1. Introduction

To overcome the challenges, many efforts  have been made in the past decades. However, because numbers of assumptions have to be made in each model, no general conclusions can be drawn about satisfactory approach to deal with load combination of earthquake load and truck load. In more recent papers, a methodology is proposed by Liang and Lee [10, 11]; however, its accuracy is yet to be substantiated.

One objective of this paper is to describe a methodology to handle truck load and earthquake load combinations. Earthquake load is modeled using seismic risk analysis. Truck load is modeled using Stationary Poisson processes based on the BHMS and statistical analysis. Two numerical examples of truck load and earthquake load combinations are used to illustrate the methodology.

A number of variables describe the effects that earthquakes have on bridges, such as the intensity of acceleration, the rate of earthquake occurrences, the natural period of the bridge, the seismic response coefficient, and the response modification factor. In order to explain the methodology of load combinations, only the intensity of acceleration and the rate of occurrence are chosen as the main variables.

Based on the Poisson process assumption, the probability of exceedance (Pe) in a given exposure time (Te) is related to the annual probability of exceedance (λ) by ,(1)Pe=1-e-λ·Te.Because the number of earthquakes varies widely from site to site, they are converted to Peak ground acceleration (PGA) and the return period curve (TR=1/λ, TR is return period). The cumulative probability of an earthquake in time T can be written as(2)P=e-T/TR.The PGA and frequency of exceedance curve can be obtained from US Geological Survey (USGS) mapping in the United States but cannot be obtained in China. Therefore, seismic risk analysis is used to calculate earthquake probability curve. The procedures are presented just as follows.

For more than one potential seismic source zone, suppose the parameters of the earthquake are random distributions and the probability over 1 year is a stable Poisson process. Based on the total probability theorem, the probability of exceeding a given earthquake intensity S0 in one site can be expressed by (3), by considering the uncertainties of occurrence and the upper limit magnitude: (3)PtS>S0=1-k=1Ki=1IWr,kiexp-j=1Jυk,ji·PkjWu,kj·t,where Pk(i) is the probability of the jth upper limit magnitude exceeding a given earthquake intensity S0 in potential seismic source k, νk,j(i) is the ith year occurrence probability of the jth upper limit magnitude in potential seismic source k, Wr,k(i) is the weight of the ith year occurrence probability in potential seismic source k, and Wu,k(j) is the weight of the jth upper limit magnitude in potential seismic source k.

The earthquake intensity could be acceleration, velocity, or displacement. For acceleration, S=lna (S is earthquake intensity; a is acceleration).

Because of the uncertainties of direction impact of potential seismic source zones, for t=1 year, the probability of exceedance is(4)P1S>S0=k=1Ki=1Ij=1Jυk(i)Pk(j)Wr,k(i)Wu,k(j),where Pk(j) is the conditional probability of the jth upper limit magnitude.

For disperse potential seismic source areas, the probability of the jth upper limit magnitude can be expressed as (5)PkjS>S0|Mu,k=k=1KPξjS>S0|RΔAξAk,where Ak is the area of the potential seismic source k; ΔAξ is the area of zone x. If the occurrence probability of 1 year is divided by the weights in each upper limit magnitude of potential seismic source zone, the exceedance probability of the jth upper limit magnitude of potential seismic source k can be given as (6)Pξ(j)=nξnKrΦn+1′′-Φn′′+e-βMnjΦn-e-βMn+1jΦn+1ξme-βMnj-e-βMn+1j+Φ0.According to the seismic belt materials and reports, the southeast coastal area of China has two I degree seismic areas, namely, the South China seismic area and the South China Sea seismic area. The seismic belt of southeast coastal areas of China is located south of the middle Yangtze River seismic belt, bordering on the seismic region of the Tibetan Plateau on the west, and includes Kwangtung province, Hainan province, most of Fujian and Guangxi provinces, and part of Yunnan, Guizhou, and Jiangxi provinces. Crustal thickness ranges between 28 and 40 km, gradually increasing from the southeast coastal area of China to the northwest mountains. An internal secondary elliptical gravity anomaly is relatively developed in the earthquake zones. There are no obvious banded anomalies except the gravity gradient zones of southeast coastal areas and Wuling Mountain. In the zones, magnetic anomalies change gently and there are no larger banded anomalies. Because the southeast coastal areas of China are in the same seismic belt and most of the areas in the zone have a PGA seismic fortification level of 0.1 g, Shenzhen city is then used for the basic earthquake probability calculation and comparison in this paper. The South China belt is shown in Figure 1. From Figure 1, it can be seen that there are many higher than Ms 6.0 earthquakes in the southeast coastal areas of China present in the seismic analysis. Earthquake load is still the main load considered for bridge design in these areas.

The 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) The southeast costal seismic belt (South China coastal seismic belt). (7) The seismic zone in the middle reach of Yangtze River. (8) Seismic area of the west of Taiwan. (9) Seismic area of the east of Taiwan. (10) Boundaries of seismic zone. (11) Boundaries of seismic belt. (12) South China coastal seismic rupture. (13) Epicenter of Ms 7 and above. (14) Epicenter of Ms 6.0–6.9. (15) Epicenter of Ms 5.0–5.9. (16) Epicenter of Ms 4.7–4.9. (17) Calculation site.

Based on (3) to (6), the annual exceedance probability of Shenzhen is shown in Figure 2.

Annual exceedance probability of PGA.

Assume that the probability density of earthquake load intensity in time T follows a distribution defined as fT(y), where y is a variable of PGA intensity. Based on the Poisson process assumptions, the cumulative probability function Ft(y) over interval t can be obtained using the following:(7)Fty=FTyt/T.The probability density function can then be derived as (8)fty=Ftyy=tT·FTyt/T-1·fTy.Note that t and T should have the same dimension.

Studies on truck load have been difficult historically, principally because weighing equipment was lacking and the data are correspondingly rare [13, 14]. Fortunately, the installation of BHMS is required on newly built long-span bridges, including the weighing-in-motion (WIM) system . Time, gross weight, axle weight, wheel base, velocity, and so forth are measured and collected. The probability model of truck load can be obtained through statistical analysis.

Nowak  indicated that at a specific site heavy trucks may have an average number of 1000, which is also discussed by Ghosn. Moses  suggested heavy trucks follow a normal distribution with a mean of 300 kN and a standard deviation of 80 kN (coefficient of variable, COV = 26.5%). Zhao and Tabatabai  discussed the local standard vehicle model, using data from about six million vehicles in Washington, which can be used as a reference for a truck load model. In this paper, the truck load model is obtained through data mining from measured WIM data from three bridge sites in Hangzhou, Xiamen, and Shenzhen (Figure 3). Truck load data of 5 axles and more in the three sites are filtered and selected. Through WIM data analysis, truck load probability characteristics of Hangzhou, Xiamen, and Shenzhen are similar, even the shape of truck load probability curves. Considering the three sites have very similar traffic flow, almost equal to 1.0, truck load probability curve of Shenzhen City is used for analysis and validity of subsequent case studies. Truck load probability curve is shown in Figures 4 and 5. The fitted curve is obtained using normal distribution, whose mean value is 294.9 kN, and the coefficient of variance is 37.4%.

Layout of monitoring sites.

Histogram of the monitoring sites.

Curve fitting of truck load density.

For a typical bridge, the truck load will consist of a varying number of trucks on the bridge. The probability function for such a bridge can be obtained using following analysis. Assume A is a set consisting of the elements A1,A2,,Am, which present m events, and the probability of A is P(A), while B is a set consisting of the elements B1,B2,,Bn, which present n events, and the probability of B is P(B); so the probability P(A+B) presents the probability of intensity (A+B). Then P(A+B) can be calculated by(9)PA+Bk=iPAi·PBk+1-i.Note that the length of P(A) is m and the length of P(B) is n. The sum is over all the values of i which lead to legal subscripts for A(i) and B(k+1-i), where k is the kth (A+B)(k), i=max(1,k+1-n):min(k,m). Equation (9) reflects the probability of combining two sets, and when it comes to a series of sets Φ=Φ1+Φ2++ΦN, (9) can be extended to N dimensions,(10)PΦϕ1+ϕ2++ϕN=PΦ1ϕ1·PΦ2ϕ2·PΦ3ϕ3·PΦNϕN,where Φi in (10) is the ith set of event and PΦi(ϕi) is the probability of set Φi.

Based on total probability theory and Poisson processes, the truck load intensity function for an interval t can be calculated using the following:(11)FΦϕt=PΦ0+NPΦ1+Φ2+ΦN·pN,where PΦ0 is the probability with no truck passing on the bridge; N=1,2, maximum number of trucks. PΦ1+Φ2+ΦN is the probability of varying number of trucks passing the bridge; p(N) is the probability of occurrence of N trucks on the bridge.

4. Model of Combination

The intensity of dead load is usually defined as a time independent variable, and that of truck load is a time dependent variable, both of whom follow normal distributions [5, 8, 18]. In this paper, a normal distribution is used for dead load, which is considered to maintain more or less the same magnitude, such that it can be treated as a random time independent variable.

Dirac Delta function is introduced to deal with the characteristics of X and Y in small t interval:(15)δx=+,x=0,0,x0.Therefore, the probability density can be illustrated as(16)fix=PXi,0δx+fix,i=1,2.Note that PXi,0 is the probabilities of Xi (i=1,2) in its “zero points”; namely, the events do not happen (e.g., the maximum trucks on the bridge are 8; PXi,0 is the probability of no trucks on the bridge). fi(x) is its probability density functions without “zero points.” Then, the cumulative probability functions of X1 and X2 can be calculated through(17)Fix=PXi,0-xδτdτ+-xfixdτ=PXi,0+-xfixdτ,where i=1,2.

Based on (16) and (17), the probability density of X then can be grouped as(18)fX1,2x=-+PX1,0δτ+f1τ×PX2,0δx-τ+f2x-τdτ=-+PX1,0PX2,0δτδx-τ+PX1,0δxf2x-τ+PX2,0δx-τf1τ+f1τf2x-τdτ=PX1,0PX2,0δx+PX1,0f2x+PX2,0f1x+-+f1τf2x-τdτ,where based on the characteristic of Dirac Delta function, which is -+δ(x)dx=1, (18) can be converted to cumulative probability function. Assume the cumulative probability function of FX1,2(x); then(19)FX1,2x=-xfX1,2xdx=PX1,0PX2,0+PX1,0F2x+PX2,0F1x+-+F1τf2x-τdτ.From (19), it is clear that the combined load probability consists of four parts: events X1 and X2 are not happening; event X1 is happening while X2 is not happening; event X1 is not happening while X2 is happening; and X1, X2 are both happening.

To further simplify the discussion without losing generality, the probability of two loads occurring simultaneously is neglected. Thus (19) is simplifying to(20)FX1,2x=-xfX1,2xdxPX1,0PX2,0+PX1,0F2x+PX2,0F1x.Then the cumulative probability function, FX1,2,max,T(x), of maximum value of load combinations, X1,2,max,T(x), in time T can be obtained,(21)FX1,2,max,Tx=FX1,2xtT/t.When the number of loads is more than two and these loads apply to bridge directly while satisfying Poisson process, (14) can be extended to n dimensions. Assume f1,f2,,fn are the probability densities functions of X1,X2,,Xn;  F1,F2,,Fn are the cumulative probability functions of X1,X2,,Xn;  f1,f2,,fn are the probability density functions of X1,X2,,Xn without “zero” points; fX(x) and FX(x) are the probability density function and cumulative probability function. Then the probability density function is deduced as(22)fXx=f1xf2xfnx,where the cumulative probability FX,max(x), similar to that given by (19), is(23)FXx=-xfXxdx=-xf1xf2xfnxdx.Equation (23) can account for load combinations of all loads, which satisfy the first three assumptions. If more than two events occurring simultaneously can be neglected, (23) reduces to(24)FXx=-xfXxdxPX1,0PX2,0PXn,0+F1PX2,0PXn,0++PX1,0PX2,0PXn-1,0Fn.Then the maximum value of X in time T can be obtained by(25)FXmax,Tx=FXxtT/t,where FX(x)t is FX(x) in t interval; FXmax,T(x) is the cumulative probability function of maximum value of X in time T.

Although in our study emphasis is given to formulate the “demand” to establish load combinations, all events must address a capacity issue of the bridge. For example, the earthquake load and truck load combination on a bridge column can either consider the vertical load or the column base shear load. Theoretically, (23) can deal with most load combinations, but as more loads are considered, a more conservative design will be adopted. Based on the methodology and assumptions described above, the maximum load can be combined and the procedures are summarized as follows:

determine truck load and earthquake load distributions over a particular period;

using Poisson processes, convert earthquake load and truck load distributions over a particular period to a sufficiently small interval t;

using (23), earthquake and truck load combinations over an interval t can be obtained;

using (25), the load combinations over an interval t can be converted to the bridge service life interval T.

5. Numerical Examples Example 1.

Using the method of load combination described in the preceding section, a simple example of horizontal load combination is presented here. Profiles of the typical bridge are shown in Figures 6 and 7. The weight of the superstructure at each column is 538 tons, the eccentricity of truck load is 5.0 meters, and the effects of soil and secondary effects of gravity are ignored. Furthermore, it is assumed that the maximum number of trucks on one lane is two. The results are given in Figures 8, 9, 10, 11, 12, 13, and 14. The interval t is 10 seconds, and the average daily truck traffic (ADTT) is about 1947.

Longitudinal profile of the typical bridge.

Transverse profile of the typical bridge.

Probability curve of each truck load effect.

Truck load probability density curve for varied number of trucks.

Probability of passing truck number simultaneously in t interval on the bridge.

Combined probability curves for truck load in earthquake load duration.

Probability curves for earthquake load in 100 years.

Probability density of load combination in 100 years.

Cumulative probability of load combination in 100 years.

Truck and earthquake load effects are the base moment caused by trucks and earthquakes, respectively. Figure 8 shows the probability curves of each truck load effect, which has a similar shape to truck load. Figure 9 shows the truck load probability density curve for varied numbers of trucks. From Figure 10 we can see that over the interval t the probability of no truck on the bridge is much larger than the other number of trucks passing. Figure 11 shows the combined probability curves for truck load over an earthquake load duration, which indicate that the truck load over an earthquake load duration is larger than each truck load.

Example 2.

Example 2 illustrates vertical load combinations. Most of the configurations are the same as used in Example 1. The difference is in the maximum number of trucks; namely, the maximum number of trucks on one lane in this example is four. The results are shown in Figures 15, 16, 17, 18, 19, and 20. Figures 13 and 14 are the results of truck load. Note that in this example the t interval is 10 seconds and T is taken as 100 years.

Vertical truck load probability density curves for varied number of trucks.

Probability of passing truck number simultaneously in t interval on the bridge with maximum number 4.

Probability density of vertical load combination in 100 years.

Cumulative probability of vertical load combination in 100 years.

Probability density of dead, truck, and earthquake load combination.

Probability curves of dead, truck, and earthquake load combination.

Figures 15 to 16 show vertical truck load probability curves for varied numbers of trucks and probabilities of passing truck numbers on the bridge, over the interval t. From Figure 16, it can be seen that the probability is very low when the maximum number of trucks on one lane is four. Figures 17 and 18 show similar curve shapes with those in Figures 13 and 14, which indicate that there is the same rule in load combinations in both the horizontal and vertical directions. Comparing Figures 17 to 20, though the maximum number of trucks is four, because dead load is combined with truck and earthquake load in the gravity direction, the dead load contributes a substantial portion in vertical load combinations.

6. Conclusions

This paper describes a method to combine earthquake load and truck load in the service life of bridges. The following conclusions can be drawn.

Given the more than 70% seismic areas in China, earthquake load is a main consideration for bridge design in the southeast coastal areas of China. The earthquake load probability curve is obtained using seismic risk analysis.

Using measured truck load data from BHMS, multimodal characteristics of truck load are analyzed. The truck load density of each truck is obtained by curve fitting. Considering that truck load may consist of varying numbers of trucks, truck load is calculated through traffic analysis.

In this method, the maximum value of combined load is defined as Xmax,T=maxT[X1+X2+Xn]t, which means (X1+X2+Xn) over the interval t is first combined and then the maximum value in the bridge service life is determined. Xmax,T is based on probability, which covers all the probability combinations of the combined situations. In this method, a Dirac Delta function is introduced to deal with X over a small interval t. To demonstrate the methodology intuitively, examples of load combinations in horizontal and vertical directions are provided.

The shape of the earthquake and truck load combination is similar to that of truck load alone, but the curve is displaced to the right, which means the mode and mean value of truck and earthquake load combination in this example is larger than that of truck load alone. This also illustrates that truck load is more sensitive to bridge design and over most ranges truck load is larger than earthquake load in this area.

The curve from direct load combined over 100 years is further away from the curve obtained using the method in the paper, with the direct combination of truck load and earthquake load over 100 years giving much larger values. It is not suggested that this method be used in bridge design considerations.

Because dead load is combined with truck and earthquake load along the direction of gravity, the dead load contributes a substantial portion in vertical load combinations.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This study is jointly funded by Basic Institute Scientific Research Fund (Grant no. 2012A02), the National Natural Science Fund of China (NSFC) (Grant no. 51308510), and Open Fund of State Key Laboratory Breeding Base of Mountain Bridge and Tunnel Engineering (Grant no. CQSLBF-Y14-15). The results and conclusions presented in the paper are of the authors and do not necessarily reflect the view of the sponsors.