5.1. Monte Carlo Simulation Calculation Model
MCS is a stochastic simulation method that uses random numbers to solve a problem by finding the probability distribution of the random variables in the problem, then using a computer to sample a large number of them to obtain an approximate solution to the problem. The main principle of the MCS for solving a problem is: sampling the variables (initial speed, end speed) that affect the limit-state function, and then substituting the extracted variable values into the limit-state function to determine whether or not the failure occurs. Finally, the failure probability can be obtained. This method becomes more accurate when the number of simulation runs increases [
35,
49].
MCS has several advantages. Firstly, there is no requirement for the probability distribution of variables and there is no limit to the number of variables. Secondly, the error of the method is independent of the dimensionality of the problem. Finally, MCS leverages computer technology, making it a more efficient and scalable approach. However, the deterministic problem needs to be transformed into a stochastic one.
MCS is applicable when the problem involves random variables with known probability distributions. In this paper, the distribution of operating speed is obtained by investigation. Moreover, in the field of freeway design, MCS is commonly used [
35,
50,
51], so this article uses MCS to study the length and slope of the deceleration lane.
By using MATLAB software to program the calculation process, the flow chart shown in
Figure 10 was prepared.
The detailed calculation steps are as follows:
Step1: Select N = 100,000 for the number of samples and enter the design value of LS (supply value) for the length and slope value of the deceleration lane;
Step2: Determine the probability distribution of the initial speed at the diverging point and the end speed at the diverging nose;
Step3: Generate random values by sampling from the probability distribution of the variables and calculate the demand value LD for the length of the deceleration lane in combination with Equation (3);
Step4: Record the number n of times LS−LD < 0 and calculate the failure probability .
5.2. Improved First-Order Second Moments Calculation Model
The FOSM model is more widely used in engineering, and the IFOSM and JC methods have been derived according to its shortcomings.
In cases where the distribution of the variable is not clear, FOSM uses two statistical indicators, the mean and the standard deviation. It linearizes the function by expanding it with a Taylor series at the mean point. This leads to a new mathematical model to solve for reliability. However, for nonlinear functions, the larger the linearization point is from the failure boundary, the larger the error becomes. This error is due to the omission of the higher-order terms after performing the Taylor series expansion. However, although the mean points tend to be within the reliability zone, the selection of mean points as linearization points can lead to large errors. IFOSM selects the design checkpoint corresponding to the maximum probability of failure of the limit-state function. This approach addresses the limitation of the traditional method and provides more reliable calculation results [
35,
36].
IFOSM is suitable for the case wherein the random variables are independent and obey normal distribution. This assumption helps ensure high accuracy in the calculation results.
The previous section has established a limit-state function
Z based on
n variables
xi (
i = 1, 2, …,
n) that affect the length of the deceleration lane.
Expand the Taylor series of the function at the design checkpoint
p*(
x1,
x2, …,
xn), keeping only the linear term.
Assuming that the initial design checkpoint is the mean of each variable (
), then obtain the mean
and standard deviation
of the limit-state function.
Calculate the reliability index
and sensitivity coefficient
of the limit-state function by Equations (10) and (11):
From the calculated mean value
, standard deviation
, and sensitivity coefficient
of the function, a new design checkpoint is calculated.
The design checkpoint is the closest point to the failure boundary, and in order to find the point accurately, a calculation accuracy
e needs to be given. Therefore, after calculating the reliability index, use Equation (13) to determine whether the accuracy is satisfied.
When the reliability index does not meet the accuracy, the IFOSM algorithm iteratively repeats to determine the final design verification point. Subsequently, the reliability index is obtained. The algorithm of the IFOSM model is shown in
Figure 11.
Finally, calculate the failure probability of the limit-state function by Equation (14).
where
denotes the probability density function of the standard normal distribution.
5.3. Sensitivity Analysis and Results
Since the calculation results of the MCS and the IFOSM are similar, we only plot the calculation results of the MCS in
Figure 12.
Figure 12 illustrates the calculation of failure probability for different mainline design speeds, lengths, and slopes of the deceleration lane using the two methods mentioned above.
Figure 12a shows that the failure probabilities are 34.47% and 51.29% for a 100 m length of the deceleration lane when the slopes are 0% and 4%, respectively. It can be seen that the failure probability of the length of the deceleration lane is greatly affected by the slope. With a constant length, the failure probability increases as rapidly as the increase of slope.
In addition, for the length of 240 m, the failure probabilities are 0.65% and 5.54% for the slopes of 0% and 4%, respectively. It can be observed that the effect of slope on the failure probability gradually decreases as the length of the deceleration lane increases.
Figure 12a,b are basically the same, that is, the failure probability is basically the same for the 120 km/h and 100 km/h cases because they are similar at the initial and end speeds.
Figure 12c,f show the failure probability of cars and trucks at the design speed of 80 km/h for the mainline.
In the case of a 6% slope and 100 m length of the deceleration lane, the failure probabilities for cars and trucks are 28.44% and 87.78% respectively. The same is true for the rest of the length and slope combinations, indicating that the truck is more sensitive to the slope.
To ensure that the calculation results have a 95% guarantee rate, the corresponding minimum length of the deceleration lane and its failure probability are listed in
Table 9 and
Table 10.
- 1.
The car is the dominant model in calculating the length of the deceleration lane.
Through the calculation of the deceleration lane lengths for both cars and trucks, we found that at mainline design speeds of 120 km/h and 100 km/h, when the slope of the deceleration lane is ≤3%, the requirement for the deceleration lane length of the car is greater than the truck. When the slope of the deceleration lane exceeds 3%, the braking performance of the truck becomes worse under the influence of the slope. In this situation, the truck demands a longer deceleration lane length than the car. When the design speed is 80 km/h, the operating speeds of cars and trucks are similar. However, due to the poorer braking performance of trucks, they need a longer length for the deceleration lane compared to cars. Therefore, the deceleration lane length mainly considers the truck. This conclusion can be verified by the Chinese “Design Specification for Highway Alignment” [
40]. It states that when the slope is ≤3%, the impact on the speed of trucks is small.
However, according to the analysis in
Section 3, the truck actually starts to decelerate from the beginning of the taper section, and the range of the deceleration length of the truck in
Table 10 is the sum of the taper section and the deceleration section. Therefore, as long as the deceleration lane length for cars is ensured, the safe deceleration of trucks can be ensured.
- 2.
Vehicles can adapt to larger slopes on downhill deceleration lanes
The slope of the vehicle can reach 4%, 5%, and 6%, respectively, when the mainline design speed is 120 km/h, 100 km/h, and 80 km/h. However, as the slope increase, the deceleration lane length needs to be extended accordingly. Consequently, the larger the slope, the greater the increase in value of the deceleration lane length. The maximum slope proposed in this paper is the same as the limiting slope of the basic section on freeways in China’s “Design Specification for Highway Alignment” [
40].
- 3.
The MCS and the IFOSM models have good applicability in calculating the deceleration lane length.
From the calculation results of the two reliability models in
Table 9 and
Table 10, it can be obtained that the results of the MCS and the IFOSM are similar, and both calculation models are reasonable. However, we found that the MCS requires a high number of operations in the calculation; otherwise, the results will fluctuate.
Through the comprehensive comparison of the MCS and the IFOSM for the calculation of deceleration lane length, we propose the recommended length of the deceleration lane at different design speeds and slopes (
Table 11).
Table 12 shows the Chinese specifications for deceleration lane length. It can be seen that when the design speed is 80 km/h, the deceleration section length recommended in this paper is basically the same as the specification value. For the design speeds of 100 km/h and 120 km/h, the deceleration section lengths proposed in this paper exceed the standard values. This is due to the high requirements for failure probability for the results of reliability calculations. Furthermore, the reliability calculation method used in this paper considers various combinations of initial and end speeds, resulting in a significant number of unfavorable combinations. In order to ensure driver comfort and safety, a comfortable deceleration is adopted, which results in a longer deceleration section length.
However, it can be observed that the length of the deceleration section proposed in this paper is similar to the sum of the deceleration section and the taper section in the specification. Therefore, it is possible to set up traffic signs to remind drivers to start decelerating in the taper section, and the standardized value can meet the recommended value proposed in this paper.
In addition, the maximum slope of the deceleration lane specified in the specifications is less than the recommended value in
Table 11. A reasonable increase in the deceleration lane slope provides a greater improvement for the design. For the design of mountainous interchanges, in order to overcome the height differences in ramp design, it is usually chosen to extend the length of the ramp. However, this leads to an increase in the project cost. Additionally, the extensive filling and excavation required for these longer sections can cause significant damage to the mountainous environment. Therefore, using the standard slope value for design purposes poses a significant disadvantage.
The length and slope of the deceleration section proposed in this paper are well applicable, which can effectively improve the current situation of terrain limitations in mountain interchange design. By specifying the failure probability at a small value and increasing the deceleration lane length, the slope of the deceleration lane can be appropriately increased within the safety range. The recommended values of deceleration lane length and slope proposed in this paper quantify the safety degree, which can effectively ensure the safety of mountainous interchange design.