Keywords

1 Introduction

Current asphalt pavement design specifications in China fail to consider dynamic load in vehicle axle load analysis. Peer studies in China focused on dynamic theories as well as the magnitude of dynamic load, and significant research has been conducted. For instance, Zhong et al. [1] analysed the relationship between vehicle dynamic load and travel speed using two degree of freedom model. Huang [2], and Sun and Deng [3] investigated the relationship between pavement roughness and dynamic load. Assuming vehicle and ground structure as a synthetic system, Deng [4] developed a mathematical model to describe the contact roughness between vehicle and pavement surface. Additional studies focused on dynamic load characters and vehicle-pavement interaction force [5,6,7,8].

Numerous studies have been conducted worldwide as well regarding this topic. Dodds and Robson [9] took pavement roughness as a function of vehicle departing off flat pavement and utilized power spectral density (PSD) to calculate dynamic load [9]. Todd and Kulakowski [10] analysed dynamic load by quarter truck model and half truck model. Hunt [11] proposed a model for pavement vibrating PSD estimation. With the development of research, Watts and Krylov [12] examined vehicle vibrating characteristics when vehicles ran over pavement distress, such as pavement cracking, rutting, pot holes. By means of a three-dimensional truck model, Javier et al. [13] found that the vertical dynamic load applied by a tyre to the pavement depends not only on the profile under that wheel but also on the profile under the other wheels, mainly under the one on the same axle. Dae-Wook and Papagiannakis [14] formulated a continuous or distributed model using a two-dimensional (2D) half-truck finite element model. Using the 2D half-truck finite element model, numerical simulations were performed to obtain the dynamic loads using various parameters such as the road roughness, vehicle speed, suspension stiffness and damping in order to evaluate their individual effects on the dynamic axle load response. All these researches have supplied basic theories for dynamic load characters.

Obtaining acceleration speed of vehicle axles and tires is the basic method of obtaining dynamic load. Only a few studies obtained dynamic load by experiment tests. Chen et al. [15] invented a vehicle-pavement test rig including a vehicle model and a distributed stiffness pavement-roadbed model, and the vehicle model was simplified as a quarter of resonance source vehicle model. The dynamic response of distributed of stiffness pavement under moving resonance load and shock excitement were analysed respectively.

In this research a new testing instrument, Vehicle Axle-tire Dynamic Load Tester is employed to test vertical acceleration speed of vehicle axles and tires in different conditions. Mean square value of acceleration speed is analysed and dynamic load is obtained. Dynamic load is taken as a series of static load following normal distribution, and equivalent axles load is calculated according the Asphalt Pavement Design Specification, and the increasing rate of equivalent axle load is analysed after dynamic load is taken into account.

2 Vehicle Vibration Model and Dynamic Load

Vehicle vibration models with varying degrees of freedom are generally built to examine the dynamic load theoretically. In this study, a single degree of freedom vehicle model on a roughness pavement is established, as is shown in Fig. 1.

Fig. 1
A circuit of vehicle vibration. It has a sprung mass, suspension stiffness, and dampness coefficient labeled with letters M, K, and C.

Vehicle model with single degree of freedom

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In this model, M is the sprung mass of vehicle; K is suspension stiffness; C is dampness coefficient of tires; the pavement profile is modelled as a one-dimensional random field and is represented by the height of pavement surface irregularities q; Z is absolute displacements of sprung mass.

The dynamic load Fd is calculated by inertia mass of M as follows

$$F_{d} = MZ^{\prime}$$
(1)

where, \({Z}^{^{\prime}}\) represents the first derivative processes of Z.

It can be known that dynamic load is linearly dependent with acceleration speed of axle-tires, as frequency of dynamic load is same with acceleration of tires. So characteristics of acceleration reflect characteristics of dynamic load.

3 Axle-Tire Acceleration Field Test Design

3.1 Testing Road and Roughness Testing

Changxiao Road was selected as a low class highway and Jinan belt expressway was select as a high class highway for the field tests. Continuous pavement roughness tester was implemented on the low class highway, and laser pavement roughness tester was utilized on the high class highway. The test results are as follows: roughness variance of Changxiao road is 1.8 m/km, International Roughness Index (IRI) of Jinan belt expressway is 1.40 m/km.

Previous studies have examined the relationship between IRI and roughness variance, that is σ, the conclusions are almost the same [16,17,18] In this article it is assumed as follows [17]:

$$\sigma = 0.6\;IRI$$
(2)

According to the two formulas and testing results in fields, roughness variance of high class road equals to 0.84, low class road equals to 1.8.

3.2 Test Vehicles

A light truck and a heavy truck were selected as testing vehicles. Vehicle Axle-tire Dynamic Load Testers were installed on the axles of the two trucks, and tests were carried out when the trucks were empty and fully loaded travelling at different speeds.

3.3 Testing Method and Data Collection

Real time vehicle travel speed is obtained from GPS modules, and vertical acceleration is obtained by three-axis acceleration sensor. In field, axles of trucks are fixed with the Axle-tire Dynamic Load Tester, and then empty trucks and fully loaded trucks travel at the speed of 40, 50, 60, and 70 km/h on the low class highway, and at 60, 80, and 100 km/h on the high class highway, travel speed of trucks and vertical vibration acceleration of axles are obtained.

4 Result Analysis

4.1 Vertical Axle-Tire Vibration Acceleration Temporal Analysis

Field tests were conducted and time travel curve of vertical vibration acceleration are obtained. Figures 2 and 3 are a part of time travel curve when the light truck ran on the low class highway with the speed of 60 km/h, and on the high class highway with the speed of 80 km/h, respectively.

Fig. 2
A line graph plots vibration acceleration versus time for an empty loaded light truck traveling on a low-class highway. The lines fluctuate between negative 1 and 4 on the y-axis.

Vertical acceleration time travel curve of empty loaded light truck traveling on low class highway (60 km/h)

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Fig. 3
A line graph plots vibration acceleration versus time for a fully loaded light truck traveling on a high-class highway. The lines fluctuate between negative 6 and 6 on the y-axis.

Vertical acceleration time travel curve of fully loaded light truck traveling on high class highway (80 km/h)

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Besides, tests are carried out in all scenarios where the heavy truck is empty or full loaded with different travel speeds, time travel curves of vertical vibration acceleration are obtained. When the tests are conducted, the trucks ran at least 3 min on the testing road with sampling frequency more than 100 Hz to ensure the enough data for analysing. Figure 4 demonstrate a part of time travel curve when the heavy truck is empty and travels at 40 km on the low class highway, and Fig. 5 illustrates a part of the travel time curve of the empty loaded heavy truck traveling at 80 km/h on the high class highway.

Fig. 4
A line graph plots vibration acceleration versus time for an empty loaded heavy truck traveling on a low-class highway. The lines fluctuate between negative 6 and 5 on the y-axis.

Vertical acceleration time travel curve of empty loaded heavy truck traveling on low class highway (40 km/h)

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Fig. 5
A line graph plots vibration acceleration versus time for an empty loaded heavy truck traveling on a high-class highway. The lines fluctuate between negative 4 and 4 on the y-axis.

Vertical acceleration time travel curve of empty loaded heavy truck traveling on high class highway (80 km/h)

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It is assumed that vehicle axle-tire vertical vibration acceleration follows a normal distribution with zero mean value, and therefore the standard variance is equal to mean square value. Since the mean square value reflects the size of vibration acceleration, standard variance also indicates vibration acceleration size patterns. Vertical vibration acceleration of different cases is listed in Table 1.

Table 1 Vehicle axle-tire vertical vibration acceleration mean square value (μ)
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As vehicle suspension, tires, loading mass, and pavement roughness vary greatly, the response of vehicles vary greatly while they are stimulated by pavement roughness. From the data, it shows that:

Mean square value of axle-tire vertical acceleration increases with the increase of vehicle travel speed for both light truck and heavy truck; the value is also the same when the trucks are empty, but vibration acceleration speed of light truck is much higher than heavy truck when they are fully loaded.

Loading mass has great influence on vehicle axle-tire acceleration, for a fully loaded truck it is obviously less than an empty truck.

The smoother the pavement is, the smaller the vibration acceleration is. Pavement roughness has a great influence on full loaded trucks while a lighter influence on empty trucks.

4.2 Axle Load Conversion Considering Dynamic Load

Vehicle vibration acceleration follows the normal distribution with zero mean value [4], and therefore vertical vibration acceleration probability distribution can be obtained following a normal distribution, shown in Table 2. (Spaces of acceleration is 0.2σ with16 spaces, and the negative sign means upward acceleration).

Table 2 Vertical acceleration distributional probability Qi
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According to asphalt pavement design code of China, equivalent standard axle load conversion can be inferred by the equation below when tensile strain at bottom of surface is checked as a design index:

$$N = \sum\limits_{i = 1}^{k} {c_{1i} c_{2i} n_{i} } \left( {\frac{{p_{i} }}{P}} \right)^{4}$$
(3)

where N is accumulated equivalent standard axle load; c1i is axle number coefficient, c2i is tire number coefficient; ni is traffic volume, veh/d; pi is axle load, KN; P is standard axle load, 100 KN.

After dynamic load is taken into account, equivalent axle load conversion formula can be expressed as follows:

$$N^{\prime} = \sum\limits_{i = 1}^{k} {\sum\limits_{j = 1}^{m} {c_{1i} c_{2i} n_{i} Q_{j} } } \left( {\frac{{p_{i} \pm F_{D} }}{P}} \right)^{4}$$
(4)

where: m is numbers of separated acceleration distribution, m = 1–16; Qj is the probability of acceleration distribution of each separated acceleration distribution.

After dynamic load is considered, equivalent standard axle load increasing rate is:

$$\gamma = \frac{{N^{\prime} - N}}{N} \times 100\%$$
(5)

Using the values listed in Tables 1 and 2, vehicle axle-tire acceleration and distributional probability of different traveling speeds can be calculated. The dynamic load and distribution probability can be obtained from Eq. (1), and the equivalent standard axle load can be calculated using Eq. (3). The increasing rate of equivalent standard axles load can be calculated using Eq. (4), and the results are listed in Table 3.

Table 3 Increasing rate of equivalent axles load times when tensile strain at bottom of surface is taken as design index
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It is shown in Table 3 equivalent standard axle load exhibits different increasing patterns for heavy truck and light truck after considering dynamic load. On high class highway it increases 8.3–14.9% for heavy truck, and 3.6–5.4% for light truck; on low class highway, it increases 6.8–17.3% for heavy truck, and 3.2–12.5% for light truck.

According to asphalt pavement design code of China, when tensile stress at the bottom of semi-rigid base is taken as a design index, equivalent standard axle load conversion can be obtained by the equation below:

$$N^{\prime} = \sum\limits_{i = 1}^{k} {c^{\prime}_{1i} c^{\prime}_{2i} n^{\prime}_{i} (\frac{{p_{i} }}{P})^{8} }$$
(6)

where Nʹ is accumulated equivalent standard axle load; cʹ1i is axle number coefficient, cʹ2i is tire number coefficient; nʹi is traffic volume, veh/d; pi is axle load, KN; P is standard axle load, 100 KN.

Using the same method, equivalent standard axle loads and the corresponding increasing rates are obtained and listed in Table 4.

Table 4 Increasing rate of equivalent axles load times when tensile stress at bottom of semi-rigid base is taken as design index
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It shows in Table 4 that equivalent standard axle load increases differently for heavy truck and light truck after considering dynamic load when tensile stress at the bottom of semi-rigid base. On high class highway it increases 34.2–64.9% for heavy truck, 14.2–21.8% for light truck; on low class highway, it increases 27.5–77.1% for heavy truck, 12.6–53.2% for light truck.

5 Conclusions

This study examines axle-tire vertical acceleration of heavy truck and light truck on low class highway and high class highway based on field tests and analyses. The main conclusions of this study include:

  1. (1)

    Travel speed, loading mass and pavement roughness have significant influence on axle-tire vertical acceleration. Mean square value of axle-tire vertical acceleration increases with the increase of vehicle travel speed; The mean square value of the vibration acceleration for a fully load truck is considerably less than that for an empty truck.

  2. (2)

    After considering dynamic load, equivalent standard load increase by vehicle traveling speed: when tensile strain at bottom of surface is taken as design index, on high class highway equivalent standard load increases 8.3–14.9% for heavy truck, 3.6–5.4% for light truck; on low class highway, it increases 6.8–17.3% for heavy truck, 3.2–12.5% for light truck; when tensile stress at the bottom of semi-rigid base is taken as checking index, On a high class highway it increases 34.2–64.9% for a heavy truck, 14.2–21.8% for a light truck; on a low class highway, it increases 27.5–77.1% for a heavy truck, 12.6–53.2% for a light truck.