Risk evaluation method for highway roadside accidents

Abstract

In order to strengthen the safety of highway roadsides, it is necessary to take targeted measures according to the roadside hazards, so there is an urgent need to develop research on risk evaluation of highway roadside accidents. Based on accident simulation analysis and the form of accident after vehicles run to the roadside, the rollover risk of roadside accident is classed into four grades, namely, no departure from the ground, slight departure from the ground, one or two turnovers, and more than two turnovers. The factors involved in the causes of roadside accidents of different rollover risks are studied, and the thresholds of driving speed, sideslope gradient, and sideslope height are given. A Bayesian network for risk evaluation of roadside accidents is constructed. Based on the factor thresholds for the causes of roadside accidents, the calculation methods of the probability of different rollover risks of roadside accidents are carried out according to a single factor, two factors, and three factors. The typical cases of roadside accidents are analyzed and the calculation results of the Bayesian network show that the probability of one or two turnover accidents is 0.939, which is consistent with the results of roadside accident simulation tests, proving the accuracy of the risk evaluation methods for highway roadside accidents.

Introduction

By the end of 2016, China’s total highway mileage reached 4,696,300 km, and roadside accidents accounted for about half of accidents, in which more than three people died. Statistics for Beijing and Guizhou area showed that roadside accidents accounted for about a quarter of all accidents, leading to a 40% mortality rate and severe injury rate of more than 50%. Identifying the sources of roadside hazards and evaluating the risk of roadside accidents have become problems to be resolved.

Zegeer et al. ¹ reported that a 27% reduction in run-off-road (ROR) crashes could be achieved by flattening a sideslope from $1\mathrm{V}:2\mathrm{H}$ to $1\mathrm{V}:7\mathrm{H}$ or more. Bella ² investigated the driver perceptions of roadside configurations on two-lane highways using simulation method. The results indicated that only cross-sections influenced driver behaviors, and the drivers did not change their behaviors on highways without barriers. In a similar study, Fitzpatrick et al. ³ explored the influence of clear zone width and roadside vegetation on driver behaviors with respect to vehicle speed and lateral position, using four combinations of clear zone widths and densities of roadside vegetation. The analysis results demonstrated that the wider the clear zone, the greater the observed driver speed. Roque et al. ⁴ collected ROR crash data on freeways in Portugal and developed multinomial and mixed logit regression models to analyze the data. The results indicated that critical slopes and horizontal curves significantly contributed to fatal ROR crashes. In another study, Zou and Tarko. ⁵ selected 2049 collision accidents related to guardrails in the United States from 2003 to 2013 and set up a multivariate logit regression model to evaluate the severity of roadside accidents. Shankar and Mannering ⁶ collected historical accident data for 5 years in Washington and adopted the multivariate logit model to categorize roadside accident hazards into five levels: property loss, possible injury, obvious injury, serious injury, and death. In literature,^7–10 related research was also carried out by similar methods. Although the statistical-based method on accident data is objective, it is usually not applicable to China, where the traffic accident data are rare complete. Ray et al. ¹¹ put forward a roadside safety evaluation framework and designed an interactive software to evaluate the roadside safety level. The probability of no severe or fatal accident in a year is used as a standard to evaluate the roadside safety level. This method can easily evaluate the roadside safety level under different highway design schemes. Jalayer and Zhou ¹² evaluated the roadside safety risk on two-lane rural highways by the reliability analysis method and showed that the slope, obstacle density, distance between the obstacle and the lane, and shoulder width are the main factors affecting roadside safety. Daniello and Gabler ¹³ put forward a classification method of roadside risk based on occupant risk; the index of occupant risk is characterized by acceleration severity index, which can only indirectly evaluate the occupant risk in the collision process. Ray and Hiranmayee ¹⁴ propose a method to evaluate the risk of damage to occupants in a collision between a vehicle and roadside fixture, the method is based on the velocity distribution on the impacted surface of the fixture. Mak and Sicking ¹⁵ propose the Likelihood Exposure Consequence method, namely the LEC method, which takes the possibility of roadside accidents, the frequency of drivers and passengers’ exposure to dangerous environments, and the severity of accidence consequences as factors affecting roadside danger, considers their evaluation scores, and calculates the risk scores of the roadside environment according to the product of the scores of the three aspects. Roque and Cardoso ¹⁶ developed a software system for roadside safety evaluation based on cost-benefit analysis and presented practical cases. Moreover, Richard and Christina ¹⁷ studied the risk of occupant injury during the collision between vehicle and roadside guardrail based on three real vehicle crash tests and 50 simulations. Wang and Gabler ¹⁸ established a finite element model for highway guardrail based on the finite element analysis theory, and the model is calibrated by the occupant damage data from real vehicle collision tests and simulations. In 2008, the Ministry of Transport developed a study on “Highway Roadside Safety Assessment and Protection Methods” ¹⁹ and suggested the classification of roadside safety into four levels according to the clean area width at the roadside, roadside depth, sideslope, and danger in the clean area. Zheng et al. ²⁰ propose a model of roadside safety assessment based on historical accident data, but many indicators are not quantified and need to be assigned based on the experience and knowledge of experts.

Objectively speaking, there is no essential difference between the division into three levels, four levels, or five levels and the division into seven levels in the United States and all divided according to the main factors affecting the level of roadside risk, which is determined by a qualitative formulation. Although they have played a certain role in engineering practice, there are still some problems, such as the blurring of the boundary between the levels and the lack of foundation of the fundamental theory. At present, quantitative measurement of roadside risk is still not mature, due to incomplete roadside accident data. The high risk of roadside accidents means that real vehicle tests are not feasible. This article uses accident simulation software to study the influence of speed and roadside environment on the rollover risk of the accident after the car leaves the lane for the roadside, determines the thresholds of roadside accident factors causing roadside accidents of different grades based on the form of roadside accident, and constructs a Bayesian network which can accurately calculate roadside accident probability, in order to realize a quantitative evaluation of the roadside accident risk.

Simulation test and influence factor analysis of roadside accident

Factors influencing roadside safety

Driving speed is the main factor leading to roadside accidents. When a vehicle makes a turn, a high speed will result in an excessively large lateral force, so the vehicle will drive to the roadside. The related research^21,22 results also showed that driving speed is not only one of the main causes of traffic accidents but also the most effective parameter for predicting highway traffic accidents. In addition, some scholars carried out some researches. Roque and Jalayer ²³ propose that single-vehicle crashes in rural areas have higher likelihood of fatality than crashes in urban areas owing to the higher speed limits. Xie et al. ²⁴ suggested that roadside accidents on interstate highways are more likely to be fatal, which can be caused by the higher average speed.

Slope is also an important factor affecting roadside accident severity. The forms of slope comprise embankments, cuttings, and semi-filled and semi-excavated embankments. Compared to cuttings, embankments (related to steep sideslopes) are more likely to lead to turnover and other severe roadside accidents. Some studies also confirmed this point, Jalayer and Zhou ¹² obtained that the segments with steep sideslopes might experience more roadside crashes by reliability analysis. Lord et al. ²⁵ proved that sideslopes have a significant impact on the frequency and severity of roadside accidents. Wang et al. ²⁶ made a statistical analysis that 54.54% of accidents occurred on a 4%–5% steep sideslope, which is about 10 times that of 3%–4% sideslopes and 20 times that of the less than 3% sideslopes. Therefore, this article mainly studies slopes in the form of embankments. Related studies ³ have categorized slopes into four types: slope >1:2 (dangerous slope), 1:2 >slope >1:3 (non-returning slope), 1:3 >slope >1:6 (returning slope), and slope <1:6 (safe slope). Slope heights are categorized into three kinds: height <4 m (low slope), 4 m <height <10 m (medium slope), and height >10 m (high slope). In general, when the slope height is fixed, the steeper the slope, the greater the turning speed of the vehicle and the more severe the roadside accident. When the slope is fixed, the higher the slope, the more likely it is to cause serious consequences in the case of a road accident.

In addition, factors affecting roadside safety include roadside obstructions that the vehicle may hit when falling down the slope. Considering the randomness of the distribution of obstacles, this article does not consider them for the time being. In addition, roadside guardrails are also seen as obstacles, and the object of this study has no roadside guardrail. Therefore, this article selects three indexes, namely driving speed, sideslope, and slope height, for the roadside accident simulation test.

Simulation test of roadside accident

PC-Crash software is widely used in traffic accident simulation and reconstruction analysis, as it has many advantages, such as abundant vehicle databases and the ability to simulate vehicle dynamics and kinematics accurately. Based on the provisions of representative vehicle types adopted in highway operating speed prediction as stipulated in “Specifications for Highway Safety Audit” (No. JTG B05-2015), the wheelbase of small vehicles is ⩽3.5 m and that of oversized vehicles is >3.5 m. From the PC-Crash vehicle database, BMW320 is selected to represent small vehicles, and the Auwaerter Eurostar Scania-1 is used to represent oversized vehicles.

In addition, the pavement adhesion coefficient, transverse crown slope, lane width, horizontal curve radius, longitudinal road slope, slope gradient, slope height, and other parameters can be determined in combination with the mechanism of occurrence of different roadside accidents based on the Technical Standard of Highway Engineering (No. JTG B01-2014) and other relevant specifications.

After the parameters of the simulation test have been determined, the values of driving speed, sideslope, and slope height are determined for the simulation test of the roadside accident in order to obtain test results on whether the vehicle will roll over after driving to the roadside, the rollover angle, and whether there will be a crash, thereby determining the indicator thresholds of different forms of roadside accidents. Based on the research results of survey data and slope on highway operation speed in the existing research, the driving speeds set in the test are 40, 60, 80, 100, and 120 km/h; the slope heights are 1.5, 3.5, 5.5, 6, 7.5, 9.5, and 11.5 m; and the slope gradients take 38 different values between 1:0.1 and 1:7.7. The number of rollovers and speed changes of the vehicle are recorded as a result of the simulation test. The experiment consists of 225 groups, the simulation scene is shown in Figure 1, and the results of the simulations are shown in Tables 1–3, respectively.

Figure 1.

Simulation test of roadside accident.

Table 1.

Statistics of number of turnovers under different slope conditions.

Number of turnovers slope gradient	0	1~2	>2	Total
$V:H⩽1:6.3$	10	0	0	10
$1:6.3<V:H⩽1:3.7$	15	0	0	15
$1:3.7<V:H⩽1:2.1$	16	4	0	20
$V:H>1:2.1$	75	66	39	180
Total	116	70	39	225

Table 2.

Statistics on number of turnovers under different slope height conditions.

Number of turnovers slope height	0	1~2	>2	Total
1.5 $(V:H⩽1:3.7)$	10	0	0	10
1.5 $(1:1.4⩾V:H>1:3.7)$	25	0	0	25
3.5	22	13	0	35
5.5	16	6	8	30
6	16	16	3	35
7.5	10	7	13	30
9.5	7	14	9	30
11.5	10	14	6	30
Total	116	70	39	225

Table 3.

Statistics on number of turnovers under different speed conditions.

Number of turnovers driving speed	0	1~2	>2	Total
40	24	21	0	45
60	40	5	0	45
80	25	6	14	45
100	15	13	17	45
120	12	27	6	45
Total	116	70	39	225

Single-factor analysis of roadside accident

On the basis of the test results provided by the PC-Crash software, the following circumstances may occur after the vehicle drives to the roadside:

1. The slope is gentle, and the roadside is shallow. The vehicle does not leave the ground after driving to the roadside, there are no casualties, and the vehicle can drive back to the highway and continue to drive as normal.

2. The slope is moderate and the roadside is moderate. The vehicle does not leave the ground after driving to the roadside and there are minor injuries. The vehicle is likely to return to normal driving.

3. The slope is steep and the roadside is deep. The vehicle has at most two rollovers (one rollover is defined as a 90° lateral turn of the vehicle) and there are serious injuries. The vehicle cannot resume normal driving.

4. The slope is very steep, and the roadside is very deep. The vehicle has at least two turnovers and death occurs. The vehicle is badly damaged.

Using SPSS statistical analysis software, correlation analysis was carried out on the test data of rollover times, slope gradient, speed change, slope height, and driving speed, and the results are listed in Table 4. In the Table 4, ** indicates a significant correlation at the 0.01 level (bilateral). Now the number of rollovers is defined as the rollover risk of the accident, it means that the more times the rollover, the greater the rollover risk is. The results show that the driving speed, the slope, and the height of the slope have a significant impact on the number of rollovers and the change in speed and it means that there is a positive correlation between the single factors and the indicators of the rollover risk of the accident. The steeper and higher the slope and the faster the driving speed, the more likely it is that the vehicle will roll over.

Table 4.

Correlation analysis of indicators of roadside accidents.

	Slope gradient	Slope height	Driving speed
Number of turnovers	0.389**	0.405**	0.242**
Speed change	0.485**	0.708**	0.112

**

indicates a significant correlation at the 0.01 level (bilateral).

Relationship between sideslope and roadside accident risk

The number of rollovers on different slopes is shown in Table 1. The analysis of the simulation test results shows that as the slope gradient gradually steepens, the number of turnovers experienced by the vehicle gradually increases, meaning an increase in rollover risk. When the slope gradient $V:H$ is smaller than 1:6.3, the vehicle will not leave the ground or crash after driving to the roadside at any speed and any slope height, and this can be regarded as the safest case. When $1:6.3<V:H⩽1:3.7$ , the vehicle may leave the ground slightly, leading to a relatively minor roadside accident. When $1:3.7<V:H⩽1:2.1$ , the vehicle begins to roll over after driving to the roadside, but it turns over at most twice; that is, the vehicle makes a complete overturn at most. When $V:H>1:2.1$ , the vehicle may suffer a serious roadside accident after driving to the roadside, and the number of turnovers may be more than two but at most seven (with a driving speed of 80 km/h, slope height of 9.5 m, and slope gradient of 1:0.16), which is the most dangerous case. In addition, at a fixed slope gradient, the rollover risk of roadside accidents increases as the slope height and driving speed increase.

Relationship between slope height and roadside accident risk

Statistics on the number of turnovers under different slope heights are shown in Table 2. The analysis of the simulation test results shows that the probability of vehicle turnover accidents increases with increasing height of the slope. In addition, at a fixed slope height, the roadside accident rollover of risk increases with increasing slope gradient and increasing driving speed.

When the slope height is $h⩽1.5\mathrm{m}(V:H⩽1:3.7)$ , the vehicle will never leave the ground at any driving speed, so this is the safest roadside environment. When $h⩽1.5\mathrm{m}(1:3.7<V:H⩽1:1.4)$ , the vehicle may leave the ground after driving to the roadside, but this is safe. When $1.5\mathrm{m}<h⩽3.5\mathrm{m}$ , the vehicle may experience at most two turnovers after driving to the roadside, that is, the vehicle experiences at most one complete overturn (four wheels in the air). When $h>3.5\mathrm{m}$ , the vehicle may suffer more than two turnovers, which is the least favorable case.

Relationship between driving speed and roadside accident risk

Statistics on the number of turnovers under different driving speed conditions are shown in Table 3. The analysis of the simulation test results shows that as a whole, the probability of vehicle turnover increases with increases in driving speed. The threshold of driving speed for vehicle departure from the ground is $V⩽40\mathrm{km}/\mathrm{h}(V:H⩽1:3.7)$ . The threshold of driving speed at which the vehicle leaves the ground but does not turn over is $V⩽40\mathrm{km}/\mathrm{h}(1:3.7<V:H⩽1:1.4)$ . The threshold of driving speed at which the vehicle experiences at most two turnovers is $40<V<80\mathrm{km}/\mathrm{h}$ . The threshold of driving speed at which the vehicle may experience more than two turnovers is $V⩾80\mathrm{km}/\mathrm{h}$ , which is the least favorable case. In addition, at a fixed driving speed, the roadside accident rollover of risk increases with increases of the slope gradient and height.

Thresholds of factors causing roadside accidents

The four degrees of rollover risk of no departure from the ground, slight departure from the ground, at most two turnovers (where a rollover is defined as a lateral turn of the vehicle by 90°), and more than two turnovers of the vehicle after driving to the roadside are defined as roadside accidents at Levels I, II, III, and IV. According to the test results, the corresponding thresholds of slope gradient, slope height, and driving speed of roadside accidents at Levels I, II, III, and IV can be obtained, as listed in Table 5. One point needs to be noted that when either slope height is $h⩽1.5\mathrm{m}$ or driving speed is $V⩽40\mathrm{km}/\mathrm{h}$ , in the case of $V:H⩽1:1.4$ , vehicles in simulation do not turn over; conversely, vehicles may experience at least one turnover when $V:H>1:1.4$ . Hence, the value 1:1.4 is used as a critical value and applied for dividing the thresholds of slope height and driving speed between Levels II and III.

Table 5.

Thresholds of indicators in roadside accidents.

Grade	Slope gradient $V:H$	Slope height h (m)	Driving speed V (km/h)
I	$V:H⩽1:6.3$	$h⩽1.5,(V:H⩽1:3.7)$	$V⩽40,(V:H⩽1:3.7)$
II	$1:6.3<V:H⩽1:3.7$	$h⩽1.5,(1:3.7<V:H⩽1:1.4)$	$V⩽40,(1:3.7<V:H⩽1:1.4)$
III	$1:3.7<V:H⩽1:2.1$	$1.5<h⩽3.5$	$40<V<80$
IV	$V:H>1:2.1$	$h>3.5$	$V⩾80$

Prediction of roadside accident risk

Using the regression analysis method, it is very difficult to deal with the complex nonlinear relationship between the roadside accident risk and the roadside environment index, but a Bayesian network has the function of backward inference. Under the premise that a roadside accident of a certain risk occurs, the trained Bayesian network can be used to carry out reverse calculation to analyze the possibilities of objective factors leading to roadside accidents. Therefore, this article uses a Bayesian network to carry out research on the risk evaluation of roadside accidents.

Application and construction of Bayesian network

Bayesian networks originated from Bayesian statistical analysis theory, which was proposed by Judea Pearl in 1986. It is a method of describing uncertain knowledge and reasoning problems. With the advancement in statistical modeling techniques and computing capabilities, Bayesian approach has been the common and rigorous approach for traffic safety research. Park et al. ²⁷ used empirical and full Bayes methods to estimate the safety effects of roadside barriers with different crash conditions. Alarifi et al. ²⁸ accurately predicted the number of collisions at intersections and segments along corridors by a Bayesian multivariate hierarchical spatial joint model. Schneider IV et al. ²⁹ applied a negative binomial model based on full Bayes’ methods and demonstrated that both horizontal curvature and passenger vehicle volumes significantly contribute to increase truck crashes. Flask and Schneider IV ³⁰ also established a full Bayesian negative binomial model with mixed effects which can account for some of the uncertainty inherent in the accident data.

The construction of a Bayesian network includes the determination of the network structure and the evaluation of conditional probabilities. The topological structure of the Bayesian network built in this article is shown in Figure 2.

Figure 2.

Topological structure of the Bayesian network.

Training of Bayesian network

The slope gradient, slope height, and driving speed are set as $B_1$ , $B_2$ , and $B_3$ ; the rollover risk of roadside accident is set as A; the slope gradient, slope height, driving speed, and rollover risks of roadside accident all have four states, which are $B_{\mathrm{ij}}(i=1,2,3;j=1,2,3,4)$ and $A_i(i=1,2,3,4)$ , respectively.

According to the Bias theorem

$$\begin{array}{c}\multicolumn{1}{c}{P\left(B_{\mathrm{ij}}|A_i\right)=\frac{P\left(A_i|B_{\mathrm{ij}}\right)P(B_{\mathrm{ij}})}{P(A_i)}}\\ \multicolumn{1}{c}{=\frac{P\left(A_i|B_{\mathrm{ij}}\right)P(B_{\mathrm{ij}})}{P\left(A_i|B_{i1}\right)P(B_{i1})+\dots +P\left(A_i|B_{i4}\right)P(B_{i4})}}\\ \multicolumn{1}{c}{=\frac{P\left(A_i|B_{\mathrm{ij}}\right)P(B_{\mathrm{ij}})}{\sum _{j=1}^{4}P\left(A_i|B_{\mathrm{ij}}\right)P(B_{\mathrm{ij}})}}\end{array}$$

(1)

The $P(B_{\mathrm{ij}})$ is known as the edge probability in Bayesian network, in the absence of other information about the variable $B_{\mathrm{ij}}$ , the $P(B_{\mathrm{ij}})$ is allowed to be given an initial value, since what we are concerned with is the updated and not the initial value, and the updated value can be obtained by a series of simulated accident data in Bayesian network; therefore, it can be regarded as $P(B_{\mathrm{ij}})=0.25$ . In this way, formula (1) can be used and $P(B_{\mathrm{ij}}|A_i)$ can be calculated by $P(B_{\mathrm{ij}})$ and $P(A_i|B_{\mathrm{ij}})$ , so that the modeling of the Bayesian network can be completed.

According to the accident simulation test results, the prior probabilities of the roadside accidents at different rollover risks according to the three indicator thresholds corresponding to Table 5 are obtained, as shown in Tables 6–8. In terms of the slope gradient, I means that $G⩽1:6.3$ , II means that $1:6.3<G⩽1:3.7$ , III means that $1:3.7<G⩽1:2.1$ , and IV means that $G>1:2.1$ . With regard to slope height, I means that $h⩽1.5\mathrm{m}(G⩽1:3.7)$ , II means that $h⩽1.5\mathrm{m}(1:3.7<G⩽1:1.4)$ , III means that $1.5<h⩽3.5\mathrm{m}$ , and IV means that $h>3.5\mathrm{m}$ . In terms of driving speed, I means that $V⩽40\mathrm{km}/\mathrm{h}(G⩽1:3.7)$ , II means that $V⩽40\mathrm{km}/\mathrm{h}(1:3.7<G⩽1:1.4)$ , III means that $40<V<80\mathrm{km}/\mathrm{h}$ , and IV means that $V⩾80\mathrm{km}/\mathrm{h}$ .

Table 6.

Prior probability according to sideslope gradient.

Grade	Slope gradient I	Slope gradient II	Slope gradient III	Slope gradient IV
I	1.0000	0.3333	0.1000	0
II	0	0.6667	0.7000	0.4167
III	0	0	0.2000	0.3666
IV	0	0	0	0.2167
Total	1.0000	1.0000	1.0000	1.0000

Table 7.

Prior probability according to sideslope height.

Grade	Slope height I	Slope height II	Slope height III	Slope height IV
I	1.0000	0.0699	0.0571	0
II	0	0.9301	0.6286	0.3806
III	0	0	0.3143	0.3806
IV	0	0	0	0.2387
Total	1.0000	1.0000	1.0000	1.0000

Table 8.

Prior probability according to driving speed.

Grade	Driving speed I	Driving speed II	Driving speed III	Driving speed IV
I	1.0000	0.0833	0.0822	0.0444
II	0	0.9167	0.5616	0.4167
III	0	0	0.3562	0.2465
IV	0	0	0	0.2924
Total	1.0000	1.0000	1.0000	1.0000

From Table 6, the probability of a Level I roadside accident is 100% at slope gradient I $(G⩽1:6.3)$ ; 33.33% at slope gradient II $(1:6.3<G⩽1:3.7)$ ; 10.00% at slope gradient III $(1:3.7<G⩽1:2.1)$ ; and 0 at slope gradient IV $(G>1:2.1)$ .

According to formula (1) and Table 6, the probability of the slope gradient falling into each of the above four levels can be calculated in the case of Level I roadside accidents

$$\begin{array}{c}\multicolumn{1}{c}{P\left(B_{11}|A_1\right)=\frac{P\left(A_1|B_{11}\right)P(B_{11})}{P(A_1)}}\\ \multicolumn{1}{c}{=\frac{P\left(A_1|B_{11}\right)P(B_{11})}{P\left(A_1|B_{11}\right)P(B_{11})+\dots +P\left(A_1|B_{14}\right)P(B_{14})}}\\ \multicolumn{1}{c}{=\frac{P\left(A_1|B_{11}\right)P(B_{11})}{\sum _{j=1}^{4}P\left(A_1|B_{1j}\right)P(B_{1j})}}\\ \multicolumn{1}{c}{=\frac{1\times 0.25}{(1+0.3333+0.1+0)\times 0.25}=0.6977}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{P\left(B_{12}|A_1\right)=\frac{P\left(A_1|B_{12}\right)P(B_{12})}{P(A_1)}}\\ \multicolumn{1}{c}{=\frac{P\left(A_1|B_{12}\right)P(B_{12})}{P\left(A_1|B_{11}\right)P(B_{11})+\dots +P\left(A_1|B_{14}\right)P(B_{14})}}\\ \multicolumn{1}{c}{=\frac{P\left(A_1|B_{12}\right)P(B_{12})}{\sum _{j=1}^{4}P\left(A_1|B_{1j}\right)P(B_{1j})}}\\ \multicolumn{1}{c}{=\frac{0.3333\times 0.25}{(1+0.3333+0.1+0)\times 0.25}=0.2325}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{P\left(B_{13}|A_1\right)=\frac{P\left(A_1|B_{13}\right)P(B_{13})}{P(A_1)}}\\ \multicolumn{1}{c}{=\frac{P\left(A_1|B_{13}\right)P(B_{13})}{P\left(A_1|B_{11}\right)P(B_{11})+\dots +P\left(A_1|B_{14}\right)P(B_{14})}}\\ \multicolumn{1}{c}{=\frac{P\left(A_1|B_{13}\right)P(B_{13})}{\sum _{j=1}^{4}P\left(A_1|B_{1j}\right)P(B_{1j})}}\\ \multicolumn{1}{c}{=\frac{0.1\times 0.25}{(1+0.3333+0.1+0)\times 0.25}=0.0698}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{P\left(B_{14}|A_1\right)=\frac{P\left(A_1|B_{14}\right)P(B_{14})}{P(A_1)}}\\ \multicolumn{1}{c}{=\frac{P\left(A_1|B_{14}\right)P(B_{14})}{P\left(A_1|B_{11}\right)P(B_{11})+\dots +P\left(A_1|B_{14}\right)P(B_{14})}}\\ \multicolumn{1}{c}{=\frac{P\left(A_1|B_{14}\right)P(B_{14})}{\sum _{j=1}^{4}P\left(A_1|B_{1j}\right)P(B_{1j})}}\\ \multicolumn{1}{c}{=\frac{0\times 0.25}{(1+0.3333+0.1+0)\times 0.25}=0}\end{array}$$

In the same way, the probability of having a Grade II, III, or IV roadside accident at a slope gradient falling into each of the above four levels can be calculated, and the calculation results are listed in Table 9.

Table 9.

Conditional probability according to sideslope grade.

	Grade I	Grade II	Grade III	Grade IV
Slope gradient I	0.6977	0	0	0
Slope gradient II	0.2325	0.3738	0	0
Slope gradient III	0.0698	0.3925	0.3529	0
Slope gradient IV	0	0.2337	0.6471	1.0000
Total	1.0000	1.0000	1.0000	1.0000

The calculation of the conditional probability according to driving speed and slope height is the same as that for the slope gradient, and the results are listed in Tables 10 and 11, respectively.

Table 10.

Conditional probability according to sideslope height.

	Grade I	Grade II	Grade III	Grade IV
Slope height I	0.8873	0	0	0
Slope height II	0.0620	0.4796	0	0
Slope height III	0.0507	0.3241	0.4523	0
Slope height IV	0	0.1963	0.5477	1.0000
Total	1.0000	1.0000	1.0000	1.0000

Table 11.

Conditional probability according to driving speed.

	Grade I	Grade II	Grade III	Grade IV
Driving speed I	0.8265	0	0	0
Driving speed II	0.0688	0.4837	0	0
Driving speed III	0.0679	0.2964	0.5910	0
Driving speed IV	0.0367	0.2199	0.4090	1.0000
Total	1.0000	1.0000	1.0000	1.0000

The results of conditional probability according to the three indicators in Tables 9–11 are put into the topological structure of the Bayesian network; that is, the modeling of Bayesian network can be completed. By setting of the range of each indicator, calculation of the probability of roadside accidents of different rollover risks can be carried out.

Inference and prediction by Bayesian network

According to the Bayesian network in this article and conditional probability theory, different values of different indicators are set to predict the probability of roadside accidents of different rollover risks.

Single-factor inference

In the case of a known state of any of the indicators slope gradient $B_1$ , slope height $B_2$ , and driving speed $B_3$ , recorded as $B_{i_0{j}_0}(i_0\in 1,2,3;j_0\in 1,2,3,4)$ . If the values of other indicators are unknown, formula (2) can be used to calculate the probability of roadside accidents of different rollover risks

$$P\left(A_i|B_{i_0{j}_0}\right)=\frac{P\left(B_{i_0{j}_0}|A_i\right)P(A_i)}{P(B_{i_0{j}_0})}=\frac{P\left(B_{i_0{j}_0}|A_i\right)P(A_i)}{\sum _{i=1}^{4}P\left(B_{i_0{j}_0}|A_i\right)P(A_i)}$$

(2)

For example, when the slope gradient is 1:3, which is within the range of Level III, and the other indexes are unknown, the process of calculation of the probability of roadside accidents of different rollover risks is as follows

$$\begin{array}{c}\multicolumn{1}{c}{P\left(A_1|B_{13}\right)=\frac{P\left(B_{13}|A_1\right)P(A_1)}{P(B_{13})}=\frac{P\left(B_{13}|A_1\right)P(A_1)}{\sum _{i=1}^{4}P\left(B_{13}|A_i\right)P(A_i)}}\\ \multicolumn{1}{c}{=\frac{0.0698}{0.0698+0.3925+0.3529+0}=0.086}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{P\left(A_2|B_{13}\right)=\frac{P\left(B_{13}|A_2\right)P(A_2)}{P(B_{13})}=\frac{P\left(B_{13}|A_2\right)P(A_2)}{\sum _{i=1}^{4}P\left(B_{13}|A_i\right)P(A_i)}}\\ \multicolumn{1}{c}{=\frac{0.3925}{0.0698+0.3925+0.3529+0}=0.481}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{P(())=\frac{P\left(B_{13}|A_3\right)P(A_3)}{P(B_{13})}=\frac{P\left(B_{13}|A_3\right)P(A_3)}{\sum _{i=1}^{4}P\left(B_{13}|A_i\right)P(A_i)}}\\ \multicolumn{1}{c}{=\frac{0.3529}{0.0698+0.3925+0.3529+0}=0.433}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{P\left(A_4|B_{13}\right)=\frac{P\left(B_{13}|A_4\right)P(A_4)}{P(B_{13})}=\frac{P\left(B_{13}|A_4\right)P(A_4)}{\sum _{i=1}^{4}P\left(B_{13}|A_i\right)P(A_i)}}\\ \multicolumn{1}{c}{=\frac{0}{0.0698+0.3925+0.3529+0}=0}\end{array}$$

Similarly, the probability of roadside accidents of different rollover risks occurring when the value of the slope gradient falls into the range of Grade I, II, or IV and the values of other indicators are unknown can be calculated. In the case in which the values of slope height and driving speed are of a certain grade and the values of the other two indicators are unknown, the calculation of the probability of roadside accidents of different rollover risks is the same as that of the slope gradient.

Double-factor inference

When a node H (with m states) has more than one argument $E_{1}E\dots E_n$ , it is assumed that the condition between the arguments is independent, and its Bainite probability can be calculated according to the following formula

$$\begin{array}{c}\multicolumn{1}{c}{P(H_i|E_{1}E\dots E_n)=\frac{P\left(E_{1}E\dots E_n|H_i\right)P\left(H_i\right)}{\sum _{k=1}^{m}P\left(E_{1}E\dots E_n|H_i\right)P\left(H_i\right)}}\\ \multicolumn{1}{c}{=\frac{P\left(E_1|H_i\right)P\left(E_2|H_i\right)\dots P\left(E_n|H_i\right)P\left(H_i\right)}{\sum _{k=1}^{m}P\left(E_1|H_k\right)P\left(E_2|H_k\right)\dots P\left(E_n|H_k\right)P\left(H_k\right)}}\end{array}$$

(3)

In the case of known states of any two indicators among the slope gradient $B_1$ , the slope height $B_2$ , and the driving speed $B_3$ , they are respectively recorded as $B_{i_0{j}_0},B_{{i\prime }_0{j\prime }_0}(i_0,i{\prime }_0\in 1,2,3;i_0\ne i{\prime }_0;j_0,j{\prime }_0\in 1,2,3,4)$ , and formula (4) can be used to calculate the probability of roadside accidents of different rollover risks

$$P\left(A_i|B_{i_0{j}_0}{B}_{{i\prime }_0{j\prime }_0}\right)=\frac{P\left(B_{i_0{j}_0}|A_i\right)P\left(B_{{i\prime }_0{j\prime }_0}|A_i\right)P\left(A_i\right)}{\sum _{i=1}^{4}P\left(B_{i_0{j}_0}|A_i\right)P\left({B_{{i\prime }_0}}_{{j\prime }_0}|A_i\right)P\left(A_i\right)}$$

(4)

For example, when the slope gradient is 1:3, the slope height is 3 m, meaning that the slope gradient and slope height fall into the range of Level III, and the value of the driving speed is unknown, the process of calculation of the probability of occurrence of roadside accidents of different rollover risks is as follows

$$\begin{array}{c}\multicolumn{1}{c}{P\left(A_1|B_{13}{B}_{23}\right)=\frac{P\left(B_{13}|A_1\right)P\left(B_{23}|A_1\right)P\left(A_1\right)}{\sum _{i=1}^{4}P\left(B_{13}|A_i\right)P\left(B_{23}|A_i\right)P\left(A_i\right)}=\frac{0.0698\times 0.0507}{0.0698\times 0.0507+0.3925\times 0.3241+0.3529\times 0.4523}=0.012}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{P\left(A_2|B_{13}{B}_{23}\right)=\frac{P\left(B_{13}|A_2\right)P\left(B_{23}|A_2\right)P\left(A_2\right)}{\sum _{i=1}^{4}P\left(B_{13}|A_i\right)P\left(B_{23}|A_i\right)P\left(A_i\right)}=\frac{0.3925\times 0.3241}{0.0698\times 0.0507+0.3925\times 0.3241+0.3529\times 0.4523}=0.438}\end{array}$$

$$P\left(A_3|B_{13}{B}_{23}\right)=\frac{P\left(B_{13}|A_3\right)P\left(B_{23}|A_3\right)P\left(A_3\right)}{\sum _{i=1}^{4}P\left(B_{13}|A_i\right)P\left(B_{23}|A_i\right)P\left(A_i\right)}=\frac{0.3529\times 0.4523}{0.0698\times 0.0507+0.3925\times 0.3241+0.3529\times 0.4523}=0.550$$

$$P\left(A_4|B_{13}{B}_{23}\right)=\frac{P\left(B_{13}|A_4\right)P\left(B_{23}|A_4\right)P\left(A_4\right)}{\sum _{i=1}^{4}P\left(B_{13}|A_i\right)P\left(B_{23}|A_i\right)P\left(A_i\right)}=\frac{0}{0.0698\times 0.0507+0.3925\times 0.3241+0.3529\times 0.4523}=0$$

Similarly, the probability of roadside accidents of different rollover risks occurring when the values of slope gradient and slope height fall into the range of Grade I, II, or IV and the value of driving speed is unknown can be calculated. In the case in which the values of the slope height and the driving speed are of a certain grade and the value of slope height is unknown, the calculation of the probability of occurrence of roadside accidents of different rollover risks is as above.

Three-factor inference

In the case of known states of the slope gradient $B_1$ , the slope height $B_2$ , and driving speed $B_3$ , they are respectively recorded as $B_{1j_0},B_{2{j\prime }_0},B_{3{j″}_0}(j_0,j{\prime }_0,{j″}_0\in 1,2,3,4)$ , and the general form of calculation for the probability of occurrence of roadside accidents of different rollover risks is as shown in formula 5

$$\begin{array}{c}\multicolumn{1}{c}{P\left(A_i|B_{1j_0}{B}_{2{j\prime }_0}{B}_{3{j″}_0}\right)}\\ \multicolumn{1}{c}{=\frac{P\left(B_{1j_0}|A_i\right)P\left(B_{2{j\prime }_0}|A_i\right)P\left(B_{3{j″}_0}|A_i\right)P\left(A_i\right)}{\sum _{i=1}^{4}P\left(B_{1j_0}|A_i\right)P\left(B_{2{j\prime }_0}|A_i\right)P\left(B_{3{j″}_0}|A_i\right)P\left(A_i\right)}}\end{array}$$

(5)

For example, in the case of a slope gradient of 1:3, a slope height of 3 m, and a driving speed of 60 km/h, the values of slope gradient, slope height, and driving speed fall into the range of Level III, and the calculation process is as follows

$$\begin{array}{c}\multicolumn{1}{c}{P\left(A_1|B_{13}{B}_{23}{B}_{33}\right)=\frac{P\left(B_{13}|A_1\right)P\left(B_{23}|A_1\right)P\left(B_{33}|A_1\right)P\left(A_1\right)}{\sum _{i=1}^{4}P\left(B_{13}|A_i\right)P\left(B_{23}|A_i\right)P\left(B_{33}|A_i\right)P\left(A_i\right)}}\\ \multicolumn{1}{c}{=\frac{0.0698\times 0.0507\times 0.0679}{0.0698\times 0.0507\times 0.0679+0.3925\times 0.3241\times 0.2964+0.3529\times 0.4523\times 0.5910}=0.002}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{P\left(A_2|B_{13}{B}_{23}{B}_{33}\right)=\frac{P\left(B_{13}|A_2\right)P\left(B_{23}|A_2\right)P\left(B_{33}|A_2\right)P\left(A_2\right)}{\sum _{i=1}^{4}P\left(B_{13}|A_i\right)P\left(B_{23}|A_i\right)P\left(B_{33}|A_i\right)P\left(A_i\right)}}\\ \multicolumn{1}{c}{=\frac{0.3925\times 0.3241\times 0.2964}{0.0698\times 0.0507\times 0.0679+0.3925\times 0.3241\times 0.2964+0.3529\times 0.4523\times 0.5910}=0.285}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{P\left(A_3|B_{13}{B}_{23}{B}_{33}\right)=\frac{P\left(B_{13}|A_3\right)P\left(B_{23}|A_3\right)P\left(B_{33}|A_3\right)P\left(A_3\right)}{\sum _{i=1}^{4}P\left(B_{13}|A_i\right)P\left(B_{23}|A_i\right)P\left(B_{33}|A_i\right)P\left(A_i\right)}}\\ \multicolumn{1}{c}{=\frac{0.3529\times 0.4523\times 0.5910}{0.0698\times 0.0507\times 0.0679+0.3925\times 0.3241\times 0.2964+0.3529\times 0.4523\times 0.5910}=0.713}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{P\left(A_4|B_{13}{B}_{23}{B}_{33}\right)=\frac{P\left(B_{13}|A_4\right)P\left(B_{23}|A_4\right)P\left(B_{33}|A_4\right)P\left(A_4\right)}{\sum _{i=1}^{4}P\left(B_{13}|A_i\right)P\left(B_{23}|A_i\right)P\left(B_{33}|A_i\right)P\left(A_i\right)}}\\ \multicolumn{1}{c}{=\frac{0}{0.0698\times 0.0507\times 0.0679+0.3925\times 0.3241\times 0.2964+0.3529\times 0.4523\times 0.5910}=0}\end{array}$$

Similarly, the probability of roadside accidents of different rollover risks occurring when the values of slope gradient, slope height, and driving speed fall into the ranges of Level I, II, and IV can be calculated.

Case analysis

Accident cases

In June 2013, at Line K13+620 of the County Highway X125 from Suo Er Ba Si Tao Scenic Area to Changji via Miaoergou Township in Xinjiang, a bus carrying 36 passengers (allowed load of 37 passengers) turned over on the left embankment, leading to 15 deaths and 21 injuries, causing direct economic losses of 8,780,000 Yuan.

Accident investigation

The inspection period of the vehicle was valid until 2016, the grade of technical evaluation was first grade, and the permitted types of driver were A1 and A2. The accident section was Grade 4 mountainous highway with asphalt pavement presenting an “S” curve. The turning radius was 20.35 m, and the longitudinal gradient was 9.2%. The shoulder was soft without any protective measures and a good line of sight. When the accident occurred, the pavement was slippery, and the speed of the bus was 42 km/h.

Cause of the accident

The direct reason was that the driver was speeding on the turning section of the steep slope in the mountainous area, an improper operation, which led to the turnover. The indirect reason was that the section does not have effective safety measures.

Risk evaluation of roadside accident

The driving speed was 42 km/h, which belongs to Level III, and the slope gradient and slope height were both in the range of Grade IV. These values were put into the Bayesian network to calculate the probability of roadside accident risks, and the probability of occurrence of roadside accidents of different grades could be obtained. The probability of the bus turning over up to twice was 0.939, while the probability of the bus coming off the ground slightly was only 0.061. It can be considered that the bus was likely to turn over when driving to the roadside, but at most twice. PC-Crash software was used to simulate the accident. The parameters were set as follows: radius of horizontal curve $R=20.35\mathrm{m}$ , pavement adhesion coefficient $\phi =0.45$ , slope height $h=11.5\mathrm{m}$ , slope width $w=1.5\mathrm{m}$ , and driving speed $V=42\mathrm{km}/\mathrm{h}$ . The results of the simulation analysis showed that the bus would experience rollover, which was consistent with the calculation results, indicating that the bus would be most likely to suffer a Grade III roadside accident (one or two rollovers).

Conclusion

Based on the accident simulation and analysis, this article studies that the driving speed, the slope, and the height of the slope have a significant impact on the number of rollovers by the regression analysis method; categorizes the rollover risk of roadside accidents; gives the thresholds of factors causing the accidents of different rollover risks by PC-Crash simulation software; then puts forward a method of evaluating the risk of highway roadside accident based on a Bayesian network; and confirms its effectiveness by a typical cases of roadside accidents. This proposed method can accurately calculate the probability of roadside accident and achieve a quantitative evaluation of the roadside accident risk. In the study on threshold categorization of factors causing roadside accidents, this study assumes that there are no roadside obstacles, and only selects the driving speed, slope gradient, and slope height to construct a Bayesian network that calculates the probability of occurrence of a roadside accident of a certain risk. Actually, there are many roadside obstacles not considered (e.g. roadside guardrails, traffic signs, utility poles, trees, walls, ditches, buildings, and culverts) also affecting roadside accident risks. Hence, the factor of roadside obstacles will be incorporated in further studies.