I. Report No. 2. Government Accession No.

TTI-04690-3 4. Title and Subtitle

EVALUATION OF ALTERNATIVE HORIZONTAL CURVE DESIGN APPROACHES ON RURAL TWO-LANE HIGHWAYS

7. Author(s)

A. Voigt 9. Performing Organization Name and Address

Texas Transportation Institute The Texas A&M University System College Station, Texas 77843-3135 12. Sponsoring Agency Name and Address

Federal Highway Administration U.S. Department of Transportation 6300 Georgetown Pike McLean, VA 22101-2296

15. Supplementary Notes

Technical Report Documentation Page

3. Recipient's Catalog No.

5. Report Date

August 1996 6. Performing Organization Code

8. Performing Organization Report No.

04690-3 I 0. Work Unit No. (TRAIS)

II. Contract or Grant No.

DTFH61-92-X -00019 13. Type of Report and Period Covered

September 1993 - May 1995

14. Sponsoring Agency Code

Agreement Officer's Technical Representative (AOTR): Jeffrey F. Paniati, HSR-20 16. Abstract

This report documents evaluations of the effects of superelevation on operating speeds and accident experience and the effects of side friction demand on accident experience at horizontal curves on rural two-lane highways. The goal of the evaluations was to glean insights about the most appropriate speed-related assumptions for alignment design

Variables considered in the operating speed analysis included degree of curvature, length of curve, deflection angle, and superelevation. Two previously developed regression models for estimating 85th percentile speeds on curves were verified. Superelevation was a statistically significant predictor ot ~5th percentile speeds on horizontal curves when added to these models.

Potential accident surrogates examined included AADT, length of curve, degree of curvature, lane width, lane plus adjacent shoulder width, total pavement width, operating speed reduction, superelevation deficiency, and implied side friction demand. The cross section width variables were not significant in this study, although some trends in the data were noted. Operating speed reduction and superelevation deficiency were statistically significant accident predictors, and implied side friction demand was the strongest accident surrogate found.

Comparisons of alternative horizonal curve design methods, with respect to the speed that should used for curve design, were made. Eighty-fifth percentile operating speed on the curve was the strongest performer of four basic curve design ideologies. These findings support the adoption of an operating-speed-based design procedure for rural two-lane highways in the United States.

17. Key Words

Design Consistency Superelevation Highway Accidents

19. Security Classif.(of this report)

Unclassified Form DOT F 1700.7 (8-72)

18. Distribution Statement

No restrictions. This document is available to the public through the National Technical Information Service, Springfield, Virginia 22161

20. Security Classif.(ofthis page)

Unclassified Reproduction of completed page authorized

21. No. ofPages 22. Price

103

APPROXIMATE CONVERSIONS FROM Sl UNITS Symbol When You Know Multiply By To Find Symbol Symbol When You Know Multiply By To Find Symbol

LENGTH LENGTH in inches 25.4 millimeters mm mm millimeters 0.039 inches in It feet 0.305 meters m m meters 3.28 feet It yd yards 0.914 meters m m meters 1.09 yards yd mi miles 1.61 kilometers km km kilometers 0.621 miles mi

AREA AREA in2 square inches 645.2 square millimeters mm2 mm2 square millimeters 0.0016 square inches in2 ft2 square feet 0.093 square meters m2 m2 square meters 10.764 square feet ft2 yd2 square yards 0.836 square meters m2 m2 square meters 1.195 square yards yd2 ac acres 0.405 hectares ha ha hectares 2.47 acres ac mi2 square miles 2.59 square kilometers km2 km2 square kilometers 0.386 square miles mi2

VOLUME VOLUME It oz fluidounces 29.57 milliliters ml mL milliliters 0.034 fluidounces It oz gal gallons 3.785 liters L L liters 0.264 gallons gal -· Ill ft1 cubic feet 0.028 cubic meters mJ m3 cubic meters 35.71 cubic feet ft1 -· ydl cubic yards 0.765 cubic meters m3 m3 cubic meters 1.307 cubic yards yd3 NOTE: Volumes greater than 1000 I shall be shown in m3.

MASS MASS oz ounces 28.35 grams g g grams 0.035 ounces oz lb pounds 0.454 kilograms kg kg kilograms 2.202 pounds lb T short tons (2000 lb) 0.907 megagrams Mg Mg megagrams 1.103 short tons (2000 lb) T

{or "metric ton") (or "t") {or "t") (or "metric ton") TEMPERATURE (exact) TEMPERATURE {exact)

OF Fahrenheit 5(F-32)19 Celcius oc oc Celcius 1.8C + 32 Fahrenheit OF temperature or (F-32)11.8 temperature temperature temperature

ILLUMINATION ILLUMINATION

fc foot-candles 10.76 lux lx lx lux 0.0929 foot-candles fc It foot-Lamberts 3.426 candela/m2 cdlm2 cdlm2 candela/m2 0.2919 foot-Lamberts fl

FORCE and PRESSURE or STRESS FORCE and PRESSURE or STRESS

lbf pound force 4.45 newtons N N newtons 0.225 poundforce lbf lbllin2 poundforce per 6.89 kilopascals kPa kPa kilopascals 0.145 poundforce per lbflin2

square inch square inch

• Sl is the symbol for the International System of Units. Appropriate (Revised September 1993) rounding should be made to comply with Section 4 of ASTM E380.

TABLE OF CONTENTS

~

10 INTRODUCTION 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0. 0 0 0 0 o 1 PROBLEM STATEMENT o 0 0 0 0 0 0. 0 0 0 0 .. 0. 0. 0. 0 .. 0 0 0 0 0 0 0 0 0 o 0 0 0 0 1 OBJECTIVES AND SCOPE 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 . 0 0 0 0 0 . 0 0 0 0 0 o o o o o o o 0 0 2 STUDY ORGANIZATION o o 0 0 0 0 0 0 . 0 o 0 0 0 o 0 o 0 . 0 0 o o 0 0 o 0 0 0 0 0 o 0 0 0 0 0 2

20 LITERATURE REVIEW o o o o o 0 0 0 0 0 . 0 0 o o • o o o 0 o 0 o 0 o o o o o o • o 0 o o o 0 0 0 0 3 CURRENT UNITED STATES HORIZONTAL CURVE

Design Approach 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 0 0 0 0 3 ALTERNATIVE HORIZONTAL CURVE DESIGN APPROACHES o 0 0 o 0 o 0 . 0 . 6

Alternative U 0 S 0 Based Methods 0 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 6 Foreign Horizontal Curve Design Approaches 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . 6

DRIVER SPEED BEHAVIOR ................................... 8 Estimating Speeds on Horizontal Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Estimating Operating Speeds on Tangents . . . . . . . . . . . . . . . . . . . . . . . . . 9

ACCIDENT STUDIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Superelevation Deficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Side Friction Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

SUMMARY ............................................. 13

3. METHODOLOGY ............................................ 14 STUDY DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 SITE SELECTION AND DATA COLLECTION . . . . . . . . . . . . . . . . . . . . . . 15

Speed-Geometry Data Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Curve-Geometry Data Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Curve-Accident Data Base .................................. 18

STATISTICAL ANALYSIS ................................... 22 Speed-Geometry Data Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Accident Surrogate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Verification of Previous Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Analysis of Superelevation Deficiency . . . . . . . . . . . . . . . . . . . . . . . . 24 Analysis of Implied Side Friction Factors . . . . . . . . . . . . . . . . . . . . . 25

4. RESULTS ................................................. 27 EFFECT OF SUPERELEV ATION ON 85TH PERCENTILE OPERATING SPEEDS ON HORIZONTAL CURVES . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . 27

Verification of Previous Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 85th Percentile Speed Estimation Equation-

Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 85th Percentile Speed Estimation Equation-

Multiple-Linear Model ................................. 29

111

TABLE OF CONTENTS (CONTINUED)

Examination of the Effects of Superelevation on 85th Percentile Operating Speeds ........................... 29

Inclusion of Superelevation into the Simple-Linear Regression Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Inclusion of Superelevation into the Multiple-Linear Regression Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Other Variables and Interactions Significantly Affecting 85th Percentile Operating Speeds on Horizontal Curves . . . . . . . . . . . . . . . . 33

Comparison of the 85th Percentile Speed Estimation Models for Horizontal Curves ............................................ 33

ACCIDENT SURROGATE ANALYSIS ........................... 36 Traffic Volume ......................................... 38 Length of Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Accident Rate Used in Analysis .............................. 40

DEGREE OF CURVATURE AND OPERATING SPEED REDUCTION AS ACCIDENT SURROGATES ...................... 41

Degree of Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Lane Width ........................................... 41 Lane Plus Paved Shoulder Width .............................. 42 Total Pavement Width .................................... 45 Operating Speed Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

SUPERELEV ATION DEFICIENCY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 IMPLIED SIDE FRICTION FACTOR (DEMAND) .................... 58 SUMMARY OF ACCIDENT ANALYSIS .......................... 60 SUMMARY ............................................. 62

Operational Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Accident Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS ............... 65 SUMMARY ............................................. 65 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 RECOMMENDATIONS ..................................... 67

REFERENCES ................................................. 69

APPENDIX A STATISTICAL ANALYSIS OUTPUT ....................... 73

APPENDIX B GROUPING ANALYSIS SUMMARY ....................... 90

lV

LIST OF FIGURES

Figure Page

1 Superelevation Rate Versus Degree of Curvature for 138 Curves in 5 States .................................................. 5

2 Speed Profile Model Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Number of Sites by Degree of Curvature ........................... 18

4 Number of Accidents Versus Degree of Curvature . . . . . . . . . . . . . . . . . . . . . 20

5 Percent of Sites With and Without Accidents by Degree of Curvature Category 21

6 Scatterplot of 85th Percentile Speed at the Midpoint of the Curve Versus Degree 28 of Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Linear Regression Equation for 85th Percentile Speed Versus Degree of Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8 Multiple-Linear Regression Equation for 85th Percentile Speeds on Curves . . . . . 31

9 Linear Including Superelevation Regression Equation for 85th Percentile Speeds Versus Degree of Curvature .............................. 34

10 Multiple-Linear Including Superelevation Regression Equation for 85th Percentile Speeds on Curves .............................. 35

11 Residual Plots of Bach 85th Percentile Speed Equation .................. 37

12 Ln (#accidents/year/site) Versus Ln (AADT) ........................ 39

13 Ln (#accidents/year/site) Versus Ln (Length of Curve) .................. 40

14 Mean Accident Rate Versus Mean Degree of Curvature .................. 43

15 Mean Accident Rate Versus Mean Degree of Curvature by Lane Width Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

v

LIST OF FIGURES (CONTINUED)

16 Mean Accident Rate Versus Mean Degree of Curvature by Lane Plus Paved Shoulder Width Category .......................... 47

17 Mean Accident Rate Versus Mean Degree of Curvature by Total Pavement Width Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

18 Mean Accident Rate Versus Mean Speed Reduction by Each Speed Estimation Model and With/Without Sight Distance Restrictions . . . . . . . . 51

19 Mean Accident Rate Versus Mean Speed Reduction For All Speed Estimation Models Without Sight Distance Restrictions . . . . . . . . . . . . . . 53

20 Relative Mean Accident Rate Versus Mean Operating Speed Reduction . . . . . . . . 54

21 Mean Accident Rate Versus Mean Superelevation Deficiency .............. 57

22 Mean Accident Rate Versus Mean Implied Side Friction ................. 61

Vl

LIST OF TABLES

1 Maximum Allowable Side Friction Factors for Various Design Speeds in the U.S., Germany, and Australia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Speed-Geometry Data Base-Site Selection Controls and Criteria . . . . . . . . . . . . 15

3 Independent Variables Considered in Modeling 85th Percentile Speeds on the Midpoint of Horizontal Curves and Tangents . . . . . . . . . . . . . . . 16

4 Curve-Geometry Data Base-Site Selection Controls and Criteria . . . . . . . . . . . . 17

5 Summary of Curve-Accident Data Base Variables ..................... 19

6 Accident Summary by Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7 Summary of Degree of Curvature Categories . . . . . . . . . . . . . . . . . . . . . . . . . 42

8 Summary of Operating Speed Reduction Calculation Methods . . . . . . . . . . . . . . 49

9 Summary of Speed Reduction Categories (Multiple-Linear . Speed Estimation Equation Without Sight Distance Restriction) ............. 50

10 Summary of Operating Speed Reduction Analysis Results . . . . . . . . . . . . . . . . 51

11 Summary of Superelevation Deficiency Analysis Results . . . . . . . . . . . . . . . . . 56

12 Summary of Implied Side Friction Calculation Methods . . . . . . . . . . . . . . . . . . 59

13 Summary of Implied Side Friction Analysis Results . . . . . . . . . . . . . . . . . . . . 60

vii

1. INTRODUCTION

PROBLEM STATEMENT

It has been recognized that accidents are more likely to occur on horizontal curves than on tangent sections of rural two-lane highways. Previous studies have found that accident frequency and severity are greater on the curve sections of rural roadways (1 ,2). For example, Glennon, Neuman, and Leisch reported that accident rates on horizontal curves may be more than three times the accident rates on tangent sections (2). Consequently, a great amount of research has focused on the operational and safety characteristics of horizontal curves. Whereas accident experience and vehicle operating speed have been linked to geometric elements (primarily degree of curvature), it was believed that other geometric elements and alignment characteristics may also affect accident experience and operations on horizontal curves.

Current horizontal curve design procedures in the United States employ the design-speed concept. This concept originated in the 1930's in response to accident problems at horizontal curves and was created to provide a consistent operating-speed profile along an alignment (3). The design-speed concept consists of two fundamental principles: first, all curves on an alignment should be designed for the same speed; and second, the design speed should reflect the speed that a very high percentage of drivers desire to maintain (4). However, the design-speed concept achieves the goal of a consistent alignment only when the operating speed of drivers does not exceed the design speed of the roadway. There has been recent evidence that a disparity between design speeds and operating speeds exists (2,4,5). The American Association of State Highway and Transportation Officials (AASHTO) Policy on Geometric Design of Highways and Streets (6) does not recognize that a discrepancy between operating and design speed may exist. Therefore, the current procedure provides no quantitative guidance to eliminate potentially dangerous inconsistencies in an alignment.

AASHTO uses the application of superelevation as the primary method to ensure speed consistency on curves that are less sharp than the maximum degree of curvature. However, this method relies on the assumption that drivers operate at the design speed along an alignment even when they feel comfortable driving the alignment at a higher speed, an assumption that is clearly flawed. An additional flaw of the concept related to design consistency is that states may employ different maximum superelevation rates. The variability in adopted maximum superelevation rates may lead to a given curve having a different rate of superelevation, depending on the maximum superelevation adopted by each state. Variation in superelevation design may lead to a lack of consistency between different states and may complicate the driver's task of selecting the appropriate speed on the curve ( 4).

This research is being funded by the Federal Highway Administration under Grant Agreement DTFH61-92-X-00019 for Highway Safety Research with the Texas Transportation Institute. Previous studies at the Institute have examined the relationships between operating speed and curve geometric features (7), and the relationships between accident experience on horizontal curves and accident surrogate measures including degree of curvature and operating

1

speed reduction (8,9). All of the basic curve design parameters, except superelevation and side friction, had been examined to determine any quantifiable effects they may have on roadway safety. From the results of previous research and inferences from recent studies at the Texas Transportation Institute (4,7,8,9), it was concluded that superelevation and side friction were important curve design parameters to be examined further.

OBJECTIVES AND SCOPE

The primary objectives of this thesis were to determine if significant relationships exist between:

• 85th percentile vehicle operating speed and superelevation; • Accident experience on horizontal curves and superelevation deficiency; and • Accident experience on horizontal curves and implied side friction based on an

estimated 85th percentile operating speed.

A secondary objective of this study was to verify the relationships developed by Ottesen (7), Anderson (8), and Fink (9). The significance of these relationships was evaluated to determine if present values used in design of superelevation and side friction should be reevaluated. Results of the analysis were used to evaluate alternative methods of horizontal curve design.

The scope of this thesis is limited to rural, two-lane highways in level or rolling terrain functionally classified as collectors or minor arterials.

STUDY ORGANIZATION

This thesis is divided into five chapters, including this introductory chapter. Chapter II reviews previous research on design consistency, operational characteristics of horizontal curves, and surrogate measures to estimate accident experience on horizontal curves. Chapter ill outlines the study design, data collection, and statistical analysis methodology. Chapter IV presents the results of the statistical analyses performed to verify previously reported relationships, and the analyses to fmd the relationships between: (1) operating speed and degree of curvature, length of curve, deflection angle, and superelevation; (2) accident rates and superelevation deficiency; and (3) accident rates and implied side friction. Chapter V presents a summary of the analysis results, conclusions drawn from the analysis, and recommendations.

2

2. LITERATURE REVIEW

Many researchers have investigated factors affecting operations and safety on rural roadways. This chapter reviews literature dealing with these factors and their relation to operations and safety on horizontal curves. This chapter begins with a review of the current U.S. horizontal curve design policy and identifies several possible flaws in this policy. A brief discussion of foreign horizontal alignment procedures highlights alternative horizontal alignment methods. The review continues with previous efforts to relate curve operating speeds and geometric elements and concludes by presenting previous attempts to quantify the effect of superelevation and side friction on the safety of horizontal curves.

CURRENT UNITED STATES HORIZONTAL CURVE DESIGN APPROACH

The current horizontal curve design procedure is presented in A Policy on Geometric Design of Highways and Streets published by AASHTO ( 6). The procedure employs the design­speed concept to provide a consistent alignment with respect to operating speeds. Design speed is defined by AASHTO as "the maximum safe speed that can be maintained over a specified section of highway when conditions are so favorable that the design features of the highway govern" (6). AASHTO states that the design speed "~hould be consistent with the speed a driver is likely to expect" and "should fit the travel desires and habits of nearly all drivers" (6).

The design-speed concept evolved in the 1930's because of safety and maintenance concerns at horizontal curves (I 0). Before 1930, horizontal curves were traditionally designed with crowned sections to provide better drainage. Horse drawn carriages had no trouble traversing crowned sections since their typical speeds never exceeded 5 to 6 kilometers per hour (3-4 mi/h). However, as more motorized vehicles began to use the roadway, increasing safety problems resulted with normal crown sections. The only way that motorized vehicles could travel at their desired speeds (typically 40-48 kilometers per hour (25-30 milh)) was to cross the centerline and use the banking effect that the crown provided on the inside of the curve. This practice caused not only obvious safety concerns but maintenance difficulties since the inside lane was used much more frequently than the outside lane (1 0). Therefore, highway engineers began to superelevate curves more frequently to help alleviate these problems.

M.C. Good (3) credits Young (11) for the first statements proposing a design-speed concept. In 1930, Young argued that roads "should be planned on a miles-per-hour basis-that is, sections, of highways, preferably between towns, should have all curves superelevated for the same theoretical speed" (11). Barnett (I 2) also made significant contributions to the design­speed concept. Barnett recognized the problems drivers face traversing curves that are sharper than expected and argued that all features on an alignment should be safe for the assumed design speed. From the statements of Young and Barnett, it is seen that the desire to provide a consistent alignment is not new. The impetus for the creation of the design-speed concept was the disparity between the speeds desired by drivers of motorized vehicles and the speeds required to safely traverse curves.

3

The design-speed concept works well when the design speed selected for a roadway adequately represents the desired speeds of drivers. However, if the selection of the design speed is in error (set too low), undesirable operating speed/design speed disparities become apparent. Recent studies have shown that a disparity may exist between design and operating speeds on rural two-lane highways. Messer, Mounce, and Brackett reported that drivers rarely exceeded the design speed on roadways that have a 112.7 km/h (70 mph) design speed, that half of the drivers exceeded the design speed on 96.9 km/h (60 mph) design speed roadways, and a great majority of drivers exceeded the design speed on 80.5 km/h (50 mph) design speed highways (I 3). Similar results were found in studies by Krammes ( 4); Chowdhury, Warren, and Bissell ( 14); and in Australia by McLean ( 15). Each of these studies suggested that 85th percentile speeds exceeded design speeds on curves with design speeds less than 90-100 km/h (56-62 milh), and that 85th percentile speeds were lower than design speeds on curves with design speeds greater than about 100 km/h (62 milh).

The discrepancy between operating and design speed reveals several flaws in current U.S. horizontal curve design procedure. First, design speed only has real meaning on curves,_n9!<Jn tangents. AASHTO provides no quantitative-guidance to establish maximum tangent lengths to c~ntrol operating speeds and encourages the use of above-minimum design values on horizontal curves. Use of above-minimum values for design may encourage operating speeds greater than the design speed of the controlling geometric element. No quantitative guidance is included in AASHTO's policy to help test for the possibility that an operating/design speed inconsistency exists (4).

The primary method AASHTO uses to ensure a consistent horizontal alignment is the application of superelevation. However, the policy for distributing superelevation on curves less sharp than the maximum degree of curvature uses a flawed assumption that drivers will operate at the design speed even on curves where they feel comfortable operating at higher speeds ( 4). Kanellaidis notes that "for a given design speed, radii of curves exceeding the minimum without any limitation seems to be one of the main reasons for the appearance of the great differences between design and operating speeds" (16). Krammes hypothesized that the "variation in superelevation rates complicates the driver's task of selecting the appropriate speed on curves" ( 4). The measured superelevation rate was plotted versus the degree of curvature for the 13 8 curves included in the speed data base used in studies at the Texas Transportation Institute. Figure 1 presents the plot of measured superelevation versus degree of curvature and confirms the variability in superelevation rates for given degrees of curvature ( 4).

Hayward notes that because of the assumptions used in developing safe side friction factors and maximum superelevation rates, the design speeds used in the AASHTO method are not representative of the maximum safe speeds derived from the basic formula relating the dynamics of vehicles on curves ( e + f = V2 /127 .5R) (17).

Hayward also states that·"more precisely, a curve with a fixed degree of curvature and superelevation rate can be considered to have different design speeds, depending on the state criteria that have been used to design the curve" (17). The design speed for an alignment on the maximum superelevation rate adopted by each state agenc~, usually based on

4

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FIGURE 1 Superelevation Rate Versus Degree of Curvature for 138 Curves in 5 States ( 4)

5

prevailing terrain and environmental conditions. The maximum superelevation rate has a definite impact on curve design elements because of the manner in which AASHTO distributes superelevation and side friction for curves below the maximum degree of curvature, given a specific design speed. The assumption is made that side friction factors vary in a curvilinear relationship with the degree of curvature between e=O.O and e=emax- Hayward states: "therefore, for different emax values, the curve takes on a different shape and hence affects the curve geometry" (17). The use of different maximum superelevation rates between states results in a consistency problem among states and may complicate the driving task of selecting the appropriate speed on a curve ( 4).

Several researchers in the U.S. and overseas have recognized that the design-speed concept cannot reliably produce consistent alignments. These recognized weaknesses of the design-speed concept have spurred several alternative approaches to horizontal curve design.

ALTERNATIVE HORIZONTAL CURVE DESIGN APPROACHES

Alternative U.S. Based Methods

Leisch and Leisch (18) and Lamm et al. {19) both presented methods of attaining geometric consistency in the United States. Leisch and Leisch developed a procedure using an operating-speed proflle similar to that used by the Swiss. The speed estimation procedures were derived from 1965 and 1973 AASHTO design policies. Leisch and Leisch {18) recommended consistency should be based on three principles: (1) average speeds along an alignment should not vary by more than 16.1 km/h (10 mi/h), (2) design speed reduction should not exceed 16.1 km/h (10 mi/h), and (3) average truck speeds should not differ from passenger car speeds by more than 16.1 km/h (10 mi/h).

Lamm et al. {19) adapted the German horizontal alignment design guidelines for United States use. The speed estimation equations were based on data from a sample of curves in New York State. This technique is similar to the Swiss procedure for consistency evaluation. Measures of consistency proposed by Lamm, et al. are: ·

1) Good:

2) Fair:

3) Poor:

~D ~5° or ~Vs5 ~ 9.7 kmlh (6 milh), 5 °< ~D ~ 1 oo or 9.7 kmlh (6 mi/h) < ~V85 ~ 19.3 km/h (12 milh), and ~D > 10° or ~V85 > 19.3 kmlh (12 mi/h) (18).

Foreign Horizontal Curve Design Approaches

Several foreign countries have addressed the disparity between design speed and operating speed in their design guidelines. The impetus for the inclusion of both design and operating speed into design was to address safety problems on rural roadways. Germany and Australia have updated their horizontal alignment design procedll!es to include checks of actual

6

driver speed behavior. Germany employs a speed~profile estimation technique for use in speed consistency checks, and Australia provides checks of operating speeds on low design speed roadways ( 4).

In Germany, the basic procedure for road planning and design requires an iterative procedure to consider "the numerous relationships between driving behavior, road design, capacity and the environment" (20). In Germany, a consistency evaluation procedure was first implemented in the 1973 edition of the German alignment guidelines published by the German Road and Transportation Research Association (20). These procedures were updated in 1984. The German guidelines incorporate the idea that when differences between operating speeds and design speeds exist, superelevation and stopping-sight distances should be based on the normally higher operating speeds (16). The operating speed is estimated using curvature change rate (essentially degree of curvature for single elements) and pavement width (20). The German guidelines require that design speed and operating speed be tuned to within certain tolerances. Operating-speed consistency is checked using acceptable ranges for successive curve radii. If these checks reveal a consistency problem, transition sections are considered to adjust operating speeds or the design speed is increased on the alignment.

German maximum superelevation rates are typically 0.07 (compared to 0.08 in the U.S.) and 0.08 in exceptional cases (compared to 0.10 to 0.12 in the U.S.). Superelevation design is based on a function of curve radius and 85th percentile operating speeds (20). Kanellaidis (I 6) noted that a convergence is occurring between German and U.S. maximum superelevation rates. An emax of 0.08 has been suggested as "the correct balance, from a driving safety point of view, between the low superelevation of0.06 and the high superelevation ofO.lO" (21).

German guidelines also have limited side friction factors to 40 percent of the maximum allowable tangential values for rural roadway design. By using only 40 percent of the side friction there is more than 90 percent available for friction in the tangential direction (22). This limitation is important because it leaves adequate tangential friction forces available for sudden acceleration, deceleration, or evasive actions. Because of the smaller values of side friction used in Germany, the minimum radii of curve are greater than those of the U.S ..

Australian guidelines also provide an iterative method of horizontal alignment design. For high speed designs(~ 100 km/h (62 mi/h)) the design-speed concept is applicable based on conclusions from operating speed studies in Australia. McLean (5) observed that 85th percentile speeds on alignments designed for speeds of 100 km/h (62 milh) or more were generally lower than the design speed. In this case no design iteration is necessary, and all elements may be designed for the design speed. However, for low-speed alignments, the estimated 85th percentile speeds are used as the design parameter on horizontal alignments (23).

In the Australian guidelines, superelevation is chosen based on safety and comfort and considers design speed (or operating speed on low speed alignments), the tendency of slow vehicles to track to the middle of the curve, the stability of high-profile vehicles, and the length available to introduce the superelevation (23). Australian guidelines for the use of side friction differ from those of AASHTO and Germany. Australian guidelines recognize that the

7

differences between the roadway and vehicle conditions assumed in the circular path formula means that side friction values used in design may not be derived directly from the known inventory of pavement resistance. The guideline recognizes that "drivers learn to assess the speed appropriate to a given curve, and this can form the basis of a design criterion" (23).

The side friction values used in design in Australia have been derived from observations of driver speed behavior on rural roadways. Side friction values for design speeds less than 90 km/h (55.9 mph) are those used by vehicles traveling at the 85th percentile speed. Maximum side friction values for design speeds higher than 90 km/h (55.9 mi/h) are more than those side friction values used by vehicles operating at the 85th percentile speed, due ·to the concerns of comfort and safety for all roadway users (23). Table 1 presents side friction factors used in the U.S., Germany, and Australia. As expected, Australian side friction factors are higher because of the friction factor being based on 85th percentile operating speeds (under 90 kmlh). German side friction factors are the lowest, providing the largest margin of dynamic safety in design.

TABLE 1 Maximum Allowable Side Friction Factors for Various Design Speeds in the U.S., Germany, and Australia.

Maximum Side Friction Factors Design Speed (km/h)

United States Germany Australia

30 0.17 0.20 -

40 0.17 0.18 -

50 0.16 0.17 0.35

60 0.15 0.15 0.33

70 0.14 0.12 0.31

80 0.14 0.10 0.26

90 0.13 0.09 0.18

100 0.12 0.08 0.12

110 0.11 0.075 0.12

120 0.09 0.07 0.11

130 - - 0.11

DRIVER SPEED BEHAVIOR

It has become more accepted that operating speed should be used in certain aspects of horizontal curve design. However, there must be an accurate method of estimating operating

8

speeds for the process to be efficient. This section reviews several efforts to quantify driver operating speeds on tangents and horizontal curves.

Estimating Speeds on Horizontal Curves

Krammes ( 4) states that the principal operating-speed-based method of evaluating alignment consistency is the change in 85th percentile operating speed between the approach tangent and horizontal curve. This method uses the estimated operating speed on the tangent and the c\rrve, and estimates the speed reduction as the .difference between curve and maximum approach tangent speeds. There have been several efforts, both in the U.S. and abroad, to develop relationships to predict speeds on horizontal curves. Krammes ( 4) identifies two basic theories proposed to predict operating speeds on curves: (1) speeds depend on local characteristics of the curve only, and (2) speeds depend on both curve-specific characteristics and the general characteristic of the alignment. Although 85th percentile speeds have been estimated with sufficient accuracy (R2 values between 0.65 and 0.84) using isolated curve features, incorporating desired speed into the prediction models reduces the unexplained variability. The problem with equations including the desired speed is that the desired speed of drivers is difficult to define and measure ( 4).

There have been several operating-speed prediction equations developed for speeds on horizontal curves. Ottesen (7), Lamm and Chouieri (24), Glennon et al. (2), and Taragin (25) have developed models based on U.S. speed behavior. McLean (5) in Australia; Lamm et al. (26) in Germany; Emmerson (27) in England; Gambard and Louah (28) in France, Lindenmann and Ranft in Switzerland (29), and Kanellaidis (30) et al. in Greece have developed models specific to their respective countries. These equations use degree of curvature as the independent variable and 85th percentile operating speed as the dependent variable.

Ottesen (7) found a relationship between speeds on horizontal curves and degree of curvature, as well as other variables. Ottesen found a statistically significant multiple-linear regression equation using the independent variables degree of curvature, length of curve and deflection angle.

Estimating Operating Speeds on Tangents

The maximum speed achieved by drivers on a tangent section depends on the tangent length, the sharpness of curve on either end ofthe tangent, and the desires ofthe driver (4,7). Figure 2 depicts the three typical cases used in speed profile modeling procedures (24,31 ,32). Each case differs by the length of the tangent between the two curves. Several assumptions are used in this speed profile technique: (1) speeds on curves are constant, (2) acceleration and deceleration rates are constant and equal in value, and (3) acceleration and deceleration occur only on the tangent sections. Although these assumptions may not be precisely correct, speed profile studies have shown that these assumptions are reasonable (33).

In case 1, the acceleration or deceleration occurs uniformly between the two curves, and the maximum 85th percentile speed occurs at one end of the tangent. In case 2, the tangent

9

length is sufficient to allow some acceleration, but not completely to the desired speed. Therefore, in case 2, the combination of tangent length and curve speeds determines the maximum 85th percentile speed on the tangent. In case 3, tangent length is sufficiently long to allow drivers to reach a desired speed and maintain that speed for some distance before decelerating, if necessary, to the next curve. In consistency evaluation procedures, the desired speed on the tangent is given an assumed value or estimated from the curve speed estimation equation where degree of curvature is equal to zero (32).

ACCIDENT STUDIES

There have been extensive efforts to quantify the safety effects of roadway and operational characteristics. Most accident surrogate analyses have concentrated on variables such as degree of curvature, lane widths, and shoulder widths. However, some have quantified the effects of superelevation and side friction on accident experience.

Superelevation Deficiency

There have been several efforts to determine relationships among accident rates on horizontal curves and independent variables believed to affect accident rates. Zegeer (34), Datta (35), Zador (36), and Terhune and Parker (37) all included superelevation, specifically superelevation deficiency or error, in accident surrogate analyses.

Zegeer (34) conducted a study published in October 1991 for the Federal Highway Administration that evaluated the cost-effectiveness of geometric improvements for safety upgrading on horizontal curves. In this study, superelevation data were collected for 732 of the 10,900 curves in the study. The variable used to quantify the effects of superelevation on safety was "superelevation deviation." Superelevation deviation was defined as "(optimal superelevation)- (actual superelevation), where optimal superelevation was defined from the AASHTO Design Guide as a function of degree of curvature and terrain type" (34). Zegeer's analysis revealed a significant relationship between total accident rate and degree of curvature, width, presence of spiral, and superelevation deficiency. Zegeer estimated the accident reduction corresponding to a 0.02 superelevation deficiency on a 3° curve with a width of 30 feet, no spiral, and 0.3 million vehicle miles, to be about 10.6 percent. Zegeer also concluded that "too much" superelevation was not associated with higher accident rates (34).

Zador (36) published a study in 1985 that examined the effects of superelevation deficiency at fatal crash sites in the states of Georgia and New Mexico. One objective ofthis study was "to determine the effect of superelevation on the incidence of fatal single vehicle rollover crashes after adjustments for both grade and curvature." Zador found that superelevation were deficient at crash sections compared to the superelevation rates applied at comparable

sections. Zador gives several explanations for the results of the study. For vehicles entering left curves, the combination of minimal distance to the pavement edge and higher speed produces run-off-the-road accidents at higher speeds, possibly due to a lack of side-friction supply. Vehicles entering right curves have the additional outside lane to use in corrective maneuvers,

10

Plan View Tangent Curve

Vr = Desired Speed (Not to Scale)

'"0 Q)

~i[-. Q)

0.. Case 1

------------------- --------~

Distance

QJ -------

~if-~---~~---QJ

o.. Case 2 Distance

"d QJ

4t~ ------------QJ

0.. .Case 3 Distance

FIGURE 2 Speed Profile Model Cases ( 4)

11

reducing the chances for run-off-the-road type accidents, but increasing the possibility of head-on collisions with vehicles on the outside lane. Zador also suggested that in many cases the design speed was set too low, so that the superelevation is "nominally adequate" but is not capable of offsetting the vehicle side forces at the higher speeds desired by drivers (36).

Datta (35) conducted a study published in 1983 for the Federal Highway Administration that identified several potential surrogates for predicting accident experience. Datta analyzed rural isolated curves on two-lane roadways, rural signalized intersections, and two-lane tangent sections in urban areas. For the rural isolated horizontal curve portion of the study, 25 horizontal curves were examined. Run-off-the-road accidents were significantly influenced by the independent variables degree of curvature and superelevation error, where superelevation error was defined as the difference between emin in prevailing conditions (given the terrain and environmental conditions) and actual superelevation. These· findings were produced from a subset of 14 of the 25 curves characterized by limited sight distance and zero or one driveways on the curve. Datta did note the limitations imposed on his conclusions because of the limited size of the sample data the limited geographical scope of the study (35).

Terhune and Parker used data collected in New York State in their "Evaluation of Accident Surrogates for Safety Analysis of Rural Highways" (37). This study focused on isolated curves and unsignalized intersections on rural two-lane roads. The general objective of the study was to "validate the use of accident surrogates at rural curves and unsignalized intersections." Seventy-eight sites were used for evaluation. Accidents were collected for a three-year period and accidents related to animals on road or pedestrians eliminated. Superelevation error was not found significant in the New York curve data base. Although superelevation error was included in the list of possible accident surrogates, it did not correlate well and was dropped after initial analysis. Analysis continued with several.independent variables, mainly degree of curvature, AADT, distance from last major event in the outside lane, and encroachment rate. Several equations were derived with these variables, and significant relationships were found using primarily degree of curvature and AADT (37).

Side Friction Demand

A study by Lamm, Choueiri, and Mailaender (38) was undertaken to determine whether AASHTO's existing side friction factors used in design provide adequate dynamic safety of driving. The study was based on geometric, operating speed, and accident data for 197 curved­roadway sections in the state of New York and included 569 accidents.

The first objective of the analysis was to examine the relationship between side friction assumed and side friction demand, or comparing side friction values used in design to actual side friction factors used by the 85th percentile driver. The sid~ friction demand was computed from the following formula:

frd = ( ( V 85 )2 * ( D I 85,660) ) - e

where frd equals side friction demand, and V 85 equals the 85th percentile operating speed on the curve (milh). Regression analysis was undertaken to obtain quan~itative estimates of the effects

12

of degree of curvature, 85th percentile operating speed, and accident rate on side friction assumed and side friction demand. Significant relationships were found between side friction assumed for design, actual side friction demand, and accident rates.

Lamm et al. concluded from this analysis that side friction assumed exceeded the side friction demand in the range of"good" design practices (where .:1D~ 5° and .:1V85 ~ 9.7 kmlh (6 mi!h)) whereas side friction demand exceeded side friction assumed in the range of "poor" design practices (where .:1D ~ 10° and .::1 V 85 ~ 19.3 km/h (12 mi/h)). These findings suggest that a danger exists when friction demand exceeds the friction assumed in higher degree of curvature classes and that higher accident danger results. The conclusion was made that "driving dynamic safety aspects have an important impact on geometric design, operating speed, and accident experience" on curved sections of rural two-lane highways. Also included in conclusions was that an "overall safety improvement, would result through an interaction between three geometric criteria: (1) achieving consistency in horizontal alignment, (2) harmonizing design speed and operating speed, and (3) providing adequate dynamic safety of driving" (38). Lamm noted that relying on only one of these criterion would not result in the improvement of horizontal alignment safety.

SUMMARY

Because of the recognized inadequacy ofthe design-speed concept to provide consistent alignments, there have been many research efforts to try to link the safety and operational characteristics of horizontal curves to surrogate measures of accident experience and speed. Previous studies at the Texas Transportation Institute have examined all of the major curve design parameters, except superelevation, for their effects on operations and safety on horizontal curves.

Previous research on the operational aspects of horizontal curves has focused on degree of curvature. Though the relationship between speed and degree of curvature explains much of the variability between speed and geometric elements, a considerable amount of unexplained variability exists. Ottesen found that the length of curve and deflection angle also significantly affect the speeds drivers choose to traverse a curve. The only other curve design parameter not previously linked to speed on curves is superelevation.

The laws of physics link speed, superelevation, side friction, and radius of curve, but the current method of superelevation application employed by the design- speed concept weakens these fundamental links. Anderson found relationships between accident experience and degree of curvature, and accident experience and operating speed reduction, but did not examine superelevation and side friction. Datta, Zegeer, and Zador found significant relationships between superelevation deficiency and accident experience, and Lamm, Choueiri, and Mailaender found significant relationships between side friction and accident rates. Because side friction and superelevation are significant variables in horizontal alignment design, more research was needed to determine if significant relationships exist between side friction, superelevation, and accident experience on rural two-lane roadways.

13

3. METHODOLOGY

This chapter outlines the basic methodology used in the study. The chapter summarizes the site selection criteria and data collection efforts for both portions of the study: operating speed and accident surrogate analyses. The chapter continues by summarizing the relationships verified from other studies at the Texas Transportation Institute, and concludes by outlining the hypotheses and relationships examined using data bases created for this study.

STUDY DESIGN

The basic study design was separated into two main tasks. The first task was to examine the effect of superelevation on 85th percentile operating speeds on curves. The second task was to examine the potential of several variables as accident surrogates.

Ottesen (7) found significant relationships between 85th percentile vehicle operating speed and several curve geometry elements, including degree of curvature and length of curve. These relationships were verified, and analyses made to determine if the inclusion of superelevation improves explanatory power of these relationships.

Anderson ( 8) and Fink (9) conducted safety analyses using a data base that included roadways from three regions of the United States. This data base included rural roadways in Texas, Washington, and New York. Several curve geometry variables and five years (1987-1991) of accident data were included in the data base for each roadway. The basic study design for this study included the addition of four more Texas roadways, collecting seven years of accident data for the new roadways ( 1987-1993 ), and collecting cross-section data for each Texas roadway in the data base. Two more years of accident data were added to the five years of accident data for the Texas roadways used by Anderson (8) and Fink (9). Only the Texas roadways used by Anderson and Fink were used in this study because of the time and cost considerations associated with the field collection of cross-section data. These data were compiled into a data base for use in statistical analysis.

Anderson ( 8) found statistically significant relationships between accident rates, curve geometry elements, and operating speed reduction. These relationships were verified and analyses made to determine if superelevation deficiency and side friction have significant effects on accident experience.

Fink (9) examined the effects of limited sight distance on accident rates. Fink concluded that incorporating sight distance information into a speed profile model would most likely not contribute significantly to the accuracy of the model. This conclusion will be indirectly verified by using speed profile models with and without sight distance limitations while verifying the relationship between operating speed reduction and accident experience.

14

SITE SELECTION AND DATA COLLECTION

Site selection and data collection were separated into two segments. Site selection criteria and data collection efforts are presented first for the speed-geometry data base and then for the accident-geometry data base.

Speed-Geometry Data Base

The speed-geometry data base used is the same database used by Ottesen (7). Spot speed data were collected on 138 curves and 78 approach tangents on rural two-lane highways in three geographic regions of the U.S.: the East (New York and Pennsylvania), the West (Oregon and Washington), and the South (Texas). Free-flow speeds were collected at the midpoint of the curve and at a point on the preceding tangent where free-flow speed was believed to be attained. All sites were selected so that the preceding tangent was of sufficient length to allow drivers to attain their desired speed. In total, 22,740 speed observations were made. Table 2 summarizes the site selection controls and criteria for the speed data collection. Table 3 summarizes the variables in the data base and the source of the data.

TABLE 2 Speed-Geometry Data Base-Site Selection Controls and Criteria

Control Criteria

Area Type Rural Administrative Classification State Functional Classification Collector and Minor Arterial Design Speed ~ 100 km/h (62.1 mi/h) Posted Speed Limit 80.5-88.5 km/h (50-55 milh) Terrain Level to Rolling Grade ~ 5 percent Traffic Volumes 750-2500 vehicles/day Lane Widths 3.05-3.66 m (10-12 ft) Shoulder Widths 0-2.44 m (0-8 ft) Degree of Curvature 1-20°(Primarily 3-12°) Length of Curve ~ 61 m (200ft) Preceding Tangent Length ~244m (800ft) Sight Distance to Curve ~ 122m (400ft)

15

TABLE 3 Independent Variables Considered in Modeling 85th Percentile Speeds on the Midpoint of Horizontal Curves and Tangents

Independent Variable Source of Data Curve Tangent

Degree of Curvature Plans X Length of Curve Plans X Deflection Angle Plans X Superelevation Rate Field X Travel-way Width Field X X Total Pavement (Lane & Shoulder) Width Field X X Sight Distance to Curve Field, Plans X X Speed on Preceding Tangent Field X Tangent Length Plans X X Speed on Preceding Curve Field X X Lane (Inside or Outside) Field X X Terrain Type (Level or Rolling) Field, Plans X X Annual Average Daily Traffic (AADT) Traffic Reports X X Geographic Region Field X X

Curve-Geometry Data Base

The roadways in the accident analysis included the same nine Texas roadways used in studies by Anderson (8) and Fink (9). In addition, four more Texas roadways were added for analysis. These roadways are located in central and eastern Texas. Table 4 summarizes the site selection controls and criteria.

All roadways in the accident study were two-lane rural roadways at least 4.0 km (2.5 mi.) in length. Curves included in the data base were outside city limits by at least 0.8 km (0.5 mi.) and at least 0.8 km (0.5 mi.) from the end of the roadway to eliminate effects of controlled speed environments. Only sections of the roadway where geometric features of the roadway controlled were desired. Curves with intersections on the curve or within 150 meters (500ft) of the curve were excluded from the data base. This restriction was imposed to eliminate curves where most accidents might be intersection-related. A total of247 curves were selected for analysis. Since each direction of each curve had differing preceding geometric characteristics, each direction was considered a separate site, resulting in a total of 494 curve sites.

16

TABLE 4 Curve-Geometry Data Base- Site Selection Controls and Criteria

Control Criteria

Area Type Rural Administrative Classification State Functional Classification Collector or Arterial Design Classification Two-Lane Design Speed ~ 88.5 km/h (55 mph) Posted Speed Limit ~ 88.5 km/h (55 mph) Terrain Level to Rolling Grade ~ 5 percent Pavement High Type Traffic Volumes 280-4500 vpd Lane Widths 2.8-3.7 m (9-12ft) Shoulder Widths 0-2.4 m (0-8 ft) RRR Improvements None in previous 7 years Length of Roadway Section ~ 4.0 km (2.5 mi) Distance from Town Limit ~ 0.8 km (0.5 mi) Distance from End of Roadway ~ 0.8 km (0.5 mi) Intersections None Within 150m of the Curve

Curve geometry data was recorded for each curve site. Geometric information was collected from plans, direct measurements in the field, and from visual inspection in the field. Site numbers were assigned to reference each site to the accident data base. Variables included in the curve-geometry data base were: AADT, roadway number, milepoint at the point of curvature, lane (inside or outside), radius (degree of curvature), length of curve, deflection angle, superelevation, preceding tangent length, preceding radius (degree of curvature), preceding sight distance, preceding length of curve, preceding superelevation, total pavement width, total travel width, shoulder width, foreslope, backslope, and roadside rating.

An attempt was made to select curves representing a wide range of values for most of the variables in the geometry data base. However, because of the small numbers of sharp curves on the selected roadways, this was not possible. The resulting distribution of degree of curvature in the data base has a greater proportion of lower degrees of curvature, as shown in Figure 3.

17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Degree of Curvature

I I I

I I

l !

______ j FIGURE 3 Number of Sites by Degree of Curvature

Curve-Accident Data Base

The data base developed by Anderson (8) and Fink (9) consisted of five years (1987-1991) of accident data for the nine Texas roadways used in the previous analyses. Accident information was updated for the years of 1992 and 1993 for these roadways. Seven years of accident data (1987-1993) were obtained for the four roadways added to the data base for this study. A seven-year period was used to include enough accidents for analysis but exclude the possible inconsistencies in roadway performance associated with the effects of resurfacing and reconstruction projects.

Individual police accident reports were obtained from the Texas Department of Public Safety for each of the roadway segments. Review of the actual reports was necessary to more accurately find the location and cause of each accident. The reports were screened to exclude accidents not curve related. Accidents were not included in the data base if their cause included any ofthe following: (1) driver asleep, (2) animal on the roadway, (3) passing, parked, or turning vehicle, (4) bicyclist or pedestrian related, or (5) mechanical defect in the vehicle. The accidents were then separated into three types: (1) run-off-the-road, (2) multiple-vehicle collision

18

in the same travel direction, and (3) multiple-vehicle collision involving vehicles in the opposite travel directions. The data base was built by characterizing each accident using variables describing the roadway condition, driver condition, and accident type and severity. Table 5 summarizes the variables included in the accident data base. Each of these variables was obtained from the police accident reports except the site number, which was included for reference to the geometry data base.

TABLE 5 Summary of Curve-Accident Data Base Variables

Variables Included in Accident Data Base

Site Number Roadway DPS Accident Number Driver Condition (sober or intoxicated) Severity (PDO, injury, fatality) Roadway Surface Condition (dry, wet, ice) Light Conditions (dawn, day, dusk, night, fog) Accident Type (single vehicle run-off-the-road, multiple-vehicle collision) Runoff Location (inside or outside of curve)

There were 238 curve-related accidents on sites included for analysis. Accidents involving motorcycles and large trucks were removed from the data base because of the small numbers of incidents (12) including these types of vehicles. Therefore, there were 226 accidents used in the analysis, involving only passenger cars and light trucks. These accidents were distributed by degree of curvature as shown in Figure 4. Figure 4 is similar to Figure 3 in that the number of accidents is greater in the lower degrees of curvature, but this is most likely due to the greater numbers of sites in those degrees of curvature. Figure 5 shows the percentage of sites with accidents and the percentage of sites without accidents for eight degree of curvature categories. This figure shows that as degree of curvature increases, the percentage of sites experiencing accidents increases. Table 6 summarizes the accidents by type.

19

40

35

30

J!l 25 c Q) "0 ·u ~ 20

N L..

0 Q) .0 E ::J 15 z

5

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Degree of Curvature

FIGURE 4 Number of Accidents Versus Degree of Curvature

100 II) Q)

90 ·;:: 0 C) Q) ......

80 ro (..) Q)

c: 70 ::I (.) -0 60 Q)

N ....... ~ C)

50 Q)

Cl c: :c 40 ...... :;: II) 30 Q) ...... i:i) - 20 0 ...... c: Q) (.) 10 .... Q) 0..

0 DC<=l l<DC<=2 2<DC<=3 3<DC<=4 4<DC<=6 6<DC<=9 9<DC<=10 DC>lO

Degree of Curvature Category

•sites With Accidents •Sites Without Accidents

FIGURE 5 Percent of Sites With and Without Accidents by Degree of Curvature Category

TABLE 6 Accident Summary by Type

Characteristic Accidents

Severity Property Damage Only = 101 Injury/Fatality= 125

Driver Condition Sober= 165 Alcohol Involved = 61

Lane Inside= 76 Outside = 150

Road Condition Dry= 153 Other= 73

Light Condition Day= 105 Other= 121

Accident Type Runoff=207 Collision in Opposite Direction= 19 Collision in Same Direction = 0

STATISTICAL ANALYSIS

Speed-Geometry Data Base

The speed-geometry data base was used by Ottesen (7) to develop regression equations for the 85th percentile speed at the midpoint of the curve and the 85th percentile speed on long tangent sections. Ottesen (7) found several statistically significant relationships between the 85th percentile operating speed on horizontal curves and several geometric variables. These relationships were the base models for which the effect of superelevation was analyzed.

Ottesen found the relationship between 85th percentile operating speed and degree of curvature in a simple linear model to be:

where:

Vs5 = 103.61- 1.95 D

= 85th percentile speed at the midpoint of curve (km/h), and = degree of curvature (0

).

Other independent variables were found significant and added additional explanatory power to the simple linear model. Ottesen found that one other additional independent variable was significant: length of curve. The interaction between length of curve and degree of curvature was also significant. This interaction is referred to as the deflection angle of the curve. The form of the multiple-linear regression is as follows:

22

V85 = 102.44- 1.57 D + 0.012 L- 0.01 ll

where: V 85 = 85th percentile speed at the midpoint of the curve (kmlh), D = degree of curvature CO), L = length of curve (m), and /::,. = deflection angle of the curve e).

Superelevation was examined for significance in the simple linear and multiple-linear regression speed models. It was hypothesized that as superelevation increases, the 85th percentile speed at the midpoint of the curve will increase. This increase may be because as more superelevation is applied to a given curve, the side forces on the driver are lessened, increasing the comfort level of the driver. The increased level of comfort may lead the driver to adjust to a higher speed on the curve.

Accident Surrogate Analysis

Verification of Previous Findings

Anderson (8) examined surrogate measures of accident experience on rural two-lane highways. Analyses were undertaken to determine if certain curve and approach variables had significant effects on accident rates on horizontal curves. Anderson found two statistically significant relationships:

where:

In (Mean Accident Rate)=/( Mean ll V85 )

In ( Mean Accident Rate ) == f ( Mean D )

ll V 85 = difference between estimated maximum 85th percentile speed on the approach tangent and the estimated 85th percentile speed at the

midpoint of the horizontal curve (kmlh), and D = degree of curvature CO).

The analysis suggested that as both operating speed reduction and degree of curvature increased, accident rates increased. Both mean speed reduction and mean degree of curvature were found to be good predictors of accident rates. These relationships will be verified using the data base created for this study.

Anderson (8) examined the effects of other variables for their effect on accident rates. These variables included: state, travel-way width, and total pavement width (travel-way width plus shoulder width). Anderson used indicator variables in the regression analysis to determine if significant differences existed when these variables were divided into categories or levels. Indicator variables were used in the regression analysis to determine if the slopes or intercepts of different categories were statistically different from each other. The form of the regression equation using indicator variables was as follows:

23

where pi= parameter estimate, D =degree of curvature, and zi =indicator variable. For example, in the travel-way width analysis, the data base was divided into thirds, based on the travel-way widths. Indicator variables were then assigned to each travel-way width category. In this case, the indicator variable z1 is set to one for the middle category and set to zero for the lower and high travel-way width category. The indicator variable q is set to one for the high travel-way width category and set to zero for the lower and middle travel-way width categories. Sites in each travel-way width category were divided by degree of curvature categories, allowing the slopes and intercepts of three travel-way width categories to be compared. If the parameter estimates P2 or P3 were significant at the 0.05 level, it could be concluded that the intercepts for the middle or high travel-way width categories were significantly different from the intercept of the low category. This would suggest that the accident rate was significantly different for the middle or high category compared to the low category. Similarly, if the parameter estimates P4

or Ps were significant at the 0.05 level, it could be concluded that the slopes of the middle or high travel-way width categories were different from the low travel-way width category, and a difference existed in the rate of change in the accident rate in each category with respect to the degree of curvature. Anderson found no significant relationships between state, travel-way width, or total pavement width and accident experience in the three state data base.

Analysis ofSuperelevation Deficiency

One objective of this study was to examine the effect of superelevation on the safety of horizontal curves. A potential accident surrogate is superelevation deficiency. Because ofthe sparsity of the data, curve sites were grouped into intervals for analysis. A mean superelevation deficiency and mean accident rate were calculated for each interval. The calculated means were used in regression analysis. The basic form of the regression equation is:

ln (Mean Accident Rate ) = Po + P1 (Mean Superelevation Deficiency )

where superelevation deficiency is the actual superelevation rate subtracted from a theoretical optimum superelevation rate. The theoretical optimum superelevation rate is calculated given the degree of curvature, assumed side friction factors for design, and design speed. For this study, superelevation deficiency was calculated based on several different horizontal curve design perspectives.

The first speed used resulted from a faithful application of the AASHTO design-speed concept. For each roadway section, the inferred design speed was found using the actual superelevation rate and degree of curvature using AASHTO minimum design values. The minimum inferred design speed on each roadway was used to design the superelevation for each curve on that roadway. For example, if 95 percent of curves on the roadway are designed for speeds more than 96.6 kmlh (60 mi/h), and only one curve on the alignment is designed as a 48 kmlh (30 mi/h) curve (defined as the controlling geometric element), superelevation for each curve on the alignment was designed using a 48 kmlh (30 mi/h) design speed.

24

The second speed used to determine "optimum" superelevation rates was 96.5 kmfh (60 mi!h). This speed was chosen because of Ottesen (7) finding that the desired speed of drivers on long tangents was approximately 97.8 kmfh (60.8 milh). The strictest interpretation of the design consistency concept is that curve design should fit the desired speed of drivers. This selection of 96.5 kmfh (60 milh) represents the strictest application of the operating-speed based consistency concept.

The third speed used in calculating "optimum" superelevation rates was the maximum 85th percentile speed on the preceding tangent. This speed is estimated on the preceding tangent using the speed profile model. This speed represents a more relaxed interpretation of the design consistency concept. This concept accounts for curves that may have their operating speeds constrained by adjoining geometric features, resulting in lower 85th percentile speeds than if the preceding alignment did not constrain speeds. Maximum 85th percentile tangent speeds were found using each 85th percentile speed estimation equation in the speed profile model, and superelevation designed using each of the resulting maximum tangent speeds.

The final speed used in the analysis was the estimated 85th percentile speed at the midpoint of the curve. This approach was inspired by the design procedures of several foreign countries that use the 85th percentile operating speed for superelevation design. The speed· estimation models found by Ottesen and those found in this study were used to calculate the 85th percentile operating speed at the midpoint of the curve. ·

This analysis was also used to evaluate the 85th percentile speed estimation equations found in the operational/speed analysis. Both the 85th percentile curve speed and maximum 85th percentile tangent speed were found using each speed estimation model, and then used to determine "optimum" superelevation rates for each curve site. A secondary hypothesis of the analysis was that the best fitting model Gudged by R2 and MSE values) may represent the most suitable speed for curve design, implying that the speed used in the best fitting model more accurately represents actual driver speed behavior.

Analysis of Implied Side Friction Factors

The second surrogate, implied side friction, is an indirect way to examine the combined effects of radius (degree of curvature), speed, and superelevation. Implied side friction is related to speed, radius, and superelevation as follows:

where: hmplied

vestimated

Ractual

eactual

J;mplted = ( V estimated)

2

l2? Ractual

= implied side friction,

- e actual

=estimated operating speed (km/h), = actual curve radius (m), and =actual measured superelevation (m/m).

25

The analysis began by dividing the curve sites by lOth percentile groups. For each group, the mean implied side friction and mean accident rate was calculated. The form of the regression equation was as follows:

In (Mean Accident Rate) = Po+ p, (Mean Implied Side Friction Factor).

It was hypothesized that as the implied side friction (or friction demand) increased, accident experience would increase. This hypothesis is explained by the relationship between side friction, superelevation, degree of curvature, and speed. As side friction demand increases, the forces on the driver and vehicle may make steering and speed adjustments more difficult. As the side friction factor increases, the amount of tangential friction supply available decreases, making speed reduction after entering the curve more difficult.

After accident surrogate relationships were found, it was possible to make relative comparisons of the various strengths of the accident surrogate measures. From this comparison, it was possible to infer which accident surrogate measure was the best predictor of accident experience. This analysis was also used to compare the 85th percentile speed estimation equations found in the operational analysis. The analysis was used to make inferences about the speed that horizontal curve design should be based, based on the goodness-of-fit measures of the various design speeds examined.

26

4. RESULTS

The first section of this chapter examines the effect of superelevation on the operating speeds of drivers on horizontal curves. The second section summarizes the verification of relationships found by Anderson (3) and Fink (9) with respect to the potential accident surrogates of degree of curvature, operating speed reduction and sight distance. The third portion of the chapter summarizes the examination of superelevation and side friction as potential accident surrogates. The last section of the chapter compares accident surrogate measures found significant and makes inferences about which speed should be used in horizontal curve design.

EFFECT OF SUPERELEVATION ON 85TH PERCENTILE OPERATING SPEEDS ON HORIZONTAL CURVES

Verification of Previous Findings

Ottesen (7) verified the statistically significant relationship between 85th percentile operating speed and degree of curvature found by previous research efforts. Ottesen also found a significant relationship between operating speed on curves, degree of curvature, and length of curve. The purpose of the first portion of this analysis was to verify Ottesen's findings. The same speed-geometry database used by Ottesen, with minor corrections, was used for this study.

85th Percentile Speed Estimation Equation- Linear Model

Figure 6 is a scatterplot of 85th percentile speeds versus degree of curvature for all of the 138 curve sites in the speed database. The 85th percentile speeds appear to decrease approximately linearly with increasing degree of curvature. Ottesen noted that the scatterplot appears flat up to 4 degrees. Ottesen observed that mean values of the 85th percentile speeds on the curves less than or equal to 4 degrees were not statistically different from each other or from the mean of the 85th percentile speeds on tangent sections (97.9 km/h (60.8 milh)). Ottesen suggested that the speeds on long tangents and curves less than four degrees may not be constrained by geometric features but by posted speed limits (7).

Ottesen developed regression equations containing degree of curvature as the independent variable and 85th percentile operating speeds as the dependent variable in four forms: linear, exponential, inverse, and polynomial. All of the models had similar goodness-of-fit measures; however, the linear model was preferred due to its simplicity, practicality, conformity to the scatter plot, and reasonableness of the intercept term. The linear model for 85th percentile speeds on horizontal curves found by Ottesen (7) and verified in this study is as follows:

V85 = 103.61 - 1.95 D

27

120

<> e ~ ~ 8 ~ <> <>

<> - 100

9 88 8 i i ~ ~ §

I : ~ <> <>

.s:.

<>

E

<>

~

<> e <>

<> <>

<>

-8 <>

"C

<> s s <>

<>

Q)

80-Q) a.

<>

C/) 0) c: .. e! 60 Q) a. 0 Q)

tv

:E 00 I <>

40 c: Q) 0 ..... Q) a. .s:.

20 .. -1.0 c:o

0 ---;--·,---r---.---t···-----,---r---r-····--r··'"·· ··t ·--- ., r - ·r f · ·r- --;-- --T- --f--,--,----,--··r--·-1

0 5 10 15 20 25 30

Degree of Curvature

FIGURE 6 Scatterplotof85th Percentile Speeds at the Midpoint of the Curve Versus Degree of Curvature

where: =85th percentile speed at the midpoint of the curve (km/h), and = degree of curvature CO).

This linear model had an R2 value of 0.80 and a root MSE value of 5.2 km/h. Figure 7 shows the simple linear model against the scatter plot of the 85th percentile speeds observed for each degree of curvature. The linear model performs very well considering its simplicity.

85th Percentile Speed Estimation Equation- Multiple-Linear Model

Ottesen (7) also examined other variables listed in Table 3 for their statistical significance on 85th percentile speeds on horizontal curves. Ottesen found degree of curvature, length of curve, and the interaction between degree of curvature and length of curve statistically significant. The interaction between degree of curvature and length of curve is related to the deflection angle of the curve (deflection angle CO) = D*L * 0.00305). The equation found by Ottesen was verified and is as follows:

V85 = 102.44- 1.57 D + 0.012 L- 0.10 L'l

where: Vss D L L'l

= 85th percentile speed on the curve (km/h), = degree of curvature CO), = length of curve (m), and = deflection angle (0

).

This equation has an R2 of 0.82 and a root MSE of 5.0 km/h. Figure 8 shows the surface represented by this equation. The graph is discontinuous at the lower right due to the danger of extrapolating beyond the range of data used to determine the model parameters.

One interesting feature of this equation is that speeds generally increase as the length of curve increases on curves less than about 4.5 degrees. For curves having degree of curvature of more than 4.5 degrees, 85th percentile speeds decrease as the length of curvature increases. Ottesen noted that "on sharp curves that are relatively short, drivers may be able to flatten the curve and decelerate less. Whereas on sharp curves that are relatively long, drivers may be less inclined to flatten the curve and also have greater length over which to decelerate before the midpoint of the curve, which results in lower speeds measured at the curve midpoint" (33).

Examination of the Effects of Superelevation on 85th Percentile Operating Speeds

. The objective of the study was to examine the effect of superelevation on operating speeds on horizontal curves. For this study, it was hypothesized that as more superelevation was introduced to a curve of specific curvature, 85th percentile operating speeds would increase. This hypothesis is explained by the relationship between superelevation and side friction. For a certain degree of curvature, at a given speed, as more superelevation is applied to the curve, the side forces decrease. This reduction in side forces results in more comfort for the driver and,

29

120 -

-r. E .llo:: -"0

80 Cll Cll a. (/) C) c:

·.;:::. co

60 .... Cll 0.. 0 Cll

~ 40

w Cll 0 () .... Cll a. r. -LO 00

20

0 -1----r··--r----, .. ·-·-r·----r· ---·r---·-r · i ----, ..... , .. t ·····,-··---f.---··-,-· · r·-----,--,----f----r--.,- --.,---,---1

0 5 10 15 20 25 30

Degree of Curvature

FIGURE 7 Linear Regression Equation for 85th Percentile Speed Versus Degree of Curvature

\.;J ......

120

110-

100

V85 (km/h) 90

80

..-- ..-- 0 0 (")

Length of Curve (m}

-~ , __ _

r··r ·1 ·-o l r-r ··l·-. m o r· (") <X) 0

-.:t r--IO

···--,

10

15

20

0

5

Degree of Curvature

11110-120

.100-110

990-100

.80-90

870-80

.60-70

FIGURE 8 Multiple-Linear Regression Equation for 85th Percentile Speeds on Curves

theoretically, the less discomfort a driver experiences traversing a curve, the less the driver is inclined to reduce his or her speed. The first step in this procedure was to include superelevation in both the simple-linear and multiple-linear regression equations.

Inclusion ofSuperelevation into the Simple-Linear Regression Equation

Statistical analysis was undertaken with 85th percentile operating speeds at the midpoint of curves as the dependent variable and degree of curvature and superelevation as independent variables. Statistical analysis showed that superelevation was statistically significant in this form. The positive sign of the superelevation parameter indicated that increasing superelevation increases the expected operating speed of drivers. The equation has the form:

where:

e

V85 = 101.98- 2.08 D + 40.33 e

=85th percentile speed at the midpoint of the curve (kmlh), = degree of curvature CO), and = superelevation rate (m/m).

The R2 for this model was 0.81 with a root MSE = 5.15 km/h. Figure 9 shows the scatter plot of the observed 85th percentile speeds for each degree of curvature, and the estimated speeds for each curve site using this model. The data points show a generally linear relationship with some variation within each degree of curvature explained by the superelevation term.

Inclusion ofSuperelevation into the Multiple-Linear Regression Equation

Superelevation was also examined for inclusion in Ottesen's multiple-linear regression equation. Statistical analysis was run with 85th percentile operating speed as the dependent variable and degree of curvature, length of curve, deflection angle, and superelevation as independent variables. The hypothesis was the same as when superelevation was added to the simple linear model: for a curve of given degree of curvature and length of curve, as superelevation is increased, vehicle operating speeds would be expected to increase. Superelevation was found to have a statistically significant effect on 85th percentile speeds on curves when added to the multiple-linear equation. The equation has the form:

where: Yss D L ~

e

V85 = 99.61- 1.69 D + 0.014 L- 0.13 ~ + 71.82 e

= 85th percentile speed on the curve (km/h), = degree of curvature (0

),

= length of curve (m), = deflection angle CO), = superelevation rate (m/m),

This model has an R2 value of 0.84 and a root MSE of 4.80 km/h. Figure 10 shows the surface represented by the model given superelevation equals 0.04. The model is similar to the Ottesen

32

multiple-linear model. However, the effect of superelevation causes a vertical shift in the surface, increasing the expected 85th percentile speed.

' Other Variables and Interactions Significantly Affecting 85th Percentile Operating Speeds on Horizontal Curves

Several variables and their interactions were examined for their effect on 85th percentile speeds on curves. The interactions between the independent variables included in the multiple­linear speed estimation model (degree of curvature, length of curve, and superelevation) were examined for their significance on 85th percentile speeds on horizontal curves. Other than deflection angle (D*L *0.00305), only the interaction between length of curve and superelevation was significant. The model form is:

where: Vss D L 1:1 e

V 85 = 102.03 - 1.39 D - 0.20 1:1 + 0.039 L *e

= 85th percentile speed on the curve (km/h), = degree of curvature (D), = length of curve (m), = deflection angle CO), and = superelevation rate (rnlm).

This equation had an R2 value of 0.84 and a root MSE value of 4. 7 kmlh. While this model was statistically promising, it was eliminated from further analysis because of concerns about the structure of the model. The interaction of length of curve and superelevation is a logical phenomenon, and can be explained by the fact that as the driver has more length of curve to adjust to an adequate amount of superelevation, he or she may adjust to a higher speed at the midpoint of the curve. While this model can be explained logically, it is very complex compared to the four other models found previously. Because of its complexity and similar goodness-of-fit measures compared to the other models, it was not used in any further analysis and is not recommended for use in practice. This model is presented for its unique, yet logical, relationship between length of curve and superelevation, and because it supports the argument for inclusion of superelevation in an 85th percentile speed estimation model.

Comparison ofthe 85th Percentile Speed Estimation Models for Horizontal Curves

Each of the speed estimation equations has similar goodness-of-fit (R2 ranging from 0.80 to 0.84). However, there are issues to be considered when evaluating the logic and practicality of each model. The simple linear model, relating 85th percentile operating speed and degree of curvature, has the advantage that it explains a great amount of variance (R2 = 0.80) while consisting of only one independent variable. However, residual analysis (examining the difference between observed and estimated 85th percentile speeds) indicated that some unexplained variation may not be attributed to random error. Figure 11 shows the residual plots for each of the regression models. The desired shape of a residual plot is a "box-like" plot centered on 0.0, which indicates that a majority of remaining variance may be attributed to random error.

33

w ~

120 ,-., ..c c.. a 100 .._, "0 cu cu ~ M .5 '§ cu c..

0 .£ ·.g cu i:! cu c.. ..c -V) 00

80

60

40

20

0

0

uu;

5

!ifil!! i ~

+ 10 15

• 0

8 e ~a •

··+· 20

Degree of Curvature

• 0

i

25 30

FIGURE 9 Linear Including Superelevation Regression Equation for 85th Percentile Speeds Versus Degree of Curves

w VI

120-

110

100

V85 (km/h) 90

80-

70

60

...- ...- 0 N 0

('t)

Length of Curve (m)

d--,__r···· r··T---·r··· ··-m o 1 ·r ('t) co 0

..q 1'--LO

10

15

20

0

5

Degree of Curvature

11110-120

111100-110

CJ90-100

11180-90

11170-80

11160-70

FIGURE 10 Multiple-Linear Including Superelevation Regression Equation for 85th Percentile Speeds on Curves

The residual plot of the simple linear model ( V85 = f(D)) shows a distinct V-shaped set of residual values. This V -shape indicates the need to include additional variables to explain additional error. The same shape is present in the linear model including superelevation ( V85 = f (D,e) ). The multiple-linear model without superelevation has less ofthe V-shape seen in the linear and linear including superelevation models. However, one concern with the multiple­linear regression model not including superelevation is that the residual values are not centered on 0.0, but are centered on -3.0 kmlh, indicating that additional variables are needed to explain the remaining variation. The multiple-linear model including superelevation is roughly centered on 0.0. This may be attributed to the intercept term of the multiple-linear model without superelevation being roughly 3 km/h less than that of the model without superelevation. This reduction in the intercept term suggests the significance of superelevation on 85th percentile operating speeds on horizontal curves. The residual plot of the multiple-linear model including superelevation has more of the desired "box-like" shape. This may imply that of the four models, the multiple-linear including superelevation model more accurately represents driver operating speeds on curves.

Practically, the R2 of the multiple-linear model including three independent variables: degree of curve, length of curve, interaction of degree of curve and length of curve (deflection angle), and superelevation; only improves on the simple linear model by four percentage points. To further examine these four speed models, each was used to determine operating speed reductions, superelevation deficiencies, and implied side friction factors used in accident surrogate analysis. It was hypothesized that one model might produce consistently better results. If this was the case, which model may be the best to use in an operating-speed based consistency design procedure could be identified.

ACCIDENT SURROGATE ANALYSIS

The dependent variable considered in the statistical analysis was accident rate on horizontal curves. Anderson (8) found operating speed reduction and degree of curvature as strong accident surrogates. This study attempted to verify these relationships and test for the significance of other variables as surrogates for accident experience. The basic model form throughout the analysis is

ln (accident rate+O.l ) = f( surrogate measure )

The natural logarithm of the accident rate was used because the frequency of accidents is assumed to be Poisson. The ln (accident rate) is assumed normally distributed, which satisfies the assumption of standard regression techniques. Since over 50 percent of the curves experienced no accidents during the study period, the quantity of 0.1 was added to each accident rate before the logarithmic transformation.

Several studies have concluded that AADT and length of curve have significant effects on accident rates. Using AADT and length of curve in the accident rate greatly simplifies the modeling process, especially given the size of the data base. The assumptions necessary for

36

"'C Q) Q) a.­(J)"'C "'C Q) Q) Q) _.a. coW E-c

:0::0 Q) (/) 2: w Q) -(/) -.a ~0

"'C "iii Q)

0:::

"'C Q) Q) o.­(J)"'C "'C Q) Q) Q) _.a. coW E-c :p Q) (/) 2: w Q) -(/) -.a ~0

"'C "iii

&

V85 = f (DC)

15

10 <>

5

0

-5~ -10 t -15 _l_ <>

85th Percentile Operating Speed on Curve (km/h)

V85 = f (DC, e)

15:

10 + <> 5 __(_

I 0 ti--~~~~~lma­

-5.W -10 + <>

I

-15 _L 0 <>

85th Percentile Operating Speed on Curve (km/h)

-g Q) a.­

(J)"'C "'C Q) Q) Q) _a. coW E-c

:0::0 Q) (/) 2: w Q) -(/) -.a ~0

"'C ·u; Q)

0:::

"'C Q) Q) o.-(J)"'C "'C Q) Q) Q) _.a. coW E-c

:0::0 Q) (/) 2: w Q) -(/) -.a ~0 "'C ·u; Q)

0:::

V85 = f ( DC, LC, Deft.)

15 T I

10 + 5 + <>

<>

<>

<>

85th Percentile Operating Speed on Curve (km/h)

V85 = f (DC, LC, Deft., e)

15-0

10 ~ <>

5_;_

0 '

-s4P -10 ~

/ <> <> 0

-15 _;__

85th Percentile Operating Speed on Curve (mph)

FIGURE 11 Residual Plots of Each 85th Percentile Speed Equation

37

AADT and length of curve to be included in the accident rate are: (1) that the relationship between accident frequency (accidents/year/site) and AADT or length of curve is linear, and (2) the relationship has a slope of 1.0. The first step in analysis was to check these implicit assumptions using the data base created for this study.

Traffic Volume

In the data base used for this study, AADT ranged from 300 to 4,690 vehicles per day. It was hypothesized that the sites with higher traffic volumes would have higher accident experience, due to more exposure. The data base was sorted into five groups, each representing 20 percent of the data. The median AADT value for the group was used in analysis as the group's AADT value. Linear regression was performed using the five AADT groupings using the following model:

#accidents In ( + O.I ) = Po + P, In ( AADT )

#years x #sites

The basic accident rate (#accidents/year/site) was used initially since no other variables had been found significant on accident experience. The regression produced the following results: R2 =

0.84, MSE = 0.09, p-value = 0.03, and j3 1 = 0.81. Appendix A presents the statistical results of the analysis.

Figure 12 shows that the relationship between ln (#accidents/year/site) and ln (AADT) does have a nearly linear relationship. The other assumption necessary to include AADT in the denominator of the accident rate is that the slope of the regression line is approximately equal to 1.0. The slope was p, = 0.81 which is close to 1.0. As a result, AADT was included in the denominator of the accident rate.

Length of Curve

In the data base used for this study, the length of curve ranged from 40 to 1,040 meters (130 to 3,410 feet). It was hypothesized that sites with greater curve length would have higher accident rates. To test for inclusion of length of curve in the denominator of the accident rate, the data base was sorted into five groups, each group containing 20 percent of the data. The median length of curve was used for each group in the analysis. Linear regression was performed using the five length of curve categories in the following model:

#accidents In ( + O.I ) Po + P, In ( Length of Curve )

#years x #sites

38

-7 T

-7.5 + ,...._ I <> ~~ "' -8 r ...... "' :a 4)

-8.5 t :lt' <> ...... u u <> CI:S I ~

-9 t -s:: <> ...l I

I -9.5 + <> i I i

-10 i

5.5 6 6.5 7 7.5 8 8.5

Ln (AADT)

FIGURE 12 Ln (#accidents/year/site) Versus Ln (AADT)

#accidents ln ( + 0.1 ) = Po + P1 ln (Length of Curve )

#years x #sites

The regression produced the following results: R2 = 0.82, MSE == 0.06, p-value = 0.03, and P1 = 0.70. Appendix A presents the results of the statistical analysis.

Figure 13 shows the linear relationship between ln (#accidents/year/site) and ln (length of curve). The slope of the regression best fit line was P1 = 0. 70, which was sufficiently close to 1.0 so that length of curve could be included in the denominator ofthe accident rate.

39

--~ "' ~

-7 T I I

I -7.5 r

-8 + 0

-3' -a.5l e i '0 I ·~ -9 + .._., :

i -9.5 + 0

I i

0 0

0

-10 ~1 ----------~----------~----------~----------~ 4 4.5 5 5.5 6

Ln (Length of Curve (m))

FIGURE 13 Ln (#accidents/year/site) Versus Ln (Length of Curve)

Accident Rate Used in Analysis

The inclusion of AADT and length of curve in the denominator of the accident rate resulted in the final form for the calculation of accident rates for the remaining analyses:

where:

#accidents x 1,000,000 Accident Rate = -----...,----'------'"---

accident rate AADT L

#years x (AADT/2) x 365 x L

=accidents per million vehicle-kilometers =annual average daily traffic (vehicles), and = length of curve (km).

AADT was divided in half to account for the directional split in traffic volumes. The 50/50 directional split is reasonable for rural two-lane highways.

40

DEGREE OF CURVATURE AND OPERATING SPEED REDUCTION AS ACCIDENT SURROGATES

Degree of Curvature

Many research efforts have identified degree of curvature as a potentially strong indicator of accident experience (2,8,18,34-37). Anderson (8) found a statistically significant relationship between mean accident rates and mean degree of curvature (R2 = 0.92). The hypothesis was that as degree of curvature increases, accident rates increase.

Degree of curvature ranged in value from 0.5° to 18° in the Texas-only data base. The data base was divided into categories to ensure that at least 30 sites were included in each category, satisfying the central limit theorem. The data base was categorized as shown in Table 7, along with the number of sites in each category.

In the analysis, the mean degree of curvature for each category was computed and regressed against the natural logarithm of the mean accident rate for all sites within each degree of curvature category. The model form used in the regression was:

In (Mean Accident Rate + 0.1) = J3o + f3 1 (Mean Degree of Curvature)

The regression analysis suggested a statistically significant relationship between the natural logarithm of the mean accident rate and mean degree of curvature when categorized by degree of curvature categories. The R2 = 0.79, MSE = 0.03, and p-value = 0.0034. The statistical output from this analysis is shown in Appendix A.

The relationship between mean accident rate and mean degree of curve resulting from this analysis is presented in Figure 14. The relationship between accident rates and degree of curve verified the strong relationship found by Anderson. After this relationship was found, other independent variables were examined to determine if they added explanatory power to the relationship found between accident experience and degree of curvature.

Lane Width

Several previous research efforts have concluded that lane width has a significant impact on accident rates. It has been hypothesized that many accidents may occur because of the driver leaving the paved lane and losing control of the vehicle on gravel or earthen shoulders. Conversely, it has been hypothesized that a roadway with sufficient lane widths provides a margin of safety for the driver, and may experience fewer accidents.

41

TABLE 7 Summary of Degree of Curvature Categories

Category Range of Degree of Curvature Number of Sites in Category

1 ~ 1 66

2 > 1-2 86

3 >2- 3 76

4 > 3-4 64

5 > 4-6 72

6 > 6-9 40

7 > 9- 10 32

8 > 10 58

The lane widths in the data base ranged from 2.9 to 3.9 meters. The data base was divided into three lane width categories, each containing 33 percent of the data, categorized as follows:

lane width~ 3.3 m (10.8 ft) 3.3 m (10.8 ft) <lane width~ 3.5 m (11.5 ft)

lane width> 3.5 m (11.5 ft).

All curve sites in each of the lane width categories were divided by degree of curvature categories and regression using indicator variables used to determine if one of the lane width categories were statistically different. No statistically significant differences were found between the slopes and intercepts of the three lane width categories. The statistical results of this analysis are presented in Appendix A. Figure 15 shows the graphical results of this analysis. This analysis verified the relationship found by both Anderson (8) and Fink (9) that lane width had no significant effect on accident experience. This result could be expected since previous research has not su~gested that significant differences exist between lane widths in the ranges examined.

Care must be taken in drawing any firm conclusions from this portion of the analysis. More than half of the categories had less than 30 sites. Because of the paucity of the data, solid conclusions cannot be made about the influence of lane width on accident rates.

Lane Plus Paved Shoulder Width

This portion of the analysis examined the effect of the combined lane and adjoining paved shoulder width. Lane width by itself does not appear to affect accident rates. However,

42

0.50 --

0.45

0.40

'E ~

0.35 > ~ u 0 0.30 ~ Q) -ro 0.25-a:: -c: Q)

~ "0 0.20 w "(3 0 <( c: 0.15 ro 0 Q)

:E 0.10 0

0

0.05

0.00 -- -j l + - ·+---·-------·--+---- ···-1

0 2 4 6 8 10 12 14 16

Mean Degree of Curvature

o Data Points -Best Fit Regression Line

FIGURE 14 Mean Accident Rate Versus Mean Degree of Curvature

E ~

> ~ 0 (.)

~ Q) -ro

..j:>.. 0:::

..j:>.. -t: Q) "0 ·u (.) <{ t: ro Q)

~

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0.4

0.3

0.2

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0

-

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5 10

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I

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+ 15

Mean Degree of Curvature

A

0 LW <=3.3 m best. fit

<> 3.3 m < LW <= 3.5 m • - • best fit

A LW> 3.5 m best fit

1.._, __ ·---·--··--·····------

- --------1

20

FIGURE 15 Mean Accident Rate Versus Mean Degree of Curvature by Lane Width Category

the benefits of adding shoulders have been recognized by several studies. Additional shoulder width provides an area for the driver to recover from errors. From this observation, it was hypothesized that as the sum of the lane width plus paved shoulder width increased, accident experience would decrease.

The lane and adjacent paved shoulder widths in the data base ranged from 2.9 to 6.1 meters (9.5 to 20ft). The data base was divided into three groups, each representing 33rd percentiles of the lane plus paved shoulder widths. The three categories were:

lane plus paved shoulder width :s: 3.4 m (11.2 ft) 3.4 m (11.2 ft) <lane plus paved shoulder width :s: 4.1 m (13.5 ft)

lane plus paved shoulder width> 4.1 m (13.5 ft).

Curve sites within of the three lane plus paved shoulder width categories were divided by degree of curvature categories to determine if one or more of the lane plus paved shoulder width categories were statistically different. The statistical output is shown in Appendix A. Figure 16 presents the relationships found in the analysis. No statistically significant differences were found among the intercepts or slopes of the three lane and adjoining paved shoulder width categories. However, this figure does show a trend that as combined lane and shoulder widths decrease, accident experience increases. The p-values for the difference in slopes of the middle and higher width categories compared to the narrowest category were significant at the 0.38 and 0.08 levels, respectively. This suggests that there may be an advantage to providing lane and shoulder widths greater than about 4 m (13ft). Because of the lack of data in some categories, care must be taken in drawing conclusions using this analysis.

Total Pavement Width

Although lane width or lane plus paved shoulder width did not have statistically significant effects on accident experience, total pavement width may have significant effects on accident rates. The total paved width of the roadway was examined in the analysis because the total width affects the space available for recovery, indifferent to which lane the vehicle is traveling or to which portion of the roadway the vehicle strays. It was hypothesized that as the width of the paved surface increased, the accident experience would decrease.

The analysis with total pavement width was similar to that of lane width and combined lane and adjoining paved shoulder width. The database was divided into three equal portions representing 33 percent of the data. The total pavement widths in the data base ranged from 5.7 to 11.7 meters (18.7 to 38.4 ft). The three categories were:

pavement width :s: 6.9 m (22.6 ft) 6.9 m (22'.6 ft) <pavement width ~ 8.1 m (26.6 ft)

pavement width> 8.1 m (26.6 ft).

Each of the total pavement width categories was divided into degree of curvature categories to determine if one or more of the pavement width categories had significantly

45

different accident rates. No statistically significant differences were found between the slopes and intercepts ofthe three pavement width categories. However, the trends indicated that accident experience may be lower on wider pavement widths. The statistical results of the analysis are presented in Appendix A. Figure 17 shows the relationship between mean accident rate versus mean degree of curvature by total paved width categories. As the room available to make corrective maneuvers increases, accident experience would decrease. Since no statistically significant relationships existed and sample sizes were too small to draw firm conclusions, total pavement width was not incorporated used in any final model.

Operating Speed Reduction

Anderson ( 8) found that operating speed reduction was a strong accident surrogate. This finding was important because it supported the foundation of the operating-speed based geometric design consistency concept. Two principles of the operating-speed based consistency concept are: (1) improper speed estimation by the driver is an important factor in many horizontal curve accidents, and (2) with more speed reduction, a greater potential exists for faulty speed estimation, therefore, the higher the expected accident experience. The hypothesis for this portion of the study was that as operating speed reduction between the approach tangent and the curve increased, accident experience would increase. A secondary objective of this portion of the study was to examine the effect of including sight distance restriction into the speed profile model. The motivation for examining a sight distance was a study by Fink (9), which concluded that it is unlikely that incorporating sight distance into a speed profile model would improve the performance of the model. If the speed reductions computed accounting for sight distance gave a significantly better goodness-of-fit, then it could be concluded that sight distance might be needed to correctly apply the speed profile model.

The operating speed analysis was also used to compare the four speed estimation equations. The comparison of goodness-of-fit measures (R2 and Mean Square Error) among the four speed equations and two sight distance cases might lead to a clearer picture of which curve speed estimation model is more appropriate to apply in practice.

The analysis began by incorporating each speed estimation equation into the speed profile model. Table 8 summarizes the eight methods used in calculating the operating speed reduction from the speed profile model.

46

-E ..:.:: > :2 0 0 ~ Q) -m a:: -c ~ Q) -.....) -o ·o ~ c m Q)

:2

1

0.9

0.8

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0.5

I

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I

I

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I <> I

I

~

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fl.

o LSW<= 3.4 m

--best fit

<> 3.1 m < LSW <= 4.1 m - - - best fit

b. LSW> 4.1 m - - best fit

"6 ~ ..

0.: ~~~~~ I .

15 4

20 0 5 10

Mean Degree of Curvature

FIGURE 16 Mean Accident Rate Versus Mean Degree of Curvature by Lane Plus Paved Shoulder Width Categories

E .::.:. > ::2 u 0 ~ Q) ...... (IJ

0::

~ c 00 Q)

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::2

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6

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<>

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• • • best fit 1

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- best fit ----·· ----------------------------

FIGURE 17 Mean Accident Rate Versus Mean Degree of Curvature by Total Pavement Width Category

TABLE 8 Summary of Operating Speed Reduction Calculation Methods

Speed Estimation Model Sight Distance Restriction?

Linear y

Linear Including Superelevation y

Multiple-Linear y

Multiple-Linear Including Superelevation y

Linear N

Linear Including Superelevation N

Multiple-Linear N

Multiple-Linear Including Superelevation N

Each grouping was divided into speed reduction ranges for use in the regression analysis. All curve sites with no speed reduction (dV85 ~ 0.0) were included in a single category, while the sites remaining were divided into categories consisting of approximately fifty sites. The speed reductions ranged from 0 to 30 km/h (0 to 18.6 milh). A typical categorization is shown in Table 9. This categorization was for the speed reductions calculated using the multiple-linear speed estimation equation including superelevation with no sight distance restriction imposed on the speed profile.

A mean speed reduction and mean accident rate was calculated for each speed reduction category. The form of the regression was as follows:

ln ( Mean Accident Rate + 0.1 ) = Po + p, ( Mean Speed Reduction ).

Appendix A presents the statistical results. Table 10 summarizes the results of the regression analysis. The results of the regression for each grouping are presented based on that particular independent variable. For example, the result of the regression of mean operating speed reduction found using the linear model with sight distance restriction versus mean accident rate represents the results when grouped by the speed reductions found using the linear model with sight distance restrictions.

49

TABLE 9 Summary of Speed Reduction Categories (Multiple-Linear Speed Estimation Equation Without Sight Distance Restriction).

Category Number Speed Reduction Included in Category (km/h)

1 ~ 0

2 < 0- 1

3 < 1-3

4 < 3-5.4

5 < 5.4-9

6 < 9- 15

7 > 15

TABLE 10 Summary of Operating Speed Reduction Analysis Results

Curve Speed Estimation Model Sight Distance R2 MSE

Restriction? (km/h)

Linear y 0.92 0.0075

Linear Includip.g Superelevation y 0.78 0.0201

Multiple-Linear y 0.79 0.0202

Multiple-Linear Including y 0.84 0.0128 S uperelevation

Linear N 0.91 0.0078

Linear Including Superelevation N 0.74 0.0234

Multiple-Linear N 0.72 0.0264

Multiple-Linear Including N 0.83 0.0128 Superelevation

The results of the analysis confirmed the results of the analysis by Anderson ( 8). Operating speed reduction had a significant effect on accident rates on horizontal curves. Figure 18 shows the relationships between operating speed reduction and accident rates using each speed estimation equation, with and without sight distance restrictions. There does not seem to be a significant difference in the slopes or intercepts of any of the models, and there is not a great difference in the R2 values between each model with and without sight distance restrictions. The

50

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~- .... ,. __.,.,.

0

0.00 5.00 10.00 15.00

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+· 20.00

--···-·-·1

25.00

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- best lit

() Linear Model Including Superelevation (SO)

·---- ....... ·-best lit

A Multi-linear Model (SO)

best lit

O Multi-linear Model Including Superelevation (SO) best lit

• Linear Model (NSO)

best lit

• Linear Model Including Superelevation (NSO) best lit

A Multi-linear Mode\ (NSD)

- -bestfit

e Multi-linear Model Including Superelevation (NSO)

--- -- best lit

FIGURE 18 Mean Accident Rate Versus Mean Speed Reduction by Each Speed Estimation Model and With/Without Sight Distance Restrictions

two strongest models, the linear and multiple-linear including superelevation models, only differ by one percent with respect to the R2 values. Figure 19 shows the relationship between operating speed reduction and accident experience found using each speed estimation equation in the speed profile model without sight distance restrictions. There is little difference in the results of the analysis using each of the speed estimation equations. Figure 20 presents the relative accident rate as compared to the ;:tccident rate corresponding to a 0 krnlh speed reduction. Figure 20 indicates that accident rates triple at a speed reduction of9 krnlh (5.6 milh) and are approximately six times the accident rate of 0 kmJh reduction at a 20 km/h (12.4 milh) reduction. These speed reduction values correspond roughly to the guidelines for "good, fair, and poor" designs as proposed by Lamm (19).

This analysis suggested that operating speed reduction is a strong predictor of accident rates. This result verifies the conclusions made by Anderson ( 8) about operating speed reduction as a strong accident surrogate. The results of this analysis also concluded that sight distance increases the predictive power of these speed profile techniques, but not significantly. This result verifies the recommendations of Fink (9). Sight distance was not considered in any further analysis. This analysis also identified the linear and multiple-linear including superelevation 85th percentile speed estimation equations as potentially the best estimation equations. These results will be compared to the results of the superelevation deficiency and implied side friction analyses.

SUPERELEV ATION DEFICIENCY

Another objective of this thesis was to determine the effects of superelevation on accident experience on horizontal curves. Previous research (34-37) has examined superelevation deficiency (or error) as a potential accident surrogate. Several of these research efforts have found statistically significant relationships between accident rates and superelevation deficiency. It was hypothesized that as the amount of superelevation deficiency from an "optimum" superelevation rate increases, accident experience would increase. The point in question is how "optimum" superelevation is defined.

Superelevation deficiency was calculated using several different methods. Each method represents a "school" of horizontal curve design. "Optimum" superelevation for each curve site was estimated using four approaches. Each of these approaches each used different "design speeds" that represented different curve design methods.

The first approach used a faithful implementation of the design-speed concept. The second approach was to design superelevation for 96.6 krnlh (60 milh). This approach was of interest because of findings that the 85th percentile operating speed of drivers on long tangents was about 97 km/h (60.8 milh) (4,7). This approach is closely related to the design consistency concept that an alignment should be designed to fit the desired speed of most drivers. The third approach was to base the design speed on the estimated 85th percentile speed at the midpoint of the curve given by each of the four 85th percentile speed estimation equations. The final approach was to base the design speed on the estimated maximum 85th percentile speed on the

52

0.3

0

0.00

·-----+--·-·-··

5.00 10.00 1

15.00 20.00

Mean Speed Reduction (km/h)

25.00

FIGURE 19 Mean Accident Rate Versus Mean Speed Reduction For All Speed Estimation Models Without Sight Distance Restrictions

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4 v. ......

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0 5 10 15 20 25

Mean Speed Reduction (km/h)

FIGURE 20 Relative Mean Accident Rate Versus Mean Operating Speed Reduction

approach tangent. This approach was used to determine if the consistency concept in its strictest interpretation might be an appropriate method for alignment design.

Superelevation deficiencies were calculated for each curve site. For each method of determining design speed, the "optimum" superelevation was calculated using the AASHTO method for distributing superelevation and side friction on curves less than the maximum degree of curvature. When the degree of curvature exceeded the maximum degree of curvature as computed by AASHTO (given the- maximum superelevation rate and maximum side friction factor) the side friction factor used was kept at the maximum and the "optimum" superelevation was calculated based on this maximum side friction factor. The maximum superelevation rate for this analysis was 0.08 and the maximum side friction factors were based on current recommended AASHTO maximum side friction factors ( 6). This method of calculating optimum superelevation resulted in superelevation values up to approximately 0.60. These extreme superelevation rates typically occurred in situations with high design speeds (60 mph), and the highest degrees of curvature (D> 16°).

To provide a common ~asis for the comparison of each method, the data base was divided into eight degree of curvature categories (see Table 7). For each method, the mean superelevation deficiency for each degree of curvature category was regressed against mean accident rate for each category. The basic regression form was:

ln (accident rate + 0.1 ) = Po+ P1 (mean superelevation deficiency).

The hypothesis of this portion of the analysis was that the best fitting model (based on R2

and MSE values) of each of the design speed approaches would best represent actual driver behavior and might imply the speed best suited for use in superelevation design. The analysis also examined each speed estimation equation to determine if one of the equations produced significantly better results. Statistically significant relationships were found in the superelevation deficiency analysis for each of the approaches except the AASHTO design-speed concept. Appendix A presents the statistical results of this analysis. Table 11 summarizes the analysis results.

Figure 21 shows the relationships between mean accident rates and superelevation deficiencies. The results of this analysis indicated that although statistically significant relationships existed between superelevation deficiency and accident experience, stronger relationships were found when the estimated 85th percentile curve speeds were used as the basis for superelevation design. This result was mildly unexpected because according to the design consistency concept, stronger relationships might exist between accident rates and superelevation deficiencies found using the approach speeds, rather than the estimated curve speeds. This may suggest that strict adherence to the consistency concept may not be necessary. The 85th percentile speed on the cur;e being the strongest model may suggest that drivers may expect that a speed reduction will be necessary on sharper curves, and this fact can be relied on in design.

Not so surprising was the result that the analysis basing superelevation design on the current AASHTO method (that the minimum curve speed on the alignment controlled) had the

55

weakest relationship. The relative weakness of a faithful implementation of the AASHTO design-speed concept strongly implied that the assumptions or values used to distribute

superelevation and side friction may not represent current driver behavior. This analysis provides strong evidence that the assumptions used in the application of superelevation on curves less than maximum currently used by AASHTO may be flawed. Practically, it is not reasonable to base any firm conclusions on this analysis due to the many assumptions necessary to develop "optimum" superelevation rates using the current method to design superelevation.

TABLE 11 Summary of Superelevation Deficiency Analysis Results

Deficiency Based On: R2 MSE p-value

AASHTO Design Speed Concept (Min. V 0 on 0.04 0.13 0.64 Roadway Controls)

96.6 km/h (60 mi/h) Design Speed on all Curves 0.56 0.06 0.034

85th %ile Speed on Curve (Linear Model) 0.82 0.02 0.0018

85th %ile Speed on Curve (Linear including 0.84 0.02 0.0013 Superelevation Model)

85th %ile Speed on Curve (Multiple-linear Model) 0.79 0.03 0.0032

85th %ile Speed on Curve (Multiple-linear including 0.81 0.03 0.0023 Superelevation Model)

85th %ile Speed on Approach (Linear Model) 0.67 0.04 0.013

85th %ile Speed on Approach (Linear including 0.67 0.04 0.014 Superelevation Model)

85th %ile Speed on Approach (Multiple-linear 0.67 0.04 0.014 Model)

85th %ile Speed on Approach (Multiple-linear 0.66 0.05 0.014 including Superelevation Model)

This analysis with superelevation deficiency cannot be used to make any firm conclusions. However, the analysis lends itself as more of an "index" of the relative strength or weakness of each of the desigp. speed selection methods. The strongest models were those based on the 85th percentile speed on the curve and the maximum 85th percentile speed on the approach tangent. The weakest relationships resulted from the use of the AASHTO design-speed concept and the 96.6 km/h (60 milh) design speed. The AASHTO and 96.6 km/h (60 milh) design concepts were far outperformed by the operating speed based methods, and were subsequently dropped from further consideration. However, because of the many assumptions

56

0.18

0.16

0.14

........ E ~

> ::E

0.12 -0 0 ~ 0.10 Q) -Cll 0:: -c:: 0.08 Vl Q)

-......l "0 ·u 0

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I

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+··. -+ 0.1 0.2

I

0.3

Mean Superelevation Deficiency

0 0

0.4

I

0.5

o AASHTO Design Speed Concept

---11- best fit

o Design Speed 96.6 km/h (60 mi/h)

- -bestfit

A 85th Percentile Curve Speed Based

- - - best fit

o Maximum 85th Percentile Approach Speed Based

best fit

'· -··- --·-·-··-- -----------

FIGURE 21 Mean Accident Rate Versus Mean Superelevation Deficiency

necessary in the superelevation deficiency analysis, a more direct relationship between superelevation

IMPLIED SIDE FRICTION FACTOR (DEMAND)

Due to its relationship with superelevation, radius (degree of curvature), and speed, side friction was determined to be a better measure of superelevation deficiency. This relationship (with variables defined as used in the analysis) is widely recognized as:

where:

e

v2 fs = 127 R

= implied side friction factor,

- e

=estimated 85th percentile operating speed (km/h), =actual curve radius (m), and = actual superelevation rate (m/m).

No assumptions about the distribution of superelevation and side friction were necessary in calculating the implied side friction. The only assumption made is the speed used in the calculation. Using the implied side friction factor is a direct way to test different speeds. The use of side friction is based on the laws of physics that bind the roadway, vehicle, and driver.

The hypothesis ofthis portion of the analysis was that as the implied side friction increased, accident experience would increase. At a given speed, as superelevation is decreased for a curve of specific radius (degree of curvature), the side friction forces experienced by the driver increase. Side friction may be thought of as a relative measure of the comfort level of the driver while traversing a curve. If the side friction forces are excessive in a curve, caused by either excessive speed or inadequate superelevation, drivers may have difficulty making the adjustments needed to keep an appropriate track through the curve, or the vehicle may slide, due to excessive friction demand.

Similar to the superelevation deficiency analysis, this analysis relied on the selection of an appropriate design speed. From the superelevation deficiency analysis, it was concluded that the two most appropriate speeds used for design may be the 85th percentile operating speed on the curve and the maximum 85th percentile operating speed on the approach tangent. This analysis incorporates both the 85th percentile curve and approach speeds estimated by each of the four speed estimation equations. A secondary hypothesis of this analysis was that the better method of design speed selection could be implied from the strength of the relationships between the implied side friction (calculated using different operating speeds) and accident rates. It was hoped that this analysis could be used to imply which speed to use for design of horizontal curves and superelevation.

The analysis began by calculating implied side friction factors using the relationship between side friction, radius, speed, and superelevation. The implied side friction was calculated eight times for each site, using the actual radius and superelevation on the curve, and the

58

estimated operating speed calculated using each speed estimation equation and the speed profile model. Table 12 summarizes the eight methods used in calculating the implied side friction.

TABLE 12 Summary oflmplied Side Friction Calculation Methods

Speed Estimation Model Curve Or Approach Tangent Speed?

Linear Curve

Linear Including Superelevation Curve

Multiple-Linear Curve

Multiple-Linear Including Superelevation Curve

Linear Tangent

Linear Including Superelevation Tangent

Multiple-Linear Tangent

Multiple-Linear Including Superelevation Tangent

The implied side friction values ranged from approximately 0.001 to 0.700. The curve sites were separated into 1Oth percentile groups based on each method used to calculate implied side friction. The mean implied side friction and mean accident rates were calculated for each 1Oth percentile. The form of the regression analysis was as follows:

ln (Mean Accident Rate+ 0.1) = ~0 + ~ 1 (Mean Implied Side Friction Factor).

Appendix A presents the statistical results of this analysis. Table 13 presents a summary of the regression analysis. Each result includes the R2, MSE, and p-values for the regression results where the independent variable was grouped by 1Oth percentile groups of that specific independent variable. For example, the results presented for the 85th percentile speed on the curve using the linear model used the groupings produced by separating the curve sites by the implied side friction values calculated using the speed estimated using the linear model on the curve.

The analysis confirmed the hypothesized relationship between mean implied side friction and mean accident rates, indicating that some of these relationships are stronger than the accident rate-degree of curvature and accident rate-operating speed reduction relationships previously found. The analysis also revealed that each model based on estimated 85th percentile curve speeds, except the linear model including superelevation, were stronger predictors of accident rates that any model based on the maximum 85th percentile approach speed. Figure 22 shows the 85th percentile curve speed relationships between mean accident rate and mean implied side friction for the linear and multiple-linear including superelevation models. Due to the many models tested, only the best fitting models, 85th percentile speed at the midpoint of the curve given by the linear and multiple-linear including superelev~tion, are included in Figure 22.

59

TABLE 13 Summary oflmplied Side Friction Analysis Results

Independent Variable R2 MSE p-value

85th Percentile Speed on Curve (Linear Model) 0.95 0.004 0.0001

85th Percentile Speed on Curve (Linear Model 0.79 0.018 0.0005 including Superelevation)

85th Percentile Speed on Curve (Multiple-Linear 0.93 0.006 0.0001 Model)

85th Percentile Speed on Curve (Multiple-Linear 0.98 0.001 0.0001 Model Including Superelevation)

Maximum 85th Percentile Speed on Approach 0.89 0.009 0.0001 (Linear Model)

Maximum 85th Percentile Speed on Approach 0.90 0.008 0.0001 (Linear Model Including Superelevation)

Maximum 85th Percentile Speed on Approach 0.91 0.007 0.0001 (Multiple-Linear Model)

Maximum 85th Percentile Speed on Approach 0.85 0.013 0.0002 (Multiple-Linear Model Including Superelevation)

SUMMARY OF ACCIDENT ANALYSIS

Several independent variables were analyzed for relationships with accident experience on horizontal curves. Variables found having significant relationships with accident rates on horizontal curves were degree of curvature, operating speed reduction, and the implied side friction factor. AADT and length of curve also had significant effects on accident experience and were included in the denominator of the accident rate.

One objective of this study was to examine each speed model in each of the accident surrogate analyses to determine if any trend existed which may have indicated the stronger model. To examine the speed model selection question and further detemline if the strength of each relationship was dependent upon the grouping used in each analysis, all models (each accident surrogate: degree of curvature, operating speed reduction, and implied side friction) were rerun using the groupings for each independent variable. The summarized output of this analysis is shown in Appendix B. This analysis indicated that some clear trends existed that were independent of the grouping used. This analysis made comparisons between MSE and R2

values of each accident surrogate relationship (and each speed model) easier, due to the common grouping of variables.

60

E .::.:.

~ --0 0 ~ <1) ...... co

0::: 0\ ......

c: - <1) "0 '(3 0 ~ c: co <1)

~

0.25 ..

0.20

0.15

0.10

0

0.05 f

I " I rre'

0.00 ~--~------- -+. -----+---

0 0.05 0.1 0.15

" 0

t.

0.2

'-

' ' ' '/"[j

0.25

Mean Implied Side Friction Factor

0

,o ,

'

0.3 0.35

o Linear Model

- ·- -- -best fit

o Multi-linear Model Including Superelevation

• • • best fit

FIGURE 22 Mean Accident Rate Versus Mean Implied Side Friction

The analysis showed that each accident surrogate continued to perform consistently well with each grouping. The analysis further suggested that the strength of each model was not dependent on the grouping used in regression. The accident surrogates from strongest to weakest were:

1) mean implied side friction using the 85th percentile speed on the curve, 2) mean degree of curvature, 3) mean implied side friction using the maximum 85th percentile speed on the

approach, and 4) mean operating speed reduction.

Within speed estimation equations, the multiple-linear models (both with and without superelevation) generally outperformed the simple linear models (with and without superelevation). This outcome might be expected since these equations incorporate more variables believed to affect operating speeds on curves. However, the simple linear model cannot be overlooked, as it performed very well while only consisting of one independent variable, degree of curvature.

SUMMARY

This chapter reported the findings of the operational and safety analysis of horizontal curves on rural two-lane highways. Several independent variables were analyzed to determine their effect on the operational and safety performance of horizontal curves.

Operational Analysis

Degree of curvature, length of curve, deflection angle, and superelevation were all found to significantly affect 85th percentile operating speeds on horizontal curves. Two regression equations found by Ottesen were verified and an additional two equations were found that included superelevation. The four models incorporating combinations of degree of curvature, length of curve, deflection angle, and superelevation resulted from linear and multiple-linear regression analyses. The four forms were linear ( V 85= f(D) ), linear including superelevation ( V85=f(D,e) ), multiple-linear ( V85 = f(D,L,~) ), and multiple-linear including superelevation ( Vss= f(D,L,~,e )).

Accident Analysis

Including the variables AADT and length of curve in the denominator of the accident rate simplified the modeling process. The implicit assumptions necessary to include AADT and length of curve in the accident rate were satisfied. Both AADT and length of curve were included in the denominator of the accident rate used in all analyses:

Accident Rate = __ #_a_cc_i_d_en_t_s_x_l..:..,o_o_o~,_oo_o __ #years x (AADT/2) x 365 x L

62

Linear regression was performed to evaluate the relationship between mean degree of curvature and mean accident experience for eight degree of curvature categories. Degree of curvature was verified as having a statistically significant effect on accident rates, having an R2

value of0.79. Other variables examined for inclusion into a final model were lane width, lane and adjoining paved shoulder width, and total pavement width. None of these variables had statistically significant effects on accident experience, possibly because site selection criteria limited the range of values.

Operating speed reduction was verified as having a significant effect on accident rates. Regression was performed using the mean accident rate and mean operating speed reduction for six or seven categories, depending on which of the four speed estimation models were used. Results indicated that operating speed reduction, independent of the speed estimation model employed, was a good predictor of accident rates, with R2 values ranging from 0.72 to 0.92. The strong relationship between operating speed reduction and accident experience suggests that operating speed reduction may be an important parameter to include in curve design procedures.

The operating speed reduction analysis also examined sight distance as a variable to determine if it is necessary for accurate performance of the speed profile model. The analysis showed only slightly better results when using sight distance in the speed profile model. Sight distance information is not recommended for inclusion into a speed profile model. The small increase in accuracy does not justify the cost of collecting sight distance data.

The examination of superelevation deficiency included analysis of several "design speeds" that represented various schools of horizontal curve design. "Optimum" superelevation rates were determined using four approaches: AASHTO design-speed concept, design speed of 96.6 km/h (60 milh), 85th percentile operating speed on the curve, and maximum 85th percentile operating speed on the approach tangent. The analysis provided evidence that the current method of curve design in the U.S. may not be representative of the behavior of drivers, based on the relative strength of each model. The AASHTO design-speed concept and the -96.6 km/h (60 mi!h) design speed were not included in further analysis because of weak performance using superelevation deficiency as an index of the strength of the concept. This analysis provided a relative index of the strength of the various design methods examined, and led to the implied side friction analysis.

Implied side friction was examined because of its relationship to other curve design parameters examined: radius (degree of curvature), operating speed, and superelevation. Linear regression was performed to evaluate the relationship between mean implied side friction and mean accident rate. The speeds used to calculate implied side friction were the 85th percentile speed on the curve and the maximum 85th percentile speed on the approach tangent, each calculated using the four speed estimation equations. Implied side friction was found to have the strongest relationship with accident experience, with R2 values ranging from 0.85 to 0.98.

To determine if the results may have been influenced by the groupings of each variable, the analysis was run using each accident surrogate when separated by other independent variable's groupings. This analysis showed that the accident surrogates were, from strongest to

63

weakest: implied side friction using 85th percentile speeds on the curve with R2 values ranging from 0.79 to 0.98; degree of curvature, with R2 values ranging from 0.71 to 0.96; implied side friction using the maximum 85th percentile speed on the approach tangent with R2 values ranging from 0.71 to 0.95; and operating speed reduction, with R2 values ranging from 0.61 to 0.92. This analysis consistently indicated that implied side friction was the strongest accident surrogate. This supports the notion that degree of curvature, operating speed, and superelevation are all important factors in the development of an operating-speed based horizontal curve design procedure. ·

64

5. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

Previous research efforts have shown that horizontal curves experience higher accident rates and severity than tangent sections of rural two-lane highways. Current horizontal curve design policy uses the application of superelevation as the basis for providing a consistent alignment. However, several flaws in the application and distribution of superelevation and side friction weaken the physical relationship between vehicle and roadway. These flaws include: (1) the assumption that drivers will operate at the design speed of a roadway when they can traverse the alignment at higher speeds, and (2) that consistency problems may result from the application of different maximum superelevation rates among different states. This thesis evaluated the effects of superelevation on 85th percentile operating speeds on curves and the effects of superelevation and side friction on the safety of horizontal curves on rural two-lane roadways.

SUMMARY

The operational analysis used speed data collected on 138 curves in three geographic regions ofthe U.S.: the East (Pennsylvania, New York), the South (Texas), and the West (Washington, Oregon). Speed models developed in earlier studies at the Texas Transportation Institute ( 4, 7) using these speed data were verified and examined for the inclusion of superelevation. It was found that superelevation had significant effects on 85th percentile operating speeds on horizontal curves. The analysis showed that as superelevation increased on a curve, operating speeds at the midpoint of the curve increased. Four speed estimation models for 85th percentile speeds on horizontal curves were found: a simple linear model V85 = f(D), and three multiple-linear models: V85 = f(D,e); V85 = f(D,L,D*L); and V85 = f(D,L,D*L,e).

Accident data were collected on 247 curves on 13 Texas farril-to-market roadways. These two-lane rural roadways were functionally classified as minor arterials or collectors. Police accident reports were used to identify the curve-related accidents during a period of seven years on these roadways. The independent variables examined in the accident surrogate analysis included AADT, deflection angle, length of curve, degree of curve, superelevation deficiency, implied side friction factor, lane width, lane and shoulder width, and total pavement width.

To simplify the accident analysis, the implicit assumptions associated with the accident rate per million-vehicles kilometers were checked. The assumptions that both AADT and length of curve were linearly related to the natural logarithm of the basic accident rate (accidents/year/site), with a slope of 1.0 were checked. These assumptions were verified and both AADT and length of curve were included in the denominator of the accident rate.

Degree of curvature was found to be a strong accident surrogate measure. Linear regression was performed on the mean accident rate and mean degree of curvature using eight categories of degree of curvature. The R2 value ofthis analysis was 0.79. Using categorizations based on other independent variables, R2 values for the relationship between mean accident rate

65

and mean degree of curvature ranged from 0. 79 to 0.96. This result verified the findings of Anderson (3).

Lane width was not found to be a significant variable in this accident-geometry database. Although past research efforts have suggested that lane width does have a significant relationship with accident experience, lane width was limited in the site selection criteria. Total pavement width and lane plus paved shoulder width did not have statistically significant effects on accident rates.

The analysis showed that operating speed reduction was an effective predictor of accident rates. Linear regression produced R2 values ranging from 0.61 to 0.92. This result verified the findings of Anderson (3). The effect of sight distance was examined by using the speed profile model with and without sight distance restrictions. The results suggested only slightly stronger results when the effect of sight distance was incorporated into the speed profile model.

Superelevation deficiency was used as an index of the relative strength of four different horizontal curve design methods: AASHTO design-speed concept, 96.6 km/h (60 milh) design speed, 85th percentile speed on the curve, and maximum 85th percentile speed on the tangent. The results showed that the strongest relationships were found using the 85th percentile speed on the curve and tangent, while the weakest relationships were found using the AASHTO and 96.6 km/h (60 mi/h) design speed methods.

Linear regression using accident experience as a function of implied side friction produced the strongest accident prediction models. This was expected due to the relationship of implied side friction with three other design parameters that were strong accident surrogates: degree of curvature, superelevation, and operating speed. Two speeds were used in calculating implied side friction: 85th percentile speed on the curve, and maximum 85th percentile speeds on the approach tangent. The results showed that the strongest relationships existed when using the 85th percentile speed on the curve. R2 values ranged from 0.73 to 0.98 when using the 85th percentile speed on the curve as the speed used in calculating implied side friction and from 0.71 to 0.95 when using the approach speed. This result may imply that the best speed to use for horizontal curve design is the 85th percentile speed on the curve.

CONCLUSIONS

The following conclusions were made from this analysis:

• Degree of curvature, length of eurve, deflection angle, and superelevation all have statistically significant effects on vehicle operating speeds on horizontal curves.

• The simple linear model and multiple-linear including superelevation model consistently performed best in the results of the operating speed reduction and implied side friction accident surrogate analyses.

66

• Implied side friction calculated using 85th percentile curve speed is the strongest accident surrogate found in this analysis, followed by degree of curvature, implied side friction calculated using maximum 85th percentile speeds on the tangent, and operating speed reduction.

RECOMMENDATIONS

The speed estimation equations developed by Ottesen (7) were verified by this study. Superelevation was found to significantly influence operating speeds on horizontal curves. There was no clear indication that any particular speed estimation equation was superior. In practice, the simple linear model or multiple-linear not including superelevation model would be appropriate for developing speed profiles for operating speed consistency checks in initial alignment design. The multiple-linear including superelevation model could be used to compute speed profiles on existing roadways and for final consistency checks on new designs.

The accident surrogate analysis showed that implied side friction was the strongest predictor of accident experience. While side friction may have a strong influence on accident rates, its components (degree of curvature, operating speed, and superelevation) also have significant relationships to accident experience.

The results of this analysis suggest that a two dimensional approach may be necessary to ensure the design of a safe alignment. First, the strength of the relationships found using the 85th percentile operating speed on the curve suggests that the 85th percentile speed on the curve should be used to design curve parameters, including superelevation. While the results of this study have identified several concerns with our current procedure, it is recognized that the design-speed concept works well when the design speed selected accurately reflects the desired speeds of drivers. Problems only arise when the speed chosen for design is lower than the expected operating speed. One interesting feature in several analyses is that when curves sharper than four or five degrees are examined (the maximum degree of curvature for a design speed of 96.6 kmlh (60 mi/h)), operating speed reductions begin to take place as well as noticeable increases in accidents. A procedure similar to Australia's, where the design-speed concept is used above 100 km/h (62 milh) and an iterative procedure is used on low-speed alignments (<100 km/h (62 milh)), would be appropriate for use in the United States.

The second facet of an updated design procedure would incorporate a check of operating speed consistency. The operating speed reduction analysis supports the idea of a design procedure that would use a feedback loop to check for and address operating speed consistency problems. Further research is needed to determine if operating speed reduction thresholds exist where accident rates significantly differ. This research should be conducted on a larger data base, with a considerable representation of each of the three speed profile cases. A set of operating speed reduction ranges, similar to one set of criteria in the German method of consistency checks would be appropriate in U.S. design procedures.

67

In conclusion, the current U.S. horizontal curve design procedure may have flaws that limit its ability to produce consistently safe alignments. This analysis suggests that an operating­speed based design procedure would provide a method of providing a consistent alignment on rural two-lane highways. It is recommended that curve parameters, including superelevation, be designed using the expected 85th percentile speed on the curve and then consistency checks, based on operating speed reduction and side friction demand, be incorporated into an operating speed based design procedure.

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1. D.W. Reinfurt, C.V. Zegeer, B.J. Shelton, and T.R. Neuman. Analysis ofVehicle Operations on Horizontal Curves. In Transportation Research Record 1318, TRB, National Research Council, Washington, D.C., 1991. pp. 43-50.

2. J.C. Glennon, T.R. Neuman, and J.E. Leisch. Saftty and Operational Considerations for Design of Rural Highway Curves. Report FHW AIRD-86/035. FHWA, U.S. Department ofTransportation, 1985.

3. M.C. Good. Road Curve Geometry and Driver Behavior. ARRB Special Report No. 15. Australian Road Research Board, Vermont South, Australia, 1978.

4. R. A. Krammes, R. Q. Brackett, M.A. Shafer, J. L. Ottesen, I. B. Anderson, K. L. Fink, K. M. Collins, 0. J. Pendleton, and C. J. Messer. Horizontal Alignment Design Consistency for Rural Two-Lane Highways. Report FHWA-RD-94-034. FHWA, U.S. Department of Transportation, 1993.

5. J. R. McLean. "Driver Speed Behaviour and Rural Road Alignment Design." Traffic Engineering & Control, Vol. 22, No. 4, 1981, pp. 208-211.

6. A Policy of Geometric Design of Highways and Streets. Washington, DC: American Association of State Highway and Transportation Officials, 1990, with an addendum on "Interim Selected Metric Values for Geometric Design," 1993.

7. J. L. Ottesen. Speed Reduction Mode/for a US. Operating-Speed-Based Design Consistency Procedure on Two-Lane Rural Highways. Master of Science Thesis. Texas A&M University, College Station, Texas, 1993.

8. I. B. Anderson. An Evaluation of Operating Speed Reduction as a Surrogate Measure for Accident Experience on Horizontal Curves on Two-Lane Rural Highways. Master of Science Thesis. Texas A&M University, College Station, Texas, 1993.

9., K. L. Fink. Tangent Length and Sight Distance Effects on Accident Rates at Horizontal Curves on Two-Lane Rural Highways. Master of Science Thesis. Texas A&M University, College Station, Texas, 1993.

10. K. Fitzpatrick and K. Kahl. Historical and Literature Review of Horizontal Curve Design. Report TX-92/1949-1. Texas Department of Transportation, College Station, Texas, 1992.

11. W.R. Young. Road Design for Uniform Speed. Engineering News Record, Vol. 105, p. 539.

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12. J. Barnett. Safe Side Friction Factors and Superelevation Design. Proceedings of the Highway Research Board, Vol. 16, 1936, pp. 69-80.

13. C. J. Messer, J. M. Mounce, and R. Q. Brackett. Highway Geometric Design Consistency Related to Driver Expectancy, Vol. Ill, Research Report. Report FHW AIRD-81103 7. FHWA, U.S. Department of Transportation, 1981.

14. M.A. Chowdhury, D. L. Warren, and H. Bissell. Analysis of Advisory Speed Setting Criteria. Public Roads, Vol. 55, No.3, December 1991, pp. 65-71.

15. J. R. McLean. Speeds, Friction Factors, and Alignment Design Standards. Research Report ARR 154. Australian Road Research Board, Sydney, Austrailia, 1988.

16. G. Kanellaidis. Aspects of Highway Superelevation Design. Journal ofTransportation Engineering, Vol. 117, No. 6, November/December 1991. pp. 624-632.

17. J. C. Hayward. Highway Alignment and Superelevation: Some Design-Speed Misconceptions. In Transportation Research Record 757, TRB, National Research Council, Washington, D.C., 1980, pp. 22-25.

18. J. E. Leisch and J.P. Leisch. New Concepts in Design-Speed Application. In Transportation Research Record 631, TRB, National Research Council, Washington, D.C.,1977, pp. 5-14.

19. R. Lamm, E. M. Choueiri, J. C. Hayward, and A. Paluri. Possible Design Procedure to Promote Design Consistency in Highway Geometric Design on Two-Lane Rural Roads. In Transportation Research Record 1195, TRB, National Research Council, Washington, D.C., 1988, pp. 111-121.

20. R. Lamm and J. G. Cargin. Translation of the "Guidelines for the Design of Roads (RAS-L-1)," Edition 1984, Federal Republic of Germany and "The Swiss Norm SN 640080a, Highway Design, Fundamentals, Speed as a Design Element" as discussed by K. Dietrich, M. Rotach, and E. Boppart in Road Design, Zurich, Switzerland: ETH, Institute ofTraffic Planning and Transport Technique, Edition 1983. May 1985.

21. J. Craus and M. Livneh. Superelevation and Curvature of Horizontal Curves. In Transportation Research Record 685, TRB, National Research Council, Washington, D.C., 1978. pp. 7-13.

22. R. Lamm. Driving Dynamic Considerations: A Comparison of German and American Friciton Coefficients for Highway Design. In Transportation Research Record 960. TRB, National Research Council, Washington, D.C., 1984. pp. 13-20.

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23. Rural Road Design: Guide to the Geometric Design of Rural Roads. Sydney, Australia. Austroads, 1989.

24. R. Lamm and E. Choueiri. Operating Speeds and Accident Rates on Two-Lane Rural Highway Curved Sections -- Investigations about Consistency and Inconsistency in Horizontal Alignment. Rural Roads Speeds Inconsistencies Design Methods, Part I, State University ofNew York Research Foundation, Albany, N.Y., July 1987.

25. A. Taragin. Driver Performance on Horizontal Curves. Public Roads. Vol. 28, No.2, June 1954, pp. 21-39.

26. R, Lamm, A.K. Guenther, and E.M. Choueiri. Safety Module for Highway Design: Applied manually of Using CAD. Paper Presented at the 71st Annual Meeting of the Transportation Research Board. Washington, D.C. January 1992.

27. J. Emmerson. A Note on Speed-Road Curvature Relationships. Traffic Engineering & Control. November 1970, p. 369.

28. J.M. Gambard and G. Louah. Vitesses Practiques et Geometrie de Ia Route. SETRA, Paris, France, 1986.

29. H.P. Lindemann and B. Ranft. Geschwindigkeit in Kurven. ITH-Honggerberg, Institut fiir Verkegrsplanung und Transporttechnik, Zurich, Switzerland, 1978.

30. G. Kannellaidis, J. Golias, and S. Efstathiadis. Drivers Speed Behaviour on Rural Road Curves. Traffic Engineering & Control, Vol. 31, No. 7/8, 1990, pp. 414-415.

31. R. Lamm, E.M. Choueiri, and A. Paluri. A Design Method to Deteremine Critical Operating Speed Inconsistencies on Two-Lane Rural Roads in the State of New York. Rural Roads Speed Inconsistencies Design Methods, Part II, State University ofNew York Research Foundation, Albany, New York, October 1987.

32. Geschwidigkeit als Projektierungselement. Schweizer Norm 640 080b. Vereinigung Schweizerischer Strassenfachleute, Zurich, Switzerland, 1991.

33. J.L. Ottesen and R.A. Krammes. Speed Profile Model for a U.S. Operating-Speed-Based Design Consistency Evaluation Procedure. Paper Presented at the 73rd Annual Meeting of the Transportation Research Board. Washington, D.C. January 1994.

34. C. Zegeer, R. Stewart, D. Reinfurt, F. Council, T. Neuman, E. Hamilton, T. Miller, and W. Hunter. Cost-Effective Geometric Improvements for safety Upgrading of Horizontal Curves. Report FHWA-RD-90-021. FHWA, U.S. Department of Transportation, 1991.

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35. T. K. Datta, D. D. Perkins, J. I. Taylor, and H. T. Thompson. Accident Surrogates for Use in Analyzing Highway Safety Hazards. Volume 2. Technical Report. Report FHWAIRD-82/104. FHWA, U.S. Department ofTransportation, 1983.

36. P. L. Zador, H. S. Stein, J. Hall, and P. Wright. Superelevation and Roadway Geometry: Deficiency at Crash Sites and on Steep Grades. Insurance Institute for Highway Safety, Arlington, Virginia, 1985.

37. K. W. Terhune and M. R. Parker. Evaluation of Accident Surrogates for Safety Analysis of Rural Highways. Volume 2. Technical Report. Report FHW A/RD-86/128. FHWA, U.S. Department ofTransportation, 1986.

38. R. Lamm, E. Choueiri, and T. Mailaender. Side Friction Demand Versus Side Friction Assumed for Curve Design on Two-Lane Rural Highways. In Transportation Research Record 1303, TRB, National Research Council, 1991. pp. 11-21.

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APPENDIX A

STATISTICAL ANALYSIS OUTPUT

73

Source

Model Error c Total

Root MSE Oep Mean c.v.

Variable OF

INTERCEP LNAOT

Source

Model Error C Total

Root MSE Oep Mean c.v.

Variable OF

INTERCEP LNLC

Analysis of Variance

Sum of Mean OF Squares Square

1 1. 38394 1 . 38394 3 0.25859 0.08620 4 1.64253

0.29359 -8.70850 -3.37133

R-square Adj R-sq

Parameter Estimates

F Value

16.056

0.8426 0.7901

Parameter Standard T for HO:

Prob>F

0.0279

Estimate Error Parameter=O Prob > ITI

-14.359410 1.41637990 -10.138 0.0020 0.806557 0.20129002 4.007 0.0279

FIGURE AI. AADT

Analysis of Variance

Sum of Mean OF Squares Square F Value Prob>F

1 0.75163 0.75163 13.654 0.0344 3 0.16514 0.05505 4 0.91677

0.23462 R-square 0.8199 -8.70921 Adj R-sq 0.7598 -2.69396

Parameter Estimates

Parameter Standard T for HO: Estimate Error Parameter=O Prob > ITI

-12.094192 0.92205196 -13.117 0.0010 0.701460 0.18983295 3.695 0.0344

FIGURE A2. Length of Curve

74

Source

Model Error C Total

Root MSE Oep Mean c.v.

Variable OF

INTERCEP MEANOC

OF

1 6 7

Analysis of Variance

S1.111 of Mean Squares Square

0.63265 0.63265 0.17256 0.02876 0.80521

0.16959 -1.89865

R-square Adj R-sq

-8.93197

Parameter Estimates

F Value

21.998

0.7857 0.7500

Parameter Standard T for HO:

Prob>F

0.0034

Estimate Error Parameter=O Prob > ITI

-2.280883 0.10117623 -22.544 0.0001 0.064016 0.01364904 4.690 0.0034

FIGURE A3. Mean Accident Rate Versus Mean Degree of Curvature.

Analysis of Variance

S1.111 of Mean Source OF Squares Square F Value Prob>F

Model 5 6.68785 1.33757 12.169 0.0001 Error 17 1.86861 0.10992 C Total 22 8.55646

Root MSE 0.33154 R-square 0.7816 Oep Mean -1.53813 Adj R-sq 0. 7174 c.v. -21.55472

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP 1 -2.083820 0.20161486 -10.336 0.0001 MEANOC 1 0.105029 0.02810336 3.737 0.0016 Z1 1 -0.120112 0.28229521 -0.425 0.6758 Z2 1 -0.274214 0.27653074 -0.992 0.3353 XZ1 1 -0.001860 0.03842033 -0.048 0.9620 XZ2 1 0.031303 0.03731043 0.839 0.4131

FIGURE A4. Lane Width (2.9 m~LW~3.3 m, 3.3 m<LW~3.5 m, 3.5<LW~3.9)

75

Analysis of Variance

Sllll of Mean Source OF Squares Square F Value Prob>F

Model 5 8.21501 1.64300 . 16.516 0.0001 Error 18 1. 79064 0.09948 C Total 23 10.00565

Root MSE 0.31540 R-square 0.8210 Oep Mean -1.47847 Adj R-sq 0.7713 c.v. -21.33311

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP 1 -2.288748 0.19341540 -11.833 0.0001 MEANDC 1 0.165337 0.02716629 6.086 0.0001 Z1 1 0.078692 0.27297637 0.288 0. 7764 Z2 1 0.023879 0.26754974 0.089 0.9299 XZ1 1 -0.033764 0.03804247 -0.888 0.3865 XZ2 1 -0.067949 0.03646139 -1.864 0.0788

FIGURE A5. Lane and Adjoining Paved Shoulder Width (2.9 m~L+SW~3.4 m, 3.4 m<L+SW~4.1 m, 4.1 <L+SW~6.1)

Source

Model Error C Total

Root MSE Dep Mean c.v.

Variable OF

INTERCEP MEANDC Z1 Z2 XZ1 XZ2

OF

5 23 28

Analysis of Variance

Sum of Mean Squares Square

19.04503 3.80901 5.25877 0.22864

24.30380

0.47817 -1.23005

R-square Adj R-sq

-38.87370

Parameter Estimates

F Value

16.659

0.7836 0. 7366

Parameter Standard T for HO:

Prob>F

0.0001

Estimate Error Parameter=O Prob > ITI

-2.178091 0.28752836 -7.575 0.0001 0.152297 0.04197444 3.628 0.0014

-0.355643 0.38538942 -0.923 0.3657 0.095669 o.3849m3 0.249 0.8059 0.074700 0.05192190 1.439 0.1637

-0.052786 0.05152548 -1.024 0.3163

FIGURE A6. Total Paved Width (5.7 m~PW~6.9 m, 6.9 m<PW~8.1 m, 8.1 <PW~11.7)

76

Source

Model Error C Total

Root MSE Oep Mean c.v.

Variable OF

INTERCEP MLNREONS

Analysis of Variance

Sum of Mean OF Squares Square

1 4 5

0.31534 0.31534 0.03118 o.oon9 0.34652

0.08828 -1.94442

R-square Adj R-sq

-4.54039

Parameter Estimates

F Value

40.459

0.9100 0.8875

Parameter Standard T for HO:

Prob>F

0.0031

Estimate Error Parameter=O Prob > ITI

-2.136088 0.04697916 -45.469 0.0001 0.028946 0.00455065 6.361 0.0031

FIGURE A 7. Accident Rate vs. Speed Reduction Linear Speed Model-No Sight Distance Restrictions

Source

Model Error c Total

Root MSE Oep Mean c.v.

Variable OF

INTERCEP MLNREOSO

Analysis of Variance

Sum of Mean OF Squares Square F Value Prob>F

1 4 5

0.33014 0.02984 0:35998

0.33014 0.00746

44.257 0.0027

0.08637 -1.94150 -4.44854

R-square Adj R-sq

Parameter Estimates

0.9171 0.8964

Parameter Standard T for HO: Estimate Error Parameter=O

-2.138000 0.04599656 -46.482 o.029969 o.oo45o4n 6.653

Prob > ITI

0.0001 0.0027

FIGURE A8. Accident Rate vs. Speed Reduction Linear Speed Model-Sight Distance Restrictions

77

Analysis of Variance

Sun of Mean Source OF Squares Square F Value Prob>F

Model 1 0.33199 0.33199 14.184 0.0131 Error 5 0.11703 0.02341 c Total 6 0.44903

Root MSE 0.15299 R-square 0.7394 Oep Mean -1.85213 Adj R-sq 0.6872 c.v. -8.26038

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.048950 0.07794218 -26.288 0.0001 MLNERDNS 0.036431 0.00967334 3.766 0.0131

FIGURE A9. Accident Rate vs. Speed Reduction Linear Including Superelevation Speed Model-No Sight Distance Restrictions

Analysis of Variance

Sun of Mean Source OF Squares Square F Value Prob>F

Model 1 0.36295 0.36295 18.067 0.0081 Error 5 0.10045 0.02009 C Total 6 0.46340

Root MSE 0.14174 R-square 0. 7832 Oep Mean -1.85000 Adj R-sq 0. 7399 c.v. -7.66141

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O 'Prob > IT I

INTERCEP -2.056899 0.07238259 -28.417 0.0001 MLNERDSO 0.039074 0.00919270 4.251 0.0081

FIGURE AlO. Accident Rate vs. Speed Reduction Linear Including Superelevation Speed Model-Sight Distance Restrictions

78

Source

Model Error c Total

Root MSE Oep Mean c.v.

Variable OF

INTERCEP MMLREONS

Analysis of Variance

OF Sum of

Squares Mean

Square

1 4 5

0.26879 0.10544 0.37423

0.26879 0.02636

0.16236 -1.85681

R-square Adj R-sq

-8.74407

Parameter Estimates

F Value

10.196

0.7182 0.6478

Parameter Standard T for HO:

Prob>F

0.0331

Estimate Error Parameter=O Prob > ITI

-2.072644 0.09466838 -21.894 0.0001 0.028059 0.00878706 3.193 0.0331

FIGURE All. Accident Rate vs. Speed Reduction Multiple-Linear Speed Model-No Sight Distance Restrictions

Analysis of Variance

Sum of Mean Source OF Squares Square F Value Prob>F

Model 1 0.31018 0.31018 15.395 0.0172 Error 4 0.08059 0.02015 c Total 5 0.39078

Root MSE 0.14194 R-square 0.7938 Oep Mean -1.84948 Adj R-sq 0. 7422 c.v. -7.67479

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.082256 0.08293159 -25.108 0.0001 MMLREOSO 0.031130 0.00793387 3.924 0.0172

FIGURE A12. Accident Rate vs. Speed Reduction Multiple-Linear Speed Model-Sight Distance Restrictions

79

Source

Model Error C Total

Root MSE Oep Mean c.v.

Variable OF

INTERCEP MMLERONS

Analysis of Variance

Sl.lll of Mean OF Squares Square

1 5 6

0.31383 0.31383 0.06412 0.01282 0.37795

0.11324 -1.86563

R-square Adj R-sq

-6.07005

Parameter Estimates

F Value

24.471

0.8303 0.7964

Parameter Standard T for HO:

Prob>F

0.0043

Estimate Error Parameter=O Prob > ITI

-2.064467 0.05871683 -35.160 0.0001 0.028964 0.00585502 4.947 0.0043

FIGURE A13. Accident Rate vs. Speed Reduction Multiple-Linear Including Superelevation Speed Model­

No Sight Distance Restrictions

Analysis of Variance

Sl.lll of Mean Source OF Squares Square F Value Prob>F

Model 1 0.33566 0.33566 26.288 0.0037 Error 5 0.06384 0.01277 c Total 6 0.39950

Root MSE 0.11300 R-square 0.8402 Oep Mean -1.85917 Adj R-sq 0.8082 c.v. -6.07783

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.064834 0.05859255 -35.241 0.0001 MMLEROSO 0.030576 0.00596345 5.127 0.0037

FIGURE A14. Accident Rate vs. Speed Reduction Multiple-Linear Including Superelevation Speed Model­

Sight Distance Restrictions

80

Source

Model Error c Total

Root MSE Oep Mean c.v.

Variable OF

INTERCEP 1 MEANMNVD 1

Analysis of Variance

Sllll of Mean OF Squares Square F Value Prob>F

1 0.03201 0.03201 6 o.m19 0.12887 7 0.80521

0.35898 -1.89865

-18.90702

R-square Adj R-sq

Parameter Estimates

Parameter Standard Estimate Error

-1.920363 0.13418591 -17.574859 35.260n417

0.0398 -0.1203

T for HO: Parameter=O

-14.311 -0.498

0.248 0.6359

Prob > ITI

0.0001 0.6359

FIGURE A15. Mean Accident Rate vs. Mean Superelevation Deficiency AASHTO Design Speed Concept

Source

Model Error C Total

Root MSE Oep Mean c.v.

Variable OF

INTERCEP MEANAS60

Analysis of Variance

Sl.lll of Mean OF Squares Square F Value Prob>F

1 0.44754 0.44754 6 0.35766 0.05961 7 0.80521

0.24415 -1.89865

-12.85927

R-square Adj R-sq

Parameter Estimates

Parameter Standard Estimate Error

-2.079241 0.10860566 1.937319 0.70704160

0.5558 0.4818

T for HO: Parameter=O

-19.145 2.740

7.508 0.0337

Prob > ITI

0.0001 0.0337

FIGURE A16. Mean Accident Rate vs. Mean Superelevation Deficiency 96.6 km/h (60 mi/h) Design Speed on all Curves

81

Analysis of Variance

Sl.ln of Mean Source OF Squares Square F Value Prob>F

Model 1 0.66372 0.66372 28.147 0.0018 Error 6 0.14148 0.02358 C Total 7 0.80521

Root MSE 0.15356 R·square 0.8243 oep Mean -1.89865 Adj R-sq 0. 7950 c.v. -8.08780

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.260711 0.08720528 -25.924 0.0001 MNLMCV 6.174248 1.16376601 5.305 0.0018

FIGURE A17. Mean Accident Rate vs. Mean Superelevation Deficiency 85th Percentile Speed on the Curve-Linear Model

Source

Model Error C Total

Root MSE Oep Mean c.v.

Variable OF

INTERCEP MNLMECV

OF

1 6 7

Analysis of Variance

Sum of Mean Squares Square

0.67987 0.67987 0.12534 0.02089 0.80521

0.14453 -1.89865

R-square Adj R-sq

-7.61230

Parameter Estimates

F Value

32.546

0.8443 0.8184

Parameter Standard T for HO:

Prob>F

0.0013

Estimate Error Parameter=O Prob > ITI

-2.271881 0.08301338 -27.368 0.0001 6.512398 1.14153378 5.705 0.0013

FIGURE A18. Mean Accident Rate vs. Mean Superelevation Deficiency 85th Percentile Speed on the Curve-Linear Model Including Superelevation

82

Analysis of Variance

Sun of Mean Source OF Squares Square F Value Prob>F

Model 1 0.63486 0.63486 22.361 0.0032 Error 6 0.17035 0.02839 C Total 7 0.80521

Root MSE 0.16850 R-square 0.7884 oep Mean -1.89865 Adj R-sq 0.7532 c.v. -8.87460

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.242092 0.09393551 -23.868 0.0001 MNMLCV 5.757548 1.21756808 4.729 0.0032

FIGURE A19. Mean Accident Rate vs. Mean Superelevation Deficiency 85th Percentile Speed on the Curve-Multiple-Linear Model

Analysis of Variance

Sun of Mean Source OF Squares Square F Value Prob>F

Model 1 0.65156 0.65156 25.443 0.0023 Error 6 0.15365 0.02561 C Total 7 0.80521

Root MSE 0.16003 R-square 0.8092 Oep Mean -1.89865 Adj R-sq 0. 7774 c.v. -8.42838

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.252140 0.09006733 -25.005 0.0001 MNMLECV 6.049516 1.19931591 5.044 0.0023

FIGURE A20. Mean Accident Rate vs. Mean Superelevation Deficiency 85th Percentile Speed on the Curve-Multiple-Linear

Including Superelevation Model

83

Analysis of Variance

Sum of Mean Source OF Squares Square F Value Prob>F

Model 1 0.53772 0.53772 12.062 0.0133 Error 6 0.26748 0.04458 c Total 7 0.80521

Root MSE 0.21114 R-square 0.6678 Oep Mean -1.89865 Adj R-sq 0.6124 c.v. -11.12058

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.127327 0.09953886 -21.372 0.0001 MNLMAPNS 1.979486 0.56996176 3.473 0.0133

FIGURE A21. Mean Accident Rate vs. Mean Superelevation Deficiency Maximum 85th Percentile Speed on the Approach-Linear Model

Analysis of Variance

Sum of Mean Source OF Squares Square F Value Prob>F

Model 1 0.53617 0.53617 11.957 0.0135 Error 6 0.26904 0.04484 C Total 7 0.80521

Root MSE 0.21175 R-square 0.6659 Dep Mean -1.89865 Adj R-sq 0.6102 c.v. -11.15288

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.125444 0.09953157 -21.354 0.0001 MNLMEANS 1.919748 0.55517061 3.458 0.0135

FIGURE A22. Mean Accident Rate vs. Mean Superelevation Deficiency Maximum 85th Percentile Speed on the Approach-Linear Including

Superelevation Model

84

Analysis of Variance

Sun of Mean Source OF Squares Square F Value Prob>F

Model 1 0.53595 0.53595 11.943 0.0135 Error 6 0.26925 0.04488

.c Total 7 0.80521

Root MSE 0.21184 R-square 0.6656 Oep Mean -1.89865 Adj R-sq 0.6099 c.v. -11.15728

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.126611 0.09980223 -21.308 0.0001 MNMLAPNS 1.969212 0.56981219 3.456 0.0135

FIGURE A23. Mean Accident Rate vs. Mean Superelevation Deficiency Maximum 85th Percentile Speed on the Approach-Multiple-Linear Model

Analysis of Variance

Sun of Mean Source OF Squares Square F Value Prob>F

Model 1 0.53432 0.53432 11.835 0.0138 Error 6 0.27088 0.04515 C Total 7 0.80521

Root MSE 0.21248 R-square 0.6636 Oep Mean -1.89865 Adj R-sq 0.6075 c.v. -11.19102

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.126855 0.10021782 -21.222 0.0001 MNMLEANS 1.996068 0.58021357 3.440 0.0138

FIGURE A24. Mean Accident Rate vs. Mean Superelevation Deficiency Maximum 85th Percentile Speed on the Approach-Multiple-Linear

Including Superelevation Model

85

Analysis of Variance

SI.Jll of Mean Source OF Squares Square F Value Prob>F

Model 1 0.56191 0.56191 150.622 0.0001 Error 8 0.02984 0.00373 c Total 9 0.59175

Root MSE 0.06108 R-square 0.9496 Oep Mean -1.91707 Adj R·sq 0.9433 c.v. -3.18602

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.277640 0.03515952 -64.780 0.0001 MFILMCV 2.717941 0.22146012 12.273 0.0001

FIGURE A25. Mean Accident Rate vs. Mean Implied Side Friction 85th Percentile Speed on the Curve-Linear Model

Analysis of Variance

Sum of Mean Source OF Squares Square F Value Prob>F

Model 1 0.56084 0.56084 31.900 0.0005 Error 8 0.14065 0.01758 C Total 9 0.70149

Root MSE 0.13259 R·square 0. 7995 Oep Mean -1.91479 Adj R·sq 0. 7744 c.v. -6.92470

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.283721 0.07761991 -29.422 0.0001 MFILMECV 2.868376 0.50785378 5.648 0.0005

FIGURE A26. Mean Accident Rate vs. Mean Implied Side Friction 85th Percentile Speed on the Curve-Linear Including Superelevation Model

86

Analysis of Variance

Sllll of Mean Source OF Squares Square F Value Prob>F

Model 1 0.58018 0.58018 104.987 0.0001 Error 8 0.04421 0.00553 C Total 9 0.62439

Root MSE 0.07434 R-square 0.9292 Oep Mean -1.91484 Adj R-sq 0.9203 c.v. -3.88225

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP 1 -2.276586 0.04241579 -53.673 0.0001 MFIMLCV 1 2.699057 0.26341737 10.246 0.0001

FIGURE A27. Mean Accident Rate vs. Mean Implied Side Friction 85th Percentile Speed on the Curve-Multiple-Linear Model

Analysis of Variance

Sllll of Mean Source OF Squares Square F Value Prob>F

Model 1 0.65699 0.65699 509.179 0.0001 Error 8 0.01032 0.00129 C Total 9 0.66731

Root MSE 0.03592 R-square 0.9845 Oep Mean -1.92383 Adj R-sq 0.9826 c.v. -1.86714

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.312137 0.02061929 -112.135 0.0001 MFIMLECV 2.938581 0.13022744 22.565 0.0001

FIGURE A28. Mean Accident Rate vs. Mean Implied Side Friction 85th Percentile Speed on the Curve-Multiple-Linear Including Superelevation Model

87

Source

Model Error C Total

Root MSE Oep Mean c.v.

Variable OF

INTERCEP MFLMAPNS

OF

1 8 9

Analysis of Variance

S1.111 of Mean Squares Square

0.59390 0.59390 0.07149 0.00894 0.66540

0.09453 -1.91941

R-square Adj R-sq

-4.92519

Parameter Estimates

F Value

66.456

0.8926 0.8791

Parameter Standard T for HO:

Prob>F

0.0001

Estimate Error Parameter=O Prob > ITI

-2.186684 0.04436903 -49.284 0.0001 1.536628 0.18849598 8.152 0.0001

FIGURE A29. Mean Accident Rate vs. Mean Implied Side Friction Maximum 85th Percentile Speed on the Approach-Linear Model

Analysis of Variance

S1.111 of Mean Source OF Squares Square F Value Prob>F

Model 1 0.58511 0.58511 72.406 0.0001 Error 8 0.06465 0.00808 C Total 9 0.64976

Root MSE 0.08989 R-square 0.9005 Oep Mean -1.92031 Adj R-sq 0.8881 c.v. -4.68124

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.183416 0.04200208 -51.984 0.0001 MFLMEANS 1.497413 0.17597659 8.509 0.0001

FIGURE A30. Mean Accident Rate vs. Mean Implied Side Friction Maximum 85th Percentile Speed on the Approach-Linear Including Superelevation Model

88

Analysis of Variance

Sllll of Mean Source OF Squares Square F Value Prob>F

Model 1 0.60210 0.60210 80.946 0.0001 Error 8 0.05951 0.00744 C Total 9 0.66160

Root MSE 0.08625 R-square 0.9101 Dep Mean -1.91995 Adj R-sq 0.8988 c.v. -4.49205

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.189160 0.04048668 -54.071 0.0001 MFMLAPNS 1.541880 0.17137671 8.997 0.0001

FIGURE A31. Mean Accident Rate vs. Mean Implied Side Friction Maximum 85th Percentile Speed on the Approach-Multiple-Linear Model

Analysis of Variance

Sum of Mean Source OF Squares Square F Value Prob>F

Model 1 0.57270 0.57270 44.784 0.0002 Error 8 0.10230 0.01279 C Total 9 0.67500

Root MSE 0.11308 R-square 0.8484 Oep Mean -1.92276 Adj R-sq 0.8295 c.v. -5.88135

Parameter Estimates

Parameter Standard T for HO: Variable OF Estimate Error Parameter=O Prob > ITI

INTERCEP -2.184172 0.05295946 -41.242 0.0001 MFMLEANS 1.513004 0.22608926 6.692 0.0002

FIGURE A32. Mean Accident Rate vs. Mean Implied Side Friction Maximum 85th Percentile Speed on the Approach-Multiple-Linear Including Superelevation

Model

89

APPENDIXB

GROUPING ANALYSIS SUMMARY

90

Mean Accident Rate Versus: Speed Model Sight Distance

R-square MSE

Restriction? (acc/MVkm)

Mean Degree of Curvature -- -- 0.79 0.029

Mean Speed Reduction Linear y 0.75 0.033 Linear + e y 0.73 0.036 Multiple-Linear y 0.77 0.031 Multiple-Linear +e y 0.75 0.034 Linear N 0.74 0.035 Linear+ e N 0.72 O.Q38

Multiple-Linear N 0.76 0.032

\0 Multiple-Linear +e N 0.74 0.036 - Mean Implied Side Friction Linear -- 0.88 0.016

Using 85th Percentile Linear+ e -- 0.89 0.015

Curve Speed Multiple-Linear -- 0.87 0.017 Multiple-Linear +e -- 0.88 0.016

Mean Implied Side Friction Linear y 0.78 0.030

Using Maximum 85th Linear+ e y 0.77 0.031

Percentile Approach Multiple-Linear y 0.78 0.030

Speed Multiple-Linear +e y 0.78 0.030 Linear N 0.77 0.031 Linear+ e N 0.77 0.031 Multiple-Linear N 0.77 0.031 Multiple-Linear +e N 0.77 0.031

FIGURE Bl. Summary of Regression Results When Grouped by Degree of Curvature Categories

\0 tv

Grou :d By: Linear Model Linear + e Model Multi-Linear Model Multi-Linear + e Model

-. Sight Distance Mean Accident Rate Versus: Speed Model Restriction? R-square

MSE (acc/MVkm) R-square MSE (acc/MVkm) R-square MSE (acc/MVkm) R-square MSE (acc/MVkm)

Mean Degree of Curvature -- -- 0.96 0.004 0.79 0.019 0.79 0.020 0.87 0.010

Mean Speed Reduction Linear y 0.91 0.007 0.73 0.024 0.73 0.026 0.81 0.014

Linear+ e y 0.91 0.008 0.74 0.023 0.73 0.025 0.81 0.014 Multiple-Linear y 0.92 0.007 0.75 0.023 0.73 0.025 0.82 0.013 Multiple-Linear +e y 0.92 0.007 0.77 0.021 0.74 0.024 0.84 0.012 Linear N 0.91 0.008 0.73 0.024 0.71 0.027 0.80 0.015

Linear+ e N 0.91 0.008 0.74 0.023 0.72 0.026 0.81 0.014

Multiple-Linear N 0.91 0.007 0.75 0.023 0.72 0.026 0.82 0.014

MultiPle-Linear +e N 0.91 0.007 0.76 0.022 0.73 0.025 0.83 0.013

Mean Implied Side Friction Linear -- 0.96 0.004 0.89 0.010 0.95 0.005 0.95 0.004

Using 85th Percentile Linear+ e -- 0.95 0.004 0.88 0.011 0.95 0.004 0.95 0.003

Curve Speed Multiple-Linear -- 0.96 0.003 0.88 0.011 0.94 0.005 0.95 0.004 Multiple-Linear +e -- 0.96 0.003 0.88 0.011 0.95 0.005 0.95 0.004

Mean Implied Side Friction Linear y 0.94 0.005 0.79 0.019 0.78 0.021 0.86 O.Q\1

Using Ma~imum 85th Linear+ e y 0.94 0.005 0.79 0.019 0.77 0.021 0.86 0.011

Percentile Approach Multiple-Linear y 0.94 0.005 0.79 0.019 0.78 0.021 0.86 0.011

Speed Multiple-Linear +e y 0.94 0.005 0.79 0.019 0.78 0.021 0.86 0.010 Linear N 0.94 0.005 0.79 0.019 0.77 0.022 0.86 0.011

Linear+ e N 0.94 0.005 0.78 0.019 0.77 0.022 0.86 O.Q\1

Multiole-Linear N 0.94 0.005 0.79 0.019 0.77 0.022 0.86 0.011 Multiple-Linear +e N 0.94 0.005 0.79 0.019 0.77 0.022 0.86 O.Oll

FIGURE B2. Summary of Regression Analysis When Grouped by Operating Speed Reduction Categories No Sight Distance Restrictions in the Speed Profile Model

\0 w

Grou >ed Bv:

Linear Model Linear + e Model Multi-Linear Model Multi-Linear + e Model

Mean Accident Rate Versus: Speed Model 5if:iJ?c~ R-square MSE (acc/MVkm) R-square MSE (acc/MVkm) R-square MSE (ace/MVkm) R-square MSE (acc/MVkm)

Mean Degree of Curvarure -- -- 0.96 0.003 0.84 0.015 0.86 0.014 0.88 0.010

Mean Speed Reduction Linear y 0.92 0.007 0.77 0.021 0.79 0.021 0.82 0.015 Linear+ e y 0.92 0.007 0.78 0.020 0.79 0.020 0.82 0.014 Multiple-Linear y 0.92 0.007 0.78 0.020 0.79 0.020 0.83 0.014 Multiple-Linear +e y 0.92 0.007 0.80 0.019 0.80 0.019 0.84 0.013 Linear N 0.92 0.007 0.78 0.021 0.79 0.020 0.82 0.015 Linear+ e N 0.92 0.007 0.79 0.020 0.80 0.020 0.82 0.014 Multiole-Linear N 0.92 0.007 0.79 0.020 0.79 0.020 0.83 0.013 MultiPle-Linear + e N 0.92 0.007 0.80 0.018 0.80 0.019 0.84 0.013

Mean Implied Side Friction Linear -- 0.95 0.004 0.91 0.008 0.97 0.002 0.93 0.006

Using 85th Percentile Linear + e -- 0.95 0.004 0.91 0.009 0.98 0.002 0.93 0.006

Curve Speed Multiple-Linear -- 0.96 0.004 0.91 0.008 0.97 0.003 0.93 0.006 Multiple-Linear +e -- 0.96 0.004 0.91 0.009 0.97 0.003 0.93 0.006

Mean Implied Side Friction Linear y 0.95 0.004 0.84 0.015 0.84 0.016 0.87 0.010

Using Maximum 85th Linear + e y 0.95 0.005 0.83 0.016 0.84 0.016 0.87 0.010

Percentile Approach Multiole-Linear y 0.95 0:004 0.83 0.015 0.84 0.016 0.87 0.010

Speed Multiole-Linear +e y 0.95 0.004 0.83 0.015 0.84 0:016 0.87 0.010

Linear N 0.95 0.004 0.84 0.015 0.84 0.016 0.87 0.010

Linear + e N 0.95 0.004 0.83 0.015 0.84 0.016 0.87 0.010 Multiple-Linear N 0.96 0.004 0.83 0.015 0.84 0.016 0.87 0.010 Multiple-Linear +e N 0.95 0.004 0.83 0.015 0.84 0.016 0.87 0.010

FIGURE B3. Summary of Regression Analysis When Grouped by Operating Speed Reduction Categories With Sight Distance Restrictions in the Speed Profile Model

\0 ~

Mean Accident Rate Versus:

Mean Degree of Curvalllre

Mean Speed Reduction

Mean Implied Side Friction Using 85th Percentile

Curve Speed

Mean Implied Side Friction

Using Maximum 85th Percentile Approach

Speed

~-

Grouped By:

Linear Model Linear + e Model Multi-Linear Model Multi-Linear + e Model

Speed Model Sight Distance

R-square MSE (acc/MVkm) R-square MSE (acc/MVkm) R-square MSE (acc/MVkm) R-square MSE (acc/MVkm) Restriction?

.. .. 0.92 0.006 0.71 0.026 0.92 0.006 0.96 0.004

Linear y 0.85 0.011 0.62 0.034 0.86 0.011 0.90 0.008

Linear + e y 0.85 0.011 0.63 0.033 0.86 0.011 0.90 0.008

Multiple-Linear y 0.86 0.010 0.63 0.032 0.87 0.010 0.90 0.008

Multiple-Linear +e y 0.87 0.010 0.66 0.030 0.88 0.010 0.92 0.007

Linear N 0.85 0.011 0.61 0.034 0.86 0.011 0.90 0.009

Linear+ e N 0.85 0.011 0.63 0.033 0.86 0.011 0.90 0.008

Multiple-Linear N 0.86 0.010 0.63 0.033 0.87 0.010 0.90 0.008

Multiple-Linear +e N 0.87 0.010 0.65 0.031 0.88 0.010 0.91 0.007

Linear .. 0.95 0.004 0.80 0.017 0.93 0.006 0.98 0.002

Linear + e -- 0.97 0.004 0.80 0.018 0.93 0.006 0.98 0.002

Multiple-Linear .. 0.95 0.004 0.80 O.QI8 0.93 0.006 0.98 0.001

Multiple-Linear +e -- 0.95 0.004 0.79 0.018 0.93 0.005 0.98 0.001

Linear y 0.91 0.007 0.72 0.025 0.91 0.007 0.95 0.004

Linear + e y 0.91 0.007 0.71 0.025 0.91 0.007 0.95 0.004

Multiple-Linear y 0.91 0.007 0.72 0.025 0.91 0.007 0.95 0.004

Multiple-Linear + e y 0.91 0.007 0.72 0.025 0.91 0.007 0.96 0.004

Linear N 0.91 0.007 0.71 0.025 0.91 0.007 0.95 0.004

Linear + e N 0.91 0.007 0.71 0.026 0.91 0.007 0.95 0.004

Multiple-Linear N 0.91 0.007 0.71 0.025 0.91 0.007 0.95 0.004

Multiple-Linear +e N 0.91 0.007 0.71 0.025 0.91 0.007 0.95 0.004

FIGURE 84. Summary of Regression Analysis When Grouped by Implied Side Friction Categories Based on the 85th Percentile Speed on the Curve

\0 Vl

Mean Accident Rate Versus:

Mean Degree of Curvarure

Mean Speed Reduction

Mean Implied Side Friction Using 85th Percentile

Curve Speed

Mean Implied Side Friction

Using Maximum 85th Percentile Approach

Speed

Grou :d By: Linear Model Linear + e Model Multi-Linear Model Multi-Linear + e Model

Speed Model Sight Dis1ance

R-square MSE (acc/MVkm) R-square MSE (acc/MVIcm) R-square MSE (acc/MVIcm) R-square MSE (acc/MVkm) Restriction?

-- -- 0.90 0.008 0.91 0.007 0.92 0.007 0.85 0.012

Linear y 0.82 O.QI5 0.83 0.014 0.84 0.013 0.76 0.020 Linear+ e y 0.83 0.014 0.83 0.013 0.85 0.013 0.78 0.019 Multiple-Linear y 0.83 0.014 0.84 0.013 0.86 0.013 0.77 0.019 Multiple-Linear +e y 0.84 0.013 0.85 0.012 0.83 0.012 0.79 0.017

Linear N 0.81 0.015 0.82 0.014 0.84 0.014 0.76 0.020 Linear + e N 0.82 0.015 0.83 0.014 0.84 0.013 0.77 0.019 Multiple-Linear N 0.82 0.015 0.83 0.014 0.85 0.013 0.77 0.019 Multiple-Linear +e N 0.84 0.014 0.85 0.012 0.85 0.012 0.79 0.018

Linear -- 0.93 0.006 0.95 0.004 0.95 0.004 0.89 0.009

Linear+ e -- 0.93 0.006 0.94 0.005 0.94 0.005 0.89 0.010

Multiple-Linear -- 0.94 0.005 0.95 0.004 0.95 0.004 0.89 0.009

Multiple-Linear +e -- 0.93 0.006 0.95 0.004 0.95 0.004 0.88 0.010

Linear y 0.90 0.009 0.91 0.008 0.91 0.007 0.85 0.013

Linear+ e y 0.89 0.009 0.90 0.008 0.91 0.007 0.85 0.013

MultiPle-Linear y 0.90 0.009 0.91 0.008 0.91 0.007 0.85 0.013

Multiple-Linear +e y 0.90 0.009 0.91 0.008 0.91 0.007 0.85 0.013

Linear N 0.89 0.009 0.90 0.008 0.91 0.007 0.85 0.013

Linear + e N 0.89 0.009 0.90 0.008 0.91 0.007 0.85 0.013

Multiple-Linear N 0.89 0.009 0.90 0.008 0.91 0.007 0.85 0.013

Multiple-Linear +e N 0.89 0.009 0.90 0.008 0.91 0.007 0.85 0.013

FIGURE B5. Summary of Regression Analysis When Grouped by Implied Side Friction Categories Based on the Maximum 85th Percentile Operating Speed on the Approach Tangent

No Sight Distance Restrictions in the Speed Profile Model

I

1.0 0'1

Mean Acddent Rate Versus:

Mean Degree of Curvarure

Mean Speed Reduction

Mean Implied Side Friction Using 85th Percentile

Curve Speed

Mean Implied Side Friction Using Maximum 85th Percentile Approacb

Speed

Groulcd By: Linear Model Linear + e Model Multi-Linear Model Multi-Linear + e Model

Speed Model Sigbt Distance

R-square MSE (acc/MVkm) R-square MSE (acc/MVkm) R-square MSE (acc/MVkm) Restriction?

R-square MSE (acc/MVkm)

-- -- 0.91 0.008 0.90 0.008 0.90 0.009 0.85 0.014

Linear y 0.83 0.014 0.82 0.014 0.82 0.015 0.77 0.021

Linear + e y 0.83 0.014 0.83 0.014 0.83 0.014 0.78 0.020

MultlDle-Linear y 0.84 0.014 0.83 0.014 0.83 0.014 0.78 0.020

Multinle-Linear +e y 0.85 0.013 0.85 0.012 0.85 0.013 0.80 0.017

Linear N 0.83 0.015 0.82 0.015 0.82 0.015 0.76 0.021

Linear + e N 0.83 0.014 0.83 0.014 0.83 0.014 0.77 0.020

Multio1e-Linear N 0.84 0.014 0.83 0.014 0.83 0.014 0.77 0.020

Multiole-Linear +e N 0.85 0.013 0.84 0.013 0.84 0.013 0.79 0.019

Linear -- 0.93 0.006 0.95 0.004 0.93 0.006 0.88 0.011

Linear+ e -- 0.92 0.006 0.95 0.004 0.92 0.007 0.87 0.011

Multiole-Linear -- 0.93 0.006 0.95 0.004 0.93 0.006 0.88 0.011

Multiple-Linear +e -- 0.93 0.006 0.95 0.004 0.93 0.006 0.87 0.011

Linear y 0.90 0.008 0.90 0.008 0.90 0.009 0.85 0.013

Linear + e y 0.90 0.008 0.90 0.008 0.90 0.009 0.85 0.014

Multiole-Linear y 0.90 0.008 0.90 0.008 0.90 0.009 0.85 0.013

Multiole-Linear +e y 0.90 0.008 0.90 0.008 0.90 0.008 0.85 0.013

Linear N 0.90 0.008 0.90 0.008 0.90 0.009 0.85 0.014

Linear+ e N 0.90 0.009 0.90 0.008 0.89 0.009 0.85 0.014

Multinle-Linear N 0.90 0.008 0.90 0.008 0.90 0.009 0.85 0.014

Multiple-Linear +e N 0.90 0.008 0.90 0.008 0.90 0.009 0.76 0.014

FIGURE B6. Summary of Regression Analysis When Grouped by Implied Side Friction Categories Based on the Maximum 85th Percentile Operating Speed on the Approach Tangent

With Sight Distance Restrictions in the Speed Profile Model

l