DESIGNING HORIZONTAL CURVES

DESIGNING HORIZONTAL CURVES FOR LOW-SPEED ENVIRONMENTS

J. L. Gattis, Ph.D., P.E., B. Finley Vinson III,

Mack-Blackwell National Rural Transportation Study Center, and

Lynette K. Duncan, Center for Statistical Consulting,

University of Arkansas

CHAPTER 1

INTRODUCTION

This project was a pilot study to explore alternative criteria for the geometric design of low-speed urban horizontal curves. The researchers collected data and then developed alternative low-speed urban horizontal curve design paradigms. The study considered factors such as curve radius, pavement cross slope, vehicle speed within the curve, and vehicle speed in advance of the curve. The methods derived and values found were compared with the practices in the current American Association of State Highway and Transportation Officials (AASHTO) A Policy on Geometric Design of Highways and Streets (Green Book). Following Green Book (2001, p. 72) practice, low-speed was defined as 70 kilometers per hour (km/h), or 45 miles per hour (mph), or less.

BACKGROUND

Many factors are considered when designing a horizontal curve. One of these factors is the minimum acceptable radius of the curve, or “what is the smallest acceptable radius”? The minimum radius of a curve is normally equal to minimum radius that allows the driver to comfortably traverse the curve at the designated design speed.

When a vehicle traverses a curve, the driver evaluates his or her speed with respect to the radius of the curve. All other things being equal, the smaller the radius, the more likely it is that a driver will choose a lower speed. Two other factors affect this relationship between curve radius and speed: side friction and cross slope (or superelevation).

Side Friction

Side friction is the friction force created by the contact between a vehicle’s tires and the road. It is this force that counteracts the centrifugal force and keeps the vehicle on the road. A coefficient called side friction factor (f ) is used to quantify this force. The side friction factor is a unitless value and is equal to the friction force required by the vehicle divided by the component of the vehicle’s weight that is perpendicular to the pavement surface.

Observers have noted that drivers typically do not operate at the speed at which side slip is impending, but rather operate vehicles in curves at speeds well below the threshold of impending side slip. The friction factor used for design of horizontal curves is based on this threshold of driver discomfort rather than the point of impending slip of the vehicle. Tables in the 2001 Green Book list, for a given velocity, the design side friction factors above which a driver is no longer comfortable traversing a curve (pp. 145, 197, 201).

Cross Slope

Cross slope is the slope of the pavement surface perpendicular to the direction of travel. The superelevation of a road refers to a cross slope that has been modified from its normal “shape”, to aid the vehicle in negotiating the curve successfully.

The Green Book suggests that superelevation not be used in low-speed urban areas. This means that the design criteria for low-speed urban curves call for a vehicle’s centrifugal force to be completely counteracted by side friction until the maximum side friction value has been reached; only then would superelevation be used. This method is chosen because in many urban environments, superelvation can create a number of aesthetic and operational problems. The maximum value assumed for safe side friction factors is pivotal in the design of low-speed urban curves.

Relating Factors to Design a Curve

The current edition of the AASHTO Green Book contains an equation (pp. 133 ff.) that can be used to calculate the minimum radius of a curve, based on the design speed. This equation relates the velocity of the vehicle (V), the curve radius (R), side friction (f), and superelevation or cross slope (e).

(metric)

(standard)

The “friction factors” are listed in tables in Chapter 3 of the Green Book. One table contains friction factors for urban, low-speed situations, while another table applies to rural and high-speed urban situations.

Alternative Method

The second method for assessing the speed suitable for a given curve is that of using a ball bank indicator. A ball bank indicator is a device that consists of a steel ball and some damping fluid located inside of a sealed glass tube. Both ends of the tube are curved upward so that the position of the ball in the tube can be converted to an effective angle in degrees (^o). The device is used by mounting it inside the car and measuring the maximum angle that the steel ball reaches while the vehicle is within the curve. The ball bank indicator has been used to determine the advisory speeds posted below warning signs in advance of curves.

GOALS OF THIS PROJECT

The maximum side friction factors listed in the current Green Book are based upon research that was performed decades ago. Given the fact that vehicle components (i.e., suspension components) are constantly being improved, drivers may be willing to accept much higher side friction factors than those listed in the Green Book. Furthermore, the emphasis of much of the research was on high-speed environments such as highways and arterials, and not on low-speed urban situations.

This research was conducted to reconsider the side friction factors for low-speed, urban, horizontal curve design. In addition, alternative design approaches were also investigated.

Design Speed Concept

The selection of any side friction factor reflects implicit assumptions about design. Many design decisions in recent decades have been predicated on an assumed design speed. That is, based on informed experience, the designer identified the speed at which drivers were likely to want to drive the roadway being designed. Then, various design elements such as the horizontal curve radius were selected so that drivers could safely maintain this speed.

But in reality, the combination of different drivers in vehicles with different characteristics and capabilities results in vehicles operating over a range of speeds for any given situation. If a design is created so that almost all drivers can operate at the design speed, then many vehicles will be traveling in excess of the design speed. This philosophy seems appropriate on busy highways where the roadway will at some times be operating near capacity, for if only one vehicle cannot maintain a minimum speed, then traffic flow on the facility may break down. However, on lower speed, lower volume urban roadways, there may be merit to designing so that most vehicles will not exceed a certain speed, and will fall within a desired range of speeds.

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CHAPTER 2

LITERATURE REVIEW

Methods to design horizontal curves for railroads and roadways have been discussed for decades. There is a growing interest in reexamining the design of horizontal curves on roadways.

EARLIER RESEARCH

One of the first resources on the topic of horizontal side friction (Barnett, 1936) assumed a safe side friction factor of 0.16 for all speeds up to and including 60 mph. For speeds above 60 mph, the friction factor decreased by 0.01 for every 5 mph increase above 60 mph. This friction factor was found by determining the safe speed around various curves, where the safe speed was defined as “…the minimum speed at which the centrifugal force, created by the movement of a vehicle around the curve, causes the driver or passenger to feel a side pitch outward.”

Another source (Moyer, 1940) described a different method of determining safe side friction factors. Moyer used the ball bank indicator to determine safe side friction factors. By surveying all of the existing 48 states, he found that most engineers considered a maximum ball bank indicator reading of 10^O to be satisfactory. However, their research indicated that this could lead to unsafe friction factors for speeds above 60 mph due to small path variations or driver error. Furthermore, for speeds below 30 mph, a ball bank angle of 12^O to 14^O was recommended due to the fact that control was easier to maintain at lower speeds. The safe speed values listed were all intended for favorable street conditions. The author, however, expected that drivers would realize the need to lower their speed under wet or icy conditions.

RECENT RESEARCH

The current AASHTO Green Book lists low-speed urban side friction factors ranging from 0.16 to 0.31, depending upon design speed (30 km/h to 70 km/h, or 20 mph to 45 mph, respectively).

In 1983, a study was released (McLean, 1983) that criticized the use of the friction factor as a design criterion. This study noted that the friction factor’s relationship with speed is only valid for vehicles driving at or below the design speed. Furthermore, the study suggested that friction factor had no direct influence on a driver’s curve speed.

A recent work (Mudry, 1999) recognized the lack of research conducted in a low-speed environment, and conducted a study of observed friction factors on low-speed urban curves. The study used twenty-one sites, each with between 70 and 120 vehicle observations. Using a magnetic speed measuring device, vehicle speeds were measured at the point of curve (PC), the midpoint, and the point of tangency (PT). It was assumed that the 85th percentile side friction factors were representative of driver comfort. Mudry found that in most cases (56 out of the 63 test sites) the 85th percentile friction factor exceeded the AASHTO friction factor design values for low-speed urban streets.

Other recent literature also recognized the shortcomings of the current AASHTO standards. (Bonneson, 1999) found a correlation between side friction factor and vehicle approach speed, indicating that drivers will accept higher side friction factors on curves with higher speed reductions. This suggests that current Green Book standards may be overly conservative. Bonneson also referred to the Green Book background literature, pointing out that there was little agreement upon what driver reaction constituted a maximum side friction factor. This maximum friction factor has definitions ranging from the point at which drivers become aware that they are on a curve, to the point of impending slip.

In a recent National Cooperative Highway Research Program (NCHRP) report (Bonneson 2000), Bonneson formulated a new model for side friction factor recognizing the following phenomena. There is a decrease in side friction demand with an increase in approach speed, and there is an increase in side friction demand with an increase in speed reduction. In developing the model, the testing included a range of approach speeds from 40 km/h (25 mph) to 120 km/h (75 mph). Computer monitored sensors on the pavement were used to determine the vehicle’s speeds. A laser-gun was used, however, when traffic volume was too heavy to install the pavement sensors. The speed, leading headway, following headway, and vehicle classification were recorded. Linear regression analysis was used to arrive at the equation.

where:

f_{D,
95, PC} = maximum design side friction factor

V_α,
95 = 95th percentile approach speed, km/h

dv₉₅ = 95th percentile speed reduction

V_c,95= 95th percentile curve speed

I_TR = indicator variable (1 for turning roadways; 0 otherwise).

The report acknowledged that both the 85th percentile and the 95th percentile values could be reasonable for use in curve design. However, the 95th percentile was recommended for use in developing maximum side friction factors. The reason for this was that side friction factor is dependent upon only one variable (speed). This is in contrast to something such as stopping distance, where many variables (reaction time, deceleration rate, and speed criteria) must all be at or below their worst-case in order for failure to occur. Therefore, failure is more likely to occur in maximum side friction factor design.

Fitzpatrick (2000) measured drivers speeds through a horizontal curve. She found that for most drivers, the speed through the curve reached its nadir somewhere between the half-way and two-thirds point along the length of the curve.

SUMMARY

There is a lack of current information on the design of low-speed urban horizontal curves. The Green Book assumes maximum side friction factors that are based on research from the 1940’s. Even Moyer recognized in 1940 that newer cars could take curves faster than old cars. Interestingly enough, he was able to detect a noticeable difference between cars with only a one-year age difference. One can only imagine how this difference might be detected for cars of a fifty-year age difference.

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CHAPTER 3

SELECTING AND SURVEYING TEST SITES

To conduct the field data collection, suitable data collection sites had to first be identified. Based on a knowledge of the street designs in Fayetteville, Fort Smith, Little Rock, and North Little Rock, and input from city engineers, potential study sites were nominated. Subsequent field inspections and surveys helped determine which sites were suitable for collecting speed data on the horizontal curves.

CRITERIA FOR A SUITABLE TEST SITE

In order for a site to be suitable, it was desirable to have the following attributes.

∙ Be in a low-speed environment

∙ Have moderate traffic volume (not too heavy or too light)

∙ Have a suitable place to position data collectors (either have on-street parking or sufficient right-of-way width)

∙ Have an unobstructed line of sight from observers to vehicles

∙ Be a true circular curve (undistorted)

∙ Have little variation in cross slope within the curve

∙ Have a relatively flat grade

∙ Be preceded by a long tangent

∙ Not immediately follow a major intersection

∙ Have vehicles traveling on the inside of the curve, to eliminate the possibility of cutting the curve

IDENTIFYING POSSIBLE TEST SITES

A number of curves were mentioned as candidate study sites, but were removed from consideration after a cursory examination of relevant factors, such as traffic volume, proximity to nearby intersections, pavement width, or abutting land use. After eliminating the curves with obvious shortcomings, a total of eleven locations remained to be considered as possible test sites. These curves are listed in Table 3-1.

Eliminating Unsuitable Sites

Many of the curves listed in Table 3-1 were eliminated due to physical imperfections or problems that were apparent once they had been visited and closely inspected. The first location that was

Table 3-1 Sites Considered for Study

Street name City

Breckenridge Drive Little Rock

Brooken Hill Drive Fort Smith

Cliff Drive Fort Smith

Gary-Independence Fort Smith

Indian Trail-Richwood Little Rock

Kavanaugh Boulevard Little Rock

Mabelvale Pike Little Rock

Mall-Shiloh Fayetteville

Meandering Way Fort Smith

Pine Valley Road Little Rock

Salem Road Fayetteville

eliminated was the curve on Cliff Drive east of Gary Street. Initial measurements revealed that curve had dramatically different tangent lengths, and was more of a spiral than a true circular curve.

Two separate curves at the connection of Gary Street and Independence Street were eliminated based on visual observation of the road surface. The road surfaces were rough and the cross slopes appeared to fluctuate greatly.

The curve at the junction of Indian Trail and Richwood Road in Little Rock was eliminated due to some geometric imperfections. The curve’s PT was very difficult to locate due to the fact that the center pavement markings did not follow the centerline of the road. This problem was aggravated by the fact that the opposite sides of the street seemed to begin curving at different points. After surveying this curve, the tangent lengths appeared to be different, and the curve non-symmetrical.

The curve at Kavanaugh Boulevard just south of Cantrell Road appeared to be suitable at first. After surveying the curve, however, it was discovered that the cross slope fluctuated dramatically between the beginning and ending points of the curve.

One curve on Mabelvale Pike was eliminated because inspection revealed that pavement width and cross slope varied, and the curve was not symmetrical.

Two curves on Meandering Way were eliminated. Inspection of a curve near the Village Road intersection revealed that the cross slope fluctuated near a drain opening on the curve’s outside edge. A long curve near Rogers Avenue contained a pavement marking transition, with the number of lanes changing from two to three. Therefore, the curb radius was not the same as the pavement marking radius.

Two curves were considered on Salem Road, both of which are located between Vassar Street and Mica Street. One of these curves, the south curve, was eliminated because of a grade at the south end of the curve.

Suitable Study Sites

Table 3-2 lists the five curves that were selected to be used as study sites in the data collection process.

Table 3-2 Suitable Study Sites

Street name City

Breckenridge Drive Little Rock

Brooken Hill Drive Fort Smith

Mall-Shiloh Fayetteville

Pine Valley Road Little Rock

Salem Road Fayetteville

The curve on Breckenridge Drive located near Danbury Circle had one undesirable trait -- traffic to be measured was on the outside of the curve -- but was otherwise suitable.

The curve on Brooken Hill near Londonderry Road had an acceptable length and the cross slope appeared to be relatively constant.

The curve at the connection of Shiloh Drive and Mall Avenue was constructed less than a year before our examination of it. This made it a desirable location because the geometry of the curve was very symmetrical.

The curve at Pine Valley Road south of Dalewood Street was probably the sharpest curve that was considered as a test site. Despite the old appearance of the pavement, the curve proved to be fairly symmetrical and was chosen as a test site.

The cross slope of the north curve on Salem Road did fluctuate some; however, it was minimal in comparison to some of the curves that were eliminated in the previous discussion.

GENERAL SURVEYING PROCEDURE

Three types of information were needed from each of the curves: the tangent length (T), the central angle (Δ), and the elevation of the roadway surface at multiple points. The tangent length and the central angle were used to compute the radius (R) of the curve. The elevation points were used to compute the cross slope (e) of the curve. Using the equation from the Green Book, R, e, and measured vehicle velocities (V) were used to calculate the side friction factors (f ) experienced on each curve at the measured velocities. Table 3-3 presents a summary of the surveyed and calculated curve data.

Table 3-3 Summary of Curve Data

Street name Radius Δ e Length

Centerline Lane,

per plans calculated avg min calculated

(m) (ft) (m) (ft) (º) (%) (%) (m) (ft)

Breckenridge ? ? 93.1 305.3 46.9 -3.10 -5.35 75 245.3

Brooken Hill 97 319.1 105.5 346.0 45.9 1.63 1.31 87 283.8

Mall-Shiloh 61 200.0 57.9 189.9 67.8 2.69 2.26 71 231.3

Pine Valley ? ? 42.0 137.9 79.2 -4.37 -5.12 56 183.8

Salem 70 229.2 70.2 230.3 59.9 2.87 2.15 75 247.6

NOTES: The Mall-Shiloh radius was adjusted by subtracting 5 ft for half-width of the center two-way left turn lane. At Brooken Hill, the centerline marking differed from the curb alignment. m = meter ft = feet

Locating the Point of Intersection

The first step in surveying the curve was to locate the centerline of the pavement along the tangent to the curve on each end. This was accomplished by measuring the width of the street at various points and then dividing by two in order to find the offset to the centerline from each edge. Lumber crayon was used to mark the centerline of the street at three or more points past each end of the curve. The point of intersection (PI) of the curve, the point where the two tangents intersect, was located by one of two methods.

In most cases, a nylon measuring tape was used to locate the PI by lining the tape up with the points marked along each tangent, then projecting the tangent lines toward the PI. Lumber crayon and chaining pins were used to mark points along the measuring tape at about three foot intervals. The lumber crayon was used for marking on the asphalt surface, and chaining pins were used for marking on soil. After completing this process for both tangents, the PI could be found by stretching a piece of twine between two of the points on each tangent to form an X. The point where the two pieces of twine crossed was the PI.

Another method for locating the PI involved the use of a total station. Because it was impractical, however, to set up a total station in the middle of the street, an offset line was constructed parallel to each tangent, toward the center of the curve. A convenient offset distance (about one-half the width of the street) was chosen and recorded. New points were then marked with lumber crayon at this offset distance from the original points along the tangent line.

Once the new points had been marked on the offset tangent, a total station was set up over one of the points. The point over which the total station set up was the middle point. In other words, there was one point marked along the tangent line on each side of the total station. The total station was aligned with one of the points along the tangent. Then, the sight was turned vertically to look at one of the points on the opposite side. If the line tangent to the street had been perfectly straight, then the second point would have lined up perfectly. This was not the case, so the total station was adjusted to split the difference.

Once the total station had been aligned with the tangent, the line was extended by sighting the total station at various distances and placing chaining pins along the tangent line. These pins were placed in the area where the PI was expected to fall. While only two pins are actually necessary, at least one pin had to be on either side of where the PI actually was. For this reason, four chaining pins were used as a failsafe.

Once the process of identifying the tangent line, marking the offset, and placing the chaining pins had been completed for both tangents, the PI could be identified. A piece of twine was stretched between two of the chaining pins for both of the tangent lines. The point where the two pieces of twine crossed was identified with a final chaining pin as the PI of the curve.

Extending the tangent line with a total station was the more accurate method of locating the PI. It was, however, very tedious and troublesome. It was for this reason that this method was only used on the first curve that was surveyed.

Determining Other Curve Parameters

Once the PI had been located, the next step was to find the location of the point of curvature (PC) and the point of tangency (PT). The location of these two points occurs where the centerline of the curve deviates from the tangent lines. These two points were located by stretching a measuring tape between a point on the tangent line and the PI. The PC and PT could then be visually identified as the points where the centerline (either a striped centerline or the measured center of the pavement) of the curve diverged from the measuring tape, which served as the tangent line. Once these two points were located, the tangent lengths could be measured and recorded as the distance between each of the two points and the PI.

In order to measure the internal angle of the curve, the total station was placed over the PI of the curve. Chaining pins were held at the PC and the PT. Measuring the angle between these two chaining pins revealed the internal angle of the curve. The central angle is equal to the difference between 180^O and the internal angle.

Measuring Relative Elevation

The last step in surveying the curve was to measure the relative elevation of the street at multiple points. The first two obvious points were at the PT and the PC. In addition to these two points, the elevation was measured at intermediate points through the curve. Elevations were measured at each edge of the lane in which data were to be collected. These elevations were computed in the following manner. A level was set up in an area where the entire curve could be seen. A level rod was held over each point of interest and the elevation was recorded. Using the relative change in these elevations, the cross slope of the street could be computed.

Most of the curves that were surveyed closely followed the general procedure outlined above. The surveying procedures for each curve evolved and slightly changed as experience was gained.

SURVEYING INDIVIDUAL SITES

The surveying procedures were tailored to each site. Remarks about the surveying from both used and unused sites follow.

Breckenridge Drive

The curve at Breckenridge Drive followed the general surveying procedures very closely. The elevation measurements, however, were only recorded on the south side of the curve. This was done because a connecting street on the north side created a large hump in the pavement, which would have adversely affected westbound traffic. For this reason, data collection could only be conducted on the eastbound traffic. Based on observations of traffic paths, the outside-of-lane elevations were taken at an eleven-foot offset from the centerline.

Brooken Hill Road

For the curve at Brooken Hill, elevation measurements were taken every 50 feet rather than every 100 feet. This was done for two reasons. First, the curve was relatively short in length. The second reason was to eliminate errors in the cross slope data. Many of the other curves that had been previously surveyed had extremely varying cross slopes. Taking elevation measurements more frequently would give more frequent slope data along the curve, revealing the degree of variability.

Furthermore, elevation measurements were not taken on the north side (i.e., westbound direction) of the street, because of the curve’s proximity to another curve to the east. It was decided that only eastbound traffic data should be collected because the westbound traffic would have already decelerated in the preceding curve. The outside elevations were measured at about one inch from the gutter edge.

Indian Trail -- Richwood

The curve at the junction of Richwood and Indian Trail was one of only two curves that were eliminated from the final site list after the surveying had already taken place. This curve was relatively sharp, which caused a problem. It could be seen from the start that the PI of the curve was located in the middle of a very large shrub, so a more complicated field measurement process was followed.

Once the PC and the PT had been located, the curve elevations were recorded. Elevations were only taken on one side of the curve because of a large incline that could bias the speed of oncoming traffic from the east end of the curve.

Kavanaugh Boulevard

The curve at Kavanaugh Boulevard was the other curve that was eliminated after it was surveyed. A total station was used to measure all of the lengths and angles for this curve.

Three setup points were needed, and these were offset from the centerline. The first and second points were offset from the PC and PT of the curve. Because the PI of the curve was inaccessible, a third setup point, C, that could be seen from both the PT and the PC was set.

Mall Avenue -- Shiloh Drive

Mall Avenue and Shiloh Drive both consist of a through lane in each direction, plus a two-way left turn lane (TWLTL). The curve at the connection of Mall Avenue and Shiloh Drive was the last curve that was surveyed, but it proved to be the easiest. The PC, PT, and PI locations were all clearly marked from an unknown previous survey; these locations were verified in the field. Elevation shots were taken at the inner and outer edges of the inside lane.

Pine Valley

The surveying procedure used at Pine Valley Road was very similar to the general procedures, but there was some difficulty in determining the tangent lengths. As was the case with most of the curves that were surveyed, the geometry did not prove do be exact. As a result of overlays and re-striping, more effort was required to identify the PC and the PT. Elevation readings were taken on the outside of the curve, 10 feet to the right of the centerline.

Salem Road

For the curves on Salem Road, a total station was used to project the tangent lines. It could be seen by looking down the centerline of each tangent that the PI (the point where the two tangents meet) of the south curve would end up behind a fence in a private back yard. For this reason, an offset line was constructed parallel to each tangent, toward the center of the curve, so that the PI would not fall on private property. The maximum offset that was allowed by the width of the street was 13 ft. Lumber crayon was used to mark new points 13 ft closer to the inside of the curve from the original points. Care was taken to ensure that the thirteen-foot offset was measured perpendicular to the centerline of the street.

As mentioned before, this method of extending the tangent lines was difficult and time consuming. It was decided after this measurement that using a nylon tape to project the tangent lines would introduce only second-degree errors and was accurate enough for our purposes. This method was used for the second curve, which was located only a little over 30 m (100 ft) north of the first.

CHAPTER 4

DATA COLLECTION

Equipment used in the collection of field data included Lidar guns, a laptop computer, two-way radios, and camcorders with the ability to display the time to seconds. Before collecting field data during each session, the laptop computer and the camcorder times were synchronized.

GENERAL PROCEDURE

Two people were required to collect data at each curve. One person positioned well in advance of the curve measured speeds of vehicles approaching the curve. The second person was situated to record speeds of vehicles within the curve. Both measured speeds with continuously recording Lidar guns.

In addition to the Lidar gun, video cameras were positioned adjacent to each data collector to record the action of the vehicles. The person in advance of the curve described each approaching vehicle and called out its speed and distance from the observer. The person recording speeds within the curve also called out the speed and distance readings. This was done so that later, while reducing data, the advance speeds for each vehicle could be properly matched with its in-curve readings.

The data collection continued until the one of the following three occurrances: the laptop’s battery power was exhausted, the sun set, or data collection was deemed futile due to low traffic volume. Due to the fact that the videotapes being used were two hours in length, data collection efforts were normally attempted in two-hour increments.

COSINE EFFECT

Cosine effect is the term used to describe the problem that arises if the target is moving at an angle relative to the Lidar gun’s laser beam. If this occurs, then the speed that is displayed by the Lidar gun is not the vehicle’s actual speed. Instead, it is the vehicle’s speed multiplied by the cosine of the angle in effect. The problem of cosine effect was alleviated by ensuring that the angle between the traveling vehicle and the Lidar gun’s laser beam was as small as possible.

DISTANCES FROM BEGINNING OF CURVE

In most cases, the first observer was positioned in the driver’s seat of a parked vehicle on the side of the road in advance of the curve. The distance from the first observer to the beginning of the curve was measured for future reference. This was useful in assuring that the vehicle was neither too close to the observer nor too close to the beginning of the curve when the speed was measured.

It was undesirable to record a speed too close to the observer in order to avoid the cosine effect. Any readings within 80 feet of the observer were later discarded. For a vehicle shifted laterally one lane or about 12 feet from the parked car in which the first observer sat, distances of 75 feet or more yield very small cosine effects, less than the + 1 mph tolerance limits of the Lidar gun.

On the streets being studied, a majority of speeds are less than 35 mph. Roughly 2 seconds of travel distance at 35 mph is 100 feet. To eliminate speed readings that reflect driver’s deceleration in anticipation of the curve immediately ahead, it was decided to eliminate all speeds measured within 100 ft of the beginning of a curve.

DISTANCES FROM POINTS WITHIN THE CURVE

The second person positioned near the end of the curve continuously recorded each vehicle’s speed through the curve. In order to minimize the cosine effect on the Lidar gun reading, the second person was positioned so that they were aligned with the trajectory of the oncoming vehicle when that vehicle was between the one-half and two-thirds of the way through the length of the curve. Thus, vehicles traveling within these two points were aligned at a very small angle with respect to the Lidar gun’s laser beam. (A previously mentioned study had found that most drivers reach their minimum speed between the mid and three-quarter point of the curve.) The data collector’s distance from these two points of interest was measured so that all of the speed readings collected outside of the two points of interest could later be omitted. Distances from the first observer to the beginning of the curve (PC) and from the second observer to the curve mid-point are presented in Table 4-1.

Due to the fact that the second data collector was located in the line of sight of the oncoming vehicles, there was concern that some of the passing motorists would slow down out of curiosity. Preliminary data collections at Salem Road confirmed this suspicion. At each site, various methods of camouflage were used in an attempt to minimize this problem.

THE DATA COLLECTION PROCESS

Despite the similarities between the processes as each site, the data collection procedure evolved from one site to the next as mistakes were made and the process was refined. In the following sections, some of these refinements are listed for each of the respective sites. Figure 4-1 shows data collection in progress.

Table 4-1 Distance from Observer to Curve Reference Point

Data Collection Distance from observers to

Street name Date PC(m) Mid-point(m) PC(ft) Mid-point(ft)

Breckenridge Dec 20, 01 109 43 356 140

Dec 21, 01 107 43 350 140

Brooken Hill Nov 17, 01 128 51 420 167

May 3, 02 131 52 430 170

Mall-Shiloh June 5, 02 181 51 595 168

Pine Valley Dec 20, 01 99 30 325 100

Salem Feb 28, 02 98 50 320 164

Mar 8, 02 97 48 318 157

NOTE: observers may move a few feet during course of data collection

Figure 4-1 Data Collection

Preliminary Data Collection

A preliminary data collection effort was conducted on Salem Road in order to determine the best locations for the data collectors and to identify any potential problems. The first problem that was encountered was equipment failure. After numerous problems with the first data collector’s Laser gun, it was determined that either overhead or underground wires might have caused the failure. Changing the data collector’s position from a private yard to a vehicle on the side of the street alleviated this problem. The next problem that was observed was in camouflaging the data collectors from oncoming traffic. The first data collector did not call as much attention to himself due to the fact that he was not facing oncoming traffic. The second data collector, however, presented a larger problem due to the fact that he was sitting directly in the oncoming vehicle’s line of site. Various methods were used at each site in an attempt to alleviate this problem. The first data collection effort took place at the Salem Road site on November 1, 2001. The first set of data was a failure, however, because the program that was used to capture the speed data had an inadequate buffer size. For this reason, most of the data collected was lost.

After the failure on November 1, a new computer program was created with a much larger buffer size. This new computer program was successfully used in all subsequent data collection efforts. In addition to the larger buffer size, the new program attached a timestamp to each speed and distance reading making it much easier to combine each vehicle’s set of in-curve and advance speeds.

Breckenridge Drive

On December 20 and 21, 2001, data were collected at the Breckenridge site. In an attempt to make the data collector less obvious to passing motorists, the video camera was hidden behind a roadside mailbox, and the data collector sat behind a walker covered by a brown tarpaulin. The walker helped, but did not completely remove the potential problem of motorists slowing down out of curiosity.

Brooken Hill

The walker and the brown tarp were also used as camouflage at the Brooken Hill test site. In contrast to driver’s reactions at the Breckenridge site, however, almost none of the passing drivers seemed to notice the second data collector at all. This was probably due to the data collector’s surroundings, which consisted of overgrown shrubs and large trees that provided shade and concealment.

Mall Avenue -- Shiloh Drive

The traffic volume at the curve on Mall Avenue was heavier than at any of the other test sites. As a result, more data were collected at this site than any of the other sites. The heavy traffic made the data collection process go very smoothly, but there were drawbacks to such heavy volume. There was more platooning of vehicles at this site due to the heavy volume. Every effort was made to exclude any vehicles that were slowing down for traffic in front of them.

Pine Valley Drive

Data were collected at the Pine Valley site on December 21, 2001. The site topography allowed the second data collector to position himself behind a tree in such a way that the vehicle’s speeds were recorded as they were moving away from him. This made the data collector nearly invisible to the passing driver’s.

The traffic volume at this site was very low. A second attempt was made to collect data at this site on May 3, 2002. However, bad weather prevented any data collection on that morning. It was felt that it would be difficult to record a large number of vehicles at this site.

Salem Road

Despite the fact that both data collectors were sitting in parked cars, it was felt that passing drivers sometimes took notice of the second data collector. As a countermeasure, a piece of cardboard was placed over the rear seat window and a coat was draped over the equipment. This helped somewhat.

On February 28, 2002, a police officer pulled into the driveway near the second data collector and asked for an explanation of his suspicious appearance. The officer said that an off-duty police officer had called in and requested that a unit come out take a look. This does suggest that some passers-by may have noticed the second data collector.

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CHAPTER 5

DATA REDUCTION AND ANALYSIS

The data were collected with the objective of producing values for the side friction factor (f). This value represents the imposed friction force divided by the mass of the vehicle perpendicular to the pavement. A simplified equation for f is shown below.

(metric)

(standard)

This equation is dependant upon the radius of the curve (R), the superelevation (e) of the curve, and the velocity (V) of the vehicle. In addition to reevaluating low-speed urban side friction factors, the analysis also consisted of exploring other approaches to relate the speed in a curves to a given radius.

CURVE CALCULATIONS

The curve radii were calculated from the tangent length (T) and the internal angle (Δ) of the curve. Sample calculations for the Salem Road curve are shown below.

The calculation for average tangent length (T_avg) was necessary only if the curve’s two tangent lengths were not the same. This equation gives the radius of the centerline of the road. For each of the five sites, the roadway centerline radius was adjusted to reflect the radius of the center of the lane that traffic speeds were collected in. The final adjusted radii are shown in Table 5-1.

The cross slope of each of the curves was calculated from the simple equation slope = rise / run, where the slope is the cross slope of the roadway within the curve, rise is the change in elevation between the inner and outer lane edges, and the run is the lane width. For all of the roadways studied, the cross slope of the street varied within the curve. For this reason, the average, maximum, and minimum cross slopes for each curve are listed in Table 5-1. The most dramatic fluctuation in cross slope was encountered on Breckenridge Drive. This curve contained one very low measurement in the immediate vicinity of a drainage inlet that lowered the average significantly.

Table 5-1 Curve Radii and Cross Slopes

Street name Radius (m) Radius (ft) Cross Slope

at Center at Center Average Maximum Minimum

of Lane of Lane

Breckenridge Dr 93.1 305.3 -3.10 -1.81 -5.35

Brooken Hill Rd 105.5 346.0 1.63 2.09 1.31

Mall-Shiloh 57.9 189.9 2.69 3.17 2.26

Pine Valley Dr 42.0 137.9 -4.37 -2.05 -5.12

Salem Rd 70.2 230.3 2.87 3.42 2.15

The average cross slope is actually a weighted average. The first and last measurements, which correspond to the beginning and end of the curve, were each weighed by 0.5, and therefore each holds half as much weight in the calculation as the middle measurements. Below is an example calculation for the average cross slope of Breckenridge Drive:

Using the calculated cross slope for each site, the side friction factor for each vehicle was calculated in the following manner.

(metric)

(standard)

A small difference in the two friction factors will result from round-off error.

The Breckenridge site was chosen for the sample calculation above because the curve at this site had the largest fluctuation in cross slope. For this reason, this site can be considered the worst case when considering the fluctuation in friction factor as a result of cross slope variation. The resulting variation in friction factor based on the maximum and minimum cross slope for the Breckenridge site was found to be 0.0002. This relatively small number shows that even the largest recorded variation in cross slope for the five curves tested had a very small effect on the friction factor.

DATA REDUCTION

The raw data that were collected by the in-curve observer at each site were automatically uploaded from the Lidar gun to text file on the laptop computer. Each text file consisted of three columns of continuous data. Figure 5-1 shows a sample of the data from Brooken Hill.

+ 30 223 16:41:19.304

+ 30 223 16:41:19.344

+ 30 223 16:41:19.384

+ 30 223 16:41:19.425

+ 30 223 16:41:19.465

+ 31 211 16:41:19.605

+ 31 211 16:41:19.645

+ 31 211 16:41:19.685

+ 31 211 16:41:19.725

+ 31 199 16:41:19.905

+ 31 199 16:41:19.935

+ 31 199 16:41:19.975

+ 31 187 16:41:20.176

+ 31 187 16:41:20.216

+ 31 176 16:41:20.416

+ 31 176 16:41:20.456

+ 31 176 16:41:20.496

+ 31 165 16:41:20.696

+ 31 165 16:41:20.726

+ 30 154 16:41:20.937

+ 30 154 16:41:20.967

+ 30 143 16:41:21.177

+ 30 143 16:41:21.207

+ 30 132 16:41:21.407

+ 30 132 16:41:21.447

+ 30 121 16:41:21.648

+ 30 121 16:41:21.688

Figure 5-1 Raw Data Sample

The sample above represents one vehicle’s speed, distance and time through a curve. The column on the left represents the speed, in mph, of the vehicle being tracked. The plus sign indicates that a positive speed is being recorded. All of the speeds recorded at Mall-Shiloh and Pine Valley were negative because the observer was recording the vehicles as they were traveling away from him rather than toward him. Speeds at the other three sites were positive because the vehicles were recorded as they were approaching the observer.

The second column of numbers is the distance, in feet, between the observer and the vehicle. As indicated by the numbers above, at the Brooken Hill site this distance decreased as the vehicle approached the observer. This distance increased at the sites where the observer recorded the vehicles as they were departing.

The last column is the timestamp for each individual recording in the format hour:minute:second, where the seconds (sec) are recorded to three decimal places.

Importing and Formatting Data

The first step in reducing the data was to import it into spreadsheet format. By doing this, each of the three columns could be physically separated into columns rather than just spaced. The decimal places in the timestamp were removed because they were deemed unnecessary. A heading was added to each spreadsheet that included the following information for future use: the site name, direction of traffic flow recorded, date of data collection, distance between first observer and curve PC, maximum usable vehicle distance from first observer, minimum usable vehicle distance from first observer, distance from the second observer to the curve’s midpoint, and the distance from the second observer to the curve’s 2/3 point.

After the data had been transferred to spreadsheet format, the rows of data for each vehicle had to be separated from the rows of data for the preceding and succeeding vehicles by inserting a blank row between the data for each vehicle. This was accomplished by observing the distances on each row. Depending on the direction of capture, the distances associated with a given vehicle either increased or decreased steadily. The interface between data rows for two successive vehicles was accompanied by an abrupt discontinuity in the distance readings. Another method for distinguishing between separate vehicles was to simply look for a discontinuity in the timestamp associated with each reading.

Speed

Once the data for each vehicle had been separated, the minimum in-curve speed was determined. This was done in two steps. First, a column was created to display a “Y” for each distance that corresponded to a vehicle being located between midpoint and 2/3 of the distance through the curve, and an “N” for distances outside this range. Then, the minimum speed of each vehicle was found from among those speeds falling within the “1/2" and the “2/3" limits. For the Mall--Shiloh and Pine Valley sites, the less-than and greater-than signs in the formula were reversed because vehicles were moving away from the Lidar gun.

Once the minimum in-curve speed was found for each vehicle, the videotapes were reviewed to identify each vehicle’s speed in advance of the curve. The advance speed of each vehicle was entered in an adjacent column labeled “Upstream Speed”. The upstream speed was subtracted from the minimum in-curve speed. This speed change, usually a negative value, was entered in a column labeled “Slowdown”.

Vehicle Type

In addition to identifying upstream speed, the videotapes were viewed to identify vehicle type. This was entered into a column labeled “Vehicle Type”.

A total of eight vehicle categories were used: bus, coupe, pickup, sedan, SUV, truck, van, and station wagon. All of the vehicle types are self explanatory with the exception of the coupe and the sedan. The sedan was defined as a four-door notchback passenger car (as opposed to the wagon which is a four-door hatchback), and the coupe was defined as a two-door passenger car. For purposes that are explained later, the sedan and the wagon vehicle types were combined in some cases. In this case, the sedans and the wagons were labeled as 4-door passenger cars, and the coupes were labeled as 2-door passenger cars.

Cutting the Curve

A new column labeled “Cut” was inserted to indicate whether or not a vehicle cut the curve by crossing over the center line. This entry was applicable only for the Breckenridge and Pine Valley sites. The reason that this process was only necessary for two of the five sites is that only curves to the left (i.e., “outside” curves) give vehicles the opportunity of cutting the curve. The videotapes for these two sites were watched and the word “Cut” was inserted in this column for any vehicle that cut the curve.

Finalizing the Data

At this point, all of the required information for each vehicle had been entered into the spreadsheets. The next step was to condense all of the data into one complete spreadsheet. In order to do this, all unnecessary data were removed and only the pertinent data were retained. To begin with, each vehicle was represented on the new sheet by only one row of data rather than a string of data as before. Secondly, data for all of the vehicles that were unusable either due to unusable distances, cutting the curve, or inadequate data were removed. Thirdly, all of the columns containing vehicle distances, time, usability, continuous speed, or cut information were removed.

Columns were added to include other important information. Columns labeled “Site”, “Radius”, “Ave e”, and “Min e” were added with the appropriate street name, curve radius, average cross slope, and minimum cross slope respectively next to each vehicle. Two more columns were added, called “Ave f” and “Max f”, to calculate the friction factors that each vehicle experienced. The average friction factor was calculated based upon the average cross slope with the following equation.

(metric)

(standard)

The maximum friction factor was calculated based upon the minimum cross slope.

(metric)

(standard)

DATA ANALYSIS

The formatted data were analyzed to investigate relationships among speed, radius, cross slope, and side friction. A separate analysis was performed to examine the effects of vehicle types.

Speed Comparisons

The first step in the analysis was to observe any trends between the upstream speed, minimum in-curve speed, and speed change at each of the sites. While it may seem intuitive, an obvious trend can be seen between advance speed and in-curve speed on the graph shown in Figure 5-2.

Simply put, this graph shows that the vehicles recorded were likely to travel the curve faster if their upstream speed was higher. Linear regression was used to fit trend lines to each curve. The equations and corresponding coefficients of determination (R²) values for each curve are shown on the graph.

Another trend can be seen between advance speed and in-curve speed on the graph shown in Figure 5-3. This trend shows that the vehicles recorded were found to slow down more for the curve as their upstream speed increased.

Figure 5-2 Advance Speed vs. In-Curve Minimum Speed

Figure 5-3 Advance Speed vs. Speed Change

Figure 5-4 plots the ratio of the speed change divided by the advance speed, vs. advance speed. This plot increased the spread of the data and reduced the R² values.

Figure 5-4 Advance Speed vs. Speed Change in Percent

Figure 5-5 shows the final speed comparison. The data in this graph, which shows minimum speed vs. speed change, appear to be completely random. No obvious trends were apparent.

Figure 5-5 Minimum Speed vs. Speed Change

Next, the minimum in-curve speeds for each site were sorted and plotted according to rank (Figure 5-6). It was determined by observation that the “breaking points” for the distribution of the minimum in-curve speeds were approximately at the 10th and 90th percentile speeds.

Figure 5-6 Sorted Speed Data

Table 5-2 shows the design, average, 90% and 10% in-curve speeds for each site.

Table 5-2 Design Speeds vs. Recorded Speeds

Street Name Design Speed Observed In-Curve Speeds

10% Average Speed 90%

km/h mph km/h mph km/h mph km/h mph

Breckenridge 47.8 29.7 42 26 47.7 29.7 55 34

Brooken Hill 54.3 33.7 42 26 49.6 30.8 55 34

Mall-Shiloh 44.1 27.4 37 23 40.2 25.0 48 30

Pine Valley 35.6 22.1 27 17 32.1 19.9 35 22

Salem 47.5 29.5 39 24 42.9 26.6 50 31

Looking for Trends Among R, V, e, and f

Based on the breaking points found from the sorted speed dispersion, a graph of radius vs. e+f was created to show the 10th percentile, average, and 90th percentile ranges for each of the five sites. This graph is shown in Figure 5-7.

Inspection of Figure 5-7 reveals trends corresponding to the 10% and 90% e+f values for each site. The only exception to the fit of these curves occurs with the Pine Valley site. For some reason, the values of e+f for this site did not fit the trend exhibited by the other sites. One possible explanation may be that vehicles traveled slower on this road relative to the other sites due to the presence of a low rock wall at the edge of the road. The rock wall could have made this curve look much more threatening than the other curves. Another reason for this may be the fact that Pine Valley has the lowest average cross-slope, which lowers the value of e+f.

By disregarding the data from the Pine Valley site, the relationship between radius and e+f can be further analyzed. Figure 5-8 shows another graph of radius vs. e+f , and includes a linear trend line formed using the ranges of e+f values for each site that corresponded to the 85th to 90th percentile range of values. The regression process, using the 85% to 90% values of e+f , produced the following equation, where e_AVG is the average cross slope.

e_AVG + f = - 0.0019 R + 0.425 (metric)

e_AVG + f = - 0.0006 R + 0.423 (standard)

In the metric version, radius R is in meters, and in the standard version, the radius is in feet.

Figure 5-7 Radius vs. e+f

Figure 5-8 Radius vs. e+f (Linear)

Comparison with Green Book f-Values

By comparing the friction factors calculated for each vehicle with the Green Book’s suggested friction factor values, the total number of vehicles that exceeded the Green Book’s recommended f values could be found. In doing this, it was found that an average of 32.3% of the vehicles recorded at all sites exceeded the Green Book’s f values. Table 5-3 gives a breakdown of the proportion of vehicles that exceeded the Green Book’s f values.

Table 5-3 Percentage of Vehicles That Exceeded Green Book f-Values

Street Name Total # Exceeded % Exceeded

Breckenridge 107 57 53.3%

Brooken Hill 116 18 15.5%

Mall-Shiloh 301 117 38.9%

Pine Valley 37 4 10.8%

Salem 152 34 22.4%

Total 713 230 32.3%

This supports the hypothesis that a segment of the driving population will exceed the low-speed urban design speeds computed from the factors in the current Green Book. Also note that the percentage of vehicles exceeding the Green Book friction factors varied noticeably from one site to the next.

A graph of the 90th percentile in-curve minimum speed vs. corresponding friction factor, shown in Figure 5-9, was created to analyze the Green Book’s relationship of maximum friction factor decreasing with increasing vehicle speed. Even though the plot of these four data points (one per site) did not reveal a strong trend, a trend line may still be created in order to observe the relative difference between the Green Book f-values and the calculated f-values corresponding to the 90th percentile in-curve minimum speed. Using this trend line as a reference point, the friction factors derived from the observations seem to be about 0.05 higher than the values in the Green Book. The equation corresponding to this line is:

f SUB{90} ~=~-0.0032`V SUB{90}~+~0.424

(metric)

where V₉₀ is the 90th percentile in-curve minimum speed value in km/h and f₉₀ is friction factor corresponding the 90th percentile speed. In standard units, the equation takes the form of:

Figure 5-9 Curve Speed vs. Friction Factor

f SUB{90} ~=~-0.0052`V SUB{90}~+~0.422

(standard)

where V₉₀ is in mph. Table 5-4 compares minimum radii calculated using the Green Book’s recommended f values with the f₉₀ values calculated from these equations.

Table 5-4 Comparison of f₉₀ Values

METRIC

V AASHTO Observed

(km/h) e f R(m) f₉₀ R (m)

30 0 0.312 23 0.328 22

40 0 0.252 50 0.295 43

50 0 0.214 92 0.263 75

60 0 0.186 152 0.231 123

STANDARD

V AASHTO Observed

(mph) e f R(m) f₉₀ R (m)

20 0 0.300 89 0.319 84

25 0 0.252 165 0.293 142

30 0 0.221 271 0.268 224

35 0 0.197 415 0.242 338

40 0 0.178 599 0.216 494

NOTE: radius R calculated with e = 0

While the preceding equations may be used for comparison purposes, there are obviously not enough data points to ensure that the equation is representative of f₉₀. Nevertheless, this data suggests that for speeds of 60 km/h (40 mph) and below, the minimum radii recommended by the AASTHO Green Book may be overly conservative, in that in some cases a sizeable portion of drivers exceeded them. Furthermore, this does not prove that f₉₀ values are accurate when calculated based on in-curve speed alone. For this reason, the possibility of calculating maximum friction factor based on multiple inputs should be examined.

Regression Analysis Based on All Speeds

Regression analyses were performed with the SAS^® statistical analysis package in an attempt to hypothesize models relating the in-curve minimum speed, radius, and average e. The equations determine the needed radius, based on the independent variables of speed and cross slope. The friction factor was not included in the models because it is a calculated value rather than a physical value that can be directly measured. The first regression analysis performed yielded the following equation:

R~=~0.358~-~236.63`e~+~1.70`V SUB{MIN}

(metric)

where V_MIN is the minimum in-curve speed in km/h, R is the curve radius in m, and e is the average cross slope of the curve, or

R~=~1.326~-~775.91`e~+~8.95`V SUB{MIN}

(standard)

where V_MIN is in mph and R is in ft. The model was developed with the 2-door, 4-door, pickup, sedan, SUV, and van vehicle types at all five curve sites. No trucks were recorded during the data collection. Only two buses were recorded, but they were excluded from the analysis. Therefore, the analysis results reflect the passenger car design vehicle. This first regression analysis had an R² value of only 0.37. The p-value for the regression was less than 0.0001, indicating the model was significant.

The second regression analysis was performed in a manner similar to that of the first one, except vehicle speed in advance of the curve (V_ADV) was added to the model. This analysis had a much better R² value of 0.54 for the following equations. The p-value indicating significance for the regression was less than 0.0001.

R~=~24.60~-~174.61`e~+~2.56`V SUB{MIN}~+~1.16`V SUB{ADV}

(metric)

R~=~80.86~-~572.45`e~+~13.50`V SUB{MIN}~+~6.14`V SUB{ADV}

(standard)

The second regression analysis shows that an estimate of speed within the curve is greatly improved by considering the average speed in advance of the curve in addition to the radius and cross slope of the curve. However, including advance speed in the model increases the complexity of the equation. For this reason, both models -- with and without advance speed -- were considered.

It was hypothesized that eliminating Pine Valley data from the data set would increase the R-squared value of the regression because the Pine Valley data seemed to be very different from the other sites, based on the graph of “Radius vs. e+f ”. It was found, however, that eliminating Pine Valley from the data set reduced the R² value. For this reason, the Pine Valley data points remained in the data set.

These first four equations were linear. However, based on the underlying equation of physics, the current Green Book design procedure uses in-curve velocity squared (V² ) to predict radius. Given how the equations were developed, one would expect that a linear model for radius would return inaccurate radius values for speeds at the high or low end of the data range. Therefore, the values of V² for each vehicle were used in the next regression analysis to produce the following curvilinear equations.

R~=~38.09~-~220.39`e~+~0.019`V SUP{2}

........................................................................................................................................................................ (metric)

R~=~79.97~-~152.04`e~+~0.028`V SUP{2}~-~1.13`V SUB{ADV}

R~=~125.01~-~722.67`e~+~0.158`V SUP{2}

..................................................................................................................................................................... (standard)

R~=~262.41~-~498.43`e~+~0.238`V SUP{2}~-~5.98`V SUB{ADV}

The R² values of the curvilinear equations were 0.36 without V_ADV included in the model and 0.52 with V_ADV included in the model. The p-values for all four equations was less than 0.0001, indicating that the shapes of the curves were significant.

Regression Analysis Based on Low and High Percentile Speeds

The equations developed in the preceding section were regressed on the entire range of minimum speeds within the curves; therefore, the output radius is conceptually based on the “average” minimum speed within the curve. As an alternative, the next step involved developing relationships for the radius based on the 90th and 10th percentile minimum speeds within the curves. (Previously, an inspection of cumulative plots had indicated that the “break points” in the speed distributions were approximately at the 10th and 90th percentile points.) A statistical method called “bootstrapping” (Efron, 1993) was used to determine 95% confidence intervals about the 10th and the 90th percentile minimum speed values for each horizontal curve. The bounds for these confidence intervals are shown in Table 5-5.

Table 5-5 Confidence Intervals About the 10% and 90% In-Curve Minimum Speeds

Lower Limit Average Upper Limit

Street Name km/h mph km/h mph km/h mph

Breckenridge 90%-ile 53.1 33 55.1 34.2 57.9 36

10%-ile 38.6 24 41.0 25.5 43.4 27

Brooken Hill 90%-ile 53.1 33 55.4 34.4 57.9 36

10%-ile 41.8 26 42.6 26.5 43.4 27

Mall-Shiloh 90%-ile 48.3 30 48.9 30.4 49.9 31

10%-ile 35.4 22 36.7 22.8 38.6 24

Pine Valley 90%-ile 33.8 20 36.3 22.6 37.0 23

10%-ile 25.7 16 26.7 16.6 29.0 18

Salem 90%-ile 48.3 30 50.0 31.1 51.5 32

10%-ile 37.0 23 38.1 23.7 40.2 25

After identifying the confidence limits of both the 10th and the 90th percentile points, the rows of data for all vehicles with an in-curve minimum speed outside of this range were removed. (Note that for each vehicle remaining in the data set, its advance speed also remained in the data set.) The regression analyses were performed on these reduced data sets. The resulting equations follow.

......................................................................................................................................................... (90th % - metric)

.............................................................................................................................. (90th % - standard)

......................................................................................................................................................... (10th % - metric)

...................................................................................................................................... (10th % - standard)

The R² values for these 90th percentile models without and with V_ADV included were 0.77 and 0.84, respectively. The R² values for these 10th percentile models without and with V_ADV included were 0.78 and 0.82, respectively. The p-values for all were less than 0.01, with the exception that the p-value for the intercept of the 10th percentile model with upstream speed was 0.21.

Figure 5-10 shows a graph of radius vs. in-curve minimum speed for the 90th percentile and 10th percentile values, along with the AASHTO values, for no cross slope (i.e., e = 0). The values at both ends of the plot were extrapolated outside of the range of the data collected in this study, but terminated at 60 km/h (40 mph) so as to limit the extrapolation. From this graph of the equations, one would estimate that a curve having a radius of 91 m (300 ft) with no cross slope would be driven by most drivers at speeds ranging from 42 to 54 km/h (26.3 to 33.6 mph). The AASHTO design speed for this curve is approximately 50 km/h (31.1 mph).

DATA ANALYSIS BY VEHICLE TYPE

While analyzing vehicle speeds, it was noticed that there were differences among speeds of each of the vehicle types recorded. Table 5-6 shows the mean, standard deviation, and total number recorded for each of the five vehicle types used. The maximum difference between the mean speeds of the vehicle types was over 3 km/h (2 mph).

Table 5-6 In-Curve Speed by Vehicle Type

Vehicle Mean Std Dev Count

Type (km/h) (mph) (km/h) (mph)

2-door 45.66 28.37 6.74 4.18 73

4-door 44.05 27.37 5.93 3.68 330

Pickup 44.85 27.87 6.75 4.19 119

SUV 45.88 28.51 5.83 3.62 130

Van 42.58 26.46 6.71 4.17 59

Because of the difference of average in-curve minimum speeds by vehicle type, it is possible that mean speeds could be different between areas with different vehicle mixes. For instance, it is possible that the mean in-curve minimum speed could be higher in an area predominated with SUV’s than in an area with mostly 4-door passenger cars.

The bus and truck vehicle types were eliminated because not enough of those vehicles were recorded to get a representative sample. A two way analysis of variance (ANOVA) was performed on the vehicle types listed as well as four of the test sites (Pine Valley was excluded from the analysis due to small sample size) in order to determine if there were significant differences among the average speeds recorded for the different vehicle types. The procedure found no statistically significant differences among average in-curve minimum speeds recorded for each of the different vehicle types. Any significant difference that was found could be attributed to differences in the study sites.

Figure 5-10 In-Curve Speed vs. Radius

CHAPTER 6

SUMMARY AND CONCLUSION

The intent of this project was to serve as a pilot study to investigate whether the minimum radii calculated from the Green Book design procedure are larger than necessary for low-speed urban horizontal curves. Two approaches were used in an attempt to answer this question. First, friction factors were calculated for each site based upon actual vehicle speeds and compared with the Green Book friction factors. Second, speed data were used to investigate an alternative method of calculating minimum curve radii. These radii were compared with the Green Book minimum radii.

SUMMARY OF PROCEDURES

After examining a number of potential sites, some were excluded because they possessed geometric irregularities. The sites selected for data collection were Breckenridge Drive and Pine Valley Drive in Little Rock, AR, Salem Road and Mall Avenue--Shiloh Drive in Fayetteville, AR, and Brooken Hill Drive in Fort Smith, AR. The radii of the test sites used ranged from 42 m (138 ft) to 106 m (346 ft). Over 100 vehicles were recorded at each of the sites, with the exception of Pine Valley. The average speeds of traffic recorded ranged from 32 km/h (20 mph) to 50 km/h (31 mph).

After recording radii, cross slopes, and speeds of vehicles traversing the horizontal curves, the Green Book equation relating R, V, e, and f was used to calculate side friction values accepted by drivers. In order to evaluate the relationship between speed and radius, regression analyses were performed. They produced a number of possible equations to estimate the minimum radius of a curve as a function of in-curve speed, advance speed, and cross slope.

OBSERVATIONS AND QUESTIONS

While conducting the research, a number of observations were made and questions raised. These should be considered by future researchers.

When collecting data, attention should be given to concealment of data collectors. The degree to which drivers slowed in reaction to the presence of a data collection was not known.

In selecting study sites and evaluating data, researchers should be cognizant of site geometric irregularities. The degree to which irregular curve geometry and fluctuating cross slopes were encountered was not expected. From this, possible inferences could range from:

1. determining the need for greater adherence to the construction plans when building roadways, to

2. incorporating factors of safety in the horizontal curve design equations to account for construction or after-construction imperfections.

For any given study sample, variations in vehicle type may affect the findings. Although the differences were not statistically different for the sample sizes in this study, differences of up to 3.3 km/h (2.1 mph) in the mean speeds were found among the different types of vehicles. Data samples that have differing proportions of vehicle types (e.g., vans, two-door coupes, etc.) may produce different speed profiles. This suggests that a similar study in a different location with a different mix of vehicle types could produce somewhat different results.

This study raised a question: if there is a large variation in cross slope within a curve, which cross slope are drivers reacting to? For the purposes of this study, a weighted average was used. Arguments could also be made that the maximum or minimum would be more appropriate. However, the effects of cross slope variation upon chosen speed or on computations may be small.

For any given speed, the sensitivity of the final regression models developed herein to the cross slope is constant. In other words, changing the value of e for a curve with a relatively low speed has the same effect on the change of the value of radius as it does for the same change in e for a curve with a high speed. This is in contrast to the Green Book procedure, where a change in cross slope at higher speeds will produce a greater change in the resulting radius value.

The analysis suggests that a driver’s speed in advance of a curve can influence speed within the curve. This agrees with findings from other recent research. This is noteworthy because advance speed is not differentiated from in-curve speed in the Green Book’s horizontal curve design equations.

CONCLUSION

This pilot study did show that the methods employed could be used to reevaluate side friction factors for the design of low-speed urban horizontal curves. In addition, other methods were developed which may show promise in the design of horizontal curves.

In conclusion, the results of this pilot study support the hypothesis that the minimum radii calculated with the Green Book design procedure may be larger than necessary, in that drivers can be expected to exceed the intended design speeds associated with the Green Book’s low-speed urban side friction factors. The magnitude of the differences found would have a greater impact on designs at the higher end of the ”low speed” range, i.e, speeds above 50 km/h (30 mph). A larger study could be performed to reevaluate the current Green Book friction factors. Also, alternatives to the current design equation could be considered.

REFERENCES

American Association of State Highway and Transportation Officials (AASHTO). (2001). A Policy on Geometric Design of Highways and Streets, Washington, D.C.

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