Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 8438921, 17 pages
https://doi.org/10.1155/2018/8438921

Research Article

Modeling of Lateral Stability of Tractor-Semitrailer on Combined Alignments of Freeway

Guixian Qu,¹ Yulong He,¹ Xiaoduan Sun,² and Jingjing Tian¹

¹College of Metropolitan Transportation, Beijing University of Technology, 100 Pingleyuan, Chaoyang, Beijing 100124, China
²Department of Civil Engineering, University of Louisiana at Lafayette, Lafayette, LA 70504, USA

Correspondence should be addressed to Yulong He; ylhe@bjut.edu.cn

Received 9 November 2017; Accepted 1 March 2018; Published 18 April 2018

Academic Editor: Xiaohua Ding

Copyright © 2018 Guixian Qu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The objective of this research was to investigate the impact of roadway geometrics and speed on lateral stability of tractor-semitrailer on combined alignments of freeways since the current design guidelines of combined alignments are not available in China. A closed-loop vehicle dynamic simulation model was established using TruckSim multibody dynamics software. The maximum wheel side friction demand and lateral load transfer ratio were used to measure the skidding and rollover risks, and the variation laws of each indicator were determined by developing the controlled variable simulation scenarios. Based on the theory of statistical analysis, an orthogonal test was designed, and the effects of various impact factors on the lateral stability of the tractor-semitrailer were analyzed. The models for lateral stability indicators were developed by multiple linear regression analysis and applied to evaluate the driving risk of tractor-semitrailer on a wet road surface. The results showed that the radius and speed have significant effects on the lateral stability while the effect of the downgrade is of general significance. In addition, lower safe speed should be adopted on wet road surfaces of curved downgrade. This paper proposed a surrogate approach to road safety analysis and the models can be used for building the freeway driving security system.

1. Introduction

At present, tractor-semitrailer safety on combined alignments of freeways has been a serious concern in China due to the overloading and relatively poor vehicle performances. According to the survey of serious and major traffic accidents of national highways (2010 to 2014), on freeways of China, the tractor-semitrailer accounted for 38% of all accident vehicles, killing 1,520 people with a mortality rate of 32%. Rear-end collision, fixed-object accident, and rollover accident are the main accident types of tractor-semitrailer with the percentage of 73.17% [1]. In the current Chinese highway design standard, the tractor-semitrailer is the newly added design vehicle. No detailed specifications on the configurations of tractor-semitrailer were provided, which required being further studied [2]. Currently, the maximum allowable weight for six-axle tractor-semitrailer is 49000 kg, much higher than the medium truck used in existing highway design and other types of tractor-semitrailer [3]. Meanwhile, the Chinese highway design standard did not specify the design criteria for the combined alignments of freeways [2, 4]. Compared with other truck types, the complex articulated structure, high center of gravity, and the heavy load of six-axle tractor-semitrailer can lead to more severe forms of losing lateral stability when cornering on combined alignments. Therefore, a study is needed to analyze the variation laws of lateral stability of six-axle tractor-semitrailer on combined alignments of freeways.

In previous road safety studies, the impact of road alignment parameters on vehicle safety is mainly based on the point-mass model and vehicle dynamics simulation. The side friction factor is commonly used as the indicator of skidding, and the lateral acceleration and lateral load transfer ratio are adopted to measure the risk of rollover. In AASHTO green book, the point-mass model mainly considers the effects of radius and speed on the side friction factor of vehicle on horizontal curves [5]. Since the effect of the longitudinal grade is negligible, Dunlap et al. (1978) developed the equation of side friction factor by analyzing the forces acting on a vehicle on combined grade and horizontal curvature [6]. Eck and French (2002) introduced the wheelbase and wheel angles of tractor and semitrailer to the equation of side friction factor to estimate the required additional superelevation for the tractor-semitrailer at the low radius [7]. However, the principle of the point-mass model is based on the steady-state cornering performance of the rigid vehicle, ignoring the vehicle configuration and characteristics, such as the suspension system, which have a great effect on vehicle rollover as well as the distribution of forces acting on different tires, since friction can vary significantly between tires when cornering [8–10].

The vehicle dynamics model has the advantages of reflecting the real-time response of the vehicle to evaluate the vehicle operational characteristics under the specific road environments, and the multiple regression analysis was adopted to develop the mathematical models of lateral stability indicators. Kontaratos et al. (1994) developed an enriched bicycle model to study the effects of grade and speed on lateral stability of passenger cars. Since the tractive forces expend some of the lateral friction, the results revealed that the upgrade can have a significant effect on side friction supply and the minimum radius should be increased on upgrades at higher vehicle speeds [11]. Bonneson (2000) found that the downgrade depletes the margin of safety for heavy trucks traveling on downgrade horizontal curves owing to the great weight and high peak side friction demands of heavy trucks [12]. You and Sun (2013) developed the driver-vehicle-road closed-loop dynamic simulation model of a two-axle convention truck and found that the rollover is the main concern for trucks on horizontal curves, which is affected by the vertical curve, small radius, and steep slope [13]. Kordani et al. (2014, 2015) used CarSim and TruckSim to examine the effect of combined alignments on lateral stability of passenger cars and two-axle conventional trucks and presented a series of formulas based on the multiple regression analysis. They found that a higher maximum wheel side friction demand is produced on steep downgrades and the truck is more likely to roll over [14, 15]. Shin and Lee (2015) investigated the lateral stability of heavy trucks by TruckSim and the study suggested that the rollover is highly sensitive to vehicle speed and steering angle, whereas skidding is vulnerable to the change in tire–road friction coefficient [16]. Zhang et al. (2015) studied the lateral stability of passenger cars on horizontal curves by CarSim and developed the mathematical models of lateral stability with the consideration of curvature radius, longitudinal grade, speed, and brake pressure [17]. Wang et al. (2015) adopted the multiple linear regression models to analyze the effects of the combined alignments on the lateral acceleration of passenger cars [18].

By focusing on the dynamic performances of trucks, a series of researches were conducted to put forward the recommendations to the existing roadway design policies. Considering the driver comfort levels and necessity to maintain the vehicle safety, the effect of vertical alignment on minimum radius requirements was evaluated using computer simulation and multiple linear regression method was employed to calibrate the minimum radius requirements for combined alignments based on the AASHTO green book [19–21]. Torbic et al. (2014) used a combination of field data and analytical and vehicle dynamics simulation models to evaluate geometric design criteria for sharp horizontal curves on steep grades and put forward the limit of maximum superelevation rate on a simple horizontal curve combined with vertical alignments [22]. Li and He (2016) demonstrated the safety margins of lateral tire forces for radius, operating speed, and superelevation rate on horizontal curves to guarantee good vehicle lateral reliability and ride comfort [23].

The previous studies mainly concentrated on lateral stability of passenger cars and two-axle conventional trucks according to the AASHTO roadway design policy but paid little attention to the lateral stability of six-axle tractor-semitrailer on combined alignments of freeways. To analyze and reveal the traffic accident mechanism of rollover and skidding of six-axle tractor-semitrailer, the present study is organized as follows. In Section 2, a driver-vehicle-road closed-loop model considering three-dimensional alignment was established by TruckSim. In Section 3, to demonstrate the effects of speed and roadway geometrics on lateral stability of the vehicle, the controlled variable method and orthogonal test method were used to develop the simulation scenarios, and the one-way analysis of variance was adopted to analyze the effects of various factors quantitatively. In Section 4, the simulation results were used to establish the models for lateral stability indicators and put forward a speed limit for six-axle tractor-semitrailer on the wet road surface.

2. Materials and Methods

TruckSim multibody vehicle dynamics software was used to establish the driver-vehicle-road dynamic simulation model. It consists of three essential components: the driver control model, the vehicle model, and the 3D road model. A typical six-axle tractor-semitrailer was used to evaluate the stability performance on different terrains at various speeds. The detailed parameter settings and establishment process of the model were described in this section. Based on force analysis of tractor-semitrailer on combined alignments, the evaluation indicators of lateral stability were determined to measure the risk of skidding and rollover.

2.1. Driver-Vehicle-Road Dynamic Simulation Model

2.1.1. Vehicle Model

The investigated tractor-semitrailers selected are Suzuki QL4250SKFZ three-axle tractor and Huajun ZCZ9390 three-axle semitrailer, which are commonly used in China. The drive type of the tractor is 6 × 4. The gross vehicle weight is 49000 kg with a rated load of 33000 kg. The vehicle model includes four-wheel drive powertrain, air pressure braking system, Ackermann steering mechanism, solid axle suspensions of steering axle, drive axles, and trailer axles. According to the structural parameters of the investigated tractor-semitrailer and the field test to calibrate the vehicle model, the main parameters of tractor-semitrailer were determined, as shown in Table 1. The vehicle configuration interfaces in TruckSim are shown in Figure 1.

Table 1: Main parameters of tractor and semitrailer.

Category
Parameter
Value

Size and weight of tractor
Gross vehicle weight (kg)
25000

Unladen mass (kg)
7880

Maximum towed mass (kg)
39600

Length (mm)
6895

Width (mm)
2465

Height (mm)
3060

Wheelbase (mm)
3400 + 1300

Front track (mm)
2060

Rear track (mm)
1855/1855

Size and weight of semitrailer
Length (mm)
13000

Width (mm)
2500

Wheelbase (mm)
6831 + 1310 + 1310

Sprung mass (mm)
6250

Maximum load (kg)
33000

Powertrain
Rated power/engine speed (kW/rpm)
265/2000

Maximum torque (Nm/rpm)
1422/1400

Compression ratio
17.5 : 1

Suspension
Unsprung mass (kg)
Steering axle: 570
Drive axles: 785
Trailer axles: 690

Axle roll and yaw inertia (kg⋅m²)
Steering axle: 335
Drive axles: 305
Trailer axles: 266

Spring stiffness (N⋅mm⁻¹)
Steering axle: 250
Drive axles: 700
Trailer axles: 2500

Damping coefficient (N⋅s⋅mm⁻¹)
Steering axle: 15
Drive axles: 30
Trailer axles: 30

Brake torque
Steering axle (kN⋅m)
7.5

Drive axles (kN⋅m)
10

Trailer axles (kN⋅m)
10

Tires
Tire size
11.00R20

Effective rolling radius (mm)
528

Unloaded free radius (mm)
542

Maximum allowed force (kN)
100

(a) Configuration interface of tractor

(b) Configuration interface of semitrailer

Figure 1: Vehicle configuration interfaces in TruckSim.

2.1.2. 3D Road Model

The 3D road model in TruckSim simulation software defines the geometric alignments and friction of the ground relative to a 3D reference line. The geometric alignments consist of the horizontal alignment, vertical alignment, and cross section. In TruckSim, the “- coordinates of centerline” module defines the horizontal alignment of the road, using a table of - coordinates of the investigated road segment calculated by multiquadric (MQ) interpolation method. For the vertical alignment design, the “centerline elevation: versus ” module is available for specifying the elevation for the reference line through inputting the elevation of the starting and ending points of the road segment. The superelevation rate of the cross section is set by modifying the elevations of both sides of the road segment at the feature points such as the straight-spiral point, spiral-curve point, curve-spiral point, and spiral-tangent point. For the dry asphalt pavement, the typical value of friction coefficient is 0.8 (Yu, 2011) [24]. Figure 2 shows an example of the detailed settings of 3D road model in TruckSim simulation.

(a) Settings of horizontal alignment

(b) Settings of vertical alignment

(c) Settings of cross section

Figure 2: Settings of 3D road model in TruckSim simulation.

2.1.3. Driver Control Model

The closed-loop driver control model contains four parts: speed control, brake control, gear-shifting control, and steering control. The vehicle travels at a constant target speed. Automatic gear-shifting control was adopted using the closed-loop shift control at all gears. For the downgrades, the step brake control was used to maintain the constant descending speed, and the values of braking force varied along with the simulation step size. In steering control, the key control parameters are the target path of the vehicle and the preview time. The vehicle travels along the road centerline as the target path with no offset. The preview control looks ahead a distance determined from the specified time and the current speed, to compare the current vehicle trajectory to the target path. A realistic value for the preview time is about 1.5 seconds, which can produce smoother steering to keep the vehicle on the target winding paths.

2.1.4. Experimental Validation

To further investigate the accuracy of the driver-vehicle-road dynamic simulation model, the field test was conducted on the Hunchun-Dunhua freeway, which is a segment of G12 National Highway, in Jilin province, China. The tractor-semitrailer was controlled by a driver with more than 10 years of experience. The speed and lateral acceleration were recorded by VBOX and gyroscope simultaneously, and the experimental setup and test instruments are shown in Figure 3. One typical segment with curve radius of 490 m and average gradient of 3.1% (−3.1% for downgrade) was selected, and the lengths of the spiral transition curve and circular curve section are 120 m and 305 m, respectively. In the process of field test, owing to the effect of real-time traffic status, the initial speeds for upgrade and downgrade were 86.3 km/h and 67.9 km/h, respectively. The comparisons of simulated and measured results on curved upgrade and downgrade are shown in Figure 4.

Figure 3: Experimental setup and test instruments.

(a) Speed and lateral acceleration on curved upgrades

(b) Speed and lateral acceleration on curved downgrades

Figure 4: Comparison of simulated and measured results of typical freeway section.

As shown in Figure 4(a), on the curved upgrade, with the increase of the length of longitudinal slope, the simulated speed and measured speed decrease gradually. The simulated and measured lateral acceleration reaches 0.11 g at the spiral-curve point. On the circular curve section, the lateral acceleration decreases slowly; after entering the spiral transition curve, the lateral acceleration decreases rapidly with the decrease of curvature. As shown in Figure 4(b), on the curved downgrade, the simulated and measured speeds increase slightly with the increase of longitudinal grade. On the circular curve section, the lateral acceleration obtained from the simulation and field test increases slowly, reaching the maximum value of 0.08 g at the curve-spiral point. After entering the spiral transition curve, the values of lateral acceleration decrease rapidly. It can be concluded that the dynamic responses of the simulation model were in good agreement with those of field tests, indicating that the driver-vehicle-road dynamic simulation model was sufficiently accurate to analyze the lateral stability of the tractor-semitrailer.

2.2. Force Analysis of Combined Alignments

A three-dimensional reference system of the road was developed to analyze the force condition under the influence of cross slope and the longitudinal slope. The original reference system is defined as the reference system of the horizontal curve without superelevation, where is the coordinate origin fixed on the ground. Then, by rotating coordinators with the degree of around the axis, the reference system on vertical alignment was developed, and is the angle of vertical slope. Finally, through rotating the coordinators with the degree of (the angle of cross slope) around the axis, the three-dimensional reference system for combined horizontal and vertical alignments was developed, as shown in Figure 5. The reference transformation formula from the original reference system to the three-dimensional reference system can be expressed as

Figure 5 shows the three-dimensional reference system and the force analysis of tractor-semitrailer cornering on combined horizontal and vertical alignments.

(a) Three-dimensional reference system

(b) Force analysis of combined alignments

Figure 5: Force analysis of tractor-semitrailer cornering on combined alignments.

In Figure 5, is the centrifugal force (N), is the vehicle gravity (N), is the total lateral tire force (N), is the total vertical tire force (N), , , , and are lateral forces of the outer and inner tires of the left and right th axle (N), , , , and are vertical forces of the right th axle (N), is the gravity acceleration (m/s²), is the speed (km/h), is the radius of curve (m), is the angle of cross slope (°), and is the angle of vertical slope (°).

A general formula of lateral tire forces and vertical tire forces acting on six-axle tractor-semitrailer can be written as

2.3. Evaluation Indicator of Lateral Stability

The skidding and rollover are the main hazardous situations of the six-axle tractor-semitrailer losing lateral stability on the curved upgrades and downgrades. The vehicle skids when the centrifugal force on the mass center of the vehicle exceeds the lateral forces between the tire and the pavement and rolls over when the centrifugal force moment exceeds the gravity moment. The high speed, heavy load, and decreased road friction coefficient are the primary cause of the skidding and rollover.

According to Figure 5, the maximum wheel side friction demand was selected as the evaluation indicator of skidding. The definition of maximum wheel side friction demand is the absolute value of the maximum ratio of total lateral forces to total vertical forces of each side of axle, denoted by . It is also the maximum wheel side friction demand by the vehicle moving on the curved road without skidding, which can be given by [17]

where denotes the maximum wheel side friction demand, and are lateral forces of the outer and inner tires of the left th axle (N), and are lateral forces of the outer and inner tires of the right th axle (N), and are vertical forces of the left th axle (N), and and are the vertical forces of the right th axle (N). The vehicle will skid if the maximum wheel side friction demand exceeds the supplied road friction coefficient , so the value of road friction coefficient is the safety threshold of maximum wheel side friction demand .

The vehicle rollover is one of the most frequent single vehicle accidents, especially for six-axle tractor-semitrailers with a high gravity center. The truck experiences vertical load transfer from the tires on one side of the vehicle to those on the other side when traveling on curved road sections. The lateral load transfer ratio denoted by is used to measure the risk of the rollover of the heavy truck [25, 26]. It is defined as the ratio of the difference between the sum of right wheel loads and the sum of left wheel loads to the sum of total wheel loads [27], as shown in

where denotes the lateral load transfer ratio, and are vertical forces of the left th axle (N), and and are the vertical forces of the right th axle (N).

Based on the research conducted by Fan et al. (2016), the risk level of can be classified into three states [28]:

(1)When , the vehicle has a modest potentiality in occurrence of rollover, which is defined as the warning state of rollover.(2)When , the vehicle has a definite potentiality in occurrence of rollover; this state is defined as dangerous state of rollover.(3)When , the wheels of one side leave the ground, and the vertical load of one side transfers to the wheels on the other side; this state is defined as “critical state of rollover.”

3. Simulation Process and Driving Dynamics Analysis

This section describes the criteria used for the selection of road alignments; a series of controlled variable simulation scenarios and orthogonal tests were designed. The variation laws of lateral stability indicators in controlled variable simulation scenarios and the ANOVA results of orthogonal tests regarding the roadway geometrics and speeds were presented.

3.1. Design of Controlled Variable Simulation Scenarios

To explore the variation laws of and along the stations under the influence of roadway geometrics and speeds, respectively, a series of controlled variable simulation scenarios were developed. For the determination of the road alignments, according to the Chinese standard Technical Standard of Highway Engineering (JTGB01-2014) [2] and Design Specification for Highway Alignment (JTGD20-2006) [4], the freeway with a design speed of 80 km/h was considered. For the horizontal alignment, the ultimate minimum radius is 250 m with a maximum superelevation rate of 8%. To compare the impact of the radius on the lateral stability, a control group with the radii of 450 m, 650 m, and 850 m were selected. The total length of the simulated segment is 800 m: the length of the approach and departure tangent is 100 m, the length of the spiral transition curve is 200 m, and the length of the circular curve is 200 m. For the vertical alignment, the maximum allowable longitudinal grade of freeway with the design speed of 80 km/h is 6%, and therefore the upgrades of 3%, 4%, 5%, and 6% and downgrades of −3%, −4%, −5%, and −6% were selected. In the standard, the critical slope length increases with the decreasing longitudinal grade. To simplify the influence of the slope length, the longitudinal slope starts at the distance of 150 m and ends at the distance of 650 m with a total length of 500 m. The road alignment parameters of scenarios are shown in Figure 6, and the detailed design of controlled variable simulation scenarios is presented in Table 2.

Table 2: Design of controlled variable simulation scenarios.

Road type
Scenarios
Radius (m)
Grade (%)
Speed (km/h)
Simulation purpose

Curved upgrades
1
250, 450, 650, 850
4
80
Study the effect of radius

2
450
3, 4, 5, 6
80
Study the effect of grades

3
450
4
60, 70, 80, 90
Study the effect of speeds

Curved downgrades
1
250, 450, 650, 850
−4
80
Study the effect of radius

2
450
−3, −4, −5, −6
80
Study the effect of grades

3
450
−4
60, 70, 80, 90
Study the effect of speeds

Figure 6: Road geometric parameters of scenarios.

3.2. Simulation Results of Controlled Variable Simulation Scenarios

Figure 7 shows the simulation results of the variation laws of and on curved upgrades. It can be found that and present the same variation law. On the approach tangent (from 0 to 100 m), the values of and are approximate to 0. As the vehicle enters the spiral transition curve (from 100 m to 300 m), the centrifugal force increases as the curvature radius reduces gradually, and the values of and increase dramatically. As the length of longitudinal slope increases, the speed decreases on upgrades, and the values of and decrease slightly. On the circular curve (from 300 m to 500 m), owing to the smallest curvature radius, the constant maximum centrifugal force causes the maximum roll motion of the vehicle, and therefore and reach the peak values. However, the constant horizontal radius makes constant and limited contribution to the roll motion of the vehicle; due to the speed loss of the upgrades, the values of and decrease sharply. On the spiral transition curve (from 500 m to 700 m), as the curvature radius increases gradually and the ascending speed decreases, the centrifugal force on the truck slowly decreases. Therefore, the values of and show a dramatic reduction. After entering the departure tangent (from 700 to 800 m), the values of and are approximate to 0, without the influence of horizontal curve and upgrades. The variation laws reveal that the values of and are in direct proportion to the speed while being in inverse proportion to the radius and upgrade.

(a) Variation laws of and LTR of different curvature radii

(b) Variation laws of and LTR of different grades

(c) Variation laws of and LTR of different speeds

Figure 7: Variation laws of and LTR on curved upgrades.

Figure 8 shows the simulation results of the variation laws of and on curved downgrades. The values of and on curved downgrades have a similar tendency to the curved upgrades except in the circular curve section (from 250 m to 450 m). Instead of the decreasing values of and on upgrades, and reach the peak values after entering the circular curve, as the braking forces act on the vehicle to maintain a constant speed, and there is no speed loss on downgrades; the values of and remain constant. Particularly, on the circular curve, there is an increase in values of and on downgrades of −5% and −6%, indicating that, on downgrades steeper than −4%, the descending speed still increases slightly though the braking maneuvers are applied. The variation laws reveal that, in braking cases, the values of and are in direct proportion to speed and downgrade while being in inverse proportion to the radius.

(a) Variation laws of and LTR of different curvature radii

(b) Variation laws of and LTR of different grades

(c) Variation laws of and LTR of different speeds

Figure 8: Variation laws of and LTR on curved downgrades.

3.3. Design and ANOVA Results of Orthogonal Tests

The orthogonal test method is commonly used in experimental study of multilevel factors by effectively simplifying the design of experiment test. In this research, it can be used to quantitatively analyze the effects of multilevel factors on the lateral stability of six-axle tractor-semitrailer on curved upgrades and downgrades. The orthogonal array was selected to analyze the three factors with four levels for curved upgrades and downgrades, respectively. Factors and corresponding levels are shown in Table 3. In the column of grade, the negative gradients in brackets denote the downgrades.

Table 3: Factors and corresponding levels.

Level
Radius (m)
Grade (%)
Speed (km/h)

1
250
3 (−3)
60

2
450
4 (−4)
70

3
650
5 (−5)
80

4
850
6 (−6)
90

-test was used for significance testing of factors in the variance analysis. The significance level includes 0.01, 0.05, and 0.10 for distribution. For the factor , if , the factor has a high significant effect on the dependent variables; if , the factor has a significant effect on the dependent variables; if , the factor has a general significant effect on the dependent variables; if , the factor has no significant effect on the dependent variables.

Tables 4 and 5 show the variance analysis of factors influencing and on curved upgrades. It can be found that, on curved upgrades, the order of primary and secondary factors affecting and is radius, speed, and grade according to the order of sum of squares. The values of radius and speed are greater than the corresponding critical values, and therefore the radius has a highly significant effect on the lateral stability of tractor-semitrailer, and speed has a significant effect whereas the effect of the grade is not significant.

Table 4: Variance analysis of factors influencing on curved upgrades.

Source
Sum of squares
df
Mean square
value
Critical value

Corrected model
0.186
9
0.021
32.686

Intercept
1.731
1
1.731
2731.354

Radius
0.146
3
0.049
77.008

Grade
0.001
3
0.000
0.414

Speed
0.039
3
0.013
20.637

Error
0.004
6
0.001

Total
1.921
16

Correct total
0.190
15

Table 5: Variance analysis of factors influencing the LTR on curved upgrades.

Source
Sum of squares
df
Mean square
value
Critical value

Corrected model
0.243
9
0.027
34.915

Intercept
3.042
1
3.042
3926.216

Radius
0.178
3
0.059
76.416

Grade
0.001
3
0.000
0.507

Speed
0.065
3
0.022
27.821

Error
0.005
6
0.001

Total
3.291
16

Correct total
0.248
15

Tables 6 and 7 show the variance analysis of factors influencing and on curved downgrades. It can be found that, on curved downgrades, the values of sum of squares decrease gradually from the radius to speed to grade, indicating that the order of primary and secondary factors affecting and is the radius, speed, and grade, respectively. The values of the radius, speed, and grade are greater than the corresponding critical values, and therefore the effect of the radius is of high significance, the effect of the speed is of significance, and the effect of the grade is of general significance. This is mainly due to the fact that the radius and speed can influence the roll motion of vehicle and centrifugal force directly when cornering on curves. The grade can change the component of force, as its value is relatively low, so the effect of the grade is less significant than radius and speed.

Table 6: Variance analysis of factors influencing on curved downgrades.

Source
Sum of squares
Df
Mean square
value
Critical value

Corrected model
0.325
9
0.036
21.211

Intercept
1.951
1
1.951
1146.316

Radius
0.208
3
0.069
40.808

Grade
0.035
3
0.012
6.873

Speed
0.081
3
0.027
15.953

Error
0.010
6
0.002

Total
2.286
16

Correct total
0.335
15

Table 7: Variance analysis of factors influencing the LTR on downgrades.

Source
Sum of squares
df
Mean square
value
Critical value

Corrected model
0.480
9
0.053
23.119

Intercept
4.197
1
4.197
1819.425

Radius
0.301
3
0.100
43.538

Grade
0.052
3
0.017
7.524

Speed
0.127
3
0.042
18.294

Error
0.014
6
0.002

Total
4.691
16

Correct total
0.494
15

4. Results and Discussion

Following the consideration of the simulation results, the multiple linear regression analysis was conducted to develop new models of and for six-axle tractor-semitrailer by selecting the radius, longitudinal grade, and vehicle speed as the independent variables. Based on the models, the risk analysis of vehicle on the wet road surface was conducted and the safe speed limits were put forward.

4.1. Calibrated Mathematical Model

A total of 56 observations of curved upgrades and downgrades were used to calibrate the mathematical model. Many independent variables were examined, and the final models were calibrated based on the following criteria [29]:

(1)The coefficient of determination must be significant at the 0.95 confidence level.(2)Each of the independent variables used in the model must have a coefficient that is significantly different from zero at the 0.95 confidence level.(3)The algebraic signs of the coefficients of the independent variables must have a logical explanation.

The final models of and on curved upgrades and downgrades were presented in (5)-(6).

where is the maximum wheel side friction demand; is the radius (m); is the grade (%); is the speed (km/h).

where is the lateral load transfer ratio; is the radius (m); is the grade (%); is the speed (km/h).

From Tables 8 and 9, the value of all independent variables is significantly different from zero at the 95% confidence level, so (5)-(6) have logical explanations for the effect of each independent variable on dependent variables, and . The positive sign for the coefficient of means that and increase with the increase in . The negative sign for the coefficient of means that, on curved upgrades, and decrease as grade increases. On curved downgrades, and increase when downgrade increases (as the signs for downgrades are negative).

Table 8: Results of statistical analysis for cornering on upgrades and downgrades.

Road type
Independent variable

value

Curved upgrades
Constant
0.194
26.441
0.001

0.015
63.064
0.001

−0.013
−7.976
0.001

Curved downgrades
Constant
0.022
2.635
0.015

0.018
68.886
0.001

−0.020
−11.326
0.001

Table 9: Results of statistical analysis for LTR cornering on upgrades and downgrades.

Road type
Independent variable
B
t
value

Curved upgrades
Constant
0.281
60.183
0.001

0.017
113.230
0.001

−0.014
−14.143
0.001

Curved downgrades
Constant
0.112
8.347
0.001

0.022
50.799
0.001

−0.025
−8.792
0.001

4.2. Risk Analysis of Vehicle on Wet Road Surface

On wet road surface, the road friction coefficient is greatly decreased by rainwater. Meanwhile, the rainwater that accumulated in the tread pattern may lead to a decrease in the contact area between the tire and the pavement. As a result, brake failure and inflexible steering can greatly reduce the lateral stability of the vehicle, and the high speed can lead to the increase in centrifugal force, increasing the probability of traffic accidents. Therefore, by conducting the risk analysis of the vehicle on a wet road surface, the safe speed limit is put forward.

The supplied friction coefficient of wet road surface at different speeds can be determined as follows (Wambold et al., 1984) [30]:

The lateral stability of the vehicle on curved upgrades and downgrades can be analyzed by utilizing (5)-(6). By comparing the values of and with the available friction coefficient and the rollover risk level, respectively, the risk analysis on wet road surface was conducted. For the freeway with the design speed of 80 km/h, the ultimate minimum horizontal curve radius of 250 m and general minimum horizontal curve radius of 400 m with the superelevation rate of 8% were selected as the evaluating section. The upgrade ranges from 3% to 6%, and the downgrade ranges from −3% to −6%. The six-axle tractor-semitrailer travels at the speed of 70 km/h to 90 km/h. The risk analyses of vehicle on upgrades and downgrades combined with the ultimate minimum horizontal curve radius and the general minimum horizontal curve radius were presented in Tables 10 and 11, respectively.

Table 10: Risk Analysis of Tractor-semitrailer on the Ultimate Minimum Horizontal Curve Radius Combined with Upgrades and Downgrades.

i/%
V/(km/h)

Risk status
i/%
V/(km/h)

Risk status

3
70
0.66
0.45
0.57
Safe
−3
70
0.66
0.43
0.62
Warning state of rollover

75
0.65
0.49
0.62
Warning state of rollover
75
0.65
0.49
0.68
Warning state of rollover

80
0.64
0.54
0.67
Warning state of rollover
80
0.64
0.54
0.75
Warning state of rollover

85
0.63
0.59
0.73
Warning state of rollover
85
0.63
0.60
0.82
Dangerous state of rollover

90
0.61
0.64
0.79
Skidding, warning state of rollover
90
0.61
0.67
0.90
Skidding, dangerous state of rollover

4
70
0.66
0.44
0.56
Safe
−4
70
0.66
0.45
0.64
Warning state of rollover

75
0.65
0.48
0.61
Warning state of rollover
75
0.65
0.51
0.71
Warning state of rollover

80
0.64
0.53
0.66
Warning state of rollover
80
0.64
0.56
0.78
Warning state of rollover

85
0.63
0.58
0.72
Warning state of rollover
85
0.63
0.62
0.85
Dangerous state of rollover

90
0.61
0.63
0.78
Skidding, warning state of rollover
90
0.61
0.69
0.92
Skidding, dangerous state of rollover

5
70
0.66
0.42
0.54
Safe
−5
70
0.66
0.47
0.67
Warning state of rollover

75
0.65
0.47
0.59
Warning state of rollover
75
0.65
0.53
0.73
Warning state of rollover

80
0.64
0.51
0.65
Warning state of rollover
80
0.64
0.58
0.80
Dangerous state of rollover

85
0.63
0.56
0.70
Warning state of rollover
85
0.63
0.64
0.87
Skidding, dangerous state of rollover

90
0.61
0.62
0.76
Skidding, warning state of rollover
90
0.61
0.71
0.95
Skidding, dangerous state of rollover

6
70
0.66
0.41
0.53
Safe
−6
70
0.66
0.47
0.67
Warning state of rollover

75
0.65
0.45
0.58
Safe
75
0.65
0.53
0.73
Warning state of rollover

80
0.64
0.50
0.63
Warning state of rollover
80
0.64
0.60
0.83
Dangerous state of rollover

85
0.63
0.55
0.69
Warning state of rollover
85
0.63
0.66
0.90
Skidding, dangerous state of rollover

90
0.61
0.60
0.75
Warning state of rollover
90
0.61
0.73
0.97
Skidding, dangerous state of rollover

Table 11: Risk analysis of tractor-semitrailer on the general minimum horizontal curve radius combined with upgrades and downgrades.

i/%
V (km/h)

Risk status
i/%
V (km/h)

Risk status

3
70
0.66
0.34
0.45
Safe
−3
70
0.66
0.30
0.46
Safe

75
0.65
0.37
0.48
Safe
75
0.65
0.34
0.50
Safe

80
0.64
0.40
0.51
Safe
80
0.64
0.37
0.54
Safe

85
0.63
0.43
0.55
Safe
85
0.63
0.41
0.58
Safe

90
0.61
0.46
0.58
Safe
90
0.61
0.45
0.63
Warning state of rollover

4
70
0.66
0.33
0.43
Safe
−4
70
0.66
0.32
0.48
Safe

75
0.65
0.35
0.46
Safe
75
0.65
0.36
0.52
Safe

80
0.64
0.38
0.50
Safe
80
0.64
0.39
0.56
Safe

85
0.63
0.41
0.53
Safe
85
0.63
0.43
0.61
Warning state of rollover

90
0.61
0.45
0.57
Safe
90
0.61
0.47
0.66
Warning state of rollover

5
70
0.66
0.31
0.42
Safe
−5
70
0.66
0.34
0.51
Safe

75
0.65
0.34
0.45
Safe
75
0.65
0.38
0.55
Safe

80
0.64
0.37
0.48
Safe
80
0.64
0.41
0.59
Safe

85
0.63
0.40
0.52
Safe
85
0.63
0.45
0.63
Warning state of rollover

90
0.61
0.43
0.56
Safe
90
0.61
0.49
0.68
Warning state of rollover

6
70
0.66
0.30
0.41
Safe
−6
70
0.66
0.36
0.53
Safe

75
0.65
0.33
0.44
Safe
75
0.65
0.40
0.57
Safe

80
0.64
0.36
0.47
Safe
80
0.64
0.43
0.61
Warning state of rollover

85
0.63
0.39
0.50
Safe
85
0.63
0.47
0.66
Warning state of rollover

90
0.61
0.42
0.54
Safe
90
0.61
0.51
0.71
Warning state of rollover

From Tables 10 and 11, the following can be concluded.

(1)To avoid the skidding risk, the safe speed on the curved upgrade is higher than that on curved downgrades. For the radius of 250 m combined with upgrades of 3%, 4%, and 5%, the skidding occurs at 90 km/h. There is no skidding risk of tractor-semitrailer at 90 km/h on upgrade of 6%. On curved downgrades, the skidding occurs at 90 km/h on downgrades of −3% and −4% and 85 km/h on downgrades of −5% and −6%.(2)From the viewpoint of the rollover risk, the risk levels on the downgrades are higher than on upgrades. On curved upgrades, for the ultimate minimum radius of 250 m combined with upgrades of 3%, 4%, and 5%, at the speed of 70 km/h, the vehicle is in the safe status. However, when the speed reaches 75 km/h or more, the vehicle remains in the warning state of rollover. On the upgrade of 6%, the vehicle is in the warning state of rollover within a range from 80 km/h to 90 km/h. On curved downgrades of −3% and −4%, the vehicle is in the warning state of rollover within a range from 70 km/h to 80 km/h and dangerous state of rollover within a range from 85 km/h to 90 km/h. On curved downgrades of −5% and −6%, the vehicle remains in the warning state of rollover within a range from 70 km/h to 75 km/h and dangerous state of rollover within a range from 80 km/h to 90 km/h.(3)The lateral stability of tractor-semitrailer improves markedly on the general minimum radius of 400 m. On upgrades, the values of and are much lower than the threshold values, so the six-axle tractor-semitrailer remains in safe status at the speed of 90 km/h. On downgrades of −3%, −4%, −5%, and −6%, the speed causing the warning state of rollover is 90 km/h, 85 km/h, 85 km/h, and 80 km/h, respectively. Therefore, the general minimum radius can basically satisfy the lateral stability requirements of six-axle tractor-semitrailer at the design speed of 80 km/h. Besides, it is necessary to limit the speed on the ultimate minimum horizontal curve radius combined with upgrades and downgrades.

5. Conclusions

To investigate the lateral stability of six-axle tractor-semitrailer on combined alignments of the freeway, the closed-loop vehicle dynamic simulation model using TruckSim multibody software was established, and the following remarks are offered:

(1)The variation laws of the maximum wheel side friction demand and lateral load transfer ratio on curved upgrades are consistent with the curved downgrades except in circular curve section due to the speed loss on curved upgrades. The steeper upgrade causes a reduction in and whereas the steeper downgrade can lead to an increase in and , showing that more side friction demand and roll motion are created on steep downgrades compared with upgrades and mild downgrades.(2)On both curved upgrades and downgrades, the order of primary and secondary factors affecting and is radius, speed, and grade. The effect of the horizontal radius is of high significance, and the effect of speed is of significance. On curved downgrades, the grade has the general significant effect on lateral stability of six-axle tractor-semitrailer. Therefore, more concerns should be attached to the safety issues of curved downgrades.(3)The horizontal radius, speed, and grade were introduced to calibrate the mathematical models of the lateral stability. Skidding and rollover risks were analyzed on the wet road surface of the curved upgrades and downgrades, respectively. This study recommends that, to ensure the lateral stability of six-axle tractor-semitrailer on the wet road surface, low speed should be adopted to ensure the lateral stability of tractor-semitrailer. From the comprehensive perspective of preventing the skidding risk and rollover risk, for the ultimate minimum radius, the safe ascending speed and descending speed should better not exceed 75 km/h and 70 km/h, respectively; for the ultimate minimum radius of 400 m, a safe descending speed should better not exceed 80 km/h.(4)The results and recommendation models can provide guide for both combined alignments design and safety improvement of existing roads. However, this research only considered the lateral stability of six-axle tractor-semitrailer on the simple horizontal curves combined with vertical slopes. Therefore, the lateral stability related to more alignment combinations such as the compound horizontal curves and reverse horizontal curves on upgrades and downgrades should be investigated in further research.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors would like to appreciate the support from the Ministry of Transport of China for the National Science & Technology Support Plan Project (Grant no. 2014BAG01B01) and Key Special Project of National Key Research and Development Program (Grant no. 2017YFC0803903) for this work.

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