2001

[20] J Zhao and H Tabatabai ldquoEvaluation of a permit vehiclemodel using weigh-in-motion truck recordsrdquo Journal of BridgeEngineering vol 17 no 2 pp 389ndash392 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

16000

05

1

15

2

2000 4000 6000 8000 10000 12000 14000

Prob

abili

ty d

ensit

y

times10minus3

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN-M)

Figure 13 Probability density of load combination in 100 years

05 1 15 2

05

055

06

065

07

075

08

085

09

095

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Cum

ulat

ive p

roba

bilit

y

times104Intensity (kN-M)

Figure 14 Cumulative probability of load combination in 100 years

design in the southeast coastal areas of China Theearthquake load probability curve is obtained usingseismic risk analysis

(2) Using measured truck load data from BHMS multi-modal characteristics of truck load are analyzed Thetruck load density of each truck is obtained by curvefitting Considering that truck load may consist ofvarying numbers of trucks truck load is calculatedthrough traffic analysis

(3) In thismethod themaximumvalue of combined loadis defined as 119883max119879 = max

119879[1198831+ 1198832+ sdot sdot sdot 119883

119899]119905

which means (1198831+ 1198832+ sdot sdot sdot 119883

119899) over the interval 119905

is first combined and then the maximum value in the

0 2000 4000 6000 8000 100000

05

1

15

2

25

3

35

4

1 truck passing2 trucks passing

3 trucks passing4 trucks passing

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 15 Vertical truck load probability density curves for variednumber of trucks

0 1 2 3 40

01

02

03

04

05

06

07

08

Truck number

Prob

abili

ty

Figure 16 Probability of passing truck number simultaneously in 119905interval on the bridge with maximum number 4

bridge service life is determined 119883max119879 is based onprobability which covers all the probability combi-nations of the combined situations In this method aDiracDelta function is introduced to deal with119883 overa small interval 119905 To demonstrate the methodologyintuitively examples of load combinations in horizon-tal and vertical directions are provided

(4) The shape of the earthquake and truck load com-bination is similar to that of truck load alone butthe curve is displaced to the right which means themode and mean value of truck and earthquake loadcombination in this example is larger than that oftruck load aloneThis also illustrates that truck load is

Shock and Vibration 11

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 17 Probability density of vertical load combination in 100years

0 1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Figure 18 Cumulative probability of vertical load combination in100 years

more sensitive to bridge design and over most rangestruck load is larger than earthquake load in this area

(5) The curve from direct load combined over 100 yearsis further away from the curve obtained using themethod in the paper with the direct combinationof truck load and earthquake load over 100 yearsgiving much larger values It is not suggested that thismethod be used in bridge design considerations

(6) Because dead load is combined with truck and earth-quake load along the direction of gravity the dead

0 2000 4000 6000 8000 10000 12000 140000

1

2

3

4

5

6

7

8

Dead loadTruck and earthquakeTruck and earthquake and dead load

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 19 Probability density of dead truck and earthquake loadcombination

0 2000 4000 6000 8000 10000 12000 140000

02

04

06

08

1

12

14

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Dead loadTruck and earthquakeTruck and earthquake and dead load

Figure 20 Probability curves of dead truck and earthquake loadcombination

load contributes a substantial portion in vertical loadcombinations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Shock and Vibration

Acknowledgments

This study is jointly funded by Basic Institute ScientificResearch Fund (Grant no 2012A02) the National NaturalScience Fund of China (NSFC) (Grant no 51308510) andOpen Fund of State Key Laboratory Breeding Base of Moun-tainBridge andTunnel Engineering (Grant no CQSLBF-Y14-15) The results and conclusions presented in the paper areof the authors and do not necessarily reflect the view of thesponsors

References

[1] P R China Ministry of Communications ldquoGeneral code fordesign of highway bridges and culvertsrdquo Tech Rep JTG D60-200 China Communications Press Beijing China 2004

[2] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[3] C J Turkstra and H O Madsen ldquoLoad combinations incodified structural designrdquo Journal of Structural Engineeringvol 106 no 12 pp 2527ndash2543 1980

[4] ACI Publication SP-31 Probabilistic Design of Reinforced Con-crete Buildings American Concrete Institute Detroit MichUSA 1971

[5] B Ellingwood Development of a Probability Based Load Cri-terion for American National Standard A58-Building CodeRequirements forMinimumDesign Loads in Buildings andOtherStructures US Department of CommerceNational Bureau ofStandards 1980

[6] Y K Wen Development of Reliability-Based Design Criteria forBuildings Under Seismic Load National Center for EarthquakeEngineering Research Buffalo NY USA 1994

[7] Y-K Wen ldquoStatistical combination of extreme loadsrdquo Journalof Structural Engineering vol 103 no 5 pp 1079ndash1093 1977

[8] M Ghosn F Moses and J Wang ldquoDesign of highway bridgesfor extreme eventsrdquo NCHRP 489 Transportation ResearchBoard National Research Council Washington D C USA2003

[9] S E Hida ldquoStatistical significance of less common load combi-nationsrdquo Journal of Bridge Engineering vol 12 no 3 pp 389ndash393 2007

[10] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loadsmdashpart I comprehensive reliability and par-tial failure probabilitiesrdquo Journal of Earthquake Engineering andEngineering Vibration vol 11 no 3 pp 293ndash301 2012

[11] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loads Part II Examples for live and earthquakeload effectsrdquo Journal of Earthquake Engineering and EngineeringVibration vol 11 no 3 pp 303ndash311 2012

[12] E EMatheu D E Yule and R V Kala ldquoDetermination of stan-dard response spectra and effective peak ground accelerationsfor seismic design and evaluationrdquo Tech Rep no ERDCCHLCHETN-VI-41 2005

[13] HN Li TH YiMGu and LHuo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009

[14] B Chen Z W Chen Y Z Sun and S L Zhao ldquoConditionassessment on thermal effects of a suspension bridge basedon SHM oriented model and datardquo Mathematical Problems inEngineering vol 2013 Article ID 256816 18 pages 2013

[15] T-H Yi H-N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013

[16] Y Xia B Chen X Q Zhou and Y L Xu ldquoFieldmonitoring andnumerical analysis of Tsing Ma suspension bridge temperaturebehaviorrdquo Structural Control and HealthMonitoring vol 20 no4 pp 560ndash575 2013

[17] G Fu and J You ldquoExtrapolation for future maximum loadstatisticsrdquo Journal of Bridge Engineering vol 16 no 4 pp 527ndash535 2011

[18] A S Nowak ldquoCalibration of LRFD bridge design coderdquoNCHRPReport TransportationResearchBoard of theNationalAcademies Washington DC USA 1999

[19] F Moses ldquoCalibration of load factors for LRFD bridge evalua-tionsrdquo NCHRP Report 454 Transportation Research Board ofthe National Academies Washington DC USA 2001

[20] J Zhao and H Tabatabai ldquoEvaluation of a permit vehiclemodel using weigh-in-motion truck recordsrdquo Journal of BridgeEngineering vol 17 no 2 pp 389ndash392 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 17 Probability density of vertical load combination in 100years

0 1000 2000 3000 4000 5000

01

02

03

04

05

06

07

08

09

1

Amdashtruck onlyBmdashearthquake only

Cmdashtruck and earthquakeDmdashtruck and earth T-Direct

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Figure 18 Cumulative probability of vertical load combination in100 years

more sensitive to bridge design and over most rangestruck load is larger than earthquake load in this area

(5) The curve from direct load combined over 100 yearsis further away from the curve obtained using themethod in the paper with the direct combinationof truck load and earthquake load over 100 yearsgiving much larger values It is not suggested that thismethod be used in bridge design considerations

(6) Because dead load is combined with truck and earth-quake load along the direction of gravity the dead

0 2000 4000 6000 8000 10000 12000 140000

1

2

3

4

5

6

7

8

Dead loadTruck and earthquakeTruck and earthquake and dead load

times10minus3

Prob

abili

ty d

ensit

y

Intensity (kN)

Figure 19 Probability density of dead truck and earthquake loadcombination

0 2000 4000 6000 8000 10000 12000 140000

02

04

06

08

1

12

14

Intensity (kN)

Cum

ulat

ive p

roba

bilit

y

Dead loadTruck and earthquakeTruck and earthquake and dead load

Figure 20 Probability curves of dead truck and earthquake loadcombination

load contributes a substantial portion in vertical loadcombinations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

12 Shock and Vibration

Acknowledgments

This study is jointly funded by Basic Institute ScientificResearch Fund (Grant no 2012A02) the National NaturalScience Fund of China (NSFC) (Grant no 51308510) andOpen Fund of State Key Laboratory Breeding Base of Moun-tainBridge andTunnel Engineering (Grant no CQSLBF-Y14-15) The results and conclusions presented in the paper areof the authors and do not necessarily reflect the view of thesponsors

References

[1] P R China Ministry of Communications ldquoGeneral code fordesign of highway bridges and culvertsrdquo Tech Rep JTG D60-200 China Communications Press Beijing China 2004

[2] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[3] C J Turkstra and H O Madsen ldquoLoad combinations incodified structural designrdquo Journal of Structural Engineeringvol 106 no 12 pp 2527ndash2543 1980

[4] ACI Publication SP-31 Probabilistic Design of Reinforced Con-crete Buildings American Concrete Institute Detroit MichUSA 1971

[5] B Ellingwood Development of a Probability Based Load Cri-terion for American National Standard A58-Building CodeRequirements forMinimumDesign Loads in Buildings andOtherStructures US Department of CommerceNational Bureau ofStandards 1980

[6] Y K Wen Development of Reliability-Based Design Criteria forBuildings Under Seismic Load National Center for EarthquakeEngineering Research Buffalo NY USA 1994

[7] Y-K Wen ldquoStatistical combination of extreme loadsrdquo Journalof Structural Engineering vol 103 no 5 pp 1079ndash1093 1977

[8] M Ghosn F Moses and J Wang ldquoDesign of highway bridgesfor extreme eventsrdquo NCHRP 489 Transportation ResearchBoard National Research Council Washington D C USA2003

[9] S E Hida ldquoStatistical significance of less common load combi-nationsrdquo Journal of Bridge Engineering vol 12 no 3 pp 389ndash393 2007

[10] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loadsmdashpart I comprehensive reliability and par-tial failure probabilitiesrdquo Journal of Earthquake Engineering andEngineering Vibration vol 11 no 3 pp 293ndash301 2012

[11] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loads Part II Examples for live and earthquakeload effectsrdquo Journal of Earthquake Engineering and EngineeringVibration vol 11 no 3 pp 303ndash311 2012

[12] E EMatheu D E Yule and R V Kala ldquoDetermination of stan-dard response spectra and effective peak ground accelerationsfor seismic design and evaluationrdquo Tech Rep no ERDCCHLCHETN-VI-41 2005

[13] HN Li TH YiMGu and LHuo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009

[14] B Chen Z W Chen Y Z Sun and S L Zhao ldquoConditionassessment on thermal effects of a suspension bridge basedon SHM oriented model and datardquo Mathematical Problems inEngineering vol 2013 Article ID 256816 18 pages 2013

[15] T-H Yi H-N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013

[16] Y Xia B Chen X Q Zhou and Y L Xu ldquoFieldmonitoring andnumerical analysis of Tsing Ma suspension bridge temperaturebehaviorrdquo Structural Control and HealthMonitoring vol 20 no4 pp 560ndash575 2013

[17] G Fu and J You ldquoExtrapolation for future maximum loadstatisticsrdquo Journal of Bridge Engineering vol 16 no 4 pp 527ndash535 2011

[18] A S Nowak ldquoCalibration of LRFD bridge design coderdquoNCHRPReport TransportationResearchBoard of theNationalAcademies Washington DC USA 1999

[19] F Moses ldquoCalibration of load factors for LRFD bridge evalua-tionsrdquo NCHRP Report 454 Transportation Research Board ofthe National Academies Washington DC USA 2001

[20] J Zhao and H Tabatabai ldquoEvaluation of a permit vehiclemodel using weigh-in-motion truck recordsrdquo Journal of BridgeEngineering vol 17 no 2 pp 389ndash392 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

Acknowledgments

This study is jointly funded by Basic Institute ScientificResearch Fund (Grant no 2012A02) the National NaturalScience Fund of China (NSFC) (Grant no 51308510) andOpen Fund of State Key Laboratory Breeding Base of Moun-tainBridge andTunnel Engineering (Grant no CQSLBF-Y14-15) The results and conclusions presented in the paper areof the authors and do not necessarily reflect the view of thesponsors

References

[1] P R China Ministry of Communications ldquoGeneral code fordesign of highway bridges and culvertsrdquo Tech Rep JTG D60-200 China Communications Press Beijing China 2004

[2] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[3] C J Turkstra and H O Madsen ldquoLoad combinations incodified structural designrdquo Journal of Structural Engineeringvol 106 no 12 pp 2527ndash2543 1980

[4] ACI Publication SP-31 Probabilistic Design of Reinforced Con-crete Buildings American Concrete Institute Detroit MichUSA 1971

[5] B Ellingwood Development of a Probability Based Load Cri-terion for American National Standard A58-Building CodeRequirements forMinimumDesign Loads in Buildings andOtherStructures US Department of CommerceNational Bureau ofStandards 1980

[6] Y K Wen Development of Reliability-Based Design Criteria forBuildings Under Seismic Load National Center for EarthquakeEngineering Research Buffalo NY USA 1994

[7] Y-K Wen ldquoStatistical combination of extreme loadsrdquo Journalof Structural Engineering vol 103 no 5 pp 1079ndash1093 1977

[8] M Ghosn F Moses and J Wang ldquoDesign of highway bridgesfor extreme eventsrdquo NCHRP 489 Transportation ResearchBoard National Research Council Washington D C USA2003

[9] S E Hida ldquoStatistical significance of less common load combi-nationsrdquo Journal of Bridge Engineering vol 12 no 3 pp 389ndash393 2007

[10] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loadsmdashpart I comprehensive reliability and par-tial failure probabilitiesrdquo Journal of Earthquake Engineering andEngineering Vibration vol 11 no 3 pp 293ndash301 2012

[11] Z Liang and G C Lee ldquoTowards multiple hazard resilientbridges a methodology for modeling frequent and infrequenttime-varying loads Part II Examples for live and earthquakeload effectsrdquo Journal of Earthquake Engineering and EngineeringVibration vol 11 no 3 pp 303ndash311 2012

[12] E EMatheu D E Yule and R V Kala ldquoDetermination of stan-dard response spectra and effective peak ground accelerationsfor seismic design and evaluationrdquo Tech Rep no ERDCCHLCHETN-VI-41 2005

[13] HN Li TH YiMGu and LHuo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009

[14] B Chen Z W Chen Y Z Sun and S L Zhao ldquoConditionassessment on thermal effects of a suspension bridge basedon SHM oriented model and datardquo Mathematical Problems inEngineering vol 2013 Article ID 256816 18 pages 2013

[15] T-H Yi H-N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013

[16] Y Xia B Chen X Q Zhou and Y L Xu ldquoFieldmonitoring andnumerical analysis of Tsing Ma suspension bridge temperaturebehaviorrdquo Structural Control and HealthMonitoring vol 20 no4 pp 560ndash575 2013

[17] G Fu and J You ldquoExtrapolation for future maximum loadstatisticsrdquo Journal of Bridge Engineering vol 16 no 4 pp 527ndash535 2011

[18] A S Nowak ldquoCalibration of LRFD bridge design coderdquoNCHRPReport TransportationResearchBoard of theNationalAcademies Washington DC USA 1999

[19] F Moses ldquoCalibration of load factors for LRFD bridge evalua-tionsrdquo NCHRP Report 454 Transportation Research Board ofthe National Academies Washington DC USA 2001

[20] J Zhao and H Tabatabai ldquoEvaluation of a permit vehiclemodel using weigh-in-motion truck recordsrdquo Journal of BridgeEngineering vol 17 no 2 pp 389ndash392 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of