Modeling of Required Preview Sight Distance

MODELING OF REQUIRED PREVIEW SIGHT DISTANCE

By Y. Hassan1 and S. M. Easa,2 Member, ASCE

(Reviewed by the Highway Division)

ABSTRACT: Poor coordination of horizontal and vertical alignments can create locations where the availablesight distance drops below the required sight distance. Therefore, current design guides have recommended anumber of guidelines to enhance the alignment coordination. A better and more quantified approach for alignmentcoordination can be achieved using a concept, called sight distance red zones, based on 3D analysis. A red zone,based on preview sight distance (PVSD), is defined as a section of the road where a horizontal curve shouldnot start relative to a vertical curve. This paper presents a framework to estimate the required PVSD, which isthe sight distance required to see, perceive, and react to a horizontal curve before its beginning. The requiredPVSD consists of two parts: PVSD on tangent and PVSD on curve. A simple analytical model of PVSD ontangent is presented based on the laws of kinematics. The PVSD on curve was investigated empirically usingphysical modeling and computer animation. Curves with different radii (500–2,000 m), turning directions (leftand right), and configurations (with and without spirals) were simulated. Using the collected data, the effect ofcurve parameters was examined, regression models for the required PVSD on curve were developed, and pre-liminary design values for the required PVSD are presented.

INTRODUCTION

A safe and efficient highway facility should meet the driv-ers’ expectations and thus reduce their workload. On the otherhand, ambiguous or confusing information can cause driversto perform hazardous and erratic maneuvers. Although driversuse most of their senses to perceive the required informationto perform the driving tasks, about 90% of this information isperceived visually (Alexander and Lunenfeld 1986). There-fore, current design practices require highway designers toprovide motorists with a stopping sight distance (SSD) enoughto stop before hitting an unexpected object on the road. Pro-vision of enough passing sight distance (PSD) on frequentstretches of two-lane highways reduces driver frustration anderratic passing maneuvers. In situations where information isdifficult to perceive or where stopping is not the appropriateaction, a decision sight distance (DSD) should be provideddepending on the complexity of the situation. A special caseof DSD, called preview sight distance (PVSD), has been sug-gested by Gattis and Duncan (1995) to enable drivers to seethe road features ahead. In this paper, PVSD is used to referto the sight distance required to perceive the existence of ahorizontal curve ahead and react properly to it.

Current design guides and previous research work have ad-dressed the different types of highway sight distance. Mathe-matical models and design values for SSD, PSD, and DSDhave been presented in the design guides (Manual 1986; Apolicy 1994). Further revisions of the SSD model have beensuggested by Olson et al. (1984), Neuman (1989), and Fambroet al. (1997). Also, several revisions of the PSD model havebeen suggested, based on the concept of critical sight distance,by Lieberman (1982), Saito (1984), Glennon (1988), Rilletet al. (1990), and Hassan et al. (1996). Gattis and Duncan(1995) provided values of the required PVSD on separate hor-

1Asst. Prof., Public Works Dept., Cairo Univ., Giza, Egypt; currently,Visiting Prof., Dept. of Civ. and Envir. Engrg., Carleton Univ., Ottawa,ON, Canada K1S 5B6. E-mail: [email protected]

2Prof., Dept. of Civ. Engrg., Lakehead Univ., Thunder Bay, ON, Can-ada P7B 5E1. E-mail: [email protected]

Note. Discussion open until July 1, 2000. To extend the closing dateone month, a written request must be filed with the ASCE Manager ofJournals. The manuscript for this paper was submitted for review andpossible publication on October 22, 1998. This paper is part of the Jour-nal of Transportation Engineering, Vol. 126, No. 1, January/February,2000. qASCE, ISSN 0733-947X/00/0001-0013–0020/$8.00 1 $.50 perpage. Paper No. 19495.

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izontal and crest vertical curves based on field measurementsby one driver.

Nonetheless, the Canadian design guide (Manual 1986)states that ‘‘a section of road might be designed to meet thesestandards, yet the result can be a facility exhibiting numerousunsatisfactory or displeasing characteristics.’’ Therefore, gen-eral guidelines have been introduced to help designers avoidsuch unsatisfactory characteristics. Because current designpractices address a separate design element at a time (e.g., acurve), the guidelines have addressed mainly successive ele-ments of a horizontal or a vertical alignment and the coordi-nation of both alignments. Using a 3D analysis of sight dis-tance, Hassan and Easa (1998a,b) developed a concept, calledsight distance red zones, to identify the locations where a hor-izontal curve should not start relative to (or overlap with) avertical curve. The concept was illustrated using hypotheticalvalues of the required PVSD, and no real values have beenpresented. The analysis of red zones has shown that a highwaydesigned according to current 2D practice may exhibit inade-quate sight distance at some locations. Using the 3D analysis,the locations where the available sight distance is less than therequired value are identified.

Two types of sight distance red zones were defined: SSDred zones and PVSD red zones. In either type, the location ofthe vertical curve is assumed fixed, and the location of thehorizontal should be shifted to avoid the red zone. BecauseSSD is required on every point of the alignment, any overlapbetween horizontal and vertical curves should be avoidedwithin an SSD red zone. On the other hand, because PVSD isrequired only for the beginning of a horizontal curve, the de-signer should avoid starting the horizontal curve within aPVSD red zone. If the curve starts before the red zone, furtheroverlap between the horizontal and vertical curves is not aconcern. Although much research work has been conducted onthe required SSD, very little work has been done on the re-quired PVSD. Using 2D analysis, Gattis and Duncan (1995)studied the PVSD on a crest vertical curve at daytime and ona simple horizontal curve at nighttime. The required PVSDwas measured by only one driver who exercised a high levelof attention, whereas most drivers usually perform severaltasks while driving. Therefore, the obtained values could notbe generalized to highway design, and more research was rec-ommended to determine desirable values of PVSD. This paperfocuses on the required PVSD to identify its components anddevelop relevant design values.

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FIG. 1. Driver’s View of Horizontal Curve following Crest Ver-tical Curve on Highway 11, Ontario, at Three Successive Loca-tions on Tangent

The following sections present practical aspects of PVSDand a framework for determining the required PVSD, followedby a description of the modeling of the PVSD on a horizontalcurve (which is required to perceive its existence) and the datacollection. The results of the required PVSD on curve are thenpresented to study the effect of curve parameters on the re-quired value and to develop regression models for the requiredPVSD.

PRACTICAL ASPECTS OF PVSD

When a horizontal curve starts near the top of a crest ver-tical curve (or near the bottom of a sag vertical curve), thehorizontal curve may not be visible to drivers long enoughbefore the start of the horizontal curve to adjust their speedsaccording to the safe speed on the curve. For illustration, Fig.1 shows three successive snapshots for the driver’s view of ahorizontal curve that starts near the top of a crest vertical curveon Highway 589 (Ontario, Canada). As shown, even a flatcrest vertical curve may cause the horizontal curve to be in-visible to drivers until they are very close to it.

This problem has been spotted by the Ministry of Trans-portation, Ontario (MTO) on some segments, where relativelyhigh collision rates were recorded. For example, nine colli-sions were reported within two back-to-back horizontal curves(radius = 350 m) on Highway 11, Ontario, between May 1989and January 1993. Seven of these collisions were attributed tothe road conditions with reduced sight distance being the maincontributor. This reduction in sight distance is caused by a rockcut in the inside of a horizontal curve and is aggravated by acrest vertical curve that is immediately before the same hori-zontal curve. The low visibility combined with a high oper-

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ating speed in the westbound direction, due to a 2.4-km tan-gent section before the curve (despite the 80-km/h advisoryspeed posted on the curves), caused difficulty for some driversto realize the severity of the upcoming horizontal curve (MTO1995).

To avoid such combined alignment problems, the AmericanAssociation of State Highway and Transportation Officials(AASHTO) design guide (A policy 1994) states that ‘‘sharphorizontal curvature should not be introduced at or near thetop of a pronounced crest vertical curve . . . or near the lowpoint of a pronounced sag vertical curve.’’ The same generalguidelines are also stated in the Canadian design guide (Man-ual 1986). A less subjective guideline in the French designguide (Highway 1995) requires the point of curve (PC) of ahorizontal curve to be visible 3 s before PC. Thus, the requiredPVSD equals the design speed multiplied by 3 s. Although the3-s value is recommended to enable drivers to see and perceivethe curve, it is not expected that drivers can perceive the hor-izontal curve just by seeing PC. Intuitively, a specific distanceafter the beginning of the curve should be seen by drivers toperceive the horizontal curvature.

Similar to other sight distance types, PVSD is measured byextending a line from an initial point, referred to as origin, toa target point, referred to as object. According to the Frenchguide (Highway 1995), the PVSD is measured from a drivereye of 1.0-m height to the pavement surface with zero height.Because drivers depend on the highway marking to perceivethe geometry of the highway (Teply 1987), the pavementmarking of zero height is the appropriate object for PVSD.However, two driving conditions must be considered for theorigin of PVSD, namely, daytime and nighttime driving. Atdaytime, the sight distance should correspond to a sight linefrom the driver eye to the pavement marking. Thus, 1.05 m(Manual 1986), 1.07 m (A policy 1994), or 1.0 m (Highway1995) is an appropriate origin height. However, at nighttime,the sight distance is limited by the light beams of the vehicle’sheadlights. Thus, the available sight distance would corre-spond to a light ray from the vehicle’s headlights to the pave-ment marking. In this case, a vehicle’s headlight height of 0.6m is an appropriate origin height (Manual 1986; A policy1994). Clearly, consideration of nighttime driving provides aconservative condition and should be used in defining PVSDred zones.

FRAMEWORK FOR REQUIRED PVSD

The PVSD is defined as the sight distance required to seeand perceive a horizontal curve ahead and react properly to it.Because tangents usually have higher operating speeds thanhorizontal curves, drivers may have to adjust their speeds bydecelerating from the tangent’s operating speed to the curve’soperating speed before reaching the circular curve. Therefore,the PVSD should have a provision for the distance traveledduring the perception-reaction (PR) time and during deceler-ation. The distance traveled during the PR time (simply, thePR distance) must lie on the tangent and end before the cir-cular curve. Also, if the circular curve is introduced directlywithout a spiral curve, the distance traveled during decelera-tion (simply, the deceleration distance) must lie fully on thetangent. However, because a spiral curve before the circularcurve provides a smooth transition from the tangent to thecircular curve, the deceleration distance can lie on the spiralcurve and can extend on the tangent, if necessary. The totalnet distance that lies on the tangent will be referred to asPVSD on tangent S1.

In addition, drivers can recognize the existence of the hor-izontal curve by seeing a bent or a deflection in the highway(i.e., by seeing a specific distance on the curve). Thus, a sec-ond component of the PVSD is a distance seen on the curve,

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FIG. 2. Components of Required PVSD: PVSD on Tangent S1

and PVSD on Curve S2

referred to as PVSD on curve S2, that should be visible todrivers to recognize the existence of the curve. The beginningof S2 is the first point of the horizontal curve, which is the PCif a spiral curve is not used or the tangent-to-spiral (TS) pointif a spiral curve is used. Therefore, the use of a spiral curvewill result in an overlap between the deceleration distance andS2. The PVSD should be given as the two separate compo-nents, S1 and S2, where the total range of S2 should be visibleto the driver from a distance S1 before the beginning of thehorizontal curve (Fig. 2).

PVSD on Tangent

As explained earlier, the PVSD on tangent S1 is the totalnet distance traveled on the tangent, and consists of the PRdistance and a portion of the deceleration distance that dependson the configuration of the horizontal curve (simple or spi-raled). That is

L 1 L simple curve1 2

S = L 1 L 2 l spiraled curve, L > l (1)1 1 2 s 2 sHL spiraled curve, L # l1 2 s

where L1 = PR distance; L2 = deceleration distance; and ls =spiral length. For example, consider a horizontal curve wherethe PR distance L1 equals 35 m and the deceleration distanceL2 equals 65 m. If no spiral is used, S1 will equal 100 m (S1

= L1 1 L2). On the other hand, if a 60-m spiral is introducedbefore the circular curve, S1 will equal 40 m (S1 = L1 1 L2 2ls). However, if the spiral length is 70 m, S1 will equal 35 m(S1 = L1).

The PR distance, which is the distance traveled during thePR time, is an integral part of the PVSD on tangent. Duringthe PR time, drivers should be able to perceive the change ofthe geometry, decide on the proper action (deceleration), andinitiate this action. Generally, the PR time can vary signifi-cantly from one driver to another depending on age, experi-ence, and any kind of impairment. For a specific driver, thePR time can also vary depending on the complexity of thesituation and the information required to make the decision.According to the North American design guides (Manual1986; A policy 1994), a PR time of 2.5 s is recommended forSSD. However, because the deceleration required for thechange of alignment is less hazardous than the complete stoprequired in the case of SSD, a lower value might be used. ThePR distance L1 can be written as follows:

L = 0.278PV (2)1 T

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FIG. 3. Typical Values of Required PR and Deceleration Dis-tances

where VT = tangent operating speed (km/h); and P = PRtime(s).

The deceleration distance is the distance traveled during de-celeration from the tangent operating speed to that of thecurve. Because horizontal curves usually have lower operatingspeeds than those of tangents, drivers are expected to reducetheir speeds when they perceive the existence of a horizontalcurve. The reduction in speed depends on many factors, in-cluding the length of the preceding tangent and the curve ra-dius. Generally, long tangents would encourage a high oper-ating speed, particularly on downgrades. On the other hand,the sharper the horizontal curve (smaller radius), the lower theoperating speed on the curve. Several models have been de-veloped and can be used to estimate the operating speed onhorizontal curves or the change in operating speed from a tan-gent to a curve (Gibreel et al. 1999). Using the estimated op-erating speed, the deceleration distance L2 can be calculatedas follows:

2 2V 2 VT CL = (3)2 25.92d

where VC = curve operating speed (km/h); and d = decelerationrate (m/s2). For illustration, Fig. 3 shows the values of L1 fordifferent tangent operating speeds and L1 1 L2 for combina-tions of tangent and curve operating speeds, assuming a 2.5-sPR time and a 0.85-m/s2 deceleration rate (Collins and Kram-mes 1996). It should be noted that although the figure showshigh values of L1 1 L2 that can reach 300 m, these values areassociated with considerably large drops in the operating speedfrom the tangent to the curve, which are unlikely to occur.

PVSD on Curve

The PVSD on curve, the portion of the PVSD that lies onthe horizontal curve, must be seen by the driver to perceivethe existence of the curve. The PVSD on curve depends onthe radius of the horizontal curve and its configuration (simpleor spiraled). Generally, the sharper the horizontal curve, thesmaller the distance required to produce a specific deflection,and in turn, the smaller the distance S2. Thus, it would beexpected that S2 will be greater if a spiral curve is introducedbefore the circular curve. However, in this case, the portion ofS2 on the spiral curve can be utilized in the deceleration pro-cess in addition to its role in providing the driver with a visualcue for the horizontal curve ahead, as indicated by (1). Todetermine the required PVSD on curve, experimental workwas conducted and used to develop analytical models for de-sign purposes.

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MODELING PVSD ON CURVE

Model Development

Two modeling procedures were used to estimate the valueof PVSD on curve: physical modeling and computer anima-tion. These modeling procedures were investigated in two sep-arate undergraduate degree projects at Lakehead University(Barber and Van Duuren 1997; Gibbons and Ruddy 1998).Additional data were subsequently collected for the more com-prehensive analysis presented in this paper. It should be notedthat the experimental work was designed based on modeling2D horizontal curves only. The results of this modeling shouldalso be valid to 3D curves if the perception of horizontalcurves is independent of the overlapping vertical alignment.Investigating the validity of this assumption is beyond thescope of this paper.

The physical modeling involved using AutoCAD to drawand plot highway segments at 1:20 scale, where each drawingconsisted of a tangent followed by a horizontal curve. Thedrawing sheets were laid on a special table (2.4 3 0.9 m), andthe relative movement between the road and the driver wassimulated by rolling the sheet. A barrier was added near theend of the table to limit the sight distance of the highwaydrawing. The roadway surface (two lanes and a shoulder) waspainted in black, and the delineation was painted in yellow.Drivers were asked to view the road drawings from a smallopening simulating the windshield, where the barrier wouldinitially allow only the tangent to be seen. As the drawing wasrolled, the curve would start to be visible until the driver couldsee the curve and determine its turning direction. The requiredPVSD on curve was measured as the distance between thebeginning of the curve and the barrier.

For the computer animation, several computer programswere used to create a realistic animation of the road. An ele-ment net of the road surface, the side slopes, and the deline-ation was created using a computer program MARKC devel-oped by Hassan et al. (1997). The element net was in the formof a ‘‘script’’ file readable by AutoCAD. The element net wasimported into AutoCAD to create a ‘‘drawing’’ file. The draw-ing file was then imported into Modelview (by Intergraph) tocreate an animation as a camera view of the road. The ani-mation was saved as separate frames, taken at 0.5-m intervalsof camera movement, using a shareware program of the In-ternet called Dave’s TGA Animation Program. Finally, anothershareware package, called Powerflic, was used to run theframes into an animation at approximately 25 frames/s.

The camera was used to simulate the driver’s eye and wasmounted at a 1.05-m height from the road surface. The relativemovement between the road and the driver was created bydisplaying successive frames of the camera view, where eachframe represented a 0.5-m movement. The sight barrier wassimulated by a fog that limited the available sight distance to75 m. By attaching a 100-m tangent to the beginning of thecurve, drivers could initially see only a straight segment of theroad. As the animation ran, the curve started to become visible(Fig. 4). Once the driver could see the curve and identify theturning direction, the driver pressed ‘‘X’’ to stop the animationand display the number of frames used in the animation nF.Because the number of frames accounts for the 25 m on thetangent (which was not initially visible) plus the distance onthe curve used to perceive it, then

S = 0.5(n 2 1) 2 25 (4)2 F

The computer animation provided a better modeling that iscloser to reality and would provide more reliable results thanthe physical modeling. Based on the evaluation of the researchteams and the participants, the frequent warping in the drawingsheets of the physical modeling was a problem, and the setting


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FIG. 4. Successive Driver View of Horizontal Curve in Com-puter Animation

of the computer animation was more comfortable. The re-mainder of this paper addresses the results of the computeranimation modeling only.

Data Collection

Before data collection, the curve parameters were defined.A total of 40 curve combinations were animated using thefollowing curve parameters:

• The highway was a two-lane facility with 3.7-m lanewidth and 2.5-m shoulder width. According to the mark-ing practice in Ontario, road delineation was 10 cm wide(Manual 1995).

• A 110-km/h design speed was selected as a guide tochoose other curve parameters. The design speed for ma-jor two-lane highways in Ontario is 110 km/h.

• The curve radius ranged from a relatively sharp (500 m)to a considerably flat (2,000 m); it should be noted thatthe minimum radius corresponding to a 110-km/h designspeed and 6% superelevation rate is 600 m. Intermediatevalues were also included to investigate the trend ofPVSD on curve, resulting in a total of 10 curve radii (500,600, 700, 800, 900, 1,000, 1,250, 1,500, 1,750, and 2,000m).

• The horizontal curve configuration can be simple or spi-raled. For spiraled curves, the value of the spiral param-eter A was selected according to the Canadian designguide (Manual 1986). Thus, A was equal to 220, 220, 220,220, 235, 245, 275.8, 305, 327.5, and 350 m for the 10curve radii, respectively.

• The turning direction can be right or left.

The data collection required testing a sample of drivers thatis a good representation of the general driver population, freeof trends, and normally distributed. Two sets of data were col-lected: (1) A control group that consisted of 15 drivers (11males and 4 females) ranging in age from 22 to 45, who haddifferent education levels, professions, and driving experience;and (2) a random group that consisted of 56 volunteers rangingin age from 18 to 60. In addition to being a part of the overallsample of drivers, the control group served as a reference to

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validate the modeling procedures. Because the members of thecontrol group were accessible even after the experiment wascompleted, any problem that would arise could be correctedand the experiment could be rerun. All volunteers were alsoasked to participate in a survey containing questions related todriving history, age, gender, residence, and occupation. A totalof 1,584 data points were collected, where each point includeda unique combination of a measurement of the PVSD oncurve, curve parameters, and driver characteristics.

RESULTS AND ANALYSIS

A detailed statistical analysis was performed on the col-lected data using the computer package SPSS. The objectivesof this analysis were (1) to study the effect of the curve pa-rameters; and (2) to develop regression models for PVSD oncurve. For easy reference, the curve radius, turning direction,and curve configuration will be referred to as RC, TURN, andCONFIG, respectively. The level of significance a used in theanalysis is 5%, unless otherwise stated. To compare two ormore sets of results, two types of statistical tests were used.The first type was the analysis of variance (ANOVA) that cantest the significance of the difference in the mean of two ormore samples and the interaction between two or more param-eters. The second type was nonparametric tests that use thedata of the whole sample rather than the mean and standarddeviation only. Two nonparametric tests were used: (1) TheMann-Whitney test for comparing two samples; and (2) theKruskal-Wallis test for comparing n samples. In both tests, theconclusion of accepting or rejecting the hypothesis that thesamples’ means are equal is based on testing the ranking ofthe samples data points. Thus, the nonparametric tests mightbe better if a value other than the mean (e.g., the 85th per-centile value) is of concern. A detailed description of the AN-OVA and nonparametric tests can be found in the statisticstextbooks (Romano 1977).

Effect of Curve Parameters

A visual comparison between the results showed that therequired PVSD on curve generally increases as the curve ra-dius increases and if a spiral curve is used. As for the turningdirection, the values for right- and left-turn curves were gen-erally close. To study the significance of the effect of the curveparameters, the ANOVA test for the effect of all parameterstogether was performed. The test showed the RC and CONFIGhave a significant effect on S2, and the difference in S2 due toTURN is insignificant (Table 1). That is, S2 depends on RCand CONFIG but is independent of TURN. The two-way in-teraction shows that the difference in S2 due to either RC orCONFIG is independent of each other. Since the deflectionangle corresponding to a distance on the horizontal curve de-creases as the curve radius increases, it has an opposite trendof S2. Therefore, it was hypothesized that if S2 is convertedinto a deflection angle d the effect of RC and CONFIG mightvanish. Table 1 also shows the ANOVA results for the signif-icance of the effect of RC, CONFIG, and TURN on the de-flection angle d. As shown, the hypothesis is rejected, and ddepends on RC and CONFIG but not on TURN. In addition,the two-way interaction shows a significant effect for the in-teraction between RC and CONFIG, where the change in ddue to either RC or CONFIG depends on the other parameter.

To confirm the finding that RC has a significant effect onS2 and d, the nonparametric Kruskal-Wallis test was used tocheck the null hypothesis that all samples corresponding to the10 curve radii (with the same TURN and CONFIG) belong tothe same population. For the four combinations of TURN andCONFIG, the calculated values of x2 were high enough toreject the null hypothesis (a9 < 0.05). Also, the nonparametric

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FIG. 5. Scatter Diagram of Deflection Angle Corresponding toPVSD on Curve

TABLE 1. ANOVA Results of Effect of Curve Parametersa

Curveparameter

(1)

PVSD on Curve S2

F b

(2)a9c

(3)

Deflection d

F b

(4)a9c

(5)

(a) Main effects

Combined 268.541 0.000 749.936 0.000RC 146.911 0.000 808.922 0.000CONFIG 1,611.515 0.000 988.225 0.000TURN 0.483 0.487 1.963 0.161

(b) Two-way interactions

Combined 1.452 0.094 41.600 0.000RC ?CONFIG 1.466 0.155 86.441 0.000RC ?TURN 0.895 0.529 0.931 0.497CONFIG ?TURN 6.166 0.013 3.287 0.070

aResults are calculated based on multiway interaction among curveparameters.

bCalculated value of F-statistic.cLevel of significance corresponding to calculated value of F. Effect

is significant if a9 < 0.05.

Mann-Whitney test was performed to confirm the significanteffect of TURN and CONFIG. The first null hypothesis wasthat the results for right- and left-turn curves belong to thesame population if all other parameters (RC and CONFIG) arethe same. The calculated values of the U-statistic were lowenough to accept the null hypothesis for 16 cases out of the20 combinations of CONFIG and RC (a9 > 0.05). The secondnull hypothesis was that the results for simple and spiraledcurves belong to the same population if all other parameters(RC and TURN) are the same. The calculated values of theU-statistic were high enough to reject the null hypothesis forall 20 combinations of TURN and RC (a9 < 0.05). The samelast finding was valid when right- and left-turn curves werecombined together, where a9 < 0.05 for all 10 combinationsof RC.

Regression Models

Regression models were developed to predict the requiredPVSD on curve. According to the common practice of high-way geometric design, the sample’s 85th percentile value wasselected as the required PVSD on curve S2 (or the correspond-ing deflection angle d) rather than the sample’s average. Be-fore the regression analysis was performed, the scatter diagram



TABLE 2. Summary of Estimated Regression Models

Dependentvariable

(1)

Independentvariables

(2)

Regression Results

Coefficient(3)

t(4)

a9(5)

ANOVA

F(6)

a9(7)

Coefficient ofdetermination r 2

(8)

S2 (m) const. 61.506 11.937 0.000 21.187 0.000 0.714R 0.00888 6.116 0.040 21.187 0.000 0.714Ga 23.480 2.230 0.000 21.187 0.000 0.714

const. 80.932 15.202 0.000 20.532 0.000 0.7071/R 28,788.413 22.118 0.049 20.532 0.000 0.707Ga 23.480 6.048 0.000 20.532 0.000 0.707

const. 4.086 0.135 0.894 21.192 0.000 0.714log R 22.397 2.230 0.039 21.192 0.000 0.714Ga 23.480 6.116 0.000 21.192 0.000 0.714

const. 81.614 15.184 0.000 21.188 0.000 0.7142R/1,000e 228.000 22.230 0.040 21.188 0.000 0.714

Ga 23.480 6.116 0.000 21.188 0.000 0.714

d (degrees) const. 6.858 14.629 0.000 44.153 0.000 0.710R 20.00259 26.645 0.000 44.153 0.000 0.710Ga —b —b —b 44.153 0.000 0.710

const. 1.337 3.035 0.007 34.165 0.000 0.8011/R 2,740.463 7.982 0.000 34.165 0.000 0.801Ga 20.690 22.148 0.046 34.165 0.000 0.801

const. 24.601 9.968 0.000 36.366 0.000 0.811log R 26.751 28.239 0.000 36.366 0.000 0.811Ga 20.690 22.203 0.042 36.366 0.000 0.811

const. 1.228 2.811 0.012 36.715 0.000 0.8122R/1,000e 8.450 8.279 0.000 36.715 0.000 0.812

Ga 20.690 22.211 0.041 36.715 0.000 0.812aG = 0 for simple curve and G = 1 for spiraled curve.bExcluded as a9 = 0.065.

of the required values of S2 and d were plotted against thecurve radius to determine the possible trend between S2 or dand the curve radius R. No clear trend was noted between S2

and R, whereas the trend between d and R was close to thelogarithmic (log R) or the inverse of exponential (e2R) shape(Fig. 5).

A number of stepwise linear regression attempts were per-formed for S2 and d as dependent variables and R, CONFIG,and TURN as independent variables, where index values wereassigned to CONFIG (G = 0 for simple curve, and G = 1 forspiraled curve) and TURN (T = 0 for right-turn, and T = 1 forleft-turn). In these attempts, R was expressed as R, 1/R, log R,and . The parameter TURN was always found to be2R/1,000einsignificant and was removed from the regression equations,confirming the previous finding that the turning direction doesnot affect the value of the required PVSD on curve. Therefore,S2 and d were taken as the 85th percentile value of the mea-surements for right- and left-turn curves combined, and theparameter TURN was no longer considered.

The resulting regression models including the dependentvariable, independent variables, their coefficients, and the cal-culated value of t-statistic for each independent variable withthe corresponding level of significance a9 are summarized inTable 2. The t-statistic and the corresponding a9 test the nullhypothesis that the coefficient of a specific independent vari-able is equal to zero. If a9 is greater than the assumed levelof significance (5%), the null hypothesis is accepted and thevariable is excluded from the regression model. In addition,the table shows the ANOVA results of testing the null hypoth-esis that the values of the independent variable belong to thesame population (regression model is insignificant) as well asthe coefficient of determination r 2.

As noted in Table 2, the nonlinear regression of S2 basedon 1/R, log R, and instead of R does not improve r 2.2R/1,000eOn the other hand, the nonlinear regression of d improves r 2.Therefore, the following formulas for S2 and d are suggested:



2S = 61.506 1 0.00888R 1 23.480G, r = 0.714 (5)2

2d = 24.601 2 6.751 log R 2 0.690G, r = 0.811 (6)

where R = radius of horizontal curve (m); and G = 0 for simplecurves and 1 for spiraled curves.

Clearly, the model of (6) has the advantage that its r 2 ishigher than that of (5). Moreover, by calculating the deflectionangle and converting it into a distance on the curve, the modelof (6) accounts indirectly for the spiral curve length and thespiral parameter. Both factors could not be considered in theanalysis as the spiral parameter was taken as the design valuecorresponding to each curve radius. Thus, the nonlinear modelof (6) might be used for calculating d and then S2 as follows.

For simple curve:

dpS = R (7)2 180

For spiraled curve:

dpA , S < l2 sÎ90

S = (8)2 H dp lsl 1 2 R, S > ls 2 s*S D180 2R

where A = spiral parameter (m); and ls = spiral length (m).As an illustration of using the presented models in estimat-

ing the required PVSD on curve, Fig. 6 shows typical valuesof S2 based on the nonlinear regression of (6). From the figure,it is evident that S2 for simple curves increases as R increases.However, for spiraled curves, as R increases, the spiral lengthls corresponding to a specific spiral parameter A decreases.Although a larger R requires a larger S2, the smaller ls requiresa smaller S2. The net result might be smaller S2 for larger R.For the same R of spiraled curves, S2 increases as A increases.

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FIG. 6. Typical Required PVSD on Curve

This can be attributed to the resulting increase of ls. Thus, thefigure clearly shows that the nonlinear regression of (6) hasaccounted for the value of A and hence ls.

Two points related to spiraled curves can be noted in Fig.6. First, for large R and small A, the required S2 resulting fromthe regression can be smaller than that for the simple curve.However, since the spiral curve increases the flatness of thehorizontal curve, S2 for spiraled curves should always behigher than that for simple curves. Thus, the values corre-sponding to simple curve should be a logical minimum limitof required S2. Second, for small R and large A, the requiredS2 is considerably large and can even be greater than ls. Forexample, for R = 400 m and A = 350 m, ls is 306.3 m and S2

is 164.7 m. As stated earlier, the large value of required S2

results from the long spiral curve. However, it is not possiblefor the driver to perceive the 400-m curve by seeing only abouthalf of the spiral curve while not seeing the circular curve.Thus, the required S1 and S2 should be adjusted to enable thedriver to recognize the existence of the curve and decelerateslightly, while estimating the circular curve radius and the fulldeceleration process can take place as the vehicle travels onthe spiral. It should be noted that these two cases of large Rwith small A and small R with large A are not recommendedin the design and were not included in the experiment.

PRELIMINARY DESIGN VALUES

Preliminary design values of the required PVSD were es-tablished in Table 3 in terms of the two components S1 and S2

using the models presented earlier. The table is based on thefollowing assumptions:

1. The curve operating speed VC (km/h) is calculated basedon the curve radius R (m), using the model by Lammand Choueiri (1987) as

3188.9V = 94.378 2 (9)c

R

2. Substituting R → ` in (9) for the tangent operating speedVT is 94.4 km/h.

3. The PR time equals 2.5 s similar to that of SSD (A policy1994).

4. The deceleration rate is 0.85 m/s2 (Collins and Krammes1996).

5. The value of S2 is calculated using the nonlinear regres-sion of (6).

6. The required S2 for simple curves is a minimum limit ofS2 for spiraled curves.

7. If ls is greater than S2, the PVSD requirements are ad-justed through an iteration process so that both S1 and S2

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TABLE 3. Preliminary Design Values of Required PVSD

R(m)(1)

REQUIRED PVSD (m)a

Simple Curve

S1

(2)S2

(3)

Spiraled Curve

A = 100 m

S1

(4)S2

(5)

A = 200 m

S1

(6)S2

(7)

A = 300 m

S1

(8)S2

(9)

400 131 50 107 57 66 93c 66 119c

600 110 62 94 63 66 88 66 119c

800 99 70 87 70b 66 86 66 1171,000 93 76 83 76b 66 84 66 1091,200 88 80 80 80b 66 83 66 1031,400 85 83 78 83b 66 83b 66 981,600 83 83 77 83b 66 83b 66 921,800 81 83 76 83b 66 83b 66 862,000 80 81 75 81b 66 81b 66 81b

aValues are rounded to next integer.bMinimum value.cMaximum value.

correspond to a radius that requires a spiral length equalto S2. Further recognition of the sharper curvature andthe required deceleration are assumed to take place whiletraveling on the spiral curve. Still, the PR distance is aminimum value of S1 and depends only on VT.

CONCLUSIONS

This paper has presented an analytical procedure for deter-mining the required PVSD. The PVSD is an important com-ponent of the sight distance red zones concept that addressessight distance inadequacy resulting from introducing a hori-zontal curve near the high point of a crest vertical curve orthe low point of a sag vertical curve. The required PVSD isdivided into two components: PVSD on tangent and PVSD oncurve. The first component can be estimated based on realisticvalues of PR time, deceleration rate, and operating speeds onthe tangent and the curve, whereas the second component wasmodeled based on the experimental data.

Two modeling procedures were used to determine the re-quired PVSD on curve: physical modeling and computer an-imation. Randomly selected drivers participated in the exper-iment, and both research teams and experiment participantsagreed that computer animation provided a better modelingenvironment and more comfortable setting than the physicalmodeling. Therefore, the results here were based on the com-puter animation. The horizontal curve radius and configurationwere found to have a significant effect on the required PVSDon curve and the corresponding deflection angle. Generally,the value of the required PVSD on curve increases as the curveradius increases or when a spiral curve is used. The turningdirection was found to have an insignificant effect on the re-sults.

Regression models were developed to estimate the value ofthe required PVSD on curve and the corresponding deflectionangle. A nonlinear model for the deflection angle was foundto have a high coefficient of determination relative to othermodels for the PVSD on curve. The developed models areapplicable only to the range of horizontal curve radii used inthe experiment (500–2,000 m). Although flatter curves shouldnot constitute a safety concern, sharper curves can be encoun-tered on highways with lower design speeds. Also, because asingle value of the spiral parameter was used with each curveradius, the effect of the spiral parameter (and spiral length)could not be studied. Therefore, expanding the experiment byusing more curve radii and varying the spiral parameter iswarranted.

It should be noted that although models are already avail-able, to estimate the operating speed on horizontal curves, they


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are limited only to separate horizontal curves on level grades(Gibreel et al. 1999). Because PVSD red zones, by definition,will develop due to combined horizontal and vertical curves,existing operating speed models may be inaccurate. Therefore,there is a need to develop 3D operating speed models appli-cable to combined alignments. In addition, it is recommendedthat the effect of other curve parameters, such as supereleva-tion rate, on the required PVSD on curve should be examined.The perception of the horizontal curve when combined with acrest or a sag vertical curve should also be investigated, as theanimations used in this study involved only a horizontal curveon a flat vertical tangent. Finally, a close and detailed inves-tigation of the behavior of drivers as they approach horizontalcurves with limited PVSD should be studied.

ACKNOWLEDGMENTS

The writers are grateful to D. Schutte of MTO, Thunder Bay District,for this thoughtful comments and technical assistance. The financial sup-port by the Natural Sciences and Engineering Research Council of Can-ada is gratefully acknowledged.

APPENDIX. REFERENCES

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Collins, K. M., and Krammes, R. A. (1996). ‘‘Preliminary validation ofa speed-profile model for design consistency evaluation.’’ Transp. Res.Rec. 1523, Transportation Research Board, Washington, D.C., 11–21.

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Gibreel, G. M., Easa, S. M., Hassan, Y., and El-Dimeery, I. A. (1999).‘‘State of the art of highway geometric design consistency.’’ J. Transp.Engrg., ASCE, 125(4), 305–313.

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Hassan, Y., Easa, S. M., and Abd El Halim, A. O. (1998a). ‘‘State-of-the-art of three-dimensional highway geometric design.’’ Can. J. Civ.Engrg., Ottawa, 25(3), 500–511.

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Modeling of Required Preview Sight Distance

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