# Reliability Analysis of Intersection Sight Distance at Roundabouts

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{nc}) and different coefficients of variations. For the special case of single-lane symmetrical roundabouts, which have a well-defined geometry, the lateral clearance needs are established. The sensitivity analysis shows that ISD is very sensitive to both the mean and variance of the critical headway. The results show that the deterministic method results in ISD values that correspond to a very small P

_{nc}, indicating that the method is very conservative. The proposed method, which provides flexibility in selecting ISD for any given P

_{nc}, should be of interest to highway designers and practitioners to promote roundabout safety.

## 1. Introduction

## 2. Deterministic ISD with Nonlinear Deceleration

#### 2.1. Modeling Nonlinear Deceleration Profile

_{e}to the circulating speed v

_{c}. The analysis concept is based on a comparison of the kinematic equations associated with the deceleration rate. The concept is like that given by Wortman and Fox [29], who developed the deceleration profiles for stopping vehicles (v

_{c}= 0). The stopping deceleration profile is a special case of the presented general formulation of deceleration profiles. Let the deceleration distance and deceleration time be denoted by d and t, respectively. Three basic equations are used to calculate the deceleration rate, where each equation is a function of two of the variables (v

_{e}, v

_{c}, d, and t), as follows:

_{1}, a

_{2}, and a

_{3}denote the deceleration rate (m/s

^{2}) that corresponds to Equations (1)–(3), respectively; v

_{e}and v

_{c}denote the entry and circulatory speeds (m/s); d denotes the deceleration distance (m); and t denotes the deceleration time (s).

_{1}= a

_{2}= a

_{3}. For nonlinear deceleration, however, the deceleration rates of Equations (1)–(3) will be different. The different cases of the deceleration profiles are quantified as follows:

_{1}and a

_{2}are used because for nonlinear deceleration profiles they represent limiting values, where a

_{3}always lies between them. That is, there are two possible inequalities a

_{1}< a

_{3}< a

_{2}and a

_{2}< a

_{3}< a

_{1}, which correspond to r < 1 and r > 1, respectively. For linear deceleration profiles, r = 1.

_{1}. Figure 1b,c represent the nonlinear deceleration rate when r < 1 and r > 1, respectively, with the corresponding areas d

_{2}and d

_{3}(d

_{2}> d

_{1}> d

_{3}). Note that the deceleration profile of Figure 1b starts with a gentle deceleration rate followed by a more aggressive rate, covering a larger distance (during the same time t) than that of linear decelerate rate. This behavior would be expected at a low entry speed with a large circulatory speed. On the other hand, the deceleration profile of Figure 1c starts with a more aggressive rate followed by a gentler rate, covering a smaller distance (during the same time t) than that of the linear decelerate rate. This behavior is expected at a high entry speed with a low circulatory speed.

_{1}and a

_{2}into Equation (4), then:

#### 2.2. Formulas for Sight Distance Legs

_{min}= 15 m) and at the yield line. At each entry, the sight distance triangle has two conflicting approaches that must be checked independently: (1) conflicting vehicles within the circulatory roadway (left-turn movements from the opposing entry approach) and (2) conflicting vehicles from the immediate upstream entry (left-turn or through movements).

_{min}from the yield line, the approach leg is aC. For the conflicting-entry vehicle, the conflicting leg is CD (or D

_{1}) and the third leg is the sightline aD. For the conflicting-circulating vehicle, the approach leg is bC, the conflicting leg is CE (or D

_{2}), and the third leg is the sightline bE. The conflicting leg of the sight triangle should follow the conflicting-vehicle curved path; see Rodegerdts et al. [1,30,31].

_{1}, is given by (Figure 2b):

_{cir}denotes the distance along the circulatory part of the path; d denotes the distance during deceleration, given by Equation (7); and d

_{e}denotes the distance along the entry curve. The distance d

_{e}equals the entry speed v

_{e}multiplied by the time t

_{e}, which corresponds to d

_{e}.

_{cir}depends on the minimum radius of the circulatory roadway R (which depends on v

_{c}), and the central angle $\theta $ suspended by this portion of the path (assumed to be 30°, based on the visual inspection of numerous roundabouts). The minimum radius was calculated based on the lateral friction coefficients for urban low speeds and e = 0.02 [3]. For the reliability analysis, based on AASHTO data, the relation between R and v

_{c}was established using regression analysis as:

_{c}= the design speed of the circulatory roadway (m/s). Then, the portion of the entry vehicle path on the circulatory roadway, which equals $R\theta $ (where the angle is in radians), is given by:

_{cir}

_{/}v

_{c}. Thus,

_{1}, as shown in Figure 3. The three cases are formulated as follows:

_{c}≤ t

_{cir}). In this case, D

_{1}is given by:

_{cir}< t

_{c}< t

_{cir}+ t). To calculate D

_{1}in this case, let the time corresponding to the part on the deceleration profile be denoted as t’ and the corresponding speed of the entry vehicle at the end of D

_{1}be ${v}_{e}^{\prime}$. Then,

_{1}can be derived as:

_{1}on the deceleration profile (d’).

_{c}≥ t

_{cir}+ t). The time on the entry part equals the critical headway time t

_{c}minus the time spent on the deceleration and the circulatory parts. Thus, D

_{1}can be derived as:

_{1}for the linear deceleration rates.

_{2}denotes the length of the sight distance leg for the conflicting circulating vehicle (m), t

_{c}= the critical headway for entering the roundabout (s), and v

_{c}= the design speed of the conflicting circulating vehicle (km/h). Note that Equation (20) is equivalent to the sight distance leg for the entry vehicle of Case 2.

_{1}was calculated for different values of r (0.5, 1, and 1.5) for the deterministic case. Different combinations of entry speed (V

_{e}= 30 km/h to 70 km/h) and circulatory speed (V

_{c}= 20 km/h to 60 km/h) and the 90th percentile value of t

_{c}and the 10th percentile value of a were used. The results are presented in Table 2. As noted, for r = 0.5 the difference in D

_{1}between the linear and nonlinear profiles reaches up to 19.3%, and for r = 1.5 the difference reaches up to −8.1%. Obviously, when V

_{e}= V

_{c}, there is no deceleration, and the difference equals zero.

## 3. Proposed Reliability Method

#### 3.1. FOSM Reliability Method

_{i}, i = 1, 2, …, n. Then,

_{xi}= the mean of the random variable x

_{i}, σ

_{xi}

^{2}= the variance of the random variable x

_{i}, σ

_{xi}= the standard deviation of the random variable x

_{i}, and ρ

_{xi}

_{, xj}= Pearson’s coefficient of the correlation between x

_{i}and x

_{j}. Furthermore, the coefficient of variation of the random variable x

_{i}, CV

_{xi}, which is a dimensionless measure of dispersion, is given by:

_{nc}can be estimated as:

_{nc}= 5%. The larger β is, the smaller the P

_{nc}. If SM is linear, the mean and standard deviation will be accurately assessed using the FOSM method. However, if SM is nonlinear, the FOSM analysis may introduce errors. For more details about reliability analysis, the reader is referred to Benjamin and Cornell [32], Smith [33], and Haukaas [34].

#### 3.2. Reliability Analysis of ISD

#### 3.2.1. Distance D_{1}

_{1}supplied (m) and ${D}_{1}$ = the length of D

_{1}demanded (m), which equals D

_{1}from Equations (13), (16), and (17) for Cases 1–3, respectively. In general, the expected value and variance of the SM

_{1},$E\left[S{M}_{1}\right],$ and var[SM

_{1}], based on Equations (22) and (23), are given by:

_{1}, which is obtained by substituting the mean values of the component random variables into the D

_{1}equation of the respective case. Note that Equation (29) includes the correlations between several variables: (v

_{e}, t

_{c}), (v

_{e}, a), (v

_{e}, r), and (v

_{c}, a). The first three correlations are positive, while the fourth is negative. The first derivatives are given next for the three cases.

_{1}from Equation (13) into Equation (27), the first derivatives of $S{M}_{1},$ with respect to ${v}_{c}and{t}_{c,}$ can be derived as:

_{e}’ from Equations (14) and (15) and d

_{cir}and t

_{cir}from Equations (11) and (12) into Equation (16), then substituting D

_{1}into Equation (27), the first derivatives of $S{M}_{1}$ with respect to ${v}_{c},a,{t}_{c},\mathrm{and}r$ can be derived as:

_{cir}from Equations (5) and (11) into Equation (17), then substituting D

_{1}into Equation (27), the first derivatives of $S{M}_{1}$ with respect to ${v}_{e},{v}_{c},a,{t}_{c},\mathrm{and}r$ can be derived as

_{1supply}is calculated for various values of P

_{nc}. Note that the conditions for the three cases presented previously involves random variables, and therefore these conditions also involve uncertainty. The probabilistic equivalent of the deterministic conditions for the three cases are presented in Appendix A.

#### 3.2.2. Distance D_{2}

_{2}supplied (m) and ${D}_{2}$ = the length of D

_{2}demanded (m), which equals D

_{2}from Equation (19). Then,

_{2},$E\left[S{M}_{2}\right],$ and var[SM

_{2}], based on Equations (21) and (22), are given by:

_{c}and t

_{c}are given by the right sides of Equations (30) and (31), respectively. Based on Equations (25) and (44), then:

_{2supply}is calculated for various values of P

_{nc}.

#### 3.3. Verification

_{1}and D

_{2}were verified using Monte Carlo simulation. The probability distribution of the safety margin of the mathematical method, based on $E[S{M}_{1}$], var[SM

_{1}], and $E[S{M}_{2}$], var[SM

_{2}] was compared with that of the simulation. The mean values of the random variables were as follows. For D

_{1}(Case 1), µ

_{ve}= 12.85 m/s and µ

_{vc}= 10.28 m/s, and for D

_{1}(Case 2), µ

_{ve}= 12.85 m/s and µ

_{vc}= 7.71 m/s. For both cases, µ

_{tc}= 5 s, µ

_{a}= 1.3 m/s

^{2}, and µ

_{r}= 0.5. The coefficient of variation was 5% for all variables. The mean entry and circulatory speeds correspond to the design speeds of 50 km/h and 40 km/h, respectively, which were assumed to represent the 95% percentile speed, as discussed later in data preparation. For D

_{2}, µ

_{vc}= 7.71 m/s and µ

_{tc}= 5 s with coefficient of variation (CV) = 5%, where the mean circulatory speed corresponds to a design speed of 30 km/h. All the random variables were assumed to be normally distributed in the simulation. For simplicity, the variables were assumed to be uncorrelated. Note that the FOSM reliability method requires no assumptions about the distributions of the input random variables.

_{1}(v

_{e}, v

_{c}, t

_{c}, a, and r) and D

_{2}(v

_{c}, t

_{c}). The probability distributions of the component random variables were assumed to be normal, with the means and standard deviations calculated from the data. The generated random values of the component variables were then substituted into the respective equations, resulting in 30,000 values of the dependent random variable. Using these values, their means and standard deviations were calculated along with their histograms. The reliability index was β = 1.64 (P

_{nc}= 5%). For D

_{1}(Case 2), the mean and standard deviation of the safety margin of the mathematical model were $E[S{M}_{1}$], = 6.818 m and σ

_{SM1}= 4.157 m, compared with the simulation values of 6.761 and 4.155 (Figure 4a). The results for Case 3 were also very close (Figure 4b). For D

_{2}, the mathematical and simulated values were (4.469, 2.725) and (4.512, 2.718), respectively (Figure 4c). As noted, the means and standard deviations of the simulation and mathematical formulas show excellent agreement. In particular, the close standard deviations of the mathematical and simulation methods verify the rather complex first derivatives of Cases 2 and 3 of D

_{1}. In addition, the simulation results show that the probability distributions of the design variables are very close to the normal distribution.

## 4. Application

#### 4.1. Data Preparation

_{Xi}denotes the mean of the random variable x

_{i}, E

_{xi}denotes the extreme value corresponding to a certain percentile value of the random variable x

_{i}, z denotes the number of standard deviations of the normal distribution corresponding to a certain percentile value, and CV

_{Xi}denotes the coefficient of variation of the random variable. Note that z is positive for variables for which the extreme values are based on a high percentile value and negative when low percentile values should be used. For example, the value of z for a random variable with respect to its 95th percentile value is 1.64 and −1.64 for the 5th percentile.

_{e}, V

_{c}, and t

_{c}. On the other hand, the 10th percentile value represented the design value for a, since smaller values of the deceleration rate result in larger values of D

_{1}. For the deceleration shape parameter, there is currently no guidance for the roundabout deceleration profile (without stopping). However, previous research on vehicle stopping [29] showed that the value of r ranged from 0.4 to 1.7, where the approach speed for r = 1 was 77.2 km/h (48 mph). Speeds less than this limiting value corresponded to r < 1, and speeds larger than this value corresponded to r > 1. Since the entry speeds for roundabouts are typically less than 70 km/h, two values were selected for establishing the design aids: r = 0.5 (nonlinear deceleration) and r = 1 (linear deceleration).

#### 4.2. Design Values of ISD

_{1}and D

_{2}were established using the reliability-based method of Equations (41) and (46), respectively, for CV = 5% and 10% and P

_{nc}= 1%, 5%, and 10%. For D

_{1}, the values were established for different combinations of V

_{e}and V

_{c}and for linear (r = 0.5) and nonlinear (r = 1.0) deceleration profiles, as shown in Table 4. As noted, larger design values are required for a larger CV and smaller P

_{nc}. The deterministic values are also shown in the table. A comparison of the deterministic and reliability-based values for V

_{e}= 60 km/h of Table 4 are shown graphically in Figure 5. As noted, the deterministic values correspond to a low probability of non-compliance (about P

_{nc}= 1%), indicating that such values are conservative. This finding is consistent with that of Easa [23], where the deterministic values for uncontrolled intersections corresponded to a very small probability of non-compliance. This is not surprising, since the deterministic method uses very high or very low percentile values (depending on the nature of the random variable). Note also that the trends of the design values for the linear and nonlinear deceleration profiles are similar, except that the values for the nonlinear profile (r = 0.5) are greater, as previously discussed.

_{2}are shown in Table 5 for V

_{c}ranging from 20 to 60 km/h. As noted, similarly to D

_{1}, the deterministic values also lie close to the reliability-based values for P

_{nc}= 1%. Note that the design values of D

_{1}and D

_{2}do not depend on any specific geometric configuration of the roundabout, and therefore they are applicable to all types of roundabouts. Only D

_{1}depends on the minimum radius of a circulatory roadway.

#### 4.3. Lateral Clearance Needs: Special Case

_{1}= the entry radius (m), R

_{c}= the circulatory roadway radius (m), w

_{c}= the circulatory roadway width, α = the angle between the y-axis and the line connecting the centers of the entry and inscribed circle curves (45°), and w

_{1}= the distance from the curb to the centerline of the road (m). The inscribed circle radius R

_{n =}R

_{c}+ w

_{c}. Given R

_{c}and w

_{1}, R

_{1}is calculated using Equation (48).

_{d}, is determined at the tangential point d. For the conflicting circulating vehicle, the maximum lateral clearance for the sight line ab is determined. For the conflicting-entering vehicle, after the formulation of the general lateral clearance C

_{f}, the maximum lateral clearance is determined using optimization as follows:

_{f}denotes the distance of Point f on the central island curb (decision variable), measured from the road centerline (Point u), and d

_{L}and d

_{U}denote the arbitrary lower and upper limits of the decision variable, respectively, that cover the possible range of lateral clearance and can be set up in different ways.

_{1}of Table 4) and to the circulating vehicle (based on D

_{2}of Table 5) are presented in Table 6 and Table 7. For D

_{1}, the values of Table 6 correspond to the lateral clearance, C

_{d}, at the tangent point d for the sight line ab’ (Figure 6a). For D

_{2}, the values of Table 7 correspond to the maximum lateral clearance on the central island for the sight line eb (Figure 6b).

#### 4.4. Sensitivity Analysis

_{1}, the input data for the base case were V

_{e}= 50 km/h, V

_{c}= 30 km/h, t

_{c}= 5 s, a = 1.3 m/s

^{2}, and r = 1, where a capital letter V indicates the speed in km/h. The coefficient of the variation of all variables was 5%, and the correlations were ${\rho}_{ve,tc}$ = 0.5, ${\rho}_{ve,tc}$ = 0.5,${\rho}_{ve,a}$ = 0.5,${\rho}_{ve,r}$ = 0.5, and ${\rho}_{vc,a}$ = −0.5. The probability of non-compliance was P

_{nc}= 5%. One variable of the base scenario was changed at a time, while keeping all other variables at their base values. The mean values were increased by 10%, the CV was increased to 10%, and the correlations were changed to 0.8.

_{1}is very sensitive to the mean value of t

_{c}and somewhat sensitive to that of V

_{e}. The value of D

_{1}is also very sensitive to the coefficient of variation of both V

_{c}and t

_{c}and somewhat sensitive to that of V

_{e}. The sight distance is somewhat sensitive to the values of the correlation coefficients. In addition, the results show that D

_{2}is sensitive to the mean values of V

_{c}and t

_{c}and to the coefficient of variation of t

_{c}. Clearly, the mean value and variability of t

_{c}substantially affect the ISD and should be accurately determined. In addition, the variability of the circulatory speed is an important variable. Other variables including the correlations have little or moderate effects on the calculated sight distance.

## 5. Conclusions

_{1}, the only requirement is that the central island should be circular, since the design values are based on a distance around circular central island. The values were based on the minimum radius that corresponds to the design speed of the circulatory roadway. However, the design values can perhaps be used with flatter radii of the central island than the minimum values, since they correspond to smaller lateral clearance needs.

_{1}from the conflict point to the entry vehicle should be calculated. The Australian guide assumes that the entry vehicle travels at the entry speed for the entire leg [5]. Rodegerdts et al. [1] implemented a more realistic assumption by considering that the entry vehicle travels to the conflict point at the average of the Ve and V

_{c}. Both assumptions would result in conservative later clearance needs. This assumption was subsequently revised by Easa [35] by assuming that the entry vehicle travels at the average of the entry and circulatory speeds prior to reaching the circulatory path. In this case, the use of the average speed would indirectly account for the deceleration that occurred before reaching the circulatory path. The present paper has further improved the modeling of the entry-vehicle deceleration by explicitly considering the vehicle deceleration profile of the entry vehicle. Note that setting up the sightline based on the proposed method would be conservative for other entry vehicles from the left that stop at the yield line and 15 m from the yield line.

_{e}and V

_{c}is being established using measurements from drone and video-based trajectory data.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

a | Deceleration rate of linear profile (m/s^{2}) |

a_{1}, a_{2}, a_{3} | Deceleration rates of nonlinear profile (m/s^{2}) |

CV | Coefficient of variation of all random variables |

CV_{xi} | Coefficient of variation of random variable xi |

d | Deceleration distance (m) |

d_{cir} | Distance along the circulatory part of the path (m) |

d_{f} | Decision variable for determining maximum lateral clearance |

d_{L}, d_{U} | Lower and upper limits of the decision variable df |

${d}_{e}$ | Distance of the sight distance leg on the entry approach (m) |

d’ | Deceleration distance corresponding to Case 2 (m) |

D_{1} | Sight distance leg of the entry vehicle |

D_{2} | Sight distance leg of the circulating vehicle |

${D}_{1supply}$ | Length of D_{1} supplied |

${D}_{2supply}$ | Length of D_{2} supplied |

e | Superlevation of circulatory roadway |

$E\left[{D}_{1}\right]$ | Expected value of D_{1} |

E_{xi} | Extreme value corresponding to a certain percentile value of random variable xi |

E[SM] | Mean of random variable SM |

P_{nc} | Probability of non-compliance |

Probability distribution function | |

r | Deceleration shape parameter |

R | Minimum radius of the entry or circulatory roadway |

R_{1} | Entry radius (m) |

R_{c} | Circulatory roadway radius (m) |

SM | Safety margin |

${t}_{cir}$ | Time spent by the entry vehicle on the circulatory roadway (s) |

t | Deceleration time (s) |

t_{c} | Critical headway (s) |

t’ | Deceleration time corresponding to d’ (s) |

t_{e} | Time on the entry approach correspond to de (s) |

var[SM] | Variance of random variable SM |

v_{e}, v_{c} | Design speeds of entry and circulatory roadways, respectively (m/s) |

V_{e}, V_{c} | Design speeds of entry and circulatory roadways, respectively (km/h) |

w_{c} | Circulatory roadway width (m) |

w_{1} | Distance from the curb to the centerline of the road (m). |

x_{i} | Random variable, i = 1, 2, …, n. |

z | Number of standard deviations of the normal distribution for a certain percentile value |

Z | Objective function for lateral clearance |

α | Angle between the line joining entry curve and inscribed circle centres and y-axis (45°) |

$\beta $ | Reliability index |

Φ(−β) | Area under the standard normal variate PDF from −∞ to −β |

ρ_{xi,xj} | Pearson’s coefficient of correlation between xi and xj. |

$\theta $ | Central angle suspended by the entry vehicle path along the circulatory roadway |

μ_{xi} | Mean of random variable xi |

σ_{xi} | Standard deviation of random variable xi |

${\sigma}_{xi}^{2}$ | Variance of random variable xi, |

## Appendix A. Probabilistic Conditions for Cases 1–3 of D_{1}

_{c}$\le $t

_{cir}or t

_{c}– t

_{cir}$\le $ 0. Therefore, the probabilistic equivalent at the 95% confidence level is given by:

_{c}$\ge $t

_{cir}+ t or:

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**Figure 1.**Illustration of linear and nonlinear deceleration profiles. (

**a**) Linear (r = 1); (

**b**) Nonlinear (r < 1); (

**c**) Nonlinear (r > 1).

**Figure 2.**Illustration of the intersection sight distance. (

**a**) Approach and conflicting vehicles. (

**b**) Geometry of intersection sight distance (ISD).

**Figure 3.**Geometry of the three cases for calculating D

_{1}. Note: C is the conflict point, CA is the distance along the circulatory roadway, AB is the deceleration distance, and BD is the distance along the entry approach (see Figure 2b).

**Figure 4.**Comparison of the first-order second-moment method (FOSM) results of D

_{1}and D

_{2}by the analytical method and Monte Carlo simulation: (

**a**) D

_{1}(Case 2, V

_{e}= 50 km/h, V

_{c}= 40 km), (

**b**) D

_{1}(Case 3, V

_{e}= 50 km/h, V

_{c}= 30 km), and (

**c**) D

_{2}(V

_{c}= 30 km).

**Figure 5.**Comparison of the deterministic and reliability-based values of required sight distance D

_{1}for linear and nonlinear deceleration profiles (V

_{e}= 60 km/h). (

**a**) Nonlinear deceleration rate (r = 0.5). (

**b**) Linear deceleration rate (r = 1).

**Figure 6.**Geometry of the lateral clearance of Cases 1 and 2 of ISD [35]. (

**a**) Approach vehicle at 15 m from the yield line. (

**b**) Approach vehicle at the yield line.

Reference, Date | Features | Reliability Method | Design Element |
---|---|---|---|

Navin [8], 1990 | Reliability theory of highway geometric design. | FOSM | SD on HC and VC |

Easa [9], 1993 | Probabilistic method for determining intergreen intervals at signalized intersections. | FOSM | SSD |

Easa [10], 1994 | Probabilistic method for SD design at railroad crossings. | AFOSM | SD at railroad crossings |

Easa [11], 2000 | Reliability of ISD design that replaced the extreme values with the moments of the probability distributions. | FOSM | ISD at uncontrolled intersections |

Richl and Sayed [21], 2006 | Reliability analysis of a series of horizontal curves with varying horizontal SD restrictions. | FORM, FOSM | SD on HC |

El-Khoury and Hobeika [24], 2007 | PSD distribution that accounts for the variations in the contributing random PSD design variables. | MC simulation | PSD on two-lane roads |

Sarhan and Hassan [25], 2008 | Reliability analysis to estimate the probability of hazards from the insufficiency of SD. | MC Simulation | SD on 3D alignments |

Ibrahim et al. [14], 2012 | Reliability analysis to optimize cross-sections with restricted SD. | FORM | SD on cross sections |

Llorca et al. [23], 2014 | Reliability analysis to evaluate the risk associated with PSD standards. | FORM | PSD on two-lane roads |

Easa and Hussain [18], 2016 | Reliability analysis to estimate left-turn SD at stop-control intersections. | FOSM | ISD at stop-control intersections |

Hussain and Easa [19], 2016 | Reliability analysis to estimate left-turn SD at signalized intersections. | FOSM | SD for left turn at signalized intersections |

Osama et al. [20], 2016 | Reliability analysis framework to evaluate the risk of limited SD for permitted left-turn movements. | FORM, FOSM | SD at signalized intersections |

De Santos-Berbe et al. [22], 2017 | Reliability analysis to evaluate the risk level of limited SD for three ASD modelling methods. | FORM | SD on HC |

Wood and Donnell [26], 2017 | Reliability analysis to improve consistency between the ASD and SSD criteria in design policy. | MC Simulation | SD on HC |

Faizi and Easa [28], 2018 | Probabilistic method for the determination of SSD at roundabouts. | FOSM | SSD at roundabouts |

Andrade-Catano et al. [27], 2020 | Probabilistic approach to evaluate the risk level associated with sag curve designs with headlight SD. | MC simulation | SD on VC |

Ve (km/h) | V_{c}(km/h) | R^{a}(m) | d_{cir}^{b}(m) | Required Sight Distance D_{1} (m) | ||||
---|---|---|---|---|---|---|---|---|

Linear | Nonlinear | |||||||

r = 1 | r = 0.5 | r = 1.5 | ||||||

Value | Diff (%) ^{c} | Value | Diff (%) | |||||

30 | 30 | 24 | 12.4 | 45.1 | 45.1 | 0.0 | 45.1 | 0.0 |

20 | 9 | 4.2 | 39.8 | 42.2 | 6.0 | 38.8 | −2.5 | |

40 | 40 | 50 | 26.7 | 60.2 | 60.2 | 0.0 | 60.2 | 0.0 |

30 | 24 | 12.4 | 52.8 | 55.4 | 4.9 | 51.8 | −1.9 | |

20 | 9 | 4.2 | 43 | 51.3 | 19.3 | 39.5 | −8.1 | |

50 | 50 | 94 | 48.3 | 75.2 | 75.2 | 0.0 | 75.2 | 0.0 |

40 | 50 | 26.7 | 65.3 | 68 | 4.1 | 64.3 | −1.5 | |

30 | 24 | 12.4 | 54.3 | 60.9 | 12.2 | 51.6 | −5.0 | |

60 | 60 | 149 | 78.5 | 90.2 | 90.2 | 0.0 | 90.2 | 0.0 |

50 | 94 | 48.3 | 77.4 | 79.4 | 2.6 | 76.7 | −0.9 | |

40 | 50 | 26.7 | 65.6 | 69.9 | 6.6 | 63.9 | −2.6 | |

70 | 60 | 149 | 78.5 | 90.5 | 90.8 | 0.3 | 90.4 | −0.1 |

50 | 94 | 48.3 | 77.4 | 79.4 | 2.6 | 76.7 | −0.9 | |

40 | 50 | 26.7 | 65.6 | 69.9 | 6.6 | 63.9 | −2.6 |

^{a}Minimum radius for V

_{c}based on the American Association of State Highway and Transportation Officials (AASHTO) lateral friction coefficients for urban low speeds and e = 0.02;

^{b}Length of entry vehicle path on a circulatory roadway based on Equation (11) (assuming θ = 30 degrees);

^{c}Difference between the values for the linear and nonlinear deceleration profiles.

Variable | Unit | Extreme Value | Percentile ^{a} | z | Mean Value ^{b} |
---|---|---|---|---|---|

V_{e} | km/h | 30 to 70 | 95% | 1.64 | 25.8 to 60.1 |

V_{c} | km/h | 20 to 60 | 95% | 1.64 | 17.2 to 51.5 |

t_{c} | s | 5.8 | 95% | 1.64 | 5 |

a | m/s^{2} | 1.2 | 10% | −1.28 | 1.3 |

^{a}Percentile values are based on the literature or are assumed;

^{b}The mean values are calculated assuming CV = 10% for all variables.

**Table 4.**Reliability-based design values of the required sight distance for entry vehicle D

_{1}for linear and nonlinear deceleration profiles.

V_{e}(km/h) | V_{c}(km/h) | Deterministic D_{1}(m) | Reliability-Based D_{1} (m) | |||||
---|---|---|---|---|---|---|---|---|

CV = 5% | CV = 10% | |||||||

P_{nc} = 1% | P_{nc} = 5% | P_{nc} = 10% | P_{nc} = 1% | P_{nc} = 5% | P_{nc} = 10% | |||

(a) Deceleration Shape Parameter, r = 0.5 | ||||||||

30 | 30 | 46 | 46 | 44 | 43 | 49 | 46 | 43 |

20 | 43 | 43 | 41 | 40 | 47 | 43 | 41 | |

40 | 40 | 61 | 61 | 58 | 57 | 80 | 74 | 71 |

30 | 56 | 57 | 54 | 53 | 72 | 66 | 63 | |

20 | 52 | 55 | 52 | 51 | 63 | 57 | 55 | |

50 | 50 | 76 | 75 | 72 | 70 | 89 | 82 | 79 |

40 | 69 | 72 | 69 | 67 | 80 | 74 | 71 | |

30 | 61 | 64 | 61 | 60 | 72 | 66 | 63 | |

60 | 60 | 91 | 91 | 87 | 85 | 98 | 90 | 88 |

50 | 80 | 81 | 77 | 76 | 89 | 82 | 79 | |

40 | 70 | 72 | 69 | 67 | 80 | 74 | 71 | |

70 | 60 | 91 | 91 | 87 | 85 | 98 | 91 | 88 |

50 | 80 | 81 | 77 | 76 | 89 | 82 | 79 | |

40 | 70 | 72 | 69 | 67 | 80 | 74 | 71 | |

(b) Deceleration shape parameter, r = 1. | ||||||||

30 | 30 | 46 | 46 | 44 | 43 | 49 | 46 | 43 |

20 | 40 | 41 | 39 | 38 | 44 | 41 | 39 | |

40 | 40 | 61 | 61 | 58 | 57 | 80 | 68 | 65 |

30 | 53 | 54 | 51 | 50 | 62 | 57 | 55 | |

20 | 44 | 46 | 43 | 42 | 51 | 47 | 45 | |

50 | 50 | 76 | 75 | 72 | 70 | 89 | 78 | 75 |

40 | 66 | 67 | 64 | 63 | 73 | 68 | 65 | |

30 | 55 | 56 | 54 | 53 | 62 | 57 | 55 | |

60 | 60 | 91 | 91 | 87 | 85 | 98 | 90 | 86 |

50 | 78 | 78 | 75 | 73 | 84 | 78 | 75 | |

40 | 66 | 67 | 64 | 63 | 73 | 68 | 65 | |

70 | 60 | 91 | 91 | 87 | 85 | 97 | 90 | 86 |

50 | 78 | 78 | 75 | 73 | 84 | 78 | 75 | |

40 | 66 | 67 | 64 | 63 | 73 | 68 | 65 |

**Table 5.**Reliability-based design values of the required sight distance for circulating vehicle D

_{2}.

V_{c}(km/h) | Deterministic (m) | Reliability-Based D_{2} (m) | |||||
---|---|---|---|---|---|---|---|

CV = 5% | CV = 10% | ||||||

P_{nc} = 1% | P_{nc} = 5% | P_{nc} = 10% | P_{nc} = 1% | P_{nc} = 5% | P_{nc} = 10% | ||

20 | 33 | 31 | 30 | 29 | 34 | 31 | 30 |

25 | 41 | 39 | 37 | 36 | 42 | 39 | 37 |

30 | 49 | 47 | 45 | 43 | 51 | 46 | 44 |

35 | 57 | 54 | 52 | 50 | 59 | 54 | 51 |

40 | 65 | 62 | 59 | 58 | 68 | 62 | 59 |

45 | 73 | 70 | 66 | 65 | 76 | 69 | 66 |

50 | 81 | 78 | 74 | 72 | 84 | 77 | 73 |

55 | 89 | 85 | 81 | 79 | 93 | 84 | 80 |

60 | 98 | 93 | 88 | 86 | 101 | 92 | 88 |

**Table 6.**Reliability-based maximum lateral clearance needs for the circulating vehicle for single-lane symmetrical roundabouts (approach vehicle at yield line)

^{b}.

V_{c}(km/h) | Rc_{min}(m) | Maximum Lateral Clearance, C_{m} (m) | ||||
---|---|---|---|---|---|---|

Deterministic ^{a} | CV = 5% | CV = 10% | ||||

P_{nc} = 5% | P_{nc} = 10% | P_{nc} = 5% | P_{nc} = 10% | |||

20 | 8.1 | 7.1 | 5.9 | 5.4 | 6.4 | 5.9 |

25 | 14.6 | 6.5 | 4.8 | 4.4 | 5.6 | 4.8 |

30 | 23.8 | 5.3 | 4.0 | 3.3 | 4.3 | 3.6 |

35 | 35.9 | 4.3 | 3.0 | 2.5 | 3.5 | 2.7 |

40 | 51.2 | 3.5 | 2.2 | 2.1 | 2.8 | 2.2 |

45 | 70.0 | 2.8 | 1.7 | 1.5 | 2.2 | 1.7 |

50 | 92.7 | 2.0 | 0.8 | 0.5 | 1.3 | 0.7 |

^{a}Based on the design speed V

_{c}and the 95th percentile of the critical headway t

_{c}.

^{b}Other roundabout dimensions are w

_{1}= 6 m and w

_{c}= 5 m.

**Table 7.**Reliability-based maximum lateral clearance needs for the entry vehicle for single-lane symmetrical roundabouts, CV = 10% (approach vehicle 15 m from the yield line)

^{a}.

Entry Radius R _{1}(m) | Lateral Clearance, C_{d} (m) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Ve = 30 km/h | Vc = 40 km/h | Vc = 50 km/h | ||||||||||

Vc = 20 km/h | Vc = 30 km/h | Vc = 20 km/h | Vc = 30 km/h | Vc = 30 km/h | Vc = 40 km/h | |||||||

P_{nc} = 5% | P_{nc} = 10% | P_{nc} = 5% | P_{nc} = 10% | P_{nc} = 5% | P_{nc} = 10% | P_{nc} = 5% | P_{nc} = 10% | P_{nc} = 5% | P_{nc} = 10% | P_{nc} = 5% | P_{nc} = 10% | |

(a) Deceleration Shape Parameter, r = 0.5 | ||||||||||||

20 | 8.1 | 8.4 | 9 | 8.4 | 11.7 | 11.3 | 13.4 | 12.9 | 13.4 | 12.9 | 14.6 | 14.2 |

30 | 4.3 | 4.6 | 5.3 | 4.6 | 8.7 | 8.1 | 10.8 | 10.2 | 10.8 | 10.1 | 12.4 | 11.8 |

40 | 1.6 | 1.8 | 2.3 | 1.8 | 5.1 | 4.5 | 7.7 | 6.7 | 7.7 | 6.8 | 9.7 | 8.9 |

50 | 0.3 | 0.4 | 0.6 | 0.4 | 2.4 | 2 | 4.4 | 3.7 | 4.4 | 3.7 | 6.5 | 5.7 |

(b) Deceleration Shape Parameter, r = 1.0 | ||||||||||||

20 | 7.5 | 6.8 | 9 | 8.1 | 9.3 | 8.7 | 11.7 | 11.2 | 11.7 | 11.2 | 13.7 | 13.2 |

30 | 3.6 | 3.0 | 5.2 | 4.3 | 5.6 | 4.9 | 8.7 | 8.1 | 8.6 | 8.1 | 11.2 | 10.6 |

40 | 1.2 | 0.9 | 2.2 | 1.6 | 2.5 | 2 | 5.1 | 4.5 | 5 | 4.5 | 8.2 | 7.4 |

50 | 0.1 | 0.0 | 0.6 | 0.3 | 0.7 | 0.5 | 2.4 | 2 | 2.4 | 2 | 4.9 | 4.2 |

^{a}Other roundabout dimensions are w

_{1}= 6 m and w

_{c}= 5 m.

Changed Variable | Value | D_{1} or D_{2} ^{a}(m) | Diff (%) |
---|---|---|---|

(a) Sensitivity of D_{1} | |||

V_{e} | 55 km/h | 45.6 | −4.8 |

V_{c} | 33 km/h | 46.7 | −2.5 |

t_{c} | 5.5 s | 54.9 | 14.6 |

a | 1.43 m/s^{2} | 48.7 | 1.7 |

r | 1.1 | 48.4 | 1.0 |

CV_{ve} | 10% | 44.1 | −7.9 |

CV_{vc} | 10% | 55.0 | 14.8 |

CV_{tc} | 10% | 53.5 | 11.7 |

CV_{a} | 10% | 47.4 | −1.0 |

CV_{r} | 10% | 47.8 | −0.2 |

${\rho}_{xi,xj}$ | 0.8 | 45.7 | −4.6 |

(b) Sensitivity of D_{2} | |||

V_{c} | 33 km/h | 50.9 | 9.9 |

t_{c} | 5.5 s | 50.9 | 9.9 |

CV_{vc} | 10% | 46.8 | 1.1 |

CV_{tc} | 10% | 50.4 | 8.9 |

${\rho}_{vc,tc}$ | 0.8 | 47.0 | 1.5 |

^{a}The input variables for the base scenario are V

_{e}= 50 km/h, V

_{c}= 30 km/h, t

_{c}= 5 s, a = 1.3 m/s2, and r = 1. The corresponding values of D

_{1}and D

_{2}are 47.9 m and 46.3 m, respectively.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Easa, S.M.; Ma, Y.; Liu, S.; Yang, Y.; Arkatkar, S.
Reliability Analysis of Intersection Sight Distance at Roundabouts. *Infrastructures* **2020**, *5*, 67.
https://doi.org/10.3390/infrastructures5080067

**AMA Style**

Easa SM, Ma Y, Liu S, Yang Y, Arkatkar S.
Reliability Analysis of Intersection Sight Distance at Roundabouts. *Infrastructures*. 2020; 5(8):67.
https://doi.org/10.3390/infrastructures5080067

**Chicago/Turabian Style**

Easa, Said M., Yang Ma, Shixu Liu, Yanqun Yang, and Shriniwas Arkatkar.
2020. "Reliability Analysis of Intersection Sight Distance at Roundabouts" *Infrastructures* 5, no. 8: 67.
https://doi.org/10.3390/infrastructures5080067