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17 two-way rural highways. They used the model to create a pro- that influences driver behavior and has the most potential cedure for designing horizontal curves that would accommo- for crashes." They added that "research has indicated that date vehicles transitioning from high speeds to a stop. Based the average accident rate for horizontal curves is about three on speed profile data from 15 study sites in Nebraska, the times the average accident rate for highway tangents and researchers concluded that posted speed, median type, pres- the average run-off-the-road crash rate for highway curves ence of rumble bars, roadway surface condition, and degree is about four times that of highway tangents." They stated of rutting did not significantly affect the vehicle speed profiles that many curve-related crashes were the result of drivers at these sites at a 95% confidence level. They also concluded approaching and entering the curve at a speed that was too that the intercepts of the regression lines for approaches with fast for the alignment. A study of driver behavior and errors and without horizontal curves were significantly different in on a selection of horizontal curves led them to conclude the the case of heavy vehicles. The speed of heavy vehicles on tan- following: gent approaches was generally about 8 mph higher than on sites that exhibited horizontal curvature, although the rate of decel- Drivers approaching curves routinely exceeded the eration remained almost the same until vehicles were near the posted speed limit as well as the posted advisory speed, stop. Passenger cars exhibited no statistically significant differ- where applicable. ence between curved and tangent alignments. Researchers used Drivers had more errors at curves where they had lim- the results of the study to develop a procedure for determining ited or no visibility of the curves when the traffic control the minimum curve radius appropriate for a roadway alignment devices (TCDs) were first visible. approaching a stop ensuring that (1) the visual expectations of Drivers made more errors on horizontal curves that the driver were met, (2) the comfort of the passengers within were adjacent to vertical curves, particularly crests that the vehicle was optimized, (3) the curve design used a simple obscured a downstream horizontal curve. curve with no spirals, (4) the vehicle speed within the limits of There were increased errors when curves were com- the curve were reasonable, (5) sufficient braking distance to the bined with other elements, especially intersections. stop was available, and (6) deceleration rates were reasonable. Many design standards recommend the use of spiral curves Cafiso et al. (2005b) sought to determine design inconsis- in the transition design. Perco (2006) conducted a study to tencies on existing two-lane rural roads in Italy with the use evaluate effects of a long spiral transition on the driver's of actual driving behavior and to verify their agreement with curve perception and safety. He analyzed driving paths on a consistency evaluation model. They developed a data col- 12 transitions with and without spiral curves, and concluded lection method using a sample of test drivers operating an that the results confirmed a negative effect of excessive spiral instrumented vehicle on a pre-determined route. From this length on driver behavior. His analysis results showed that study they concluded the following: the most desirable spiral length, which offered advantages in comparison with a tangent-to-curve transition, was equal to A coordinate sequence of curves did not produce an the distance traveled during the steering time. He developed a unexpected driving event even if short bending radii model to estimate the desirable spiral length for transitions of were adopted. sharp horizontal curves on two-lane rural roads, based on the Geometric inconsistency produced by a sharp curve data collected in three studies. Starting from the radius of the following a long tangent produced tense driving behav- impending curve, the model calculated the desirable spiral ior, as observed on curves with radii of 120 m (394 ft) length and provided a description of actual driver behavior, as and 80 m (262 ft). observed in field surveys. Perco concluded that the choice of Driving inconsistencies were highlighted by high- the spiral length based on this model was useful because the speed gradients of about 2 m/s2 (6.5 ft/s2), transversal estimated length was consistent with the real distance traveled accelerations of 0.3 g, and local maximum curvatures by the vehicle during the steering action, which ensured opti- of the car path higher than those required by horizontal mal operating conditions for drivers. alignment. These values of deceleration reached with a light braking action were higher than the 0.80 to 0.85 m/s2 (2.62 to 2.79 ft/s2) generally assumed with regard to Vertical Alignment driving behavior in speed profile diagrams. Maneuvers were caused by the driver's need to suddenly Hassan (2004) described the development of two models to correct his or her driving behavior owing to an unexpected determine the required SSD on crest and sag vertical curves. alignment and could produce a dangerous situation if bad By comparing profiles of available SSD and required SSD on pavement conditions or unexpected events occur. examples of vertical curves, Hassan concluded that current The lack of transition curves was also a contributing North American design practices might yield segments of the factor in geometric inconsistency. vertical curve where the driver's view is constrained to a dis- tance shorter than the required SSD. He developed new models Lyles and Taylor (2006) stated that, "historically, the hor- based on longitudinal friction and on acceleration, then devel- izontal curve is the most critical geometric design element oped an alternative design procedure based on the models,

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18 Horizontal Plane left point Hh lane width center point right point S g2 (actual grade h is immaterial) =Hv h + S tan g1 L (L>S) Vertical Plane FIGURE 3 Illustration of headlamp analysis approach (Hawkins and Gogula 2008). which he used to determine recommendations for minimum lamps, as illustrated in Figure 3. The results of their analysis lengths of crest and sag vertical curves. Depending on the indicated that modern headlamps provided significantly less approach grade, the new values of minimum curve length light above the horizontal than sealed-beam headlamps, indi- could be greater than or less than values obtained through cating a potential need to modify the design equations for conventional design procedures; design aids were therefore sag vertical curves. According to their theoretical analysis, provided in tabular form to facilitate use by designers. the upward divergent headlamp angle used in the sag curve design equation should be reduced from 1 to between 0.75 Torbic et al. (2005) conducted a study to determine the and 0.90. They stated that results from field analysis indi- distribution of truck weight/power ratios in the current truck cated a significant difference in illuminance levels from the fleet in several regions of the United States, and compare theoretical analysis, but also indicated a need to reduce the them with the 120 kg/kW (200 lb/hp) value recommended headlamp angle used in sag curve design. in the 2001 Green Book. The researchers collected data on truck crawl speeds at locations in California, Colorado, and Easa (2008) developed a single-arc unsymmetrical verti- Pennsylvania and concluded that a "weight/power ratio of cal curve that takes the form of a cubic instead of parabolic 102 to 108 kg/kW (170 to 180 lb/hp) would be appropriate function. The curve has a rate of change in grade that gradu- for freeways in California and Colorado, and a weight/power ally varies between the start and end of the vertical curve, ratio of 126 kg/kW (210 lb/hp) would be more appropriate in which eliminates the sudden change in curvature of tradi- Pennsylvania." They also determined that truck performance tional two-arc unsymmetrical vertical curves. He developed on two-lane highways was sufficiently different from free- sight distance relationships for the new single-arc crest curve, ways to recommend different ratios for those roads: a 108 which established the sight distance profile for the new curve kg/kW (180 lb/hp) design vehicle in Colorado, and 150 to and shows a substantial improvement over the abrupt-type 168 kg/kW (250 to 280 lb/hp) for California and Pennsyl- sight distance profiles of two-arc curves. Included in the vania. According to the researchers, all of these ratios rep- description of the single-arc curve characteristics are length resented the 85th percentile of the truck population that was requirements to satisfy AASHTO stopping, passing, and deci- studied; therefore, most of the truck population performed sion sight distance guidelines. substantially better. The Highway Safety Manual (AASHTO 2010) provides Motivated by changes in headlamp design in recent guidance on the effect of grades on expected safety of road- decades, Hawkins and Gogula (2008) reviewed existing sag way segments. The base condition for grade is a generally curve design criteria to determine if revisions to the design level roadway. Table 5 presents the crash modification factors procedure were appropriate. They compared theoretical and (CMFs) for grades based on an analysis of rural two-lane, field measurements of the levels of illuminance falling across two-way highway grades in Utah. The CMFs in the table are the road surface, provided by sealed-beam and modern head- applied to each individual grade segment on the roadway being Table 5 Crash Modification Factors for Grade of Roadway Segments Approximate Grade Level Grade Moderate Terrain Steep Terrain (%) (3%) (3% 6%) CMF 1.00 1.10 1.16 Source: AASHTO (2010).