Experimental study on the identification of dynamic axle loads of moving vehicles from the bending moments of bridges
Introduction
The dynamic axle loads of vehicles have concerned by engineers for many years, since they are important parameters for the design of new pavements and bridges, the assessment of existing pavements and bridges, traffic studies, design code calibration, pavement and bridge maintenance planning, etc. The direct measurement of the dynamic axle loads of vehicles seems to be very expensive and difficult [1]. Moreover, the obtained axle loads might exhibit biased information, since their measurement is limited only to instrumented vehicles. Therefore, the identification of dynamic axle loads from bridge responses becomes a more attractive alternative, since it is much cheaper and provides unbiased load information.
Based on vehicle–bridge interaction models, the dynamic axle loads of a vehicle moving on a bridge can be identified from the bridge’s strains, displacements, accelerations or bending moments [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Many identification methods have been proposed and studied. Some early works employed the time domain method, the frequency and time domain method or the modal method. The time-domain method represents the axle loads on the bridge using a set of second order differential equations and identifies the axle load histories by convolution in time-domain [2]. The frequency and time domain method performs a Fourier transformation of the load–response relationship and identifies the axle load histories directly using the least-squares method [3]. The modal method expresses the bridge model in modal coordinates and obtains the axle load histories by solving uncoupled equations of the vehicle–bridge interaction [4], [5]. To obtain the identified axle loads, a solution method using either a pseudo-inverse or singular value decomposition technique is adopted [6], [7]. However, it is found that all three methods exhibit large fluctuations of the identified loads due to the measurement noises and possess numerical ill-conditions when the axle load is on the bridge supports. In addition, the methods consume excessive computing time due to the need for the inversion of large system matrices. Therefore, the least-squares method with a smoothening term, named the regularization method, is employed [8], [9]. The discrete version of the method is also utilized, using the dynamic programming technique [10]. Besides the efficiency of computation, this method eliminates the ill-conditioned problem and provides better identified axle loads under noisy inputs. Unfortunately, the method needs an optimal regularization parameter in the identification process. To obtain the parameter, a significant computational effort is required. Moreover, it was found that the optimal regularization parameter depends greatly on vehicle properties such as the vehicle mass, moving speed, vehicle configuration, etc. Therefore, in actual applications, only a sub-optimal regularization parameter can be determined.
To overcome this problem, a regularization method with an iterative technique called the updated static component (USC) technique has been proposed [11]. This method decomposes the axle loads into static and dynamic components and keeps updating the static component through the regularization of the associated dynamic component until a convergent solution is achieved. It has been shown by numerical simulations that the accuracy of the identified loads obtained from the regularization with USC technique is substantially improved compared to those from the conventional regularization. Although a broad spectrum of the vehicle parameters and bridge properties can be extensively studied through numerical simulations, it is known that real vehicle–bridge interaction behaviors might significantly differ from those obtained from the mathematical model assumed in the simulations.
This paper experimentally studies the identification of the dynamic axle loads of a moving vehicle from the bending moment histories of the bridge. Although some researchers have previously conducted experiment studies [12], [13], [14], [15], [16], [17], [18] or the actual field tests [19] for this purpose, they measured only the weights or the static axle loads of vehicles. Therefore, a comparison between the actual and the identified dynamic axle loads becomes impossible. Unlike previous investigations, the actual time-histories of the dynamic axle loads of the vehicle are measured in this study, and are directly compared with those obtained from the identifications. Identification methods using conventional regularization and that with USC are considered. The bridge is modeled by a uniform-thickness steel plate sitting on roller supports, while the vehicle is modeled by a two-axle load moving platform. The bending moments of the bridge from three sections at and are recorded during the passages of the vehicle. The axle load identification effectiveness using both conventional regularization and the regulation with USC technique are investigated under various weights and speeds of the vehicle as well as two different roughness amplitudes of the bridge surface.
Section snippets
Bending moments of a bridge under a moving vehicle
The bending moments at given sections of a simply supported bridge under a moving vehicle are considered. For convenience, a simple vehicle–bridge system that neglects the dynamic interaction between the vehicle and the bridge is employed. The moving vehicle is thus replaced by two moving concentrated loads traversing a bridge as in Fig. 1. The moving loads and represent front and rear axle loads of the vehicle, respectively. The and are their corresponding positions
Identification of dynamic axle loads
In this section, the identification of the dynamic axle loads of a moving vehicle on the bridge from the bending moments of the bridge is outlined. Although the relationship between the dynamic axle loads and the bending moments of the bridge can be established as in Eqs. (1), (2), (3), Box I, and (4), load identification from the given histories of bending moments by directly solving the equations of the system seems to be impractical: it involves the inversion of a large system matrix that
Experimental set-up
The experimental simulation of a vehicle moving on a bridge is conducted to investigate the effectiveness of the previously outlined identification methods for dynamic axle loads from the bending moments of the bridge. The experimental set-up and its photograph are shown in Fig. 4a, Fig. 4b, respectively.
A vehicle model having two axles with a spacing of 28 cm and two rubber wheels for each axle is employed. In order to measure the dynamic axle loads of the vehicle, strain gauges are installed
Experimental results
In the following experimental investigations, the identification of the dynamic axle loads of a two-axle vehicle moving on a bridge is considered. During the passage of the vehicle, the bending moments of the bridge at three sections, i.e. and as well as the position of the vehicle, are measured and are used as the inputs for axle load identification. The identification utilizes the vehicle–bridge model outlined in Eq. (1). To obtain the accurately identified results, the bridge is
Conclusion
The identification of the dynamic axle loads of a moving vehicle from the bending moments of a bridge is experimentally studied. Unlike in previous investigations, the dynamic axle loads of the vehicle are directly measured and are used in comparisons with the loads obtained from the identification via bending moments. The bridge is modeled by a uniform thickness steel plate sitting on roller supports, while the vehicle is modeled by a two-axle moving platform. During the passage of the
Acknowledgement
The authors are very grateful to the Chulalongkorn University for a Ratchadaphiseksomphot research grant for this project.
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