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Application of reliability theory to highway geometric design Zheng, Zhimei Ronda 1997

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APPLICATION OF RELIABILITY T H E O R Y T O H I G H W A Y G E O M E T R I C DESIGN By Zhimei Ronda Zheng M . B . A . , Northern Jiaotong University, China, 1984 B. Sc., Northern Jiaotong University, China, 1981 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F CIVIL E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A April 1997 © Zhimei Ronda Zheng, 1997 In presenting this thesis in a partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the library shall make it freely available for reference and study. I further agree that permission for extensive coping of this thesis for scholarly purpose may be granted by the Head of my Department or by his or her representatives. It is understood that coping or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Civil Engineering The University of British Columbia 2324 Main Mall Vancouver, B.C. Canada V6T 1Z4 April, 1997 A B S T R A C T To improve road safety and to reduce accident costs, roadway geometries are frequently modified based on the relationship between deficient geometric elements and the recorded accidents. This remedial approach requires fatalities and injuries to occur before the needs and priorities of roadway improvements may be identified. Rather than adopt this approach, the present research took a preventive initiate in highway geometric design stage to detect, quantify and correct hazards prior to their being constructed. Highway geometric design is proposed to be performed in relation to vehicle dynamics. This is a major contribution from this research. The methodology was a reliability-based highway geometric design, similar to "limit state" design in structural engineering. It used probability of non-compliance with design criteria to identify drivers' behaviour responses to different design alternatives. This research focused on the development of a more correct representation of the driver-vehicle-roadway interaction -the Moving Coordinate System Design (MCSD) model, the establishment of an evaluation process for the safety level of an entire highway, the application of the "racing car model" as the upper operating limit, the development of VHVIS.PAC algorithm to utilize photolog data as the geometric design input for reliability analysis, and the incorporation of the growing knowledge of human factors into highway geometric design. The research applied reliability theory to geometric design for an entire highway by 11 incorporating vehicle dynamic characteristics, operational experience, human factor consideration and the ultimate limit of road-vehicle interaction. It provides the designers with opportunities to visualize as quantitatively as possible how the design will be experienced by individual drivers under the range of speeds and other operating conditions that will occur when the highway is actually built and put into use. The resulting safety performance measures from all available design alternatives could eventually lead to a cost-effective geometric design process. The following questions are addressed by this research: • How do different groups of drivers, such as design drivers, normal drivers and expert drivers operate on the same roadway? What kind of safety performance can be expected from each group? • Is the difference in the margin of safety among different driver groups significant? • What is the cost-effectiveness associated with each modification of a particular design? • What is the connection between design and traffic operation criteria? How does every design criterion reflect the anticipated operation of the highway? How does the proposed model fit into the existing geometric design process? iii T A B L E OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES vii LIST OF FIGURES viii ACKNOWLEDGEMENT x 1.0 INTRODUCTION TO RESEARCH 1 1.1 Traditional Approach in Highway Design 2 1.2 Study Objective, Goals and Methodology 4 1.3 Study Location 10 1.4 Thesis Structure 15 2.0 LITERATURE REVIEW 19 2.1 Developments in Geometric Design Process 19 2.2 Current Geometric Design Practice 25 2.3 Geometric Design and Safety 27 2.4 Human Factors and Safety 30 3.0 SEA-TO-SKY HIGHWAY PERFORMANCE ANALYSIS 34 3.1 Accident Rates 34 3.2 Accident Severity 36 3.3 Spacial Distribution 38 3.4 Time Distribution 42 3.5 Accident Types and Contributing Causes 45 3.6 Findings/Summary 49 4.0 SEA-TO-SKY HIGHWAY GEOMETRY ANALYSIS 51 4.1 Photolog Files 51 4.2 Highway Alignment Database from Photolog Data 53 4.3 Methodology and Programming Technique 55 iv 4.3.1 Identification of vertical deficiency segments 56 4.3.2 identification of horizontal deficiency segments 59 4.4 Validation of the Software VHVIS.PAS 66 4.4.1 Data input for different years 67 4.4.2 Data input for different directions 69 4.4.3 Photolog films review 70 4.5 Evaluation of the Existing Geometry on the Sea-to-Sky Highway 73 5.0 INVESTIGATION OF THE LINK BETWEEN HIGHWAY PERFORMANCE AND GEOMETRY 75 5.1 Correlation Analysis 75 5.2 High Accident Locations 77 5.3 Quality of the Sea-to-Sky Highway 77 5.3 Findings/Summary 80 6.0 RELIABILITY-BASED HIGHWAY DESIGN THEORY AND PRACTICE 94 6.1 Reliability Theory 94 6.1.1 Reliability in highway geometric design 95 6.1.2 Margin of Safety and Reliability Index 96 6.1.3 Probability of non-compliance 98 6.2 Highway Design Practice 102 6.2.1 Stopping sight distance design 102 6.2.2 Horizontal curve design -R Values 111 6.2.3 Vertical curve design - K Values 120 6.3 Findings/Summary 128 7.0 A MORE FUNDAMENTAL HIGHWAY DESIGN MODEL - Moving Coordinate System Design 130 7.1 Conceptual Development 131 7.1.1 TDvsTs- Demand traction vs. available traction 132 7.1.2 fxvsfy- Braking coefficient vs. cornering coefficient 133 7.1.3 Friction Circle - Relationship offx and fy 136 7.1.4 Race car vs. road car 139 7.2 Model Formulation 140 7.2.1 Mathematic Model 140 7.2.2 Civil-Mechanical Engineering Application 141 7.3 Model Programming 145 8.0 SEA-TO-SKY HIGHWAY APPLICATION 149 v 8.1 Model Variables 149 8.1.1 Slipangle,a 149 8.1.2 Speed variables, variation and distribution 154 8.1.3 Lateral acceleration versus speed 155 8.1.4 Speed versus radius 157 8.2 Data Preparation 164 8.2.1 PTEC experiments 164 8.2.2 Sea-to-Sky Highway curves 167 8.2.3 Longitudinal friction test 171 .8.2.4 Design standards 171 8.2.5 Roadtest 174 8.3 Fitting Equations to Data 178 8.3.1 Curve fitting 178 8.3.2 Fitting equation to data 179 8.3.3 Curve-fitting analysis 182 8.4 Input Database for RELAN 182 9.0 SAFETY MEASURES ON SEA-TO-SKY HIGHWAY 188 9.1 Four Driving Scenarios 189 9.2 "Profile" Analysis 192 9.3 Performance Measure 197 9.3.1 Capacity ratio 197 9.3.2 Operation ratio 199 9.4 Operation Deficiency 205 9.4.1 Roadway analysis - "all case" study 205 9.4.2 Dangerous location analysis 211 9.5 Design Improvements 215 9.5.1 Importance measures 215 9.5.2 Sensitivity analysis 216 9.5.3 P versus individual design variables 218 9.5.4 Location improvements 221 9.5.5 Omission analysis 224 9.6 Summary 225 10.0 CONCLUSIONS AND FURTHER RESEARCH 229 10.1 Conclusions 229 10.2 Reliability-based MCSD Process 231 10.3 Further Research 233 BIBLIOGRAPHY 238 DISKETTE ZHENG.PHD (APPENDIX) 242 vi LIST O F T A B L E S 1.1 Study location 11 3.1 Motor vehicle accident rate (Three year period) 36 3.2 Accident severity 36 3.3 Accidents by locations 38 3.4 Histogram of accident frequency (Horseshoe Bay - Squamish) 40 3.5 Histogram of accident frequency (Squamish - Whistler) 41 3.6 Accidents by contributing causes 46 3.7 "Off-road right" accidents 47 3.8 "Off-road left" accidents 47 3.9 "Rear-end" accidents 48 3.10 "Head-on" accidents 48 3.11 "Side-swipe" accidents 48 4.1 Deficient curves for different years (Northbound, Horseshoe Bay to Whistler) 68 4.2 Deficient curves for different years (Southbound, Squamish to Horseshoe Bay) 68 4.3 Deficient curves for different directions (Horseshoe Bay to Squamish) 69 4.4 Curve radii and degree of tightness (MoTH) 70 4.5 Comparisons of Re and R 73 5.1 Frequency of deficient SSD segments 78 6.1 Probability of rocks in shoulder lane and median lane 106 6.2 Probability of non-compliance vs. operating speeds in curve design 116 6.3 Probability of non-compliance vs. lateral frictions in horizontal curve design 117 6.4 Probability of non-compliance and excavation requirement vs. K values 127 8.1 fy demanded by normal drivers in PTEC curves 166 8.2 f reached by experts in PTEC curves 167 8.3 fy demanded by normal drivers on Sea-to-Sky Highway 168 8.4 fy reached by experts on Sea-to-Sky Highway 170 8.5 Stopping sight distance 172 8.6 Lateral friction, superelevation and radius 173 8.7 Roadtest summary 175 8.8 Experiment data summary - fy versus curve radii 178 8.9 Input database for RELAN 185 9.1 All case study - Comparison of p values for four driving scenarios 207 9.2 Dangerous locations if P*= 3.5 212 9.3 Dangerous location for specific O-ratio values 213 9.4 Importance measures 216 9.5 Key variables in base case 217 9.6 Tests on LKI 10.74 - LKI 10.78 218 vii LIST OF FIGURES 1.1 Study Location 12 1.2 Thesis Structure 16 2.1 Attribution of accident responsibility 33 3.1 Accident severity 37 3.2 Pie-chart of accident locations 39 3.3 Monthly accident distribution 42 3.4 Hourly accident distribution 43 3.5 Accidents by manner of collision 45 4.1 Photolog digital display panel 52 4.2 Road angle a vs. object angle P 57 4.3 Vertical sight distance 58 4.4 Horizontal sight distance 60 4.5 Middle ordinate, d 63 4.6 Three dimensional angle searching zone 64 5.1 Safe performance and geometry integration (Horseshoe Bay - Squamish) 84 5.2 Safe performance and geometry integration (Squamish - Whistler) 89 5.3 Relationship of accident rate/km and percent of limited SSD 79 6.1 Demand vs. supply 95 6.2 Reliability Index, p 97 6.3 Probability of non-compliance vs. operating speeds in SSD design 107 6.4 Probability of non-compliance vs. perception-reaction times in SSD design 108 6.5 Probability of non-compliance using decision sight distance 109 6.6 Probability of non-compliance vs. percentage of SSD 110 6.7 Probability of non-compliance vs. operating speeds in horizontal curve design 119 6.8 Probability of non-compliance vs. lvalues in vertical curve design 123 6.9 Probability of non-compliance vs. operating speeds in vertical curve design 124 6.10 Probability of non-compliance vs. friction factors in vertical curve design 125 6.11 Probability of non-compliance vs. lvalues and operating speeds 126 7.1 Demand-supply traction diagram 133 7.2 "Friction Circle or Ellipse" 137 7.3 Dynamic characteristics in three-dimension 142 7.4 Moving Coordinate System Design flowchart 147 8.1 Geometry of a turning vehicle 149 8.2 Tire cornering force properties 150 8.3 Lateral acceleration and speed for PTEC curves using Herrin et al. model 156 viii 8.4 Empirical speed versus radii (Taragin, DMR and Emmerson) 159 8.5 Curve speed prediction relationship (McLean) 160 8.6 Comparison of speed versus radii models (McLean and Brenac) 161 8.7 Lateral Friction versus radii 163 8.8 Data Preparation 165 8.9 fy collected on Sea-to-Sky Highway by g-Analyst 170 8.10 Maximum lateral acceleration that different cars experienced on a 91.4 m diameter skid pad 177 8.11 Curve-fitting 181 9.1 System reliability index distribution for 10 km of roadway 193 9.2 System reliability index distribution for section of LKI 5.58 - LKI 8.0 194 9.3 System reliability index distribution for section of LKI 8.0 - LKI 11.9 195 9.4 System reliability index distribution for section of LKI 11.9- LKI 15.32 196 9.5 "Capacity ratio" distribution curve 200 9.6 "Capacity ratio" cumulative probability curve 201 9.7 "Operation ratio" distribution curve 203 9.8 "Operation ratio" cumulative probability curve 204 9.9 Point analysis at LKI 10.74 219 9.10 Point analysis at LKI 14.06 220 9.11 Improved design at LKI 10.74 222 9.12 Improved design at LKI 14.06 223 9.13 Improved P distribution curve (points adjusted) 226 9-14 p i m p r o v e d vs. p e x i s t i n g (speed fixed) 227 9.15 Integration of the existing design process with MCSD Model 228 ix A C K N O W L E D G E M E N T This research owes its beginning to my supervisor Dr. Frank Navin of University of British Columbia. He focused the research on improvement of the driver-vehicle-roadway dynamic model, ensuring that advances were made where they were most needed. Our monthly meetings were never dull as we tried to come up with new ideas when the previous one proved fruitless. His guidance and advice were much appreciated. More importantly, his encouragement for putting the research model into the highway design practice not only provides me with a sense of accomplishment, but also allows me to go a few steps beyond. The work was also supported by the Ministry of Transportation and Highways of British Columbia, through the Research Project T H 2769. The project provided me with financial support and offered me a unique and enriching research experience. I would like to express my thanks to the Ministry of Transportation and Highways, particularly to Mr. R. Voyer, Manager of Geometric Design Standards and Specifications, for his thoughtful comments and great help throughout my research. Along the way, the ideas of other noteworthy people helped shaped the final product. I would like to thank Mr. F . Emmanuel for his help in collecting field data and to Mr. G . Garlick for his review of the software package and helpful suggestions for improving the package. I am also indebted to Miss N . Navin and Mr. L . Chen for their many helpful comments as they read and evaluated this report. Finally and most specially, I would like to thank Mr. F. Yao for fielding a thousand questions and for being my cohort in fine tuning the Moving Coordinate System Design model. I would like to dedicate this work to my parents Wayne and Janet, my husband Sunny, my son and daughter Matthew and Fiona, whose love sustained me in my endeavours over the years and continues to do so. C H A P T E R 1 INTRODUCTION T O R E S E A R C H Statistics in every part of the world have shown that the economic losses and human suffering resulting from road accidents are large. Jefferies (1996) estimated that since 1896 about 35 million people have been killed by motor vehicles and probably over 100 million injured. Canadian roads, in 1994 claimed the lives of about 3,260 people, and another 244,975 were injured in some 169,502 accidents (All reported crashes with fatalities or injuries). Significant property-damage-only accidents probably numbered more than a million. The cost to Canada was between 8-9 billion dollars. Injuries and accidents are increasing in spite of greater investments in road safety. To improve road safety and to reduce accident costs, roadway geometry may be modified. Such actions are usually justified by the relationship between deficient geometric elements and the recorded accidents. Sayed et al (1995) have shown that one third of total accidents have a factor related to road environment. The problem with this remedical engineering approach is that it requires fatalities and injuries to occur before the needs and priorities of roadway improvements may be correctly identified. Unlike the remedial approach, the research of this thesis looks for cause-and-effect relationships to correct hazards prior to their being constructed. The proposed reliability-1 based highway geometric design is similar to "limit state" design from structural engineering. It uses probability of non-compliance with design criteria to identify behavioural responses to different design problems. The method follows systematic procedures for evaluating existing geometric conditions, it may compare these results with those conclusions derived from previous accident experiences or acceptable levels of safety, and it can detect safety needs and analyse a range of options to meet these needs. 1.1 Traditional Approach in Highway Design When designing a highway, transportation engineers refer to an accepted set of standards. In North America, the most widely accepted geometric design standards are those developed by the American Association of State Highway and Transportation Officials (AASHTO) and by the Transport Association of Canada (TAC). Although every province, many cities and other government bodies in Canada have developed their own standards, they are largely dependent on AASHTO or TAC. These standards are based on research work dated back to the 1930s and 1940s and updated to reflect current practice and the latest technology in transportation engineering. They reflect judgements about the effects of changes in the design variables. They provide guidance to designers by referencing a recommended range of values for critical dimensions. The design standards and practice, therefore, determine to great extent the nature and quality of resulting roads. Safety considerations in highway design standards have not been ignored. For example, AASHTO road geometry policies recommend minimum design values and desirable design 2 values for all of the geometric design elements, such as lane widths, shoulder widths, horizontal and vertical curves, sight distance, superelevation at curves, etc. The problems are: 1) These standards tend to be implicit rather than explicit criteria. Designers have only been provided an empirical and not a scientific approach to build "safety factors" into design. For example, standards suggest that all geometric design elements be determined based upon the performance of a "design driver"or an "average driver". Because a "design driver" is assumed to be slow and not overly attentive, it is accepted that if a roadway can accommodate a design driver, it will provide a sufficient margin of safety to all drivers. This implicit assumption results in the safety factor built into a roadway design being unknown. 2) The safety of the facility depends to a great degree on how well the designers implement these standards and how the priority they give to safety criteria. For example, if a road is intended for high-speed driving, a designer may propose wider lanes and shoulders, longer sight distance, and more gentle curves. However, what additional benefit would be provided through his "generous" design is unknown. 3) Standards are upgraded based largely upon operational failures. If a facility does not operate as expected, engineers attempt to find out why. If a design does meet minimum standards, it may be necessary to improve geometric features for that particular design because no design can be completely evaluated until it is subject to traffic. 4) Many safety standards require additional costs. However, the traditional design 3 approach does not allow designers to know to what extent these improvements would save lives and reduce injuries. Designers would like to consider cost-effectiveness not only in deciding whether to widen a road but also in deciding the extent to which it should be widened. Unfortunately, not enough is known about what and to which degree safety factors are built into design, and not enough is known about safety gains that will occur after the geometry of an existing highway is improved or other safety-oriented improvements are made. Because of the shortcomings of the existing geometric design process, there is a need to develop a consistent procedure to estimate the performance of a road in a manner appropriate for the different driver-vehicle combinations prior to its operation, to evaluate the importance of all significant parameters for the geometric feature under consideration, and to maintain the safety of the design. 1.2 Study Goal, Objectives and Methodology The goal of this research is (1) to establish, in probabilistic terms, the relationship between the geometric characteristics of a highway and its safety performance; (2) to expand the range of safety research associated with geometric design from accident analysis to potential accident analysis; and (3) to bring operational performance evaluation process into the design stage. This research develops a highway geometric design procedure that is based on the ideas found in reliability analysis as used in structural engineering. The objective is to develop meaningful 4 engineering road safety measures which can be used in such a way that: 1) a reasonably consistent level of safety may be calculated at any location along a road or over the entire road; 2) the safety consequences of existing conditions and potential improvements may be evaluated; and 3) safety factors pertaining to each specific design may be explicitly addressed. This reliability-based design process will allow and encourage designers to make a more deliberate selection of geometric design values and explicitly address issues of safety and overall highway consistency in geometric design. The methodology of this research is to evaluate the margin of safety for a particular roadway geometric design. It accounts for the uncertainties in the variables so that changes in vehicles, the drivers, and the pavement surface can be directly studied. It calibrates the basic highway alignment design equations so that continuity exists with the older procedures. It follows a consistent approach to estimate probability of non-compliance (or reliability index) so that safety conditions at any points along a roadway or an overall safety level for the entire roadway can be well defined. This is meaningful because the established quantitative measure allows for the comparison of different design alternatives so that dollar values can be assigned to each specific design. The major contributions of this research include: 5 1) Develops the Moving Coordinate System Design (MCSD) Model Recognizing that highway geometric design must be performed in relation to vehicle dynamics, the model focuses on the basic "contact patch" existing at the interface between the tire and the road and adopts the "Friction Circle" concept to simultaneously study the control and stability of a vehicle, i.e., longitudinal and lateral frictions. The model proposes that highway geometric design be performed to include not only Civil, but also Mechanical engineering factors and that the reliability analysis be expanded from a single design element to multiple design components. This is meaningful because the model permits control of the inputs from driver, vehicle and roadway and measures the effect of the vehicle-roadway combination experienced by drivers. The model illustrates how the vehicle dynamic characteristics can be incorporated into a highway design stage such that the effect of the roadway-vehicle combination experienced by drivers can be measured. The model also demonstrates how to relate a driver's actions to the vehicle's performance capability, including the effects of roadway surface and roadway design on overall system behaviour. 2) Proposes an Evaluation Process on the Safety Level for an Entire Highway The research considers an entire highway as a unit and puts forward a consistent design approach to each design element that comprises a highway system and, more importantly, allows a check and feed-back of operational information for an entire highway at the geometric design stage. Performance functions are established for all design parameters and human factors are taken into account in relation to vehicle-roadway interaction. 6 Safety consequences of existing conditions and potential improvements can be evaluated in terms of probability of non-compliance, or reliability index, p. The evaluations are performed by using a customized RELAN program for both the overall system and the design components. The former is characterized by system probability of non-compliance, Pnc, while the latter are characterised by the probability of non-compliance in stopping sight distance for horizontal curve design, Pnc (SSDH), the probability of non-compliance in lateral fraction design, Pnc (fy), the probability of non-compliance in longitudinal friction design, Pnc (fx) and the probability of non-compliance in stopping sight distance for vertical curve design, Pnc (SSDV). The designers, therefore, can visualize as quantitatively as possible how the design will be experienced by individual drivers under the range of speeds and other operation conditions that will occur when the highway is actually built and put into use. Applies the Racing Car Model as the Upper Operating Limit This research considers tire coefficients of friction reached by expert drivers (such as police) as the operational limits. In order to obtain longitudinal friction, fx and lateral friction, f at any points along a roadway in a format of supply versus demand, the research identifies demand by normal drivers, supply by design standards, and capacity by expert drivers. Their relationships are further established in the performance functions of SSDH, SSDV,fxmdfy Because race drivers are capable of reaching the vehicle's limit, a boundary can be drawn around their fx and fy points in a "g-g diagram". This diagram provides a means to move the traditional highway geometric design into the area of handling task performance, quantifies the manoeuvre envelope of the car, and demonstrates how much of this capability is utilized by the driver. As the objective in motor racing is to win races, race cars should have a large "g-g" manoeuvre diagram throughout their performance envelope and race drivers should operate close to the diagram boundaries. The application of race cars/drivers characteristics is important to highway geometric design. The research suggests that it may provide a new method to perform highway geometric design. In the existing design process, a "design driver" is used who is assumed to be not very attentive and slow, but no one knows what safety margin is being built into the design with such a driver. An expert driver is more predictable than a "design driver", his performance can be quantified by a "g-g" diagram. Therefore, it is possible to consider that the capability of an expert driver as the maximum representative of all drivers. This capability can then be consistently factored back in order to propose various geometric design values. In addition, there is no real lower limit as to how bad a driver-vehicle may be, but there is an upper limit as to how good a driver-vehicle may be. Unlike the existing "working up from the worst" design process, the proposed "working down from the best" design process proposes a real departure from the history of highway geometric design, creating a design process that is more controllable and more reliable. Develops the VHVIS.PAC Algorithm to Utilize Photolog Data Each year, the Ministry of Transportation and Highways of BC (MoTH) uses a monitoring vehicle equipped with a camera and various measuring devices to collect 8 electronic information on various highways. The recorded information is then compiled into a photolog which can be retrieved to provide visual information along a highway. In the past, the photolog served as a "video tape" which was displayed to view the roadway itself and the roadway environment. This research shows how to operationalize the photolog information for analytical purpose. It develops a dynamic processing procedure to successfully convert the photolog information into a highway alignment database. This database is then used as an input for reliability analysis. 5) Incorporates the Growing Knowledge of Human Factors into Highway Geometric Design The reliability-based design approach considers human factors or driver expectations and allows significant changes in design alternatives to accommodate drivers. The following questions are addressed by this research: • How do different groups of drivers, such as design drivers, normal drivers and expert drivers operate on a same roadway? • What kind of safety performance can be expected? • Is the difference in the margin of safety among different driver groups significant? • What are the design alternatives to promote the overall consistency for an entire highway? • How does this research fit into the current highway geometric design process? 9 In short, the research applies reliability theory to geometric design for an entire highway by incorporating vehicle dynamic characteristics, operational experience, human factor consideration and the ultimate limit of road-vehicle interaction. The most challenging part of this research is to deal with driving dynamics, i.e., uncertainties in longitudinal and lateral frictions that drivers would experience. 1.3 Study Location The Sea-to-Sky Highway on Highway 99 between Horseshoe Bay and Whistler is the study location. As shown in Figure. 1.1, it extends from the junction with Highway 1 at Horseshoe Bay to the landmark of Whistler Road in Whistler. The study location, identified by the Ministry's Landmark Kilometre Inventory (LKI), is listed in Table 1.1. 10 Table 1.1 Study location Location Landmark LKI * SEGMENTS Horseshoe Bay Hwy 1 & Hwy 99 0.00 2920 Squamish Cleveland Ave. intersection 43.97 2930 Whistler Whistler Road 98.54 * LKI - Landmark Kilometre Inventory The total length of the study highway is approximately 100 km. As shown in Photos 1 and 2, cliffs and mountains are located on one side of the highway and the ocean is located on the other side. The Sea-to-Sky Highway is predominantly a two-lane rural highway with one travel lane in each direction (Photo 3). Photo 1 - Highway 99 between Horseshoe Bay and Whistler is known as the Sea-to-Sky Highway 11 T O P C M B C R T O N Figure 1.1 Study Location 12 Photo 2 The Sea-to-Sky Highway stretches approximately 100 km with cliffs/mountains on one side and the ocean on the other side The 44 km of roadway from Horseshoe Bay to downtown Squamish is a winding two lane mountain highway first built in the mid-1950's. In 1983 and 1991, two Vancouver-Squamish Corridor studies were carried by the Design and Survey Branch of the Ministry of Transportation and Highways of B .C . and by Klohn Leonoff Limited (19 . The studies identified that Highway 99 has long stretches of difficult alignment totalling 25.7 km (Photo 4) and long sections totalling 5.9 km where rock slides are a problem. Three of these section starting at km 1.0, 14.5, and 18.0, are over 1 km in length. In 1989, a Vancouver-Squamish Highway Planning and Pre-design Engineering Study was carried out by G.D. Hamilton & Associates Consulting Limited (2). The study indicated that continued traffic growth would result in traffic demand increasing beyond the capacity of the highway. Due to the presence of numerous sharp curves, the alignment of the Sea-to-Sky Highway is judged to be very difficult to drive by many motorists 14 In addition, the number of motor vehicle accidents recorded on that highway identified the need for highway improvement to accommodate adverse weather conditions, excessive speeding and increased traffic density (Photo 5). Photo 5 - Traffic volume on the Sea-to-Sky Highway has been increasing dramatically for the past 5 years 1.4 Thesis Structure The study is divided into three major parts: Part 1 includes the first two chapters, Part 2 includes chapter 3, 4 and 5 and Part 3 includes chapter 6, 7, 8 and 9. A flow chart of the study is shown in Figure 1.2. 15 Current design practice Research & development Parti Single parameter design - SSD, R values, K values Photolog database 5-year Accident database - Traffic volume - Speed - Specific locations Part 2 Developing VHVIS.PAS software Cross-component design I "Friction circle" theory More fundamental model Highway alignment] information I [Accidents by - severity - locations -time - types - causes Geometric design deficincies - R , SSD for H & vf curves Part3 Figure 1.2 Thesis Structure 16 CALIBRATION Assign values to parameters in design code Area analysis Part 1 describes the background of the research work, a brief review of the existing highway geometric design process, the challenge to the future design, and the principle of reliability-based design. Part 2 focuses on the studies of accident experience and highway geometric design. It begins with motor vehicle accident review, followed by the examination of existing highway geometric design. By developing a computer software program, digitized photolog data was transferred into a highway alignment data base. The goal of Part 2 is to identify design characteristics and operational deficiencies through the evaluation of the road performance and its geometry. The systematic approach taken is described as follows: 1) Assembling information on measured safety parameters, which includes collecting data on highway inventories, accident summary reports, photolog data files, posted speeds, bridge code numbers and widths, as well as lane and shoulder widths; 2) Analysing accident records by type of accident, severity, environmental conditions, contributing factors, and time period; 3) Combining highway inventory information with accident frequency to formulate Accident Spot maps; 4) Developing a computer software program to transform photolog data into highway alignment data; 5) Testing the reliability and validity of the transformation software by using data for different years and different directions, and by observing the photolog films; 17 6) Identifying trouble sites which have certain deficient geometric design problems; 7) Investigating the link between safe performance of the Sea-to-Sky Highway and highway geometric design standards; 8) Identifying the needs for improvements in highway design. Part 3 focuses on a preventive design consideration. It starts with the description of a reliability-based highway design framework, followed by the highway design practice of the Ministry of Transportation and Highways with a focus on single parameter application, such as stopping sight distance design, horizontal curve design and vertical curve design. To allow highway geometric design to be performed in relation to vehicle dynamics, particularly in sharp curves, a more fundamental model, a Moving Coordinate System Design (MCSD) model, is developed and the RELAN program is customized to measure the effect of the roadway-vehicle combination experienced by drivers. A total of 10 km of the Sea-to-Sky Highway between Horseshoe Bay and Squamish was tested with various operation conditions. The reliability-based Highway Design Process is documented in the end. 18 C H A P T E R 2 L I T E R A T U R E REVIEW This chapter starts with a review of the current highway geometric design process, followed by research on the impacts of engineering factors (i.e., geometric design) and human factors on highway safety. Shortcomings in the existing design process are identified and possible improvements are proposed. 2.1 Development in Geometric Design Process The term "geometric design" pertains to the dimensions and arrangements of the physical features of a highway. This includes pavement widths, horizontal and vertical alignment, grades, channelization, interchanges, and other features of the design that significantly affect highway operation, safety and capacity. The science of high-speed highway geometric design has evolved over many years. Once the objective was simply to provide a traversable way between two points. In the 1930s and 1940s, national design standards, policies and guidelines were introduced into highway geometric design to promote consistency and clarity throughout the entire highway. In the 1970's, emphasis shifted to "dynamic design for safety" (Leisch et al, 1972). It was suggested that highway facility designs conform with the characteristics and behaviour of drivers and vehicles. In other words, a good highway design should be safe for a wide variety 19 of drivers under a wide variety of operating conditions. The design elements, such as location, alignment, profile, cross section and intersections of the highway, should properly reflect drivers' safety, desire, comfort and convenience. The assumption was that roadway users must be provided with well designed equipment and surroundings if they were to be able to function effectively. Some of the basic considerations were as follows: 1. Communication It was suggested that the designed highways be able to communicate with their users through the forms and features that drivers see and experience. The information communicated by the appearance of the highway ahead includes two parts: traffic signs and highway geometric layout. Traffic signs are important in conveying needed information to motorists. Highway geometric layout is important in providing motorists with a more accurate impression of the characteristics and requirements of the roadway ahead. A highway can only be communicated if these two important features are clearly delineated. 2. Consistency It was believed that a driver should be led to expect conditions ahead that are consistent with the conditions he has experienced on the roadway previously. Consistency in the operational characteristics and driving experience along the highway may be accomplished through expansion and refinement of the design speed concept. 20 3. Three-Dimensions It was recognized that if designers are to fit the highway to the capabilities and expectations of drivers, they have to anticipate how the highway will be experienced in a three-dimensional space. The treatment of horizontal alignment, vertical alignment and cross-section as separate two-dimensional elements made it difficult to coordinate these elements into an integrated highway design. In 1987, the Transportation Research Board (TRB) conducted a study to determine the safety cost-effectiveness of geometric design criteria of standards currently in effect for construction and reconstruction of highways. The study resulted in Special Report 214 (1987), a signal document to integrate safety into the road design process. The report introduced the principle of safety cost-effectiveness and used this principle in developing many of its detailed recommendations. It discussed the concept of a safety-conscious design process and suggested that highway agencies review and revise their practice to incorporate this process, i.e., to assess current conditions, determine project scope, document the design practice and evaluate the design. As a way to improve design decisions, greater attention to safety, along with greater documentation of the design process was promoted. Research studies conducted during the past two decades have led many experts to believe that highway geometric design should be "the design of the visible dimensions of a highway with the objective of forming the facility to properly reflect driver safety, driver desire, driver comfort and convenience" (Woodridge, 1994). Highway geometric design should, therefore, 21 address at least three design issues to gain direct or indirect safety advantages (Lamm, 1989): 1) achieve consistency in horizontal and vertical alignments; 2) harmonize design speed and operating speed; and 3) provide adequate dynamic safety of driving. The achievement of geometric consistency was the subject of several studies. Wilson (1981) defined geometric consistency in rural highway design as a combination of geometric features that are similar in size or in magnitude and that meet driver expectations from the road. Glennon and Harwood (1983) argued that driver performance was directly affected by driver expectation. If driver expectancy was met, driver performance would tend to be error free. The concept of highway consistency was further expanded by McLean (1987) who presented a methodology based on driver behaviour principles associated with his workload - potential ratings for different geometric features. For example, because sharper curves are generally more troublesome, the driver's workload increases with the degree of curvature. AASHTO also developed the Driver Expectancy Checklist (1984). Design consistency was a major parameter in the list. The AASHTO study pointed to the fact that good driver communication, and therefore good design consistency, are achieved only by proper coordination among all roadway and terrain features. Another concept of consistency was introduced by (Hirsh,1985), who suggested that reliability is an indication of the operational consistency of a highway facility over a period of time. In his study, he assessed consistency by comparing daily performance measures, and then related these measures to safety level. He proposed 22 three horizontal curve measures. The first measure is the average curvature, defined as the ratio between the sum of deflection angles of two consecutive tangents and the length of the section. The second measure is the ratio of the minimum radius to the maximum radius of an alignment, which measures horizontal consistency in terms of the use of similar horizontal radii along the road. The third measure is the average radius which provides the relationship with the design speed. This may also provide information to drivers who tend to build up expectations of what the upcoming roadway will be like based on their immediate previous driving experience. The harmonization of design and operating speed was the subject of several reports, publications and presentations (Lamm, 1973, Leisch, 1977 and Bock, 1979). These investigations included processes for evaluating design speed and operating speed differences, relationships between geometric design parameters and operating speeds or accident rates and recommendations for achieving good and fair practice. Studies showed that the driving behaviour on an observed road section often exceeds the design speed by substantial amounts, especially at lower design speed levels. The provision of adequate dynamic safety of driving was the subject of a comparative analysis performed by Choueiri et al (1994) for tangential and side friction factors in the highway design guidelines of Germany, France, Sweden, Switzerland and the United States. These design guidelines determined the type of relationships that exist between friction factors and design speed as well as overall relationships between friction factors and design speed. The 23 resulting overall relationship was compared to actual pavement friction inventories in New York State and Germany. Analyses indicated that friction factors derived from the New York 95th percentile level distribution curve (i.e. 95 percent of wet pavements could be covered by using the 95th percentile level distribution curve as a driving dynamic basis for design purpose) coincided with the friction factors derived from the German 95th percentile level distribution curve. Based on these results, recommendations were provided for minimum sight distances and minimum radii of curves. The idea was that, in order to have higher dynamic traffic safety built into a design, when differences between operating speeds and design speeds exist, superelevation rates and stopping sight distances should be based on the normally higher operating speeds, which was expressed as the 85th percentile speeds of passenger cars under free flow conditions. Research studies conducted during the past two decades have also focused on design compatibility and selection of alternatives. Trietsch (1987) pointed out that this issue is rather complex and had at least three facets: 1) the question of the compatibility of existing standards with theoretical considerations of vehicle dynamics and performance characteristics; 2) the question of the sensitivity of the designed road to various driver groups. Drivers in different age groups and in different sex groups may have different characteristics; 3) the variance in the use of standards by various designers. The confusion emerges from the selection of the design values between the minimum and desirable design. 24 Obviously, many factors have to be considered in designing a highway: safety, economy, environmental concerns, energy conservation, and social effects. Geometric design, therefore, becomes a dynamic area of highway engineering and traffic engineering, which, in its true sense and broad application, translates advance technology, research and operational experience into a physical highway plan. Past work in geometric design took some significant steps forward by introducing such concepts as workload, driver expectancy, highway reliability, and general design criteria that are functionally related to highway consistency. However, no overall model has been developed for highway consistency. The fact is that drivers expect consistency in all alignment standards and that designers are provided with little quantitative information as to how this might be achieved. 2.2 Current Geometric Design Practice Current geometric design process is based heavily on design standards and the following basic design process is used. First the highway section to be designed is classified into one of the several functional classes (e.g., freeway, arterial, local). Then a design speed is selected for the highway on the basis of its classification and local conditions. After highway classification and design speed have been specified, design values for the various highway elements are selected from a set of predefined design standards. The design practice has two major advantages. First, the design concepts are transparent. 25 This enables highway engineers to be trained easily and quickly. Second, this practice supports consistent design. For example, geometric elements of freeways in different locations designed for 100 km/h will have the same design values. The design practice, however, has the following shortcomings: 1) It does not allow the designer to use his own judgement in special cases in which deviations from the standards are clearly justified (Crowel, 1989). The practice is not always sensitive to important factors such as traffic volume and construction cost. That is, once the highway class and design speed have been selected, the minimum design value of a horizontal curve, for example, is fixed. 2) The inflexible geometric design standards tend to be based mainly on safety considerations, which results in excessively high design standards in many situations (Nusbaum, 1985). For example, in the design of a vertical curve, the relevant inputs are the driver's perception-reaction time, the speed, the friction factor and the driver eye height. The design standards specify safe values for all these factors. For example, a perception-reaction time of 2.5 sec. is used because it is valid for a large percentage of the population. Thus, the sight distance based on these values would result in a costly design. 3) Highway alignment methodologies describe the three dimensional alignment on two dimension media, paper. The common practice is to split the alignment into two-dimensional projections - plan and profile, and analyse the alignment from these 26 orthogonal viewpoints. Straight lines and constant radius of curves are quickly produced on paper and the heritage of these techniques can be seen in the analysis methodology of today. Both horizontally and vertically, an alignment is compared to tangents and arcs. Sight distance is found by considering the horizontal distance that a line of sight diverges from the horizontal projection of the alignment. Designers and engineers have continuing interest in various alignment characteristics and many complex and ingenious mathematical methodologies are in use to calculate these values from the horizontal and vertical projections. The complex analysis is due to the limited computing power (and time) of the engineer. Although all the factors involved in the geometric design process (e.g., speed, friction, reaction time) are stochastic in nature and are distributed among the road users, the current approach is based on a single, arbitrarily chosen value to represent each factor. The failure to account for the stochastic nature of these factors is likely to lead under some circumstances to poor designs. That is, in some cases the combination of deterministic values may not represent road user population, and yield an unsuitable design of a highway section. 2.3 Geometric Design and Safety While exploring relationships between highway geometric design and its performance, Zegeer (1989) suggested that the occurrence of traffic accidents be considered as the probability of a failure by the road, the vehicle and the driver, separately or jointly. 27 Previous studies appear to have followed two distinct approaches: comparison approach and simulation approach. The comparison approach involves obtaining inventories of geometric and other variables (e.g. traffic variables) for a number of road segments in an area, and relating accident data for those segments to those variables. Olson (1984) performed a statistical analysis on ten pairs of sites that were matched for similarity - except for their sight distance. He concluded that the sites with limited sight distance had 50% higher accident rates than the locations with adequate sight distance. The simulation approach involves developing a simulation model to represent a real world situation by capturing all of the major factors that affect traffic operations. For example, Farber (1988) employed a Monte Carlo simulation technique to investigate accident potential for a deficient stopping sight distance situation. He investigated the hypothetical situation of a left turning vehicle downstream from a sight-distance-limiting crest vertical curve. He was able to draw conclusions about accident potential as a function of traffic volume, sight distance and other related factors. Research summarised in TRB Special Report 214 (1987) shows the correlation between accident rates and the different geometric elements that composed the road. For a two-lane rural highway, the degree of curvature has been found to be the strongest geometric variable related to accident rates. The accident rate in curves is roughly six (6) times greater than on corresponding tangent sections. With the exception of these three studies, there does not appear to be any work that conclusively defines the relationship between deficient geometric design and safety. The 28 understanding of the effects of geometric design on safety has not been adequate to predict traffic accidents effectively in response to individual geometric design element changes. The reasons might be as follows: 1) Accidents are complex phenomena. The problems at different locations may likely be different or site specific. The availability of sites necessary to the design of meaningful comparison studies is limited because of the need to control all elements at or near the sight distance restriction. 2) Accident data that are not recorded with enough precision to allow association between particular accidents and the short length of roadway that exhibit geometric deficiencies. 3) The current highway design process is aimed at ensuring consistency of geometric variables (e.g. small radius curves are often associated with low sight distances and widths), making it difficult to accurately identify the effect of each variable separately. As a result, many highway features are difficult to research directly with a neat package of statistically significant results in terms of reduced accidents. 4) The available geometric drawings may not represent the existing roadway layout and conditions, as there are considerable uncertainties over the stages of design, construction and operation. 5) A combination of geometric design elements and road user behaviours are generally considered to be the major causes in about a third of all road fatalities. Accident statistics, in general, do not provide much direction or guidance in determining their interactions. 29 A guaranteed accident-proof highway is probably unattainable, although Sweden has adopted a target of zero fatalities as its safety goal. It is believed that better safety may be made through route continuity, lane balance, alignment coordination and the communicative features of design, and more importantly, may be checked through feed-back of operational information on completed facilities. The problem is how this can be formalized, organized, and introduced as a continual part of highway design. 2.4 Human Factors and Safety Human factors that are considered in transportation engineering include human characteristics, expectations and behaviour. It is not only concerned with the person themselves, but also with the vehicle driven, the roadway travelled, and the surroundings and people that they interact with. Human factors play an important role in highway safety. The knowledge of human factors as it affects and is affected by highway design has increasingly been recognized as the basis for the development of designs that better meet the needs and expectations of the driver. Michon (1975) pointed out that in order to predict a realistic reliability of a facility, the human reliability must be included. As humans are part of the system, the reliability of the total system cannot be estimated unless the reliability of human operation is known. Hulbert (1992) further stated that the human element of any system is largely "given" and the other elements must be designed and operated around the human element. 30 Studies of human factors in highway engineering indicate the need for a high degree of consistency in geometric design. The drivers expectations are largely based on the quality of road just travelled. Any drastic change in roadway may be sufficient to confuse the driver and lead to unsafe conditions. On the other hand, traffic safety problems are often caused by limited human capabilities in reacting to a wide range of traffic situations. At higher speeds with increasing traffic densities, the road user cannot always evaluate all information correctly. Therefore, it is important to become familiar with the functional limitations of the individual road user. Past research work on human factors also took some significant steps to define and quantify human errors. Sussman (1986) defined human errors as events of different types, made by different people and that different actions are required to prevent them from happening again. Different types can be slips and lapses, mistakes, violations, errors of judgement, mismatch and ignorance of responsibilities. Different people can be managers, designers, drivers, construction workers, maintenance workers, etc. Different actions can be influenced in some cases by better training or instructions, in other cases by better enforcement of the rules, in most cases by a change in the work situation. Further, a system failure can be defined as the production of undesirable outputs of the system which may be measured in two ways: one is to compare the inputs to the system with its outputs and thus to measure its efficiency; the other is to compare the outputs with the objective of the system and thus measure its effectiveness. Whether a particular case 31 is deemed to be a success or a failure is a matter of judgement, and in looking for failures the basic activity behind the idea of judgement is a comparison in which we compare an ideal or a goal with what we see as the output from an activity. Failure is judged to occur when the comparison shows a shortfall. Williams (1988) tried to quantify the probability of human errors. He pointed that every person is different and no-one can estimate the probability that they will make a mistake. However, the probability that mistakes would be made, if a large number of people are in a similar situation, can be estimated. Treat, Sabey (1980) and Wong et al (1992) found that human factors are involved in 95% of accident cases and are the most influential. This is supported by T. Sayed (1995), he presented that human factors are involved in 93% of the accident cases in British Columbia (Figure 2.1), while the road environment is involved in 30% of the cases. Despite more than one-half century of modern road building, the knowledge of safety consequences of highway design decisions is limited. Previous studies have reported widely different results and little is known about the effects of geometric design on highway performance. 32 Road Environment Figure 2.1 Attribution of accident responsibility (Source: T. Sayed, 1995) The recognition and application of human factors can produce significant changes in design and evolve more appropriately designed highways to accommodate the driver. The questions are: • How can growing knowledge of human factors be incorporated effectively to produce a safer highway design? • How can opportunities for errors in the future operations be removed from the present design stage? It is necessary to develop a new approach which would allow designers and engineers to "visualize" future performance and incorporate the "operation" into the design stage. 33 C H A P T E R 3 HIGHWAY P E R F O R M A N C E ANALYSIS Two large databases were assembled to evaluate the effect of geometric design features on safety performance. One database consists of approximately 1,000 study segments (0.1 km each), the other database represents nearly 2,300 accidents between January 1986 and December 1990 on the Sea-to-Sky Highway. Characteristics such as accident type, severity, contributing factors, environmental conditions, and time-related data are examined in this chapter. Accident Summary Reports were obtained from the Highway Safety Branch of the Ministry of Transportation and Highways of British Columbia (MoTH). The reports include the following information: • Accident location, using the Ministry's Landmark Kilometre Inventory (LKI). • Accident data, including time, day, month, and year. • Accident severity, indicating fatalities, injuries, and property damages. • Contributing causes of the accidents. • Major accidents, indicating types of the accidents. 3.1 Accident Rates Accident rate is a fraction measurement where the numerator is defined as the number of accidents of a particular type and the denominator defined as the risk exposure. The risk 3 4 exposure refers to the characteristics of the amount of travel, the conditions of travel, and the characteristics of the driver and vehicle undertaking the travel. The rates most commonly used are: the number of accidents per Million Vehicles Kilometres of travelling per year (A/MVK) for roadway segments and the number of accidents per Million Vehicles Entering the intersection per year (A/MVE) for intersections. The mathematic definitions are: A * 1,000,000 A/MVK = N * 365 * ADT * L AIMEV - A * 1 ' 0 0 0 ' 0 0 0 N * ADTE * 365 where, A = number of accidents per year N = number of years L = length of highway in kilometre, km ADT = Average daily traffic, veh./day ADTE = ADT entering intersection, veh./day A total of 2,298 accidents was recorded within the study area during the period of 1986 to 1990. Using the Annual Average Daily Traffic Volumes (AADT or ADT) listed in Table 3.1, the accident rate from Horseshoe Bay to Squamish was found to be 1.38 and the accident rate from Squamish to Whistler was found to be 2.70. It was also noted that the accident rate in winter was greater than in summer. According to information provided by the Ministry, the average accident rate on North American two-lane highways with AADT volumes of more than 5,000 vehicle per day is 0.8 accidents per million vehicle kilometres travelled. The Sea-to-Sky Highway accident rate is much higher than the North American average. 35 Table 3.1 Motor vehicle accident rates (Three-year period) Periods Accident Rates 1986- 1988 1.47 Horseshoe Bay to Squamish 1987 - 1989 1.42 1988 - 1990 1.38 1986 - 1988 2.69 Squamish to Whistler 1987 - 1989 2.61 1988 - 1990 2.70 3.2 Accident Severity An accident severity study was conducted to identify the number of occurrences of fatalities, injuries and property damages. The results of the analysis were summarized in Table 3.2 and Figure 3.1. Table 3.2 Accident severity Fatality Injury P. D. O. Total No % No. % No. % No. % 1986 2 0.09 152 6.61 229 9.97 383 17 1987 4 0.17 151 6.57 275 11.97 430 19 1988 9 0.39 167 7.27 280 12.18 456 20 1989 5 0.22 175 7.62 287 12.49 467 20 1990 7 0.30 178 7.75 377 16.41 562 24 Total 27 1.17 823 35.81 1448 63.01 2298 100 36 600 500 H 400 LU o o O 300 < L L o O 200 100 EMI T O T A L [YEI D A M A G E ESD INJURIES • F A T A L I T I E S 1986 1987 1988 YEAR 1989 1990 Figure 3.1 Accident severity As shown in Figure 3.1, the total number of annual accidents increased consistently from 385 in 1986 to 562 in 1990. This trend indicated a growth of 45 percent in the number of accidents between 1986 and 1990, an 8 percent annual increase mostly due to increasing property damage only accidents. As shown in Table 3.2, 1.17 percent of accidents involved fatalities. This percentage had increased greatly between 1986 (0.09 percent fatal accidents) and 1990 (0.30 percent fatal accidents). In the five year period between 1986 and 1990, approximately 35.8 percent of accidents involved injuries, and 63 percent involved property damage. The number of injury 3 7 and property damage accidents had also increased from 152 to 178 (17% increase), and from 229 to 377 (65% increase), respectively. 3.3 Spatial Distribution The spatial distribution of motor vehicle accidents was identified using the Ministry's Location Code, which categorized the accidents by types and locations. As shown in Table 3.3 and Figure 3.2, the majority of the accidents occurred on the segments of the highway between intersections. Table 3.3 Accidents by locations Locations Fatality Injury P.D.O Total Percentage Unknown 0 35 39 74 3.22 At intersection 2 143 258 403 17.54 Between intersections 18 550 984 1552 67.54 In driveway 0 11 20 31 1.35 Bridge 1 32 43 76 3.31 Ferry/dock 0 0 1 1 0.04 Tunnel 1 0 0 1 0.04 Exit deceleration lane 0 1 1 1 2 0.09 Enter acceleration lane 0 4 2 6 0.26 Ramp entrance 0 0 1 1 0.04 Intersection entrance 0 0 1 1 0.04 Off highway 3 25 49 77 3.35 Single/multilevel parking lot 0 1 7 8 0.35 Crossing 1 7 27 35 1.52 Other 1 14 15 30 1.31 Total 27 823 1448 2298 100.00 38 17.5% Figure 3.2 Pie-chart of accident locations Accident spot maps with intervals of 0.1 km were formulated for both sections of Horseshoe Bay to Squamish and Squamish to Whistler, respectively, in which accident locations were specified to the Ministry's Landmark Kilometres Inventory (LKI). Accident locations were then re-arranged by ranking number of accidents from the highest to the lowest, as shown in Tables 3.4 and 3.5. For example, Table 3.4 indicate that LKI 12.1, LKI 4.9, LKI 18.7, LKI 44.3 and LKI 7.3 are the first five (5) high accident locations between Horseshoe Bay and Squamish. Table 3.5 indicate that LKI 40.1, LKI 41.1, LKI 23.7, LKI 13.1 and LKI 15.1 are the first five (5) high accident locations between Squamish and Whistler. These locations may warrant detailed analysis. 39 Table 3.4 Histogram of accident frequency (Horseshoe Bay to Squamish) KM .MARK (+0.lkm) HISTOGRAM O F ACCIDENT FREQUENCY F A T I N I P D O T O T A L C L E V . I N T N nimiiunnippppppppppppppppppppppppppppppppppp 0 15 36 SI SOUTHEND uiiiiiiiiippppppppppppppppppppp 0 11 21 32 26 12.1 IIIIIIPPPPPPPPPPPPPPPPPPPP 0 6 20 4.9 miippppppppppppppp 0 5 15 20 18.7 • FFUIUUPPPPPPPP" 2 7 8 17 4 4 3 unnnppppppppp 0 8 9 17 7.3 nnppppppppp 0 4 9 13 13 26.8 mnpppppppp 0 5 8 39.9 nnnppppppp 0 6 7 13 12 « 12 6.7 nnnnpppp 0 8 4 12.0 nmppppppp 0 5 7 32.9 HIPPPPPPPPP 0 3 9 13.5 inrappppp 0 6 5 11 13.7 UIIPPPPPPP 0 4 7 11 10 22.4 rrpppppppp 0 2 8 5.5 IIPPPPPPP 0 2 7 9 29.7 HPPPPPPP 0 2 7 9 31.4 iiiiiiipp 0 7 2 9 43.5 raippppp 0 4 5 9 0.6 IPPPPPPP 0 1 7 g 9.7 IIIPPPPP 0 3 5 8 16.3 IPPPPPPP 0 1 7 8 17.2 IPPPPPPP 0 1 7 8 20.5 raiippp 0 5 3 8 27.2 uniipp 0 6 2 8 29.4 IIPPPPPP 0 2 6 8 32.0 nupppp 0 4 4 8 32.3 Iipppppp 0 2 6 8 32.8 DIPPPPP 0 3 5 8 43.0 npppppp 0 2 6 8 0.8 IIIPPPP 0 3 4 7 7.2 UPPPPP 0 2 5 7 14.4 UPPPPP 0 2 5 7 14.9 IIIPPPP 0 3 4 7 17.3 IIIPPPP 0 3 4 7 31.1 IIIPPPP 0 3 4 7 37.1 IIIIPPP 0 4 3 7 8.2 IIPPPP 0 2 4 6 8.3 PPPPPP 0 0 6 6 8.5 IIIIPP 0 4 2 6 10.5 PPPPPP 0 0 6 6 12.5 HPPPP 0 2 4 6 13.0 IPPPPP 0 1 5 6 13.2 IPPPPP 0 1 5 6 13.4 FHIPP 1 3 2 6 14.1 IIPPPP 0 2 4 6 25.8 IIPPPP 0 2 4 6 35.5 HIPPP 0 3 3 6 L E N G T H 3 180 326 509 4.6 km 40 Table 3.5 Histogram of accident frequency (Squamish to Whistler) K M . M A R K ( + 0 . 1 k m ) H I S T O G R A M O F A C C I D E N T F R E Q U E N C Y F A T INJ P D O T O T A L 4 0 . 1 IIIIIIIIIIIIIPPPPPPPPPPPPPPPPPPPPPPPPP 0 13 2 5 3 8 4 1 . 1 rarmmmpppppppppppppppppp 1 12 18 31 2 3 . 7 u r m r a i m p p p p p p p p p p p p p 0 13 13 2 6 1 3 . 1 IIIIIIIIIIPPPPPPPPPPPPPP 0 1 0 14 2 4 1 5 . 1 u n n i p p p p p p p p p p 0 7 1 0 17 1 2 . 5 F F n n n r p p p p p p p 2 7 7 16 7 . 0 i m m p p p p p p p p 0 7 8 15 4 0 . S F n i U D l P P P P P P 1 8 6 15 1.7 HIIPPPPPPPPPP 0 4 1 0 14 8 . 9 IIPPPPPPPPPPP 0 2 11 13 3 9 . 7 i n n p p p p p p p p 0 5 8 13 1 4 . 5 IIIPPPPPPPPP 0 3 9 12 . 4 1 . 0 IUIIPPPPPPP 0 5 7 12 1 4 . 4 UPPPPPPPPP 0 2 9 11 1 9 . 7 IIIPPPPPPPP 0 3 8 11 4 0 . 8 IHPPPPPPPP 0 3 8 11 AS IUPPPPPPP 0 3 7 10 1 3 . 6 FIIIPPPPPP 1 3 6 10 1 5 . 6 IUPPPPPPP 0 3 7 10 2 8 . 2 n p p p p p p p p 0 2 8 10 2 8 . 3 n i p p p p p p p 0 3 7 10 3 3 . 3 IHIPPPPPP 0 4 6 10 1 7 . 2 HPPPPPPP 0 2 7 9 1 7 . 4 IIIIIPPP 0 5 3 8 2 4 . 4 IIIIPPPP 0 4 4 8 3 5 . 9 IPPPPPPP 0 1 7 8 4 0 . 1 IIIPPPPP 0 3 5 8 13.1 IIPPPPP 0 2 5 7 1 4 . 2 IIIPPPP 0 3 4 7 1 6 . 0 IIPPPPP 0 2 5 7 1 8 . 4 IIPPPPP 0 2 5 7 2 4 . 6 IIPPPPP 0 2 5 7 3 1 . 9 FIPPPPP 1 1 5 7 4 0 . 7 IIPPPPP 0 2 5 7 4 1 . 6 IIPPPPP 0 2 5 7 2 . 9 IIIIIP 0 5 1 6 13 .3 IPPPPP 0 1 5 6 1 3 . 9 F P P P P P 1 0 5 6 1 6 . 4 IIPPPP 0 2 4 6 2 0 . 4 IIPPPP 0 2 4 6 2 5 . 4 IIPPPP 0 2 4 6 3 2 . 3 IIPPPP 0 2 4 6 4 2 . 5 I P P P P P 0 1 5 6 L E N G T H 4 . 3 lent 7 168 3 0 9 4 8 4 41 3.4 Time Distribution An accident analysis by time period was conducted to assist in identifying safety deficiencies by defining the time period patterns of the accident events. For example, the accident characteristics occurring during a certain time period may determine the possible accident causes to be considered in later studies. In addition, the occurrence of the accident types under a night time condition may identify "inadequate lighting or delineation" as a possible accident cause. The monthly and hourly distributions of accidents are shown in Figure 3.3 and Figure 3.4, respectively. 350 300 250 CO g 200 O o < u_ O 150 ci 100 50 Fatalities Injuries Damage Total / / / / / / / v . ...» y *-*••/ ^ • i i 1 2 3 4 5 6 7 8 9 10 11 12 MONTH Figure 3.3 Monthly accident distribution 42 6:00-10:00 10:00-14:00 14:00-18:00 < Q u. O 18:00-22:00 UJ 2 H 22:00 - 2:00 2:00 - 6:00 Unknown D u — D H ' h 0 10% 20% 30% 40% PERCENT OF ACCIDENTS Fatalities f H Injuries H Damages Figure 3.4 Hourly accident distribution Conclusions can be drawn as follows: • More accidents occurred in December and January than in any other months, and the fewest accidents occurred between April and August; • More accidents occurred between 14:00 and 18:00 than during any other periods. Accidents occurred in the 12-hour period of 18:00 to 6:00 represented slightly more than one-third of all accidents, but were 40 percent of the severe or fatal injury accidents. 43 3.5 Accident Types and Contributing Causes The study of accident types identified patterns of accidents based on the specific type and location. This serves as the major indicator for the possible causes of the specific safety problems at a specific site. It also identified factors such as direction of the involved vehicles, intended movements of the involved vehicles, and the number of involved vehicles. With the review of these factors, an accident analysis by contributing causes was further conducted to develop a preliminary list of possible accident causes. (1) Accident types The different types of accidents which occurred on the Sea-to-Sky Highway in the period between 1986 and 1990 were analysed, using the "Primary Accident Occurrence" information available from the Ministry's databases. As shown in Figure 3.5, the results of the analysis were as follows: • The most common accident type are "Off Road" accidents (44 percent), in which approximately 27 percent were classified as "Off Road Right", while approximately 17 percent were classified as "Off Road Left". • Other common accidents types included "Rear-End" (14 percent), "Side Swipe" (6 percent) and "Head-On" (5 percent). 44 1200 1000 800 + P 2 LU Q O O £ 600 f O rr UJ m 400 200 FATALITY INJURY mm PDO V :.: TOTAL Figure 3.5 Accident by manner of collision (2) Contributing causes The accident contributing causes were analysed, and the most common contributing causes are listed in Table 3.6. The results of the analysis indicated that "Driving without due care" and "Unsafe speed" were the most common accident causes, each comprising approximately 14 percent of all accidents. The third most common accident cause was "Weather", which was involved in 11 percent of all accidents. Other accident contributing causes accounted in less than ten percent of accidents. 45 Table 3.6 Accidents by contributing causes CONTRIBUTING FACTORS SEVERITY Fatality Injury Property Damage Total No. % No. % No. % No. % Not applicable 5 0.22 138 6.01 272 11.84 415 18.06 Driving without care 6 0.26 135 5.87 175 7.26 316 13.75 Unsafe speed 6 0.26 127 5.53 173 7.53 306 13.32 Weather 1 0.04 64 2.79 196 8.53 261 11.36 Other 1 0.04 54 2.35 98 4.26 153 6.66 Alcohol involvement 4 0.17 84 3.66 60 2.61 148 6.44 Driver inexperience 0 0.00 28 1.22 70 3.05 98 4.26 Wild animal 0 0.00 10 0.44 83 3.61 93 4.05 Follow too close 0 0.00 18 0.78 29 1.26 47 2.05 Failing to yield 0 0.00 22 0.96 21 0.91 43 1.87 Driving on wrong way 1 0.04 20 0.87 14 0.61 35 1.52 Total No < 30 331 14.4 TOTAL 2298 100.0 (3) Major type of accidents with contributing causes Major accident types of "Off Road Right", "Off Road Left", "Rear-End", "Head-On", and "Side Swipe" accidents with contributing causes were listed in Tables 3.7, 3.8, 3.9, 3.10 and 3.11, respectively. "Driving without care" and "Unsafe speed" are the most influential factors to cause the accidents. These factors can also be considered as the results from drivers interacting with the highway. 4 6 Table 3.7 "Off Road Right" accidents PERCENT CONTRIBUTING FACTORS No. Fatality Injury Property % % Damage • % Driving without care 117 5.09 37.03 18.72 Unsafe speed 108 4.7 0 35.29 17.28 Weather 85 3.70 32.57 13.60 Alcohol involvement 60 2.61 40.54 9.60 Driver inexperience 42 1.83 42.86 . 6.72 Obstruction/debris 14 0.61 26.92 2.24 Fell asleep 13 0.57 48.15 2.08 Road maintenance 13 0.57 43.33 2.08 Wild animal 13 0.57 13.98 2.08 Avoiding veh/ped 12 0.52 46.15 1.92 Tires failure 12 0.52 44.44 1.92 Other 40 1.74 26,14 6.40 Not applicable 29 1.26 6.99 4.64 Total No < 10 67 10.7 TOTAL 625 100.00 Table 3.8 "Off Road Left" accidents PERCENT CONTRIBUTING No. FACTORS Fatality Injury Property Damage Unsafe speed 106 4.61 4.64 26.57 Weather 65 2.83 24.90 16.29 Driving without care 53 2.31 16.77 13.28 Alcohol involvement 43 1.87 29.05 10.78 Driver inexperience 19 0.83 19.39 4.76 Wild animal 12 0.52 12.90 3.01 fell asleep 12 0.54 44.44 3.01 Other 19 0.83 12.42 4.76 Not applicable 17 0.74. 4.10 4.26 Total No < 10 53 13.3 TOTAL 399 100.0 47 Table 3.9 "Rear End" accidents PERCENT CONTRIBUTING No. FACTORS Fatality Injury Property Damage Driving without care 48 2.09 15.19 15.29 Following too close 40 1.74 85.11 12.74 Weather 25 1.09 9.58 7.96 Other 22 0.96 14.38 7.01 Not applicable 136 5.92 32.77 43.31 Total No < 10 43 13.7 TOTAL 314 13.66 Table 3.10 "Head On" accidents PERCENT CONTRIBUTING No. FACTORS Fatality Injury Property Damage Unsafe speed 18 0.78 5.88 14.63 Driving without care 14 0.61 4.33 11.38 Wild animal 12 0.52 12.90 9.76 Driving on wrong 10 0.44 28.57 8.13 Not applicable 32 Total No < 10 37 30.1 TOTAL 123 5.35 Table 3.11 "Side Swipe" accidents PERCENT CONTRIBUTING No. FACTORS Fatality Injury Property Damage Unsafe speed 17 0.74 5.56 11.81 Driving without care 17 0.74 5.38 11.81 Weather 16 0.70 6.13 11.11 Driving on wrong 14 0.61 40.00 9.72 Not applicable 46 2.00 11.08 31.94 Total No < 10 34 23.6 TOTAL 144 6.27 48 3.6 1. Findings/Summary There were 2,298 total reported accidents on the Sea-to-Sky Highway during the five year period between January 1986 and December 1990. 2. The average accident rate was found to be 1.38 accidents per million vehicle kilometres on the Horseshoe Bay to Squamish section, and 2.70 accidents per million vehicle kilometres on the Squamish to Whistler section. Accident rates were higher in winter than in summer. 3. A review of the accident data by type reveals the most frequently reported accidents are "Off Road" accidents (44 percent), followed by "Rear End" accidents (14 percent), "Side Swipe" accidents (6 percent) and "Head On" accidents (5 percent). 4. The most common accident contributing cause was "Driving Without Due Care" and "Unsafe Speed", each involved in approximately 14 percent of all accidents. The third most common cause is "Weather", which involved 11 percent of all accidents. 5. In terms of accident severity, injury accidents accounted for 35.8 percent of the total accidents, property damage accidents accounted for 63 percent and fatality accidents accounted for 1.2 percent. 49 6. Accident spot maps with intervals of 0.1 km for the Horseshoe Bay - Squamish section and Squamish - Whistler section were formulated. The identified high accident locations were summarized in Tables 3.4 and 3.5. They indicated that LKI 12.1, LKI 4.9, LKI 18.7, LKI 44.3 and LKI 7.3 are the first five (5) high accident locations between Horseshoe Bay and Squamish and LKI 40.1, LKI 41.1, LKI 23.7, LKI 13.1 and LKI 15.1 are the first five (5) high accident locations between Squamish and Whistler. 7. The analysis of monthly distribution indicated that more accidents occurred in December and January than in any other months, and the fewest accidents occurred between April and August. The analysis of hourly distribution showed that more accidents occurred between 14:00 and 18:00 than any other periods. 50 C H A P T E R 4 EXISTING G E O M E T R Y ANALYSIS "Existing geometry" refers to what is actually built into a highway. The Sea-to-Sky Highway was built in the 1950s using a variety of standards. Due to the variation and uncertainty involved in different stages of the highway's design, construction and operation, no one actually knows the exact geometry of the Sea-to-Sky Highway. 4.1 Photolog Files Each year, the British Columbia Ministry of Transportation and Highways (MoTH) uses a monitoring vehicle equipped with a camera and various measuring devices to collect electronic information on a number of highways at 10 m intervals in urban areas and 20 m intervals in rural areas. The recorded information is then compiled into a photolog which can be retrieved to provide visual information and engineering details along a highway. The photolog shows a view of the roadway in the lower 75 percent of the screen and a digital display panel of certain road measurements in the upper 25 percent of the screen. As shown in Figure 4.1, the panel displays the following engineering details: odometer (km), speed (km/h), grade (%), transverse slope, side friction, altitude (m), and compass readings (degree). A plus or minus sign is specified for the odometer readings, according to the direction of travel. If one direction is listed as a plus, the opposite direction will have a minus sign. The bearing of the vehicle is automatically determined by a gyro compass. Percent 51 grade of the road is recorded by using an electrically powered gyro as a reference platform. Transverse slope is recorded by using the grade gyro with a sensor perpendicular to the grade sensor. This sensor indicates the total transverse slope of the road, plus any slope contributed by the vehicle due to static or dynamic loading. - R O U T E " -CONTROC-U-YR - D A T E ' go | O | D | O l-o-l o l o | o ^ o l o l o | o l o l o M o | o l o l o l o | o 7"io it a 13 M a w rr w » 20 21 22 *s 24 » 26 zr te » so 31 sa riMe-* «|<3ce— o\o\o 10 1 oIoj0jo 1 2 3 4 9 6 7 8 - O O O M E T E R -K m K S P C E D H * C R A O E -k m / h percent -TRANSVERSE-SLOPE —soe—' FRICTION t l o l o l . |o ' t |o | . |o | . | o | o l t l o | o l • I o I ± I o I -Tol . ' . i . m IUI 11 m m M « 1 M BT M M BO M « « M - C U R M H T U R C --Howr-• |o|0|oI.|o |o (0 o o 33 34 33 36 37 36 39 40 41 42 43 44 48 46 47 46 49 00 61 K M M M M W M i t W M « 63 - R O U O H N E S S -SHOPT-—j J * — L O W — j r ^ « ^ 7 o S . ' 7 2 ' r 3 S 4 ' 7 0 ' 7 6 V , T 8 V 8 0 ^ , , 6 2 l r 8 3 V 80 . 6 67 66 69 90 9, 92 . 3 94 90 Figure 4.1 Photolog digital display panel In the past, the photolog served as a "video tape" which could be displayed to view the roadway as well as the roadway environment. Although it provides some engineering measurements, none of them could be directly related to the existing geometric design process urn - A L T I M E T E R -metres MS 0 0 h - C O M P A S S - H degree I K - F t L M M 0 . - H o I o Io M o I ofo -EVENT MAIMER- 1 olo|o|o|o|olo|o without further processing. In this study, the photolog for the Sea-to-Sky Highway was used. The section of the highway is mainly a two lane road with a posted speed of 80 kilometres per hour. 52 4.2 Highway Alignment Database from Photolog Data The purpose of developing a highway alignment database is to obtain engineering information which is related to certain design features. This database would then be used to identify highway geometric design deficiencies along the Sea-to-Sky Highway. In order to create a highway alignment database from the photolog, a computer algorithm is developed based on the following considerations: the appropriate objectives, the constraints on what can be done, the interrelationships between the photolog and the highway alignment database and the possible alternative course of actions. Highway performance analysis indicated that 45 percent of accidents along the Sea-to-Sky Highway are "off-road" accidents. This provided insight about the role of various geometric and roadway features relevant to accidents, especially sight distance problems. Field investigation along the Sea-to-Sky Highway also supported the notion that sight distance on certain sections of the roadway may not be adequate. Complaints made by users of the Sea-to-Sky Highway was the third source of evidence. The complaints showed that there was a strong need to evaluate sight distance. It is clear that the objective should be developing computer software by which the design parameters, such as radii on horizontal curves, k (curvature) values on vertical curves and sight distances at each point along the highway can be evaluated. Highway alignment is largely based on the requirement of stopping sight distance. By 53 definition, stopping sight distance is the length of roadway that is visible to the driver. The stopping sight distance available to drivers should be long enough for a vehicle travelling at the design speed to stop just before reaching a stationary object in its path. Deficient stopping sight distance, therefore, can be defined as not meeting the current policy. The percentage of deficient stopping sight distance varied greatly depending on the criteria used to define adequate stopping sight distance. One of the criteria used by MoTH is that if the design speed is 80 km/h, less than 140 m sight distance is considered as inadequate stopping sight distance. This is based on the driver eye height of 1.05 m and the object height of 0.15 m. The terminology of "deficient sight distance segments" will hereafter be used to indicate the portions of the highway that have less sight distance than that specified in TAC. The possible course of action is mostly determined by the nature of the data. As the photolog information was obtained by a dynamic method - using a monitoring vehicle to collect on-line data, a dynamic method of processing this data would be expected. In addition, it is believed that the task for identifying deficient sight distance segments should be accomplished by making a sequence of interrelated decisions, because the driver is a critical component in the road system and he has to recognize visual information needed for the driving task. For drivers to actually see an important clue ahead, they must search a field of view for the object, focus on the object, and then recognize the object. The drivers' main concern is how far they can see and how wide a field of view they have from each point along the highway. It is clear that this research should not only focus on the sight distance available at each point, 54 but also study the rate of change in sight distances from point to point. This is a dynamic consideration and consistent with the driving task. To achieve this goal, the following developments need to be considered: • an advanced technique for dealing with a sequence of interrelated decision-making, • a systematic procedure for identifying situations that generate the maximum demand for stopping sight distance, and • an appropriate recursive equation for describing relationships of all individual driving situations. 4.3 Methodology and Programming Technique When designing a highway, stopping sight distance is the most basic criterion to be met. This is because stopping sight distance requirements affect all geometric elements - horizontal alignment, vertical alignment and cross section. This section is devoted to the development of a computer algorithm to transfer photolog data into a highway alignment database. This database forms the main part of the existing highway geometry. A three dimensional search technique together with vector analysis is used to develop the VHVIS.PAS (Vertical Horizontal Visibility written in PAScal language). The software can be used to simulate real-world traffic operations and identify deficient stopping sight distance segments along horizontal curves, vertical curves, or both. The VH VIS. PAS program first goes through all of the records in the photolog to identify deficient vertical curves on the Sea-to-Sky Highway, then goes through all of the records again 5 5 to search for deficient horizontal curves on that highway. If the segments appear in both vertical and horizontal search loops, it means that both horizontal and vertical deficiencies exist in those segments. At each point, the coordinate (x, y) must be known. These values are obtained through compass and odometer readings: x = ODM * cos0 v = ODM * sin0 4.3.1 Identification of vertical deficiency segments Assume that the road angle, a, is an angle from the horizontal to the road when sighted from the eye position. The object angle, p, is an angle from the horizontal to the object when sighted from the eye position. The difference between road angle and object angle might be positive or negative at a trial solution, as shown in Figure 4.2. The following conditions must be met: where, ODM = 6 odometer readings, km compass reading, radian a - P > 0 =» downhill a - P < 0 =» uphill road angle object angle 56 eye height = 1.05 m Figure 4.2 Road angle a vs. object angle p Let E be the position of the eye above the roadway point, as shown in Figure 4.3, P and O be the object sitting at point Q some distance D along the road, q is the vector from E to O and m is the vector from E to an arbitrary point M on the road between P and Q. Interpolating along q the distance I m I we find B and compare it to M. 57 Figure 4.3 Vertical sight distance If Bz > Mz for all points M between P and Q, then P has a vertical sight distance of D (measured along the curve). Conversely, if for some points Q, there exists an intermediate point M such that Bz < Mz , then there is a vertical sight distance deficiency. Inclusion of the physical width of the alignment and associated obstacles such as mid-highway barriers is considered by checking Bz against the elevation of the appropriate point on the surface in question. 58 Vertical visibility at each point on the road may be calculated by the following algorithm: 1) Set eye position at eye height defined by user above point (X;, Y„ Z,); 2) Calculate the relative position of the road X metres further on; 3) Set object position at object height defined by user above road; 4) Calculate the elevation of the road and the object above it relative to eye position; 5) Update the maximum road elevation encountered as necessary; 6) If the object elevation is less than the maximum road elevation then the object is "behind" a previous section of road and point (X„ Y„ Z,) has a vertical deficiency; 7) If the object elevation is still greater than the maximum road elevation then return to step 2 with X = X + 20, where 20 m is the interval that electronic information is being collected during the photolog survey; 8) If X reaches the minimum interval (say 160 m) then no vertical deficiency will be found and the program restarts calculations at the next point on the road. 4.3.2 Identification of horizontal deficiency segments The stopping sight distance profiles on horizontal curves have different characteristics from vertical curves because the sight obstruction is off the highway instead of being the highway alignment itself. Horizontal visibility can be detected by the developed algorithm in two ways: (1) the Middle Ordinate Analysis and (2) the Three Dimensional Angle Search. 59 Middle Ordinate Analysis This method is tied to the current design practice on sight distances for horizontal curves. Sight distance at a horizontal curve is measured along the centre of the lane of travel. The sightline is a straight line that connects the driver's eye position to the end of the required sight distance, as shown in Figure 4.4. An algorithm was, therefore, designed to check if an obstruction blocks this sightline. The required middle ordinate is a distance from the sight line to the arc of the vehicle path. It varies according to the driver's position and the radius of the horizontal curve. If the calculated (or the demanded) middle ordinate, M, is greater than a user-specified maximum allowable lateral clearance on the curve, then the current highway location is considered to have deficient sight distance. Figure 4.4 Horizontal sight distance 60 To determine horizontal sight distance deficient segments, the VH VIS. PAS program adjusts the middle ordinate to account for the position of the photolog vehicle in the centre of the inside lane. As shown in Figure 4.4, the sightline and its perpendicular line are described by v, = a + kxx y2= b + k2x (4.1) where, a = intercept, assume a = 0 k, = slope of the sightline k2 = slope of the perpendicular line to the sightline Notice that the line y7 intersects the line y2 at the point (x, y), the following condition met: yx - y2 (4.2) Substituting back to Equation 4.1 yields fcjX = b + k2x (4.3) Thus b x = kl k2 kxb kt kz (4.4) 61 The available middle ordinate, M, therefore, can be calculated by k, b y3)2 (4-5) M = 1 Where, k2 y2l x2 -1/kj = Another way to calculate the middle ordinate, d, is to perform a vector analysis. Let k be the vector from P, to Pk and j be the vector from P, to Pj, the distance or the middle ordinate, d, from P, to a straight line between P, and Pk is As shown in Figure 4.5, given a position of P,on the road, we say that P, has sight distance D if the straight line connecting P, and Pk (a point at distance D measured along the road from P,) has a maximum lateral separation (or middle ordinate), d, from the road no greater than, say, 5 m. Since the physical position of all parts of the alignment are obtained directly from the composite spline functions, this formula can be applied at any point on the alignment. d = (4.6) 62 Vector PiPH Vector PtPi=k Figure 4.5 Middle ordinate, d The algorithm used to locate horizontal sight distance deficient segments is as follows At each point (X/5 Y,) on the road 1) Calculate road position X metres further along from (Xk, Y*); 2) For each (X7, Yj) between (X„ Y,) and (Xk, Y*) calculate the perpendicular distance from (X,, Y,) to a straight line from (X„ Y;) to (Xk, Yk), call it distance, d; 3) The maximum of these distances is called the maximum distance; 4) If the maximum distance is greater than the maximum allowable M defined by the user then a horizontal deficiency is found for point (X;, Y,); 5) If the maximum distance is less than the maximum allowable M then return to step 1 with X = X + 20; 63 6) If X reaches the minimum interval (say 160 m) then no horizontal deficiency is found and the program restarts calculations at the next point on the road. An alternative method of measuring sight distance is to consider how far from directly ahead the driver must look to see an object some distance down the road. This leads to an angle analysis. 2. Angle analysis Angle analysis is based on the idea that most driving information comes to drivers visually in a stream of changing scenes from which they must make decisions about where they are going to be in the next few seconds. This is a spacial commitment zone that is fan-shaped and extends in front of the moving vehicle, as shown in Figure 4.6. The short term view is t and the longer term view is k. P. Vector PiP. = t Vector PA = k Figure 4.6 Three dimensional angle searching zone 64 The angle 6, subtended by the vector t and k is 0 = ArcCosine t k (4.7) This angle 0, defined by Equation 4.7, needs to be checked from point to point. If it is outside some acceptable limits, a sightline problem may exist at that point of the alignment. The exact configuration of this fan-shaped zone will depend on vehicle speed, turning radius and stopping distance which is related to the driver's perception-reaction time. The critical value of 0 represents the "awareness angle" of the driver. That is to say that the driver is able to quickly interpret information presented within 0 degree from straight ahead. In three dimensions we could consider a driver's "awareness cone" representing the solid cone centred upon the driver's line vision. This cone could have various cross sections: circular for equal awareness at all divergent angles, elliptical for enhanced awareness in the horizontal or vertical directions and axial for awareness biased away from the diagonal. Since all the vectors are available from digital analysis we might easily link sight distance to projected vehicle speeds. This illustrates that digital alignment information will allow the formulation of more intricate, subtle and responsive analysis techniques. 65 A dynamic search technique and vector analysis are used in the VHVIS.PAS to calculate three-dimensional angles at each point. If the calculated angle is greater than a maximum user specified angle, deficient stopping sight distance is defined at that point. At each point (X„ YJ on the road, the algorithm is as follows: 1) Store the current vehicle direction as the eye direction; 2) Calculate the relative position of the road X metres further on; 3) Calculate the direction to this point, call it the object direction; 4) If the angle between eye direction and object direction is greater than the maximum off-line angle defined by user, then the road is said to be horizontally deficient at point (X;, Y;); 5) If the angle is still too small to be a problem then return to step 2 with X = X + 20; 6) If the X reaches the minimum interval (say 160 m) then no horizontal deficiency is found and the program restarts calculations at the next point on the road. The VHVIS.PAS software program is included in the diskette ZHENG.PHD. 4.4 Validation of the Software VHVIS.PAS Validation is a process of assessing the extent to which an instrument measures what it purports to measure. The criteria used in this study for judging the validity of the 66 VHVIS.PAS software are 1) to check whether it produces the known facts with sufficient accuracy so that a sound comparison can be made when photolog data were input for different years and different directions; 2) to examine whether the output from the model behaves in a plausible manner when varying the input parameters or variables; and 3) to verify whether the calculated results are close to the results viewed from the photolog film. The tests involve using historical data for different years and different directions, and observing "situations" from the photolog films. Comparisons are made to determine whether using the VHVIS.PAS tends to yield significantly consistent results. The northbound and southbound photolog data for the years of 1988, 1989, 1990 and 1991 were used as input for VHVIS.PAS software. Both segment 2920 (Horseshoe Bay to Squamish) and segment 2930 (Squamish to Whistler) were tested. To validate the VHVIS.PAS software, comparisons were made in the Sections 4.4.1, 4.4.2 and 4.4.3. 4.4.1 Data input for different years The following two issues were addressed when photolog data obtained from different years were input into VHVIS.PAS program: 67 1. Number of curves with deficient stopping sight distance As shown in Tables 4.1 and 4.2, VHVIS.PAS produces an average variation less than eight percent in identifying the number of deficient curves by using photologs obtained from different years. Table 4.1 Number of deficient curves for different years (Northbound: Horseshoe Bay to Whistler) Number of vertical curves Number of horizontal curves Average variation HBtoS Sto W HBtoS Sto W 1988 48 47 73 34 0.076 1990 43 49 67 36 Variation = 11988-19901 - 1990 0.116 0.041 0.090 0.056 HB = Horseshoe Bay, S = Squamish, W = Whistler Table 4.2 Number of deficient curves for different years (Southbound: Squamish to Horseshoe Bay) Number of vertical curves Number of horizontal curves Average variation 1989 43 78 0.079 1991 40 85 Variation = 11989-19911 H- 1991 0.075 0.082 68 2. Deficient stopping sight distance For each small segment of 20 m, the slope estimation is made by the VHVIS.PAS software to approximate the total length of the curve. Due to the process of this estimation, the identified deficient stopping sight distances are slightly different among the different years. 4.4.2 Data input for different directions The output from northbound and southbound photolog data were compared and the results are shown in Table 4.3. The results indicate that VHVIS.PAS produces an average variation less than ten percent in identifying the number of deficient curves by input photolog data collected for different directions. It is noted that four horizontal curves and two vertical curves have been upgraded since the year of 1989. Table 4.3 Deficient curves for different directions (Horseshoe Bay to Squamish) Year Number of vertical curves Number of horizontal curves Southbound 1989 43 78 1991 40 85 S/B variation = 11989-19911 -H 1991 0.075 0.082 Northbound 1988 48 73 1990 43 67 N/B variation = |1988-1990|- 1990 0.116 0.089 Average variation 0.0955 0.0855 69 4.4.3 Photolog films review Observations of the photolog films provide a review of the hazardous locations or situations "in the field". They can be used to verify or supplement the findings of using VHVIS.PAS. The films were reviewed in a manner similar to the field method. A checklist of questions was used to evaluate the field operations and conditions. For effective results, the roadway was initially viewed at a normal travel speed to experience the location as a driver would. Subsequently, the roadway was viewed at a lower speed to permit recording data while observing the physical characteristics of the site. Radii of curvature are the typical measures for determining the tightness of the curves. Based on the design standards established by the British Columbia Ministry of Transportation and Highways, the following relationships between the radii of curvature and the degree of tightness exist: Table 4.4 Curve radii and degree of tightness Radii of curvature (m) Degree of Measurement < 230 Substandard Curve 230 < R < 350 Tight Curve 350 < R < 1,000 Good Curve > 1,000 Generous Curve 70 The procedure for checking the effective radius, Re, from the photolog film is to "drive" on a curve, locate the point of curve PC, continue to "drive" to the end of the curve PT, record the compass readings from the screens along with PC and PT, and use the following equation to calculate Re L = ( ODMpT - ODMpc ) * 1000 R = L I ( CpT - Cpc ) * (—) ± 1.8 (4.8) c 180 length of curve (m) odometer readings at the PT and PC compass bearing at the PT and PC, respectively radius (m) The effective radii, Re, calculated by Equation 4.8 for randomly selected 30 curves from photolog film were then compared with radii, R, calculated by VHVIS.PAS. As shown in Table 4,5, the variances are negligible. where, ODMpt, ODMpc C C R 71 Table 4.5 Comparison of R^n and Re for the Sea-to-Sky Highway Curves Categories # of curves Odometres (m) C„-C„ (m) R.*(m) R.(m) 1 0.08 - 0.34 347 - 284 223.3 < 238.3 2 2.44 - 2.62 045 - 000 229.2 < 231.0 3 5.88-5.96 012-345 195.7 > 171.6 4 10.74 -10.82 017-353 191.0 < 192.8 R<230m 5 13.12-13.22 366 - 70 176.3 > 170.0 6 14.32-14.54 009-310 106.3 < 115.5 7 15.20-15.28 313-291 168.5 < 210.1 8 18.50-18.60 348-281 99.0 < 104.4 9 20.48 - 20.64 037-350 220.4 > 196.8 10 21.88-21.98 037-010 218.3 > 214.0 1 2.10-2.18 030-016 370.8 > 329.2 2 2.68 - 2.70 045 -014 229.2 < 260.6 3 2.84 - 2.92 047 - 032 286.5 < 307.4 4 4.92 - 5.00 033 -016 312.6 > 291.5 230 <R<350 5 6.28 -6.44 020 - 341 252.1 > 236.9 6 8.04-8.12 013-350 220.4 > 201.1 7 12.74-12.86 348 -314 260.5 > 204.0 8 23.16-23.28 052 - 020 245.6 > 216.7 9 26.42 - 26.60 036-016 245.6 < 288.3 10 30.54-30.90 098-014 289.5 > 245.6 1 1.04-1.06 046-015 473.0 > 408.4 2 3.92-4.22 034 - 003 542.9 < 556.0 3 8.44 - 8.50 347 - 341 573.1 < 574.8 4 8.84 - 9.00 354-338 573.1 < 574.8 5 11.12-11.20 355 -348 636.7 > 654.9 350 <R< 1000 6 18.14-18.22 304 - 299 955.1 > 916.8 7 19.96-20.06 017-008 687.7 > 638.5 8 21.06-21.14 030 -021 458.5 < 511.1 9 22.46 - 22.52 043 -035 491.2 > 431.6 10 25.32-26.66 030-010 532.1 < 517.5 1 17.28-17.38 355 -350 1146.1 < 1147.8 R> 1000 m 2 35.94 -36.32 040 - 026 1146.K 1155.3 72 4.5 Evaluation of the Existing Geometry on the Sea-to-Sky Highway Geometric conditions on the Sea-to-Sky Highway were identified and summarized as follows: 1) Stopping sight distance Stopping sight distances available at each point along the Sea-to-Sky Highway were calculated for both northbound and southbound directions. • About one-third of the entire highway is deficient in stopping sight distances at the time of analysis in 1990. • Horizontal deficiencies are present in about two-thirds of the total deficiencies, while vertical deficiencies account for the other one-third. • Deficiencies on both horizontal and vertical curves are present for one-sixth of the entire highway. 2) Horizontal curve radii Since horizontal curves constitute critical points along the Sea-to-Sky Highway, radii on horizontal curves for the first 10 km of the Sea-to-Sky Highway from LKI 5.68 to LKI 15.32 were calculated. The following conclusions can be drawn: • There are a total of 25 curves on the first 10 km of the Sea-to-Sky Highway from LKI 5.68 to LKI 15.32. • There are fifteen (15) horizontal curves, two (2) vertical curves and eight (8) both horizontal and vertical curves. They account for 60%, 8% and 32% among the total number of curves, respectively. • Based on Table 4.4, the percentages of the substandard curves, the tight curves, 73 the good curves and the generous curves are 32%, 24%, 36% and 8%, respectively. In other words, over 50% of the curves are characterized as substandard or "tight" curves. Superelevation and grade Superelevation, e, range from 2 m/m to the maximum 0.06 m/m, while grade, G, rang from 3% to the maximum 10%. 74 C H A P T E R 5 INVESTIGATION O F T H E LINK B E T W E E N HIGHWAY P E R F O R M A N C E AND EXISTING G E O M E T R Y The purpose of this chapter is to investigate the link between highway performance as measured by highway accidents and existing geometry on the Sea-to-Sky Highway. 5.1 Correlation Analysis Assuming that patterns of accident types are associated with possible causes then the need for specific engineering improvements can be inferred from an analysis of probable accident causes. An accident pattern table can be developed to show the frequency of occurrence of different types of accidents in different geometric situations. The accident experience between 1986 and 1990 were integrated with the roadway geometry on the Sea-to-Sky Highway to facilitate such a correlation analysis. Figures 5.1 and 5.2 show the safety performance and geometry integration between Horseshoe Bay and Squamish, and between Squamish and Whistler, respectively. Accident experience includes the number of fatal, injury and property damage accidents; The roadway geometry includes roadway profile, gradient (%), horizontal curvature (R), vertical curvature (K), forward sight distance (m) and backward sight distance (m). The forward sight is assumed for the northbound direction and the backward sight is then considered for the southbound direction. This assumption makes it possible to calculate sight distances for both northbound and southbound traffic using one photolog survey. The Ministry of Transportation and Highways collects photolog information 75 once every other year and each time only one direction of photolog survey is conducted. To perform the correlation analysis, the following three steps are taken: 1) determination of curvature and grade percentile distributions; 2) the weights of all sections with restricted curvatures and grades; 3) accident probability distribution of curvature values and of sight distance values. From the correlation analysis, the following general safety observations for the Sea-to-Sky Highway can be made: 1) On a two-lane rural highway, a driver is at a higher risk of having an accident in a horizontal curve than on a straight portion of the same highway. 2) Curve sections with bad geometry are more common to have fatal off-road accidents than the curve sections with good geometry, particularly when motorists are driving at speeds higher than the speeds posted for these bad curves. 3) The most successful parameters in explaining the variability in accident rates was curve radius. The sharper the radius of curve, the higher the accident rate. 4) Gradients of up to about 6 percent have a relatively small effect on the accident rate. A sharp increase in the accident rate was noted on grades of more than 6 percent. 5) About 17 percent of the accident segments on the Sea-to-Sky Highway have curvatures sharper than 350 m and grades steeper than 7 percent; 76 5.2 High Accident Locations Chapter 3 identified the high accident locations. As shown in Tables 3.4 and 3.5, LKI 12.1, LKI 4.9, LKI 18.7, LKI 44.3 and LKI 7.3 are found to be the first five (5) high accident locations between Horseshoe Bay and Squamish; LKI 40.1, LKI 41.1, LKI 23.7, LKI 13.1 and LKI 15.1 are found to be the first five (5) high accident locations between Squamish and Whistler. However, they are not necessarily associated with difficult geometric conditions. Figure 5.1 shows that some accidents occurred at locations with insufficient sight distance, such as LKI 12.1, LKI 18.7 and LKI 44.3 between Horseshoe Bay and Squamish; some accidents occurred at locations that sightline is not a problem at all, for example, LKI 4.9 is an intersection and LKI 7.3 is a narrow bridge. In addition, Figure 5.1 shows that passing lane sections from LKI 14.98 to LKI 15.8, and vehicle pullouts at LKI 10.2 also appear to be high accident locations on the Sea-to-Sky Highway between Horseshoe Bay and Squamish, which are not necessarily related to any geometric design deficiencies. 5.3 Quality of the Sea-to-Sky Highway The 44 km of the roadway from Horseshoe Bay to Squamish is mostly a winding two lane mountain highway. There are two measures to assess the quality of the highway: stopping sight distance and horizontal and vertical alignment. 1. Stopping sight distance The effect of limited sight distance is examined in this section. A criterion of 140 m stopping sight distance based on a speed of 80 km/h is used for the examination. The terminology 77 "percent limited stopping sight distance" is hereafter used to indicate the percent of the roadway that has less than the specified SSD based on the current TAC driver eye height (1.05 m) and object height (0.15 m). The 44 km of the roadway is divided into 20 m intervals and 1,309 out of the 2,200 (60 %) road segments between Horseshoe Bay and Squamish on the Sea-to-Sky Highway have no sight distance problem according to this standard. Of 44 km of the Sea-to-Sky Highway between Horseshoe Bay and Squamish, the highway has long continuous sections where sightlines are a problem, about 16 km in total. Ten of these sections starting at LKI 9.26, LKI 13.60, LKI 14.18, LKI 15.42, LKI 17.50, LKI 22.96, LKI 29.64, LKI 30.38, LKI 35.06, LKI 39.0 are over 0.5 km in length. Stopping sight distance deficiencies vary from 60 m to 140 m. Table 5.1 shows that there are 897 roadway segments with limited sight distance due to horizontal curvature, vertical curvature or both horizontal and vertical curvature. Table 5.1 Frequency^  of deficient SSD segments Available sight distance (m) Horizontal Vortical No. of segments %of segments No. of segments %of segments <60 0 - 0 -60 0 - 2 0.5 % 80 52 10% 11 3.5 % 100 115 21 % 63 17% 120 195 37% 141 38% 140 168 32% 150 41 % Total 530 100% 367 100 "» 78 Figure 5.3 examines the possible relationship between accident rate per million vehicle kilometre per year and the measurement of limited sight distance. No strong relationship can be seen between the average accident rates and percent limited SSD. 9, 1 1 o o 6 o 3 *o a a o a, 4 V • • • • • • 10 ^0~ 30 40 Percent of Umited stopping sight distance < 140 m Figure 5.3 Relationship of accident rate/km and percent of limited SSD 2. Horizontal and vertical alignment The alignment is noted as being either easy or difficult to drive. This factor refers to the relative difficulty of.driving the highway. A section termed difficult is judged to have a greater number of more severe curves than a section termed easy. An alignment with numerous sharp curves is judged to be more difficult to drive, and thus of a lower quality than a straighter alignment. A 79 severe curvilinear alignment is also more costly to drive due to greater fuel consumption and vehicle wear. As identified by VHVIS.PAS software, there are 43 vertical curves and 67 horizontal curves in the 44 km of the roadway, 35% of them fall into the difficult alignment category. 5.4 Findings and Summary The above analysis shows that road sections with deficient sight distances or difficult alignment do not necessarily show a significant adverse accident experience. The reasons may be as follows: 1) The task of identifying locations with high accident history is greatly facilitated by the available computerized accident data and highway geometry inventory. Although police accident summary reports can be used to identify target groups, such as accident types, high-risk drivers, dangerous vehicles, hazardous locations, etc., they are not recorded with enough geographical precision to allow for an establishment of the relationship between accident occurrence and various geometric design factors. The records do not allow for an examination of changes in road safety and the reasons for such changes. 2) A site is typically deemed to be hazardous if its accident history exceeds some specified level or an average level. However, the accuracy with which road safety will be measured depends not only on the count of reported accidents but also on the knowledge of the proportion of accidents that is reported. 80 3) The Sea-to-Sky Highway has a large proportion of injuries and fatal accidents in the two-lane rural highway sections every year. Since the majorities of motor vehicle accidents occur in daylight on dry roads with sound vehicles, the causes seem to be more related with the drivers and the way they interact with the highway. In fact, the Sea-to-Sky Highway Committee (1994) reported that speeding violation is the major factor that leads to most recorded serious motor vehicle accidents. 4) The AASHTO stopping sight distance design model alone is not a good indicator of highway safety performance. Thus, adherence to the model alone in designing projects may not result in safe and efficient highways. 5) Stopping sight distance-related accidents are event-oriented. The presence of a segment of highway with inadequate stopping sight distance does not guarantee that accidents occur. Inadequate stopping sight distance is simply one item in a chain of events that may lead to an accident. 6) There are many minor, uncontrollable factors that contribute to accidents at limited stopping sight distance locations. These minor factors such as worn tire tread, deficient vehicle breaking characteristics, irregular pavement, impaired driver, etc., may become more important when the driver enters a critical situation. 7) The safety effect of an individual curve is influenced not only by the curve's geometric characteristics, but also by the geometry of the adjacent highway segments. There is a need to consider an entire highway as a system and then evaluate its performance. It can be concluded that despite extensive studies in this research, the understanding of the 81 detailed effects of geometric design on safety is not adequate to predict motor vehicle accidents as a function of change in any individual geometric design element. Even the fairly large data base of accident experience and roadway geometry of the Sea-to-Sky Highway was not enough. With the tested analytical techniques, it is very difficult to reach a solid conclusion as to which locations are actually hazardous and what countermeasures can be implemented to improve the safety conditions. Since the road safety has a magnitude and is subject to measurement, some other systematic approach is required to be able to forecast accident potential and geometric design elements. Further, it is clear that 1) The effects of sight distance restrictions and the tightness of horizontal curves may only be seen through their interaction with other geometric features. 2) The probability of occurrence of stopping sight distance-related accidents are specific to critical driving conditions. It is a function of traffic volumes, roadway geometry, vehicle characteristics, and other factors. Given that the correlation between accidents and geometric design features is so poor, a dynamic approach to the road safety design process must be introduced. This approach should address drivers' expectations on consistency throughout an entire highway and provide designers with quantitative information as to how this might be achieved. The approach should also focus on the highway design process as the minimum horizontal curve radius does not remain constant along a roadway but changes with operating speed and grade of alignment. 82 The above requirements can be accomplished by the reliability-based highway design process suggested by Navin (1987) and practised by the author. The process requires that probabilistic concepts be consistently applied so that comparable levels of calculated reliability can be provided in a format that all design results are equitably accounted for in the system analysis. Reliability analysis also allows highway geometric design to be performed in a clear, concise, and accurate manner. In order to achieve reliability in design, the design process must correctly address, identify, and account for appropriate areas of variability and evaluate variability between designs in the same manner. In the following sections, Chapter 6 describes reliability-based highway design theory and practice. To demonstrate the reliability analysis procedure.for performing highway geometric design, case studies with the Ministry of Transportation and Highways on single geometric design parameters are provided. Chapter 7 focuses on the development of a Moving Coordinate System Design (MCSD) model. Chapters 8 and 9 consider the application of MCSD model on the Sea-to-Sky Highway and provide reliability measure on cross-component geometric design. Chapter 10 provides conclusions and recommends further research. 83 C H A P T E R 6 RELIABILITY-BASED HIGHWAY DESIGN T H E O R Y AND P R A C T I C E The proposed reliability-based highway design methodology evaluates safety consequences to account for uncertainty, permits a definition of tolerable risk, and allows for the direct input of changes relative to the drivers, the vehicles, and the road surface. 6.1 Reliability Theory A given design situation can be described mathematically in terms of a performance function: G (X) = G (xr x2 xN) where X = (xux2, xN) is an N-dimensional vector of design variables. Some of these variables will influence the demand D on the system, and others will characterize the supply 5 to withstand the demand. By convention, the performance function G is written as Equation 6.1: G = S - D (6.1) Thus non-performance, or non-compliance, or failure to meet some criteria of the system will be indicated by all those combinations of the variables resulting in G < 0 (demand exceeds supply). Conversely, all those combinations resulting in G > 0 correspond to performance as intended or survival of the system. The situation G = 0 is then a "limit state" between 94 survival and failure, and all those variable combinations satisfying G = 0 are said to belong to this limit state. 6.1.1 Reliability in highway geometric design The reliability of a highway is the probability that it wil l perform as intended in a given situation. The design of such a system is normally controlled by several intervening variables, some representing the physical properties of the highway and others characterizing the driver-vehicle demands. Normally, these variables are uncertain (random) and can be described in statistical terms and obtained by surveys. The "limit state" approach requires the designer to think of the demand of driver-vehicles for a particular highway design parameter and of the supply provided by the current highway design standard. The general arrangement for such a system is shown in Figure 6.1. Expected Value Figure 6.1 Demand vs. Supply 95 As shown in Figure 6.1, the shaded area is where the supply S may be exceeded by the demand D. This area may represent system failure, non-performance or non-compliance. The objective of reliability-based design is to reduce this overlapping area to some acceptably small value. \ 6.1.2 Margin of Safety and Reliability Index The Margin of Safety, M, is the difference between the expected value of the supply and that of the demand; the Reliability Index or the Safety Index, p, is defined as the ratio of the Margin of Safety, M, and the combined variance. They are as follows: M = E(S) - E(D) (6.2) P = M (6-3) sjVar(S) + Var(D) where, E (S) — Expected value of Supply E (D) = Expected value of Demand p = Reliability Index M = Margin of Safety Var (S) = Variance of supply Var (D) = Variance of demand Equations 6.2 and 6.3 are consistent with Equation 6.4 only if S and D are normal and uncorrected. Figure 6.2 is a graphical representation of the reliability index, p. 96 Pnc=P[G<0] E[G] Figure 6.2 Reliability Index, p Equation 6.1 shows that the probability of failure can be obtained by calculating the probability of the event G < 0. However, in general, many random variables are involved in the G function and the calculation requires the joint probability density function and integration over the failure region G < 0. This exact procedure can rarely be applied because the required joint probability density function is not known or is difficult to obtain. As an alternative, approximate methods have been developed to estimate the probability of non-compliance. These, known as FORM/SORM procedures (First Order or Second Order Reliability Methods), are based on a reliability index p, from which the probability of failure P7can be estimated using the standard normal probability distribution function <&(P): Pf = $(-P) (6 .4) 97 6.1.3 Probability of non-compliance The objective of applying the reliability theory to highway geometric design is to establish and promote a more consistent and more reliable design process. Navin (1990) took the first initiative by introducing the "limit state" concept into highway geometric design for isolated sections, such as horizontal curves. Later in 1991, he designated the probability of failure as used in structural engineering to be the probability of design non-compliance for highway geometric design. The probability of design non-compliance or simply non-compliance occurs when the constructed highway does not meet the design standards, or the built-in safety margin does not meet drivers expectation. As a result, the constructed highway can not function as well as intended because it "supplies" lower driving conditions at certain locations than the roadway users expect. The probability of non-compliance can, therefore, serve as a predictor for expectation or design failures resulting from either selecting improper design parameters, or from underestimating drivers' demand, or from making wrong decisions about what is supposed to be built into the highway. These "design errors" come from an inappropriate match between the roadway design and the driver's capabilities. For example, designers mis-perception may create task requirement that are incompatible with driver's capabilities, both during normal and emergency situations. Roadway users can affect the consequences of the "design errors" in two ways: (1) they cause system behaviour to transcend acceptable boundaries, therefore, 98 they can continue their driving task; and (2) they fail to maintain acceptable system behaviour due to insufficient capabilities. The implementation of the probability of non-compliance into highway geometric design opens up a unique way to study "events" that occur frequently, that can be clearly defined in engineering terms, that can be reliably measured, and that are related to traffic operations. All highway design parameters can be considered as such "events". They can end up into such situations that the highway does not comply with its operations, i.e. highway supply is exceeded by driver demand. Traffic unsafety is the result of the various road users' needs not being met. In order to control risk, one has to know first which dangerous decisions take place and in which situation or what circumstances these decisions are taken. It is important to reduce the level of danger that drivers may have to tolerate during the highway design stage. The potential critical situations would not necessarily result in accidents, but may require motorists to take evasive actions or to adjust their speeds enough to avoid unsafe driving conditions. The severity of the deficiencies in geometric design varies according to circumstances. Serious deficiencies in design may be associated with very difficult driving conditions, which require motorists to give increased attention to their driving tasks. Slight deficiencies in design may be associated with uncomfortable driving conditions, which require motorists to make extra efforts to perform their driving task. No matter if the deficiency is serious or slight, a vehicle action or manoeuvre that exceeds a predefined limit is considered as non-compliance, and the probability of non-compliance suggests that the design of the 9 9 roadway is inadequate to some degree. The concept of the probability of non-compliance is a unique way of looking at the "unsafety" of particular locations or situations in traffic. Unsafety is not visible in the traditional approach, but can be quantified by the probability of non-compliance used in a reliability-based highway design process. If a system has a large number of non-compliance sites, it may also experience a large number of near accidents and real accidents. This notion is based on the assumption that there exists a continuum of events, which range from normal safe driving conditions through accident and injury conditions. The continuum lies on what might be termed the critical dimension, since it moves from basic manoeuvres to proximity, precautionary manoeuvres, encounters, conflicts, serious conflicts, collisions. With the events that are deemed to be close to collision, then the probabilities depend on both site factors and human factors. The non-compliance concept can be considered as a means for identifying some of the hazardous locations and locations where the drivers behaviour may be erratic. The responsibility of a highway designer is to remove those unsafe driving conditions and address potential human driving failures. The following conclusions can be drawn: • The probability of design non-compliance implies that reliability-based design is applicable to traffic safety. It requires that highway operations be considered at the highway geometric design stage. • The probability of design non-compliance can be used in two ways: the detection of the 100 potentially hazardous locations and the diagnosis of the safety problems at both design and operation stages. The probability of design non-compliance provides a "uniform" method to quantify deficiencies in geometric design. This probability measure is compatible among different design alternatives as well as different sections of the roadway. It implies that more comprehensive evaluations can be made in relation to the geographical locations and to the motorists' reactions. The probability of design non-compliance offers the possibility of gaining more understanding of the road system, as it suggests that the design of the roadway is inadequate to a certain degree, i.e., no road is perfectly safe. The probability of design non-compliance allows a feedback at the design stage, therefore, potential unsafe operating conditions can be removed at the design stage. The probability of design non-compliance is grounded on the assumption that non-compliance in design is related to operational problems. Although the design deficiency may or may not result in accidents, the probability of non-compliance may provide a quantitative way to estimate the cost associated with each sub-standard design alternatives. 101 6.2 Highway Design Practice Applications of reliability theory in highway geometric design are described in the following sections. The author (1991 and 1992) summarized her research work with the B.C. Ministry of Transportation and Highways (MoTH) and presented the reports on the stopping sight distance design, the horizontal curve radii, R design and the vertical curvature, K value design. The problems were initiated either by district or regional engineers of the Ministry or by designers or developers of the outside agencies. They requested that the Highway Engineering Branch, Headquarters of the Ministry of Transportation and Highways, provide direction or guidance on these specific design problems. They are particularly interested in the safety and the cost impact if several design alternatives are available to them because the simple application of the existing design standards are not able to address their concerns. All design practices are documented into the following format: (1) Problem statement; (2) Performance function; (3) Reliability analyses; and (4) Recommendations. 6.2.1 Stopping Sight Distance This example deals with a typical trade-off problem related to 'Stopping Sight Distance' design. The determination of proper stopping sight distance is one of the fundamentals of highway engineering. Shorter sight distance cannot be used by drivers to evaluate the situation ahead and take preventive action. Longer sight distance is obviously better for the interest of safety, but there is a point of diminishing economic returns which has not been very well 102 established except through experience. 1. Problem statement A two-lane highway, currently has adequate stopping sight distance on certain horizontal curves, but would not provide adequate stopping sight distance if widened to the standard multi-lane cross-section with a concrete median barrier. Because the concrete median barrier itself becomes a lateral obstruction limiting visibility for drivers on the lane adjacent to the median barrier. In some situations a wide depressed median and/or sufficiently large radii can be used to provide full stopping sight distance, but in constrained conditions providing complete stopping sight distance for the median lane may be expensive and thought not to be cost-effective. One of the alternatives is to provide complete (100%) stopping sight distance for the shoulder lane and three quarter (75%) stopping sight distance for the median lane. Several questions are raised with respect to: what design factors should be taken into account; what changes in accident rates can be expected if different stopping sight distances are used; what would be the consequences for safety and highway conditions of an alternative design when budget is limited; and what would be the most cost-effective safety solution. 103 2. Performance function Based on reliability theory, how a designed stopping sight distance will be experienced by drivers can be described as follows: G = SSDh - SSDv (6.5) Where SSDh is stopping sight distance supplied by the highway, SSDV is stopping sight distance demanded by the driver-vehicle combination, which depends on the speed that the driver selected, his perception-reaction time, and the stopping capabilities of the vehicle. The performance function becomes Vt V2 G = SSD. - ( — + — ) (6.6) * 3.6 254 / J X where, V = assumed initial speed, km/h t — perception-reaction time, sec. fx = average coefficient of longitudinal friction 3. Reliability analyses The probability of non-compliance in the Stopping Sight Distance (SSD) design, Pnc, is defined as the chance that the SSD required by drivers exceeded the SSD supplied by the highway. It is evident that there are many locations where it would be prudent to provide either shorter sight distance or longer sight distance. For example, in constrained conditions shortening the sight distance to, say 75%, for the median lane may be a cost-effective alternative; while for the locations characterized by conditions that create the potential need for 104 drivers to make complex or instantaneous decisions, lengthening the sight distance to, say decision sight distance, may be required. The issue is to determine the most cost-effective safety solution. The following tests were designed to accomplish this task and the reliability analyses results are as follows: a) For design speed V=100 km/h, assume that the highway provides 100% SSD, i.e. the SSDh is 200 m, substituting this value into Equation 6.6 yields Pnc = 0.05. Assume that the highway provides 75% SSD, i.e. the SSDh is 150 m, substituting this value into Equation 6.6 yields Pnc = 0.48. This means that the probability of design non-compliance was increased by approximately 10 times when SSDh is reduced from 200 m to 150 m. However, the above calculation was based on the assumption that an obstacle on the roadway is constant throughout so that drivers are required to have emergency stops all the time. Obviously, this is unrealistic because objects such as fairly large rocks on the road are very rare events. Further, it was found that the accident rates for both arrangements (100% SSD for shoulder lane and 75% SSD for median lane) would be consistent if the probability of a large rock falling in the shoulder lane is approximately ten times greater than that in the median lane. Such a relationship as indicated in Table 6.1 has to be maintained in order to achieve a consistent probability of failure for an entire road. 105 Table 6.1 Probability of rocks in shoulder lane and median lane Relation Number 1 2 3 4 5 6 Prob. of rock in shoulder lane IO6 IO5 IO"4 103 io-2 10"1 Prob. of rock in median lane IO"7 IO'6 IO"5 IO"4 IO"3 IO"2 For example, if the relationship in Column 4 exists, probabilities of failure for vehicles operating in the shoulder lane and in the median lane are: 0.05 x 0.001 = 0.00005 for shoulder lane 0.48 x 0.0001 = 0.000048 = 0.00005 for median lane i.e. the probability of failure for both shoulder lane and median lane is the same. b) Assume that 100% SSD and 75% SSD are provided, what are the probabilities of design non-compliance with operating speeds ranging from 70 km/h to 110 km/h? Substituting SSDh with values of 200 m and 150 m, Equation 6.6 yields the probabilities of non-compliance for various operating speeds. As shown in Figure 6.3, the following conclusions can be made: • The probabilities of design non-compliance increase as the operating speeds increase. • Up to an operating speed of 90 km/h, the probabilities of design non-compliance increase at a rate of 1% if the highway provides 100% SSD and increase at a rate of 11 % if the highway can only provides 75% SSD. • When operating speed exceeds 90 km/h, the probabilities of design non-106 compliance increase at a rate of 10% if the highway provides 100% SSD; increase at a rate of 30% if the highway can only provides 75% SSD. (Assume design speed is 100 km/h) Operating speed (km/h) Figure 6.3 Probability of non-compliance vs. operating speeds in SSD design c) Assume that 75 % SSD is provided by the highway, speed and coefficient of friction remain the same, what value of perception-reaction time is required in order to keep the same probability of design non-compliance as that using 100% SSD? Substituting SSDh with values of 200 m and 150 m, Equation 6.6 yields the 107 probabilities of design non-compliance with various perception-reaction times ranging from 2.5 sec. to 0.4 sec. As shown in Figure 6.4, the following conclusions can be drawn: • To keep the probability of design non-compliance for both SSD arrangements lower than 0.05, i.e., Pnc = 0.05, drivers, who are provided with 75% SSD, have to react to their driving conditions 1.5 sec. quicker than those who are provided with 100% SSD; • If the probability of design non-compliance for both SSD arrangements is allowed to be greater than 0.05, say, P^ . = 0.12, 0.25 and 0.45, drivers, who are provided with 75% SSD, have to be able to detect their driving conditions 1.0 sec. quicker than those who are provided with 100% SSD because their perception-reaction time is only 1.5 sec. (Assume design speed is 100 km/h) •< %i I IS I O 75% SSD + 1<XK SSD Perception-reaction (sec.) Figure 6.4 Probability of non-compliance vs. perception-reaction times in SSD design 108 Assume that the decision sight distance can be provided in certain locations, how much is highway safety improved by the reduction of the probability of design non-compliance? Substituting SSDh with the Decision Sight Distance (DSD) of 300 m and 390 m, Equation 6.6 yields the probabilities of design non-compliance. As shown in Figure 6.5, increasing sight distance supply will reduce the probability of design non-compliance significantly. If sight distance can be increased from 200 m (SSD) to 300 m (the lower value of DSD), the safety can be improved by reducing the probability of design non-compliance of 4.8% (from 0.05 to 0.0013). Additional 90 m (upper value of DSD) sight distance increase from 300 m would only result in a marginal reduction of probability of design non-compliance (from 0.0013 to 0.00037). (Using decision sight distance) 0.06 -i — — — — 1 DSD-390 DSD-3O0 SSD-ZOO Sight distance (m) Figure 6.5 Probability of non-compliance using Decision Sight Distance 109 Assume that the values of speed, perception-reaction time and coefficient of friction are the same, if SSD provided by the highway is reduced from 200 m to 120 m in 5% intervals, what are the probabilities of design non-compliance, respectively? Substituting SSDh with values of 200 m, 190 m, 180 m, 170 m, 160 m, 150 m, 140 m, 130 m, and 120 m, respectively, Equation 6.6 produces the probabilities of design non-compliance. As shown in Figure 6.6, the bigger reduction in stopping sight distance supply, the larger the probability of design non-compliance. With the above SSDh inputs, the probabilities of design non-compliance are found to be 5%, 8%, 14%, 22%, 33%, 47%, 62%, 76% and 87%, respectively. The conclusions are: Up to 15% reduction in SSD: every 5% reduction in SSD from 200 m to 170 m would result in a 5% increase in the probability of design non-compliance. Beyond 15% reduction in SSD: every 5% reduction in SSD from 170 m to 120 m would result in a 10% increase in the probability of design non-compliance. (Assume design speed is 100 km/h) 1 0 o « 3 * 0 t S * 0 » T O < 3 < 0 Percentage of stopping sight distance (m) Figure 6.6 Probability of non-compliance vs. percentage of SSD 110 4. Recommendations The proposed alternative of providing complete (100%) stopping sight distance for the shoulder lane and three quarter (75%) stopping sight distance for the centre lane is not recommended as it would result in a significant safety impact on the roadway users. The probability of design non-compliance would be increased by about 10 times. As shown in Figure 6.6, the maximum compromise or reduction in sight distance should be limited within the range of 15% of the complete sight distance, i.e, stopping sight distances supplied by the highway should never be shorter than 170 m at any point along the highway. 6.2.2 Horizontal Curve Design - R Values This example deals with a specific safety problem related to horizontal alignment. It is recognized that vehicle run-off-road accidents are more likely to occur on horizontal curves than on straight segments of roadway. It is also recognized that various design elements affect one another in combination. If some accidents can be said to be caused by highway design features, most of those accidents are caused not by a single feature alone, but combinations of factors. 1. Problem statement When a vehicle moves around a curve the change of direction produces a lateral force which is offset by the superelevation and the friction between the tire and pavement surface. There is evidence, however, that accidents will still occur where vehicles leave the roadway on curves 111 that were designed according to the criteria. From a qualitative point of view, single vehicle run-off-road accidents, particularly common at horizontal curves, can be attributed to increased demands placed on the driver, the vehicle and. the traction between the tires and pavement. Horizontal alignment features affect safety by influencing the ability of the driver to maintain vehicle control and identify hazards, and by affecting the behaviour and attentiveness of the driver, especially the choice of travel speed. From a quantitative point of view, the relationship between safety and horizontal alignment are not clearly understood. However, available data demonstrates that highway design strongly influences highway safety. From the laws of physics, when a vehicle is travelling along a circular curve, the centripetal force acting toward the centre of the circle at the road surface generates an equal centrifugal force acting on the vehicle at the centre of gravity outwards from the centre of the circle. These two equal and opposite forces produce a momentum which tends to overturn the vehicle. The relation between design speed and curvature and their joint relations with superelevation and side friction is established by: V2 e + / y = (6.7) y 127 R where, e = rate of roadway superelevation, m/m fy = coefficient of side friction factor V = vehicle speed, km/h R — radius of curve, m 112 In addition to capturing the major factors of a horizontal curve and their proper interrelations, the equation indicates the requirements of the following logical controls: • To achieve balance in highway curve design all geometric elements should, as far as economically feasible, be determined to provide safe, continuous operation at a speed under the general conditions for that highway. For the most part this is done through the use of design speed as the overall control. Under this control, proper values are assigned to superelevation, side friction factor, and radius of curve. • To achieve compliance with design in highway operation the capability and physical geometry constraints that the highway offers to road users should be properly posted. Obviously, the more the drivers obey the posted speed, the more the operation is in compliance with the designed situation, and the better the safety. Thus running speed, as a major variable, should be studied and controlled in an operation phase. • Another factor needing to be treated as a variable is the coefficient of side friction^. The friction factor at which skidding is imminent depends on the speed of the vehicle and the type and condition of the roadway surface and the tires. With wide variation in vehicle speeds on curves, different pavement textures, weather conditions and tire conditions, the maximum side friction factors range greatly. • Apart from the need to study running speed and side friction, there are situations in which drivers consistently overdrive the roadway alignment, i.e. they travel at speeds higher than the alignment's design. It would be desired for designers to be able to "watch" the performance of such a driver-vehicle-road system in the design stage, to know as much as possible about the safety consequences that would occur after the 113 geometry of an existing highway is improved or other safety-oriented improvements are made. 2. Performance function Applying reliability theory to horizontal curve design, the situation can be described by the following performance function, G: V2 G = * m i n - ( 6 - 8 ) m m 127 (e+fy) The first term, as a single value, represents the supply provided by highways, while the second term, as a function of the variables V (speed) and fy (side friction factor), indicates demand generated by the driver-vehicle combinations. The superelevation, e, is considered as a constant for this particular curve. On a given curve, the superelevation, e, and radius, R, are built into the road. The maximum rate of superelevation used in British Columbia is 0.06 m/m. For a rural highway with a design speed of 100 km/h, the minimum radius for a limiting value of e — 0.06 m/m is found to be 440 m in the Ministry's design manual. The speeds that drivers will judge to be possible under the prevailing traffic conditions are uncertain, and the side friction available to assist a vehicle negotiating a circular curve varies with different vehicles travelling on different road surfaces. The performance function is then developed as: V2 G = 440 - (6.9) 127 (0.06 + / ) 114 The safety index becomes sfVar(RV) M - 440 - E(RV) where, E(RV) and Var(RV) are the Expected Value and Variance of R in demand side. The reason for selecting the radius is that it is a physical design parameter that is easily related to the vehicle's speed and acceleration. The lower the radius demanded by the driver-vehicle combination the safer the turning manoeuvre. When the radius demanded by the driver-vehicle exceeds that provided by the highway, the curve may fail to keep a vehicle on track and is said not to comply with the design standard. Whether a serious outcome will happen or not depends on actual circumstances. 3. Reliability analyses a) Assume that the operating speed V — 112 km/h and side friction factor fy - 0.35, the probability of design non-compliance is P„c = 0.26680 x 104, i.e. the probability of a vehicle leaving the curve is two in one hundred thousand if no emergency action can be taken. This is also the built-in probability of failure associated with the curve design. It results from dynamics and uncertainties of vehicles on a circular curve. b) Suppose that the assumption of the operating speed of 112 km/h does not hold all the times. Some drivers travel at a higher speed up to 120 km/h and others travel at a 115 lower speed of 85 km/h. The safety estimates for different groups of drivers are shown in Table 6.2: Table 6.2 Probability of non-compliance vs. operating speeds in horizontal curve design Input Output / V (km/h) P 0.35 120 3.011 0.13008 x IO"2 100 5.834 0.27093 x 10"8 85 8.473 0.11959 x IO 1 6 The following conclusions can be drawn: • If drivers obey the design speed of 100 km/h, the probability of a vehicle leaving the curve will be three in one billion. • If drivers decide to operate vehicles at a speed of 120 km/h, 20 km/h higher than the design speed, their exposure to the risk is one in a thousand. An acceptable risk level can then be established based on this probability value if the "operating speed 20 km/h higher than the design speed" can reasonably represent the outcome of the driver-vehicle combinations. • If drivers operate their vehicle at a speed of 85 km/h, their safety conditions are improved considerably. If e, V, and R have values such that^  exceeds the value that the pavement and tires can supply, lateral stability is lost and the vehicle will move away from the centre of the curve. Table 6.3 shows the results varying the side friction factors from 0.30 to 0.10. 116 Table 6.3 Probability of non-compliance vs. lateral friction in horizontal curve design Input Output / v V (km/h) P Pnc 0.30 120 2.089 0.18347 x10 1 100 4.784 0.86037 x 10"6 85 7.444 0.48978 x 1013 0.25 120 1.105 0.13450 x 10° 100 3.635 0.13889 x 10"3 85 6.236 0.22437 x 10"9 0.20 120 0.043 0.48294 x 10° 100 2.359 0.91710 x 10"2 85 4.807 0.76683 x 10'6 0.15 120 -1.107 0.86585 x 10° 100 0.946 0.17217 x 10° 85 3.155 0.80246 x 10"3 0.10 120 -2.355 0.99074 x 10° 100 -0.616 0.73121 x 10° 85 1.272 0.10164 x 10° As shown in Figure 6.7, the following conclusions can be drawn: • The reliability of the designed curve is decreased with the decrease of the side friction factor from 0.30 to 0.10. Table 6.3 shows that if vehicles operating at a speed of 120 km/h, the side friction factors of 0.30, 0.25, 0.20, 0.15 and 0.10 would result in probabilities of design non-compliance of 2%, 13%, 48%, 86% and 99%, respectively. This means that the side friction factor is a very sensitive variable in curve radii design; • If the side friction factor available to certain vehicles can be represented by a 117 single value, say,fy of 0.30, as the operating speed decreases, the reliability of the designed curve increases; Assume P = 0, then no safety margin is provided in the design and the probability of failure is 0.5, i.e. the designed curve is subject to 50% chance of failure due to uncertainty. Figure 6.7 shows that as the operating speeds decreases from 120 km/h to 85 km/h, the side friction factors available to vehicles are 0.30 to 0.10. This relationship is similar to that defined by the TAC design manual. The reliability-based design method shows that when variables in the demand side take the same values as those in the supply side, there is no reliability at all in the designed system because p = 0. Since fy is never lower than 0.10, the value of P is much greater than zero. In other words, the current standards on horizontal curve design do provide a relatively large margin of safety. 118 8.0 _ 3 0 _j ( : ( 1 1 1 80.0 90.0 100.0 110.0 120.0 130.0 Operating Speed (km/h) Figure 6.7 Probability of non-compliance vs. operating speeds in horizontal curve design 4. Recommendations Based on Equation 6.7, a change in speed, or a change in curve radius, affects lateral acceleration on horizontal curves. The change will be proportional to l^/R, where Vis speed and R is the radius of the turn. The equation shows that doubling the speed on any curve will quadruple the resulting lateral acceleration, while tightening the radius by half will only double lateral acceleration. Thus, the cornering limit is much more sensitive to speed than it is to tightness of the turn. To avoid vehicle run-off road accidents, the speed limits on tight horizontal curves should be clearly defined. 119 6.2.3 Vertical Curve Design - K Values This example deals with different K values for crest vertical curves on the Island Highway -Malahat Section. The length of a section of curve measured horizontally over which there is a change of grade of 1 % is a constant for the curve and is referred to as the K value. K value is a measure of the flatness of a vertical curve, the larger the K value, the flatter the curve. In the same way that radius is a measure of the flatness of a horizontal curve. The economic design of highways involves a balance of earthwork between cut and fill as well as the safe and efficient operation of vehicles with varying physical characteristics. Nowhere in the design of highways is the interplay between safety and economics more pronounced than in the selection of K values for crest vertical curve. 1. Problem Statement Cost estimations on all of the vertical alignment options for the Island Highway - Malahat Section are available at the Highway Engineering Branch. The Highway Engineering Branch is expected to provide quantitative measures to evaluate the trade-off between safety and performance against cost, and to draw conclusions about the safety-cost effectiveness of different standards. 2. Performance function Probability of non-compliance is defined as the chance that the lvalue required by drivers exceeds the K value supplied by highway. Based on reliability theory and existing design procedure, how a designed crest vertical curve will be experienced by drivers can be described 120 as follows: G = K - (SSD^ when L > SSD (6.10) 398.745 The first term of the equation is the standard K value for design which represents the highway supply, while the second term is a function of stopping sight distance which characterizes the demand generated by driver-vehicle combinations. The demanded stopping sight distance depends on the speed that the driver selected, his or her perception-reaction time, and the stopping capabilities of the vehicles. The performance function, therefore, becomes c ^ • f 3.6 254 / G = K - - (6.11) 398.745 where, V = assumed initial speed, km/h t — perception-reaction time, sec. fx = average coefficient of friction 3. Reliability analyses Four reliability analyses are carried out for the Island Highway - Malahat Section with an intention to observe • what changes in probability of non-compliance can be expected if different standard K values are built into the highway, if the highway is subjected to different operating speeds, or if different friction coefficients are used. • what would be the consequences for safety and highway condition of an alternative crest vertical curve design, and 121 what would be the optimum safety benefit solution. Assume the driver-vehicle demand on the Malahat Section are as follows: operating speed is 100 km/h, perception reaction time is 2.5 sec. and coefficient of friction is 0.29. If lvalues of 90, 95, 100, 105, 110, 115, 120, 125, 130 are alternatively used in the design section, what are the probabilities of non-compliance? Figure 6.8 shows the results of probability of non-compliance versus K values. As the K value increases, the probability of non-compliance decreases. The probability of non-compliance is the most sensitive where the K values change from 90 to 95, the less sensitive part is where the K values range from 95 to 105, and the least sensitive or benefit part is where the K values go beyond 105. The instantaneous line of sight from the driver's eye to the top of an object is tangential to the curve. The crest curve should be flat enough so that the distance from the driver to the object is at least equal to the minimum stopping sight distance. The minimum K value is, therefore, a function of the minimum stopping sight distance, the driver's eye height and the height of the object. Above 80 km/h, stopping sight distance on crest curves have been increased to accommodate an observed increased perception-reaction time. This may explain why the 'probability of non-compliance versus K values' line has greater slope about K values from 90 to 95 and lesser slope about K values from 95 to 105. 122 Vertical curve - K values Figure 6.8 Probability of non-compliance vs. K values in vertical curve design Assume that value is constant, perception reaction time is 2.5 sec. and coefficient of friction is 0.29, what are the probabilities of non-compliance with operating speeds ranging from 50 km/h to 120 km/h? Figure 6.9 shows the results of probabilities of non-compliance versus operating speeds. The probability of non-compliance increases with the increases of the operating speed. A dramatic increase in the probability of non-compliance can be found when operating speeds are greater than 90 km/h. 123 (For vertical curve L > SSD only) o.ia i — — Operating speed (km/h) Figure 6.9 Probability of non-compliance vs. operating speeds in vertical curve design Assume that the built-in K value is 90, operating speed is 100 km/h, perception-reaction time is 2.5 sec, what are the probabilities of non-compliance with an average coefficient of friction of 0.29, 0.30, 0.31, 0.32, 0.34 or 0.36? Figure 6.10 shows the results of probabilities versus coefficient of friction. The probability of non-compliance decreases as the coefficient of friction increases. Unlike the previous two analyses, the parameter,/^ , affects the margin of safety in a moderate way. 124 (For vertical curve L > SSD only) 0.29 0.30 0.31 0.32 0.34 0.36 0.30 Friction factors,/ Figure 6.10 Probability of non-compliance vs. friction factors in vertical curve design Assume that K values of 90, 95, 100 and 105 can be built into the design, perception-reaction time is 2.5 sec. and coefficient of friction is 0.29, what are the probabilities of non-compliance with running speeds of 50 km//h to 120 km/h for each category of K value? 125 Figure 6.11 shows the results of probabilities of non-compliance versus speeds for each category of K value. The calculation is based on the assumptions that the posted speed is the same as the design speed and the operating speed is about 10 km/h higher than the posted speed with a variation of plus or minus 10 km/h. When K values of 90, 95, 100, and 105 are alternatively used in design, the changes in probabilities of non-compliance are shown in Table 6.3: (For vertical curve L > SSD only) v o 5 ex 6 o 0 1 a o c <4-l o & a O K=90 + JC=95 O K=100 a K=105 Operating speed (km/h) Figure 6.11 Probability of non-compliance vs. values 126 Table 6.4 Probability of non-compliance and excavation requirement versus K values (For vertical curve L > SSD only) Design Speed (km/h) Post Speed (km/h) Running Speed (km/h) K Values Probability of Non-compliance Excavation (m3) Mass ordinate 90 90 100 90 0.066 22,300 (constant) (constant) (constant) 95 0.018 40,300 100 0.006 -105 0.001 -120 - 199,900 Recommendations From a safety point of view, it is significantly better to choose a K value of 95, instead of 90, as the absolute reduction in probability of non-compliance is 0.048, which is greater than any other absolute reductions. The ultimate decision should be made based on both technical evaluation and economic analysis. It was estimated that the cost difference is $120,000 between a 22,300 m3 excavation for K = 90 and a 40,300 m3 excavation for K = 95. Crest vertical curves with a K value of 95 is, therefore, recommended by Highway Engineering Branch to 127 build into the Island Highway - Malahat Section. This decision was based on the premise that obtaining about fourfold decrease in probability of non-compliance for an increased construction cost of $120,000 was acceptable. 6.3 Findings and Results The examples use the basic highway alignment equations to estimate the margin of safety for stopping sight distance, curve radii and K values. They provide for a safety impact evaluation in which the safety consequences of the selected design and potential improvements are evaluated. The following findings and results can be summarized: • The examples require highway designers to seek opportunities specific to each design project and apply sound safety and traffic engineering principles. If it is necessary, an exception to a design standard can be explicitly addressed in terms of the expected safety consequences along with cost. • The stopping sight distance provided to road users should be close to the design standards since the probability of design non-compliance is very sensitive to the reduction of stopping sight distance. • The standards for horizontal curve design provide relatively larger margins of safety than that for stopping sight distance design. This may be attributed to the fact that vehicle run-off-road accidents on curves usually result in consequences so serious that a more conservative design is required. In addition, lateral friction is proportional to J^/R, i.e., doubling the speed on any curve will quadruple the resulting lateral 128 acceleration, while tightening the radius by half will only double lateral acceleration. Thus, the cornering limit is much more sensitive to speed than it is to tightness of the turn. To avoid vehicle run-off road accidents, the speed limits on tight horizontal curves should be clearly defined. In conclusion, the above studies put useful information in the hands of designers which permit the evaluation of the safety consequences for different designs and operations and include the limit state provided by the highway, the degree to which design is not appropriate to operation, and the seriousness of non-performance as represented by design failure and the availability of countermeasures. 129 C H A P T E R 7 A M O R E F U N D A M E N T A L H I G H W A Y DESIGN M O D E L - Moving Coordinate System Design (MCSD) Reliability-based design practice provides a powerful tool in quantifying safety factors associated with each specific highway design. However they are to a large degree restricted to single design parameters, such as stopping sight distance, horizontal curve radius and vertical curve K value. As a result, they are unable to explain cross-component effects, such as roadway-vehicle interaction experienced by drivers. The driving task has three performance levels: control, guidance and navigation. Control is the immediate vehicle-road interaction, guidance is staying safely within a lane and navigation is getting through the road network to a desired destination. Control relates to a driver's interaction with the vehicle. The vehicle is controlled by speed and direction through the steering wheel, accelerator and brake. Information about how the driver has controlled the vehicle comes primarily from the vehicle-road interaction and the vehicle's display. In such a road-vehicle-driver system, motorists must detect and select information from the general environment and translate the decisions into a set of actions. Through the feedback, there is a continuous interaction among the road, the vehicle and the driver. Highway geometric design must be performed in relation to vehicle dynamics. This leads to the Moving Coordinate System Design model ( M C S D ) suggested by the author. In this chapter a 130 brief analysis of the entire system of tires, roads, weather, vehicles, and vehicle operations is presented. The "contact patch" that exists at the interface between the tire and the road is described and from this, the M C S D model is set forth. The model comes from the reliability-based highway design framework described in Chapter 6, further developed with the vehicle dynamic considerations, quantified by two performance functions for braking and cornering manoeuvres, and solved for the joint probability of non-compliance in design for the moving coordinated system. The model illustrates how the vehicle dynamic characteristics can be incorporated into the highway design stage such that the effect of the roadway-vehicle forces experienced by drivers can be measured. As a new technique, it stands between those concerned with geometric design (civil) and those involved in vehicle dynamics (mechanical). The model demonstrates how to relate a driver's actions to the vehicle's performance capability, including the effects of roadway surface and roadway design on overall system behaviour. 7.1 Conceptual Development As a more fundamental model, the M C S D focuses on the basic "contact patch" which exists at the interface between the tire and the road. It is traction that brings together the road, the tire, the vehicle and the driver into a dynamic system. In order to develop a meaningful assessment of tire traction performance, an understanding of tire mechanics is useful. The conceptual development involves an in-depth study of the following three "versus" issues: 131 • TD vs Ts - Demand traction vs. available traction, which leads to a unique way of handling vehicle dynamics in the highway design stage with the provision of the reliability-based highway design framework. • fxysfy- Braking coefficient vs. cornering coefficient, which provides the basic elements for designing a moving coordinated system. It is the "Friction Circle" or "Friction Ellipse" concept that establishes a relationship between braking and cornering manoeuvres. • Race car vs. road car, which distinguishes the "theoretical limit" or "capacity" from the "operational limit" of a vehicle. 7.1.1 Demand traction vs. available traction TD vs Ts, i.e., demand traction vs. available traction, leads to a unique way to "handle" vehicle dynamics in the highway design stage. The subject of "vehicle dynamics" is concerned with the movement of vehicles on a road surface. The movements of interest are acceleration, braking and turning. Dynamic behaviour is determined by the forces imposed on the vehicle from the tires, gravity and aerodynamics. In handling a road-vehicle-driver dynamic system, the driver seeks to control both the speed throughout the manoeuvre and to control the path of the vehicle. If control of either is lost, the vehicle is in a skid, i.e., the driver fails to maintain "vectorial trajectory control". In any given manoeuvre, there is a certain traction demand and there is a certain amount of available traction, depending upon local conditions at that time. If demand exceeds available traction (supply), loss of control will occur. 132 The situation can be analysed for some given control manoeuvre, which involves cornering and braking, and can be illustrated in Figure 7.1. Demand traction TD Available traction Ts Figure 7.1 Demand-Supply Traction Diagram By forming a ratio of the supply traction, Ts, to the demand traction, Tp, the outcome of the intended manoeuvre can be determined. 7.1.2 Braking coefficient vs. cornering coefficient fx vs fy, i.e, braking coefficient vs. cornering coefficient, provides basic elements for designing a moving coordinate system, instead of separating braking and cornering manoeuvres. The coefficient of friction is a measure of the tire-pavement traction ability. It is defined as the ratio 133 of the force resisting the motion of a normal force. It is an overall measure of the vehicle-roadway interaction experienced by drivers. Much of highway design is concerned with curve sections. Vehicle requirements on a curve are best expressed in terms of acceleration, because the vector velocity is constantly changing when negotiating with the curve, and acceleration is by definition the change in vector velocity with time. The coefficient of friction is a measure of the tire-road traction ability. Assume the braking coefficient is fx and the cornering coefficient is fy, the following equations exist: a = f S X J x° a = f s y Jyb ax and ay have the same units of the gravity factor, g, m/sec2. The following factors influence the coefficient of friction: • Tire-pavement friction levels differ in wet and dry conditions. • Roads differ in their surface characteristics (macro texture vs. micro texture); differ in their layout, such as curves, bends, and road crown; and differ in the traffic density that they sustain and in the demands on vehicle traction that they impose. • Weather, mainly rainfall, varies considerably in time and location. • Vehicles differ in their mechanical design characteristics, such as the degree of understeer designed into the vehicle. • Vehicle operations differ among drivers. A prudent driver attempts to make allowance 134 for the road-vehicle effects when driving on a slippery road. If the driver is successful, vehicle control is maintained. If not, a skid often results. Veith (1993) summarized the most important components that determine Ts and TD in a generalized equation form: TD =f(V, Rc'\ TD, VU'\ AS-1) (7.1) Ts =f(V-\ Tx, D w \ pt) (7.2) where, TD = demand traction 7*5 = available (supply) traction V = speed Rc - radius of curvature of manoeuvre TJD = traffic density VU = degree of vehicle understeer AS = alertness and skill of driver Tx = combined macro-micro texture of pavement Dw = water depth p, = certain tire wet traction performance factors (tread compound, pattern, etc.) Equation 7.1 indicates that the demand for traction will be high for: high speeds, low radius or abrupt manoeuvres, high traffic density (the need to avoid other vehicles), vehicles with little understeer or actual oversteer tendencies (spinout possible) and lack of driver alertness to impending control inputs. Equation 7.2 indicates that available traction will be high for: low speeds, high texture 135 pavements, and low water depths on the pavement. By forming the Ts/ TD ratio and adopting constant values for TJD, VU, and AS since they are equivalent to a standard vehicle, driver, and traffic density, we get T* 1 V'\ T , D _ 1 , p, = ( — - — - x x : ') (7 .3) TD TD, VU'\ AS V , R~cl Collecting TJD, VU and AS together and denoting k to these standard components, the Equation 7.3 becomes: T„ T , R — = H-^—±)Pl (7 .4) TD V2, D The main functions of tires is to maintain the ratio expressed in Equation 7.4 greater than 1. This equation may be used to indicate how certain values of Tx, Rc, V, and Dw will influence this ratio. It is apparent that the ratio will decrease and approach to a value less than 1 at high values of V and Dw and at low values of Tx and Rc. Such a situation is defined as a critical condition, i.e. a section of highway that has lost or never had skid resistance and that is located where traction demand is nominally high. 7.1.3 Friction circle - relationship of fx and fy The Moving Coordinate System Design requires an understanding of the relationship between 136 the braking coefficient,/^ , and the cornering coefficient,/^ ,. The "Friction Circle or Ellipse" concept establishes a relationship between braking and cornering forces. The combined longitudinal and lateral force capabilities of a rubber tire may be visualized by considering an idealized tire with similar traction properties in the x and y direction. Experiments have shown that the peak or maximum instantaneous braking coefficient is nx and peak instantaneous cornering coefficient is juy. Assume that the accelerating vehicle may fully utilize tire road traction so that it has the same traction accelerating and braking. The rotation and length changes of the resultant acceleration vector as the vehicle progresses along a curve has led to the concept of the "Friction Circle or Ellipse" or the "g-g" diagram, shown in Figure 7.2. Figure 7.2 "Friction Circle or Ellipse" 137 In combined cornering and braking, the resulting instantaneous coefficient of friction is given by: In actual fact, the longitudinal and lateral tire properties are not identical due to tire construction and tread design. An ellipse is a better definition of the "Friction Circle", and re-writing the ellipse equation with the appropriate fractional values gives: The idea of a friction circle is that no matter what combination of steering and braking is applied to a wheel, the maximum horizontal force that the tire can produce is limited by the tire-road friction coefficient multiplied by the load on the wheel; The "g-g" diagram (Friction Circle) is a method for characterizing the performance of the driver-vehicle-tire system, including the influence of roadway design and surface conditions. The advantage of the "g-g" diagram presentation is that it moves away from the limitations of roadway geometry and vehicle stability into the area of vehicle handling task performance. Thus it is not only a means of quantifying the manoeuvre envelope of the vehicle but also demonstrating how much of this capability is utilized by the driver in the performance of the driving task. 138 7.1.4 Race car vs. road car Race car vs road car distinguishes the "theoretical limit" or "driver-vehicle-road-tire capacity" from the "operational limit" of a vehicle. The objective in motor racing is to win races. At the very heart of this activity is the problem of achieving maximum performance from the road-tire-vehicle-driver combination with the particular environment. The race car is essential to represent the "theoretical" limit or capacity of the vehicle for any given operating conditions. The term "theoretical" denotes the ultimate vehicle limit or the limit with a "best" driver and may be treated as the limiting supply capacity of the system. The objective in racing is the achievement of a vehicle configuration, which can traverse a given course in a minimum time when operated manually by a driver utilizing techniques within his/her capabilities. A suitable performance margin must be available for dealing with traffic, environmental factors such as wind and surface conditions, driver fatigue and emergencies. In other words, race cars should have large "g-g" manoeuvre diagrams throughout their performance envelope and that race drivers should operate close to the boundaries of the "g-g' diagram. Since these drivers are capable of reaching the vehicle's limit, a boundary can be drawn as the probable manoeuvre acceleration capability of the vehicle if ideally driven, i.e. the maximum potential of the vehicle. The problem imposed by racing may be summarized as one of spending as much time as possible as close as possible to potential vehicle "g-g" boundary. It follows that the basic design requirements of a race car are: 139 • The provision of the largest vehicle "g-g" manoeuvre areas throughout the range of operating conditions. • The provision of vehicle control and stability characteristics that enable a skilled driver to operate at or near these acceleration limits. Although the focus in racing and road cars is different, the same engineering fundamentals apply. It is important to know the difference between the maximum technological capabilities of vehicles and the practical design levels necessitated by considerations of passenger safety and comfort. 7.2 Model Formulation 7.2.1 Mathematic model A general tire-road-vehicle-driver interaction model can be described as follows: { SSDH zf(R, 0, iix, iiy, V,a,T)}U{R zf(R, 0, u,, \iy, V,a)} U{ SSD V ^ /( ux, y, V, T)} where, SSDH = Stopping sight distance in horizontal curve R = Horizontal curve radii, m 0 = Angle in cross-section, degree (tand = e, Superelevation, m/m) jux = Maximum instantaneous braking coefficient juy = Maximum instantaneous cornering coefficient V = Operating speed, km/h T = Perception-reaction time, second a = Angle in plane, steer angle, degree SSDV = Stopping sight distance in vertical curve Y - Angle in profile, degree (tcmy = G, Grade, m/m) SSD, R, e and G are determined by roadway conditions, px and fiy are determined by vehicle-tire-140 road conditions, and V, a and T are determined by drivers conditions while a is related to curve radii and can be simplified as 1/R. 7.2.2 Civil-mechanical engineering application The development of a road-vehicle-driver interaction model starts with the current highway design practice, shown in Equations 7.5 and 7.6. Based on the idea of a dynamic design for safety, a moving coordinated system model is developed using the "Friction Circle" concept, shown in Equation 7.7. Failure functions are then established for braking and cornering manoeuvres on tight horizontal curves, shown in Equation 7.8. As for a generous curve (say R > 350m), the available sight distance becomes the primary concern. The failure function is then formulated in Equation 7.9 for non-compliance in horizontal sight distance and in Equation 7.10 for non-compliance in vertical sight distance. The joint performance function, shown in Equation 7.11, is solved for the system probability of non-compliance. Three angles were used in developing the model to describe the dynamic characteristics of the interaction in three dimensions. They are the angle in plane (steer angle), a, the angle in cross-section, 8, and the angle in profile, y, as shown in Figure 7.3. 141 Radius of ^ T ^ < O mg tngcosd Y angle in plane, a angle in cross-section, 6 angle in profile, y Steer angle vehicle in horizontal curve, vehicle on grade, superelevation, e = 0 G^tany Figure 7.3 Dynamic angle characteristics in three-dimension If SSD, R, Q, y and T, are known (see Chapter 4, Highway Geometry Analysis), assuming V,a, jux j and ny are random variables with Mean Values and Standard Deviations, then: 1. Horizontal Stopping Sight Distance Design -• Demanded value of fx for a specific SSD V2 v SSDH = VT + — (7.5) 2a fxmgcosQ = max fx V2 2gcosQ(SSDH-VT) 142 2. Curve Radii Design -*• Demanded value offy for a specific R v a = — (7.6) fymgcosQ + mgsinQ = mo^ / v gRcosQ tanO 3. Friction Ellipse -> The maximum friction coefficient /and ^  which can be generated by a specific vehicle at a steer angle a (7.7) u tana = — 2 6' a1 + (tana)2 / = u = J a 2 (tana)2 + b2 Jy,max ry V v / 4. Non-compliance Functions, as shown in Equation 7.8 below: - (f) capacity ^ x 'demand 2 b-a1 + \ (tana)2 - 2gcosQ(SSDH-VT) G(y) = C Q y 'capacity ^y'demand (fv)d I V = V a2(tana)2 + b2 - + tan0 g.Rcos0 143 5. For a horizontal curve with R > 300 m (Equation 7.9): G(h) = (SSDH) . - (SSDH). . v ' K 'supply K 'demand (SSDH) , - (VT + K 'supply K v2 2& 6. For vertical curve (Equation 7.10):, G(v) = (SSDV)supply - (SSDV), demand V2 = (SSDV) . - (VT + 2g(fx+G) ) 7. Joint Performance Function, as shown in Equation 7.11 below: G(x, y, h, v) G(h) R>300m G(x)l)G(y) R<300m G(v) tany^O G(h)l)G(v) R>300m, tany*0 G(x)UG(v)UG(v) i?<300m, tany^O Mode 3 Mode 1,2 Mode 4 Mode 3,4 Mode 1,2,3,4 where, h = horizontal curve v = vertical curve To compute the probability of an undesirable event, a joint performance function, as a combination of several limit state functions, is defined. If the function G (x, y, h, v) is negative, it means that at least one of the design parameters does not comply with its design standard. Among the input variables, some are known with certainty, some are random or uncertain. They are treated as random variables and a probabilistic characteristics are selected-. To each failure 144 mode arising from the failure of an element, possibly in one of several failure modes, a safety margin can be calculated in terms of the basic variables. The system fails if at least one safety margin is negative. The joint probability of failure is derived based upon the individual failure modes and their contributions to the system failure. 7.2.3 Model Programming R E L A N is a general Reliability Analysis F O R T R A N Program written by Dr. Foschi, et al (1993). It can be used to determine the probability of exceedance for engineering problems. Calculation of the probability of failure and the reliability index , P follows from consideration of the failure function, G, of the problem. To perform the quantification task outlined in the last section, this research made significant changes in the R E L A N program. Figure 7.4 is a flow diagram which shows the procedure to modify the R E L A N program. It indicates the work to be done and the decision to be made at each stage. Four types of failures are defined: Modes 1 and 2 are related to failures in longitudinal friction and lateral friction on a tight horizontal curve, respectively; Modes 3 and 4 are related to failures in stopping sight distance on horizontal curves and vertical curves, respectively. As a maximum of four types of failures may occur at a particular point, the program would have to have the capability to handle up to four failure modes. For example: • if it is on a tangent section and there is sufficient sightlines, no failure could be found in relation to geometric design deficiency; 145 • if it is on a horizontal curve section with R < 300 m, mode 1 and 2 could happen; • if it is on both horizontal and vertical curve sections, any of the all modes (mode 1 to 4) could happen. For an entire highway, we want to obtain a "profile" which indicates how the probability of system failure would change over each section of the highway. The probabilities of system failures are assumed to be independent on each section of the highway. They are determined by geometric design, i.e., limited by what has been built into the highway. Further, we are able to show that, on each section of the roadway, there is a limit in terms of the combined probability of failure among all of the failure modes. This limit is determined by whatever mode becomes a governing factor at the study section and can be represented by the probability of system failure. The customized RELAN program can perform the following task: • Calculate Probability of Non-compliance, Pnc, or Reliability Index, p, for each failure modes • Calculate Probability of System Non-compliance 146 Sea-to-Sky Highway Horseshoe Bay - Squamish (44 km) Define 4 failure modes: mode 1 - £ mode 2 - fy mode 3- SSDH mode 4- SSDy _ L _ Section i tatty = 0 Calc. Calc. mode 3 mode 1 & 2 Section i + 1 Figure 7.4 Moving Coordinate System Design Flowchart tat*Y'0 tarty *0 R = 0 R*o 1 Horizontal curve Vertica curve Both horizontal and vertical curve Calc. mode 4 End of roadway ? Calc. Calc. mode mode 3 & 4 1,2, 3 and 4 For each section, output: - Reliability index - Prob. of non-compliance for each failure mode - Prob. of non-compliance for whole system - Sensitivity factors 147 In addition, the RELAN program is modified such that a series of runs of the MCSD model can be accommodated and the Reliability Indices for the entire roadway can be calculated in one pass. As the program can automatically generate an output database, the results can be easily plotted and displayed as a "profile" in which the changes in the Probability of non-compliance throughout the entire highway can be observed. Another way to obtain a safety status for an entire highway is to determine an average probability of system failure. By formulating a cumulative reliability function curve, the mean value of the reliability index for a specific group of drivers on the entire highway can be determined. This indicator is meaningful and can be used to be compared with other design alternatives. The subroutines are also developed in this chapter which calls for the RELAN program to perform the above analysis. The customized RELAN program and the subroutines are included in the diskette ZHENG.PHD. A more fundamental highway design model - the MCSD model was presented in this chapter. How is it combined with vehicle, driver and the roadway environment characteristics to form an integrated dynamic vehicle system? What would be required as the input to the MCSD model? The next chapter will describe the MCSD analysis method, with the main emphasis on the required input to the analysis. The information available for each variable will be collected and organized and interrelations and correlations will be described. 148 C H A P T E R 8 SEA-TO-SKY H I G H W A Y APPLICATION This chapter applied the Moving Coordinate System Design model to the Sea-to-Sky Highway. To obtain the model inputs, this chapter started with the identification of all important variables, followed by data preparation through designed experiments and tests. All data were then fit into equations in order to assign values to each variables necessary to run the model. 8.1 Model Variables 8.1.1 Slip angle, a The cornering behaviour of a motor vehicle is an important performance mode. At low speeds (parking lot manoeuvres) the tires do not need to develop significant lateral forces. Thus they roll with little or no slip angle, and the vehicle must negotiate a turn as shown in Figure 8.1. Figure 8.1 Geometry of a turning vehicle (Source: Fundamentals of Vehicle Dynamics) 149 For the geometry in the turn (assuming small angles), the steer angles are given by: where 6G 6, r R + til K - ill Angle of the outside front wheel Angle of the inside front wheel Distance between the front wheels For convenience, the two front wheels can be represented by one wheel at a steer angle, 6, with a cornering force equivalent in both wheels. The same assumption is made for the rear wheels which produces the "bicycle" model. At high speeds, the turning equations differ because lateral acceleration will be present. To counteract the lateral acceleration, the tires must develop lateral forces, and slip angles will be present at each wheel. Under cornering conditions, the tire will experience lateral slip as it rolls. The angle between the tire's direction of heading and its direction of travel is known as slip angle, a, as shown in Figure 8.2. 800 Direction itf heading Direction! of Travel ti 2 4 6 8 10 12 Slip Angle, a (deg) Figure 8.2 Tire cornering force properties (Source: Fundamentals of Vehicle Dynamics) Slip Angle (-) 150 At a given tire load, the cornering force increases with slip angle. At low small slip angles (5 degrees or less) the relationship is linear, hence, the cornering force is described by: F = Ca y « (8.1) where Fy C„ Cornering force Cornering stiffness, defined as the slope of the curve for Fy vs a at a = 0 The steady-state cornering equations are derived from the application of Newton's Second Law and another equation describing the geometry in turns (modified by the slip angle conditions necessary on the tires). For a vehicle travelling forward with a speed of V, the sum of the forces in the lateral direction from the tires must equal the mass multiplied by the centripetal acceleration. I X = F Y f + F Y r = M V 2 / R (8.2) where F, yf F y r M V R Cornering force at the front axle Cornering force at the rear axle Mass of the vehicle Forward speed Radius of the turn Also, for the vehicle to be in moment equilibrium about the centre of gravity, the sum of the moments from the front and rear lateral forces must be zero. FYfb - FYRc = 0 Where b c Distance from front wheel to the centre of mass Distance from rear wheel to the centre of mass 151 Thus F yf Substituting back into Equation 8.2 yields: F = MbIL (V2IR) (8.3) Equation 8.3 says that the lateral force developed at the rear axle must be Wr/g multiplied by the lateral acceleration at that point. Solving for Fyf in the same fashion will indicate that the lateral force at the front axle must be W/g multiplied by the lateral acceleration. With the required lateral force known, the slip angles at the front and rear wheels are also established from Equation 8.1. That is af = WfV2/(CfgR) a = WrV2l(CrgR) (8.4) (8.5) With consideration of the geometry of the vehicle, it can be seen that: 6 = 57.3 — + a , + a R f (8.6) Substituting for af and ar from Equations 8.4 and 8.5 gives: 8 = (8.7) 152 Assume Thus Wf = 0.S5W W = 0A5W r Cf = Cr = 175 lb/degree S = 5 7 . 3 ^ + O ^ Z l ( 8 . 8 ) R 175 gR where 8 = Steer angle at the front wheels (deg) L = Wheel base (ft) R = Radius of turn (ft) V = Forward speed (ft/sec) g = Gravitational acceleration constant = 32.2 ft/sec2 Wf = Load on the front axle (lb) Wr = Load on the rear axle (lb) Cf = Cornering stiffness of the front tires (lb y /deg) Cr = Cornering stiffness of the rear tires (lb y /deg) Converting the Equation 8.8 from the British system into the metric system and substituting L, the average wheelbase, with 3.5 m, and W, the average load on tires, with 600 kg, the Equation 8.8 becomes 6 = 5 7 . 3 1 ^ + ° - 2 2 * 6 0 0 ° - 3 0 4 F 2 (8.9) R 175 9.SR Equation 8.9 shows that steering angle, 6, is proportional to the square of speed, V, and inversely proportional to the curve radius, R. 153 8.1.2 Speed variables, variations and distribution 1) Speed variables The selection of a speed when driving on a curve is influenced by a great number of factors: the driver, the curvature, the approach speed, the friction factor, the lane width, the traffic volume, the skid resistance factor and the superelevation. Taragin (1954) identified the degree of curvature as the most important factor in determining speed on curves. All following research shows that the radius of a curve is the most important geometric feature that influences driver's speed. For compound curves, Badeau et al. (1995) found a high correlation between the driving speed in the curve and the "curves' first encountered radius". Lin (1990) explains that curvatures account for 87% of the variation in the following 85th percentile speed equation: Vg5 = 61.9 - 1.906D + 0.032£>2 (8.10) degree of curvature, degree 85th percentile speed, km/h 2) Speed variation The common conclusion is that speed adjustment is made prior to the curve, and speed in the curve is constant thereafter. Taragin (1954) suggested that drivers do their speed adjustments before entering a road curve and concluded that the speed in the curve is mostly constant. Mintsis (1988) explained that speed variations are highly dependent on the degree of curvature, and for high curvatures, speeds were changing considerably. Badeau (1995) found that where D = V85 = 154 minimum speeds were at the middle of the curve. 3) Speed distribution Mintsis (1988) investigated the probability distribution which best represents the speed population in horizontal curves. He concluded:"the normal distribution was found to provide a statistically significant description of the speed,..., on the approach to curves,..., as well as the entry, middle and exit points and for cars and goods vehicles." McLean (1970) also suggested that passenger car and heavy goods vehicle speed distributions at the curve entrance showed no significant departures from the normal distribution, with a coefficient of variation of about 0.14. 8.1.3 Lateral friction versus speed Craus et al. (1979) determined a linear relationship between decreasing lateral acceleration with increasing travel speed in the curve, as shown in Equation 8.11. This decrease is the result of the "driver being overcautious when evaluating driving risk, which increases with increasing travel speed." In the relationship between the lateral acceleration and the curve speed, the low speed driver type of the entire population is closer to the mean than the fast driver type. a — = 0.262 - 0 . 0 0 1 8 2 F o g (8.11) where ay lateral acceleration, g design speed, km/h D Herrin et al. (1974) hypothesized an exponential relationship between speed and lateral 155 acceleration. Based on the trade-off idea, he proposed a car-driving behaviour model in the following expression: y - HYa - v) (8.12) ay = lateral acceleration A j- lateral acceleration tolerance V = speed, km/h vA = aspiration velocity, km/h p parameter to represent the "expedience" of driver's strategy: Large negative values of P would indicate unwillingness to trade speed for acceleration, while a value P = 0 would produce a linear speed-acceleration trade-off. The model is used by Felips (1996) to represent the trade-off between speed and lateral acceleration for PTEC data. He assumed A, = 0.4 g and VA = 55 km/h for operating a comfortable speed on PTEC curves; and A, = 0.76 g and VA = 85 km/h for operating a maximum safe speed on PTEC curves. The results are summarized in Figure 8.3. 0.80 - , 0.70 H .<o 0.60 H J 0.50 H Scenario: Comfortable speed VI R! - 16m Q o R2-2Sm 10 20 30 40 50 60 70 60 Speed (km/h) Figure 8.3 Lateral acceleration and speed for PTEC curves using Herrin et al. Model (Source: E. Felips,! 996) 156 8.1.4 Speed versus radii There have been a number of relationships proposed to describe the variation of speed with minimum curve radius. They are as follows: 1. Taragin (1954) presented measurements of speed at the point of minimum sight distance on 35 two-lane highway curves with radii ranging from 61 m to 634 m (200 ft - 2080 ft). From least-squares fits of a straight line, hypobulia and parabola to the data, he concluded that a linear variation of speed with degree of curvature D represented the data best. The regression for the mean speed was V = 46.26 - 0.74D (y =0.74) Taragin concluded that curvature was far more important than sight distance in determining speeds on curves, and that the curve superelevation had no effect at all. 2. The Department of Main Roads (1969) measured the speeds near the centre of 21 curves on rural two-lane highways, with radii ranging from 76 m to 457 m (250 ft - 1500 ft). Within the range of radii studied, the observed 85th percentile speeds varied approximately linearly with the radius R, as shown below: 1 7 6 7 R = ±ULL± + 33.7 . ( v=0.83) 8 5 100 3. Speeds at the centre of 12 curves with radii between 2 1 m and 457 m (70 ft - 1500 ft) were observed by Emmerson (1979). He found a curvilinear relationship between the 157 mean speed and curve radius, with little change in vehicle speeds for radii greater than 200 m or speeds were relatively unaffected by degree of curvature less than about 8 (R >200 m). Emmerson proposed an exponential relationship which had the "advantages"that the speed is zero for zero radius, and tends to be a constant value at large radii. The equation to the least-square fit obtained by Emmerson for the mean speed V (km/h) was V = 74 (1 - e " 0 0 1 7 * ) where R is the curve radius in metres. McLean (1974) attempted to resolve some questions as to the most appropriate form of relationship between speed and radius, by comparing the variance accounted for by linear regressions of speeds on: a) radius b) square root of radius c) curvature, and d) an exponential relationship of the form suggested by Emmerson The data of Taragin, the Department of Main Roads and Emmerson was used in the comparison. As shown in Figure 8.4, the results show that, in terms of the proportion of variance accounted for by the various models, the model originally proposed for each data set gives the best fit. However, McLean also made tests to compare the contributions to the unexplained variance from 'lack of fit' and from 'pure error', for Taragin data only. Whereas the curvature model 'explained' the greatest proportion of 158 the variance, only the exponential model was free from statistically significant 'lack of fit'. The exponential model, therefore, provided a better representation of the variation of speed with radius. 5. The French experience was described by Brenac (1990) for rural highways. Brenac found that, for French roads, the only factors of significance in vehicle speed were: radius of curve, grade, lane width and number of lanes. All other factors were insignificant. He developed a single relationship between speed, radius of curve, number of lanes and lane width. The equations have a simple quadratic form in grade, stratified by the number of lanes and lane width. Figure 8.4 Empirical speed versus radii for data of Taragin, D M R and Emmerson (source: McLean 1974) 159 McLean (1981) summarized the Australian experience of vehicle speeds on curves. He showed that the expected value of the speed in a curve is not only dependent on geometry variables, but also on the driver's desired speed. The desired speed is defined as the 85th percentile speed that drivers would maintain in the least constraining portion of the road (tangents). After extensive research at 120 curve sites, McLean proposed the following regression for the 85th percentile car speed in curves: K„ = 53.8 + 0 . 4 6 4 F „ - l ™ + * * ° ° ° (8.13) 85 R R where V( R 85 85th percentile car curve speed, km/h curve radius, m desired speed of 85th percentile car, km/h To better represent the 85th car speed in the curve, McLean partitioned the data into groups about the same desired speed and developed a "family of curve speed prediction relations" as shown in Figure 8.5 below: 120 200 400 600 800 Horizontal Curve Radius (m) 1000 Figure 8.5 Curve speed prediction relationship (source: McLean, 1981) 160 A comparison of McLean's results (1981) with Brenac's observation (1990) is summarized in Figure 8.6. The curves are very similar. Ho.8 120 / ( 1 «• S48/KMJ!) 120.0 i Figure 8.6 Comparison of speed versus radii models (source: McLean, 1981 and Brenac, 1990) The following conclusions can be drawn from Figure 8.6: 1) Speed versus radii models are in favour of a 2-region system, separated by the minimum radius Rt: Region 1 (R < R,): speed increase as radii increase Region 1 (R > R,): the change of the speed, V, could be negligible as radii increase 161 2) The threshold of R, varies from 50 m to 350 m with a design speed increase from 60 km/h to 100 km/h. For example, if the design speed is V= 80 km/h, the threshold R, = 300 m . The design speed is the governing factor, no matter the curve radii, motorists cannot increase their speeds as the curve radii increase. In such a situation the operating speed is no longer sensitive to curve radii in the region of R > R,. 3) Speed is highly influenced by curve radii between 50 and 350 m. When a radius is larger than 350 m, there is little influence on the driving speeds. 8.1.5 Lateral friction versus speed Lateral acceleration depends upon not only speed, but also tightness of the turn. The resulting g experienced tells drivers exactly what they need to know and how hard they are cornering, therefore, how hard the tires must work to maintain traction. Based on Equation 8.14, a change in speed, or a change in curve radius, affects lateral acceleration in a predictable way. The change will be proportional to l^/R, where Vis speed and R is the radius of the turn. The equation shows that doubling the speed on any curve will quadruple the resulting lateral acceleration, while tightening the radius by half will only double lateral acceleration. Thus, the cornering limit is much more sensitive to speed than it is to tightness of the turn. fyg = (8-14) If a vehicle demands more side friction than the pavement/tire interface can provide, the vehicle 162 will skid off the roadway. It follows that driving on a tight curve (R<350 m), generally the available lateral friction becomes a governing factor; while driving on a generous curve (R>350 m), the available sight distance is of the primary concern. Figure 8.7 plots the relationship between the lateral friction and curve radii based on the McLean study. It shows evidence of log curvature. Figure 8.7 Lateral Friction versus radii (source: McLean, 1981) 163 8.2 Data Preparation Figure 8.8 is a flow diagram of the procedure for collecting and processing the data used in this study. To obtain fx and fy at any point along a roadway in a format of supply versus demand, data collection was performed with a focus on fx and fy "demanded" by normal drivers, "supplied" by what was built into a highway or "provided" by design standards, and "capacity" reached by expert drivers. Demand by normal drivers could be visualized as an ordinary driver operating a passenger car on an existing roadway, which is determined by geometric layout. Capacity is represented by the limit of the vehicle with an expert driver for given operating conditions. For example, expert drivers can stop faster, negotiate tighter curves, reach the maximum fx and fully utilize the maximum f. A racing car with an expert driver would be an ideal for the study. The overall objective is that the race car should have a large "g-g" manoeuvre diagram throughout its performance envelope and that race driver should operate close to the diagram boundaries which represents the limit of the driver-vehicle-tire-road combination. 8.2.1 PTEC experiments An experiment was designed by Felips (1996) and conducted at the Pacific Traffic Education Centre (PTEC). Four curves with radii of 16 m, 25 m, 60 m and 100 m at zero superelevation were delineated by traffic cones. Two groups of people - normal drivers and expert drivers (policemen), were invited to participate in the test in order to obtain lateral friction values, fy, on the four curves (Photo 6). 164 Photo 6 - Four curves with radii of 16, 25, 60 and 100 m were delineated by traffic cones on the Airport. Lateral friction values were obtained by both expert and normal drivers operating on them at the maximum and comfortable speeds 1) Demand by normal drivers Listed in Table 8.1 are the7 ,^ values experienced by normal drivers while driving on the P T E C curves with comfortable speeds and maximum speeds, where the speed is the average over eight runs. Table 8.1 fv demanded by normal drivers in PTEC curves Radii Comfortable Speed Lateral Friction Maximum Speed Lateral Friction (m) (km/h) (km/h) 16 24.96 0.32 31.24 0.49 25 30.31 0.28 38.39 0.45 60 39.91 0.23 52.86 0.35 100 45.56 0.19 59.37 0.3 166 2) Capacity by expert drivers Listed in Table 8.2 are the fy values reached by expert drivers while driving through the PTEC curves when the pavement is dry and wet: Table 8.2 fy reached by experts in PTEC curves Radii (m) Maximum Speed before Sliding (km/h) Lateral Friction fy dry wet dry wet 16 39.62 37.11 0.72 0.63 25 49.31 45.38 0.66 0.59 60 68.92 67.4 0.56 0.52 100 83.49 79.58 0.49 0.47 8.2.2 Sea-to-Sky Highway curves 1) Demand by normal drivers To measure the amount of side friction generated by tire/roadway interaction, a g-Analyst was installed in the vehicle while driving through the Sea-to-Sky Highway curves. Curvatures ranged from 170 m to 3500 m and the maximum superelevation rates ranged from 2 to 8 percent. The g-Analyst was used to measure the normal driver's ability to utilize the performance inherent in the car or willingness to exploit that performance. Listed in Table 8.3 are the lateral friction values,/?,, recorded by the g-Analyst while normal drivers negotiated the curves on the Sea-to-167 Sky Highway. A total of four runs were completed: two for the northbound direction and two for the southbound direction at an operating speed of 70 km/h (Photo 7). Table 8.3 f demanded by normal drivers on Sea-to-Sky Highway 1 KI K (m) Superelevation c Lateral friction A 5.76-5.96 172 0.07 0.207 6.22-6.38 237 0.06 0.146 6.48 -6.52 170 0.07 0.207 6.64-6.90 290 0.05 0.135 7.98-8.08 201 0.03 0.142 8.20-8.26 300 0.06 0.129 9.20-9.86 180 0.05 0.145 10.50-10.58 350 0.05 0.135 11.76-11.86 655 0.02 0.067 12.46-12.56 450 0.04 0.141 12.90-12.96 500 0.05 0.08 13.02-13.20 170 0.06 0.207 13.36-13.48 350 0.04 0.116 13.60-13.88 500 0.03 0.08 14.90-14.94 350 0.02 0.121 15.20-15.32 290 0.05 0.143 17.28-17.38 1147 0.01 0.038 18.14-18.22 917 0.02 0.049 19.96-20.06 639 0.03 0.067 Two qualities of the g-Analyst were used to facilitate revisiting past manoeuvres. First, event markers were used for labelling moments of interest as they happened. For example, flags were 168 put into the data while passing all of the bridges between Horseshoe Bay and Squamish. Later, when scrolling through the memory, the g-Analyst automatically stops at these flags for quicker location references. Second, as the g-Aanlyst operates in real time, "time tick" of a tenth of a second is used to display on a horizontal axis to facilitate finding any manoeuvre of interest. Photo 7 - The g-Analyst was installed in the testing vehicle to measure a normal driver's ability on car handling - a total of four runs were completed: two for the northbound direction and two for the southbound direction at an operating speed of 70 km/h Figure 8.9 shows the fy values experienced by a normal driver while travelling through one of the Sea-to-Sky Highway curves (LKI 9.20 - 9.86 at R = 90m). The g-Analyst assigns numbers on car handling and plots every combination of turning and braking, and turning and accelerating. 169 •0.30 -I •0.40 J Figure 8.9 fy collected on Sea-to-Sky Highway by g-Analyst 2) Capacity by expert drivers Listed in Table 8.4 are fy values likely reached by expert drivers while negotiating the four Sea-to-Sky Highway curves. These values are obtained by extending the curve derived from the PTEC experiment. Table 8.4 fy reached by experts on Sea-to-sky Highway Radii Average/;. Superelevation T o t a l / (m) (without e) e (with e) 120 0.45 0.07 0.52 450 0.31 0.04 0.35 180 0.41 0.03 0.44 90 0.47 0.04 0.51 170 8.2.3 Longitudinal Friction Test Stopping sight distance tests were conducted according to the A S T M Standard Method (1981) for measurement of stopping distance on paved surfaces using a passenger automobile. For the passenger car tire at the stopping sight distance site, the results of the regression of the longitudinal friction data produce the following average friction-sliding speed relationship: fx = 0.0874 e - ° 0 0 3 9 3 v (8.15) with a correlation coefficient (y2) of 0.96. 8.2.4 Design Standards 1) Stopping sight distance (SSD) Listed in Table 8.5 are the design standards for SSDs used by the Ministry of Transportation and Highways of British Columbia (MoTH). The following assumptions are made to develop the table.: • Driver's perception-reaction time is 2.5 seconds • Operating speed is the same as design speed up to 90 km/h, and progressively less at higher speeds • Longitudinal friction values, fx, for wet pavement are as noted 171 Table 8.5 Stopping sight distance (1) Design Speed km/h (2) Friction,/, wet pavement SSD (m) (3) Horizontal Curve (4) Crest Curve (5) Grade Correction Decrease for upgrade of Increase for downgrade of 3.0% 6.0% 9.0% 3.0% 6.0% 9.0% 40 0.38 45 45 - - 5 - - -50 0.36 65 65 5 5 10 - 5 10 60 0.34 85 85 5 5 10 5 10 15 70 0.32 110 110 5 10 15 5 10 20 80 0.31 140 140 10 15 20 10 15 30 90 0.3 170 190 10 20 25 10 20 40 100 0.3 200 220 10 20 - 15 30 -110 0.29 220 245 Design values in Table 8.5 are subject to the following adjustments: 1) On crest vertical curves, the perception/reaction time is greater by as much as one second at higher speeds; Column (4) reflects this. 2) When braking on a grade, the effect of a downgrade increases the braking distance and an upgrade decreases the braking distance. Column (5) indicates adjustments on SSD for grade. 3) For design speeds of 60 - 90 km/h and radii that are no more than 110% of the minimum, the SSD should be increased by 5%. 2) Horizontal alignment 172 Maximum superelevation, e, of 0.06 m/m is used for all rural roads in British Columbia. Table 8.6 lists lateral friction values, superelevation and radius for various design situations. Due to variation in qualities of tires, pavement and driver comfort, low lateral friction values,/^ ,, are utilized. Table 8.6 Lateral friction, superelevation and radius (1) Design Speed km/h (2) Max. Friction fy Minimum Radius (m) (3) e = 0.06 m/m (4) Normal Crown (5) Reverse Crown (6) Average 40 0.17 55 700 475 587.5 50 0.16 90 1100 745 872.5 60 0.15 135 1600 1080 1340 70 0.15 190 2100 1470 1785 80 0.14 250 2800 1930 2365 90 0.13 340 3500 2460 2950 100 0.12 440 4400 3060 3730 110 0.1 600 5300 3770 4535 Design values in Table 8.6 are based on the following assumptions: 1) In milder climates, higher superelevations are appropriate, whereas in areas subject to surface icing, lower values are applied. The minimum radii, listed in the column (3), are determined by setting the friction,^ ,, and superelevation, e, to their maximum values. 173 2) When the curve radius change from the "normal crown" to a "reverse crown", a value of 0.018 for e+fy will give the same degree of comfort and safety that the motorists expect. Where e+fy is less than or equal to 0.018, normal crown is used. Where e+fy is greater than 0.018, the reverse crown is used. The minimum radii for normal crown and reverse crown are listed in the columns (4) and (5), respectively. 8.2.5 Road test 1) Distribution offx and fy The limit of a car's capacity in accelerating and braking are obtained by experiments. When the drive wheels spin, drivers cannot get any more power to the ground, and acceleration is limited accordingly. During braking, when tires lose their grip and start to slide, drivers cannot control the vehicle, and braking is limited accordingly. Similar limits occur in cornering. Car magazines frequently list cornering and braking capability of vehicles. A roadtest was published in Car and Driver Digest (1995) which summarized the performance of 126 cars and car-vans. Table 8.7 shows the performance of each car in terms of their longitudinal accelerating/decelerating frictions, fx and lateral frictions,/^ : 174 Table 8.7 Road Test Summary Longitudinal Acceleration Braking Capability Lateral Acceleration MODEL 1M Mile 0-60 mph tx dis. -fx fy k sec sec 0 ft 0 o Acura Integra OS-R 15.6 7.0 0.39 185 0.88 0.81 0.92 Acuta Integra LS 15.0 7.5 0.36 189 0.86 0.82 0.96 Acura NSX-T 13.8 5.2 0.52 173 0.94 0.95 1.01 Audi A6 Quattro 21.2 18.1 0.15 253 0.64 0.62 0.97 Audi Cabriolet 17.2 9.7 0.28 189 0.86 0.73 0.85 BMW M3 14.0 5.3 0.51 165 0.98 0.88 0.90 BMW 3251 16.0 7.7 0.35 182 0.89 0.82 0.92 BMW 540i Six-Speed 14.3 5.7 0.48 186 0.87 0.78 0.90 BMW740i 16.6 8.4 0.32 181 0.90 0.79 0.88 BMW 840CI 15.5 7.1 0.38 170 0.95 0.83 0.87 BMW 850 CSI 13.9 5.3 0.51 167 0.97 0.85 0.88 Buick Park Avenue Ultra 15.5 7.0 0.39 216 0.75 0.74 0.99 Buick Regal Gran Sport 16.5 8,3 0.33 202 0.80 0.79 0.98 Buick Riviera 16.3 8.2 0.33 199 0.81 0.75 0.92 Cadillac Eldorado Touring Coupe 14.8 6.4 0.43 184 0.88 0.80 0.91 Cadillac SLS 15.0 6.7 0.41 195 0.83 0.77 0.93 Chevrolet Blazer LT 17.0 9.1 0.30 218 0.74 0.67 0.90 Chevrolet Camaro 15.9 7.7 0.35 192 0.84 0.83 0.98 Chevrolet Camaro Convertible 17.0 9.3 0.29 195 0.83 0.84 1.01 Chevrolet Camaro Z28 14.1 5.5 0.49 162 1.00 0.86 0.86 Chevrolet Cavalier 16.7 8.8 0.31 209 0.78 0.76 0.98 Chevrolet Corvette 13.7 5.1 0.53 166 0.98 0.85 0.87 Chevrolet Impala SS 15.0 6.5 0.42 179 0.91 0.86 0.95 Chevy Suburban Turbodiesel 19.2 13.6 0.20 218 0.74 0.69 0.93 Chevrolet Lumina LS 16.3 8.0 0.34 205 0.79 0.79 1.00 Chevrolet Monte Carlo LS 16.7 8.7 0.31 197 0.82 0.80 0.97 Chevrolet Tahoe LT 18.2 11.3 0.24 198 0.82 0.68 0.83 Chrysler Circus LXi 17.0 9.0 0.30 202 0.80 0.76 0.95 Dodge Avenger ES 17.0 9.1 0.30 190 0.85 0.80 0.94 Dodge Intrepid 17.2 9.5 0.29 195 0.83 0.74 0.89 Dodge Neon Sport Coupe 16.0 7.6 0.36 207 0.78 0.82 1.05 Dodge Stratus ES 16.8 8.9 0.31 197 0.82 0.78 0.95 Dodge Viper RT/10 12.8 4.3 0.63 180 0.90 0.98 1.09 Eagle Talon TSi AWD 15.3 6.6 0.41 169 0.96 0.85 0.89 Ford Contour SE 16.9 8.8 0.31 183 0.89 0.82 0.93 Ford Crown Victoria LX 16.8 8.8 0.31 190 0.85 0.81 0.95 Ford Explorer XLT 17.9 10.7 0.25 199 0.81 0.67 0.82 Ford Mustang Cobra R 14.0 5.4 0.50 165 0.98 0.89 0.91 Ford Mustang Gt SVO GT40 14.2 5.5 0.49 170 0.95 0.85 0.89 Ford Probe GT 15.9 7.4 0.37 164 0.99 0.86 0.87 Ford Windstar 18.3 11.4 0.24 195 0.83 0.72 0.87 Honda Accord LXV-6 17.0 8.8 0.31 182 0.89 0.75 0.84 Honda Civic DX 16.7 8.7 0.31 198 0.82 0.72 0.88 Honda Odyssey EX 18.1 10.7 0.25 192 0.84 0.73 0.87 Honda Passport EX 18.0 10.4 0.26 189 0.86 0.70 0.82 Honda Prelude VTEC 15.1 6.5 0.42 181 0.90 0.86 0.96 Hyundai Elantra GLS 17.3 10.0 0.27 193 0.84 0.81 0.96 Hyundai Sonata GLS 17.1 9.3 0.29 189 0.86 0.77 0.90 Infintti G20t 16.5 8.3 0.33 176 0.92 0.83 0.90 Infiniti 130t 16.1 7.7 0.35 189 0.86 0.78 0.91 Infinrti Q45 16.1 7.8 0.35 187 0.87 0.69 0.80 Isuzu Trooper LS 18.1 10.9 0.25 193 0.84 0.69 0.82 Jaguar XJR 14.8 6.4 0.43 162 1.00 0.80 0.80 Jaguar XJS 4.0 Convertible 16.4 8.3 0.33 194 0.84 0.81 0.97 Jaguar XJ6 16.3 8.1 0.34 189 0.86 0.74 0.86 Jeep Grand Cherakee Limited 16.3 8.0 0.34 180 0.90 0.75 0.83 KiaSephiaGS 17.6 10.2 0.27 197 0.82 0.75 0.91 Land Rover Discovery 18.3 10.9 0.25 204 0.79 0.67 0.84 Lexus LS400 15.8 7.8 0.35 182 0.89 0.74 0.83 Lexus SC300 15.4 6.8 0.40 178 0.91 0.84 0.92 Lincoln Continental 15.7 7.6 6.36 178 0.91 0.78 0.86 Lincoln Town Car Skinature Series 16.9 9.0 0.30 186 0.87 0.79 0.91 175 Table 8.7 Road Test Summary Longitudinal Acceleration Braking CaDabilitv Lateral Acceleration MODEL 1/4 Mile 0-60 mph tx dis. -fx fy a k SGC fl ft a Ungenfelter Firebird 383 12.6 4.1 0.66 150 1.08 0.95 0.88 Mazda Miata R Edition 16.4 8.2 0.33 169 0.96 0.86 0.90 Mazda 626ES 16.8 8.4 0.32 189 0.86 0.80 0.93 Mercedes-Benz C36 14.6 6.0 0.45 163 0.99 0.83 0.84 Mercedes-Benz C220 16.9 8.8 0.31 178 0.91 0.80 0.88 Mercedes-Benz E420 15.7 7.3 0.37 185 0.88 0.73 0.83 Mercedes-Benz S320 16.4 8.1 0.34 178 0.91 0.72 0.79 Mitsubishi Eclipse GS 16.7 8.6 0.32 189 0.86 0.82 0.96 Mitsubishi Galant GS 16.1 7.8 0.35 207 0.78 0.77 0.98 Mitsubishi Montero SR 17.9 10.5 0.26 191 0.85 0.66 0.78 Mitsubishi 3000GT VR4 13.9 5.1 0.53 164 0.99 0.92 0.93 Nissan Maxima SE 15.9 7.3 0.37 187 0.87 0.75 0.87 Nissan Quest GXE 18.3 11.2 0.24 178 0.91 0.71 0.78 Nissan Sentra GXE 16.7 8.5 0.32 208 0.78 0.78 1.00 Nissan 2O0SX SE-R 16.1 8.0 0.34 182 0.89 0.85 0.95 Nissan 240SXSE 16.0 7.7 0.35 170 0.95 0.89 0.93 Nissan 300ZX Turbo 14.2 5.6 0.49 175 0.93 0.89 0.96 Nissan 300ZX 2+2 15.3 6.8 0.40 177 0.92 0.87 0.95 Oldsrnobile Aurora 16.4 8.2 0.33 185 0.88 0.78 0.89 Oldsmobile Eighty Eight LSS 15.5 7.0 0.39 198 0.82 0.80 0.98 Pontiac Bonneville SE 15.5 7.0 0.39 186 0.87 0.81 0.93 Pontiac Grand Am GT Coupe 17.0 9.5 0.29 185 0.88 0.79 0.90 Pontiac Sunfire GT 16.9 8.9 0.31 183 0.89 0.79 0.89 Pontiac Trans Am 25th Anniversary 14.6 6.1 0.45 166 0.98 0.84 0.86 Range Rover 4.0 SE 17.9 10.5 0.26 185 0.88 0.73 0.83 Rolls-Royce Silver Spur III 17.0 9.3 0.29 191 0.85 0.70 0.83 Saab900S 16.1 7.6 0.36 188 0.86 0.75 0.87 Saab 900SE Convertible 16.7 8.6 0.32 184 0.88 0.77 0.87 Saab 900SE Turbo 15.3 6.5 0.42 171 0.95 0.82 0.87 Saab 9000CDE 16.3 8.3 0.33 178 0.91 0.75 0.82 Saleen Mustang S351 13.9 5.1 0.53 171 0.95 0.87 0.92 Saturn SC2 16.0 7.6 0.36 174 0.93 0.84 0.90 Subaru Legacy L 17.9 10.4 0.26 176 0.92 0.75 0.81 Subaru Legacy Outback Wagon 17.9 10.6 0.26 166 0.98 0.82 0.84 Subaru SVXLS 15.9 7.6 0.36 174 0.93 0.83 0.89 Toyota Avalon xL 15.7 7.5 0.36 175 0.93 0.81 0.87 Toyota Camry 18.1 10.7 0.25 196 0.83 0.75 0.91 Toyota Camry SE 16.2 7.9 0.34 177 0.92 0.78 0.85 Toyota Celtea GT 16.5 8.0 0.34 173 0.94 0.87 0.93 Toyota Celtea GT Convertible 16.9 9.0 0.30 175 0.93 0.85 0.92 Toyota Corolla DX 17.1 9.3 0.29 197 0.82 0.77 0.94 Toyota Land Cruiser 17.9 10.7 0.25 178 0.91 0.72 0.79 Toyota Supra 15.0 6.5 0.42 173 0.94 0.92 0.98 Toyota Tercel 17.0 9.2 0.30 195 ' 0.83 0.75 0.90 Toyota T100 16.8 8.9 0.31 201 0.81 0.69 0.86 Volkswagen Cabrto 17.6 10.3 0.26 182 0.89 0.81 0.91 Volkswagen GTIVR6 15.2 6.6 0.41 179 0.91 0.78 0.86 Volkswagen Jetta III GLX 15:5 6.9 0.39 208 0.78 0.78 1.00 Volkswagen Passat GLX 16.5 8.6 0.32 178 0.91 0.81 0.89 Volvo 850 Turbo 14.6 6.1 0.45 172 0.94 0.81 0.86 vorvo850T-5R 15.2 6.7 0.41 181 0.90 0.79 0.88 Volvo 960 16.6 8.6 0.32 190 0.85 | 0.80 0.94 Max. value 0.66 1.08 0.98 1.09 Min. value 0.15 0.64 0.62 0.78 Geometric mean 0.35 0.88 0.79 0.90 Standard, deviation 0.09 0.07 0.06 0.06 176 In Table 8.7: • Column 2 indicates longitudinal acceleration capabilities of each car when time elapsed from 0 to 60 mph and through a quarter-mile distance using full-throttle acceleration; • Column 3 lists braking capabilities of each car when vehicles come to complete stops from 70 mph to reach a minimum stopping distance without sliding; and • Column 4 lists the maximum lateral acceleration that each vehicle can generate when cornering a 300 ft (91.4 m) asphalt diameter skid pad. The mean value and standard deviations are as follows: / = 0.88 of = 0.07 f = 0.79 O. = 0.07 Figure 8.10 shows the distribution curves,/^ ,, with the mean value is 0.8g and the standard deviation of 0.07g. 0.20 0.40 0.60 0.80 1.00 1.20 Maximum lateral acceleration (g's) Figure 8.10 The maximum lateral acceleration that different cars experienced on a 91.4 m diameter skid pad 177 8.3 Fitting Equations to Data 8.3.1 Curve fitting To define a relationship between the lateral friction,/^ ,, and the curve radii, R, curve-fittings are performed based on the available experimental data listed in Table 8.8: Table 8.8 Experiment data summary - fy versus curve radii Design Normal Drivers Expert Drivers R speed @ Comfor. V ©Max V ft © Max V fy @MaxV Remark faafi Standards JllRtrfl fan*. fanft <*Y fanfi wet 16 2 4 . 9 6 032. 3 1 2 4 0 .49 3 9 . 6 2 0 .72 37.11 0 .63 2 5 30 .31 0 . 2 8 3 8 . 3 9 0 . 4 5 49.31 0 .66 4 5 . 3 8 0 .59 PTEC 6 0 39 .91 0 . 2 3 5 2 . 8 6 0 .35 6 8 . 9 2 0 . 5 6 6 7 . 4 0 0 .52 1 0 0 4 5 . 6 6 0 .19 5 9 . 3 7 0 .30 8 3 . 4 9 0 .49 7 9 . 5 8 0 .47 9 0 0 .47 Sea-to-sky 1 2 0 0 .45 (erfenfionof 1 8 0 0.41 P T E C curve) 4 5 0 0.31 5 8 7 . 5 4 0 0 . 1 7 8 7 2 . 5 5 0 0 . 1 6 Design 1 3 4 0 6 0 0 .15 Standards 1 7 8 5 7 0 0 .15 2 3 6 5 8 0 0 .14 2 9 5 0 9 0 0 .13 3 7 3 0 100 0 .12 4 5 3 5 110 0 .10 All data were plotted on logarithmic paper and each group of data fall on a straight line showing evidence of log curvature. This finding is supported by McLean's research, as shown in Figure 8.7. i.e. the relationship of the lateral friction versus radii presents log curvature. 178 8.3.2 Fitting equations to data The best way to summarize large amounts of multifactor data is by a set of equations. The purpose of fitting equations are: • to summarize the large amount of data in order to obtain interpolation formulas or calibrated curves; • to confirm a theoretical relation; to compare several sets of data in terms of the constants in their representative equations; to aid in the choice of a theoretical model. Data was then divided into four groups: • Group 1: PTEC experiment (curve 1-4) • Group 2: Expert drivers on the Sea-to-Sky Highway (curve 5) • Group 3 : Normal drivers on the Sea-to-Sky Highway (curve 6) • Group 4: Design standards (curve 7) The best fitting lines are as follows: Curve 1 - Expert drivers on dry pavement: f = -0.11215Zn(fl) + 1.02191 Curve 2 - Expert drivers on wet pavement: / = -0.0943253Ln(fl) + 0.897075 Curve 3 - Normal drivers @ max. speed: 179 f = -0M5064Ln(R) + 1.03196 Curve 4 - Normal drivers @ comfort speed: f = -0.0683346Xn(i?) + 0.505976 Curve 5 - Expert drivers on the Sea-to-Sky Highway: f = -0.10123 5Ln(R) + 0.931095 Curve 6 - Normal drivers on the Sea-to-Sky Highway f = -0.0963308L n(#) + 0.685983 Curve 7 - Design standards: f = -0.0252853Zn(/?) + 0.331216 Figure 8.11 shows the above seven fitting lines. The Least Square method was applied to find the values of the constants in the chosen equations such that the sum of the squared deviations of the observed values from those predicted by the equation was minimized. This would result in each curve either going through all the relevant data or fitting the data as close as possible. In short, we obtained all the information possible from the data - both about the equation constants estimated and about the limitations on future use of the equation with these constants. 180 on dry pov«m«nt • • • I l I l l | l 'i-100.0 200.0 0.0 l I | I I 300.0 ' M i ' -400.0 ' I ' 500.0 Radius (m) i " i i i i i i i 600.0 700.0 • i | i i i i I 800.0 900.0 Figure 8.11 Curve-fitting 181 8.3.3 Curve-fitting analysis The following conclusion can be drawn from Figure 8.11: • Lateral friction values on dry and wet pavements will be no different if the radius of a horizontal curve is greater than 600 m; • Lateral friction values reached by expert drivers on the Sea-to-Sky Highway can be considered as an extension of lateral friction values experienced by expert drivers on wet pavement on the PTEC curves; • Lateral friction values resulting from highway design standards remain the least changed as the curve radii increase when compared to other experiment curves; • Lateral friction values generated by normal drivers on the Sea-to-Sky Highway lie between those generated by normal drivers at maximum speed on the PTEC curves and those normal drivers at comfortable speeds on the PTEC curves. 8.4 Input Database for RELAN Table 8.9 is a summary of the most important parameters required by performing the MCSD model. Eight steps are followed in determining the mean value and standard deviations of each variable. Step 1: R - Curve radii, m obtained from Chapter 4 - Highway Geometry Analysis Step 2: a - Max. lateral friction that an expert driver can utilize, g obtained from curve 5 in Figure 8.12 (8.3.3 Curve-fitting) Step 3: b - Max. longitudinal friction that an expert driver can reach, g 182 assumed fxy /0.9 (8.2.5 Roadtest) » Step 4: V- operating speed, km/h Expert drivers: obtained from PTEC experiment (8.2.1 PTEC) Normal drivers: McLean model, verified by speed study on Sea-to-Sky Highway Design drivers: same as design speed (MoTH) assumed same speed while entering into or exiting from a curve, 5 km/h lower in the middle of a curve Step 5: T - perception-reaction time, sec. Expert drivers: 1 sec. (Racing cars by Williams) Normal drivers: 1.6 sec. (Olsen) Design drivers: 2.5 sec. (AASHTO) Step 6: fx - Longitudinal friction, g Expert drivers: fx max, obtained from Equation 8.15 (ASTM Standard Method) Normal drivers: 62% of fx max (NCHRPR 270) Design drivers: obtained from Table 8.5 (MoTH) Step 7: SSDH - Horizontal stopping Sight Distance available, m obtained from Chapter 4 - Highway geometry Analysis. If a section was identified as an "adequate sight distance"section, 200 m sight distance is assumed for drivers to stop their vehicles. Step 8: SSDV- Vertical Stopping Sight Distance available, m obtained from Chapter 4 - Highway Geometry Analysis. If a section was identified as an "adequate sight distance"section, 200 m sight distance is assumed 183 for drivers to stop their vehicles. If the roadway is on a tangent section, or if the roadway has more than two lanes, "0" values are assigned to make the distinction between sets of data with or without engineering factors involved. A large amount of data is required to perform reliability analysis. Great efforts were made to obtain data characterized driver-vehicle behaviour on road curves, particularly for low-speed curves, where the greatest lateral demands are made. Some well-controlled experiments, such as P T E C , g-Analyst and A S T M Method were directed at finding the relationships between roadway geometry and driver-vehicle behaviour. Some empirical information was used for developing curve-fitting equations. This chapter provided the first comprehensive collection of detailed information on driver-vehicle behaviour over a range of curve geometries and made it ready to run the Moving Coordinated System Design model. 184 Table 8.9 Input database for RELAN Normal drivers on Sea-to-Sky Highway at comfortable speed LKI Vcomfort T sac. Mx My R m X(6) fx X0> SSDH m m SSDV m e tan Q G tanY Tangent 5.68 48 1.6 0.KK 0.45 120 0.45 140 200 0.07 0.00 5.70 48 1.6 0.50 0.45 120 0.45 140 200 0.07 0.00 5.72 48 1.6 0.50 0.45 120 0.45 140 200 0.07 0.00 5.76 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 40m 55 1.6 0.47 0.42 172 0.44 200 140 0.07 0.03 5.78 55 1.6 0.47 0.42 172 0.44 200 140 0.07 0.03 5.80 55 1.6 0.47 0.42 172 0.44 200 140 0.07 0.03 5.82 55 1.6 0.47 0.42 172 0.44 140 140 0.07 0.00 5.84 50 1.6 0.47 0.42 172 0.45 120 200 0.07 0.00 5.86 50 1.6 0.47 0.42 172 0.45 100 140 0.07 0.04 5.88 50 1.6 0.47 0.42 172 0.45 80 120 0.07 0.04 5.90 55 1.6 0.47 0.42 172 0.44 80 100 0.07 0.04 5.92 55 1.6 0.47 0.42 172 0.44 80 120 0.07 0.04 5.94 55 1.6 0.47 0.42 172 0.44 100 100 0.07 0.04 5.96 55 1.6 0.47 0.42 172 0.44 200 80 0.07 0.04 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 260m 622 65 1.6 0.42 0.38 237 0.42 120 200 0.06 0.00 624 65 1.6 0.42 0.38 237 0.42 100 200 0.06 0.00 626 65 1.6 0.42 0.38 237 0.42 100 200 0.06 0.00 6.28 65 1.6 0.42 0.38 237 0.42 100 200 0.06 0.00 6.30 60 1.6 0.42 0.38 237 0.43 100 200 0.06 0.00 6.32 60 1.6 0.42 0.38 237 0.43 100 200 0.06 0.00 6.34 65 1.6 0.42 0.38 237 0.42 80 200 0.06 0.00 6.36 65 1.6 0.42 0.38 237 0.42 80 200 0.06 0.00 6.38 65 1.6 0.42 0.38 237 0.42 100 200 0.06 0.00 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 100m 6.48 55 1.6 0.46 0.41 170 0.44 140 200 0.07 0.00 6.50 55 1.6 0.46 0.41 170 0.44 140 200 0.07 0.00 6.52 55 1.6 0.46 0.41 170 0.44 140 200 0.07 0.00 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 120m 6.64 70 1.6 0.41 0.37 290 0.41 140 200 0.05 0.00 6.66 70 1.6 0.41 0.37 290 0.41 140 200 0.05 0.00 6.68 70 1.6 0.41 0.37 290 0.41 120 200 0.05 0.00 6.70 70 1.6 0.41 0.37 290 0.41 120 200 0.05 0.00 6.72 70 1.6 0.41 0.37 290 0.41 120 200 0.05 0.00 6.74 65 1.6 0.41 0.37 290 0.42 140 200 0.05 0.00 6.76 65 1.6 0.41 0.37 290 0.42 200 200 0.05 0.00 6.78 65 1.6 0.41 0.37 290 0.42 140 200 0.05 0.00 6.80 65 1.6 0.41 0.37 290 0.42 140 200 0.05 0.00 6.82 70 1.6 0.41 0.37 290 0.41 120 200 0.05 0.00 6.84 70 1.6 0.41 0.37 290 0.41 120 200 0.05 0.00 6.86 70 1.6 0.41 0.37 290 0.41 100 200 0.05 0.00 6.88 70 1.6 0.41 0.37 290 0.41 100 200 0.05 0.00 6.90 70 1.6 0.41 0.37 290 0.41 200 200 0.05 0.00 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 580m 7.48 80 1.6 0.37 0.33 450 0.40 140 200 0.06 0.00 7.50 80 1.6 0.37 0.33 450 0.40 120 200 0.06 0.00 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 160m 7.66 80 1.6 0.30 0.27 1000 0.40 200 140 0.02 0.04 7.68 80 1.6 0.30 0.27 1000 0.40 200 120 0.02 0.04 7.70 80 1.6 0.30 0.27 1000 0.40 200 120 0.02 0.04 7.72 80 1.6 0.30 0.27 1000 0.40 200 140 0.02 0.04 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 100m 7.82 80 1.6 0.30 0.27 1000 0.40 200 200 0.00 0.03 7.84 80 1.6 0.30 0.27 1000 0.40 200 200 0.00 0.03 7.86 80 1.6 0.30 0.27 1000 0.40 200 200 0.00 0.03 7.88 80 1.6 0.30 0.27 1000 0.40 200 120 0.00 0.03 7.90 80 1.6 0.30 0.27 1000 0.40 200 100 0.00 0.03 7.92 80 1.6 0.30 0.27 1000 0.40 200 140 0.00 0.03 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 7.98 60 1.6 0.45 0.40 201 0.43 140 200 0.03 0.00 8.00 60 1.6 0.45 0.40 201 0.43 120 200 0.03 0.00 8.02 55 1.6 0.45 0.40 201 0.44 100 200 0.03 0.00 8.04 55 1.6 0.45 0.40 201 0.44 100 200 0.03 0.00 8.06 60 1.6 0.45 0.40 201 0.43 100 200 0.03 0.00 8.08 60 1.6 0.45 0.40 201 0.43 120 200 003 0.00 185 8.20 0 70 0.0 1.6 0.00 0.40 8.22 70 1.6 0.40 6.24 70 1.6 0.40 8.26 70 1.6 0.40 0 0.0 0.00 9.74 55 1.6 0.46 9.76 55 1.6 0.46 9.78 50 1.6 0.46 9.80 50 1.6 0.46 9.82 55 1.6 0.46 9.84 55 1.6 0.46 9.86 55 1.6 0.46 0 0.0 0.00 10.50 75 1.6 0.38 10.52 75 1.6 0.38 10.54 75 1.6 0.38 10.56 75 1.6 0.38 10.58 75 1.6 0.38 0 0.0 0.00 10.62 95 1.6 0.33 10.64 95 1.6 0.33 10.66 95 1.6 0.33 10.68 95 1.6 0.33 10.70 90 1.6 0.33 10.72 90 1.6 0.33 10.74 90 1.6 0.33 10.76 95 1.6 0.33 10.78 95 1.6 0.33 10.80 95 1.6 0.33 10.82 95 1.6 0.33 10.84 95 1.6 0.33 0 0.0 0.00 11.76 100 1.6 0.31 11.78 100 1.6 0.31 11.80 95 1.6 0.31 11.82 95 1.6 0.31 11.84 100 1.6 0.31 11.86 100 1.6 0.31 0 0.0 0.00 1246 90 1.6 0.34 12.43 90 1.6 0.34 12.50 85 1.6 0.34 12.52 85 1.6 0.34 1Z54 90 1.6 0.34 12.56 90 1.6 0.34 0 0.0 0 12.70 60 1.6 0.44 12.72 60 1.6 0.44 12.74 60 1.6 0.44 12.76 55 1.6 0.44 12.78 55 1.6 0.44 12.80 60 1.6 0.44 1Z82 60 1.6 0.44 12.84 60 1.6 0.44 0 0.0 0.00 12.90 95 1.6 0.33 12.92 95 1.6 0.33 12.94 95 1.6 0.33 12.96 95 1.6 0.33 0 0.0 0.00 13.02 55 1.6 0.46 13.04 55 1.6 0.46 13.06 55 1.6 0.46 13.08 50 1.6 0.46 13.10 50 1.6 0.46 13.12 50 1.6 0.46 13.14 55 1.6 0.46 13.16 55 1.6 0.46 13.18 55 1.6 0.46 13.20 55 1.6 0.46 0 0.0 0.00 13.36 75 1.6 0.38 13.38 75 1.6 0.38 13.40 75 1.6 0.38 13.42 70 1.6 0.38 0.00 0 0.00 0 0 0.00 0.00 0.36 300 0.41 140 200 0.06 0.00 0.36 300 0.41 140 200 0.06 0.00 0.36 300 0.41 120 200 0.06 0.00 0.36 300 0.41 120 200 0.06 0.00 0.00 0 0.00 0 0 0.00 0.00 0.41 180 0.44 140 200 0.05 0.00 0.41 180 0.44 120 200 0.05 0.00 0.41 180 0.45 120 200 0.05 0.00 0.41 180 0.45 100 200 0.05 0.00 0.41 180 0.44 60 200 0.05 0.00 0.41 180 0.44 80 200 0.05 0.00 0.41 180 0.44 80 200 0.05 0.00 0.00 0 0.00 0 0 0.00 0.00 0.34 350 0.40 140 200 0.05 0.00 0.34 350 0.40 120 200 0.05 0.00 0.34 350 0.40 100 200 0.05 0.00 0.34 350 0.40 100 200 0.05 0.00 0.34 350 0.40 100 200 0.05 0.00 0.00 0 0.00 0 0 0.00 0.00 0.30 493 0.37 200 140 0.00 0.05 0.30 493 0.37 200 140 0.00 0.05 0.30 493 0.37 200 120 0.00 0.05 0.30 493 0.37 160 140 0.04 0.05 0.30 493 0.38 140 140 0.04 0.05 0.30 493 0.38 120 200 0.04 0.00 0.30 493 0.38 100 200 0.04 0.00 0.30 493 0.37 80 200 0.04 0.00 0.30 493 0.37 80 200 0.04 0.00 0.30 493 0.37 200 200 0.00 0.00 0.30 493 0.37 200 140 0.00 0.03 0.30 493 0.37 200 120 0.00 0.03 0.00 0 0.00 0 0 0.00 0.00 0.28 655 0.37 140 200 0.02 0.00 0.28 655 0.37 120 200 0.02 0.00 0.28 655 0.37 120 200 0.02 0.00 0.28 655 0.37 120 200 0.02 0.00 0.28 655 0.37 120 200 0.02 0.00 0.28 655 0.37 140 200 0.02 0.00 0.00 0 0.00 0 0 0.00 0.00 0.31 450 0.38 140 200 0.04 0.03 0.31 450 0.38 120 140 0.04 0.03 0.31 450 0.39 120 140 0.04 0.03 0.31 450 0.39 120 120 0.04 0.03 0.31 450 0.38 120 120 0.04 0.03 0.31 450 0.38 100 120 0.04 0.03 0 0 0.00 0 0 0.00 0.00 0.40 204 0.43 140 200 0.06 0.05 0.40 204 0.43 120 140 0.06 0.05 0.40 204 0.43 100 120 0.06 0.05 0.40 204 0.44 100 120 0.06 0.05 0.40 204 0.44 100 100 0.06 0.05 0.40 204 0.43 80 140 0.06 0.05 0.40 204 0.43 80 200 0.06 0.05 0.40 204 0.43 100 200 0.06 0.05 0.00 0 0.00 0 0 0.00 0.00 0.30 500 0.37 140 200 0.05 0.03 0.30 500 0.37 120 140 0.05 0.03 0.30 500 0.37 120 140 0.05 0.03 0.30 500 0.37 120 140 0.05 0.03 0.00 0 0.00 0 0 0.00 0.00 0.41 170 0.44 140 140 0.06 0.04 0.41 170 0.44 120 120 0.06 0.04 0.41 170 0.44 100 120 0.06 0.04 0.41 170 0.45 100 120 0.06 0.04 0.41 170 0.45 100 140 0.06 0.04 0.41 170 0.45 80 200 0.06 0.04 0.41 170 0.44 100 200 0.06 0.04 0.41 170 0.44 100 200 0.06 0.04 0.41 170 0.44 100 200 0.06 0.04 0.41 170 0.44 120 200 0.06 0.04 0.00 0 0.00 0 0 0.00 0.00 0.34 350 0.40 140 200 0.04 0.00 0.34 350 0.40 120 200 0.04 0.00 0.34 350 0.40 120 200 0.04 0.00 0.34 350 0.41 120 200 0.04 0.00 120m 1.38 km 44ane 640m 40m 920m 600m 240m 60m 60m 160m 186 13.44 70 1.6 0.38 0.34 350 0.41 120 200 0.04 0.00 13.46 75 1.6 0.38 0.34 350 0.40 120 200 0.04 0.00 13.48 75 1.6 0.38 0.34 350 0.40 120 200 0.04 0.00 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 13.60 95 1.6 0.33 0.30 500 0.37 140 200 0.03 0.00 13.62 95 1.6 0.33 0.30 500 0.37 140 200 0.03 0.00 13.64 95 1.6 0.33 0.30 500 0.37 120 200 0.03 0.00 13.66 95 1.6 0.33 0.30 500 0.37 120 200 0.03 0.00 13.68 90 1.6 0.33 0.30 500 0.38 140 200 0.03 0.00 13.70 90 1.6 0.33 0.30 500 0.38 120 200 0.03 0.00 13.72 95 1.6 0.33 0.30 500 0.37 120 200 0.03 0.00 13.74 95 1.6 0.33 0.30 500 0.37 120 200 0.03 0.00 13.76 95 1.6 0.33 0.30 500 0.37 120 200 0.03 0.00 13.78 95 1.6 0.33 0.30 500 0.37 120 200 0.03 0.00 13.80 95 1.6 0.33 0.30 500 0.37 120 200 0.03 0.00 13.82 95 1.6 0.33 0.30 500 0.37 140 200 0.03 0.00 13.84 95 1.6 0.33 0.30 500 0.37 140 200 0.03 0.00 13.86 95 1.6 0.33 0.30 500 0.37 120 200 0.03 0.00 13.88 95 1.6 0.33 0.30 500 0.37 120 200 0.03 0.00 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 13.88 80 1.6 0.25 0.23 1000 0.40 200 140 0.00 0.05 14.00 80 1.6 0.25 0.23 1000 0.40 200 120 0.00 0.05 14.02 80 1.6 0.25 0.23 1000 0.40 200 100 0.00 0.05 14.04 80 1.6 0.25 0.23 1000 0.40 200 80 0.00 0.05 14.06 80 1.6 0.25 023 1000 0.40 200 60 0.00 0.05 14.08 80 1.6 025 023 1000 0.40 200 120 0.00 0.05 14.10 80 1.6 025 0.23 1000 0.40 200 120 0.00 0.05 14.12 80 1.6 0.25 0.23 1000 0.40 200 120 0.00 0.05 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 14.18 50 1.6 0.50 0.45 115 0.45 140 200 0.06 0.00 1420 50 1.6 0.50 0.45 115 0.45 120 200 0.06 0.00 14.22 50 1.6 0.50 0.45 115 0.45 100 200 0.06 0.00 14.24 50 1.6 0.50 0.45 115 0.45 100 200 0.06 0.00 14.26 50 1.6 0.50 0.45 115 0.45 100 200 0.06 0.00 14.28 50 1.6 0.50 0.45 115 0.45 100 200 0.06 0.00 14.30 50 1.6 0.50 0.45 115 0.45 140 200 0.06 0.00 14.32 45 1.6 0.50 0.45 115 0.45 100 200 0.06 0.00 14.34 45 1.6 0.50 0.45 115 0.45 80 140 0.06 0.04 14.36 45 1.6 0.50 0.45 115 0.45 80 140 0.06 0.04 14.38 45 1.6 0.50 0.45 115 0.45 80 140 0.06 0.04 14.40 50 1.6 0.50 0.45 115 0.45 140 200 0.06 0.00 14.42 50 1.6 0.50 0.45 115 0.45 120 200 0.06 0.00 14.44 50 1.6 0.50 0.45 115 0.45 120 200 0.06 0.00 14.46 50 1.6 0.50 0.45 115 0.45 100 200 0.06 0.00 14.48 50 1.6 0.50 0.45 115 0.45 80 200 0.06 0.00 14.50 50 1.6 0.50 0.45 115 0.45 80 200 0.06 0.00 14.52 50 1.6 0.50 0.45 115 0.45 80 200 0.06 0.00 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 14.62 80 1.6 0.25 0.23 1000 0.40 200 140 0.00 0.03 14.64 80 1.6 0.25 0.23 1000 0.40 200 140 0.00 0.03 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 14.90 70 1.6 0.38 0.34 350 0.41 140 200 0.02 0.00 14.92 70 1.6 0.38 0.34 350 0.41 140 200 0.02 0.00 14.94 70 1.6 0.38 0.34 350 0.41 140 200 0.02 0.00 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 15.10 75 1.6 0.36 0.32 400 0.40 140 200 0.02 0.00 15.12 75 1.6 0.36 0.32 400 0.40 120 200 0.02 0.00 15.14 75 1.6 0.36 0.32 400 0.40 140 200 0.02 0.00 0 0.0 0.00 0.00 0 0.00 0 0 0.00 0.00 15.20 65 1.6 0.40 0.36 290 0.42 140 200 0.05 0.00 15.22 65 1.6 0.40 0.36 290 0.42 120 200 0.05 0.00 15.24 65 1.6 0.40 0.36 290 0.42 100 200 0.05 0.00 15.26 60 1.6 0.40 0.36 290 0.43 80 200 0.05 0.00 15.28 60 1.6 0.40 0.36 290 0.43 80 200 0.05 0.00 15.30 65 1.6 0.40 0.36 290 0.42 80 200 0.05 0.00 15.32 65 1.6 0.40 0.36 290 0.42 100 200 0.05 0.00 120m 100m 60m 100m 260m 160m 60m 187 C H A P T E R 9 S A F E T Y MEASURES ON SEA-TO-SKY H I G H W A Y Chapter Eight (8) assigned the Mean Values and the Standard Deviations to the input variables of the MCSD model. In this chapter, the MCSD model is used to examine the geometric design parameters, such as stopping sight distance, curve radius, lateral friction and longitudinal friction, together with their traffic operation characteristics. The Sea-to-Sky Highway performance is evaluated based on these parameters individually or jointly. The input of the MCSD model are as follows: • the highway alignment information, including horizontal curve radii, available stopping sight distances, superelevation and grades, obtained from Chapter Four (4). • the driver-vehicle combination effects, including experiences by expert drivers, normal drivers and design drivers, obtained from various tests outlined in Chapter Eight (8). The performance of the Sea-to-Sky Highway is quantified by the Reliability Index, p, or by the Probabilities of non-compliance resulting from various geometric design parameters. Two Performance Measures (PM) are developed after four driving scenarios are tested. The P profile and the PMs along the roadway are then used to identify unwanted locations for scrutiny. Contingency factors and a number of useful important sensitivity measures are applied and analysed for improving the problematic designs. Suggestions for making the roadway designs more consistent follow. 188 9.1 Four Driving Scenarios The M C S D model is used to determine the system probability of non-compliance for about 10 km of the Sea-to-Sky Highway between L K I 5.58 and L K I 15.32. A total of 185 runs of the model are completed within three minutes. The M C S D model calculates the probabilities of non-compliance in 20 m intervals. At each point, there may exist in parallel up to four non-compliance modes (each mode corresponding to each design parameter). The type of non-compliance and the number of non-compliance modes are determined by the roadway geometry. The probability of non-compliance includes the probability of system non-compliance and the probability of mode non-compliance depending on each individual case. In order to test a design in a simple and informative way, the following four driving scenarios are being tested in this research: • Design drivers on the Sea-to-Sky Highway at design speed • Expert drivers on the Sea-to-Sky Highway at maximum speed • Normal drivers on the Sea-to-Sky Highway at a comfortable speed, defined as the 50 t h percentile survey speed • Normal drivers on the Sea-to-Sky Highway at maximum speed, defined as the 85 t h percentile survey speed Scenario 1 - Design drivers on the Sea-to-Sky Highway at design speed The design speeds for various curves on the Sea-to-Sky Highway range from 30 km/h to 90 km/h. Design drivers are those specified by T A C or A A S H T O . They are considered 189 to be slow and inactive. Their perception-reaction time is 2.5 sec. They operate the Sea-to-Sky Highway at speeds the same as the design speeds of each section along the roadway. While negotiating horizontal curves on the Sea-to-Sky Highway, they limit their lateral friction,^,, to a certain percentage of the maximum lateral friction as shown in Table 8.6, the fy values ranging from 0.17 to 0.10. This ensures that design drivers have enough friction for other manoeuvres such as braking. Design drivers can only reach longitudinal friction,/ , ranging from 0.38 to 0.29, as shown in Table 8.5. Although conservative, this approach represents the current design practice as specified by T A C and A A S H T O . Scenario 2 - Expert drivers on the Sea-to-Sky Highway at maximum speed Expert drivers represent the best drivers and their performance represents the ultimate capability that a driver can reach. This may be thought of as a race car model, i.e., the road is used to its maximum. Expert drivers are capable of handling their vehicle performance in a most efficient, but safe way. They only need 1 sec. to detect and react to any critical driving conditions, such as obstacles in the roadway. While negotiating the horizontal curves on the Sea-to-Sky Highway, they can fully utilize their peak lateral frictions, fy, as indicated by Curve 5 in Figure 8.11. They can also reach their peak longitudinal friction, fxmax, calculated by Equation 8.15. To estimate the margin of safety a curve provided against skidding, the lateral friction at impending skid conditions, which is determined by pavement supply, and the peak lateral friction demanded, which is determined by expert drivers, must be known. 190 Scenario 3 - Normal drivers on the Sea-to-Sky Highway at comfortable speed The 50th percentile of the survey speeds, which were collected on certain sections of the Sea-to-Sky Highway, are considered in this research to be a comfortable speed for normal drivers. Normal driver's ability to handle vehicle performance is considered to be intermediate as it lies between expert drivers and design drivers. They need 1.6 sec. to detect and react to any critical driving conditions (Olson, 1978). To measure the amount of lateral friction demanded by normal drivers and supplied by the pavement, a test was conducted on the Sea-to-Sky Highway from Horseshoe Bay to Squamish and the lateral friction factors were collected by using the g-Analyst, as shown by Curve 6 in Figure 8.11. The longitudinal friction used by normal drivers are about 62 percent of the maximum values (NCHRPR 270). Scenario 4 - Normal drivers on the Sea-to-Sky Highway at maximum speed This scenario is similar to scenario 3, except the 85th percentile of the survey speed on the Sea-to-Sky Highway is considered to be the maximum speed to normal drivers. This group of drivers can be defined as "normal faster drivers", distinguishing them from those who operate the highway at a comfortable speed. The results of the above four scenarios are included in the diskette ZHENG.PHD. It should be noted that the MCSD model only calculates the horizontal and/or vertical curve sections and assesses their functionalities. For tangent sections and four-lane roadway sections, the program does not perform any evaluations. 191 9.2 "Profile" Analysis Figure 9.1 shows an overall "profile" of the probabilities of system non-compliance for the 10 km test section of the Sea-to-Sky Highway. It displays the changes in probabilities of system non-compliance section by section. Figure 9.2, 9.3 and 9.4 is the breakdown of the overall profile, which shows the results of each subsections of L K I 5.58 - L K I 8.0, L K I 8.0 - L K I 11.9, and L K I 11.9 - L K I 15.32, respectively. 192 CM t o CL >-o D ) >-(/I I O T D cu CO c o O) ]> Q Vi o l_ o c cu o (/) 3 o rTTr o * o v *>XI CL (0 X I (/) « o <u Ji> t/i .9-J - O C 3 2 o o x o ° °-5 «/i w M P> I - »- M £ -c ~ > C O O-f-°>E E • W k- I- Q_ <U O O X Q Z Z U ' I ' ' ' ' I ' ' ' I I ' I I I | I I I I | I I I I | I I I I | I I I I | I I I I | I I I I | l l l o o o o o o o o o o • • • • • 0 xepu| Xliuqoney U I O J S A S r m r - o o L CN i n 6 p o o o o o i n cn o cn i n q T q cd r W CO I o CO TH e o M JJ c M s •E •a q oo s 03 C <u u CO s co o\ <u u a W) C! ON TJ O 0_ o D) >-to I o o <u CO c o o> c "C Q q o c fl) O o - > > « C O O -f-5>£ E • M l _ t_ Q . o) o o x o z z u i I i i i i | i i i i | i o o o oo i i i I i i i i | i i i i | i i i i | i i i i | i i i i | i i i o o o o o o • • « • • « <o m •<* K ) C M £f xepu| X A M i q D n e y t u e i s / s q "•o m o 2 i n C o o o q i n q m I i i i i | i i i i o o o ~ d 7 I Ml W is* I O W) J .5 . > ON _t JO o « e O U s o fa OS u s Ml s to ts ON 9.3 Performance Measures Performance Measure (PM) is developed to facilitate the following tasks: 1) to compare what is demanded by normal drivers with what expert drivers can actually reach; 2) to compare what was designed into a roadway with how normal drivers would operate the roadway; 3) to evaluate a roadway on a continuous basis; 4) to examine the consistency of roadway designs in a simple way. Two Performance Measures (PM) are developed in this research. They are the Capacity Ratio and the Operation Ratio. After the PMs at each point along a roadway are calculated, PMs distribution curve and cumulative distribution curve are plotted. The distribution curve indicates what drivers experience or encounter while driving the roadway; and what the road does to each driver or what the road looks like as a whole. 9.3.1 Capacity ratio A "capacity ratio" is a proposed performance measure developed to compare the reliability index, P, for the following two groups of drivers: • Normal Drivers on the Sea-to-Sky Highway at maximum (the 85 t h percentile survey) speed • Expert drivers on the Sea-to-Sky Highway at maximum speed 197 The "capacity ratio" (C-ratio) is defined as: 6 ., n ,. ' Normal driver Capacity Ratio = — "Expert driver This equation indicates that • The "Capacity Ratio" is a ratio of "operation" versus "limit state" at a particular point of interest along the roadway; • The ratio is the Reliability Index, p, (or probability of non-compliance, P„c) for normal faster drivers at the 85 t h percentile survey speed versus the Reliability Index for expert drivers at the maximum speed on the Sea-to-Sky Highway. It defines physical and operational limits. • The closer the ratio is to 1, the more predominant the engineering factors become. When the ratio equals 1, then the contribution of the engineering factors to the probability of non-compliance would be 100%. Even for the expert drivers with a high driving ability, Pnc could not be reduced; or p could not be increased. • When the ratio is smaller than 1, human factors become the governing factors in the system performance. To obtain the capacity ratio for each point of interest (20 m interval) along the 10 km of test section of the Sea-to-Sky Highway, the comparisons are made on P values for normal drivers on the Sea-to-Sky Highway at maximum speed and expert drivers on the Sea-to-Sky Highway at maximum speed. 198 A computer software program Ratio.for is developed to facilitate the above comparisons and the ratio distribution curve is shown in Figure 9.5. It indicates the change of the ratios every 20 m. The cumulative probability curve for ratios is shown in Figure 9.6. It indicates that the mean value and standard deviation of the ratios are pi Cratlo = 0.85303 and o c.ralio = 0.17885, respectively. 9.3.2 Operation ratio An "operation ratio" is another performance measure developed to permit the comparison of the reliability index, P, between the following two groups of drivers: • Design drivers on the Sea-to-Sky Highway at design speed • Normal drivers on the Sea-to-Sky Highway at a comfortable (the 50th percentile survey) speed The "Operation Ratio" (O-Ratio) is defined as: Operation Ratio = ^De"8" d n v e r "Normal driver This equation indicates that • The "Operation Ratio" is a ratio of "design" versus "operation" at a particular point of interest along the roadway; • The ratio is the reliability index, P, for design drivers at design speed versus the reliability index for normal drivers at comfortable speed on the Sea-to-Sky Highway. It quantifies margin of safety built into the design. 199 H 3 0> P* B £ £ ••-» o #> **« Q t: a-CA VI U Q "« E J -o ti I s u e o •(-> 3 o o CN « CU U • ON ti U s • When the ratio is less than 1, the closer the ratio is to 1, the safer the design becomes. • When the ratio equals 1, there is no variation between the design and its operation. • When the ratio is greater than 1, the larger the ratio is, the more margin of safety is built into the design and the more cost is involved in the design as well. To obtain the operation ratio for each point of interest (20 m interval) along the Sea-to-Sky Highway, the comparisons are made on p values for both design drivers on the Sea-to-Sky Highway at design speed and normal drivers on the Sea-to-Sky Highway at comfortable speed. A computer software RATIO.FOR was developed to calculate the ratio at each point in order to obtain the ratio distribution curve. As shown in Figure 9.7, the ratio distribution curve indicates the change of the ratios every 20 m. A computer software of RANK.FOR was also developed to rank the ratios in order to obtain the cumulative probability curve for ratios. As shown in Figure 9.8, the cumulative curve indicates that the mean value and standard deviation of the rations are M o-Ratio = 0.94551 and a 0 . R a t i 0 = 0.18876, respectively. Both RATIO.FOR and RANK.FOR are included in the diskette ZHENG.PHD. It can be seen that driving at a design speed, the design drivers react to the highway differently from normal drivers. If the roadway is built to accommodate expert drivers, the reliability of the assumed design drivers is much lower than the normal drivers. The P profile and the two PM presented in this section will be used to identify the operation deficiencies experienced by drivers on the 10 km of test section of the Sea-to-Sky Highway. 202 q CN • 0) W <D > CL 00 T J 0) 0) c ° .2>E OT o H c o .2>E OT l -0) O Q 2 ts "liiriiiiiiiiiiiiiiin II D 1 » a £ .5* o «8 jf5 S> «*- 7 3 £ ® .2 " .9 .5 I T ) »o Hllllllllfri 1 O N o II a II / r-vo >T) o I •'»'I'1111111111111111111111111111111111111111111111111111111111111111111111II1111111111111111 o c n o o r ^ c D i O ' ^ - K ) ( N T -• • • • • « . . . . • • - 0 0 0 0 0 0 0 0 0 uoiinquisjfj aAjiDinLuno C O .CN O Q0_ UP 00 o* CD O o" q o 9.4 Operation Deficiency The following two sections focus on a "roadway analysis" and "dangerous location analysis". The "roadway analysis" implies an analysis from point to point suggesting an overall evaluation. It chose a 3 km of roadway as its study object which covers all failure modes. Some sections may have one failure mode and some sections may have up to 4 parallel failure modes. The "dangerous location analysis" implies a scrutiny for unsafe or unwanted designs. It chose all locations which have the lowest safety level as its study objects. 9.4.1 Roadway analysis - "All case" study This analysis focused on one section of the roadway (LKI 5.68 - L K I 7.92) which covers all geometric conditions and all potential non-compliance failure modes resulting from geometric designs. Within the following 2.4 km of the roadway from L K I 5.68 to L K I 7.92, drivers are exposure to various geometric design conditions: some curves are tight, some curves are generous; some are horizontal curves, some are vertical curves; some are on tangent, some are on both horizontal and vertical curves; some provide sufficient sight distance, some supply inadequate sight distance. Some are sub-standard but thought to be appropriate at the date of construction. Due to roadway constraints, some sections of the roadway do not have any potential geometric hazards, such as tangent sections or 4-lane roadways (assumed, since this research focuses on two-lane roadways only). Some sections may potentially result in "off-road" accidents due to the 205 presence of very tight horizontal curves that drivers have to negotiate, particularly, when drivers do not have sufficient sightline to react to "unexpected" driving conditions. Drivers may have to take every effort to maintain their lateral friction fy and/or longitudinal friction fx lower than what the curves supply to them, otherwise, non-compliance failure mode 1 and/or mode 2 may result (refer to Chapter 7 for failure mode definition). Some sections of the roadway may potentially result in "rear-end" accidents, because of very steep vertical curves (mode 4). The driver's visibility is impaired by the presence of the uphills; or "off-road" and "read-end" accidents due to the presence of both horizontal and vertical curves (mode 3 and mode 4). To study the changes in p values in relation to the four driving scenarios, comparisons of p values are made. As shown in Table 9.1, the "Rank" column displays the comparison results among all P values and their relationship in terms of safety level. The comparisons are made at 20 m intervals. 206 Table 9.1 All case study - Comparisons of P values for four driving scenarios Comparisons of the reliability indexes, p, for four 4 driving scenarios ® l.kl I N O I l - C O I l i p i h l l l t C : Scenario 1 Scenario 2 Scenario 3 Scenario 4 Rank Modes P, P, P3 P4 5.68- 5.72 9.70553 9.52232 9.26599 8.54007 P,>P;>Pi>P< 1,2 40m tangent 5.76- 5.80 7.49740 7.71743 6.89419 7.20758 p2>Py>P,>Pi 5.82 9.68218 9.52247 9.29307 7.03076 P;>P;>Pi>P, 5.84 9.72211 9.58263 9.37618 6.73580 5.86 6.90559 7.06057 5.94302 6.30407 P,>P/>fc>P* 5.88 - 6.07327 6.26247 4,92814 5.03602 1,2, 3,4 5.90 5.45295 5.80759 4.49292 4.70585 5.92 5.45354 5.80815 4.49367 4.70602 5.94 6.28396 6.59551 5.44611 5.71238 5.96 5.45354 5.82053 4.50149 4.98276 260m tangent 6.22 9.62971 9.42143 5.71315 6.11531 P/>P;>P<>Pi 6.24- 6.28 5.44095 5.95016 4.86630 5.32157 P*>P/>P^>Pi 6.30- 6.32 9.64817 6.31528 5.22372 5.61956 P/>P;>P,>PJ 1,2 6.34- 6.36 4.35147 4.93649 3.71203 4.22681 P2>P/>P<>Pj 6.38 5.44905 5.95016 4.86630 5.32157 100m tangent 6.48- 6.52 9.67297 9.50727 9.35735 7.01326 P/>P 2 >P3>P< 1,2 120m tangent 6.64- 6.66 9.63694 9.44460 6.05144 9.54049 P/ > > P2 >Pi 6.68- 6.72 9.53946 6.33369 5.37884 5.92757 P, > P2 > 0, >p3 6.74 9.69440 9.53674 6.33498 6.68075 6.76 9.69889 9.56793 9.39026 9.06108 P/>P;>P;>P, 1,2 6.78- 6.80 9.69440 9.53674 6.33498 6.68075 P / > P 2 > P , > P 3 6.82- 6.84 9.53946 6.33369 5.37884 5.92757 6.86- 6.88 4.95151 5.57080 4.49385 5.10965 p2>p,>p/>p5 6.90 9.65989 9.51441 9.31799 8.99994 p;>p2>p3>p, 580m tangent 7.48 5.55003 6.312971 5.10149 6.78411 Pv>P;>P, >P. 3 7.50 4.82288 5.67983 4.33650 6.20647 160m tangent 207 7.66 7.68 - 7.70 7.72 5.55001 4.82288 5.55001 6.31967 5.67983 6.31967 5.10174 4.33050 5.10174 6.99908 6.45374 6.99908 P,>P2>P, >p, 3,4 100m tangent 7.82 - 7.84 7.86 7.88 7.90 7.92 6.94551 6.94551 4.82288 3.88283 5.55003 7.50444 7.50444 5.67983 4.82877 6.31971 6.82899 7.06277 4.99653 3.70246 5.39208 8.12160 8.26341 6.94100 5.98927 7.20769 P,>P;>P/>P. P,>p2>Pj>Py P,>P2>P;>P5 4 Note®: Scenario 1: Design drivers at design speed on the Sea-to-Sky Highway Scenario 2: Normal drivers at comfortable (the 50*%-ile survey) speed on the Sea-to-Sky Highway Scenario 3: Normal drivers at maximum (the 85th%-ile survey) speed on the Sea-to-Sky Highway Scenario 4: Expert drivers at maximum speed on the Sea-to-Sky Highway L K I 5.68 - 5.72: Operating conditions between L K I 5.68 and 5.72 are as follows: • A l l drivers are negotiating a tight horizontal curve (R = 120m) and the sight distance available to them is 140 m. • Design drivers are operating at a design speed of 33 km/h with a perception-reaction time of 2.5 sec. • Normal drivers are operating at a comfortable speed of 48 km/h and a maximum speed of 62 km/h, respectively, and their perception-reaction time is 1.6 sec. • Expert drivers are operating at a maximum speed of 86 km/h with a perception-reaction time of 1.0 sec. System performance over this section of the Sea-to-Sky Highway is estimated to be: • The highest safety levels are reached by design drivers who operate at design speeds which are very low. 208 • Safety level increases in a consistent way from normal faster drivers (P^ to normal drivers operating at a comfortable speed (p2) given that the available sight distance just meets the design standard. • There is a significant difference in the safety level between normal drivers and expert drivers when both are driving at the maximum speed. This is because the normal faster drivers operate their vehicles at a speed of 12 km/h lower than expert drivers. L K I 5.76 - 5.96 After a 40 m tangent section, all drivers are entering the second curve. The curve is not only tight (R = 172m), but also on an uphill (G = 0.03). Sight distances available to drivers vary from 200 m to 80 m depending on which sections they are driving. Operating conditions between 5.76 and 5.96 are as follows: • Design drivers are operating at a design speed of 40 km/h with a perception-reaction time of 2.5 sec. • Normal drivers are operating at a comfortable of 55 km/h and a maximum speed of 70 km/h, respectively, and their perception-reaction time is 1.6 sec. • Expert drivers are operating at a maximum speed of 95 km/h with a perception-reaction time of 1.0 sec. System performance for this section of the Sea-to-Sky Highway is estimated to be: • Safety levels decrease dramatically from the previous section of L K I 5.68 - 5.72 to this 209 section of L K I 5.76 - 5.96. This may be because the presence of both horizontal and vertical curves which makes the driving task more difficult. Unlike dramatic changes in the safety level for both design drivers and normal drivers, expert drivers can "minimize" this change by the control they have of their vehicles. • From 5.82 to L K I 5.84, the highest safety levels are again reached by the design drivers as they operate the curve at the lowest speed. From L K I 5.76 to L K I 5.80 and from L K I 5.86 to L K I 5.96, the safety levels for normal drivers at a comfortable speed (P2) are greater than design drivers at the design speed (P,). This is probably due to the limited sight lines associated with uphill operation and normal drivers reaction time which is 0.7 seconds quicker than the design drivers. Similar conclusions can be made that the safety levels reached by expert drivers (P4) are greater than those of normal drivers (P5) when both operate at maximum speeds. Also expert drivers' reaction time is 0.6 sec. shorter than those of normal drivers. L K I 7.82 - L K I 7.90 Unlike the previous two sections, expert drivers in this section reached a higher maximum safe performance level than any other group of drivers. In this section, the curve radius is about 1000m, and the only problem that drivers have to face is the vertical curve which results in insufficient sight distance. Expert drivers also have much better skills than those of any other driving groups when emergency stops are required. The safe performance level, in this case, depends more on human skills and less on site factors. 210 It can be concluded that • when a curve is built with insufficient sight distance either horizontally or vertically, or both, the highest safety levels are to be reached by the expert drivers because they have the best strategy and capability to handle the emergency stops when necessary. • when a curve does not present any sightline problems to the drivers, the highest safety levels are to be reached by the design drivers because they attempt to use the lowest speeds to negotiate the curve. • when a curve is built with a small radius, say less than 300 m, the highest safety levels are reached by expert drivers because of their good driving skills. 9.4.2 "Dangerous location" analysis Dangerous locations referred to those where the safety level reached is the lowest. For example, the Reliability Index of p value for the system is lower than 3.5. p of 3.5 corresponds to the probability of design non-compliance, P„c, of 0.23 X 10~4 (or 23 times in a million occurrence). Figure 9.1 shows that 10 out of 185 locations within 10 km of test section of the Sea-to-Sky Highway fall into this category. The locations are listed in Table 9.2. 211 Table 9.2 Dangerous Locations if P *=3.5 Locations (LKI) P values reached by v Geometric Features Group 1 Group 2 Group 3 Group 4 10.74 2.97 3.37 R = 393m, SSDH = 100m 10.76 1.28 2.46 1.74 R = 393m, SSDH = 80m 10.78 1.28 2.46 1.74 11.78 3.20 3.35 R = 655m, SSDH = 120m 11.84 3.20 3.35 R = 655m, SSDH = 120m 12.56 2.97 3.36 R = 450m, SSDH= 100m, SSDV = 120m 14.02 3.37 R= 1000m, SSDV= 100m 14.04 2.65 2.11 R= 1000m, SSDV = 80m 14.06 0.98 2.06 0.46 3.49 R= 1000m, SSDV= 60m 15.30 3.28 R = 290m, SSDH = 80m Note®: Group 1: Design drivers at design speed on the Sea-to-Sky Highway Group 2: Normal drivers at comfortable (50th %-ile survey) speed on the Sea-to-Sky Highway Group 3: Normal drivers at maximum (85th %-ile survey) speed on the Sea-to-Sky Highway Group 4: Expert drivers at maximum speed on the Sea-to-Sky Highway Dangerous locations can also be identified by Performance Measure (PM), for example, by Operation ratio (O-ratio). A computer software POINT.FOR was developed to facilitate the task of selecting the points whose O-ratios lower than a specified value. Table 9.3 lists all points which have the O-ratio less than 0.7, 0.8 and 0.9, respectively. The software POINT.FOR was included in the diskette ZHENG.PHD. 212 Table 9.3 Dangerous Locations for Specific O-ratio Values O-ratio <0.8 O^ >Ko<0,9 LKI p value LKI p value LKI P value 10.74 0.52 10.66 0.77 6.34 0.88 10.78 0.52 10.72 0.80 6.36 0.88 14.06 0.48 10.74 0.73 6.86 0.89 10.76 0.52 6.88 0.89 10.78 0.52 7.48 0.88 10.84 0.77 7.50 0.85 11.76 0.79 7.66 0.88 11.78 0.74 7.68 0.85 11.80 0.77 7.70 0.85 11.82 0.77 7.72 0.88 11.84 0.74 7.88 0.85 11.86 0.79 7.90 0.80 12.48 0.80 7.92 0.88 12.54 0.78 1.05 0.90 12.56 0.73 1.05 0.88 12.92 0.77 1.05 0.84 i 12.94 0.77 1.06 0.84 12.96 0.77 1.06 0.84 13.64 0.77 1.06 0.82 13.66 0.77 1.06 0.82 13.70 0.80 1.07 0.77 13.78 0.77 1.07 0.82 13.80 0.77 1.07 0.83 13.86 0.77 1.07 0.80 13.88 0.77 1.07 0.73 14.04 0.72 14.06 0.48 Total Locations 3 27 87 2 1 3 At each point, where a "significant" departure from an acceptable reliability level, say (3* = 3.5 as shown in Figure 9.1, or from a desirable performance measure, say O-ratio = 0.8, as shown in Figure 9.7, typical non-compliance are characterized as follows: • Deficient geometric design, such as curve radius less than 150 m, result in abrupt changes in operating speeds, therefore, resulting in unsatisfactory safety performance. • Road sections representing strong inconsistency in vertical curve, combined with those breaks in the speed profile, lead to critical driving manoeuvres. • Sharp curves are introduced at the end of long tangent. Sudden change, therefore, from the areas of flat curvature to areas of sharp curvature can not be avoided. • Design speed and the 85 t h percentile speed are not well balanced resulting in poor tuning between road characteristics, driving behaviour, and driving dynamics. • Superelevation rate is not adjusted to the expected operating speed to accommodate the minor inconsistency in horizonal alignment. 214 9.5 Design Improvements The following examples show that the reliability analysis for the M C S D leads to a more uniform dynamic highway design for safety. This section does not attempt to propose a procedure for implementing such methodologies. It demonstrates how a more consistent and reliable highway design process can be promoted through the fine-tuning of the undesirable locations. The examples also show how the criteria applied to a specific site could be modified in a reliable format to reflect the anticipated highway operations, and where the M C S D model could fit into the existing geometric design process, therefore, achieving a more uniform and reliable highway design process. A summary of the above work is provided at the end of this section. 9.5.1 Importance measures The reliability analysis for M C S D provides the "design point" z*, which gives the most likely values of the input variables i f the performance function for one or more design parameters is not fulfilled. This is also the point of highest probability density in the failure domain. z / = Fz~l (0(P<x,)) Where Fz ( ) is the distribution function for z and the vector a characterizing the design point has unit length. It can be shown that a,, the i'h component in this vector, gives information on the relative importance of the uncertainty arising from variable i. a? can then be interpreted as the fraction of the total uncertainty arising from uncertainty in variable i. 215 Importance measures obtained from MCSD analysis indicates that some variables are not sensitive to some specific failure modes. As shown in Table 9.4, the variables in column 2 do not make great contributions to the failure modes listed in column 1 since their important measures are close to 0. For example, the variable X(8) is the available stopping sight distance on vertical curves. This variable has a very minor influence on the failure modes 1 and 2 because increasing or reducing X(8) would not necessarily reduce the demand (or increase the supply) for the longitudinal friction,/, and for the lateral friction,/^ ,. Similar to the variable X(3) which is the maximum longitudinal friction achieved by expert drivers. As shown in Table 9.4, X(3) has minor influence on the failure modes 3 and 4 because changing X(3) would not necessarily reduce the demand (or increase the supply) for stopping sight distance on horizontal curves. The above findings would facilitate the task of simplifying the MCSD model and adjusting the important design variables to achieve a more consistent highway geometric design. Table 9.4 Importance measures Non-compliance Mode Insensitive Variables Variables can be eliminated 1 X(4), X(5), X(6), X(8) X(4), X(6),X(8) 2 X(2), X(4), X(6), X(7), X(8) 3 X(3),X(4),X(5),X(8) X(3), X(4),X(5) 4 X(3), X(4), X(5), X(7) 9.5.2 Sensitivity analysis When an evaluation is made by using the MCSD model and a reliability analysis is presented for 216 the Sea-to-Sky Highway, the question will be asked: "what effect has a change in this parameter for the result?" Such questions are easily answered by using the sensitivity measures, which give the change in probability of non-compliance to a unit change in any input variables. The analysis below is based on the driving scenario of normal drivers operating the Sea-to-Sky Highway at the maximum (the 85th percentile survey) speed, i.e., normal faster driver group. The MCSD output shows that horizontal sight distance design (Mode 3) between LKI 10.74 and 10.78 fails to meet operational requirements and the following design variables play key roles in resulting the relatively lower reliability measures. The variables are X(l), the operating speed (km/h); X(2), perception-reaction time (sec); and X(6), the longitudinal friction factor. The degree of the contribution of the variables to mode 3 follows the order of X(l) > X(2) > X(6). The "normal faster driver" group is considered as a base case. They operate the roadway in a way different from what it was designed for, as shown in Table 9.5: Table 9.5 Key variables in base case Speed, km/h Perception-reaction time, sec. Longitudinal friction design operation design operation design operation LKI 10.74 75 100 2.5 1.6 0.30 0.37 LKI 10.76 80 105 2.5 1.6 0.30 0.36 LKI 10.78 80 105 2.5 1.6 0.30 0.36 The resulting reliability measures for 3 sites are listed in Table 9.5 beside Test 1. To analyse the sensitivity of each individual variable, the following tests are made based on modifying one 217 variable at one time, as shown in Table 9.6. For example, Test 2 used an operating speed of 10% lower than that in the base case; Test 3 assumed that driver's perception-reaction time can be decreased by 10%; and Test 3 used a longitudinal friction value of 10% higher than that in the base case; other variables are the same as those in the base case. Table 9.6 Tests on LKI 10.74 - LKI 10.78 Test Variable Reliability Index, P LKI 10.74 LKI 10.76 LKI 10.78 1 Base case 3.37 1.74 1.74 2 X (1) - 10% decrease in speed, km/h 4.05 2.47 2.47 3 X (2) - 10% decrease in perception-reaction time, sec. 3.81 2.21 2.21 4 X (6) -10% increase in longitudinal friction 3.47 1.86 1.86 9.5.3 P vs. individual design variables To investigate the relationship between the Reliability Index, P, and the individual design variables, a software P-VAR was developed to produce p values corresponding to 5 points of a specific variable. For example, corresponding to variable 1 operating speed, the mean value (p) at L K I 10.74 is 100 km/h, the p-VAR software will give P values for the following operating speeds: p., p. ± lo, p. ± 3o. Where o is the Standard Deviation, which is about 10% of the mean value. Figure 9.9 shows P versus Variables 1, 2 and 6 at L K I 10.74. Figure 9.10 shows P versus Variables 1, 2 and 6 at L K I 14.06. Figure 9.11 shows p versus Variables 1, 3 and 6 at L K I 15.30 Note that the failure modes at these three locations are different: at L K I 10.74, stopping sight 218 if 7.0 6.0 H S.0 4.0 3.0 3 2.0 VartabU 1 -Spwd For th« point 10.74 Vo1 17o,.o"tieo'.b'i,,'w.o ,0(1.0 u U ' v ^ n v * V,-7Skm* V.-lHkm* Speed, km/h if Variable 2 -*ereepdo**e«cIlM«Im« For th« point 10.74 03 VI >f OS e 03 -«-> a *S P* 5, T,-1.5 tec Perception-Reaction Time, sec. if c I Variable 6 -LoottaidliulMaioo For 1h« point 10.: Longitudinal Friction Factor Vartabl* 1 -SpMd For tho point 14.06 -2.0 0.0 1 eo:0"'Vo'o'-^b1-1^ ,oo.o ,,o:o i" ij 2 0: 0 i-^:o-^ 0 Vt-tSlu*k V.-It* km* Speed, km/h 3.0 Variable 2 - rerceptioa-rejictioa time For th« point 14.06 Perception-Reaction Time, sec. IV-ZJi Vartabl* 6 - Lootftudlrud friction For th« point 14.06 f.-tJT Longitudinal Friction Factor distance on a horizontal curve fails to meet the operational requirement; at LKI 14.06, stopping sight distance on a vertical curve fails to meet the operational requirement; at LKI 15.30, both longitudinal and lateral frictions exceed the supply by the highway. The software P-VAR is included in the diskette ZHENG.PHD. 9.5.4 Location improvements If a P value is the result of using the design value for a design variable, call it as Prf; If a P value is the result of using the operation value for a design variable, call it as P„. The following criterion is met: If Po < Pd, variable justification is required If Po > Prf, no variable justification is required As analysed above, LKI 10.74, LKI 14.06 and LKI 15,30 have the lowest safety levels over the 10 km of the Sea-to-Sky Highway between LKI 5.58 and LKI 15.32: LKI 10.74 TMode 4): As shown in Figure 9.9, p0 < $d exists for variable 1; P0 > prf is met for both variables 2 and 3. For variable 1, as the operation speed of 100 km/h exceeds the design speed of 75 km/h, justification is made by reducing the operation speed to 80 km/h. Re-run the P-VAR software for point 10.74 yields the improved P-VAR curves, as shown in Figure 9.11. For speed, PG - $d due to the operating speed being reduced by 20 km/h. This change also results in significant improvements in reliability associated with the variables 2 and 3. 221 7.0 Variable 1 - S p w i Speed, km/h Longitudinal Friction Factor Speed, km/h Perception-Reaction Time, sec. Longitudinal Friction Factor LKI 14.06 (Mode 3): As shown in Figure 9.10, P0 < P,, exists for variable 1; P0 > prf is met for both variables 2 and 3. For variable 1, as the operation speed of 100 km/h exceeds the design speed of 65 km/h, justification is made by reducing the operation speed to 70 km/h. Re-run the P-VAR software for point 14.06 yields the improved p-VAR curves, as shown in Figure 9.12. For speed, P„ - Prf due to the operating speed being reduced by 25 km/h. This change also results in significant improvements in reliability associated with the variables 2 and 3. Replacing the operating speeds with 80 km/h at LKI 10.74 and 70 km/h at LKI 14.06 and increasing curve radius at LKI 15.30 to 350 m (from 290 m), re-run the MCSD model for the entire 10 km of the Sea-to-Sky Highway yields a more smooth P distribution curve, as shown in Figure 9.13. The system reliability index P corresponding to these points are improved. All P values are greater than P* = 3.5. 9.5.5 Omission analysis Lin (1989) showed that the reliability index is increased by a factor 1/M 1 - a, 2 , called omission sensitivity factor, if the uncertainty in variable / is ignored and the variable is replaced by its median value (50%). In a process of deciding on variables to reduce the overall uncertainty, these parameters obviously give very useful guidance. The omission sensitivity factors can be used to calculate the effect of replacing a random variable by a fixed value. As shown in the previous analysis, variable 1, vehicle operating speed, is the 224 most important factor contributing to all failure modes with respect to the final result. For example, replacing variable X ( l ) at each point on the Sea-to-Sky Highway with their Mean Values yields a improved P n e w distribution curve. Figure 9.14 shows the comparison between P existing &Tid P new • 9.6 Summary Figure 9.15 is the flowchart, which outlines how the design criteria, when applied to a specific site, could be modified in a reliable format to reflect the anticipated highway operations. The following conclusions can be made: • The reliability analysis method is well suited as an evaluation tool for highway performance to be taken in the face of many uncertainties. • The reliability analysis is a valuable method as it automatically provides importance and sensitivity factors together with contingency factors. • The reliability analysis is ideal in a study of the effect of different design alternatives and changes in input parameter values. Such effects are determined directly by simple hand calculations, whereas other design methods would require a complete analysis. • The method is very useful in forcing the analyst to be systematic in uncertainty assessment and thus reducing the risk of overlooking important items. • Scenarios corresponding to a specified "worse case", 'expected case" and "best case" are easily determined based on probability levels corresponding to the scenarios. 225 xopuj X | | | iqDi|oy tuejsAs Highway geometric design parametres - S S D at each point - Curve radii ^ujaej^ejevatiorusnd^rate Operation characteristics - operating speed - friction factor demands Run MCSD model Perform overall roadway evaluation Reliability Index profile for an entire highway Performance measures - Operation Ratio (O-ratio) - Capacity Ratio (C-ratio) Set up a criterion for an acceptable design, p"*=3.5, orO-ratio =0.8 List all unwanted designs Modify unwanted design Sensitivity I Omission analysis 1 analysis i Check M C S D output for - design point on the failure surface - key variables and their contribution to system failure Contingency factor analysis Identify the effect of replacing a random variable by a fixed value Quantify the margin of safety associated with design alternative Figure 9.15 Integration of the Existing Design Process with MCSD Model AW varaibles C H A P T E R 10 CONCLUSIONS AND F U R T H E R R E S E A R C H In the driver-vehicle-roadway dynamic system, because of the difficulty of representing the driver in an analysis, and the driver's characteristics which are not subject to direct control, this research developed an alternative model, called the MCSD (Moving Coordinate System Design) model. The braking and cornering friction generated between vehicle and roadway are used to analyse a vehicle's stability, control and performance in the highway geometric design stage. The model originates from the existing geometric design process, further developed with vehicle dynamic considerations, quantified by a joint performance function characterizing cornering manoeuvres, and solved for the probability of design non-compliance. As a technique, it links the concerns of highway geometric design and vehicle dynamics. This chapter provides comments on the applications of the MCSD model and summarizes its merits and shortcomings. Further research follows the proposed the reliability-based MCSD geometric design process. 10.1 Conclusions There are many issues concerning the development, presentation, adequacy, and usability of design standards for highway geometric design. This research not only addresses a majority of these issues, but also provides different perspectives on them. The methodology is a step 229 towards harmonizing the design approach and, more importantly, allowing a check and feed-back of highway operational information at the highway geometry design stage. Unlike the existing geometric design procedure, the new approach requires that design input be explicitly associated with a mean and a variance. It incorporates the dynamic features of driver, vehicle and roadway and provides a consistent approach to design reliability. Comparable levels of reliability can be calculated in a format so that all design results are equitably accounted for in the system analysis. The advantages of incorporating the MCSD model into highway geometric design are as follows: • It formalizes and organizes the ideas of dynamic highway geometric design for safety. • It makes the best use of the existing photolog information through dynamic data processing procedure for highway rehabilitation. • It allows changes in parameters which are related to driver, vehicle and roadway. Human factor consideration, operational experience and geometric design values are applied in the MCSD model. • It attempts to describe both vehicle mechanics and civil engineering characteristics. The model is represented by a set of equations which permit a control of the input from driver, vehicle and roadway. It measures the effect of the vehicle-roadway combination experienced by drivers. • It provides a means to identify the important factors, the way in which they operate, and under what conditions, so that changes necessary to reach a given performance goal can be identified at any specific point along the highway. 230 MCSD is a complex model because of the large number of variables that are involved. It forces the designer to confront all the variables that may influence the performance, and recognize all important factors that may play a key role in his design. 10.2 Reliability-based MCSD Process The reliability-based Moving Coordinate System Design process can be summarized as follows: 1. Assess Highway Performance • Analyse accident and traffic data to identify any accident patterns that may exist. • Determine the probable causes of accidents with respect to drivers, highways and vehicles. 2. Evaluate the Existing Geometry • Determine the existing geometry through the development of the methodology and computer software necessary to transform the digitized roadway data into a highway alignment data base. Provide on-line information on the radius, length, superelevation, etc. through the developed software. • Plot highway geometric design profile versus safety performance. • Identify the location of segments with sight distance restrictions due to horizontal and vertical alignment. 3. Collect Data Necessary to run MCSD Model • Perform an in-depth study of the random variables in all geometric design 231 parameters. • Provide estimates of the mean value and the standard deviation on each design variables. • Collect data for each design parameter in a format of supply versus demand. Demand is generated by normal drivers, supply is provided by design standards, and capacity is reached by expert drivers. 4. Perform MCSD Analysis • Input site specifications to MCSD model to identify features that pose safety hazards under common operating conditions. • Check the prevailing speed at approaches to horizontal curves or hill crests with possible SSD restrictions. 5. Evaluate Safety Consequences of Various Situations to Account for Uncertainty • Determine the most probable relationship between safety and a specific geometric feature. • Perform sensitivity analysis to investigate how the design elements affect one another, and to what degree they affect the margin of safety associated with each design. • Calculate the margin of safety or probability of non-compliance for the existing condition. • Propose alternative countermeasures to alleviate the deficiencies. • Calculate the margin of safety for each improved design. • Analyse safety and cost trade-off. 232 Design criterion should reflect the anticipated operation of the highway, the reliability-based Moving Coordinate System Design process, therefore, should be documented as follows: • List all feasible design options for correcting safety problems. A design option becomes feasible only when it is subject to the anticipated operations of the highway. • Evaluate safety impact on different design options and quantify the costs associated with each design alternatives. • Recommend the optimal solution for implementing the design. 10.3 Further Research Quantifying and analysing variability of geometric design parameters and operation characteristics are fundamental concerns in developing the reliability-based Moving Coordinate System Design model. To evaluate reliability on a system performance, the model requires tremendous data input. Although many challenges have been addressed, there are still hurdles to overcome with highway designs and further research needs to be undertaken. One should: 1. Assign Coefficient of Variation to Design Input Currently, there is limited data available for quantification of variations associated with construction activities, traffic operations, and environment. Therefore, previous experience, along with experiments is used to assign the mean values and the standard deviations to the variables. This immediately introduces a possible bias when 233 attempting to compare highway geometric designs. 2. Compare Design Alternatives Every attempt is made to apply design concepts consistently between design alternatives; however, no two design alternatives would be alike because of inherent differences in theory. For example, how is a designer to know whether the probabilities of non-compliance between two designs have equivalent variableness? An even more complicated question arises: How is a designer to know whether the probabilities of non-compliance in stopping sight distance are the same as probabilities of non-compliance in curve tightness? This question, which present a very complex issue, currently has no answer with design procedures used by the industry. 3. Assign Non-Compliance Criteria for Condition Measurements Designers, currently, have no knowledge on the predicted performance between two design alternatives. For example, if one design fails in providing sufficient stopping sight distance and another design fails in meeting the minimum radii requirement for horizontal curves, how does the designer know that the predicted performance between two design types are equivalent? 4. Determine the Wanted Reliability Level for Roadway Design The question of an acceptable chance of design non-compliance is at the root of the entire reliability method. Structure design according to Hart (1981) accepts a chance of 1 in 10,000 for an ultimate failure and 1 in 100 for a service failure. 5. Develop an Algorithm to Produce a Three Dimensional Alignment Automatically This will open new possibilities for a quick design analysis, such as MCSD analysis. 234 Algorithms can be used to calculate important alignment characteristics such as sight distance, curvatures and friction factors. Algorithm can also be used to calculate projected reliability for various roadway parameters, an aesthetic rating of a section of highway, overall design consistency ratings, three dimensional viewing of the alignment, and correlation coefficients between various alignment aspects since all necessary data is available immediately. The designer will be able to experiment with different alignments quickly to compare various factors, monitor the influence that modifications to the alignment have upon these characteristics. The investigations should also include: a. processes for evaluating horizontal design consistency and inconsistency and processes for evaluating design speed and operating speed differences. b. driver speed behaviour on horizontal curves and associated road geometry. A review of past work on this subject should be undertaken before commencing experimental design and data collection. This would highlight the problem areas most in need of research and give an indication of the direction the research program should take. c. speeds, involving approach speeds and curve speeds. These studies are required to ascertain whether current practice leads to high speed curves having a discrepancy between design speed and free operating speed. d. differences in driver behaviour for curves of different severity. This study is required to clearly identify driving dynamics in relation to specific geometric design. The verification of this identification could lead to a greater understanding of driver behaviour 235 as it pertains to road design. The further development of a design tool utilising the model developed in this research is a project of large proportions and possibilities. Project design is important to ensure the continuity of research and development as well as setting realistic intermediary goals to test and refine the concepts. An approach to further development could be: Preliminary development: produce a tool that could be used to analyse and enhance existing alignments and yield insight into current design methodologies; Secondary development: produce a tool that could be used to evaluate the existing or newly produced alignments. The MCSD model forms the basis for this development. Tertiary development: produce a tool that could be relied upon to produce usable designs. It seems clear that the methodology developed in this research represents a fundamentally different approach to the current design process. The real power and usefulness of this approach will only emerge after a working design tool is developed. This research suggests: 1. The principle of dynamic geometric design for safety is that geometric design should minimize the total expected cost of a highway, being the sum of certain initial cost and the expected cost of design non-compliance. 236 2. Geometric design should be based on traffic operation characteristics and driver's expectations. 3. The human element in the system is largely a "given" around which the other elements must be designed and operated. 4. The most effective way to improve traffic safety is to increase the perceived danger in driving situations and to reduce the level of danger that the driver will tolerate. The latter refers to highway geometric design improvement. A well-designed research program should be developed using standard methodologies. This would facilitate the task of implementing the updated research work into the current design process. It would also form a representative body of knowledge about dynamic highway geometric design, which would promote optimization, design consistency, and operational effectiveness. 237 BIBLIOGRAPHY AASHTO: A Policy on Geometric Design of Highways and Street, American Association of State Highway and Transportation Officials. Washington, D .C, 1994. Ang. A.H. and W.H. Tang: Probability Concepts in Engineering Planning and Design, Vol.1, John Wilev & Sons. Inc.. New York, 1975. Badeau, N. and Bass, K.G.: Safe Speed Determination in Curve, Proceedings of the Canadian Multidisciplinary Road Safety Conference IX. 1995. BC Design Manual, Ministry of Transportation and Highways of British Columbia. 1993. 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Felipe, E.: Reliability-based Design for Highway Horizontal Curves, Master degree thesis. 1996. Fitzpatrick, K. : Horizontal Curve Design: An Exercise in Comfort and Appearance. Transportation Research Record No. 1445. Transportation Research Board. Washington, D.C, 1994. 238 Foschi, R. O., Folz, B. R. and Yao, F. Z.: RELAN: RELiability Analysis user's manual. Department of Civil Engineering, The University of British Columbia, Vancouver, B.C., 1993. Foschi, R. O., Folz, B. R. and Yao, F. Z.: Reliability-Based Design of Wood Structures. Structure Research Series, Report No.34, Department of Civil Engineering, The University of British Columbia, Vancouver, Canada, 1989. Gillespie. T.D.: Fundamentals of Vehicle Dynamics, Society of Automobile Engineers. Inc.. 1992. Glennon, J. and Harwood, D.W.: Identification, Quantification, and Structuring of Two-Lane Rural Highway Safety Problems and Solutions, Report No. FHWA/83/021. 1983. Harr, M. E.: Reliability Based Design in Civil Engineering, McGraw Hill. New York, 1987. Herrin, G. D. and Neuhardt, J. B.: An Empirical Model for Automobile Driver Horizontal Curve Negotiation, Human Factors. 16 (2), 1974. Hulbert, S.: Human Factors in Transportation, Transportation and Traffic Engineering Handbook. 1992. Jackson, L. B.: Designing Safer Roads, Practice for Resurfacing, Restoration and Rehabilitation (RRR), Transportation Research Board Special Report 214. 1987. Lamm, R. et al: Recommendations for Evaluating Horizontal Design Consistency Based on Investigations in the State of New York, Transportation Research Record. No. 1122, 1989. Lamm, R.: Driving Dynamics and Design under Special Consideration of Operating Speeds, Publications of the Institute of Highway and Railroad Design and Construction. University of Karsruhe, Vol.11, Federal Republic of Germany, 1973. Leisch, J. E. and Associates: Dynamic Design for Safety, ITE Seminar Series. 1972. Leisch, J. P.: New Concepts in Design Speed Application, Transportation Research Record. No.631, 1977. Lin, F.B.: Flattening of Horizontal Curves on Rural Two-lane Highways. Journal of Transportation Engineering. Vol.116. 1990. Madsen, H. O., Krenik, S. and Lind, N. C : Methods of Structure Safety. Prentice-Hall, Inc., 1986. 239 McLean J. R.: An International Comparison of Curve Speed Prediction Relations. The paper is based on a literature review commissioned, through University of Calgary, by the Alberta Department of Transportation and Utilities. 1990. McLean, J. R.: Driver Behaviour and Rural Road Alignment Design, Traffic Engineering and Control. Vol. 34, 1987. McLean J. R.: Driver Behaviour on Curve - A Review, Australia Road Research Board. Vol.7, 1974. Michon, J.: Should Drivers Think?, Traffic Research Centre and Institute for Experimental Psychology. University of Groningen, The Netherlands, 1975. Mintsis, G.: Speed Distributions on Road Curves, Transportation Research Group. Department of Civil Engineering, University of Southampton, 1988. Navin, F.P.D.: Reliability Indices for Roadway Geometric Design - Can They be Estimated?, Transportation Research Record 1280. Transportation Research Board, Washington, D.C, pp. 181 - 189, 1990. Navin, F.P.D and Zheng, R.: Geometric Road Design as Limit State. Paper presented at the 6th International Conference on Application of Statistics and Probability in Civil Engineering. Mexico, 1990. Nusbaum, R.: Upgrading Deficient Highways to Expedient Standards Can Be Cost Effective, Public Works. 1985. Olson, P.: Parameters Affecting Stopping Sight Distance, National Cooperative Highway Research Program Report 270. 1984. Raff, M. S.: Interstate Highway Accident Study, Bulletin 173, HRB, National Research Council. Washington, DC, 1993 Sayed, T: and Navin, F.P.D: Identifying accident-prone locations using fuzzy pattern Recognition, Journal of Transportation Engineering, pp. 352 -359, 1995. Sussman, E. et al: Driver Inattention and Highway Safety, Transportation Research Record. No. 1247, 1986. TAC: Highway Geometric Design Manual, Transportation Association of Canada. 1991. Taragin, A: Driver Performance on Horizontal Curves, Highway Research Board. Annual Meeting, 1954. 240 Trietsch, D.: A Family Methods for Preliminary Highway Alignment, Transportation Science. Vol.21, 1987. Williams, J.: A Human Factors Data-Base to Influence Safety and Reliability, Human Factors and Decision Making - Their Influences on Safety and Reliability. 1988. Wilson, T. D.: Road Safety by Design, The Journal of Highway Engineers. Vol.25, 1981 Woodridge, M. D.: Design Consistency and Driver Error , Transportation Research Record. No. 1445, 1994. Wong,Y. et al: Driver Behaviour at Horizontal Curves: Risk Compensation and the Margin of Safety, Accident Analysis and Prevention. Vol.24, 1992. Zegeer C : Safety Effects of Cross-Section Design for Two-Lane Roads, Transportation Research Board. No.1195, 1989. Zheng, R.: Literature Review on Parameter Affecting Stopping Sight Distance, Internal B C MoTH Report. 1991. Zheng, R.: Probability of Design Failure Using Different Stopping Sight Distance, Internal B C MoTH Report. 1991. Zheng, R.: Margin of Safety on Horizontal Curve Design, Internal B C MoTH Report. 1991. Zheng, R.: Evaluation of Margin of Safety on the Island Highway - Malahat Section for different Crest Vertical Curves, Internal B C MoTH Report. 1992. Zheng, R.: How to Enhance Safety through Highway Design and Operation - Sea-to-Sky Highway Application, Internal B C MoTH Report. 1992. 241 / mi-O l£tl Q W a. M SJ w H H C/3 4H4>K 181 Ci — fa NO fa < fa M > > O S Q U -a fl O fa M W CN 00 0 fa CN 1 > 1 CO o 'S fl <u o CO OX) fl k 4 I-I fl f o 13 -o o B Q on U H ON CN O CN fa Pi o fa d i—i o CN O CN fa o fa 1) > u u 1/3 9> 1) 5 ii *a co ii CO o C ' S &, o u O H CN CN fa O fa O fa CO 3 fl , CO ID T3 13 I > T3 .fl co fl co > CO CQ-I •8 o 13 o o H CN CN o fa CQ-I CN CN 

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