Modeling of Unsymmetrical Single-Lane Roundabouts Based on Stopping Sight Distance Said M. Easa Professor of Transportation Engineering Department of Civil Engineering Ryerson University, Toronto, ON Canada M5B 2K3 Email:
[email protected] Note: paper accepted for publication on KSCE J. of Civil Engineering. Abstract. Evaluation of the adequacy of sight distance at roundabout is usually performed
graphically as no quantitative guidelines are available for the lateral clearance needs. This paper presents analytical lateral clearance models to satisfy stopping sight distance (SSD) at unsymmetrical single-lane roundabouts (with different entry and exit radii). The models are developed for three critical locations: approach sight distance, sight distance to exist crosswalk, and circulatory roadway sight distance. Two cases of approach sight distance are presented: sight distance to the crosswalk and sight distance to the yield line. The required lateral clearance at any specified point along the curve can also be determined. Design aids for the required maximum lateral clearance are established for the three locations of sight distance based on accurate SSD formula that accounts for effect of the radial deceleration component of the vehicle. For the circulatory roadway sight distance, infeasible combinations of the circulatory roadway radius and speed are identified based on acceptable driver’s field of peripheral vision. The models and results of this paper, which are applicable to both symmetrical and unsymmetrical roundabouts, should be of interest to highway engineers and designers. Keywords: Unsymmetrical roundabouts, stopping sight distance, lateral clearance, pedestrians, crosswalk.
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1. Introduction Sight distance at roundabouts is a key geometric element that has been addressed in the literature by various North American and overseas transportation organizations, such as the Transportation Research Board (Rodegerdts et al., 2007), Federal Highway Administration (2006), American Association of State Highways and Transportation Officials (AASHTO, 2011), Transportation Association of Canada (TAC, 2016), and government of Queensland (2013). There are generally two types of sight distances that should be checked at roundabouts (Rodegerdts et al., 2010): stopping sight distance (SSD) and intersection sight distance (ISD). Stopping sight distance allows drivers to perceive and react to an object in the roadway and brake to a complete stop before hitting the object. This distance should be provided at every point in the roundabout, especially on the roundabout approach, exit maneuver, and circulatory roadway. Intersection sight distance ensures that a driver without the right of way has enough sight distance to perceive and react to the presence conflicting vehicles. This distance should be checked at two conflicting traffic streams at each entry (entering stream and circulating stream). Many analytical models for sight distance analysis at conventional intersections (such as stop-controlled, yield-controlled, and uncontrolled) have been developed (Easa et al., 2017; AASHTO, 2011; Ali et al., 2009; Easa and Ali, 2006). However, analytical sight distance models for roundabouts are lacking. The analysis of sight distance at roundabouts is somewhat challenging since the sight triangle involves compound and reverse horizontal curves, and stopping sight distance involves tangential and radial components. Existing methods for evaluating the adequacy of roundabout sight distance to meet the requirements are graphical in nature and involve a trial procedure. Although these methods have performed well and allowed the designers to pictorially check the adequacy of sight distance, they are considered time consuming. In an attempt to eliminate the trial approach, a study for the City of Calgary has presented an analytical model for the design of the radius of the central island to address stopping sight distance needs (City of Calgary, 2010). The model can assess the impact of sight distance and the effect of placing urban features in the central island. The study, however, did not address sight distance at two critical locations at the roundabout: approach and exit sight distances. Recently, analytical models based on SSD have been presented for three critical locations at symmetrical roundabouts (with a single entry-exit curve). The three locations included approach sight distance, sight distance to exist crosswalk, and circulatory roadway sight distance (Easa, 2017a). The design aids developed in that paper were approximate as the SSD formula corresponded to straight roads. Another paper that brought sight distance analysis at roundabouts one step further has focused on ISD at symmetrical roundabouts (Easa 2017b). The analysis considered two critical locations of an approach vehicle to the roundabout: a vehicle at the yield line and a vehicle at a specified distance from the yield line (typically 15 m). Design guidelines were established for the radius of the central island, entry-curve lateral clearance, exit crosswalk location, and field of peripheral vision for a driver stopped at the yield line. This paper presents SSD-based analytical models for unsymmetrical roundabouts. Specifically, the purpose of the paper is threefold: (1) to present an accurate formula of SSD that takes into account the effect of curvature on vehicle deceleration, (2) to present analytical lateral clearance models for unsymmetrical single-lane roundabouts (with different entry and exit radii), and (3) to establish design guidelines using the developed models based on the improved SSD formula. The next section presents fundamental relations, including the improved SSD formula and geometric characteristics of unsymmetrical roundabouts. The following section presents 2
lateral clearance formulation for three critical locations of the roundabout. Design aids and application examples are then presented, followed by concluding remarks. 2. Fundamental Relations 2.1 Stopping Sight Distance on Curves Stopping sight distance along a straight road is defined as (AASHTO, 2011) 𝑉2
𝑆 = 0.278 𝑉 𝑡 +
(Straight roads)
25.92 𝑎
(1)
where S = stopping sight distance (m), V = initial speed (km/h), t = perception-brake reaction (PBR) time (s), and a = deceleration rate (m/s2). The design values of PRT and deceleration rate are 2.5 s and 3.4 m/s2, respectively, as recommended by AASHTO (2011). The preceding value of deceleration rate is applicable to straight roads. For curved roads, the actual deceleration rate used for stopping (tangential deceleration) will be less than this value. When a vehicle decelerates on a curved road, there are two components of the deceleration rate (tangential and radial). The radial deceleration rate which acts toward the curve center is given by 𝑎𝑛 =
𝑉2
(2)
12.96 𝑅
where an = radial deceleration rate (m/s2) and R = vehicle path radius (m). Since the tangential and radial deceleration components are perpendicular, the resultant deceleration is the vector sum of the two components which is assumed to be equal to a. Thus, tangential deceleration rate can be obtained as 2
𝑉 𝑎𝑡 = √𝑎2 − (12.96 𝑅)
2
(3)
Substituting for at from Eq. (3) into Eq. (1), the exact formula for SSD on curves is given by 𝑆 = 0.278 𝑉 𝑡 +
𝑉2 2 𝑉2 25.92√𝑎 2 −( ) 12.96 𝑅
(Curved roads)
(4)
Note that for R = ∞ (straight roads), Eq. (4) is reduced to Eq. (1). For the case of SSD on two consecutive curves with radii R1 and R2, then S = S1 + S2, where S1 = PRT and braking distances on Curve 1 and S2 = braking distance on Curve 2 (assuming the length of Curve 1 is greater than the PRT distance). Given S2, the speed at the beginning of Curve 2, V1, based on Eq. (4), can be derived as 4
672 𝑎 2 𝑆2 2
𝑉1 = √ 1+672(
(5)
2 𝑆2 ) 12.96 𝑅2
Then, S1 is given by 𝑆1 = 0.278 𝑉 𝑡 +
𝑉 2 −𝑉1 2
(6)
2 𝑉2 25.92√𝑎 2−( ) 12.96 𝑅1
3
The initial speed V is determined based on the roundabout element being analyzed. For approach sight distance, the initial speed is based on the speed of the road element prior to the entry curve and for exit sight distance it is based on the fastest path speed. For the circulatory roadway sight distance, the initial speed is based on a typical speed of circulating traffic. Note that when vertical alignments or cut slopes restrict the sightline, SSD should be measured from a driver’s eye height of 1.08 m to an object height of 1.08 m. 2.2 Geometry of Unsymmetrical Roundabout Consider the unsymmetrical single-lane roundabout shown in Fig. 1. The entry and exit radii are R2 and R3, respectively. A Cartesian coordinate system is set such that its origin is the center of the exit curve O3, the y-axis passes through the center of the central island circle On, and the positive x-axis is in the right direction. The angle between Road 1 and the y-axis, ϕ1 (degrees), is referred to as approach deflection angle. Thus, the angle between Road 2 and the y-axis is (90 – ϕ1). Since the angle at u of ∆O2Onu equals 90 degrees, the relation between entry and inscribed circle radii can be derived as 𝑅2 =
𝑤1 − 𝑅𝑛 sin (𝜙1 )
(7)
sin(𝜙1 )− 1
where R2 = entry radius (m), Rn = inscribed circle radius (m) and w1 = distance from the curb to the centerline of Road 1 (m). Similarly, from ∆O3Onv, the relation between exit and inscribed circle radii can be obtained as 𝑅3 =
𝑤2 − 𝑅𝑛 sin (90 − 𝜙1 )
(8)
sin(90− 𝜙1)− 1
where R3 = exit radius (m) and w2 = distance from the curb to the centerline of Road 2 (m). Note that Eqs. (7) and (8) have four variables. Given ϕ1 and Rn, for example, R2 and R3 can be determined. For ϕ1 = 45 degrees and w1 = w2, the equations yield R2 = R3 and the roundabout becomes symmetrical. Thus, the approach deflection angle and approach road widths are the main geometric element that distinguishes symmetrical and unsymmetrical roundabouts. 3. Lateral Clearance Formulation The critical locations at which SSD should be checked at the roundabout are presented in the National Cooperative Highway Research Program (NCHRP) Report by Rodegerdts et al. (2010), as follows: (a) approach sight distance, (b) entry vehicle sight distance to exit crosswalk, and (c) circulatory roadway sight distance. The key step in modeling lateral clearance is to determine the formulas for the coordinates of the driver and the object (crosswalk, yield line, or vehicle). In all cases, the stopping sight distance is measured along the vehicle path.
3.1 Approach Sight Distance
4
There are two cases of approach sight distance that should be checked: SSD to the crosswalk and SSD to the yield line. Consider the single-lane roundabout approach shown in Fig. 2. The roundabout approach has a horizontal curve with curb radius R1 that is curved in the same direction as the entry curve which has a curb radius R2. For the purpose of sight distance analysis, the vehicle’s travel path is assumed to be the location of the driver’s eye, which is located at a distance A from the curb. The crosswalk is assumed to be perpendicular to the curve. The lateral clearance formulation for each case of the approach sight distance is presented next. Case 1: SSD to Crosswalk In this study, the pedestrian is assumed to be located at the far edge of the crosswalk, which ensures that all pedestrians at the crosswalk will be visible to the driver. However, the distance S is measured to the near edge of the crosswalk along the vehicle path from the front of the vehicle (distance ch). Thus, all pedestrians on the crosswalk are visible and are provided adequate SSD. Note that this argument is true only if the curvature of the approach curve is in the same direction as the entry curve (compound curve). If the approach curve is in the opposite direction (reverse curve), a pedestrian located at the near edge of the crosswalk may control sight distance. The pedestrian is located at an offset distance, according to Washington State Department of Transportation (WSDOT 2016), which is the distance from the ramp landing area to the curb, typically 2 m (Boodlal, 2004; Easa, 2016). Consider a Cartesian coordinate system whose origin is O1 (center of the approach curve), the y-axis passes through the driver’s eye (Point a), and the x-axis is in the direction of travel. The line of sight from the driver to the crosswalk is ab. The lateral clearance for an arbitrary Point f on the curb is Cf (distance fg). The formulation of the lateral clearance Cf and the maximum lateral clearance, Cm, are presented next. From the geometry of Fig. 2, the distance from the far edge of the crosswalk to the start of the entry curve, measured along the centerline of the splitter island, Lp, and the corresponding subtended angle at O2, α, are given by 𝐿𝑝 = (𝑅2 + 𝑅𝑛 ) cos(𝜙1 ) − 𝐿𝑚𝑖𝑛 𝛼 = 𝑎𝑡𝑎𝑛 𝑅
(9)
𝐿𝑝
(10)
2 + 𝑤1
where Lmin = minimum distance from the far edge of the crosswalk to the inscribed circle, measured along the centerline of the splitter island (typically 6 m). The subtended angle from the driver’s eye to the start of the entry curve, I, can be derived as 𝐼=
𝑑𝑒 + 𝑆 + 𝑤𝑐 − 𝛼 (𝑅2 + A)
(11)
(𝑅1 + 𝐴)
where wc = crosswalk width (m). The coordinates of the driver and pedestrian, (xa, ya) and (xb, yb), respectively, are given by 𝑥𝑎 = 0
(12)
𝑦𝑎 = 𝑅1 + 𝐴
(13)
5
𝑥𝑏 = (𝑅1 − 𝑅2 ) sin(𝐼) + (𝑅2 − 𝑓𝑝 ) sin(𝐼 + 𝛼)
(14)
𝑦𝑏 = (𝑅1 − 𝑅2 ) cos(𝐼) + (𝑅2 − 𝑓𝑝 ) cos(𝐼 + 𝛼)
(15)
where fp = offset distance of the pedestrian from the curb (m). Then, the slope of the sightline from the driver to the pedestrian, P, is given by 𝑃=
(𝑦𝑏 −𝑦𝑎 )
(16)
(𝑥𝑏 −𝑥𝑎 )
To formulate the lateral clearance, consider an arbitrary Point f on the curb (xf, yf). This point may lie on the approach or entry curve. The respective slopes between Point f and the curve centers O1 and O2, respectively, are given by 𝑃𝑓 = 𝑃𝑓 =
𝑦𝑓
,
𝑥𝑓 𝑦𝑓 − 𝑦𝑂2 𝑥𝑓 − 𝑥𝑂2
,
xf ≤ xu
(17)
xf > xu
(18)
where xu = x-coordinate of the common tangent point u. Solving the equations of the two lines ab and O1f or ab and O2f, the coordinates of the intersection Point g (xg, yg) can be derived as 𝑥𝑔 =
𝑦𝑎 −𝑦𝑓 −𝑥𝑎 P+𝑥𝑓 𝑃𝑓
(19)
𝑃𝑓 −𝑃
𝑦𝑔 = 𝑦𝑓 + (𝑥𝑔 − 𝑥𝑓 )𝑃𝑓
(20)
Then, the lateral clearance at Point f, Cf, equals the difference between R1 and the length O1g or between R2 and the length O2g (Fig. 2), Cf = R1 – (xg2 + yg2)0.5,
xf ≤ xu
Cf = R2 – [(xg – xO2)2 + (yg – yO2)2)]0.5,
(21) xf > xu
(22)
The closed-form solution of Eqs. (21) and (22) directly provides the required lateral clearance at any point along the sightline between the driver and the pedestrian. The maximum lateral clearance, Cm, can be determined using Excel Solver (2010), where the decision variable is xf. That is, Maximize Z = Cf
(23)
Subject to xa ≤ xf ≤ xb
(24)
where Cf of Eq. (23) is calculated using Eqs. (21) and (22). Solver provides the maximum lateral clearance, Cm, and its distance from the x-axis, dm. 6
Case 2: SSD to Yield Line The formulation of the lateral clearance for this case is similar to that of Case 1. The sightline extends from the driver’s eye of the vehicle (Point a’) to the yield line (Point b’), as shown in Fig. 2. Point a’ lies at a distance L from its previous location of Case 1 (Point a), which equals the distance from the near edge of the crosswalk (Point h) to the yield line (Point b’), both are measured along the vehicle path. The point on the yield line to be seen by the driver (Point b’) is assumed to be the intersection of the vehicle path and the yield line. Thus, the corresponding distance at the yield line that is visible to the driver will be almost equal to a lane width. This is consistent with some design guides that recommend a visible distance of 3.65 m, such as the Department for Regional Development of Northern Ireland (DRD, 2007). The coordinates of Point b’ can be easily determined from ∆O2Onb’. Since the three sides of the triangle are known, the angle K is first determined, followed by the angle I1, yielding 𝐼1 = 𝑎𝑠𝑖𝑛 {𝑅
𝑅𝑛
𝑛
+ 𝑅2
𝑠in [𝑎𝑐𝑜𝑠 (
𝑅𝑛 2 + (𝑅2 +𝐴) 2 − (𝑅𝑛 + 𝑅2 )2 2𝑅𝑛 (𝑅2 +𝐴)
)]}
(25)
From the geometry of Fig. 2, the distance L = hb’ along the vehicle path can be written as L = wc + (R2 + A)(90 - ϕ1 – α – I1)
(26)
Thus, the angle β on the approach curve that corresponds to L is given by 𝛽=
𝐿
(27)
(𝑅1 +𝐴)
The coordinates of the driver’s eye and the yield-line point, (xa’, ya’) and (xb’, yb’), respectively, are given by 𝑥𝑎′ = (𝑅1 + 𝐴) sin(𝛽)
(28)
𝑦𝑎′ = (𝑅1 + 𝐴) cos(𝛽)
(29)
𝑥𝑏′ = (𝑅1 − 𝑅2 ) sin(𝐼) + (𝑅2 + 𝐴) sin(𝐼 + 90 − 𝜙1 − 𝐼1 )
(30)
𝑦𝑏′ = (𝑅1 − 𝑅2 ) cos(𝐼) + (𝑅2 + 𝐴) cos(𝐼 + 90 − 𝜙1 − 𝐼1 )
(31)
Then, the formulation of the lateral clearance is the same as that of Case 1, Eqs. (17)-(24). 3.2 Sight Distance to Exit Crosswalk The geometry of the sight distance to the exit crosswalk (exit sight distance) of unsymmetrical single-lane roundabouts is shown in Fig. 3. The radii of the entry curve R2 and exit curve R3, are given by Eqs. (7) and (8), given the inscribed circle radius Rn. The beginning of the exit curve is PC and its end is PT. The front of the entry vehicle (Point c) is assumed to lie on the yield line at its intersection with the vehicle path and the speed of the entry vehicle equals the entry speed. 7
The driver’s eye lies at a distance de from the front of the vehicle and at a distance A from the curb. The pedestrian lies at Point b (or Point b’) which corresponds to a sightline that ends on the exit curve (or on the tangent). The distance from the front of the entry vehicle to the near edge of the crosswalk (Point d or d’), measured along the vehicle path, equals S. The required sight line (ab or ab’, for a far edge of the crosswalk that lies on the curve or the tangent, respectively) defines the required location of the far edge of the crosswalk (Point h or h’). The required location is compared with the available location of the far edge of the typical crosswalk (Point v) that lies at a minimum distance of (Lmin + wc = 9 m) from the yield line. The available location is determined using the coordinates of Point u and the radial line uv (Fig. 3). If Pont h lies before Point v, the location of the typical crosswalk is adequate. Otherwise, the location of the typical crosswalk is inadequate. In this case, to satisfy SSD requirement, Lmin should be increased by relocating the crosswalk farther from the yield line or by changing the geometry of the roundabout. Similar to approach sight distance, the first step of modeling lateral clearance is to determine the locations of the driver and the pedestrian. Consider a Cartesian coordinate system whose origin is O3 (center of the exit curve), the y-axis passes through On, and the x-axis is in the direction of travel. To determine the coordinates of Point c, the angle I1 is first determined from the triangle O2Onc using Eq. (23), as previously explained. The angle I, subtended by de, equals de / (R2+ A). Then, the coordinates of the driver’s eye (xa, ya) are given by xa = - (R2 + A) sin(I + I1)
(32)
ya = (R2 + A) cos(I + I1)
(33)
The distance travelled before the y-axis is L2 = (R2 + A)I1. The far edge of the required crosswalk may lie on the curve or the tangent (Fig. 3). If it lies on the curve, the corresponding angle from the y-axis is given by 𝐽1 =
𝑆−𝐿2 + 𝑤𝑐
(34)
(𝑅3 +𝐴)
The coordinates of the pedestrian (Point b), (xb yb), are then given by xb = (R3 - fp) sin(J1)
(35)
yb = (R3 - fp) cos(J1)
(36)
The coordinates of the driver’s eye and pedestrian define the sightline (ab). For the situation where the far edge of the required crosswalk lies on the tangent, the distance on the tangent and the corresponding angle J2 are first determined, then the coordinates of Point b’ are easily established. The sightline in this case is ab’. In each situation, the formulation of the lateral clearance is the same as that of Case 1, Eqs. (17)-(24). 3.3 Circulatory Roadway Sight Distance The geometry of sight distance for the circulatory roadway is shown in Fig. 4. The lateral clearance for the circulatory sight distance can be solved directly using the following formula (AASHTO, 2011) 8
𝑀 = (𝑅𝑐 + 𝑓) [1 − 𝑐𝑜𝑠
90 (𝑆+ 𝑑𝑒 ) 𝜋 (𝑅𝑐 +𝑓)
]−𝑓
(37)
where M = required lateral clearance (mid-ordinate), radial distance measured from the curb of the central island to the chord (m), Rc = radius of the central island (m), f = offset of the vehicle path (driver’s eye) from the curb of the central island (m), and S = stopping sight distance (m), calculated using Eq. (4). Given R and S, the required lateral clearance can be determined. The clear zone implies that the height of landscaping and other objects around the outer edge of the central island will be restricted. Note that the roundabout cross slope (+0.02 and −0.02) does not directly affect sight distance. It affects the speed–radius relationship which can be used to predict circulating speeds of left-turn and through movements. 4. Design Aids Using the presented analytical models, design values for the maximum lateral clearance were developed. For approach and exit sight distances, the design aids were based on the following assumptions: (a) The distance from the far edge of the crosswalk to the inscribed circle, Lmin = 6 m, (b) Crosswalk width wc = 3.0 m, (c) Width of approach roads, w1 = w2 = 6.0 m. (d) The driver’s eye lies at a distance A = p + w – z from the right curb, where p = 0.53 m is the offset distance from the right side of the vehicle to the curb, w = 3 m is vehicle width, and z = 0.5 m is the distance from the left side of the vehicle to the driver’s eye, yielding A = 1.77 m (Hussain and Easa, 2016; Harwood et al., 1996). (e) For stopping sight distance, t = 2.5 s and a = 3.4 m/s2, (f) The distance from the driver’s eye to the front of the vehicle, de = 2.4 m. Table 1 presents design values of the maximum lateral clearance for the cases of sight distance to the crosswalk and to the yield line, respectively. The values correspond to approach speed, Vapp, ranging from 40 km/h to 70 km/h and R1 = 100 m to 1000 m and R2 = 20 m to 40 m. The required SSD is calculated using the two-curve formulas of Eqs. (5) and (6). Comparing the values in Table 1, it is noted that the maximum lateral clearance required for the yield line is greater than that required for the crosswalk when the approach speed and approach/entry radius are large. In design, it would be desirable to implement the larger requirement for the entire approach curve within SSD. For evaluation, however, where the lateral clearance at a specific point is required, the analytical models should be used. In this case, the lateral clearances corresponding to both sightlines are calculated and the larger value should be used. For exit sight distance, the maximum lateral clearance, Cm, and the difference between the locations of the far edge of the available (typical) and required crosswalks, Dar, are shown in Table 2. The values are presented for two cases of roundabouts: symmetrical (ϕ1 = 45 degrees) and unsymmetrical (ϕ1 = 40 degrees). A positive value of Dar means that the typical crosswalk satisfies SSD and a negative value means that it does not. The bold values indicate that the required crosswalk lies on the tangent. The range of values for the entry radius and initial speed at the yield line are based on the literature (Rodegerdts et al., 2011). The typical entry radius for urban single-lane roundabouts ranges from 10 m to 30 m or more at rural locations, as previously mentioned. The literature also 9
suggests that the desirable maximum entry speed is 30 km/h to 40 km/h. Based on these guidelines, the ranges of the entry radius and speed were 20 m to 50 m and 20 km/h to 40 km/h, respectively. With the specified ϕ1 and R2, Rn and R3 are easily determined using Eqs. (7) and (8), respectively. Note that for the case of ϕ1 < 45 degrees, R3 is always greater than R2, which is required for safe operations. For intermediate radii or entry speeds, the design values can be interpolated. For values outside the ranges used in the table, the analytical models should be used. What is interesting in Table 2 is that for unsymmetrical roundabouts, the typical crosswalk satisfies SSD requirements for all entry radius-speed combinations, except for the combination (R2 = 50 m, Venter = 40 km/h) which has a negative value of Dar. In comparison, for symmetrical roundabouts, the location of the typical crosswalk is inadequate for combinations of small radii and high entry speeds. Although exit sight distance can be improved by relocating the crosswalk farther from the yield line, this action will also increase the distance from the yield line where vehicles on the near approach should stop for the crossing pedestrians. The increase of this distance, although increases ISD, may adversely affect roundabout safety if excessive ISD is provided (Rodegerdts et al., 2011). Alternatively, the designer may consider implementing a staggered crosswalk, a flatter radius, or some strategies to reduce entry speed. For the circulatory roadway sight distance, an important issue is to ensure that the object (Vehicle 2 in Fig. 4) lies within the driver’s field of vision (City of Calgary, 2010). This is a special requirement for circulating traffic as SSD could circle more than half the central island and violates the adequate field of peripheral vision. From the geometry of Fig. 4, the driver’s field of peripheral vision is given by 𝜓=
𝑆+ 𝑑𝑒
(38)
𝑅𝑐 +𝑓
where ψ = central angle subtended by S. According to the International Council of Ophthalmology (ICO, 2006), an angle of vision of 120 degrees is considered adequate in many countries. To determine the geometric conditions corresponding to adequate field of peripheral vision, ψ of Eq. (38) was determined for Rc = 10 m to 50 m and Vcir = 20 km/h to 40 km/h, as shown in Fig. 5. Any combination of Rc and Vcir above the line corresponding to the maximum field of peripheral vision, ψmax = 120 degrees, would be inadequate and should not be used in design. As noted, this infeasible region (dashed lines) occurs for small R and large Vcir. Based on Eq. (37), the required maximum lateral clearance on the central island is presented in Table 3. The table also shows the combinations for which insufficient tangential deceleration is available or the peripheral vision requirement of the circulating drivers is not satisfied. .
5. Application All the mathematical models presented in this paper were verified graphically by comparing their results with an actual drawing of the roundabout on Excel. To illustrate the application of the design aids, consider a symmetrical single-lane roundabout with approach radius R1 = 100 m, entry radius R2 = 40 m, and approach speed Vapp = 50 km/h. The crosswalk width is 3 m. The far edge of the crosswalk is located at a distance Lmin = 6 m from the inscribed circle (Fig. 2). Thus, the roundabout has the same characteristics as those used in establishing the design table. Determine the maximum lateral clearance Cm and the required lateral clearance at a distance 40 m from the start of the entry curve. 10
From Table 1, the maximum lateral clearance to the crosswalk is 7.3 m and that to the yield line is 7.9 m. Therefore, for new design a lateral clearance of 7.9 m should be implemented along the entire length of the stopping sight distance, S = 75.7 m. To determine the required lateral clearances at a distance 40 m from the start of the entry curve, the analytical model should be used. Using Eq. (11), I = 0.668 (radians). Therefore, the length of the vehicle path corresponding to I = 0.668 (100 + 1.77) = 68.0 m. Then, the distance between the y-axis and the required point equals 68.0 – 40.0 = 28.0 m. Using this distance, the analytical model gives lateral clearances of 6.3 m and 3.3 m, for the crosswalk and yield line, respectively. Therefore, the larger value of 6.3 m is used. The solution of this example is shown graphically in Fig. 6. As another example, consider an unsymmetrical single-lane roundabout with ϕ1 = 40 degrees and entry radius R2 = 50 m. Determine the required maximum lateral clearance for exit sight distance and check whether a typical crosswalk that satisfies the minimum distance of 6 m from the yield line is adequate for entry speeds of 25 km/h and 40 km/h. From Table 2, for Venter = 25 km/h and R2 = 50 m, from Table 2, Cm = 2.1 m and Dar = 20.5 m. Since Dar is positive, the required crosswalk lies before the typical crosswalk, and therefore the typical crosswalk is adequate. For Venter = 40 km/h and R2 = 50 m, from Table 2, Cm = 4.1 m and Dar = -3.0 m. Since Dar is negative, the location of the typical crosswalk is inadequate and should be relocated farther from the yield line by 3.0 m to satisfy SSD. This means that the distance from the yield line to the near edge of the crosswalk will be 9.0 m instead 6.0 m. 6. Concluding Remarks This paper and recent research by the author have focused on developing analytical models of sight distance for roundabouts in an effort to eliminate the need for evaluating sight distance graphically. This paper has presented SSD-based models and design aids for unsymmetrical single-lane roundabouts for three critical locations: approach sight distance, sight distance to exit crosswalk, and circulatory roadway sight distance. Based on this research, the following comments are offered: 1. The evaluation of sight distance adequacy at roundabouts is currently performed graphically and involves a trial process. The presented analytical models may be integrated in current roundabout design software to automate the evaluation process. In addition, the design aids can be used to determine directly the required maximum lateral clearance that satisfies both sight distances to the crosswalk and to the yield line. This process is not only more accurate than the graphical process, but also more efficient. 2. Approach sight distance was found to be insensitive to the approach deflection angle ϕ1. This can be explained as follows. For a given entry radius, different values of ϕ1 affect exit and inscribed circle radii. However, approach sight distance does not depend on exit radius and variation of the inscribed circle radius has little effect on the intersection point of the yield line and approach vehicle path. Thus, the design aids for approach sight distance were developed assuing a symmetrical roundabout and are applicable to both types of roundabouts. On the other hand, for exit sight distance the deflection angle has a significant effect on sight distance, as expected, since it directly affects entry and exit radii. Design tables were presented for ϕ1 = 40 degrees (unsymmetrical) and 45 degrees (symmetrical). For other deflection angles, the analytical model should be used. The design table for circulatory roadway sight distance is obviously applicable to both types of roundabouts.
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3. The model for exit sight distance provides information about the required maximum lateral clearance and the corresponding location of the crosswalk to satisfy SSD needs. This location was compared with a typical crosswalk location that satisfies the minimum distance from the yield line. Comparing both locations, the developed design table provides useful information on the adequacy of the typical crosswalk location and whether improvements are needed to satisfy sight distance to exit crosswalk. The results show that the typical crosswalk is inadequate for large entry speeds and small entry radii. The analysis also shows that a small variation in the deflection angle can be very beneficial in improving sight distance to the exit crosswalk. 4. Design guidelines are provided for the feasible combinations of central island radius and circulatory roadway speed. The results show that some combinations of small radii and large speeds will produce a field of peripheral vision that is inadequate for the drivers. Such combinations should be avoided in design. 5. The SSD analytical models for the unsymmetrical single-lane roundabout presented in this paper are applicable to the symmetrical one, which is a special case (ϕ1 = 45 degrees and w1 = w2). It should be pointed out that the results of this special case would be more accurate and efficient than those of a previously published paper (Easa 2017a). The reason for this is that the models of the current paper account for the effect of the radial deceleration rate on SSD. In addition, the model for sight distance to the yield line formulates the location of the intersection point of vehicle path and inscribed circle, which was previously a user input. Therefore, for SSD of single-lane symmetrical roundabouts, the special case of this paper is recommended for practical implementation. 6. Current research on roundabout ISD by the author has shown that the required maximum lateral clearance on the central island is controlled by ISD of a stationary vehicle at the yield line or by SSD of a circulating vehicle. The research shows that ISD of a vehicle located 15 m from the yield line is always less than that for a stationary vehicle at the yield line. Since the current Second Edition of the Roundabout Information Guide (Rodegerdts et al., 2011) considers ISD only for a vehicle located 15 m from the yield line, it is recommended that the planned third edition of the guide should consider both vehicle locations in ISD analysis (Easa, 2017b). 7. The models presented in this paper provide useful geometric relations that can be easily applied to address sight distance issues for other types of roundabouts, including multi-lane roundabouts, staggered roundabouts, and complex multi-leg roundabouts. Acknowledgements This research is financially supported by a Discovery Grant and a Discovery Accelerator Supplement from the Natural Sciences and Engineering Research Council of Canada (NSERC). References Ali, Z., Easa, S.M., and Hamed, M. (2009). Stop-controlled intersection sight distance: Minor road on tangent of horizontal curve. J. Transp. Eng., ASCE, 135(9), 2009, 650-657. American Association of State Highway and Transportation Officials (2011). A Policy on Geometric Design of Highways and Streets. AASHTO, Washington D.C. Boodlal, L. (2004). Accessible Sidewalks and Street Crossings - An Informational Guide. Federal Highway Administration, U.S. Department of Transportation, Washington, D.C. 12
City of Calgary (2010). Cone-of-Vision Impacts in Roundabouts. Final Report, Calgary, Alberta. Internet: http://www.ctep.com/pdf/Bunt%20_Cone%20of%20Vision%20Final%20Report%20.pdf Department for Regional Development (2007). Geometric Design of Roundabouts. Volume 6, Section 2, Part 3 TD16/07, Belfast, Northern Ireland. Easa, S.M. (2017a). Lateral clearance needs for stopping sight distance at symmetrical singlelane roundabouts. Proc., Transportation Research Board Conference, Washington, D.C. Easa, S.M. (2017b). Design guidelines for symmetrical single-lane roundabouts based on intersection sight distance. J. Transp. Eng., Part A: Systems, ASCE, 143(10), DOI: https://doi.org/10.1061/JTEPBS.0000081. Easa, S.M. (2016). Pedestrian crossing sight distance: Lateral clearance guidelines for roadways. Transportation Research Record 2588, Journal of the Transportation Research Board, Washington, D.C., 2016, 32-42, DOI: http://dx.doi.org/10.3141/2588-04 Easa, S.M., Qu, X. and Dabbour, E. (2017). Improved pedestrian sight distance needs at railroadhighway grade crossings. J. Transp. Eng., Part A: Systems, ASCE, 143(7), DOI: https://doi.org/10.1061/JTEPBS.0000047. Easa, S.M. and Ali, Z. (2006). Three-dimensional stop-control intersection sight distance: General Model. Transportation Research Record 1961, J. of Transportation Research Board, National Research Council, Washington, D.C., 94-103. Federal Highway Administration (2006). Intersection Safety Roundabouts. U.S. Department of Transportation, FHWA, Washington, D.C. Internet: http://safety.fhwa.dot.gov/intersection/innovative/roundabouts/fhwasa10006/#s64 (accessed on July 15, 2016). Government of Queensland (2013). Road Planning and Design Manual. Chapter 14: Roundabouts, Department of Transport and Main Roads, Queensland, Australia. Harwood, D.W., Mason, J., Brydia, R., Pietrucha, M., and Gittings, G. (1996). Intersection Sight Distance. NCHRP Report 383. Transportation Research Board, Washington, D.C. Hussain, A. and Easa, S.M. (2016). Reliability analysis of left-turn sight distance at signalized intersections. J. Transp. Eng., Part A: Systems 142(3), DOI: 10.1061/(ASCE)TE.19435436.0000824. International Council of Ophthalmology (2006). Vision Requirements for Driving Safety: Visual Standards. Report presented at World Ophthalmology Congress, Sao Paulo, Brazil. Rodegerdts, L. A., Bansen, J., Tiesler, C., Knudsen, J., Myers, E., Johnson, M., Moule, M., Persaud, B., Lyon, C., Hallmark, S., Isebrands, H., Crown, R., Guichet, B., and O’Brien, A. (2010). Roundabouts: An Informational Guide. National Cooperative Highway Research Program Report 672, Transportation Research Board, Washington, D.C. Rodegerdts, L., Blogg, M., Wemple, E., Myers, E., Kyte, M., Dixon, M., List, G., Flannery, A., Troutbeck, R., Brilon, W., Wu, N., Persaud, B., Lyon, C., Harkey, D., and Carter, C. (2007). Roundabouts in the United States. National Cooperative Highway Research Program Report 572, Transportation Research Board, Washington, D.C. Microsoft (2015). Excel 2010 Solver Offers More Power for Optimization, More Help for Users. Internet: http://www.prweb.com/releases/excel2010/solver/prweb4148834.htm (Accessed on January 21, 2015). Transportation Association of Canada (2016). Manual of Uniform Traffic Control Devices for Canada. TAC, Ottawa, Ontario.
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Washington State Department of Transportation (2016). Roundabouts. Chapter 1320, WSDOT, Olympia, WA.
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LIST OF TABLES Table 1
Design values of maximum lateral clearance for approach sight distance single-lane roundabouts (symmetrical or unsymmetrical) Table 2 Design values of maximum lateral clearance for exit sight distance for unsymmetrical single-lane roundabouts Table 3 Design values of maximum lateral clearance for circulatory roadway of single-lane roundabouts (symmetrical or unsymmetrical)
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Table 1 Design values of maximum lateral clearance for approach sight distance single-lane roundabouts (symmetrical or unsymmetrical) a Approach Entry Radius, Radius, R1 (m) R2 (m)
100
200
300
400
500
20 30 40 20 30 40 20 30 40 20 30 40 20 30 40
Maximum Lateral Clearance, Cm (m) (Sight Distance to Crosswalk) Approach Speed, Vapp (km/h) b 40 50 60 70
Maximum Lateral Clearance, Cm (m) (Sight Distance to Yield Line) Approach Speed, Vapp (km/h) b 40 50 60 70
3.8 4.0 4.5 2.4 2.9 3.5 2.1 2.7 3.2 2.1 2.6 3.1 2.1 2.5 3.0
2.8 3.9 4.8 1.4 2.8 3.8 1.1 2.5 3.5 0.9 2.3 3.3 0.9 2.2 3.2
6.9 7.0 7.3 3.5 3.9 4.4 2.6 3.1 3.8 2.3 2.9 3.6 2.2 2.8 3.5
5.5 5.8 6.3 3.8 4.2 4.8 3.0 3.5 4.2 2.6 3.1 3.9
5.7 6.0 6.6 4.3 4.7 5.3 3.6 4.0 4.7
6.0 6.9 7.9 2.7 4.0 5.3 1.8 3.3 4.7 1.4 3.0 4.4 1.2 2.8 4.2
4.8 6.1 7.3 3.0 4.4 5.9 2.3 3.8 5.4 1,8 3.4 5.1
5.0 6.3 7.7 3.6 5.0 6.5 2.9 4.3 5.9
a
The crosswalk is assumed to be perpendicular to the curve.
b
For the shaded cells, R1 is less than Rmin for the respective approach speed and a superelevation of 0.06 or less.
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Table 2 Design values of maximum lateral clearance for exit sight distance for symmetrical and unsymmetrical single-lane roundabouts a Entry Speed, Venter (km/h) b Entry Radius, R2 (m)
20 c
Cm (m)
25 d
Dar (m)
Cm (m)
30 Dar (m)
Cm (m)
35 Dar (m)
Cm (m)
40 Dar (m)
Cm (m)
Dar (m)
-11.4 -7.8 -5.0
8.3
-18.7
0.7 3.3 5.8
4.1
-3.0
(a) Deflection Angle ϕ1 = 45 degrees
a
20 25 30 35 40 45 50
3.5 3.0 2.7 2.5 2.3 2.2 2.1
6.8 9.3 11.4 13.3 15.1 16.9 18.6
20 25 30 35 40 45 50
2.5 2.2 2.1 2.0 2.0 2.0 2.1
12.2 14.8 17.3 19.7 22.1 24.4 26.7
5.7 -2.4 4.5 1.4 3.9 4.1 6.1 -5.9 e 3.4 6.3 5.1 -2.5 3.1 8.3 4.5 0.1 7.0 2.9 10.2 4.0 2.4 6.0 2.7 12.0 3.7 4.4 5.3 (b) Deflection Angle ϕ1 = 40 degrees 3.3 2.9 2.6 2.4 2.3 2.2 2.1
5.5 8.3 10.9 13.4 15.8 18.8 20.5
3.4 3.1 2.8 2.6 2.5
3.6 6.2 8.7 11.2 13.6
3.7 3.4 3.1
The crosswalk is assumed to be perpendicular to the curve.
b
For the shaded area, R2 is less than Rmin for the respective entry speed and a typical cross slope. Maximum lateral clearance corresponding to minimum SSD. d Distance between the location of the far edge of the typical crosswalk and that of the required crosswalk that satisfies SSD. A positive value means that the typical crosswalk satisfies SSD and a negative value means that it does not. e Bold values means the crosswalk lies on the tangent. c
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Table 3 Design values of maximum lateral clearance for circulatory roadway of single-lane roundabouts (symmetrical or unsymmetrical) a Radius of Central Island, Rc (m) 10 15 20 25 30 35 40 a b
Maximum Lateral Clearance, Cm (m) Circulatory Speed, Vcir (km/h) 20 3.2 1.3 0.5 0 0 0 0
25 - a,b 5.3 2.6 1.6 0.9 0.5 0.2
30 - a,b -a 11.1 4.7 3.1 2.2 1.6
35 - a,b - a,b -a -a 9.0 5.5 4.0
40 - a,b - a,b - a,b -a -a -a 10.4
Available tangential deceleration is insufficient. Peripheral vision requirement of circulating drivers is not satisfied.
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LIST OF FIGURES Figure 1 Geometry of unsymmetrical roundabout with Roads 1 and 2 intersecting at right angle Figure 2 Geometry of approach sight distance for crosswalk and yield line. Figure 3 Geometry of exit sight distance for crosswalk: Required crosswalk corresponding to minimum SSD (Point h or h’) and typical crosswalk corresponding to distance Lmin from the yield line (Point v) Figure 4 Geometry of circulatory roadway sight distance Figure 5 Feasible region for the field of peripheral vision on the circulatory roadway Figure 6 Dimensions of application example for approach sight distance.
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Fig. 1 Geometry of unsymmetrical roundabout with Roads 1 and 2 intersecting at right angle
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Figure 2 Geometry of approach sight distance for crosswalk and yield line
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Figure 3 Geometry of exit sight distance for crosswalk: Required crosswalk corresponding to minimum SSD (Point h or h’) and typical crosswalk corresponding to distance Lmin from the yield line (Point v)
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Figure 4 Geometry of circulatory roadway sight distance
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Figure 5 Feasible region for the field of peripheral vision on the circulatory roadway
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Fig. 6 Dimensions of roundabout approach sight distance (application example)
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