A filtering-based bridge weigh-in-motion system on a continuous multi-girder bridge considering the influence lines of different lanes

Abstract

A real-time vehicle monitoring is crucial for effective bridge maintenance and traffic management because overloaded vehicles can cause damage to bridges, and in some extreme cases, it will directly lead to a bridge failure. Bridge weigh-in-motion (BWIM) system as a high performance and cost-effective technology has been extensively used to monitor vehicle speed and weight on highways. However, the dynamic effect and data noise may have an adverse impact on the bridge responses during and immediately following the vehicles pass the bridge. The fast Fourier transform (FFT) method, which can significantly purify the collected structural responses (dynamic strains) received from sensors or transducers, was used in axle counting, detection, and axle weighing technology in this study. To further improve the accuracy of the BWIM system, the field-calibrated influence lines (ILs) of a continuous multi-girder bridge were regarded as a reference to identify the vehicle weight based on the modified Moses algorithm and the least squares method. In situ experimental results indicated that the signals treated with FFT filter were far better than the original ones, the efficiency and the accuracy of axle detection were significantly improved by introducing the FFT method to the BWIM system. Moreover, the lateral load distribution effect on bridges should be considered by using the calculated average ILs of the specific lane individually for vehicle weight calculation of this lane.

Keywords

bridge weigh-in-motion / continuous bridge / fast Fourier transform / influence line / axle weight calculation

Introduction

Overloaded vehicles have a severe adverse impact on the service life of roads and bridges, which has raised extensive concerns worldwide [¹–⁴]. As shown in previous studies, more often, the proportion of overloaded vehicles ranged from 10% to 30%. Nevertheless, in certain countries, the percentage of overloaded trucks reached an extremely high level of 80%, which would lead to more frequent repairs and increase maintenance costs [²]. Given those concerns, with the increasing demand for transportation, the efficient and economical management of transport networks become increasingly important [¹,³]. Thus, a decision based on accurate vehicle information in the traffic flow is of vital importance for the protection and rehabilitation of the modern transportation system.

Bridge weigh-in-motion (BWIM) system is a useful tool to identify vehicle information (e.g., axle number, axle spacing, vehicle speed, and vehicle weight), which has been extensively used for bridge safety assessment and transport network management [⁵,⁶]. Compared with conventional pavement weigh-in-motion (PWIM) systems, the newly developed BWIM systems exhibit many benefits in acquiring live loads of moving vehicles. PWIM uses devices installed on the pavement to detect vehicle weights, which would inevitably cause some damages to the road. Also, the dynamic impact of the moving vehicle on the weighing devices would reduce the accuracy and service life of the axle weighing system. The recently developed BWIM system overcame the disadvantages of the PWIM systems by using the detecting sensors mounted under the bridge. The most significant characteristic of the BWIM system is that it is a non-destructive technology and can provide unbiased traffic data by continuously collecting the strains of the instrumented bridge [³,⁵,⁷].

Moreover, BWIM systems are portable devices, and the installation procedures are straightforward. Compared with other conventional weighing technologies, such as static weighing station and PWIM system, BWIM system is less time-consuming and more cost-effective. Besides, all the devices and sensors are installed under the bridge and protected by the girders of the bridge. Therefore, the BWIM system is more durable in harsh weather and invisible to drivers than the PWIM system.

For the application of BWIM systems, many researchers have made significant contributions [⁵–¹³]. Peters and his team [¹⁴] developed the first commercial BWIM in the 1980s known as the AXWAY. Later, Peters [¹⁵] developed another BWIM system named the CULWAY and applied this system to culvert. In Europe, the WAVE project [¹⁶] and the COST 323 action [¹⁷] significantly intrigued the upsurge of the study of the BWIM system, which caused the birth of the most popular commercial BWIM known as SiWIM system and largely improved the accuracy of the BWIM systems. Up to now, modern commercial BWIM systems are based on the Moses algorithm [³,⁶].

Since Moses first proposed the concept of BWIM in 1979 [¹⁸], significant development of the BWIM system was achieved in the following years [³,⁴,⁷,¹⁴,¹⁹–²¹]. Also, many new methods and technologies of BWIM system were explored. Ojio and Yamada [²²] proposed a new technique known as the reaction force method that the measured reaction of the support was employed to identify the axle weights. Bao et al. [²³] proposed a shear-strain-based BWIM system and successfully applied this method to an instrumented bridge. In 2016, Ojio and his team [⁴] invented an innovative contactless BWIM system by using digital cameras and image processing. In 2017, Sekiya et al. [¹³] developed a simplified portable BWIM system by using accelerometers.

For the Moses-algorithm-based BWIM system, the accuracy of influence lines (ILs) is of vital importance for the application of the BWIM system [²⁴]. To get accurate ILs, in 2003, McNulty and O’Brien [²⁵] proposed a point-by-point graphical method to obtain the ILs. However, this method needed a lot of manual works, which was mainly relied on the skill of the operators. In 2006, O’Brien et al. [²⁶] provided a new approach to calibrate ILs from direct measurement using the least squares method. In 2010, O’Brien and his research team [²⁷] used filtered measured ILs to improve the accuracy of vehicle weights identification. A comparison of the efficiency of the BWIM system using theoretical ILs and field calibrated ILs in an instrumented bridge indicated that field calibrated ILs had better conformity with actual structural properties [⁶].

A lot of the previous research mainly focused on the application of the BWIM system to short-span bridges and orthotropic bridges [³]. Studies about the long-span bridges, such as rigid bridges, continuous bridges, and cable-stayed bridges, were still rare. To acquire more appropriate ILs to improve the efficiency and the accuracy of BWIM system, Zhao et al. [⁵] investigated the impact of theoretical ILs and field calibrated ILs on the accuracy of BWIM system in 2015. Along this line of thinking, the paper conducted a case study on a newly constructed continuous multi-girder bridge by using field measured ILs to calculate vehicle weights. Also, the fast Fourier transform (FFT) method was introduced to the field test of the BWIM system.

BWIM system installation and field calibration

BWIM system and installation

The practice of using BWIM for vehicle information collecting has been growing in Europe and North America since the 1970s when Moses and his team first proposed the concept of the BWIM [⁵,¹⁸]. Early BWIM systems employed axle detectors to detect axle information of moving vehicles. Usually, the axle detectors were installed on the road surface. Therefore, early BWIM systems had traffic disruption issues similar to the PWIM systems. Later, it was found that the axle information could be identified by using the sensors mounted underneath the bridge [²⁸]. Recently, the concept of free-of-axle-detector (FAD) BWIM system [¹⁶], also known as nothing-on-the-road (NOR) BWIM system, has caught much attention from researchers [¹⁰]. The FAD BWIM system uses the sensors mounted under a bridge as axle detectors precisely identify axle information. Compared with the conventional BWIM system, the FAD BWIM system is more cost-efficient and durable. In 2014, Richardson et al. [²⁹] gave an overview of the development and application of BWIM systems, and Yu et al. [³] provided a comprehensive review of theories and instrumentation of BWIM systems in 2016.

The main components of the FAD BWIM system in this study included: 1) pairs of FAD sensors mounted under the deck of the bridge in the longitudinal direction to identify the number of axles, axle spacing, and vehicle speed; 2) weighing sensors installed on the bottom of each girder to detect axle weight and gross vehicle weight (GVW); 3) smart dynamic strain recorder to collect dynamic strains; 4) a portable computer to monitor and analyze the received data. The components of the test system are presented in Fig. 1.

Field tests

The bridge selected for the field tests was a continuous multi-girder bridge, with a span length combination of 3 × 45 m (147.6 ft) = 135 m (442.9 ft), located in Qingyuan, Guangdong, China. The north side of the bridge was selected for the BWIM tests to investigate the impact of boundary conditions on the bridge responses. The test span has two different boundary conditions at both north and south ends. As shown in Fig. 2, the north part of the test span has a simply-supported end, and the south part of which has a continuously supported end. Figure 3 exhibits the elevation of the instrumented bridge and sensor positions on the test span.

Three pairs of FAD sensors were mounted on the top flange, 5.0 m (16.4 ft) apart in the longitudinal direction, to detect axle spacing, axle numbers, and vehicle speed. It was pointed out that the FAD sensors should be installed near lane markings under the deck (Fig. 4), since the FAD sensors were sensitive to wheel positions. Four weighing sensors were installed in the longitudinal direction on the bottom of each girder, 1.0 m (3.28 ft) off the central line of the span, to avoid the impact of the diaphragm. Figure 4 illustrates the sensor positions on the test span.

The BWIM system presented in this paper was a FAD BWIM system or a NOR BWIM system. In Fig. 4, FAD3-1 denotes the 1st FAD sensor under lane 3, and FAD3-2 denotes the 2nd FAD sensor under lane 3. Similarly, FAD1-1 and FAD1-2 represent the FAD sensors under lane 1. FAD2-1 and FAD2-2 represent the FAD sensors under lane 2. The weighing sensors on Girder 1, 2, 3, and 4 were named as W1, W2, W3, and W4.

Two calibration vehicles employed in this test were 4-axle trucks, which were most commonly used in the local area. Table 1 presents the detailed information of the calibration vehicles. In the initial field calibration test, vehicle 1 ran ten times at the design speed of 60 km/h (37.28 mph) on each of three lanes. Figure 5 shows that the calibration vehicles were running on different lanes during the field test.

Axle detection technology

Principle of axle detection

The roughness of the road, the vehicle vibration, and electromagnetic interference had a negative influence on axle detection, sometimes the FAD signals with severe noise tend to trigger program failure and made the axle detection algorithm quite complicated. These issues made it hard to recognize axle information or even cause detection errors. The FFT filter as a useful tool to purify the original signals were used to address those problems for BWIM system in this study. Figure 6 shows a comparison between the original signal of FAD1-2 and processed signals with the FFT method. The four distinct peaks of FAD signals represented four axles of the calibration truck passing FAD1-2 at different time steps. There were three modes of FFT filter: a high-pass filter, a low-pass filter, and a band-pass filter. The high-pass filter was used in signal processing to allow only the frequency content of signals above the desired frequency to pass through the filter. The low-pass filter could be considered as the mirror of a high-pass filter. It allowed all frequency content below a selected frequency to pass through the filter. The band-pass filter allowed a selected spectrum to pass through the filter. By using a band-pass filter with a frequency range from 5.0 Hz to 30.0 Hz, the high-frequency noise and low-frequency noise were filtered out, and the jamming signals were effectively removed (Fig. 6). Thus the accuracy of axle detection was significantly improved by using the FFT method in the BWIM system.

Figure 7 illustrates the signals collected by FAD1-1 and FAD1-2 when vehicle 1 passed the bridge on lane 1. The peaks occurred when tires pressed over the FAD sensors. One could know that the peak signal of FAD1-1 occurred \mathrm{\Delta }T seconds later than that of FAD1-2, which meant those axles passed FAD1-2 and then FAD1-1. The results agreed with the fact that the calibration vehicle passed over the bridge from north to south. As shown in Fig. 4(a), the distance between two FAD sensors on the same lane is L= 5.0 m (16.4 ft). The time steps corresponding to four peak signals of FAD1-2 are t_{\mathrm{A}\mathrm{1}},t_{\mathrm{A}\mathrm{2}},t_{\mathrm{A}\mathrm{3}},t_{\mathrm{A}\mathrm{4}}, respectively, and the four counterparts for FAD1-1 are t_{\mathrm{C}\mathrm{1}},t_{\mathrm{C}\mathrm{2}},t_{\mathrm{C}\mathrm{3}},t_{\mathrm{C}\mathrm{4}}, respectively. This time interval that the vehicle passed section A, then section C is taken as \mathrm{\Delta }T.

(1)\mathrm{\Delta }T={\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{i=\mathrm{1}}^{n=\mathrm{4}}\left|t_{\mathrm{C}i}-t_{\mathrm{A}i}\right|},

where n represents axle numbers, and the vehicle speed is expressed as v that can be calculated as

(2)v={\displaystyle \frac{L}{\mathrm{\Delta }T}}.

The axle spacing is given by

(3)L_i=v·\left(t_{\mathrm{A}(i+\mathrm{1})}-t_{\mathrm{A}i}\right),i=\mathrm{1},\mathrm{2},\mathrm{3},

or

(4)L_i=v·\left(t_{\mathrm{C}(i+\mathrm{1})}-t_{\mathrm{C}i}\right),i=\mathrm{1},\mathrm{2},\mathrm{3}.

Detection results and analysis

The accuracy of axle detection is directly associated with the calibration of ILs and axle weight calculation. Thus, to acquire accurate axle numbers, axle spacing and the vehicle speed are of vital importance for the accuracy of BWIM system. Table 2 illustrates the identified results of axle spacing and vehicle speeds for three lanes. According to the results, the determined vehicle speeds of lanes 2, and 3 were quite close to the design speed of 60 km/h (37.28 mile/h). However, the identified vehicle speeds of lane 1 were lower than 60 km/h (37.28 mile/h). The reason was that the calibration vehicles were running at relatively low speeds for safety, considering the situation that lane 1 was near the sidewalk, and the calibration vehicle was heavy. As seen in Table 2, for the ten runs on lane 1, except for runs 5, and 6, the percentage errors were −0.1% to −2.0% for axle spacing L1, −2.5% to −0.9% for L2, and −2.4% to 3.6% for L3. For the ten runs on lane 2, the percentage errors of axle spacing L1 were −1.1% to 0.9%, those for L2 were −2.5% to 0.4%, those for L3 were −2.5% to 1.7%. For the ten runs on lane 2 the percentage errors of axle spacing L1 were −2.7% to 1.6%, those for L2 were −1.2% to 0.6%, those for L3 are −4.3% to −0.6%. The identified results indicated that the axle detection had a high accuracy with the FFT method, which was the prerequisite for ILs calibration and axle weights calculation.

IL calculation and field calibration

Algorithm for IL calculation

The ILs are a function of the effect upon a structural member due to a moving load as a function of the position of that load [³⁰]. In engineering practice, it is almost impossible to eliminate the difference between the actual structure and the ideal model. According to previous studies, theoretical ILs cannot accurately represent the actual boundary conditions and structural properties of the existing bridge [⁵,⁶]. To get reasonable and satisfactory ILs is of vital importance for improving the accuracy of the FAD BWIM system. Compared with theoretical ILs, the field calibrated ILs have better conformity with the actual condition and can reflect the properties of the instrumented bridge [⁵,⁶,²⁶]. In this study, the BWIM system took bending moment ILs, calibrated from field tests, as references to calculate axle weights.

To explore the general rule for different types of vehicles. An N-axle vehicle was selected as an example to explain the calibration process of ILs. At a specific time step k , the theoretical bending moment can be expressed as a function of time. The bending moment of the girder i is expressed as M_{ik}^{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}}.

(5)M_{ik}^{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}}=E_i{Z}_i{ϵ}_{ik}^{\mathrm{t}},i=\mathrm{1},\mathrm{2},\mathrm{3},...,N,

where E_i and Z_i are Young’s modulus and the geometric modulus of girder i, respectively. {ϵ}_{ik}^{\mathrm{t}} denotes the strain of the i\mathrm{th} girder collected by the weighing sensor at the time step k.

To simplify the operation of the matrix, all the girders are assumed to have the same properties as the internal girder, namely, E_i{Z}_i=EZ. Tiny differences are ignored. Thus, the hypothesis is applied to sum up the bending moment of each girder.

(6)M_k^{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}}=EZ{\displaystyle \sum _{i=\mathrm{1}}^g{ϵ}_{ik}^{\mathrm{t}}}=EZ{ϵ}_k^{\mathrm{t}},

where M_k^{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}} denotes the bending moment at mid-span, at a time step k, and {ϵ}_k^{\mathrm{t}} represents the sum of the theoretical strains on all girders at the time step k.

The bending moment can be expressed as another equation when the calibration vehicle with N axles passes through the instrumented bridge. The predicted bending moment expression is

(7)M_k^{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}}={\displaystyle \sum _{i=\mathrm{1}}^N{P}_i{I}_{\left(k-C_i\right)}},i=\mathrm{1},\mathrm{2},\mathrm{3},...,N,

(8)C_i={\displaystyle \frac{D_{i}f}{v}},

where M_k^{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}} denotes the predicted bending moment at mid-span. P_i is a point force of the i\mathrm{th} axle of the calibration vehicle. I_{\left(k-C_i\right)} is the IL of the bending moment at the position of weighing sensors, at a time step k. C_i is the frequency corresponding to D_i\left(C_{\mathrm{1}}=\mathrm{0}\right) , D_i denotes the distance between the first axle and the i\mathrm{th} axle, f is the scanning rate of the smart dynamic strain recorder DC-204R, v is the vehicle speed. Figure 8 shows the influence line ordinates corresponding to the 4-axle calibration truck at the time step k.

The relationship between M_k^{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}} and M_k^{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}} is given by

(9)M_k^{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}}=M_k^{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}}.

Thus, substitute Eqs. (6) and (7) into (9), we have

(10){ϵ}_k^{\mathrm{t}}={\displaystyle \frac{\mathrm{1}}{EZ}}{\displaystyle \sum _{i=\mathrm{1}}^N{P}_i{I}_{\left(k-C_i\right)}},i=\mathrm{1},\mathrm{2},\mathrm{3},...,N.

An error function f_{\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}} is defined as the square of the difference between the measured strains and theoretical strains.

(11)f_{\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}={\displaystyle \sum _{k=\mathrm{1}}^K{({ϵ}_k^{\mathrm{m}}-{ϵ}_k^{\mathrm{t}})}^{\mathrm{2}}},

where {ϵ}_k^{\mathrm{m}} denotes the sum of the field measured strains collected by weighing sensors and {ϵ}_k^{\mathrm{t}} is as previously defined.

Over the K time steps of data acquisition. By differentiating f_{\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}} with related IL ordinate I_R to minimize f_{\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}} , we have

(12){\displaystyle \frac{\partial \left({{\displaystyle f}}_{\mathrm{error}}\right)}{\partial {{\displaystyle I}}_R}}=\mathrm{0},R=\mathrm{1},\mathrm{2},\mathrm{3},...,K-{{\displaystyle C}}_N.

Summing up all the K−C_N equations obtained from Eq. (12), the equations can be combined into a matrix form and expressed as follows

(13){\mathit{W}}_{(K-C_N)\times (K-C_N)}\times {\mathit{I}}_{(K-C_N)\times \mathrm{1}}={\mathit{ϵ}}_{(K-C_N)\times \mathrm{1}},

where {\mathit{I}}_{(K-C_N)\times \mathrm{1}} denotes an ordinates vector of IL; {\mathit{ϵ}}_{(K-C_N)\times \mathrm{1}} is a vector related to measured strains and axle weights. The element of the vector at row R is

{{\displaystyle ϵ}}_R=EZ\left({{\displaystyle P}}_{\mathrm{1}}{{\displaystyle ϵ}}_R^m+{{\displaystyle P}}_{\mathrm{2}}{{\displaystyle ϵ}}_{R+{{\displaystyle C}}_{\mathrm{2}}}^m+\cdots +{{\displaystyle P}}_N{{\displaystyle ϵ}}_{R+{{\displaystyle C}}_N}^m\right),

(14)R=\mathrm{1},\mathrm{2},\mathrm{3},...,K-{{\displaystyle C}}_N,

where {\mathit{W}}_{(K-C_N)\times (K-C_N)} represents a sparse symmetric matrix related to the axle weights of the vehicle. Each element of the main diagonal is the sum of the squares of axle weight, which expressed as Eq. (15), the upper valid triangular elements are expressed as Eq. (16), at the same time, all the remaining items in the matrix {\mathit{W}}_{(K-C_N)\times (K-C_N)} are zero.

{{\displaystyle W}}_{R,R}={\displaystyle \sum _{i=\mathrm{1}}^N{{\displaystyle P}}_i^{\mathrm{2}}}={{\displaystyle P}}_{\mathrm{1}}^{\mathrm{2}}+{{\displaystyle P}}_{\mathrm{2}}^{\mathrm{2}}+\cdots +{{\displaystyle P}}_N^{\mathrm{2}}

(15)\left(R=\mathrm{1},\mathrm{2},\mathrm{3},...,K-{{\displaystyle C}}_N\right),

W_{R,R+\left(C_j-C_k\right)}=P_j{P}_k\mathrm{;}R+\left(C_j-C_k\right)\le K-C_N\mathrm{;}j>k,

(16)R=\mathrm{1},\mathrm{2},\mathrm{3},...,K-{{\displaystyle C}}_N\mathrm{;}j=\mathrm{1},\mathrm{2},\mathrm{3}...N\mathrm{;}k=\mathrm{1},\mathrm{2},\mathrm{3},...,N.

When the vehicle is going across the bridge, the vector of ILs ordinates is calibrated based on the Eq. (17) which derived from Eq. (13).

(17){{\displaystyle \mathit{I}}}_{\left({K-C}_N\right)\times \mathrm{1}}={{\displaystyle \mathit{W}}}_{\left({K-C}_N\right)\times \left(K-C_N\right)}^{-\mathrm{1}}\times {{\displaystyle \mathit{ϵ}}}_{\left(K-C_N\right)\times \mathrm{1}}.

Calibrated results and ILs analysis

The ILs were calculated by using the proposed calibration algorithm of ILs. During the field tests, ten repeated runs of vehicle 1 were recorded on lanes 1, 2, and 3, separately. Runs 5 and 6 of lane 1, and run 2 of lane 2 were left out due to insufficient data recording. Recorded data of runs 3, 4, and 5 of lane 3 were omitted due to serious interference of another heavy vehicle during the test. So, there were twenty-four sets of valid runs in total, and twenty-four independent ILs were obtained.

The calibrated IL of each run on lanes 1, 2, and 3 were illustrated in Figs. 9–12, separately. The calibrated ILs for each lane show significant repeatability for all runs in the specific lane, which demonstrated that the ILs calibration method was reliable based on field measured bridge responses. As shown in Fig. 9, the test span was 45 m (147.6 ft) from Pier 10 to 11, the selected ILs ranged from 30 m (98.4 ft) before the north head to 30 m (98.4 ft) after the south end. Thus the total length of the ILs was 105 m (344.5 ft) long. The missing data were replaced by zero to keep the uniformity of the ILs of each independent run. It should be pointed out that because of the roughness of the road, there was a shock effect at Pier 11 when the vehicle crossed through the bridge from north to south (Fig. 9). But the shock effect had no impact on vehicle weights calculation because only the valid 45€m ILs was used to calibrate the vehicle weights. It was noteworthy that the range of the ILs used to calculate axle weights was from the north head to the south end of the test span with the length of 45 m (147.6 ft), that is to say, the ILs ordinates of the whole test span were employed in the following axle weight calculation. The average ILs of lanes 1, 2, and 3 show highly consistent with the independent runs on lanes 1, 2, and 3, separately, which meant that the average IL of each lane could reflect the unique structural characteristics of this lane, and could be chosen as a reference to calculate axle weights of the vehicle.

For the first ten runs on lane 1, the driving direction was from north to south. However, the 2.5% longitudinal slope of the bridge results in a relatively long acceleration distance for the calibration vehicle to reach the design speed. For the following 20 runs, the vehicles were driving from south to north to shorten the acceleration distance of the calibration vehicle, and theoretically, such a change had almost no influence on the calibration tests. As shown in Figs. 9 and 10, the turning of the driving direction led to a data flipping of the ILs. Before Pier 10 with simply-supported boundary, the ordinates of the ILs were almost zero, and the ordinates of the ILs were negative between Piers 11 and 12 with continuous support, which shows high consistency with the actual boundary conditions of the bridge. Also, the calibrated ILs for lanes 2 and 3 show significant repeatability for all runs, which demonstrated that the ILs were reliable based on field measured strains.

Eight valid runs on lane 1, nine valid runs on lane 2, and seven valid runs on lane 3 were averaged for each lane, respectively, to minimize the accidental errors of the calibrated ILs. Moreover, the three averaged ILs were unified in the same direction (south to north) by using the data sorting method (Fig. 12). A comparison of the averaged ILs of lanes 1, 2, and 3, the shapes of the three calibrated ILs were roughly the same. However, a distinct difference still could be observed, which indicated that the lateral load distribution effect on the bridge should be taken into serious consideration. To further improve the accuracy of the BWIM system, the ILs of lanes 1, 2, and 3 should be calibrated separately, considering the lateral load distribution effect for the axle weights calculation. This issue was further discussed in section 5.

By using calibrated IL, the predicted strains could be calculated. The predicted strains almost coincided with the curves plotted by using actual measurement data (Fig. 13), which demonstrated the effectiveness of the ILs calibration algorithm.

Axle weight identification and field verification

Algorithm for axle weight identification

Axle weight identification can be regarded as an inverse operation of ILs calibration. Many factors can influence the accuracy of vehicle weights identification. In 2015, Zhao et al. [⁶] concluded that the accurate ILs obtained from the field calibration were the first prerequisite for axle weighing technology that should be taken into serious consideration. Based on the Moses algorithm and the collected data of actual bridge responses, acquired ILs in the field test were considered as a reference to calculate axle weights when the vehicles passed over the instrumented bridge [¹⁸]. The axle weights and GVW were obtained by using the least squares method to minimize the error function of f_{\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}.

The same as the Eqs. (9) and (10), the relationship between M_k^{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}} and M_k^{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}} can be expressed as

(18)M_k^{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}}=M_k^{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}}=EZ{ϵ}_k^{\mathrm{t}}={\displaystyle \sum _{i=\mathrm{1}}^N{P}_i}{I}_k^i,

where I_k^i is the IL ordinates for the i\mathrm{th} axle at a certain position at a time step k, and other variables are as previously defined (Fig. 8).

At the time step k , the theoretical strain {ϵ}_k^{\mathrm{t}} , at the middle of the test span where the weighing sensors located, can be derived from Eq. (18) and expressed as

(19){ϵ}_k^{\mathrm{t}}={\displaystyle \frac{\mathrm{1}}{EZ}}{\displaystyle \sum _{i=\mathrm{1}}^N{P}_i{I}_k^i},k=\mathrm{1},\mathrm{2},\mathrm{3},...,K.

From Eq. (19), there are K theoretical strains can be obtained for time steps, which can be expressed as a matrix form as follows

\left\{\begin{array}{c}\hfill {{\displaystyle ϵ}}_{\mathrm{1}}^{\mathrm{t}}\hfill \\ \hfill {{\displaystyle ϵ}}_{\mathrm{2}}^{\mathrm{t}}\hfill \\ \hfill ⋮\hfill \\ \hfill {{\displaystyle ϵ}}_{K-\mathrm{1}}^{\mathrm{t}}\hfill \\ \hfill {{\displaystyle ϵ}}_K^{\mathrm{t}}\hfill \end{array}\right\}=

(20){\displaystyle \frac{\mathrm{1}}{EZ}}

where K is the total number of strain recording, also, the total number of the time steps. Equation (20) can be written as a simple form

(21){{ϵ}^{\mathrm{t}}}_{K\times \mathrm{1}}={\displaystyle \frac{\mathrm{1}}{EZ}}{I}_{K\times N}{P}_{N\times \mathrm{1}},

where {\left\{{ϵ}^{\mathrm{t}}\right\}}_{K\times \mathrm{1}} denotes the vector of the theoretical strains, {\left[I\right]}_{K\times N} denotes the matrix of IL ordinates and {\left\{P\right\}}_{N\times \mathrm{1}} represents the vector of axle weights to be calculated.

The error function f_{\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}} and other variables are as previously defined.

(22)f_{\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}={\displaystyle \sum _{k=\mathrm{1}}^K{({ϵ}_k^{\mathrm{m}}-{ϵ}_k^{\mathrm{t}})}^{\mathrm{2}}},

The error function f_{\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}} also can be written as a matrix form as Eq. (23) and then the matrix form can be simplified as Eq. (24)

(23)f_{\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}={\left\{\left\{{ϵ}^{\mathrm{m}}\right\}-\left\{{ϵ}^{\mathrm{t}}\right\}\right\}}^{\mathrm{T}}\left\{\left\{{ϵ}^{\mathrm{m}}\right\}-\left\{{ϵ}^{\mathrm{t}}\right\}\right\},

f_{\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}

(24)-

By minimizing the error function with respected to the axle weights vector, the axle weights vector can be derived

{\displaystyle \frac{\partial \left(f_{\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}}\right)}{\partial \left\{P\right\}}}=-\mathrm{2}\left({\displaystyle \frac{\mathrm{1}}{EZ}}\right){\left[I\right]}^{\mathrm{T}}\left\{{ϵ}^{\mathrm{m}}\right\}+\mathrm{2}{\left({\displaystyle \frac{\mathrm{1}}{EZ}}\right)}^{\mathrm{2}}{\left[I\right]}^{\mathrm{T}}\left[I\right]\left\{P\right\}

(25)=\mathrm{0},

(26)\left\{P\right\}=EZ{\left[{\left[I\right]}^{\mathrm{T}}\right[I\left]\right]}^{-\mathrm{1}}{\left[I\right]}^{\mathrm{T}}\left\{{ϵ}^{\mathrm{m}}\right\}.

The axle weights could be derived based on Eq. (25), and the GVW can be obtained by summing up the axle weights

(27)GVW={\displaystyle \sum _{i=\mathrm{1}}^N{P}_i}.

Identified results and axle weight analysis

ILs have a significant influence on the accuracy of axle weights calculation. In this study, axle weights of vehicle 1 were calculated based on the averaged ILs from lanes 1, 2, and 3 (Fig. 12). For the calculation of the corresponding axle weights of the vehicle on lane 1, the average ILs from lanes 1, 2, and 3 (Figs. 9–11) were employed, respectively, for cases 1, 2, and 3. The computed results based on three averaged ILs were listed in Table 3, named lane 1-case 1, lane 1-case 2, and lane 1-case 3, respectively. Similarly, regarding the vehicle on lanes 2 and 3, the three averaged ILs were applied as references to calculate the axle weights of vehicles on lane 2 for cases 1, 2, and 3, respectively. The identified axle weights for the truck on lanes 2 and 3 were listed in Tables 4 and 5, separately.

As shown in Tables 3−5: 1) A1, A2, A3, and A4 represents the 1st, 2nd, 3rd, and 4th axle of the calibration vehicle, respectively; 2) Symbols ‘GOA1’ means the first group of axles of ‘A1+ A2’, ‘GOA2’ represents the second group of axles of ‘A2+ A3’; 3) ‘GVW’ denotes the gross vehicle weight; 4) For the mean and standard deviation of the GOA1 and 2, 16 (lane 1), 18 (lane 2) and 14 (lane 3) data were used; that for GVW, eight (lane 1), nine (lane 2), and seven (lane 3) data were used. The reason for considering axle weights calculation using group axle (GOA1 and 2) instead of the single axle (A1, A2, A3, and A4) was that the axle spacing between the first two axles and last two axles were small, and the stiffness of the bridge was high. It was hard to eliminate the coupling effect of axle loads. So A1+ A2 was considered as GOA1, also, A2+ A3 was considered as GOA2. As shown in Table 1, the static weights of vehicle 1 of A1 and A2 were 8430.0 kg (18585.0 lbs) and 9090.0 kg (20040.0 lbs), respectively, the values were closed to each other. Thus, each axle weight could be approximated as half of GOA1. Similarly, A3 and A4 were 12250.0 kg (27006.6 lbs), and 12240.0 kg (26984.6 lbs), and each single axle weight could be approximated as half of GOA2.

For vehicle 1 on lane 1, the predicted weights of GOA for case 1 had a mean of −1.2% and a standard deviation of 5.2%, and that of case 2 had a mean of −17.6% and a standard deviation of 57.9%, and case 3 had a mean of −12.7% and a standard deviation of 21.6% (Table 3). The results show that the accuracy of GOA weights calculation with using the averaged IL of lane 1 was far better than that of using averaged ILs from lanes 2 and 3. When it comes to GVW, as shown in Table 3, the accuracy of case 1 was much better than that of cases 2 and 3. The same pattern could be observed in Fig. 14 that the accuracy of case 1 was the best of all.

Similarly, for axle weight calculation of vehicle 1 of lane 2, the accuracy of case 2 was far better than that of cases 1 and 3 (Table 4 and Fig. 15), and for axle weight calculation of lane 3, the case 3 had the best accuracy out of three different cases (Table 5 and Fig. 16). As shown in Fig. 14, the box plots of three cases on lane 1 shows that the volatility of the data of case 1 was much lower than that of cases 2 and 3. The results of lane 1-case 1 was far more stable than the other two, and for a vehicle on a particular lane, the accuracy of weight calculation performed the best by using the average IL of this lane. The same conclusion could be drawn from Figs. 15 and 16. That is to say, for improving the accuracy of axle weight calculation in this type of bridge, the lateral load distribution effect should be considered by calibrating IL for each lane independently.

Furthermore, another field test was conducted to validate the applicability and accuracy of BWIM system. In the field test, vehicle 2 runs at the speed of 70 km/h on lanes 1, 2, and 3, ten runs for each lane, to simulate the real traffic. The axle weights of vehicle 2 were calculated based on the vehicle 1 calibrated averaged IL of each lane to verify the accuracy of the BWIM system. As shown in Table 6, take the result of vehicle 1 on lane 3 as an example, the predicted weights of GOA had a mean of 0.5% and a standard deviation of 9.1%, and the predicted weights of GVW had a mean of −0.3% and a standard deviation of 5.5%. The percentage error of GVW ranged from −6.5% to 7.3%. The results show that the accuracy of axle weights calculation of vehicle 2 with using vehicle 1 calibrated IL of lane 3 reduced a little bit comparing with the results of lane 3-case 3 in Table 5, however, the accuracy of GVW was still good enough, which shows great potential for the application of the BWIM system to the real traffic. The same conclusion can be drawn from the other data in Table 6.

Conclusions

Early BWIM systems that use axle detectors to identify the axle spacing, axle number, and vehicle speed are now being replaced by the FAD BWIM system. The FAD BWIM devices are mounted under the bridge to identify the vehicle information without disturbing the traffic. Most of the previous studies focused on the short span bridges or orthotropic decks, the applications of BWIM system on long-span bridges are rare. This paper takes a step further to introduce the FFT method to the signal processing of the BWIM system and extend the application of the BWIM system to concrete continuous multi-girder bridges.

The axle detection technology largely influences the accuracy of BWIM. The dynamic effect, bumping, data noise, or magnetizing have a considerable impact on the original data received from the FAD sensors, which may lead to error detection. The FFT method is a useful tool that has been extensively used in the signal processing area. By using the ‘band-pass’ of FFT filter with the range of frequency from 5.0 Hz to 30.0 Hz, the high-frequency noise and low-frequency noise are filtered out, and the jamming signals are removed by filtration effectively which significantly improved the accuracy of FAD BWIM system.

In the application of the BWIM system to continuous multi-girder bridges with multiple lanes, the lateral vehicle weight distribution effect of this type of bridge should be considered by identifying axle weights of a vehicle moving on a specific lane using the ILs calibrated on this lane. By averaging the ILs from independent runs of a particular lane, the accidental errors could be minimized, which leads to an improvement of accuracy for the BWIM system.

In the application of the FAD BWIM system to the selected bridge in the field tests by identifying vehicle moving on a specific lane using the ILs calibrated on this lane (lane 1-case 1, lane 2-case 2, lane 3-case 3). One weakness of the FAD BWIM system in this type of bridge is that the individual axle weight identification becomes very difficult for the vehicle with small axle spacing. The accuracy of single axle weights is not satisfactory because of the coupling effect of the axles. However, the accuracy of the axle weights of GOA is acceptable. The overwhelming majority of the percentage errors are less than 10%. The identification of GVW expressed superior accuracy for the application of the FAD BWIM system.

Better axle detection accuracy was achieved by using the FFT method and specific field calibrated ILs. Almost all the percentage error of calculated GVW by using calibrated ILs of a particular lane to identify vehicle weight of this lane is less than 7%, which indicates that the high accuracy of GVW is entirely satisfactory for the application of BWIM to continuous multi-girder bridges.

While the proposed method in this paper provides a useful tool for improving the accuracy of BWIM system, future work will be conducted to address the vehicle-bridge interaction problems and experimentally investigate the accuracy of BWIM system under the condition that multiple vehicles are passing the bridge.

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