Evaluation of dynamic vehicle axle loads on bridges with different surface conditions
Introduction
The bridges are dynamically loaded when vehicles travel on them. In practice, usually a moving static load is used to model the vehicle force on bridges. The static vehicle loads are, however, increased by a dynamic amplification factor (DAF) to account for the dynamic effects from the vehicle vibrations due to the interaction between the vehicle and bridge. Some codes define the DAF as a value between 0 and 0.4 for different vehicle type, such as the design code in Australia [5]. Other codes define it as a function of bridge span length, such as the AASHTO code [1]. These approaches avoid dynamic response analysis of the vehicle vibrations when moving along the bridge, and allow for a straightforward estimation of the vehicle loads in the bridge design. However, in reality the dynamic load due to the interaction between the vehicle and bridge is a complex problem and affected by many factors such as vehicle speed, road surface roughness, and dynamic properties of both the vehicle and bridge. Defining a DAF value based only on the vehicle type or the bridge span length might not lead to a reliable prediction of the dynamic loads on the bridges. In fact values of the DAF obtained from different references are quite different as discussed by Pesterev et al. [23] and Ashebo et al. [3].
Since the available field test data is limited and the test is costly, numerous theoretical derivations and numerical simulations have been carried out to calculate the bridge responses to vehicle loads. For example, with the assumption that the mass of the vehicle is small as compared to the mass of the bridge so that the interaction effect between the vehicle and bridge is insignificant, some researchers modelled the vehicle load as a static force moving along the bridge at a constant speed [26], [12], [7], [24]. Other researchers modelled the vehicle as a moving mass when the inertia force of the vehicle mass cannot be ignored [2], [6]. To account for the dynamic effects of the vehicle vibration when travelling along the bridge, dynamic vehicle models with different number of degrees-of-freedom (DOF) have also been developed to analyse the interaction between the vehicle and bridge [27], [14], [18], [19].
The vehicle dynamic response is induced when it travels along an uneven road surface. In most previous studies [27], [14], [18], [19], the road surface roughness history was simulated by inverse Fourier transform of the roughness power spectral density (PSD) function. The dynamic behaviour of the vehicle–bridge interaction was calculated using deterministic approaches. The limitation of these approaches is that the simulated roughness profile is only one realization of the randomly varying road surface roughness, which is therefore not sufficient to describe the possible road surface roughness effect on vehicle–bridge interaction. In the study by Hwang and Nowak [17], Chatterjee et al. [8], and Au et al. [4], Monte Carlo simulation was used to generate a set of road surface profiles to consider the randomness of the road surface roughness. The simulation was also carried out by inverse Fourier transform of the roughness spectral density function, and the vehicle–bridge interaction was solved in the time domain. Hwang and Nowak [17] used 2-D truck models and a continuous Euler-Bernoulli beam model in their study, and also considered the uncertainties of the truck type, total weight, axle distances, and speed. Chatterjee et al. [8] studied the coupled vertical–torsional dynamic response of multi-span suspension bridges due to vehicular movement, and considered 2D and 3D vehicle models. In the study by Chan et al. [9], [10] the shell with eccentric beam element was used to model the bridge, and a 3D vehicle model considering both pitching and twisting modes was used to simulate the moving vehicle. The results were validated with field test. Comprehensive parametric calculations were carried out. However, the vehicle–bridge interaction was also solved in the time domain with the road surface roughness profiles simulated by inverse Fourier transform of the roughness spectral density function.
The road surface roughness is essentially random in nature and is assumed to have properties of a stationary process. Therefore, the vehicle–bridge interaction is also random. A few models have been proposed for evaluating the dynamic force in random nature. Fryba [11] investigated the non-stationary random vibration of a beam subjected to a moving random force. The statistical characteristics of the first and second order moment of the deflection and bending moment of the beam were calculated by the correlation method. Zibdeh [28] investigated the random vibration of a simply supported elastic beam subjected to random forces moving with time-varying velocity. Li et al [20] developed an approach for analysing the evolutionary random response of a coupled vehicle–bridge system. The vehicle is taken as a 2-DOF mass–spring system and the bridge is taken as a simply supported uniform beam. Lin and Weng [21] presented a spectral approach to evaluate the dynamic vehicle load due to the passage of a vehicle moving at a constant speed along the bridge. A one-quarter vehicle model is used in the analysis. The effects of vehicle speed and pavement roughness on the variation of dynamic vehicle load were investigated.
In this study, an evolutionary spectral approach for evaluating the dynamic deformation of the bridge and vehicle axle loads due to the passage of a vehicle moving along a rough bridge surface is developed. The input force to the bridge is divided into a deterministic part from the gravity of the vehicle and a random part related to the pavement roughness of the bridge. The road surface roughness is simulated by the displacement PSD provided by ISO 8608 [16]. Each part is calculated separately using the Runge–Kutta method. Results from a 2-DOF vehicle model moving on a simply supported uniform beam with a smooth surface [13] are used to calibrate the algorithm. The random response calculated with this model illustrates the influence of the road surface roughness on vehicle bridge interaction analysis. Then a more realistic 4-DOF vehicle model is employed to study the dynamic response of the system. The results from the two vehicle models are compared and the reliability of each model in simulating vehicle–bridge interaction is discussed. The DAF and DLC are calculated to estimate the magnitude of the dynamic displacement of the bridge and the dynamic axle force. The effects of the vehicle speed, road surface roughness, vehicle axle spacing and bridge fundamental frequency on vehicle–bridge interaction are studied.
Section snippets
Formulation of the vehicle and the bridge model
In this study, the response of the bridge and vehicle are described by two separate sets of equations, which are coupled by the interaction force at the location of their contact point. The equations are then combined to form a fully coupled system. The system varies with time due to the vehicle moving along the bridge and vibrating in the vertical direction. The system equations are solved step by step in the time domain.
Differential equations for the coupled vehicle–bridge system
With Eq. (9), Eqs. (2), (3) can be combined together and written in a matrix form as follows:whereThe excitation f includes two parts, a deterministic excitation fd due to the moving vehicle weight and a random excitation fr due to vehicle vibration caused by the road surface roughness. They are expressed as
Example 1: quarter vehicle model
To verify the proposed method, a dynamic analysis is performed first for a vehicle–bridge system available in the literature [13] as shown in Fig. 3, which is a 1-axle 2-DOF vehicle model moving at a constant speed along a simply supported beam. The parameters of the vehicle and beam are given in Table 2.
Parametric study
The effect of the road surface roughness, vehicle speed, vehicle and bridge vibration frequencies and vehicle axle spacing on the vehicle–bridge interaction, and hence on the dynamic bridge responses are analysed in this section through two commonly used parameters in bridge design, i.e., the DAF and the dynamic load coefficient (DLC). The DLC is used to study the magnitude of the dynamic axle loads, which is defined as is the maximum value of the axle load, including the static
Comparison of DAF with code values
Fig. 17 shows different DAF values versus bridge fundamental frequencies provided by AASHTO [1], [5] and SIA 160E [25]. The AASHTO load and resistance factor design specifications recommend a DAF of 1.33 for dynamic increment. The Australian Standard [5] provides a uniform value of 1.4 for increment of static wheel and axle load. The DAF value in Switzerland standard [25] is 1.8 in the frequency range of 2–4 Hz and 1.4 when the fundamental frequency of the bridge is greater than 5 Hz.
In Fig. 13,
Conclusions
An evolutionary spectral method has been developed to analyse the dynamic response of the vehicle–bridge system. The deterministic response induced by the vehicle weight and the random response induced by the road surface roughness are calculated separately. The effects of the vehicle speed, bridge fundamental frequency, road surface roughness and axle spacing on the dynamic load are discussed. The results show that the bridge surface roughness is the main factor that causes the dynamic vehicle
Acknowledgement
The first author would like to thank the IPRS scholarship to pursue a Ph.D. study in UWA.
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