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Reliability based highway geometric design Richl, Laurel Anne 2003

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R E L I A B I L I T Y B A S E D H I G H W A Y GEOMETRIC DESIGN by L A U R E L A N N E RICHL B.Sc. (Civil Engineering) University of Alberta, Edmonton, 1988  A THESIS SUBMITTED I N P A R T I A L F U L F I L L M E N T OF THE REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF APPLIED SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Civil Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH C O L U M B I A August 2003 © Laurel Anne Richl, 2003  In presenting this thesis in a partial fulfillment 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 copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It is understood that copying 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, BC V6T 1Z4 Canada August 2003  ABSTRACT  Geometric design of roads and highways involves the calculation of the minimum stopping sight distance (SSD), minimum length of vertical curves and the minimum horizontal radius. Currently in the Transportation Association of Canada (TAC) design guidelines and in design guidelines published by similar associations, the variables to calculate each of the above listed design parameters are specified. For over ten years, the introduction of reliability theory into geometric design has been investigated (Navin, 1990 and 1992, Easa, 2000).  This process has followed the  development of limit states design in structural engineering. This research investigated the probabilities of non-compliance with the design guidelines for stopping sight distance, vertical curve length and horizontal curves. Two scenarios were developed for the application of SSD; the first was an operational condition where vehicles should be able to stop before hitting an object on the road. The second scenario was for injury prevention where vehicles should be able to decelerate to a speed at which the probability of a serious injury to a pedestrian or vehicle occupant is low. Reliability methods were used to develop probabilities of a vehicle being unable to stop within a specified SSD on a level tangent, on a curve and on a downgrade. The injury prevention condition was applied to a level tangent whereas the operational condition was applied to all SSD conditions. Similar to the SSD application, reliability methods were used to calculate the probability of non-compliance for the length of crest and sag vertical curves. For this analysis, the stopping sight distance was used as the sight distance criterion. The horizontal curve application investigated two scenarios. The first scenario calculated the probability of a passenger vehicle travelling around a horizontal curve without skidding-out. The second investigated the probability of a vehicle occupant feeling uncomfortable while travelling around a horizontal curve. Each analysis used RELAN (Foschi et. al., 2002), software developed at the University of British Columbia. The probability of non-compliance, beta and the design point were  ii  calculated for each design parameter. From these analyses, conclusions were made regarding possible applications of reliability theory to geometric design of roads.  iii  TABLE OF CONTENTS DESCRIPTION  PAGE  ABSTRACT ACKNOWLEDGEMENTS CHAPTER 1: INTRODUCTION  ii x 1  1.1  BACKGROUND  1  1.2  PROBLEM STATEMENT  3  1.3  THESIS OUTLINE  4  CHAPTER 2: RELIABILITY AND LIMIT STATES DESIGN  5  2.1  RELIABILITY THEORY  6  2.2  MEASURING SAFETY...!  8  2.3  SOLVING RELIABILITY PROBLEMS  9  2.3.1 2.3.2  Simulations First and Second Order Methods  2.3.2.1 2.3.2.2 2.3.2.3  Mean Value Methods Advanced Methods Errors Associated with FORM and SORM  2.4 RELAN CHAPTER 3: DEVELOPMENT OF RELIABILITY 3.1  DEVELOPMENT OF STRUCTURAL DESIGN EQUATIONS  3.1.1  Application of Probability in Safety Analyses  3.1.1.1 3.1.1.2 3.2  Historical Development of Target Safety Values Design Criteria Based on Performance  T H E U S E OF RELIABILITY IN GEOMETRIC ROAD DESIGN  3.2.1 General Applications 3.2.2 Stopping Sight Distance Applications 3.2.3 Horizontal Curve Application 3.2.4 Highway System Application 3.2.5 Intersection Sight Distance Application CHAPTER 4: GEOMETRIC DESIGN ELEMENTS 4.1  DESIGN SPEED  4.2  STOPPING SIGHT DISTANCE  4.2.1  4.3  4.3.3 4.4  Design Values  20 20 25  26 26 27 29 30 31 32 36 36 39 41  Current Design Practice for Horizontal Curves Horizontal Curves for Passenger Vehicles Vehicle Path  41 43 44  Factors Influencing the Safety of Horizontal Curves  VERTICAL CURVES  4.4.1  18  19  33  Stopping Sight Distance for Trucks Braking Distance on a Grade Braking Distance on a Horizontal Curve  4.3.2.1  16 18  32  HORIZONTAL CURVES  -\ 4.3.1 4.3.2  10 12 14  31  Stopping Sight Distance for Passenger Vehicles  4.2.1.1  4.2.2 4.2.3 4.2.4  9 9  45 46  Calculation of the Length of Vertical Curve iv  48  TABLE OF CONTENTS  DESCRIPTION 4.4.1.1 4.4.1.2  PAGE Length of Crest Vertical Curves Length of Sag Vertical Curves  48 50  CHAPTER 5: MEASURING SAFETY  52  5.1  COLLISION PREDICTION MODELS  52  5.2  PERCEPTION OF OVERLAPPING HORIZONTAL & VERTICAL CURVES  53  5.3  DESIGN CONSISTENCY  55  5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.3.6  Design Consistency Criteria Operating Speed Consistency Vehicle Stability Alignment Indices Driver Workload Speed Prediction Models  5.3.6.1 5.3.6.2 5.3.6.3 5.4  Speed Prediction Variables Two-Dimensional Horizontal Curve Models Three-Dimensional Models  INTERACTIVE HIGHWAY SAFETY DESIGN M O D E L  5.4.1 5.4.2 5.4.3 5.5  „  Policy Review Module Crash Prediction Model Design Consistency Evaluation  DESIGN CONSISTENCY AND SAFETY  55 55 56 57 57 58 59 60 62 63  64 64 65 66  5.5.1 Difference in Operating and Design Speed 67 5.5.2 Speed Reduction '. 67 5.5.3 Impact of Design Consistency on Safety 68 CHAPTER 6: CALCULATION PROBABILITY OF NON-COMPLIANCE & BETA..69 6.1  STOPPING SIGHT DISTANCE  6.1.1  Level Roads  6.1.1.1 6.1.1.2  6.1.3 6.1.4 6.2  70  Operational Standard Serious Injury Prevention Condition  Stopping Sight Distance on a Grade Stopping Sight Distance on a Horizontal Curve  HORIZONTAL CURVES  6.2.1 6.2.2 6.3  69  70 71  74 75 76  Maximum Speed Passenger Vehicles on a Horizontal Curve 76 Maximum Speed on a Horizontal Curve without Feeling Uncomfortable77  VERTICAL CURVES  77  6.3.1 Crest Vertical Curves 6.3.2 Sag Vertical Curves 6.4 VARIABLES USED IN THE ANALYSIS 6.4.1 Vehicle Speed 6.4.2 Perception Reaction Time.. 6.4.3 Vehicle Braking 6.4.4 Relative Truck Braking Efficiency 6.4.5 Speed of a Collision with a Serious Injury v  78 78 .........79 80 82 83 83 84  TABLE OF CONTENTS  DESCRIPTION  PAGE  6.4.6 Maximum Comfortable Lateral Friction.... 6.4.7 Height of Eye : 6.4.8 Headlight Height 6.4.9 Maximum and Minimum Values CHAPTER 7: RESULTS 7.1  STOPPING SIGHT ANALYSIS  7.1.1 7.1.2 7.1.3 7.1.4 7.2  85 85 86 87 88 88  No PDO Condition Injury Prevention Scenarios Stopping Sight Distance on a Downgrade Stopping Sight Distance on a Horizontal Curve  88 91 95 96  HORIZONTAL CURVE ANALYSIS  99  7.2.1 Maximum Speed of a Passenger Vehicle on a Horizontal Curve 7.2.2 Maximum Comfortable Speed on a Horizontal Curve 7.3 VERTICAL CURVE ANALYSIS 7.3.1 Minimum Curvature on a Crest Vertical Curve 7.3.2 Minimum Curvature on a Sag Vertical Curve 7.4  COMPARISON OF THE COMPUTATION METHODS  CHAPTER 8: CONCLUSIONS 8.1  8.2  105  ,  RESULTS OF THE ANALYSES  8.1.1 8.1.2 8.1.3 8.1.4  99 101 -..103 103 104 107 107  Stopping Distances Horizontal Curves Vertical Curves Combination of Geometric Design Elements  USE OF THE RESULTS IN GEOMETRIC DESIGN APPLICATIONS  107 109 109 109 109  CHAPTER 9: RECOMMENDATIONS BIBLIOGRAPHY  112 114  APPENDIX A - FORTRAN FILES USED FOR RELAN ANALYSIS  120  APPENDIX B - RESULTS OF CHI-SQUARED VARIABLE TESTS  136  APPENDIX C - FORM / SORM RESULTS  140  APPENDIX D - MONTE CARLO SIMULATION RESULTS  154  APPENDIX E - ADAPTIVE SAMPLING RESULTS  174  APPENDIX F - COMPARISON OF RESULTS  179  vi  2  LIST O F T A B L E S  DESCRIPTION  PAGE  Table 3.1: Societal Acceptance of Fatality Levels ....23 Table 4.1: Elements of Perception Reaction Time 33 Table 4.2: Summary of Perception Reaction Time 34 Table 4.3: Design Values Used for the Coefficient of Friction 36 Table 4.4: Maximum Values of Lateral Friction Permitted 43 Table 4.5: Changes in the Object Height and Height of Eye 50 Table 4.6 Range in Headlight Heights 51 Table 5.1: Design Consistency Criteria ; 57 Table 5.2: Regression Model Forms for 85 Percentile Speed Prediction Models 59 Table 5.3: Speed Prediction Equations - FHWA Model 63 Table 5.4: IHSDM Design Consistency Criteria 66 Table 6.1: Serious Injury Prevention Scenarios 72 Table 6.2: Vehicle Speeds 81 Table 6.3: Perception Reaction Times 82 Table 6.4: Coefficient of Friction 83 Table 6.5: Relative Braking Efficiency 84 Table 6:6: Impact Speed for Non-Serious Injury 85 Table 6.7: Maximum Comfortable Lateral Friction Around a Horizontal Curve 85 Table 6.8: Height of Eye 86 Table 6.9: Headlight Height 86 Table 6.10: Maximum and Minimum Variable Values 87 Table 7.1: Stopping Sight Analysis Completed 88 Table 7.2: Probability of Not Stopping within Minimum SSD 89 Table 7.3: Probability of a Truck Not Stopping within Minimum SSD 89 Table 7.4: Probability of Not Decelerating to a Speed Unlikely to Cause Injury within the Minimum SSD 94 Table 7.5: Probability of Not Stopping within Minimum SSD on a Downgrade. 96 Table 7.6: Probability of Not Stopping within Minimum SSD on a Horizontal Curve ....97 Table 7.7: Probability of a Car Skidding Out on a Horizontal Curve 101 Table 7.8: Probability of Feeling Uncomfortable on a Horizontal Curve 101 Table 7.9: Vertical Curve Characteristics 103 Table 7.10: Probability of Insufficient SSD on a Crest Vertical Curve 104 Table 7.11: Probability of Insufficient SSD on a Sag Vertical Curve 105 th  vii  LIST O F FIGURES DESCRIPTION  PAGE  Figure 2.1: The Reliability Index 11 Figure 2.2: Measure of Safety 12 Figure 2.3: Failure Surface in the Original Coordinates 13 Figure 2.4: Failure Surface in the Transformed Coordinates 13 Figure 2.5: FORM Failure Surface 15 Figure 2.6: SORM Failure Surface 15 Figure 2.7: FORM and SORM Approximation of a Failure Surface 16 Figure 3.1: Risk Diagram 22 Figure 3.2: Flowchart for Developing Target Reliabilities 25 Figure 4.1 Variations in the Coefficient of Friction 35 Figure 4.2: Effect of Grade on Passenger Vehicle Braking 37 Figure 4.3: Percentage of Coefficient of Friction for Passenger Vehicle on Steep Grades Compared with Flat Sections 39 Figure 4.4: Braking Forces on a Horizontal Curve 40 Figure 4.5: Friction Ellipse 41 Figure 4.6: Sliding Forces on a Horizontal Curve 42 Figure 4.7: Effective Radius on Horizontal Curves 45 Figure 4.8: Crest and Sag Vertical Curves 47 Figure 6.1: Probability of Failure for Serious Injury Scenarios 73 Figure 7.1: Probability of Being Unable to Stop 90 Figure 7.2: Probability of Being Unable to Decelerate to a Speed Unlikely to Cause Serious Injury in a Pedestrian-Car Collision 91 Figure 7.3: Probability of Being Unable Decelerate to a Speed Unlikely to Cause Serious Injury in a Truck-Car Collision 92 Figure 7.4: Probability of Being Unable Decelerate to a Speed Unlikely to Cause Serious Injury in a Car - Car Collision 92 Figure 7.5: Probability of Being Unable Decelerate to a Speed Unlikely to Cause Serious Injury in a Pedestrian - Car Collision, with Excessive Speed 93 Figure 7.6: Probability of a Passenger Vehicle being Unable to Stop Within a Specified SSD on a Downgrade.. 95 Figure 7.7: Probability of a Passenger Vehicle Being Unable to Stop Within a Specified Stopping Sight Distance on a Horizontal Curve 98 Figure 7.8: Probability of a Car Skidding Out on a Horizontal Curve 80 km/h 100 Figure 7.9: Probability of a Car Skidding Out on a Horizontal Curve 50 km/h 100 Figure 7.10: Probability of Feeling Uncomfortable on a Horizontal Curve - 80 km/h...102 Figure 7.11: Probability of Feeling Uncomfortable on Horizontal Curve - 50 km/h 102 Figure 7.12: Probability of Sufficient SSD Provided on a Crest Vertical Curve 104  viii  LIST OF FIGURES  DESCRIPTION  PAGE  Figure 7.13: Probability of Sufficient SSD Provided on a Sag Vertical Curve Figure 7.14: Comparison of the Results From FORM, SORM and Simulation  ix  105 106  ACKNOWLEDGEMENTS  This thesis research was begun under the suggestion of my thesis supervisor, Dr. Frank Navin. For many years, he has explored, with some of his earlier graduate students, the possibility of using reliability concepts in the geometric design of highways. This topic represents a small component of the possible applications in transportation engineering. I would like to thank Dr. Navin for his enthusiastic support of my research. I would like to thank my employer, U M A Engineering Ltd., for allowing me the flexibility of working odd hours so that I could attend classes or complete research part-time during the some days of the week and still work full-time. I would also like to thank U M A for their financial support during the full course of my studies. Other people that I would like to thank include: Dr. Ricardo Foschi and Dr. Felix Yau for his help and assistance with the RELAN program, Dr. Bob Sexsmith for his help with reliability theory, Dr. Tarek Sayed for drawings and Dr. Paul de Leur for data. Lastly, I would like to thank my husband, Jeff, for his continuous love and support throughout my studies. Without it, I could have never completed this thesis.  x  C H A P T E R 1:  1.1  INTRODUCTION  BACKGROUND  The geometric design process is a complex one involving explicit and implicit trade-offs between the costs of the road or highway to society and the benefits the road will provide society after it has been completed. The costs associated with transportation facilities include the capital costs of constructing and maintaining them as well as social costs such as the collisions to society. The costs of collisions to society are enormous. In order to provide society with roads and highways that maximize the benefits to society, explicit calculation of the cost and benefits is required during the planning and design process. To calculate the expected collision costs of a particular road design engineers must ask themselves, "What is the level of safety of a particular geometric element on a roadway?" For years, designers assumed that if the roadway was constructed to published standards it was safe (Navin, 1990 and 1992; Hauer, 1988). The level of safety included in those standards was never explicitly considered or discussed in the standards.  With the  publication in 1999 of the Transportation Association of Canada (TAC) Guidelines for Geometric Design for Canadian,Roads, engineers must now consider explicitly safety during the design process. While roadway designers now explicitly address safety, there is often little information to quantify the level of safety in the TAC Guidelines. Within the TAC Guidelines, the level of safety is discussed in qualitative terms with directions given to the designer in how changes to a particular geometric element is thought to increase or decrease the level of safety on the road. The TAC Guidelines also has a design domain, which provides general upper and lower limits for a geometric design element based on certain conditions. Designer engineers and road owners must decide for themselves how much safety they think a particular road requires.  With scarce resources available to construct and rehabilitate roads and  1  highways, it is important that they are used in the most cost-effective manner to achieve the maximum benefits (Sayed et. al.,1997). In the past, experts determined the level of safety of a geometric design from a review of the design concept during the road safety audit process. The experts did not necessarily agree on how an element of the geometric design should be built, which made completing the geometric design more difficult. Currently, the level of safety provided by a road can be explicitly calculated by using collision prediction models (CPM) that are modified by collision modification factors (CMF). The United States Federal Highway Administration (FHWA) developed the Interactive Highway Safety Design Model (IHSDM). One component of the IHSDM will allow the designer to predict the number of collisions based on their design (FHWA, 2003a). The IHSDM is a complex model that gives the designers a set of CPMs, CMFs and an operating speed prediction model. With these models, the designer can use it to calculate the expected number of collisions, operating speed and assess the design consistency for an alignment. Collision prediction models are developed using generalized linear regression techniques. To keep the models as generalized as possible the number of variables used in the equations is kept to a minimum (Sayed and Sawalha, 1999). As they are a regression models developed from specific data sets, they may not be applicable for every application. A complex transferability process may be required to adjust one CPM, developed from a certain set of data to be transferable to another area (FHWA, 2003a). Similarly, CMFs are developed from specific data sets gathered from before and after studies. Before and after studies are conducted to measure the effect of some change in conditions, often a road improvement that is thought to reduce the number of collisions. These studies have not always properly conducted in the past and as a result, the CMF's are not always an accurate reflection of a countermeasures' ability to reduce the number of collisions (Lin et. al., 2003; Sayed and de Leur, 2002). Even if the before and after study is properly conducted, the transferability of the counter measure from one location to another is not guaranteed. Different driver, climatic or other variables may influence  2  the effectiveness of a countermeasure.  While CMF's have been developed for a large  number of geometric design features, they do not cover all of the design possibilities. Thus, designers can find themselves unable to quantify the expected level of safety of certain design features. The application of reliability theory to geometric design can allow designers to determine the probability of non-compliance as well providing a measure of safety, called beta (Navin, 1992; Felipe, 1996; Zheng, 1997). Reliability theory assumes that a variable used to calculate a design parameter does not have one value, instead the values for that variable can range based on the probability distribution for that variable. Thus when calculating the value for a design parameter, based on many variables, there will be a level of uncertainty whether the design parameter will be exceeded under certain conditions. A performance function is used to calculate whether the design parameter will succeed or whether it will fail. Structural engineers have used reliability theory to treat uncertainties with design parameters and to help them make rational decisions (Thoft-Christensen and Baker, 1982). Using the probability of non-compliance designers could then compare the relative levels of risk associated with a design parameter for various design options.  Today, the  probability of non-compliance is not considered in the TAC guidelines. The relationship between the probability of non-compliance and beta value of a particular design parameter and collision frequency has not been determined. In addition, the relationship between non-compliance and collision frequency may vary depending on the design parameter. 1.2  PROBLEM STATEMENT  In this research, the following questions were investigated. •  Using the equations commonly used to calculate design parameters noted in the guidelines for the geometric design of highways as the basis of the failure functions, what would be the probability of non-compliance, beta and design point values? The  3  design parameters included in this research are stopping sight distance, horizontal curves, vertical curves. •  How is the probability of non-compliance, beta and design point values influenced by different design vehicles and differing pavement surfaces (i.e. wet and dry pavement)?  •  What is the calculated probability of non-compliance for the minimum TAC design value given design parameter?  •  How is the probability of non-compliance, beta and design point values influenced by the combination of design parameters (i.e. stopping sight distance and a downgrade)?  1.3  THESIS OUTLINE  The subsequent chapters of this thesis are organized in the following manner. Chapter 2 is a literature review of reliability theory and methods that have been developed to calculate the probabilities of non-compliance. The Chapter 3 examines how reliability theory was applied in structural engineering.  It also includes a literature review of  applications used in the geometric design of highways. The fourth chapter is a literature review of the geometric design concepts of highways. This chapter will provide the basis for the performance functions developed for this research as well as some background information on the values used to determine the probability distributions for each of the variables. Chapter 5 is a literature review on current methods used to measure the safety of a design. Chapter 6 describes the methodology used to calculate the probabilities of non-compliance, beta values and design points. It also includes a detailed discussion of the probability distributions and the values used to describe the probability distributions of each of the variables used in the analysis. Chapter 7 describes the results of the analyses. Chapter 8 and Chapter 9 are the conclusions and recommendations developed from this research.  4  C H A P T E R 2: R E L I A B I L I T Y A N D L I M I T S T A T E S D E S I G N  In the geometric design of roads and highways engineers calculate the minimum values for geometric design parameters using "conservative values" for the variables. Design standards and guidelines are often used as the basis for all design decisions. Over time, it has been recognized that no roadway can be made so that no collisions ever occur on it. The only completely safe road is one that does not have any traffic on it (FHWA, 2003a). Similarly, in structural design, it was recognized that some amount of risk of unacceptable performance must be considered at some probability. Absolute safety could only be accomplished by employing unlimited resources to design, construct and maintain the structure, which is highly impractical (Thoft-Christensen and Baker, 1982). The principles used in this thesis follow the limit states design approach first used by structural and geotechnical engineers. In general, there are two types of limit states, the first is the ultimate limit states, which means that the system has failed or not complied with the design parameter. The second type is a serviceability limit states, in which the function of the system is impaired. In structural engineering, a system failure means that all or part of the structure has collapsed or is unusable because of overturning or through large deformations. In the geometric design of highways and roads, the concept of a system failure is more complex in that the demand for a particular design element may have exceeded the supply but no crash results from the failure. This is because for some elements of highway design, such as stopping sight distance (SSD), the driver can continue on their trip without any adverse affects. In the event of no crash and without a detailed engineering analysis, it would likely be impossible to tell that the demand exceeded the supply. In the case of SSD, a crash only results when the driver cannot stop within the supplied SSD and there is an object in the path of the vehicle. When limit states design is applied to the geometric design of roads and highways, it requires that designers think of the demand of the driver/vehicle system for a particular design parameter and the supply provided by the highway design. When the supply  5  exceeds the demand, the system is considered to have failed or not complied with the design parameter. 2.1  RELIABILITY THEORY  Ang and Cornell developed the use of probabilistic tools in structural design in the 1960's.  Over time, reliability analysis was developed to the point that it could be  included in structural design codes. This summary of reliability theory is based on the review found in the National Bureau of Standards (NBS) Special Publication 577 on the Development of a Probability Based Load Criterion for American National Standard A58 (Ellingwood et. al.: 1980).  The variable names have been changed from load and  resistance to demand and supply to reflect a more general limit states design rather than an application for structural engineering. Equation 2.1 shows the general mathematical model that is used to represent the limit state. (2.1)  g(x ,x ,x ..jc ) = 0 1  2  3  K  Where: g = Failure surface, with failure occurring when g < 0 Xj = Supply or demand variables A level of safety for the system users can be assumed by calculating the probability of failure of the system. Using the model from Equation 2.1, the probability of failure is given by: (2.2) Where: f = Joint probability distribution function for x ,x ...x . x  }  is performed over the area where g < 0.  6  2  n  The integration  In the simplest system, there is one variable for supply and one for demand. If it is assumed that the S is the supply and D is the demand, then failure is denoted by: (2.3)  g = S-D  In the two variable system from Equation 2.3 failure occurs when S - D < 0.  The  probability of failure of the system is defined by: oo  p =p(S<D)=\F (x)f (x)dx f  s  (2.4)  D  o  Where: p = Probability of system failure f  F = Cumulative probability distribution function for the supply s  f = Probability density function for the demand D  If the both of the variables from Equation 2.4 have normal distributions, then the probability of failure can be denned as:  Pf  =o  V  S-D  2  (2.5)  2  <J +oD  s  Where: S = Mean value for supply D - Mean value for demand cr] = Variance of the supply <y  2 D  = Variance of the demand  0[ ] = Normal probability distribution function  7  While Equation 2.5 looks simple, it is not easy to use. Engineers often do not know the joint probability distribution function for all of the variables and they may not know the probability distribution functions for each variable. The only information that is known with any confidence is the mean and variance of the variables. In addition, many limit states equations are non-linear. With these limitations, the use of Equation 2.5 becomes impractical. 2.2  MEASURING SAFETY  Structural engineers have used a factor of safety in the design equations even before the development of limit states design. Before the development of the limit states design criteria, a single factor of safety was used in the design equations. With the advent of limit states design, the factor of safety used in the design equations is based on all of the supply and demand variables used to determine the value of the design parameter. In the geometric design of highways, an explicit factor of safety is not used in the design equations for a highway. Instead, "conservative" values chosen for the some of the variables, which are used in the design equations with parameters like design speed. The simplest measure of safety is the central factor of safety. It uses the average demand, D and the average supply, S as noted in Equation 2.6. This equation is rarely used.  SF,central  S  (2.6)  D  A more common measure of safety is the conventional factor of safety where the average demand is increased by some multiple of the standard deviation for demand. The average supply is decreased by some multiple of the standard deviation for supply. By using this equation, designers are implying that there is a level of uncertainty in the values for supply and demand.  To be conservative the supply is decreased and the demand is  increased. In Equation 2.7, k is some multiple of the standard deviation, a. S conventional  D  -ka  (2.7)  s  +ka  D  8  Reliability theory can be used to develop factors of safety that incorporate this uncertainty of the supply and demand variables. The resulting factor of safety is called the reliability index, B. 2.3  SOLVING RELIABILITY PROBLEMS  Exact methods to solve reliability equations are not used when there are more than two variables in the failure function. approximate methods.  These problems are solved using simulations, or  The following sub-paragraphs outline the calculation of the  probability of failure using these different methods. 2.3.1  Simulations  The probability of failure and the resulting p value can be solved using Monte Carlo simulations.  The main disadvantage of using simulations is that it can be time  consuming. This was especially true when limit states equations were first developed for structural engineering applications. With simple, limit states equations and modern computers it is not overly time intensive to use Monte Carlo simulations even if millions of simulations are required. In structural reliability problems, the limit states equations can be complex and frequently require the use of other complicated computer programs to determine the inputs for some of the random variables. In addition, the probabilities of failure tend to be very low making the Monte Carlo simulation process very time consuming. The use of adaptive sampling techniques, which are a modification of the Monte Carlo simulation process, can calculate the probability of failure and the associated 8 values. These techniques select values for the random variables that are assumed to be close to the failure surface. Thus, a smaller sample is required to determine the probability of failure. 2.3.2  First and Second Order Methods  In part because of the limitations noted in the section above, first order methods (FORM) and second order methods (SORM) were developed to solve reliability analysis problems.  9  The first and second moments i.e. the mean and the variance of the random variables are used in the analysis, which involves linearizing the limit state equation given by Equation 2.1 into Equation 2.8.  (2.8)  Where: (x*, x* ,.. .x* ) = Linearizing point 2  n  The reliability analysis is then performed using the linearized Equation 2.8. A key consideration in this analysis is the choice of the linearizing point.  The following  sections describe in two methods that have been used to choose the linearizing point. 2.3.2.1 Mean Value Methods  If the linearizing point (x\,x ,...x* )is 2  (x ,x ,...x ), ]  2  n  n  set to the mean values for that point, i.e.  then the mean Z and the standard deviation o~ can be approximated by z  the following equations. Z *  (2.9)  g(x ,x ,...x ) i  2  n  (dg Y  (2.10)  dx  K  tJ  The above approximations are exact if the function g() is linear and all of the variables are uncorrelated.  For other situations, the accuracy of Equations 2.9 and 2.10 are  dependant on the effects of ignoring the higher order terms in Equation 2.8. Equation 2.11 determines the reliability index /3, or safety index. Cornell first used this equation for the reliability index (Thoft-Christensen and Baker, 1982).  10  Beta is a measure of probability that the g() function will be less than 0. It is the distance from the origin to Z in units of standard deviations. If S - D is substituted for Z in Equation 2.11 and if -Jcs] + <j  2 D  is substituted for a  z  in Equation 2.11 then the result will  be Equation 2.12. Figure 2.1 illustrates the concept of 6 and the probability that the g( ) function will be less than 0 for the two-variable function, g = S - D.  P=  S-D 2  S  (2.12)  _1_ +  ^ r a  2  D  Figure 2.1: The Reliability Index  Ba  S-D  Source: Adapted ffomFigure 2.1 Ellingwood et. al. (1980)  Ang and Tang (1984) described a process that can be used to derive the expected value 1  and variance of a design parameter. This process can also be used to derive the measure of safety, M given in Equation 2.13. Figure 2.2 illustrates the concept of the measure of safety. M =  (2.13)  E(S)-E(D)  11  Figure 2.2: Measure of Safety  E(D)  E(S)  2.3.2.2 Advanced Methods  There are two disadvantages of using the mean value methods. The first is that the g( ) function is linearized at the mean values of the variables. When the g( ) function is non-linear the amount of error increases as the distance from the linearized point increases.  This error occurs because higher order terms are ignored. The second  disadvantage occurs because the mean value methods are sensitive to how non-linear g() functions are formulated. The beta values calculated will change depending on how the same function is formulated. The second drawback can be avoided by linearizing the g( ) at some point on the failure surface. Advanced methods for calculating FORM and SORM were developed to minimize the errors noted above. This can be accomplished by linearizing the g( ) function at some point on the failure surface where g() = 0.  This process is accomplished through a  transformation of the original limit states equation and its variables using Equations 2.14 and 2.15. (2.14)  XLZ±L  12  g (x ,x* ,...xl) t  i  l  2  (2.15)  =0  Like the mean value methods, failure occurs when gi < 0. Figures 2.3 and 2.4 graphically illustrate the transformation and show the linearized failure surface. Figure 2.3: Failure Surface in the Original Coordinates  t. x  2  Source: AdaptedfromFigure 2.2 Ellingwood et. al. (1980)  Figure 2.4: Failure Surface in the Transformed Coordinates  x  2  \  / ^  y  / / / \ \ ch \\hc<$\ \  \  \  l\  N. \ \ \ "Survival"")/  /  a?  "Failure"^  Source: AdaptedfromFigure 2.2 Ellingwood et. al. (1980)  Using advanced methods the definition of the reliability index changes from Equation 2.12 to the shortest distance from the origin to a point on the failure surface.  13  This point  (X*,X*,...JC*)  on the failure surface is referred to as the design point or the  checking point. It is found by solving a system of equations listed below.  (2.16)  a, =  (2.17) g (xl,x' ,...x ) = Q  (2.18)  ,  1  2  n  The design point (x*,x*,...x*) will fall in the lower range of the supply variables and the higher range of the demand variables. 2.3.2.3 Errors Associated with FORM and SORM  FORM and SORM are approximate methods that are used to estimate the probability of failure of a limit states equation and the reliability index. Because FORM uses a line to approximate the failure surface, there will be some error associated with the approximation unless the failure surface is actually a plane. Similarly, SORM uses a second order polynomial to approximate the failure surface. Usually there is less error associated with the SORM than FORM because the failure surface is often curved. Figures 2.5 and 2.6 show the failure surfaces and the associated errors when they are approximated by FORM and SORM.  14  Figure 2.5: F O R M Failure Surface  Sometimes the failure surface has a repeating function, similar to a sinusoidal wave, which is not well approximated by either a line or a parabola. When this occurs there should be a substantial difference between the results calculated using FORM and from SORM. In these instances, the results calculated by simulation methods are considered the most accurate. Figure 2.7 illustrates this concept. Figure 2.7: F O R M and S O R M Approximation of a Failure Surface  2.4  RELAN  Reliability Analysis Software (RELAN) was developed at the University of British Columbia (Foschi et. al, 2002).  It uses three different techniques to calculate the  probability of failure: FORM and SORM, response surface approaches and several simulations, including Monte Carlo simulation, adaptive sampling. The response surface can be developed in three ways: approximated from a neural network built from a previously trained database, interpolated locally from a database or from a response surface developed from a quadratic equation, which represents the entire failure surface. The importance sampling uses the design point calculated in the FORM analysis as the best estimate point. The most current version of RELAN, version 6.0, allows the user to use up to nine different probability distributions for the random variables, which can be modified to fit 16  specific maximum and minimum bounds set by the user. The variables may be correlated or uncorrected. The user specifies the degree of correlation between the variables in variable pairs. RELAN contains tools to allow the user to "fit" a probability distribution to a data set. The capacity of the current version is up to 50 random variables and 100 failure modes. The user enters the failure function g( ) into the sub-routine known as GFUN. The variables, which require input from the user are entered using the sub-routine called DETERM. The remainder of the information on the variables, the type of analysis and the number of failure modes are input through RELAN. Both of the sub-routines are programmed in FORTRAN. Other programs may be accessed to determine values for some of the random variables. For this research, only the basic elements in RELAN were used. A l l of the failure functions were input in the GFUN subroutine, variables that required screen input used the DETERM subroutine. The probability of failure was calculated two ways, the first using Monte Carlo simulations and the second using FORM and SORM.  17  C H A P T E R 3: D E V E L O P M E N T O F R E L I A B I L I T Y  Structural engineering was the first discipline of civil engineering to incorporate reliability concepts into design specifications. Researchers spent a great deal of time developing values of beta for different material types. The original beta values were developed for various structures often without complete agreement within the structural engineering community on the probability distributions associated with each of the variables.  In some instances, there was insufficient data available to develop the  probability distributions (Ellingwood et. al., 1980).  The beta values for the same  materials were often calculated differently and the beta values for similar structures were frequently different. Despite the differences in the calculation of the beta values, researchers found that values of beta varied depending on the material, load and load combination type. For wood concrete and steel structures and for dead and live loads due to occupancy they found that the beta values tended to vary between 2.5 and 3.5. For the full spectrum of structures and loadings, the beta values varied from approximately 1.5 to 8.0. They also found that for certain types of live loadings, wind and earthquake loads, the beta values and therefore the factor of safety was much less than for other types of live loads and dead loads (Ellingwood et. al., 1980). As a result, structural design committees decided to change the way that wind and earthquake loads were calculated. 3.1  DEVELOPMENT OF STRUCTURAL DESIGN EQUATIONS  Five materials are commonly used in structural engineering: wood, steel, reinforced concrete, reinforced masonry and aluminium (used in curtain walls). Design with each material is governed by design code that is developed by a committee specifically for that material type. The structural design committees decided to use a similar format for each material type. Despite the differences in the materials, the basic design equation for each material would remain the same. The basic format is shown in Equation 3.1 (Kulak et. al., 1985).  18  </)R>a D + y(p{a L + a Q + a T) D  L  Q  T  (3.1)  Where: < > | = Resistance factor y = Importance factor = Load combination factor a ,a ,a ,a D  L  Q  T  = Load factors for the specified loads  D, L, Q, T = Specified loads (dead load, live load, wind or earthquake load and loads resulting from temperature changes, shrinkage, creep and differential settlements) For the five different material types, the resistance factor varies from material to material. The remaining factors would remain the same for all structures regardless of the type of material. Using the same format for the basic design equation meant developing a family of curves for the <> j factors with the same values for each of the load factors, load combination factor and importance factor (Kulak et. al., 1985). This process involved reversing the processes outlined in Chapter 2 to find a probability of failure and the beta values. In this instance, target beta values were set and researchers worked backwards to calculate the factors found in Equation 3.1. 3.1.1  Application of Probability in Safety Analyses  Probability-based safety analysis in structural engineering is comprised of several activities (Sexsmith, 1999). One of the most common activities is for researchers to determine the probability of non-compliance for different structures or structural systems. A second stream of research is the calibration of probabilistic methods to design codes that will provide a consistent level of safety for all safety criteria. Equation 2.1 is the result of this type of research activity. The third area of research is the use of probabilitybased structural analysis to make design and safety decisions for specific projects. Examples of the third area of research include collision risk for ships and structures, risk  19  mitigation for structures, decisions regarding seismic retrofit and the Confederation Bridge between Prince Edward Island and mainland of Canada. 3.1.1.1 Historical Development of Target Safety Values  Senior code committee members, with little input from the engineers in the design community as well as the community at large, have made the choice of target safety factors. The basis for their decisions has largely been based on their judgement and experience rather than other more rational tools such as decision analysis (Sexsmith, 1999; Aktas et. al., 2001). In developing target safety values, the code committees have often elected to keep the levels of safety similar for different types of structures. These authors argue that the target safety values should be able to account for differences in consequences of failure as well as the costs associated with mitigating the failure. Many structural failures are caused by human error either in the design or in the construction process. Yet, structural codes do not address this problem since failure is assumed to occur only when the loads exceed the structure strength. Human error is a significant drawback to the use of probabilistic based safety analysis (Sexsmith, 1999). Sexsmith proposes one method of accounting for human error through operational controls. Material and load uncertainties, consequences cost and construction costs could be assigned using a decision analysis format. Human and construction error could be controlled by quality control processes, which would also adjudicate the amount of control through a decision analysis framework. Another method to account for problems with data collection to support the probabilities of failure or non-compliance is the use of subjective probability. It allows engineering judgement to be used when making safety decisions. Decision analysis can be supported with subjective probability (Sexsmith, 1999). 3.1.1.2 Design Criteria Based on Performance  The development of reliability design criteria is usually based on calibrating existing structures that have a history of successful performance. The structure should be able to perform well enough against the known failure modes. 20  The maximum allowable  probability of failure should be calculated such that the consequences of the failure are taken into consideration. If there will be severe consequences when a structure fails, it should have a higher target safety value than one that has limited consequences if it fails (Bhattacharya et. al., 2001; Aktas et. al., 2001; Reid, 2002). For conventional structures, the calibration of target safety performance is based on existing structures that have a successful performance history.  Setting target safety values for unusual or novel  structures is much more difficult since similar structures with comparable functions may not exist. Target reliability must be derived from other sources (Bhattacharya et. al., 2001). Reid (2002) notes that should an explicitrisk-basedapproach to standard development be implemented, it would require radical changes in the current structural codes but also in the way that structural engineering is managed. For example, it is expected that a risk analyst rather than a structural engineer would make fundamental design decisions about the adequacy of a structure (Reid, 2002). Bhattacharya et. al. developed a framework for calculating the target reliability for a large marine off-shore base for the US Navy. They used a range of concepts to develop the target reliability for the structure. This included investigating existing codes for off-shore structures, ships, structural codes from different countries and standards organizations. Figure 3.1 shows the levels of risk based on the probability of failure and the consequences of the failure. The following paragraphs describe in greater detail some of the analytical techniques that were investigated during their work.  21  Figure 3.1: Risk Diagram  Unacceptable Region 10" '. Mine Pit .N, Slopes \ ''^ x  f^rchaVt ^hipping  10"' \ V , Foundations  \  ( M O D s V , ,.  \ \  Z  ""'X  /Fixed r%\  .  V  c < .BuikJi^S/j^V^Xr-  10  Fixed P l a t f o r m s N  \ j ^ ^ y ^ ^ ,* ^ 'A.  Acceptable Region  10"  10" Lives Lost Cost in $  +1 1m  Naval C o m b a t —  10 10 m  100 100 m  ;  N >  s  Marginally Acceptable  'X>4% X **'«>!  1000 1b  10000 10 b  Consequences of Failure  Source: Adapted from Figure 2 Bhattacharya et. al. (2001) Risk Based Approach  Equation 3.2 can represent risk as noted below. Figure 3.1 is based on this equation. (3.2)  Risk = pC Where: . p = Probability of the occurrence of the event C = Consequence of the event Life Cycle Cost Analysis  Target reliabilities are chosen to minimize the expected total cost over the lifetime of the structure. Equation 3.3 is a simplified equation for the expected total cost, E[Cr]. E[C ] T  =  (3.3)  C,+C P F  f  Where: Ci = Initial costs, which usually increases as beta increases. 22  CF - Failure costs Py = Probability of failure, which is j3 = (f>~ (l - P ) 1  f  Experience and Calibration  Design standards evolve with experience.  Some factors that influence this process  include new materials, serious accidents or catastrophic events, new analytical tools and better insight into structural behaviour. Social Acceptance  Society has different acceptance levels for different hazards. The acceptable level of fatalities varies depending on the hazard and generally becomes less acceptable as fatalities increase. Table 3.1 shows the general level of societal acceptance of probability of fatalities. Table 3.1: Societal Acceptance of Fatality Levels Probability  Society's Reaction  IO" IO" IO" 10'  Level unacceptable, immediate action required to reduce hazard Society willing to spend public money to reduce risk Society still recognizes as a hazard especially for children Not of great concern to the average person  3  4  5  6  Source: Based on research conducted by Kleese & Barton, 1982. Adapted from Table 9 Bhattacharya et. al.  An annual target failure probability was developed by Flint and is noted in Equation 3.4. P=^Lp>/yr  (3.4)  Where: Ks = Social criterion factor, which accounts for the voluntary nature of the hazard or activity. A typical value is 5.0. p' = Annual basic probability of death acceptable to society, usually about 10" (based on U K data) 4  23  n = Aversion factor, which is directly proportional to the number of lives r  involved.  Some authors have proposed non-linear aversion  functions. Allan proposed a different formula for the annual target failure, which incorporated a warning of a future failure. Equation 3.5 shows this risk formula. The resulting level of acceptable risk is also shown on Figure 3.1. A  10  -5  (3.5)  Where: Py= Annual probability of failure A = Activity factor, ranges from 1.0 for normal buildings to 10.0 for high risk structures such off-shore construction W = Warning factor, ranges from 0.01 for "fail-safe" structures to 1.0 for sudden failure without warning n = Aversion factor r  Fatal Accident Rate  Another measure for hazardous activities is the fatal accident rate (FAR). Unlike the other hazard measures discussed above, this one accounts for exposure.  The FAR is  measured as the number of fatalities per 100 million hours of exposure to that activity. Equation 3.6 shows the calculation for the FAR. 10 P[F] s  FAR =  (3.6)  Where: P[F] = Probability of fatality Th = Exposure time in person hours  24  Target Reliabilities  Bhattacharya et. al developed flow chart that could be used a framework to develop target reliabilities as shown in Figure 3.2. While the reliability process has been developed for a structural engineering problem, the methodology described could be applied to other engineering systems. Figure 3.2: Flowchart for Developing Target Reliabilities Define system Define performance requirements Identify failure modes Define scope of reliability analysis  Identify system failure consequences  Analytical models Existing structures codes recommendations Socio-political considerations Engineering judgement  Select target system reliability  Define and identify structural categories Define limit states  Identify intermediate failure consequences  Derive sub-system and component reliabilities in each relevant limit state  Source: Adapted from Figure 3, Bhattacharya et. al. (2001)  3.2  T H E USE OF RELIABILITY IN GEOMETRIC ROAD DESIGN  Several authors (Navin, 1990 and 1992; Felipe, 1996; Zheng, 1997; Easa, 2000) have researched the applicability of reliability theory in geometric road design. Each author evaluated the possibility and some methodologies to include reliability into the design process for different elements of geometric design including stopping sight distance, horizontal curves, vertical curves and sight distance at intersections.  25  3.2.1  General Applications  A study by Navin (1990) investigated the possibility of development of safety measures for stopping sight distance, horizontal curves, decision sight distance, passing sight distance and vertical curves. Using values for average variable value and standard deviation, a frequency distribution and a margin of safety distribution was calculated for stopping sight distance. Margins of safety were calculated for two horizontal curve radii, the upper and lower limits of the Institute of Transportation Engineers (ITE) passing sight distance and stopping sight distance for the upper and lower speeds for an 80 km/h design speed found in 1984 edition of the AASHTO Green Book and for vertical curves. 3.2.2  Stopping Sight Distance Applications  Further research conducted by Navin (1992), investigated the possibility of applying reliability to stopping sight distance.  In this study, the probability of failure was  calculated using reliability methods for stopping sight distance for cars and trucks on wet and dry pavements for an average speed of 80 km/h. In addition, the margin of safety were calculated. The calculation of probabilities of not stopping and the margin of safety were hampered by a lack of good information on driver/vehicle performance limits. One of the general conclusions of this research was additional study into the performance limits for normal operations, human tolerances and vehicular roads limits be completed. A general equation was proposed for use with limit states design for an isolate design parameter P. (3.7)  DIM  Where: P  Geometric design parameter Highway construction and maintenance factor  x  Terrain factor  Y  Importance of the route factor  26  = Driver exposure factor £, - Environmental factor r|= Vehicle - driver mix factor PD/M = Some function of V , a™ , T D  D  Where: VD = Vehicle speed a™ = Deceleration TD = Perception reaction time 3.2.3  Horizontal Curve Application  Experiments were conducted on horizontal curves using expert and "normal" drivers, two different speed scenarios and on wet and dry pavement were investigated (Felipe and Navin, 1998; Felipe, 1996). The drivers and passengers were asked to rate the drive on a subjective four-point scale and the speeds and lateral acceleration of the vehicles were measured. The road tests, using passenger vehicles, were completed for small horizontal curve radii on a test track and for larger radius horizontal curves on an existing highway. RELAN was used to apply FORM methods to compare the expected lateral acceleration supplied by the roadway to the expected lateral acceleration demanded by the drivervehicle combination (Felipe, 1996). The probability of non-conformance and the beta was calculated for a comfortable lateral acceleration threshold for horizontal curves. On small radius curves, the tolerable acceleration was between 0.35g and 0.40g, which governs the speed selection by the driver. On large radius curves, the aspiration velocity governs driver speed. The research was concluded with a set of steps that could be followed to incorporate reliability theory into the geometric design for horizontal curves. The methodology to add reliability to horizontal curve design process was based on the development of Equation 3.8. The general design equation (Equation 3.7) developed by Navin (1992) was the basis for Equation 3.8. The design equation for horizontal curves is shown in Equation 3.8.  27  (t>R > , "rf,  ,  s  (3.8)  Where: < > j = Performance or construction factor Rs = Effective radius, which is a function of the nominal radius RH and the inner or outer lane avD = Speed factor V = Desired speed for R D  H  oify = Road friction factor ^h>cie  a  _  ]v[aximum lateral acceleration for a given R  H  e = Superelevation g = Acceleration due to gravity The recommended design process that integrates reliability analysis into geometric design for horizontal curves contains the following steps. 1. Select type of highway. 2. Select the nominal radius R . H  3. Select performance or construction factor,<j) and superelevation e. 4. Compute (j) and RH. 5. Derive effective radius, Rs, desired speed on the selected radius VD, and the maximum lateral acceleration for a given vehicle a . y  6. Select safety factors. 7. Compute probability of non-compliance and beta. If they are acceptable then a consistency check is completed. If these are unacceptable then steps 3 through 7  28  are repeated until an acceptable probability of non-compliance and beta are calculated. 8. Complete the consistency check, which is pM <  P i < Pi+i.  If it is acceptable then  continue with the design. If the result is unacceptable, then Steps 3 through 8 are repeated until the consistency check is acceptable. 3.2.4  Highway System Application  Zheng (1997) used a staged approach to apply reliability theory to geometric design. One stage investigated five years accident data and attempted to find a correlation between the accident experience and the geometric design conditions of an existing highway. More specifically, the research attempted to find a correlation between the accident rates for each highway segment that was identified as having deficient (when compared to the minimum values found in AASHTO Green Book) geometric characteristics. Geometric characteristics included in the correlation analysis were horizontal and vertical stopping sight distance, horizontal curve radii and vertical grade. Stopping sight distance was calculated for both directions. The correlation analysis found that there was not a strong relationship between the accident rate and substandard geometric characteristics such as restricted stopping sight distance and the sharpness of horizontal curve radii. In the second stage of the research, Zheng (1997) investigated preventative measures that could be used during the design of the realigned highway. This analysis used specific design situations that could apply to general highway design such as sight restrictions caused by median barrier on horizontal curves, the radius of horizontal curves and the design of vertical curves.  The probabilities of non-compliance for certain design  conditions were calculated. These models were then incorporated into a larger model called the Moving Coordinate System Design Model (MCSD model) that would calculate the probability of non-compliance for up to four different failure modes. The failure modes were longitudinal friction, tangential friction, horizontal stopping sight distance and vertical stopping sight distance. The number of failure modes that each segment of highway could fail in was related to the geometric characteristics. The probability of failure or non-compliance and beta values was calculated for each failure mode using  29  RELAN (Foschi et. al.). The probability of non-compliance for the entire system was also calculated. The data for the variables was collected by a series of road tests for expert and "normal" drivers on a test track and on an existing highway. Four driving speeds were evaluated. A sensitivity analysis was completed to determine how sensitive the results were to the values of the data. An omissions analysis was completed to determine how whether the results were sensitive to variable value fluctuation by replacing them by the mean value for that variable. The final analysis that was completed was a contingency analysis, which quantified the margin of safety for a particular design alternative. Two performance measures were developed. The first is an operation ratio and the second was the capacity ratio. These ratios were calculated using the beta values calculated for a particular design scenario. The research concluded by developing a process to integrate the MCSD model into the existing geometric design process. 3.2.5  Intersection Sight Distance Application  Easa (2000) investigated the possibility of using a reliability approach to calculating intersection sight distance for three of the five intersection sight distance design cases found in AASHTO. FORM methods were used to calculate probabilities of failure for the no control, yield control and the two-way stop control intersections. Similar to the research completed by Navin (1990 and 1992), Easa found information on the mean and standard deviation for variables difficult to find. As a result, these values were calculated based on the extreme values found in AASHTO. The sensitivity analysis found that the results were highly dependant on the values selected for the mean and standard deviation of the variables. However, Easa concluded that results indicate that the current AASHTO values used for intersection sight distance had high levels of reliability.  30  C H A P T E R 4: G E O M E T R I C D E S I G N E L E M E N T S  In the design of roads and highways, there are three main components to the design. The first component is stopping sight distance (SSD).  The second component is the  horizontal curve and the final component is the vertical curve. Traditionally the highway design standards and guidelines have specified a minimum value for each of these components for a given design speed.  In this chapter, the concepts used in the  development of design speed and the background for each design component are explored. 4.1  DESIGN SPEED  Traditionally geometric design of a road or highway has revolved around a "design speed", which governs the minimum values of horizontal radius, stopping sight distance and the length of vertical curves. It is relatively easy to arbitrarily set the design speed and complete the geometric design of the highway or roadway around the design speed. However, several authors have found fault with the logic of only setting minimum design values rather than a minimum and a maximum value (Lamm et. al., 1999). With the design speed approach, designers assume that the operating speed, which is the 85 percentile speed, does not exceed the design speed of the highway. On rural th  two-lane highways with design speeds ranging from 80 to 100 km/h, studies have shown that the 85 percentile speed ranges from 93 km/h to 104 km/h (Fitzpatrick and Collins, th  2000). When the design speed was less than 100 km/h, there was greater probability that motorists would exceed it than if the design speed was greater than 100 km/h. The problem of motorists exceeding the design speed has lead to the development of design consistency measures (Lamm et. al., 1999) and may have increased the development of speed prediction models, which predict the operating speed or the 85 percentile speed of th  a highway.  31  4.2  STOPPING SIGHT DISTANCE  The concept of SSD is central to the design of roads and highways. This distance has two components, the first is the distance traveled by the vehicle when the driver recognizes and initiates a response to the hazard. The second component is the actual distance traveled during the vehicle braking. In guidelines and standards that govern the design of roads, the SSD is calculated back from the hazard as though the hazard were a "brick wall" that cannot be hit by the vehicle. While SSD is central to the design of highways, it has yet to be shown by statistical analysis to be an important factor contributing to collisions. Studies at locations with limited SSD often do not have a statistically different accident frequency than similar locations with adequate SSD (Fitzpatrick et. al. 2000)". 4.2.1  Stopping Sight Distance for Passenger Vehicles  Stopping sight distance on a level, tangent road is defined by i  (4.1)  SSD = V T + ^j0  Where: SSD  = Stopping sight distance, m  Vo  = Initial velocity of the vehicle, m/s  T  = Perception reaction time, s  f  = Average coefficient of friction between tires and pavement,  g  = Acceleration due to gravity, m/s  x  2  Equation 4.1 is more commonly shown in geometric design guidelines and standards with the initial velocity in the units of km/h and the gravitational acceleration included in the constant that is used to convert the velocity from km/h to m/s.  32  4.2.1.1 Design Values  In highway design, the initial velocity of the vehicle is usually specified as the design speed of the highway. A strong argument could be made for using the operating speed, which is the 85 percentile speed of the highway as the initial vehicle speed. It could be th  calculated from data collected on an existing highway or it could be calculated using a speed prediction model. The other parameters included in the SSD formula are specified in the highway design standards and guidelines.  The values chosen for perception reaction time and the  coefficient of friction tend to be conservative values with a large portion of the population able to perform at a higher level than assumed in the design. Perception Reaction Time  Perception reaction time can be divided into four elements. Table 4.1 briefly outlines the each of these elements. Table 4.1: Elements of Perception Reaction Time Element  Description  Perception Intellection Emotion Volition  The time to see an object The time to understand the implication of the object's presence The time to decide how to react The time to initiate the action  Source: Transportation Research Institute Oregon State University Discussion Paper 8A Stopping Sight Distance and Decision Sight Distance  The perception reaction time used in design calculations in Canada and the United States is 2.5 s, which represents a driver that is slow to react to changing conditions on the road. Roughly, 90% of the driving population has a faster perception reaction time than the value used in design calculations (TAC, 1999). For example, alerted drivers tend to react more quickly than non-alert drivers do. Table 4.2 summarizes the results of some of the perception reaction studies.  33  Table 4.2: Summary of Perception Reaction Time  Source  85 Percentile (s)  95 Percentile (s)  1.48 1.80 1.90 1.78 1.9  1.75 2.35 2.50 2.40 1.5*  Gazis et. al. Wormian et. al. Chang et. al. Sivak et. al. Lerner  th  th  Note: * Calculated using the mean and the standard deviation Tangential Coefficient of Friction  The average coefficient of friction used in highway design represents poor tires on a wet road surface. This value varies with the initial speed of the vehicle and ranges from a high of 0.40 for 30 km/h design speeds to 0.28 for design speeds up to 130 km/h (TAC, 1999). The braking of heavy trucks is more complex and is not considered in the equation for SSD noted in Equation 4.1. The design guidelines and standards, in the past, have assumed that truck drivers can see much further because of their greater height allowing them to compensate for their poorer braking ability (AASHTO, 2001 and TRB, 2001) Unlike other countries noted in this analysis, the United States now uses a deceleration rate rather than a coefficient of friction multiplied by the acceleration due to gravity. Before the publication of the 2001 AASHTO Green Book, a coefficient of friction that changed with the design speed was used. Rather than using a coefficient of friction that changed depending on the design speed, a constant deceleration rate of 3.4 m/s is specified for all design speeds. This results in a coefficient of friction of 0.34, which most braking systems exceed on wet pavement (TRB, 2001; AASHTO, 2001). In the German design standards, the design coefficient of friction is calculated based on til  Equation 4.2 (Lamm et. al., 1999).  This equation is thought to represent the 95  percentile worst vehicle-road combination and is based on research conducted with newer tires (the PIARC standard European tire). coefficient of friction based on German tests. 34  Figure 4.1 shows the variation in the  2 ] 1 + 0.708 -0.721f liooj liooj  / , = 0-24l|{  v  (4.2)  V  Where: V = Speed in km/h Figure 4.1 Variations in the Coefficient of Friction  0  20  40  60 S P E E D V (km/h)  80  100  120  Source: Adapted from Figure 10.26 Lamm et. al., 1999  Table 4.3 summarizes the coefficient of friction values used in various countries to calculate stopping sight distance for passenger vehicles.  35  Table 4.3: Design Values Used for the Coefficient of Friction  Country & Source  Speed Range (km/h)  Coefficient of Friction Range  30-130 30-130  0.40 - 0.28 0.40-0.28**  20-130  0.34*  30-120  0.43-0.16 J  Canada (TAC 1999) United States (AASHTO, 1994) United States (AASHTO, 2001) Germany (Lamm et. al. 1999) Note:  ""Calculatedfromthe deceleration rate of 3.4 m/s AASHTO, 2001 **From Table III-l AASHTO, 1994 {From Table 10.7 Lamm et. al, 1999  4.2.2  Stopping Sight Distance for Trucks  2  The braking for heavy trucks is more complex than for passenger vehicles. Equation 4.1 will not accurately represent the distance required for trucks to stop. This equation can be modified with an additional variable, N that approximates the stopping sight distance requirements for heavy trucks with conventional braking systems.  N represents the  relative brake efficiency of a truck when compared with a passenger vehicle (Navin, 1986). Heavy trucks with anti-lock braking systems (ABS) will stop in a distance that is similar to passenger vehicles (TAC, 1999).  (4-3)  SSD = V T + - ^ 0  The values for N were developed from vehicle testing on dry pavements for a variety of different truck types, loading conditions and truck brake combinations (Navin, 1986). 4.2.3  Braking Distance on a Grade  The distance required to stop changes when a vehicle is on a grade rather than on a level surface. Figure 4.2 illustrates the results of braking distances on steep downgrades for different passenger vehicle types.  36  Figure 4.2: Effect of Grade on Passenger Vehicle Braking  On downgrades, the braking distance increases and on upgrades the braking distance decreases. Equation 4.4 can be used to calculate the braking distance on a grade (TAC, 1999). V  2  d=  ?  ,  (4.4)  254{f±G)  Where: d = Braking distance, m V= Initial velocity, km/h / = Frictional force G = Road grade in percent divided by 100 (positive values represent uphill grades while negative values represent downgrades)  37  Field experiments found that braking distances for passenger vehicles calculated using Equation 4.4 are incorrect for downgrades in excess of 5%. Experiments have shown that the braking distance of passenger vehicles is sensitive to the weight-shift that occurs when a vehicle is travelling on a downgrade.  How the vehicle size, weight, loading  configuration and the steepness of the downgrade interact is not known (Navin et. al., 1998). Equation 4.4 overestimates the frictional forces on downgrade locations.  Thus, the  braking distances calculated using Equation 4.4 would be shorter than the actual braking distance. Figure 4.3 shows the percentage of the level coefficient that was found on steep grades as compared with a level surface based on experimental results. MS Excel was used to calculate the best-fit equations for the percentage of coefficient of friction on steep grades. It found that the best-fit equations used the quadratic form rather linear or power form equations. The R of the equations ranged from 0.93 to 0.84, indicating that the 2  equations explained much of the variation due to the grade of the road. On dry pavement, Equation 4.5 represents the percentage of the dry level pavement coefficient of friction. The percentage of the level coefficient of friction is found in Equation 4.6 for wet pavements. Percent of level dry pvmt f = 0.004G + 0.0274G + 0.98556 R = 0.9376 2  2  x  (4.5)  Percent of level wet pvmt f = 0.0008G + 0.0404G + 0.9819 R - 0.8704 (4.6) 2  x  38  2  Figure 4.3: Percentage of Coefficient of Friction for Passenger Vehicle on Steep Grades Compared with Flat Sections Percentage  of Dry, Level Pavement f  = 0.0004G + 0.0274G + 0.9856 2  x  R = 0.9376 2  0.9  ..**" •. - •' .-'*  0.8  . •  - - ' '  -' •' ^'  ••<'' ^  r  0.7  .2  0.6  «  0.5  •  • Br-"""""  a  0.4  •  • a  Percentage  of Wet, Level Pavement f  = 0.0008G + 0.0404G + 0.9819 2  x  R = 0.8704 2  0.3  0.2  •  Dry Pavement  •  Wet Pavement  •h - TAC Dry Pavement — TAC Wet Pavement  0.1  Poly. (Wet Pavement) Poly. (Dry Pavement) -35  -30  -25  -20  -15  -10  Percent Grade  4.2.4  Braking Distance on a Horizontal Curve  When a vehicle is braking on a tangent section of road, it is assumed that the entire frictional force is available to be used in for braking. On a horizontal curve, some of the frictional force is used to supply the lateral force required for centrifugal acceleration. Therefore, a reduced amount of frictional force is available for braking. Figure 4.4 shows the braking forces found on a vehicle travelling around a horizontal curve.  39  Figure 4.4: Braking Forces on a Horizontal Curve  Friction for Lateral Acceleration  Available Friction  Reduced Friction for Braking  Source: Adapted from Figure X.B.2.2, RTAC (1986)  The reduced braking force can be calculated using the friction ellipse, which is representative of how the tangential and lateral frictional forces are shared. Equation 4.7 is used to calculate the frictional forces (Lamm et. al., 1999) and Figure 4.5 shows the friction ellipse.  J  1.0 <  (4.7)  R  V-^Rmax J  \frmax J  Where: / R = Available friction in the radial direction fr = Available friction in the tangential direction  40  Figure 4.5: Friction Ellipse F-Tmax  Driving • *  F Rmax  Braking  Source: Figure 10.7 Lamm et. al. (1999) 4.3  HORIZONTAL CURVES  Horizontal curves are another essential element of highway geometric design. Studies have shown that many of the collisions that occur on a road happen at horizontal curves. (Lamm, 1999; Felipe and Navin, 1998; Hauer, 1999). Improving the safety performance on horizontal curves could have significant benefits for society (Bidulka et. al., 2002). 4.3.1  Current Design Practice for Horizontal Curves  The design of horizontal curves currently involves the selection of radius or degree of curve based on a minimum radius and maximum superelevation for a given speed (Hauer, 1999). Equation 4.8 defines the minimum radius for a given speed in most design guidelines, (TAC, 1999; AASHTO, 2001). Figure 4.6 shows the sliding forces acting on a vehicle on a horizontal curve. V  2  (4.8)  Where:  41  V = Design speed in km/h emax  = Maximum allowable superelevation in m/m  fxxwx.  = Maximum allowable lateral coefficient of friction  Figure 4.6: Sliding Forces on a Horizontal Curve  ' Gravitational Force  Source: Adapted from Figure X.B.3.1.1, RTAC (1986)  The maximum allowable superelevation varies from 0.04 m/m in urban areas to up to 0.12 m/m for high speed rural highways. In Canada where ice and snow frequently occur, superelevation is generally limited to 0.06 m/m (TAC, 1999). Some areas of coastal British Columbia permit the use of 0.08 m/m superelevation rates on some roads but the majority of applications in British Columbia use a maximum superelevation rate of'0.06 m/m (BC MoT, 2001). The maximum allowable lateral friction varies with the design speed of the road TAC (1999). The TAC design guidelines permit the use of low speed urban design values for design speeds between 30 and 60 km/h in urban areas. High-speed rural designs are for speeds 40 km/h and above. Table 4.4 shows some typical values for lateral friction that are currently used in design practice today. Similar to the coefficient of friction used to calculate stopping sight distance, the maximum allowable lateral friction is for wet pavements.  42  Table 4.4: Maximum Values of Lateral Friction Permitted Design Speed (km/h)  T A C (high speed /low speed)  AASHTO  30  0.17  50  NA/0.31 0.16/0.21  80 100 130  0.14 0.12 0.08  0.14 0.12 0.08  0.16  Sources: TAC values are based on Tables 2.1.2.1 & 2.1.2.2 of TAC Geometric Design Guidelines for Canadian Roads, 1999. AASHTO values are based on Exhibit 3-14 of AASHTO Geometric Design Standards, 2001.  Using the range of superelevation values from Table 4.4 and Equation 4.8 for minimum horizontal curves different minimum radii for horizontal curves for the same design speed are predicted. Since the maximum superelevation is partly based on policy decisions of the local road authority, it is possible to have same radius of curve with different design speeds. Bonneson (2000) defines failure in horizontal curve design as occurring when the vehicle occupants experience an uncomfortable level of lateral acceleration. In this definition, the vehicle does not have to go out of control. Failure could occur if vehicles exceed the design speed of the roadway. 4.3.2  Horizontal Curves for Passenger Vehicles  The design guidelines, such as TAC and AASHTO, do not differentiate between passenger vehicles and heavy trucks for the design of horizontal curves. Equation 4.8 is implied to apply to both types of vehicles.  The actual limiting design factors for  passenger vehicles and heavy truck are quite different. Passenger vehicles are most likely to skid out on horizontal curves. It is estimated that the rollover threshold for passenger vehicles is approximately 1.2g whereas the skidding-out threshold is approximately 0.8g (Felipe, 1996; Lamm, 1999). Design equations for horizontal curves assume that passenger vehicles will skid out rather than rolling over. The design equations ignore that heavy trucks will generally rollover before they skid out on a horizontal curve. Some heavy trucks have a roll-over threshold of approximately 0.3g (Fancher Jr. and Gillespie, 1997). 43  4.3.2.1 Vehicle Path  During a road test conducted by Felipe and Navin (1998) it was noticed for the tight horizontal curves that drivers would attempt to minimize the speed change between the tangent and the curve by "cutting" the curves by driving into the shoulder or part way into the opposing traffic lane. It was thought that drivers chose to "cut" the curves so that they could achieve their desired levels of lateral acceleration and to be closer to the speed environment of the road. The results of the road tests were used to determine the maximum comfortable lateral acceleration. The research concluded that on small radius curves (up to 100 m) drivers base their speed on their comfortable lateral acceleration. On larger radius curves, drivers based their speed on their comfortable lateral acceleration and the speed environment. The vehicle path chosen by the drivers is based somewhat on the driver's own trade-off between risk, acceptable lateral acceleration levels and desired travel speed (Felipe and Navin, 1998). Drivers with higher risk tolerance may choose a path through a horizontal curve that uses part of the shoulder or even the opposing travel lane. Drivers with a lower risk tolerance would be more likely to choose a vehicle path within the travel lane. However even drivers with a low risk tolerance may choose to use the shoulder or opposing lane if they believe that they are travelling too fast to negotiate the horizontal curve safely. On curves with a small deflection angle, the opportunities to flatten the curve by driving on the shoulder are greater than for curves with large deflection angles. Figure 4.7 illustrates this point. In this figure, two simple, horizontal curves with a radius of 90 m are shown (not to scale). Curve A has a deflection angle of 10 degrees and Curve B has a deflection angle of 90 degrees. A desired path utilizing the shoulder is shown with an effective radius calculated using AutoCAD features.  44  Figure 4.7: Effective Radius on Horizontal Curves  Centreline Radius =  White edge' line ^ — E d g e  of pavement  Curve B  As shown in Figure 4.7, the greater opportunity for drivers to flatten or cut the horizontal curves is for curves with small deflection angles (Curve A). The expected curve flattening would be different for each driver and it would be unique for each design. The risk that drivers are willing to take may be based on the approach speed, lane and shoulder width, available sight distance, traffic volumes and individual driver characteristics. 4.3.3  Factors Influencing the Safety of Horizontal Curves  As noted earlier frequency of collisions on horizontal curves is greater than for tangent sections. Two components contribute to an increase in the frequency of collisions at horizontal curves (Hauer, 1999). The first is the radius used in the horizontal curve itself,  45  which relates to the ability of the driver vehicle combination to negotiate the curve. The second is the relationship between a horizontal curve and the preceding horizontal curves. Hauer's (1999) analyzed the levels of safety on horizontal curves. He found that for a given deflection angle, the larger the radius of the horizontal curve the safer the design was expected to be. When the deflection angle is large, the expected level of safety is strongly influenced by the horizontal curve radius. It is noted that none of the current design standards or guidelines use the deflection angle as part of the design criteria. Lamm et. al. (1999) has investigated how the preceding and succeeding curves influence the safety of a design. Alignments that contained tangents and curves that were similar in nature were found to be safer than those alignments that contained horizontal curves that were very different. Very long tangents combined with sharp horizontal curves were also found to be less safe than alignments with shorter tangents and similar horizontal curves. In response to these findings, three design consistency criteria were developed (Lamm et. al., 1999). 4.4  VERTICAL CURVES  Vertical curves join two segments of road of different vertical grades. Their purpose is to provide a smooth transition between the two grades. Vertical curves are typically centred on the point of vertical inflection (PVI) between two different grades. There are two types of vertical curves. Sag vertical curves are located at the bottom of grades and crest vertical curves are located at the top of a grade as shown on Figure 4.8.  46  Figure 4.8: Crest and Sag Vertical Curves  <y  / Sag Verti at Curve  — _ V  y Crest Ven ical Curve  > /'  :  /  In almost all countries, the shape of the vertical curve is based on a parabolic formula (Lamm et. al., 1999). The equation to calculate the elevation of the pavement at the centreline of the road is shown in Equation 4.9 (Davis et. al., 1981). It can be used for both sag and crest vertical curves. Y= ~° Gl  x  x +G.x 2  L  + BVC  (4.9)  Where: Y = Elevation at point x on vertical curve, m G i = Percent grade going into vertical curve G2= Percent grade going out of vertical curve L = Length of vertical curve in 100 m stations BVC = Elevation at start of vertical curve, m x = Point along vertical curve in 100 m stations  47  4.4.1  Calculation of the Length of Vertical Curve  The minimum length of vertical curve is based on the design speed and the ability of the driver to see an object of a certain height on the vertical curve at the minimum sight distance. Any type of sight distance, such as stopping sight distance, passing sight distance etc. can be used in the formula. In most of the design guidelines and standards, the minimum length of vertical curve is expressed by the factor K, which is defined as the change in vertical curvature per metre length of road. The design guidelines contain tables, which provide, for a given design speed, a minimum K value for sag vertical curves and crest vertical curves. The K value is calculated for two conditions. The first condition occurs when the length of vertical curve is shorter than the required sight distance. The second condition occurs when the vertical curve is longer than the required sight distance. The minimum K value is calculated differently depending on the condition. The length of vertical curve is calculated using Equation 4.10. (4.10) Where: K is defined from Equations 4.11 to 4.14 L = length of vertical curve, in m A = algebraic difference in grades, in percent 4.4.1.1 Length of Crest Vertical Curves  Equations 4.11, for SSD < L, and 4.12 for SSD > L, show the equations found in TAC for the minimum K value used to calculate the vertical curve length.  (4.11)  48  (4.12) Where: S = sight distance, in m hi = height of driver's eye, in m I12 = height of object, in m A = algebraic difference in grades, in percent The height of the driver's eye and the height of object are given by the design standards and guidelines. Each jurisdiction has different standards for these heights. In addition, the height of eye and the object height have changed over time. The height of eye was changed to reflect changes in the vehicle population. A study (RTAC, 1986) used vehicle heights produced by General Motors of Canada and Ford Motor Company of Canada data for the 1974 model year to determine the height of eye. The percentage of vehicles in each category was determined from 1973 sales data. It was believed that this data was representative of the vehicle population at the time. It found that the 95 percentile driver (i.e. the shortest drivers) could drive approximately th  65% of the vehicles produced by Ford and 72% of the vehicles produced by General Motors and would have a height of eye of at least 1.05 m. Conversely, the 95 percentile th  drivers would not meet the design height of eye in 35% of the Ford vehicles and 28% of the General Motors vehicles produced in 1974. The American Association of State Highway Officials (AASHTO) originally selected the object height in 1954 based on cost considerations. At that time, AASHTO decided that the 100 mm object height was the object height at which the cost of construction was equal to the benefits of seeing the object. The costs associated with the construction of vertical curves to see object heights less than 100 mm in height were considered too great for the benefits. Over time, the object height has increased. The increase in object height has been defended based on cost arguments. Table 4.5 shows the changes in the height of eye and objects over time. 49  Table 4.5: Changes in the Object Height and Height of Eye Source AASHTO, 1954 A Policy on Geometric Design of Rural Roads Manual of Geometric Design Standards for Canadian Roads and Streets, 1963 AASHTO, 1965 A Policy on Geometric Design of Rural Roads AASHTO, 1971, A Policy on Design Standards for Stopping Sight Distance TAC, 1986 & RTAC, 1976 Geometric Design Standards for Canadian Roads AASHTO, 1984 A Policy on Geometric Design of Highways and Streets TAC, 1999 Guidelines for Geometric Design of Canadian Roads BC MoT, Supplement to the TAC Geometric Design Guidelines AASHTO, 2001 A Policy on Geometric Design of Highways and Streets  Object Height (m)  Height of Eye (m)  0.1  1.37  0.1  1.37  0.15  1.14  0.15  1.14  0.38  1.05  0.15  1.07  0.38 (tail light)  1.05  0.38 (tail light) or 0.15 (rock) 0.6  1.05? 1.07  Note: The object height is for stopping sight distance. Other types of sight distance will use different object heights. A A S H T O and 1963 Design Standards were converted to metres.  4.4.1.2 Length of Sag Vertical Curves  Equations 4.13 and 4.14 calculate the K value for sag vertical curves. Equation 4.13 is used when the length of vertical curve is less than or equal to the SSD and Equation 4.14 is used when the vertical curve length is greater than the SSD.  K =  K  200(^3 +S tan  2S  (4.13)  a)  200(/j +S tan a)'  (4.14)  3  Where: I13 = Headlight height in m  50  a = Upward angle of headlight beam in degrees A = Algebraic difference in grades in percent TAC (1999) uses headlight height of 0.6 m, although this value varies from vehicle to vehicle. In addition, the headlight height on heavy trucks is higher than it is on passenger vehicles. Table 4.6 shows a range of values for headlights. The upward angle for headlights is assumed to be one degree for all vehicle types. Table 4.6 Range in Headlight Heights Descriptive Statistic  Mean High value Low value  Passenger Car (mm)  Multi-Purpose Vehicle (mm)  Heavy Truck (mm)  649 947 541  842 1174 569  1121 1351 915  Source: Adapted from Tables 31, 32 and 33 in Fambro et. al. (1997).  51  CHAPTER 5: MEASURING SAFETY The expected safety of a design can be measured using a number of tools, most of which have been developed recently. These tools include collision prediction models (CMP) that are used in conjunction with collision modification factors (CMF), design consistency models and horizontal curve perception models. Safety is quantified using collision prediction models. This chapter is a literature review of current practices used to assess the safety of roadways. 5.1  COLLISION PREDICTION MODELS  The safety of a particular design can be estimated using collision prediction models, which are then modified using collision modification factors (CMF's) to match the characteristics of the design. TAC, FHWA and other researchers have published many CMF's for various design parameters, including but not limited to, radius of curve, lane width, shoulder width or presence of climbing or passing lanes. Collision prediction models are regression models that were developed using collision data and generalized linear modeling techniques. They are for two general types; the first type is for intersections and the second is for road segments. Within these two general types are models for specific applications such as signalized urban intersections or for rural two-lane highways. Equation 5.1 shows the general form of a CMP for a road segment (Sawalha and Sayed, 1999).  E(h) = a xL xV xe " ai  a2  (5.1)  J  0  Where: E(A) = Expected number of collisions in a specified time period V = Annual average daily traffic L = Segment length Xj = Up to m variables in addition to V and L  52  ao, ai, &2, bj = Model parameters The number of variables depends on the model, for models that are applicable to a wider area the minimum number of variables need to explain the variation in the collision data are used. Most CPMs for rural highway segments use the segment length and traffic volumes as independent variables with other variables sometimes added. The models typically calculate the expected number of collisions in a specific period that ranges from one year to five years (Sawalha and Sayed, 1999). Many collision prediction models have been developed for a wide variety of uses. Since the models were developed using specific collision data and GLM techniques it is important that the model be calibrated to for the conditions that it will be used for. Some calibration factors could include climatic differences, reporting thresholds and practices and driver population differences (FHWA, 2003a). The CPM's are often general models that do not take into consideration all of the geometric characteristics of the roadway.  These characteristics often have some  influence on the expected number of collisions on the road. Their influence can be measured with collision modification factors (CMFs). Many sources, such as TAC, AASHTO, Institute of Transportation Engineers (ITE) and FHWA, have published CMF's by for a variety of conditions commonly found. Some examples of CMFs are horizontal curvature, vertical grade, lane width and shoulder width. Collision modification factors (CMF) were developed through before and after studies that investigated the effect of a particular road improvement on the collision frequency. A CMF calculated from a properly conducted before and after study should represent the change that the one variable has on the expected collision frequency (Lin et. al., 2001; Sayed and de Leur, 2002). 5.2  PERCEPTION OF OVERLAPPING HORIZONTAL & VERTICAL CURVES  Researchers such as Lamm (1999) have noted that the overlap of horizontal and vertical curves may result in an optical illusion in which the horizontal radius perceived by the  53  driver is different from the actual radius of the road. If the radius perceived by the driver is larger than the actual radius, the driver may try to negotiate the curve at a rate of speed that is too high for that curve causing a potentially unsafe situation. Research conducted by Bidulka et. al. (2002) confirmed that the overlap of horizontal and vertical curves does cause an optical illusion. When horizontal curves were combined with crest vertical curves, drivers perceived the horizontal curve to be sharper than it actually was. When horizontal curves were combined with sag vertical curves, drivers perceived the horizontal radius to be less sharp than it really was. Thus, horizontal curve - sag curve combinations could be problematic especially if the perceived radius is much larger than the actual radius. The relationship between the actual horizontal radius and perceived radius was quantified using regression analysis (Hassan, et. al., 2002). A series of equations using different explanatory variables were developed that included actual radius, type of vertical curve, algebraic difference in grades, turning direction, superelevation, rate of vertical curvature (K) and presentation background. The model chosen to quantify the relationship between perceived and actual radius is shown in Equation 4.2. R = -51.28 + 0.953^ + 132.11F+ 0.125^F P  R = 0.996 2  (5.2)  Where RP = Perceived radius in m RA = Actual radius in m V = 0 for crest vertical curves and 1 for sag vertical curves. To date no research has been published that confirms that when the perceived radius is larger than the actual radius the level of safety on the road decreases. Furthermore, the relationship between safety and perceived radius has not been determined. While no link has been made between road safety and perceived radius, Equation 5.2 identifies a potential risk. By calculating the difference between the perceived and the actual radius a level of risk can be quantified somewhat. For example, when there are small differences  54  r  between the perceived and the actual radius, the increased risk would be expected to be small. If the perceived radius were calculated to be quite a bit larger than the actual radius then the expected risk would be higher. 5.3  DESIGN CONSISTENCY  The design consistency of a roadway design is evaluated using four measures: operating speed, vehicle stability, alignment indices and driver workload (Ng and Sayed, 2003; Hassan et. al., 2001). The most commonly used measures are the design consistency measures developed by Lamm et. al. (1999). They developed three evaluation criteria for design consistency, which are sometimes referred to as the three safety criteria. The following paragraphs describe each criterion. 5.3.1  Design Consistency Criteria  The first safety criterion is to achieve design consistency. To meet this criterion the highway designer must ensure that the characteristics of the road will fit with the driving behaviour found on the road. This criterion is measured by the difference between the operating speed, the 85 percentile speed, and design speed for each alignment element th  as shown in Equation 5.3 Criterion I = V - V S5  (5.3)  D  Where: Vg5 = Operating speed in km/h VD = Design speed in km/h 5.3.2  Operating Speed Consistency  The second safety criterion is to achieve operating speed consistency. This means that the 85 percentile speed should be consistent from one alignment element to the next for th  the entire length of the alignment. This criterion is measured by the difference in the operating speed on alignment element i to the operating speed on alignment element i + 1 as shown in Equation 5.4.  55  Criterion II = AV = VS5 -VS5 S5  i  (5.4)  M  Where: V85j and V85j+i are the operating speeds on successive elements 5.3.3  Vehicle Stability  The third safety criterion developed by Lamm et. al. (1999) is for vehicle stability. This criterion measures the difference between the friction assumed in the road design and the friction demand from the driver as shown in Equation 5.5. The side friction assumed,/RD, is based on the physics equations. Lamm et. al. developed the side friction demanded,^ shown in Equation 5.6, for use in mountainous terrain. Other side friction models can be used in place of the ones noted in Equations 5.6 and 5.7. 4f* =/*-/*>  (5-5)  Where: f =0.22-1.79*10- F +0.56*10- FJ 3  R  5  D  (5.6)  V  2  fpj)  e  =  JRD  U  J  (5.7)  R  Where: fit =  Side friction assumed  fiu) = Side friction demanded VD = Design speed, km/h Vgs = Operating speed (85 percentile speed), km/h th  R=  Radius, m  e=  Superelevation rate, m/m  56  Once values have been calculated for each of the criteria, they can be compared with the threshold values for good, fair and poor design developed by Lamm et. al. (1999). The threshold values are found in Table 5.1. Table 5.1: Design Consistency Criteria  Consistency Rating  Criterion I |V -V |  Criterion II  Good Fair  A< lOkm/h 10<A<20km/h  Poor  A > 20 km/h  A< 10km/h 10<A<20km/h A > 20 km/h  8 5  |V i  d e s  85  —  V 5j+i| 8  Criterion III A/*  >0.01 0.01 > Af >-0.04 Af < -0.04 Af  R  R  R  Source: Lamm et .al. (1999)  Two other design consistency measures not included in the methodology Lamm et. al. developed are alignment indices and driver workload. Each of these measures is briefly described in the paragraphs below. 5.3.4  Alignment Indices  Alignment indices are a general quantitative measure of the alignment. They can be used to compare individual alignment elements with the overall alignment section (Hassan et. al., 2001; Ng and Sayed, 2003). One alignment index is the CRR, which is the ratio of the radius of a curve to the average radius for an entire section. Alignment indices, such as the CRR can be good indicators of alignment inconsistencies, however very flat or sharp curves can significantly influence the CRR. Therefore, these indices may not be the most appropriate design consistency measure (Anderson et. al., 1999). 5.3.5  Driver Workload  Driver workload is the amount of mental work that a driver must do to stay on the road. It is desirable to have a consistent amount of work for the driver for a segment of highway. The amount of workload for drivers should be enough so that drivers fall do not fall asleep but not so much that drivers are overwhelmed by the driving task (TAC, 1999). Currently there are few measures to quantify driver workload. Two models,  57  developed by Wooldridge et. al. (Hassan et. al., 2001), use the horizontal radius as the independent variable to describe driver workload for familiar and unfamiliar drivers. 5.3.6  Speed Prediction Models  In order to calculate the design consistency measures, the operating speed on the alignment elements must be calculated using speed prediction models. Many different speed prediction models have been developed. Most of the models were developed to estimate the 85 percentile speed on horizontal curves - in two-dimensions for rural th  two-lane highways.  Two-dimensional of models ignore the effects of the vertical  alignment, since passenger cars are generally not affected by grades of - 5% to 5% (Ottesen and Krammes, 2000). Sometimes the model specifies certain range of grades (Morrall and Talarico, 1994). Over time, researchers have developed more models that are specialized; these may be for speed prediction for heavy trucks on three-dimensional alignments or for threedimensional alignments for passenger vehicles. To estimate speeds of passenger vehicles on three-dimensional alignments two models were recently developed. The FHWA (2000) developed the first model for use with the interactive highway safety design module (IHSDM). The second model was developed using Canadian data (Gibreel et. al., 2001). Most speed prediction models in the literature are linear regression models. At least one attempt was made to develop an artificial neural network (ANN) to estimate the 85  th  percentile speed. The researchers found that an ANN could be used with the typical speed prediction model variables to estimate the 85 percentile speeds (McFadden et. al., th  2001). Some of the forms that the regression models can take are listed in Table 5.2.  58  Table 5.2: Regression Model Forms for 85 Percentile Speed Prediction Models  Model Type  Characteristics  Linear  Local only  Exponential  Local only  R j, R  Local only  B + (3 4DC  Local and general  B + &DC + B DC + B 4 DC + fi V  Nonlinear/Polynomial  0  Local and general Inverse  Model Form  0  P l D C  X  2  B^B DC  3  f  P DC +p V  +  l  4  2  3  A  f  Local only (B  B DC >) p  x +  2  Source: Table 8.3 Lamm et. al. Highway Design and Traffic Safety Engineering Handbook 1999 p 8.26 Where ft = constants, DC = degree of curve and Vy= desired speed on independent tangents i  5.3.6.1 Speed Prediction Variables i  Horizontal curvature is often expressed as the degree of curve, which is proportional to  j  1/R. Another variable used for horizontal curvature is the curve change rate for an individual element (CCRs) developed in Germany (Lamm et. al., 1999). Other variables are sometimes added to speed prediction models if they are statistically significant and increase the R value of the model. 2  i  The degree of curve has two definitions. The first is the arc definition, which defines the i  angle subtended by an arc of 100 units of length. The second is the chord definition,  J  which is the angle subtended by a chord of 100 units of length (Davis et. al., 1981). The  \'  formulae for both definitions of degree of curve, are the same in the English and the metric systems of measurement. Thus it is possible to have two different radius curves one calculated using English units and the other with metric units with the same degree of  !  curve. In the United States, it was considered desirable to develop a relationship between  j  the radius in metres and the degree of curvature calculated from English units. The  \  following formula is based on the arc definition of degree of curve (FHWA, 2000). Z ) C =  174M8  (5.8)  R  j i  59  Where: R =  Radius of horizontal curve, m  With the exception of the speed prediction model developed by Morrall and Talarico (1994), all of the models listed in this chapter using degree of curve use the degree of curve as calculated in Equation 5.8. Degree of curve and the radius of curve do not take in to consideration the spiral transition curves that are often present on either side of a circular curve. Different formulae can be used to calculate the CCRs, which takes into consideration various types of compound curves. The following paragraphs describe in detail various forms of the CCR . The basic formula for the CCRs is shown in Equation 5.9. This formula can be S  used for a horizontal curve with clothoid spiral transition curves located between the circular curve and the tangents. This curve configuration is widely used, especially on high-speed highways.  (5.9) Where: R =  Radius, m  Leu = Length of spiral curve in, m L  C R  L  C L 2  L =  = Length of circular curve, m = Length of spiral curve out, m L  C L 1  +L  C R  +L  C L 2  , m  5.3.6.2 Two-Dimensional Horizontal Curve Models Morrall & Talarico (1994) (4.561-0.0058DC  R -0.631 2  60  (5.10)  Where: 5729.58 R  DC = metric degree of curve Where: R=  Radius, m  The Morrall and Talarico model is based on two-lane rural highway speed data collected in Alberta. It is considered valid for grades between -5% and 5%. Like many speed prediction models, it defines a single speed for a circular curve and it ignores the effects of spiral curves on the 85 percentile speed. When the horizontal radius approaches th  infinity, the 85 percentile speed is equal to 95.68 km/h. th  Lamm et. al (1999)  This model developed by Lamm et. al is a modification of Morrall & Talarico's model. The model was modified to include spiral curves in the speed prediction model by adapting it to use the CCRs variable in place of the variable, degree of curve, it was originally developed with. Lamm has modified 85 percentile speed prediction models from four other countries including the United States (Lamm et. al., 1999). The Lamm model is the only model, noted in this chapter, which incorporates spiral curves. It calculates one speed for total length of spirals and circular curve(s). When the radius goes to infinity, the 85 percentile speed is 95.68 km/h. th  1/  _  (4.561-0.000527CC«  R = 0.63 2  S  (5.11)  Where: CCRs is found using Equation 5.9  I  TAC (1999)  This speed prediction model is found in the design consistency section of the 1999 Transportation Association of Canada (TAC) Geometric Design Guide for Canadian  61  Roads. It is based on US research data and uses two explanatory variables. It is not known how well this model fit the data. (8995 + 5.73Z) V„ = 102.45 + 0.0037LR  (5.12)  Where L  Length of circular curve, m  R  Radius, m  When the radius is near the minimum allowable, based on the minimum radii tables found in TAC Guide, this model calculates unusually low speeds. However, when the radius approaches infinity the 85 percentile speed is equal to 102.45 km/h. The upper th  limit of the 85 percentile speed calculated by the TAC model is somewhat greater than th  other models noted in this chapter. Thus, the speed variation calculated by this model can be quite large. 5.3.6.3 Three-Dimensional Models  Several speed prediction models have been developed for use on three-dimensional alignments. Since passenger vehicles are not as influenced by grade as heavy vehicles, separate speed prediction models are required for passenger vehicles and heavy trucks (Bester, 2000). FHWA researchers developed a three-dimensional model for passenger vehicles. It calculates the desired speed on eleven different horizontal and vertical curve combinations. This model will be used with the interactive highway safety design model (IHSDM). Table 5.3 shows the model equations.  62  Table 5.3: Speed Prediction Equations - F H W A Model  Equation No.  Alignment Condition  Speed Equation  N/A  Tangent on grade  Desired speed  1  Horizontal curve on grade - 9 % <G < -4%  2  K  85  = 102.10-  Horizontal curve on grade - 4 % <G < 0%  3 0 7 7  1 3  R  3709 90 V„ =105.98 85  3 4  Horizontal curve on grade 0% <G < 4%  F  R  85  =104.82-  Horizontal curve on grade 4% <G < 9%  3 5 7 4  R  S5  =96.61  85  Horizontal curve combined with a sag vertical curve  6  Horizontal curve combined with a nonlimited sight distance crest vertical curve  7  Horizontal curve combined with a limited sight distance crest vertical curve  5 1  2752 19 V  5  -  R  3438.19 =105.32-  V  S5  85  R  The smallest value from Eq. 1 through 4 (this table) F  85  =103.24-  -  3 5 7 6  85  5 1  R  Also check Eq. 6 (this table) 8  Sag vertical curve on a tangent  Desired speed  9  Non-limited sight distance crest vertical curve on a tangent  Desired speed  10  Limited sight distance crest vertical curve on a tangent  149.69 K  85  =105.08  85  Source: Adapted from Table 1 (Fitzpatrick and Collins, 2000) Notes: Limited sight distance is a vertical curve with a K < 43. R = radius, m and K = rate of vertical curvature 5.4  INTERACTIVEH I G H W A Y SAFETY DESIGN M O D E L  The FHWA in the US developed the interactive highway safety design model (IHSDM), which is a set of software tools that can be used by practicing design engineers to evaluate the safety and operation effects of highway geometric designs (FHWA, 2003b). The tools in the IHSDM include the following evaluation modules: policy review, crash prediction, design consistency, intersection review and traffic analysis. Each of the modules are evaluated separately from one another and the can be used on either existing highways or on proposed designs (FHWA, 2003b). The first three sections are described in more detail in the paragraphs below.  63  5.4.1  Policy Review Module  The basic function of the policy review module is to "automate" the process of checking the geometric elements of the alignment against the design policy documents (FHWA, 2003b). It compares the values for each of the alignment elements against the values found in the tables of the AASHTO design documents. The policies referenced include the AASHTO "Green Book" editions published between 1990 and 2001, the AASHTO Roadside Design Guide (1996 Edition) and the AASHTO Guide for the Development of Bicycle Facilities (1999 Edition).  The intent of the policy review module is the same as the AASHTO policy documents, which is "to provide guidance to the designer by referencing a recommended range of values for critical dimensions." (FHWA, 2003b). Other modules of the IHSDM can be used to help the designer estimate operational and safety performance. 5.4.2  Crash Prediction Model  This module estimates the crash frequency and severity for a highway based on the geometric and traffic characteristics. The methodology used for crash prediction uses CMPs to calculate the base predictions and uses CMFs to modify the base predictions for specific geometric characteristics. CMPs have been developed for highway segments and for three types of at-grade intersections. The base CMP for two-lane rural highway segments is noted in Equation 5.13 (FHWA, 2000a). The CMPs were developed using collision data from two different states. A calibration process may be required to adjust the CMP to local conditions. N = ADT *L*356*10- *exp -° 6  br  (  4865)  n  (5.13)  Where: N b r = Predicted number of total highway segment crashes per year for nominal or base conditions ADT = average daily traffi volume for a specific year in veh/day n  L = Length of highway segment in miles 64  The CMFs were developed by an expert panel that reflects their consensus on the qualitative influence each factor has on collisions (FHWA, 2003a). Nine CMFs were developed for highway segments and five were developed for intersections. The CMFs for highway segments include lane width, shoulder width and type, horizontal curve, superelevation, longitudinal grades, driveway density, passing lanes and short four-lane sections, two-way left turn lanes and roadside hazard rating. The final step in this module is the option of using empirical Bayes methods to combine the results of the CMP and CMF with existing experience. This step allows the model results to be included with existing crash experience (if available). This step is not required for certain types of analyses. 5.4.3  Design Consistency Evaluation  One of the main purposes of the design consistency module is to provide designers with a set of procedures to evaluate design consistency. This module evaluates an alignment based on two criteria. The first criterion is the difference between the operating speed (85 percentile speed) and the design speed of the highway. The second criterion th  evaluated is the reduction in speed between the approach tangent and the horizontal curve (FHWA, 2003c). These criteria are the same as the first two design consistency criteria evaluated by Lamm et. al. (1999). However, unlike the Lamm et. al. vehicle stability design consistency criterion is not evaluated. The module evaluates the design by calculating the operating speed for each alignment element and comparing it to each of the criteria. The speed prediction model uses threedimensional alignments to calculate the operating speed and prepares a speed profile for the highway. The module calculates the two design consistency criteria and compares the results to the three conditions found in Table 5.4.  65  Table 5.4: I H S D M Design Consistency Criteria  Condition  Condition 1 Green Condition 2 Yellow Condition 3 Red  Design Speed/Operating Speed Check V 5-V 8  10 km/h  20  D e  < V  km/h  s  8 5  <10  D  <  V  V85  km/h  -V es  8  5  ^0  -V  D  km/h  e  Speed Differential of Adjacent Elements  s  10 km/h  20  t a n  -V85  < V85  km/h  c u r  t a n  <  <10km/h  -V85  c u r  <20  km/h  VSStan-VSScur  Source: I H S D M Design Consistency (DCM) Engineer's Manual ( F H W A , 2003c)  It is believed that crash frequency will decrease as alignment consistency increases. Thus, the elements rated as less consistent can give designers a place to start when making improvements to an alignment or a place to begin further evaluation of the design. 5.5  DESIGN CONSISTENCY AND SAFETY  As noted in the section above, the design consistency module in the evaluates the design consistency of the alignment and another module evaluates the safety performance in the IHSDM. Since design consistency and safety performance are not linked together in the IHSDM no evaluation of safety benefits of design consistency can be made (Ng and Sayed, 2003)/;; A series of loglinear models were developed which quantified a relationship between design consistency and safety (Anderson et. al., 1999). One model used traffic volume, curve length and speed reduction on successive elements as the variables related to accident frequency. Another model related accident frequency to curve length and the ratio of the radius of a single curve to the average radius for a road (CRR). None of the models developed used the vehicle stability or driver workload as variables for consistency measures (Ng and Sayed, 2003). A series of twelve models were developed at UBC by Ng and Sayed (2003) to quantify the relationship between design consistency and safety. These models were developed  66  using generalized linear regression modelling techniques. Each of the models uses traffic volume and section length in combination with at least one other variable related to design consistency. Two types of models were developed. The first model type uses only one design consistency variable and the second type uses as many design consistency variables as found to be statistically significant. Ten models of the first type were developed. The variables included difference between design and operating speed, difference in operating speed between successive elements, vehicle stability, driver workload and the CRR. The three models related to operating speed for this type of model are shown below. 5.5.1  Difference in Operating and Design Speed  Acc/5yr = exp*" - '*Z 3  380  0892  *F ' 0  * xp  5913  ( 0 0 0 9 0 9 1  e  *  ( M ) )  (5.14)  Where L =  Length of curve, m  V =  traffic volume, AADT, veh/day  V&5 = 85 percentile speed, km/h th  VD = design speed, km/h 5.5.2  Speed Reduction  Two models were developed for speed reduction between successive elements as noted in Equations 5.15 and 5.16. The model shown in Equation 5.15 was developed using horizontal curves only whereas the data for the model shown in Equation 5.16 used horizontal curves and tangent elements. AcclSyr = exp - (  3  796)  *Z  Accl5yr = exp - - *Z (  2  281)  a8874  1096  *F ; 0  *F~ '  * exp^  5847  0 455  * xp * (/c  e  Where:  67  0428  002421  *^  *  AK85)  (5-15) (5-16)  AV =  speed reduction between successive elements, km/h  IC =  0 for tangents and 1 for horizontal curves  85  5.5.3  Impact of Design Consistency on Safety  The second type of model was developed for planning purposes and uses as many design consistency variables as was statically significant in addition to the traffic volume and segment length. Both of the models developed are shown below. The model shown in Equation 5.17 was developed using horizontal curves only and the model shown in Equation 5.18 was developed using horizontal curves and tangent elements. Accl5yr = exp Acc 15yr = exp  exp exp  (0.0049*(K -V }+0.0253* A V 85  (/C*(0.022*AK  D  85  -1.189  A5  Af )) R  -1.177 Af  R  )  (5.17) (5.18)  Where A/ = Difference in vehicle stability (See Equations 5.5 through 5.7) R  68  CHAPTER 6: CALCULATION PROBABILITY OF NON-COMPLIANCE & BETA This chapter outlines the methodology that was used to calculate the probability of non-compliance, the reliability index, B and the design point for stopping sight distance, horizontal curve and vertical curve problems.  The probability of failure and the  reliability index were calculated using RELAN (Foschi et. al., 2002)  Monte Carlo  simulations methods were used for all RELAN calculations. FORM and SORM were used to calculate the probabilities of non-compliance for all of the RELAN calculations except for the serious injury prevention scenarios and the SSD on horizontal curve scenarios. FORM was used to calculate the design point. For the simulation runs, the probabilities of failure were calculated using 100,000 to 1,000,000 simulations. All of the FORTRAN programs used in the RELAN analysis can be found in Appendix A. The probabilities of non-conformance were calculated for stopping sight distance, horizontal curve and vertical curve problems on wet and dry pavement conditions. The following sections describe the problems that were developed for each geometric design characteristic. The use of dry pavement in the analysis, while not useful for determining the values for design conditions, gives designers an opportunity to see how the upper bound of driver-vehicle combination performs. The dry pavement conditions provide the upper limit of performance that one could expect. The lower limits of a particular design would be based on wet pavement conditions, which is the current design practice today. 6.1  STOPPING SIGHT DISTANCE  This analysis investigated the probability of failure, the associated reliability index, P and the design point for two vehicle types, two pavement conditions and for two design conditions on level, tangent roads. The analysis also calculated the probability of failure and the reliability index for SSD on downgrades and on horizontal curves. The vehicle types and pavement conditions were passenger vehicles, heavy trucks, wet pavement and dry pavement.  69  6.1.1  Level Roads  The first stopping sight analysis was completed on level tangent roads. The analysis included the evaluation of stopping sight distance using an 80 km/h design speed scenario for passenger vehicles and trucks on wet and dry pavements. In the first part of the analysis, an operational standard was investigated. The second part of the analysis investigated injury prevention scenarios. 6.1.1.1 Operational Standard  The first design condition is the one in current use; it is an operational standard. The design criteria found in TAC, AASHTO and similar publications are to prevent operational problems on the roadway. Therefore, there must be no property damage only (PDO) collisions. This standard requires that the road design supply sufficient distance for the vehicle to stop before hitting an object or other vehicle on the road. This is the brick wall criterion. The limit states equation for the operational standard for a passenger vehicle is shown in Equation 6.1. The limit states equation for an operational standard for heavy trucks is shown in Equation 6.2.  GXP =  (6.1)  SSD  supply  V f  GXP = SSD  supply  Sfxj v  2  V T + —~  ^  (6.2)  0  Where: SSDsuppiy = Stopping sight distance supplied by the road, in m. This value was specified in the analysis. Vo = Random variable for the initial vehicle speed, in m/s g = Gravitational constant, in m/s TV = Random variable for the relative truck braking efficiency 70  f = Random variable for the braking frictional force. x  There was one failure mode fore each of Equations 6.1 and 6.2. Each of the random variables is described in greater detail in Section 6.4. 6.1.1.2 Serious Injury Prevention Condition  The second design condition proposed in this analysis is a serious injury prevention standard. It requires sufficient stopping sight distance for the vehicle to slow to an impact speed with only slight injuries and no serious injuries or fatalities in the resulting collision. This would be the limiting standard and would be useful under certain design conditions. Designers, for example, would want to provide operational stopping sight distance for the design or posted speed limit. They may also want to provide sufficient stopping sight distance to prevent injuries in the event of a driver exceeding the posted speed limit but not driving at excessive speed. In British Columbia, driving in excess of 40 km/h above the applicable speed limit is defined as driving with excessive speed (BC Legislature, 2001). Four serious injury prevention scenarios were investigated. The first three scenarios used random variables for initial vehicle speed, coefficient of friction, perception reaction time, the maximum speed of a collision that would cause minor or no injuries and relative braking efficiency (for truck scenarios). These scenarios included truck to car, car to car and car to person. The fourth scenario investigated the probability of being unable to decelerate to a speed unlikely to cause serious pedestrian injury in a car to pedestrian collision if the car was traveling 40 km/h over the posted speed of 50 km/h. With the exception of the initial vehicle speed, the other random variables remained the same as the other three serious injury prevention scenarios. Table 6.1 shows the area and design speed used for each of the serious injury prevention scenarios.  71  Table 6.1: Serious Injury Prevention Scenarios  Area  Posted Speed (km/h)  Urban Urban  50 50 80 50 (travel speed 90)  Vehicle M i x  Truck to car Car to pedestrian Car to car Car to pedestrian  Rural Urban  Note: Posted speed and design speed were assumed to be the same. The practice of using the design speed as the posted speed is not uncommon in British Columbia.  Non-compliance with the serious injury prevention scenario can arise under two conditions. The most common condition occurs when the vehicle is unable to decelerate to a speed unlikely to cause injury after recognizing an object on the road and trying to stop within the given stopping sight distance. The second condition occurs when the driver of the vehicle is unable to perceive and react within the given stopping sight distance. While there are two conditions when failure occurs, there is only one failure mode for the injury prevention scenario as explained in the paragraphs below. Equation 6.3 is the limit states equation used for the injury prevention scenario for a passenger vehicle and Equation 5.4 is the limit states equation for a heavy truck. GXP = V _  inj  - j(-2gf (SSD  GXP = V _  inj  - J(-2gNf (SSD  no  no  x  -V T)+V )  x  (6.3)  2  sapply  sapply  0  0  -Vj)  +V) 2  0  (6.4)  Where: The variables are as noted for Equations 6.1 and 6.2 and; V„ .i„j 0  =  random variable for a collision speed unlikely to cause serious injury or death. This variable changes for each injury prevention scenario.  72  The first non-compliance condition occurs when a vehicle-driver combination is unable to perceive and react in time in the supplied SSD. For this condition, the failure function can be reduced to Equation 6.5 for both passenger vehicles and heavy trucks because there is no opportunity for the vehicle is unable to decelerate. GXP = V.no-inj  (6.5)  0  In Equation 6.5, the probability failure is independent of the supplied SSD and will be constant for all SSD supplied conditions. If this equation is used as limit states equation for a failure mode, the probability of non-compliance will remain the same for all supplied SSD conditions. Figure 6.1 illustrates the two failure functions. The first mode of failure is based on Equation 6.5. For the speed distribution used for this research, the probability of non-compliance for the first failure mode is quite high - in the order of 98%. The second failure mode would use Equation 6.3 or 6.2 depending on the type of vehicle. The probability of failure is not independent of the supplied SSD therefore as the SSD increases the probability of failure from that mode of failure decreases. Figure 6.1: Probability of Failure for Serious Injury Scenarios  u c n a E  — N o driver reaction failure -a—  Available SSD  73  Unable to stop failure  When there is more than one mode of failure, the probability of failure from each failure mode is added together to get the combined probability of failure. If the serious injury prevention scenarios had two modes of failure, the probability of failure would be extremely high regardless of the supplied SSD. This result is incorrect and as a result, only one mode of failure was used for the injury prevention scenarios. 6.1.3  Stopping Sight Distance on a Grade  For this research, the failure function for the stopping sight distance was modified so that the coefficient of friction was a fraction of the level coefficient of friction. The fraction of the coefficient of friction found on a level surface was based on Equations 4.5 and 4.6 that are found in Chapter 4. While the fractional percentage of the coefficient of friction could be a random variable, there was insufficient data to calculate a probability distribution for the percentage of coefficient of friction on a downgrade.  Thus, this  research used a deterministic value for a particular downgrade. The failure function for stopping sight distance was modified for passenger vehicles as noted in Equation 6.6 below. Since the ability to stop on a steep downgrade is sensitive to weight-shift even in passenger vehicles, the equations used to calculate the coefficient of friction on a grade should not be applied to heavy vehicles because the weight-shift would be quite different than in passenger vehicles. Therefore, the analysis of SSD on a grade was only completed on passenger vehicles. f  GXP = SSD  sapply  v  1  VT + Q  A  (6.6)  ^  Where the variables are the same as for Equations 6.1 and 6.2 except for: P = Percentage of coefficient of friction based on Equations 4.5 and 4.6. There was one mode of failure for this analysis, which was the probability of being unable to stop within the specified SSD on the specified downgrade.  Three different  downgrades were investigated -5%, -10% and -15%. These grades cover the most commonly found steep downgrades.  74  6.1.4  Stopping Sight Distance on a Horizontal Curve  The failure function for the SSD on a grade was modified so that the available friction for stopping is reduced by the amount of lateral friction required to remain on a curve. This analysis investigated the probability of being unable to stop for passenger vehicles on three pavement conditions. The pavement conditions were wet, dry and damp. Equation 4.7 was used to calculate the tangential friction available for stopping. It was assumed that the vehicle/driver combination would use as much lateral friction as required on the horizontal curve. The remaining friction could be used for stopping. Equations 6.7 and 6.8 are the failure functions for stopping sight distance on a horizontal curve.  (6.7)  •e  GXP = f  supply  J V ^ V T + —^V P%f-em J 2  GXP = SSD  supp!y  0  (6.8)  Where all of the variables remain the same as the SSD scenario on a level tangent road except: fsupply = Random variable for coefficient of friction supplied by the road R = Radius in metres, input by the user e = Superelevation in m/m, input by the user frem = Random variable calculated using Equation 4.7 from the random variable the coefficient of friction. For the SSD on a horizontal curve analysis, there were two modes of failure. The first mode occurred when there was insufficient friction for the vehicle to remain on the horizontal curve, as calculated by Equation 6.7. The second mode of failure occurred  75  when the there was insufficient distance for the vehicle to stop within the available SSD, as calculated by Equation 6.8. 6.2  HORIZONTAL CURVES  The analysis of horizontal curves investigated two different scenarios. The first is was the speed at which a passenger vehicle would skid out on a horizontal curve. The second scenario investigated the speed at which a motorist would feel uncomfortable while travelling around a horizontal curve. The details for each analysis are noted in the paragraphs below. For the horizontal curve analyses, the centreline radius of the road was used. As noted in Chapter 4, the desired vehicle path will change from driver to driver and will be based on the geometric characteristics of the road and horizontal curve. The radius of the desired path would be a random variable. However for this research, there was insufficient data available to calculate this value and the radius of the horizontal curve was used instead. 6.2.1  Maximum Speed Passenger Vehicles on a Horizontal Curve  The analysis was done using an urban speed Of 50 km/h and rural highway speed of 80 km/h on wet and dry pavements for passenger vehicles. These speeds are the posted speed and the design speed. The limit states equation for the maximum speed on a horizontal curve without skidding-out is shown in Equation 6.9. There is one mode of failure for this limit state. V g(e + 0.925/, ) 2  GFX = R,sup ply  (6.9)  Where: = radius supplied by the road design in m. The user inputs this value during the analysis. V=  Random variable for velocity in m/s  g=  Acceleration due to gravity in m/s  76  e=  Superelevation in m/m. Assumed to be a constant of 6% for 50 km/h and 80 km/h design speeds.  f =  Random variable for the force of friction  x  The probability of failure and the reliability index were calculated using Monte Carlo simulations, FORM and SORM for a variety of supplied radii at each of the design speeds. A l l of the random variables used in the analysis would have the probability distributions, minimum and maximum values as noted in Section 6.4. 6.2.2  Maximum Speed on a Horizontal Curve without Feeling Uncomfortable  A second analysis was completed to determine the maximum speed for a given radius of a horizontal curve without an occupant feeling uncomfortable. There are two modes of failure. The first mode of failure occurs when a vehicle is travelling too fast and skids off the road. Equation 6.9 gives the limit states equation for this failure mode. The second mode of failure occurs when the lateral acceleration of the vehicle travelling around a horizontal curve exceeds the comfortable limit for the vehicle occupants and is represented by Equation 6.10. This analysis was completed for two design speeds, 50 km/h and 80 km/h. (  GFX = f  comfort  v  ^  1  \21R  S  (6.10)  j  Where: N = the portion of the frictional force that can be utilized without a feeling uncomfortable. All other variables are as per Equation 6.9. 6.3  VERTICAL CURVES  Sight distance restrictions are only found on crest vertical curves during daylight conditions. Sight distance restrictions on sag vertical curves only occur during darkness  77  and can be eliminated with street lighting. This research investigated the ability to stop on both a crest and a sag vertical curve. 6.3.1  Crest Vertical Curves  There are two conditions for the calculation of vertical curves. The first occurs when the SSD is greater than the vertical curve length, the second occurs when the SSD is less than the length of vertical curve. Equation 6.11 shows the limit states equation when the SSD is shorter than the vertical curve. Equation 6.12 represents the limit states equation when the SSD is longer than the vertical curve. 2SSD  r  GFX = VC-  200(jH~ + Jh} A  2  V  con  GFX = VC  (6.11)  2  v  *A  (6.12)  2OO(V77 + V A J Where: H = Random variable for height of eye, in metres h = Object height, assumed to be 0.38 m SSD = Stopping sight distance as calculated using the random variables found in Equations 5.1 and 5.2 depending on the vehicle type. A = Algebraic difference in grades calculated by input from the user, in percent VC = Length of vertical curve in metres. This value is input by the user during the analysis. 6.3.2  Sag Vertical Curves  Like the crest vertical curves, there are two sag vertical equations. The first occurs when the SSD is greater than the vertical curve length and the second occurs when the SSD is less than the vertical curve length. Equation 6.13 shows the limit states equation when  78  the SSD is shorter than the vertical curve. Equation 6.14 represents the limit states equation when the SSD is longer than the vertical curve. This analysis was completed for passenger vehicles and for heavy trucks on wet and dry pavement. There was one failure mode for this analysis, which was that the vertical curve would be too short to provide drivers with adequate SSD to stop. The analysis assumed that the required SSD was based on a number of variables as noted in Equations 6.1 and 6.2 depending on the variables. GFX = VC-  2SSD 200(/z + SSD * tan a} 3  v  A ccn  GFX = VC  A  (6.13)  2  -.—— ^*A 200(/* + SSD* tana)  (6.14)  3  Where: I13 = Random variable for headlight height in m a = Upwards angle of headlight, assumed to be 1 degree. . SSD = Stopping sight distance as calculated from the random variables found in Equations 6.1 and 6.2 depending on the vehicle type. VC = length of vertical curve in metres. The user in the analysis provides this distance. A = Algebraic difference between the grade in and the grade out in percent. This term is calculated based on information input during the analysis. 6.4  VARIABLES USED INTHE ANALYSIS  In transportation engineering, the type of probability distribution for each variable and the characteristic values are not well defined. Much of the literature published for transportation engineering variables has centred on providing a single, conservative value for that variable.c Researchers often do not comment on the how well the experimental  79  values fit a known distribution or even publish the complete range of values for a variable. For example in the literature for perception reaction time, values for the mean, 85 percentile and standard deviation are commonly found. There is fewer, if any, th  information on say values of the 15 percentile perception reaction time. th  The values for the variables used in the analysis represent the best estimate for the probability distribution of each of the variables and the characteristic values of that distribution. As was found in early limit states applications in structural engineering, there some instances in which the literature contains conflicting values for variables (Ellingwood et. al, 1980). Wherever possible the validity of the probability distribution was tested using the Chi Squared Test. All of the variables used in this research were assumed to be uncorrelated. Given the availability of data, there was no way to test the validity of this assumption. This section outlines the variables that were used in the analysis. Not all of the variables were used for each analysis. The preceding sections describe how each variable was used in the limit states equations. 6.4.1  Vehicle Speed  Operating speeds on roadways are highly variable, with the desired-operating speed changing from one road element, such as a horizontal curve, to the next. For a particular alignment element, drivers will select their desired operating speed for that element. The 85 percentile speed of those individual drivers would be the operating speed. Speed was th  assumed to be normally distributed variable because most 85 percentile speeds can be th  estimated by adding the mean speed to the standard deviation of the speed (TRB, 1998). Using a normal distribution, one standard deviation above the mean represents about 85 percent of the area under the normal distribution curve. On rural roads as the posted speed of a road increases, the speed dispersion tends to decrease. Similarly, as the posted speed increases the number of vehicles traveling in excess of the speed limit tends to be smaller. Research on speed prediction models has found that the 85 percentile speeds on rural two-lane highways with design speeds less th  80  than 100 km/h is greater than the design speed. It also found that when the design speed is greater than 100 km/h the 85 percentile speed tends to be less than the design speed (FHWA, 2000). On two-lane rural highways, desired speeds on long tangents tended to range between 95.5 km/h and 102.4 km/h with an overall average of 97.9 km/h. The desired speeds on tangents were found to be representative for two-lane highways with posted speeds that ranged from 80 km/h to 100 km/h (Ottesen and Krammes, 2000). For this analysis, it was assumed that trucks and cars would travel at the same speed in both wet and dry conditions. On actual roads, these assumptions are not always correct. When steep grades are present, trucks have the potential to travel more slowly than passenger vehicles. In urban areas, trucks may travel slower due to the presence of traffic signals and relatively slower acceleration to full urban speeds. FHWA research on speed prediction models has found that few drivers slow down on wet pavement when compared to travel speeds on dry pavement (FHWA, 2000). The mean and standard deviation for speed were developed for a hypothetical road in a rural and in an urban area. Table 6.2 shows the mean and standard deviation for each hypothetical road. While the speed distributions noted in Table 6.2 appear high, the data is not out of line with actual conditions. Since most automobiles can travel faster than the design and posted speeds, speed is somewhat limited by enforcement of the speed limits and geometric characteristics of the road. Table 6.2: Vehicle Speeds  Area  Urban Rural  Design Speed  Mean  Standard Deviation  50 km/h 80 km/h  50 km/h 77 km/h  16.0 km/h 16.14 km/h  Note: The design speed and the posted speed are the same.  If this analysis were completed on an actual road design, the vehicle speeds may be different between passenger vehicles and heavy trucks since trucks can be heavily influenced by steep grades. For the design for a highway, the use of existing speed distributions for local traffic conditions would be a valuable addition to the analysis.  81  6.4.2  Perception Reaction Time  Perception reaction time was assumed to be log-normally distributed for this study. The data in Olsen's studies on perception reaction time were found to be normally distributed (Olsen et. al., 1984). However, newer research suggests that perception reaction time may have a lognormal distribution (Davis et. al., 2002). The literature indicates that other probability distributions have been used to describe perception reaction time. Researchers at the University of Washington have assumed for the development of their CMP models that perception reaction time was normally distributed. To simplify their calculations they used the Weibull distribution to approximate the normal distribution (Wang et. al, 2002). The average value used in this study for the perception reaction time for passenger vehicles was 1.5 s with a standard deviation of 0.4 s. These values are based on a study conducted by Lerner. The 85 percentile perception reaction time was reported to be th  1.9 s (Fitzpatrick and Woodridge, 2001). On heavy trucks, there is a delay between the application of the brakes by the driver and the start of vehicle braking, which is referred to as brake lag. Brake lag will vary depending on the heavy vehicle type and how well the brakes are adjusted with a typical brake lag time of about 0.4 s to 0.5 s (Reust, Timothy J, 2003). Brake lag times can range from about 0.3 s to about 0.65 s (New Brunswick, 2003). To compensate for the brake lag, an additional half-second was added to the mean perception reaction time used for trucks. The standard deviation was assumed to remain the same. Table 6.3 summarizes the perception reaction times used in this research. Table 6.3: Perception Reaction Times Vehicle  Mean (s)  Standard Deviation (s)  Passenger Car  1.5  0.4  Heavy Truck  2.0  0.4  Source: Perception reaction time Lerner (Fitzpatrick and Woodridge, 2001)  82  6.4.3  Vehicle Braking  Vehicle deceleration for a given speed is based on the coefficient of friction between the tires and road. In this research, a vehicle deceleration on wet pavement was based on n  th  percentile measurements of the coefficient of friction from a study completed by K. Schulze (Navin, 1992).  The resulting distribution was assumed to be normally  distributed. This assumption was tested using a Chi-Squared test and was found to be correct at the 95% confidence level. The results of the Chi-Squared tests can be found in Appendix B. Vehicle deceleration on dry pavements was based on data found in a driver braking performance study (Fambro et. al., 2000). A normal distribution was also assumed for the coefficient of friction on dry pavements. This assumption was tested and found to be correct using a Chi-Squared test (found in Appendix B). Table 6.4 shows the mean and standard deviation values used in this analysis. This distribution was used for the urban and rural conditions. Table 6.4: Coefficient of Friction  Pavement Condition  Design Speed  Mean  Standard Deviation  Wet Wet Dry Dry  50 km/h 80 km/h 50 km/h 80 km/h  0.4829 0.3358 0.8852 0.8852  0.1207 0.1243 0.0949 0.0949  Source: Wet conditions: Olsen et. al., 1984. Dry conditions: Fambro et. al. 2000  6.4.4  Relative Truck Braking Efficiency  Truck braking follows a different model than for passenger vehicles.  As noted in  Chapter 4, an additional variable N is required to approximate the braking distance required for heavy trucks with conventional braking systems. The values for N were developed from vehicle testing on dry pavements for a variety of different truck types, loading conditions and truck brake combinations (Navin, 1986). The values for N were assumed to be normally distributed. This assumption was checked using a Chi-Squared 83  test and was found to be correct at the 95% confidence interval. The value of N for dry pavements was assumed to be representative of wet pavement conditions. Table 6.5 shows the mean and standard deviation of N. Table 6.5: Relative Braking Efficiency Pavement Condition  Design Speed  Mean  Standard Deviation  Wet & dry  50 & 80 km/h  0.599  0.102  Source: Navin, 1986  6.4.5  Speed of a Collision with a Serious Injury  Pedestrians and cyclists are among the most vulnerable of all road users they have as much as 50 times less mass than the vehicle. In a collision with a vehicle, they will absorb much of the energy. Thus, the speed at which a collision can occur without causing serious injury to the pedestrian is low (Navin, 1995). Three different types of collisions were investigated: car to pedestrian, car to car and truck to car. The impact speeds associated with each collision type unlikely to cause a serious injury are noted in the paragraphs below. Car-truck collisions are similar to car-pedestrian collisions in that the mass differential can be substantial. In this analysis, it was assumed that the impact speed that did not cause a serious injury in car-pedestrian collision would be the same as for a truck - car collision. The data on car-to car injuries came from Japanese sources. The collision speeds for a slight injury may be lower than those using North American data (Navin, 1995). Table 6.6 shows the mean and standard deviations of the impact speed of a car and pedestrian for a slight pedestrian injury and the impact speed for two cars with slight occupant injury. It was assumed that the impacts for non-serious injuries was normally distributed. This assumption was tested using a Chi-Squared test and found to be true at the 95% confidence level. The results of the Chi-Squared test are found in Appendix B.  84  Table 6.6: Impact Speed for Non-Serious Injury Collision Type  Car to pedestrian Car to car  Mean (km/h)  Standard Deviation (km/h)  24.30 26.35  12.04 17.84  Source: Navin, 1995  6.4.6  Maximum Comfortable Lateral Friction  Passenger vehicle occupants may not always feel comfortable travelling around horizontal curves even though there is sufficient frictional force to prevent a passenger vehicle from skidding off the pavement. This variable was used in one of the horizontal curve analyses. Table 6.7 shows the mean and standard deviations for frictional forces at which a vehicle occupant feels uncomfortable travelling around a horizontal curve. Table 6.7: Maximum Comfortable Lateral Friction Around a Horizontal Curve Vehicle Occupant  Passenger  Mean  Standard Deviation  0.372647  0.018851  Source: Felipe, E , 1996  It was assumed that the maximum comfortable frictional force was normally distributed, although there was not enough data to test this assumption with a Chi-Squared test. 6.4.7  Height of Eye  The height of eye is used in the calculation of the length of crest vertical curves. The height of eye varies due to the height of the driver and the height of the vehicle. In this study, two distributions were used, one for passenger vehicles and the other for heavy trucks. Table 6.8 shows the mean and standard deviation of each distribution.  85  Table 6.8: Height of Eye  Vehicle Type  Cars Multi-purpose vehicles* Heavy trucks  Mean (m)  Standard Deviation (m)  5 Percentile (m)  15 Percentile (m)  1.149 1.482  0.055 0.130  1.060 1.264  1.094 1.331  2.447  0.107  2.304  2.341  th  th  Source: Oregon Department of Transportation, Discussion Paper No. 8.A, Oregon State University, May 2002 also Tables 3 1 - 3 3 Fambro et. al. (1997) Note: T h i s vehicle class was not used in the analysis.  It was assumed for this research that the height of eye is normally distributed. However based on the research by Oregon DOT (2002) and Fambro et. al. (1997) this assumption may be incorrect for several reasons. The car vehicle type used in the analysis includes cars and multi-purpose vehicles (SUV's, pickup trucks and vans) but the "car" distribution for height of eye was used in the analysis. In addition if the height of eye were normally distributed the 15 percentile height of eye should be approximately one th  standard deviation below the mean. The 15 percentile values listed in Table 6.8 are th  close to but not exactly one standard deviation below the mean. 6.4.8  Headlight Height  One of the variables used in the calculation of sag vertical curves is headlight height. Like the height of eye, this variable is dependant on the vehicle population. Table 6.9 shows the values for headlight heights used in the analysis. Table 6.9: Headlight Height Vehicle Type  Car Heavy truck  Mean (mm)  Standard Deviation (mm)  649 1121  41 88  Source: Passenger vehicle heights Table 31 and heavy truck heights Table 33 Fambro et. al. (1997)  86  6.4.9  Maximum and Minimum Values  In RELAN, it is possible to restrict the values of the distributions between certain values. This allows the analyst to put realistic limits to the variables. Most of the variables used in the analysis were assumed to be normally distributed. Since the range of values of the normal distribution extends from negative infinity to infinity and none of the variables used in the SSD analysis have values that extend the whole range of the normal distribution, the values of each variable were limited to a specific range. Table 6.10 shows the maximum and minimum values for each of the random variables. Table 6.10: Maximum and Minimum Variable Values Variable  M a x Value  M i n Value  Perception reaction time (passenger vehicle) Perception reaction time (heavy truck) Vehicle speed (50 km/h design speed) Vehicle speed (80 km/h design speed) Coefficient of friction (wet) Coefficient of friction (dry) Relative braking efficiency, N (trucks only) Speed for slight pedestrian injury (car to pedestrian, collision) Speed for slight car occupant injury (car to car collision) Speed for slight car occupant injury (car to truck collision) Maximum tolerable lateral friction around a horizontal curve Height of eye (heavy truck) Height of eye (passenger vehicle) Headlight height (heavy truck) Headlight height (passenger vehicle)  5.0 s 5.5 s 100 km/h 200 km/h 0.8 1.0 1.0 70 km/h  0.5 s 1.0 s Okm/h Okm/h 0.001 0.001 0.001 Okm/h  130 km/h  Okm/h  70 km/h  Okm/h  0.85  0.1  2.25 m 0.90 m 0.90 m 0.50 m  2.70 m 1.70 m 1.40 m  87  1.00 m  CHAPTER 7: RESULTS The results of the RELAN analyses are in this Chapter along with a discussion of some of the results. Appendices C, D and E contain the detailed results of each of the analyses performed. 7.1  STOPPING SIGHT ANALYSIS  An analysis was completed on the available stopping sight distance for different geometric design conditions. Table 7.1 shows the different combinations that were investigated. Table 7.1: Stopping Sight Analysis Completed SSD Analysis Type  No-PDO No Injury Horizontal Curve Grade  7.1.1  Vehicle Type  Pavement Condition  Car  Truck  Wet  Dry  X  X  X  X  X  X  X  X  X X  X  Design Speed 50 km/h  80 km/h X  X  X  X  X  X  X  No P D O Condition  It was found that the distance required to stop, on a level tangent for a given speed, varied considerably depending on the vehicle type and the pavement surface. Using the variable distributions noted in the previous section, Table 7.2 shows the probability that the vehicle cannot stop based on the stopping sight distance specified in TAC.  88  Table 7.2: Probability of Not Stopping within M i n i m u m SSD  Vehicle Type & Pavement Condition  M i n . T A C SSD (m)  Truck on wet pavement Truck on dry pavement Car on wet pavement Car on dry pavement  115 - 140 115-140 115-140 115-140  Beta  -0.78 0.75 0.19 2.21  to to to to  Probability of Not Stopping in the SSD  0.78 to 0.63 0.23 to 0.080 0.43 to 0.26 0.014 to 0.002  -0.34 1.41 0.64 2.96  Note:.SSD as noted in Table 1.2.5.3 for cars and trucks with A B S (TAC, 1999).  The 1999 TAC Design Guidelines recognized that heavy trucks with conventional braking systems do not brake as effectively as passenger vehicles and trucks with ABS braking systems. They developed some SSD guidelines for heavy trucks. Table 7.3 shows the probability that a truck cannot stop based on the TAC stopping sight distance for heavy trucks. Table 7.3: Probability of a Truck Not Stopping within Minimum SSD  Vehicle Type & Pavement Condition  T A C SSD (m)  Beta  Probability of Not Stopping in the SSD  Truck on wet pavement Truck on dry pavement  155-210 155-210  -0.18to0.41 1.66 to 2.61  0.57 to 0.34 0.05 to 0.004  Note: SSD as noted in Table 1.2.5.4 for trucks without A B S (TAC, 1999).  Figure 7.1 shows the probability of being unable to stop in a given sight distance for passenger vehicles and heavy trucks on wet and dry pavement.  89  Figure 7.1: Probability of Being Unable to Stop  350 Stopping Sight Distance (m)  Note: The design speed was 80 km/h  Tables 7.2 and 7.3 show there is a significant probability of a vehicle being unable to stop within the minimum stopping sight distance found in the TAC Geometric Design Guidelines.  On our hypothetical road, we assumed that the mean speed would be  77 km/h with a standard deviation of about 16 km/h. Using a normal distribution, almost half of the traffic is likely to be exceeding the design speed.  When the SSD was  calculated in TAC, it was assumed that the operating speeds would vary between the assumed operating speed of a low volume road, which is lower than the design speed and the design speed. This range is thought to be representative of the fact that some drivers will slow down on wet pavements (TAC, 1999). The variation in the coefficient of friction and the perception reaction times may also lead to longer stopping sight distances in some cases. While the expected value for each of these variables was less conservative than assumed in TAC, the "conservative" values used in TAC would be exceeded some of the time.  90  On roads where motorists are less likely to exceed the posted speed limit, through increased enforcement or other speed management tools, the probability of being unable to stop within a specific SSD would decrease. The probabilities listed in this analysis are the probability that a vehicle-driver combination will be unable to stop in the specified SSD. This does not mean that a collision will occur. A collision would only occur if a vehicle-driver combination was traveling too fast to stop within the provided SSD, and there was an object or person on the road at the same time. 7.1.2  Injury Prevention Scenarios  Figures 7.2 through 7.5 show the results of the four injury prevention scenarios. Neither TAC nor AASHTO have developed design guidelines around this type of scenario. Figure 7.2: Probability of Being Unable to Decelerate to a Speed Unlikely to Cause Serious Injury in a Pedestrian-Car Collision  SSD from TAC at 50 km/h design speed ranges from 60 m - 65 m  8  0.7  100  150  200  Stopping Sight Distance (m)  Note: The design speed was 50 km/h  91  Figure 7.3: Probability of Being Unable Decelerate to a Speed Unlikely to Cause Serious Injury in a Truck-Car Collision  w  g 0.7 C  .2 a. E 0.6 o V  I  TAC SSD for 50 km/h design speed ranqes from 60 mto65 m  -HB— Wet Pavement,  \  0.5  o >» = 0.4  H—Dry Pavement  n n n o  \  al 0.3  \  \  .  0  50  :~"",-;;-----,-r.-^  100  1  3zr:=";:ffl  150  200  250  Stopping Sight Distance (m)  Note: The design speed was 50 km/h  Figure 7.4: Probability of Being Unable Decelerate to a Speed Unlikely to Cause Serious Injury in a C a r - C a r Collision  V\. \  f  0.6  TAC SSD for 80 km/h rangeis from 110 m to 140 m  w w \  \\  \\ \\  —•—wet pavement —B— dry pavement  a 0  20  40  60  80  100  120  Stopping Sight Distance (m)  Note: The design speed was 80 km/h  92  140  ,  160  B—i  180  200  Figure 7.5: Probability of Being Unable Decelerate to a Speed Unlikely to Cause Serious Injury in a Pedestrian - C a r Collision, with Excessive Speed  \\  ~"*\  \  S 0.7 c .2 "5. E 0.6  (  \  TAC SSD Range for 50 km/h 60 m to 65 m  V  I i  \  —•—wet pavement —B— dry pavement  — '  0  50  100  •  1  150  "  200  t—  »  250  300  350  Stopping Sight Distance (m)  Note: The design speed was 50 km/h, however the car was assumed to be traveling at 90 km/h. The probabilities of being unable to decelerate to a speed unlikely to cause serious injury can be relatively high especially for the car-truck collision scenarios. The probability of a truck being unable to decelerate to a speed unlikely to cause serious injuries to car occupants is between 10% and 14% on dry pavements using the minimum value specified in TAC for the SSD for a design speed of 50 km/h. On wet pavement, the probability of a truck being unable to decelerate to a speed unlikely to cause serious car occupant injury increases to between 29% and 34%. This analysis was completed using the Monte Carlo simulation and adaptive sampling. FORM and SORM methods did not work over the entire range of SSD evaluated for each scenario. For the lower ranges of the SSD evaluated the results of FORM/SORM and Monte Carlo simulation were similar. However once a certain point was reached and the probability of being unable to stop was less than some threshold value, the probabilities of being unable to stop became close to zero using FORM/SORM methods. To  93  overcome this problem adaptive sampling was used as a means of calculating the design point. Table 7.4 shows the probability of being unable to decelerate to a speed unlikely to cause injury for each of the injury prevention scenarios with a car for the minimum specified SSD in TAC. Table 7.4: Probability of Not Decelerating to a Speed Unlikely to Cause Injury within the Minimum SSD  Vehicle Type & Pavement Condition  Car to car on wet pavement Car to car on dry pavement Truck to car on dry pavement Truck to car on wet pavement Car to pedestrian on wet pavement Car to pedestrian on dry pavement Car to pedestrian on wet pavement (90 km/h) Car to pedestrian on dry pavement (90 km/h)  Minimum T A C SSD (m)  Beta  Probability of Not Decelerating in the Specified SSD  115-140  1.87 to 2.66  0.03 to less than 0.5%  115 - 140  2.40 to 3.22  0.01 to less than 0.05%  60-65  0.77 to 0.95  0.22 to 0.17  60-65  0.21 to 0.34  0.42 to 0.37  60-65  0.98 to 1.15  0.16 to 0.13  60-65  1.75 to 1.98  0.04 to 0.02  60-65  -3.77 to-2.12  0.99 to 0.98  60-65  -0.99 to -0.44  0.84 to 0.67  Note: SSD as noted in Table 1.2.5.3 for cars and trucks with A B S (TAC, 1999)  Like the operational condition, the probability of being unable to decelerate to a certain speed does not mean that there will be a collision. A collision will only occur if there is another object on the road at the time when the vehicle is traveling too fast to decelerate to within a speed unlikely to cause serious injury.  94  7.1.3  Stopping Sight Distance on a Downgrade  The probability of being unable to stop within a specified SSD increases for all scenarios when passenger vehicles are travelling on a downgrade. As shown in Figure 7.6 the steepness of the grade increases the probability of being unable to stop within a specified SSD. Similar to the other SSD scenarios investigated, the probability of not stopping increased on wet pavements.  The steep downgrade also appears to increase, the  probability of being unable to stop on wet pavement. This result is expected based on the known higher occurrence of collisions on downgrades in wet conditions. Figure 7.6: Probability of a Passenger Vehicle being Unable to Stop Within a Specified SSD on a Downgrade  350 Available SSD (m)  As noted in Chapter 4, there are procedures noted in TAC to adjust the stopping sight distance. The SSD can be calculated based on a coefficient of friction that has been modified for downgrades as per the TAC equations found in Chapter 4 (Equation 4.4). Table 7.5 shows the probability of being unable to stop within the minimum SSD for level ground and the SSD based grade adjusted SSD values found in TAC. 95  The  probabilities of failure, beta and the design point were calculated using FORM and SORM methods. The results can be found in Appendix C. Table 7.5: Probability of Not Stopping within Minimum SSD on a Downgrade  Vehicle Type & Pavement Condition  Car on wet pavement -5% Car on dry pavement -5% Car on wet pavement -10% Car on dry pavement -10% Car on wet pavement -15% Car on dry pavement -15% Notes:  Min. TAC SSD (m)  Adjusted SSD for Downgrade (m)  Beta  Prob. of Not Stopping in the SSD  115 - 140  125.8 to 156.3  -0.15 to 0.30  0.56 to 0.38  115-140  125.8 to 156.3  1.92 to 2.66  0.027 to 0.004  115 - 140  145.1 to 181.5  -0.45 to -0.02  0.67 to 0.51  115 - 140  145.1 to 181.5  1.65 to 2.39  0.05 to 0.008  115 - 140  177.2 to 223.5  -0.71 to -0.29  0.76 to 0.61  115 - 140  177.2 to 223.5  1.40 to 2.13  0.08 to 0.02  1  2  3  1. SSD as noted in Table 1.2.5.3 for cars (TAC, 1999) 2. Based on Equation 4.4 (TAC 1999) 3. Probability of being unable to stop and the beta are calculated based on level ground SSD  As noted in the above table, the SSD distances to provide a similar probability of not being able to stop to those of level ground for downgrades. The steeper the downgrade the more difficult it becomes to stop especially on wet pavement. 7.1.4  Stopping Sight Distance on a Horizontal Curve  The probability of being unable to stop increased on a horizontal curve compared with a tangent section. This analysis compares the probability of being unable to stop within the available SSD for several different radius horizontal curves for a design speed of 80 km/h and a tangent section. Figure 7.7 shows the probability of being unable to stop on a horizontal curve. The probability of being unable to stop while driving on a horizontal curve is not explicitly discussed in either the TAC or the AASHTO design guidelines. Instead, the  96  guidelines use low frictional values that should permit motorists to brake within the SSD on a minimum radius horizontal curve. Table 7.6 indicates the probability of failure for specified SSD while braking on a horizontal curve of 250 m with a superelevation of 0.06 m/m. Table 7.6: Probability of Not Stopping within Minimum SSD on a Horizontal Curve  Pavement Condition Car on dry pavement Car on dry pavement Car on dry pavement Car on dry pavement Car on damp pavement Notes:  1. 2. 3.  Min. TAC SSD (m)  Horizontal Radius (m)  Beta  Probability pf Not Stopping in the SSD  115 - 140  225  0.74 to 1.34  0.23-0.091  115-140  250  0.75 to 1.35  0.23-0.088  115-140  300  0.77 to 1.40  0.22 to 0.087  115 - 140  500  0.80 to 1.45  0.21 to 0.074  115 - 140  500  -0.15 to 0.89  1  2  2  3  0.56 to 0.18  3  SSD as noted in Table 1.2.5.3 for cars (TAC, 1999) Probability of not stopping and beta are the values calculated for the system i.e. both modes of failure. Probability of not stopping and beta are for SSD ranging from 100 m to 150 m.  The analysis for stopping on a horizontal curve was only completed using the Monte Carlo simulations for dry pavement. One analysis run was completed using "damp" pavement, which has coefficient of friction characteristics somewhere between wet and dry pavement conditions.  This analysis was completed this way because the wet  pavement runs did not work in RELAN. It is not known why the wet pavement analysis did not work properly, since the dry pavement analysis did work. One possibility was that there was insufficient frictional forces left for braking and as a result, failure occurred. It is unclear why the probability of not stopping could not be recorded as close to 100%. As noted in Figure 7.7 and in Table 7.6, the probability of being unable to stop within the specified SSD increases as the horizontal radius decreases. On dry pavement with a minimum radius horizontal curve, the SSD required to achieve similar probabilities of  97  non-compliance to those of tangent section on dry pavement increased to a point about half way between the SSD required to achieve the same probability of non-compliance for a tangent on wet pavement. This is a substantial increase in the SSD to achieve the same level of risk as on dry pavement. It should be noted that the speed distributions and the friction distributions remained the same for all of the SSD analyses. Therefore, the levels of risk are comparable. The results of the damp pavement analysis indicate that the probability of being unable to stop with in a specified SSD was similar to those on a tangent with wet pavement. The shape of the damp pavement curve in Figure 7.7 suggests that the probability of being unable to stop within a specific SSD was greater than for a tangent section with wet pavement when the available SSD was below 150 m. When the available SSD increased above 150 m then the probability of being unable to stop was lower on a curve with damp pavement than for a tangent with wet pavement.  These results suggest that when  traveling on wet pavement around a horizontal curve, the ability to make stop within the minimum SSD specified in TAC is limited. Figure 7.7: Probability of a Passenger Vehicle Being Unable to Stop Within a Specified Stopping Sight Distance on a Horizontal Curve  0  50  100  150  200  250  300  350  Specified SSD (m)  Notes:  The design speed was 80 km/h and the rninimum SSD is between 115 m and 140 m. The superelevation is based on T A C Table 2.1.2.6.  98  7.2  HORIZONTAL CURVE ANALYSIS  Two analyses were completed for horizontal curves. The first analysis investigated the probability of skidding out on horizontal curve by a passenger vehicle. It was assumed that all of the available friction could be utilized by the driver-vehicle combination to stay on the horizontal curve. It also assumed that passenger comfort was not an important factor.  The second analysis investigated the probability of a passenger being  uncomfortable when travelling around a horizontal curve in a passenger vehicle. Each analysis used wet and dry pavement conditions and two design speeds of 50 and 80 km/h. 7.2.1  Maximum Speed of a Passenger Vehicle on a Horizontal Curve  This analysis provides designers with information on the upper performance limits of the driver-vehicle combination. It is unlikely that the values from this type of analysis dry pavement conditions would be used directly in a design application. The wet pavement results could be used under some circumstances to evaluate risk. Figure 7.8 shows the probability of a car skidding out on a road with a given radius and a design speed of 80 km/h.  Figure 7.9 shows the probability of a car skidding out for a  given radius and a design speed of 50 km/h.  The actual travel speed around the  horizontal curve was a random variable as was the coefficient of friction that was available. It was assumed that all of the frictional forces were used to stay on the horizontal curve and that no friction was required for braking purposes. While the probability of skidding out on a horizontal curve may be an interesting performance limit to know, it is unlikely that it would be used for design purposes. In most circumstances, there should be enough friction remaining for emergency braking and travelling around a horizontal curve rather than just applying all of the friction to staying on the curve. Another factor that may make this analysis less useful for design purposes was that the comfort of the vehicle occupants was not considered. It is quite likely that the maximum travel speed around a horizontal curve would be uncomfortable for some of the vehicle occupants.  99  Figure 7.8: Probability of a Car Skidding Out on a Horizontal Curve 80 km/h  0  50  100  150  200  250  R a d i u s of C u r v e  Note: The maximum superelevation rate was 6% for all curves analyzed.  Figure 7.9: Probability of a Car Skidding Out on a Horizontal Curve 50 km/h •  — Max. Speed Dry Pavement  - • — Max. Speed Wet Pavement  0.7  0.4  0.1  10  20  30  40  50  60  70  80  90  Horizontal Radius (m)  Note: The maximum superelevation rate was 6% for all curves analyzed.  For a design speed of 80 km/h, the minimum horizontal curve radius is 250 m. As shown in Figure 7.8, the probability of a passenger vehicle skidding out on the curve is quite low 100  for the minimum radius curve. A similar situation exists for the 50 km/h design speed as well. It should be noted that about half of the vehicles were travelling faster than the design speed. Table 7.7 shows the probability of failure and associated betas for the minimum horizontal radius. Table 7.7: Probability of a Car Skidding Out on a Horizontal Curve  M i n . Radius (m)  Beta  Probability of Not Skidding Out  50 km/h dry pavement  90  3.16  0.0008  50 km/h wet pavement  90  0.082  80 km/h dry pavement  250  1.39 3.11  80 km/h dry pavement  250  1.26  Design Speed  0.0009 0.10  Note: Minimum radius calculated with 6% superelevation. 80 km/h dry pavement calculated with S O R M .  7.2.2  Maximum Comfortable Speed on a Horizontal Curve  The previous section investigated the maximum speed at which a driver could drive a horizontal curve. As was noted earlier in this research, it is likely that some of the drivers would feel uncomfortable on a horizontal curve at the maximum speed. This analysis investigated the maximum speed on a horizontal curve that drivers will likely feel comfortable driving.  Figure 7.10 shows the probability that a motorist will feel  uncomfortable on a given radius driving at a design speed of 80 km/h. Figure 7.11 shows the results for 50 km/h.  Table 7.8 shows the probability of feeling uncomfortable and  beta. Table 7.8: Probability of Feeling Uncomfortable on a Horizontal Curve Design Speed  Minimum Radius (m)  Beta  Probability of Feeling Uncomfortable  50 km/h dry pavement  90 90  1.27 1.27  0.11 0.11  250 250  2.41  0.008  1.36  0.087  50 km/h wet pavement 80 km/h dry pavement 80 km/h wet pavement  Note: The probability of feeling uncomfortable and beta were calculated using the system probabilities.  101  Figure 7.10: Probability of Feeling Uncomfortable on a Horizontal Curve - 80 km/h 1  0  50  1 00  1 50  200  250  300  Radius of Curve  Figure 7.11: Probability of Feeling Uncomfortable on Horizontal Curve - 50 km/h 1  100 Horizontal Radius (m)  102  /  Based on the variable distribution that were used, there is a relatively low probability of a vehicle occupant feeling uncomfortable on horizontal curves that meet the minimum radius requirements set out in TAC for a given design speed. It is important to note that about half of the vehicles will be driving above the posted speed limit for both design speeds. 7.3  VERTICAL CURVE ANALYSIS  The vertical curve analysis was completed for cars and heavy trucks on wet and dry pavement for crest and sag vertical curves. For this analysis, the length of the vertical curve as assumed to be long enough for motorists to see far enough ahead to come to a complete stop before they hit an object on the road. Other sight distances could have been used such as decision sight distance or the injury prevention scenarios. Table 7.9 outlines the geometric characteristics of the vertical curves. Table 7.9: Vertical Curve Characteristics Curve Type  Grade In  Grade Out  Analysis Curve Length  Sag Crest  -4%  4% -4%  Varies  7.3.1  4%  Varies  Minimum Curvature on a Crest Vertical Curve  Figure 7.12 shows the minimum length required for a crest vertical curve to provide stopping sight distance. Table 7.10 shows the probability of insufficient SSD on a crest vertical curve.  103  Figure 7.12: Probability of Sufficient SSD Provided on a Crest Vertical Curve  300  400  800  500  Length of Vertical Curve (m)  Table 7.10: Probability of Insufficient SSD on a Crest Vertical Curve Minimum V C Length (m)  Beta  Probability of Insufficient SSD  Car on dry pavement  192 to 288  2.91 to 4.15  0.002 to 0.00002  Car on wet pavement  0.59 to 1.21  0.28 to 0.11  Truck on dry pavement  192 to 288 192 to 288  1.59 to 2.49  0.056 to 0.006  Truck on wet pavement  192 to 288  -0.045 to 0.51  0.51 to 0.31  Condition  ^  Note:  Minimum vertical curve length based on minimum K value ranges found in T A C Tables 2.1.3.2. The probabilities and beta are based on values close to the minimum T A C V C length.  7.3.2  Minimum Curvature on a Sag Vertical Curve  Figure 7.13 shows the minimum length of sag vertical curve required to provide adequate stopping sight distance. Table 7.11 shows the probability of failure and the beta for the minimum vertical curve lengths specified in TAC.  104  Figure 7.13: Probability of Sufficient SSD Provided on a Sag Vertical Curve  Table 7.11: Probability of Insufficient SSD on a Sag Vertical Curve  Condition  Minimum V C Length (m)  Beta  Probability of Insufficient SSD  0.001 to 0.00005 0.26 to 0.15  Car on dry pavement  200 to 256  Car on wet pavement  200 to 256  3.00 to 3.89 0.63 to 1.05  Truck on dry pavement  200 to 256  1.29 to 1.96  0.10 to 0.03  Truck on wet pavement  200 to 256  -0.24 to 0.18  0.59 to 0.43  Note:  Minimum vertical curve length based on minimum K value ranges found in TAC Table 3.1.3.4. Headlight control is used for sag vertical curves.  7.4  C O M P A R I S O N OF THE C O M P U T A T I O N M E T H O D S  Air of the analysis were completed using FORM/SORM and Monte Carlo simulations except for the injury-prevention scenarios and the SSD on horizontal curves. The design points for the injury prevention scenarios for SSD were calculated using adaptive sampling results. Design points for the SSD analysis on horizontal curves were not calculated.  Figure 7.14 compares the results of the No-PDO analysis (shown in 105  Figure 7.1 using the Monte Carlo simulation results) with the FORM and SORM approximate methods. This figure shows that the FORM results were slightly less conservative than the results from SORM and Monte Carlo simulations. For the SSD analysis on level pavement it is likely that the results of SORM and the simulations are more accurate perhaps due to an error approximating the failure surface in FORM. Figure 7.14: Comparison of the Results from FORM, SORM and Simulation  X \  \ v.\ \ \ VV \ . V  N  |  Car Wet Pavement FORM —a— Car Wet Pavement SORM - -A- - Car Wet Pavement Monte Carlo —w— Car Dry Pavement FORM — * — Car Dry Pavement SORM • — • — Car Dry Pavement Monte Carlo  \  V ss \ .  *\ ' \  v \  \ \ \\  \ \\\  \  V\  ^ \  .  Truck Wet Pavement FORM — - — T r u c k Wet Pavement SORM — Truck Wet Pavement Monte Carlo — * — T r u c k Dry Pavement FORM m Truck Dry Pavement SORM -~ -A - Truck Dry Pavement Monte Carlo  2-  0  50  100  150  200  250  300  350  Stopping Sight Distance  Note: Design speed was 80 km/h for all analyses.  For cars on dry pavement, where the probabilities of being unable to stop within the specified SSD are the lowest, the differences between the three calculation methods are very small. In contrast, the trucks on wet pavement with the highest probabilities of being unable to stop are the highest within a specified SSD, the difference between FORM results and the results obtained by either SORM or Monte Carlo simulations is more pronounced. Appendix F contains figures, which compare the results from each of the analyses completed for the different computational methodologies.  106  !  CHAPTER 8: CONCLUSIONS This Chapter discusses the results of the analyses completed as well as some possible applications in the geometric design of roads and highways. 8.1  RESULTS OF THE  ANALYSES  All of the analyses showed that there was often a significant probability of non-compliance with the minimum design values found in the TAC Geometric Design Guidelines. Much of this disparity has to with the fact that the speed distributions that were used in the analyses. It was assumed that for a design speed of 80 km/h the mean speed would be 77 km/h with a standard deviation of about 16 km/h. For a design speed of 50 km/h, the mean speed was assumed to be 50 km/h with a standard deviation of about 16 km/h.  Using a normal distribution, almost half of the drivers would be  exceeding the design speed of the roadway. The minimum design values for a given design speed assume that drivers will not be exceeding the design speed of the roadway. While the expected values for each of the variables used in the failure functions was less conservative than assumed in TAC, the "conservative" values used in TAC would be exceeded some of the time. On roads where motorists are less likely to exceed the posted speed limit, through increased enforcement or other speed management tools, the probability of non-compliance would I  likely be less than those listed in Chapter 7. 8.1.1  Stopping Distances  When the SSD was calculated in TAC, it was assumed that the operating speeds would vary between the assumed operating speed of a low volume road that is lower than the design speed, and the design speed. This range was thought to be representative of the fact that some drivers will slow down on wet pavements (TAC, 1999). The variation in the coefficient of friction and the perception reaction times may also lead to longer stopping sight distances in some cases.  107  Trucks with conventional braking systems require a longer distance to stop than passenger vehicles. Wet pavement exacerbates this performance difference between cars and trucks. Vehicles trying to stop on downgrades or on horizontal curves will not be able to stop as quickly as vehicles on a level, tangent roadway. Based on the speed and coefficient of friction distributions used in this analysis, it is unlikely that vehicles trying to make an emergency stop on a horizontal curve with wet pavement will be able to stop quickly. This analysis suggests that there will be very little friction leftover for braking manoeuvres after the driver-vehicle combination stays on the horizontal curve when the pavement is wet. Under damp pavement conditions, braking performance on a horizontal curve appears to be similar to that of a tangent with wet pavement. The use of the injury prevention scenarios may be a useful limiting condition under certain circumstances. For example in urban areas, it may be desirable to try to reduce the injuries to vulnerable road users such as pedestrians and cyclists. In order to prevent serious pedestrian injuries from a vehicle travelling at speeds close those considered reckless under British Columbia law (40 km/h over the posted speed limit), the minimum SSD would need to be increased over what is currently provided. To reduce the occurrence of serious pedestrian injury in a collision with a car travelling 90 km/h to less than 20% of the time, the SSD would need to increase to about 170 m from the minimum range of 60 m to 65 m. The probabilities found in Tables 7.2 through 7.6 are the probability that a driver-vehicle combination will be unable to stop or decelerate to a specific speed within the specified SSD. This does not mean that a collision will occur. A collision would only occur if a driver-vehicle combination was traveling too fast to stop within the provided SSD and there was an object or person on the road at the same time. Thus, the probability of a collision would be less than the probability of non-compliance.  108  8.1.2  Horizontal Curves  The probability that a vehicle will skid-out on a minimum radius horizontal curve for a given design speed is in the order of 10% for wet pavement conditions using the probability distributions noted in Section 6.4. The probability that a vehicle occupant will feel uncomfortable on a minimum radius curve is similarly low at around 10%. The probability of skidding out on a horizontal curve is the probability of someone losing control on a horizontal curve. If there is a sufficiently large clear zone then the vehicle may be able to recover and continue undamaged on its trip. 8.1.3  Vertical Curves  The analysis of the length of vertical curves is very similar to the stopping sight analysis. This result is expected since the vertical curve length was based on stopping sight distance. The probability that there was insufficient SSD for a vehicle to stop would not mean that a collision would occur. A collision would only occur if there was a person or object in the road at the same time when a driver-vehicle combination was travelling too fast to stop. 8.1.4  Combination of Geometric Design Elements  Two of the stopping sight analyses investigated the probability of non-compliance when braking was combined with a horizontal curve and on a downgrade. In comparison with actual road conditions, the above combinations are very simple. The results indicated that the ability to stop is limited in both combinations. Design guidelines may not always address the increased risks associated with the combination of geometric elements. 8.2  U S E O F T H ER E S U L T S I N G E O M E T R I C D E S I G N A P P L I C A T I O N S  The probabilities of non-compliance found in this research are based on the probability distributions discussed in Section 6.4. A limited amount of data was used to create the probability distributions for all of the random variables except the initial speed variable, which was completely made-up. Therefore, the confidence in the random variables is  109  moderate. Even though the confidence in the random variables is not high, a sensitivity analysis could be undertaken to determine how sensitive the probabilities of non-compliance are to each variable. Using the results from the sensitivity analysis and the original results could help to give some more confidence in the probability of failures calculated by RELAN. In this instance, the designer could use the original analysis and sensitivity analysis results to develop a level of risk, measured through probability of non-compliance or beta values that is acceptable. It is possible that target beta values similar to what is used in some structural applications could be developed over time. Another use for the results is to use drawings similar to Figures 7.1 through 7.13 to determine where a design parameter is on curve. When a design parameter falls on the steep part of the curve, changing the value of that parameter slightly can change probability of non-compliance considerably. In contrast, on the flat portions of the curve, a change in the value of the design parameter will have very little change on the probability of non-compliance. The use of these curves may be an additional tool for designers to use to make better decisions. For example if Figure 7.1 were to be used for design purposes and it was known that 50% of the traffic was heavy trucks, the design team would likely want to consider increasing the available SSD to decrease the probability that a truck would be unable to stop within the SSD. The design team may also want to investigate ways that they could decrease the vehicle speed as well as increase the skid resistance of the pavement in wet conditions. The use of reliability theory to determine the probability of non-compliance can allow the designer some flexibility in the application of standards by allowing designers to quantify a risk. In this example, designers could quantify a risk by determining the values of variables performing the reliability analysis. If desired, a sensitivity analysis could be completed to determine how the probability of non-compliance and beta values react to changes in the variables. This is important since many of the variables associated with highway geometric design do not have definite probability distributions and agreed upon values.  110  The use of wet and dry pavement in the analysis gives designers the knowledge of the upper bound on design values as well as the lower bound of design values that designers most commonly work with using wet pavement. The application of reliability methods to determine the probability of exceeding the stopping sight distance is a promising concept. For simple designs, the additional work required to complete this type of analysis may not be worthwhile. On detailed design of complex road locations, this methodology may help designers make difficult decisions as to the application of design standards and guidelines.  Ill  CHAPTER 9: RECOMMENDATIONS After studying this topic, it is evident that the use of reliability theory and concepts could be used in the geometric design of roads and highways. Before this theory can be incorporated into design guidelines, further research is necessary. These areas are listed in no particular order. 1. Variables used in the geometric design of roads. The variables used in this analysis were primarily those found currently in the design guidelines. As noted in earlier chapters, there is often no agreement in the literature about the type of distributions a variable follows. In addition, there is sometimes little agreement in the literature as to the mean and standard deviation for some of the variables. This finding and recommendation is similar to those made by Navin (1992) and Easa (2000). Before reliability theory can be widely used, there should be some agreement about the type of probability distribution for each variable as well as some agreement on the values used. 2. A definition for acceptable level of risk for geometric road design that is acceptable to the transportation engineering profession and Canadian society. Meaningful discussions on levels of road risk should be undertaken with Canadian society or their representatives, since society is paying for the costs of construction and operation of roads as well as the consequences of road accidents. It is expected that the level of acceptable risk would change depending on the volume of traffic, type of roadway and the travel speed. Frameworks for the definition of acceptable risk have been made by structural engineers to define acceptable levels of risk for new types of structures. This type analysis is discussed in some detail in Chapter 3 of this research. The framework outlined by Bhattacharya et. al. (2001) is a starting point. Alternatively, some type of decision analysis as proposed by Sexsmith (1999) could be used. 3. Development of a relationship between collision frequency and probability of failure (or not meeting a design parameter) for each of type of design parameter  112  such as stopping sight distance, horizontal curves, vertical curves, grades and combinations of the above. 4. Agreement within the transportation engineering profession as to which limit states should be used in the geometric design of roads and highways. As shown in this research, the use of other guidelines other than those currently in use, such as the serious injury prevention SSD scenarios, may be beneficial. 5. Agreement within the transportation engineering profession as to how limit states design and reliability theory fit into other design and planning tools that are currently being used by the profession. 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In Journal of Transportation Engineering, Vol. 129, No. 4, American Society of Civil Engineers, 2003, pp. 377 - 384. http://facultv.washingtonedu/vinhai/wangpublication files/paper3.pdf Zheng, Zhimei Ronda, Application of Reliability Theory to Highway Geometric Design Ph.D. Thesis, University of British Columbia, Vancouver, BC, April 1997.  119  APPENDIX A FORTRAN FILES USED FOR RELAN ANALYSIS  120  C Template f o r performance f u n c t i o n subroutines C SUBROUTINE DETERM(IM0DE) C subroutine for reading deterministic variables IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/B/ A C I F ( I M 0 D E . G T . 1 ) GO TO 50 WRITE(*,100) 100 F O R M A T ( / ' ENTER DESIGN SSD ' / ) READ(*,*) A C 50 CONTINUE C WRITE ( * , 1 0 2 ) C102 F O R M A T ( / ' ENTER DESIGN S S D ' / ) C READ(*,*) A RETURN END C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) C S u b r o u t i n e t o c a l c u l a t e t h e f a i l u r e f u n c t i o n f o r A V A I L A B L E SSD NO C INJURY FOR CAR TO CAR C O L L I S I O N IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION X(N) COMMON/B/ A R D I S T = A - ( X ( 1 ) * 1 0 0 0/3 6 0 0 * X ( 2 ) ) VSQ=-2*X(3)*9.81*RDIST+(X(l)*1000/3600)**2 I F ( V S Q . L T . 0 ) VSQ=0 I F ( R D I S T . G T . 0 ) THEN VELDEM=SQRT(VSQ) GXP = ( X ( 4 ) * 1 0 0 0 / 3 6 0 0 ) - V E L D E M ENDIF I F ( R D I S T . L E . 0 ) GXP=X(4) - X (1) C RETURN END  121  C Template f o r performance f u n c t i o n s u b r o u t i n e s C SUBROUTINE DETERM(IMODE) C subroutine f o r reading d e t e r m i n i s t i c v a r i a b l e s IMPLICIT REAL*8(A-H,0-Z) COMMON/B/ A IF (IMODE.GT.l) GO TO 50 WRITE (*,100) 100 FORMAT(/' ENTER DESIGN SSD '/) READ(*,*) A C 50 CONTINUE C WRITE (*,102) C102 FORMAT(/' ENTER DESIGN SSD'/) C READ(*,*) A RETURN END C SUBROUTINE GFUN (X,N,IMODE,GXP) C S u b r o u t i n e t o c a l c u l a t e the f a i l u r e f u n c t i o n f o r AVAILABLE SSD NO C INJURY FOR CAR TO PEDESTRIAN COLLISION IMPLICIT REAL*8(A-H,O-Z) DIMENSION X(N) COMMON/B/ A RDIST=A-(X(l)*1000/3600*X(2)) VSQ=-2*9.81*X(3)*RDIST+(X(l)*1000/3600)**2 IF (VSQ.LE.O) VSQ=0 IF (RDIST.GT.0) THEN VELDEM=SQRT(VSQ) GXP=(X(4)*1000/3600)-VELDEM ENDIF IF (RDIST.LE.0) GXP=X(4)-X(1) C RETURN END  122  C Template f o r performance f u n c t i o n subroutines C SUBROUTINE DETERM(IM0DE) C subroutine for reading deterministic variables IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/B/ A C I F ( I M 0 D E . G T . 1 ) GO TO 50 WRITE(*,100) 100 • F O R M A T ( / ' ENTER DESIGN SSD ' / ) READ (*,*) A C 50 CONTINUE C WRITE(*,102) C102 F O R M A T ( / • ENTER DESIGN S S D ' / ) C READ(*,*) A RETURN END C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) C s u b r o u t i n e t o c a l c u l a t e t h e f a i l u r e f u n c t i o n f o r A V A I L A B L E SSD NO C INJURY CAR A T 90 k m / h AND PEDESTRIAN C O L L I S I O N IMPLICIT R E A L * 8 ( A - H , O - Z ) DIMENSION X(N) COMMON/B/ A RDIST=A-(90*1000/3600*X(1)) VSQ=-2*9.81*X(2)*RDIST+(90*1000/3600)**2 I F ( V S Q . L T . 0 ) VSQ=0 IF  ( R D I S T . G T . 0 ) THEN VELDEM=SQRT(VSQ) GXP=(X(3)*1000/3600)-VELDEM ENDIF IF ( R D I S T . L E . 0 ) GXP=X(3)-90 C RETURN END  123  C Template f o r performance function subroutines C SUBROUTINE  DETERM(IM0DE)  C subroutine f o r reading d e t e r m i n i s t i c v a r i a b l e s IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/B/ A  100 C 50 C C102 C  I F ( I M 0 D E . G T . 1 ) GO TO 50 WRITE(*,100) F O R M A T ( / ' ENTER DESIGN SSD READ(*,*) A CONTINUE WRITE(*,102) F O R M A T ( / ' ENTER DESIGN READ(*,*) A RETURN END  '/)  SSD'/)  C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) C subroutine t o c a l c u l a t e the f a i l u r e function f o r A V A I L A B L E SSD NO C INJURY TRUCK TO CAR C O L L I S I O N IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION X(N) COMMON/B/ A RDIST=A-(X(1)*1000/3600*X(2)) TFRIC=X(3)*X(5) VSQ=-2*TFRIC*9.81*RDIST+(X(1)*1000/3600)**2 I F ( V S Q . L T . 0 ) VSQ=0 I F ( R D I S T . G T . 0 ) THEN VELDEM=SQRT(VSQ) GXP = ( X ( 4 ) * 1 0 0 0 / 3 6 0 0 ) - V E L D E M ENDIF IF (RDIST.LE.0) GXP=X(4)-X(1) C RETURN END ,  124  C Template f o r performance f u n c t i o n subroutines C SUBROUTINE DETERM(IMODE) C subroutine for reading deterministic variables IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/B/ A , C I F ( I M O D E . G T . l ) GO TO 50 WRITE(*,100) 100  101 C 50 C C102 C  F O R M A T ( / ' ENTER DESIGN SSD ' / ) READ(*,*) A WRITE ( * , 1 0 1 ) F O R M A T ( / ' ENTER GRADE ' / ) READ(*,*) C CONTINUE WRITE(*,102) F O R M A T ( / ' ENTER READ(*,*) ***** RETURN END  *****'/)  C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) C s u b r o u t i n e t o c a l c u l a t e t h e f a i l u r e f u n c t i o n f o r A V A I L A B L E SSD NO C PDO FOR A CAR ON A DOWNGRADE ON WET PAVEMENT IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION X ( N ) COMMON/B/ A , C S D I S T = X ( 1 ) * 1 0 0 0 / 3 6 0 0 * X (2) PER=.0008*C**2+.0404*C+.9819 FRIC=X(3)*PER BDIST=(X(1)*1000/3600)**2/(2*9.81*FRIC) DDIST=SDIST+BDIST GXP=A-DDIST RETURN END  125  C Template f o r performance f u n c t i o n subroutines C SUBROUTINE DETERM(IMODE) C subroutine f o r reading deterministic variables IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/B/ A , C I F ( I M O D E . G T . l ) GO TO 50 WRITE(*,100) 100  101 C 50 C C102 C  F O R M A T ( / ' ENTER DESIGN SSD ' / ) READ(*,*) A WRITE ( * , 1 0 1 ) F O R M A T ( / ' ENTER GRADE ' / ) READ(*,*) C CONTINUE WRITE(*,102) F O R M A T ( / ' ENTER READ(*,*) ***** RETURN END  *****'/)  C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) C s u b r o u t i n e t o c a l c u l a t e t h e f a i l u r e f u n c t i o n f o r A V A I L A B L E SSD NO PDO C C O L L I S I O N FOR A CAR ON A DOWNGRADE ON DRY PAVEMENT IMPLICIT REAL*8(A-H,O-Z) DIMENSION X(N) COMMON/B/ A , C SDIST=X(1)*1000/3600*X(2) PER=.0004*(C**2)+.02 74*C+.9856 FRIC=X(3)*PER BDIST=(X(l)*1000/3600)**2/(2*9.81*FRIC) DDIST=SDIST+BDIST GXP=A-DDIST RETURN END  126  C Template f o r performance f u n c t i o n subroutines C SUBROUTINE DETERM(IM0DE) C subroutine for reading deterministic variables IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/B/ A  100  I F ( I M 0 D E . G T . 1 ) GO TO 50 WRITE(*,100) F O R M A T ( / ' ENTER DESIGN SSD READ(*,*)  C 50 C C102 C  '/)  A  CONTINUE WRITE(*,102) F O R M A T ( / ' ENTER READ(*,*) ***** RETURN END  *****'/)  C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) C s u b r o u t i n e t o c a l c u l a t e t h e f a i l u r e f u n c t i o n f o r A V A I L A B L E SSD NO PDO C C O L L I S I O N FOR A CAR ON WET OR DRY PAVEMENT IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION X(N) COMMON/B/ A SDIST=X(1)*1000/3600*X(2) BDIST=(X(l)*1000/3600)**2/(2*9.81*X(3)) DDIST=SDIST+BDIST GXP=A-DDIST RETURN END  127  C Template f o r performance f u n c t i o n subroutines C SUBROUTINE DETERM(IMODE) C subroutine for reading deterministic variables IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/B/ A I F ( I M O D E . G T . l ) GO TO 50 W R I T E ( * , 1 0 0) 100 F O R M A T ( / ' ENTER DESIGN SSD ' / ) READ(*,*) A C 50 CONTINUE C WRITE(*,102) C102 F O R M A T ( / ' ENTER * * * * * ' / ) C READ(*,*) ***** RETURN END C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) C s u b r o u t i n e t o c a l c u l a t e t h e f a i l u r e f u n c t i o n f o r A V A I L A B L E SSD NO PDO C C O L L I S I O N FOR A TRUCK ON WET OR DRY PAVEMENT IMPLICIT REAL*8(A-H,O-Z) DIMENSION X(N) COMMON/B/ A T F R I C = X ( 3 ) * X (4) SDIST=X(1)*1000/3600*X(2) BDIST=(X(l)*1000/3600)**2/(2*9.81*TFRIC) DDIST=BDIST+SDIST GXP=A-DDIST RETURN END  128  C Template f o r performance f u n c t i o n subroutines C SUBROUTINE DETERM(IM0DE) C subroutine f o r reading deterministic variables IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/B/ A , C , D C I F ( I M 0 D E . G T . 1 ) GO TO 5 0 WRITE(*,100) 100 F O R M A T ( / ' ENTER LENGTH OF VC I N m ' / ) READ(*,*) A WRITE(*,101) 101 F O R M A T ( / ' ENTER GRADE INTO V C I N PERCENT ' / ) READ(*,*) C WRITE(*,102) 102 F O R M A T ( / ' ENTER GRADE OUT OF V C I N PERCENT ' / ) READ(*,*) D C 50 CONTINUE C WRITE(*,102) C102 F O R M A T ( / ' ENTER * * * * * ' / ) C READ ( * , * ) * • * * * * RETURN END C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) C s u b r o u t i n e t o c a l c u l a t e t h e f a i l u r e f u n c t i o n f o r C r e s t VC LENGTH FOR c CAR C SSD NO PDO WET OR DRY PAVEMENT IMPLICIT R E A L * 8 ( A - H , 0 - Z ) . DIMENSION X(N) COMMON/B/ A , C , D SDIST=X(1)*1000/3600*X(2) BDIST=(X(l)*1000/3600)**2/(2*9.81*X(3)) DDIST=SDIST+BDIST ADIF=ABS(C-D) IF (DDIST.GT.A) CREST=DDIST**2/(2 00*(SQRT(X(4))+SQRT(.38))**2) I F ( D D I S T . L E . A ) THEN T1=2*DDIST/ADIF T2=(2 0 0 * ( S Q R T ( X ( 4 ) ) + S Q R T ( . 3 8 ) ) * * 2 ) / A D I F * * 2 CREST=T1-T2 ENDIF VCL=CREST*ADIF GXP=A-VCL RETURN END  129  C Template f o r performance f u n c t i o n subroutines C SUBROUTINE DETERM(IMODE) C subroutine f o r reading deterministic variables IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/B/ A , C , D C I F ( I M O D E . G T . l ) GO TO 50 WRITE(*,100) 100 F O R M A T ( / ' ENTER LENGTH OF VC I N m ' / ) READ (*,*)• A WRITE(*,101) 101 F O R M A T ( / ' ENTER GRADE INTO VC I N PERCENT ' / ) READ(*,*) C WRITE(*,102) 102 F O R M A T ( / ' ENTER GRADE OUT OF VC I N PERCENT ' / ) READ(*,*) D C 50 CONTINUE C WRITE(*,102) C102 F O R M A T ( / ' ENTER * * * * * ' / ) C READ(*,*)***** RETURN END C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) C s u b r o u t i n e t o c a l c u l a t e t h e f a i l u r e f u n c t i o n f o r C r e s t VC LENGTH FOR C TRUCK C SSD NO PDO WET OR DRY PAVEMENT IMPLICIT REAL*8(A-H,O-Z) DIMENSION X(N) COMMON/B/ A , • C , D F R I C = X ( 5 ) * X (3) SDIST=X(1)*1000/3600*X(2) BDIST=(X(l)*1000/3600)**2/(2*9.81*FRIC) DDIST=SDIST+BDIST ADIF=ABS(C-D) IE (DDIST.GT.A) CREST=DDIST**2/(2 00*(SQRT(X(4))+SQRT(.38))**2) I F ( D D I S T . L E . A ) THEN T1=2*DDIST/ADIF T2=(200*(SQRT(X(4))+SQRT(.38))**2)/ADIF**2 CREST=T1-T2 ENDIF VCL=CREST*ADIF GXP=A-VCL RETURN END  130  C Template f o r performance f u n c t i o n subroutines C SUBROUTINE DETERM(IMODE) C subroutine for reading deterministic variables IMPLICIT R E A L * 8 ( A - H , Q - Z ) COMMON/B/ A C I F ( I M O D E . G T . l ) GO TO 50 WRITE(*,100) 100 F O R M A T ( / ' ENTER DESIGN SSD ' / ) READ(*,*) A C 50 CONTINUE C WRITE(*,102) C102 F O R M A T ( / • ENTER DESIGN R A D I U S ' / ) C READ(*,*) A RETURN END C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) C s u b r o u t i n e t o c a l c u l a t e t h e f a i l u r e f u n c t i o n f o r MAXIMUM SPEED OF A C CAR ON A HORIZONTAL CURVE WITHOUT SKIDDING OUT WET OR DRY PAVEMENT IMPLICIT R E A L * 8 ( A - H , O - Z ) DIMENSION X(N) COMMON/B/ A FMAX=X(2)*.925 RMAX=((X(l)*1000/3 600)**2)/(9.81*(.06+FMAX)) GXP=A-RMAX C RETURN END  131  C Template f o r performance f u n c t i o n subroutines C SUBROUTINE DETERM(IM0DE) C subroutine for reading deterministic variables IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/B/ A C IF (IM0DE.GT.1) GO TO 50 WRITE(*,100) 100 F O R M A T ( / ' ENTER DESIGN SSD ' / ) READ(*,*) A C 50 CONTINUE C WRITE(*,102) C102 F O R M A T ( / ENTER DESIGN R A D I U S / ) C . READ(*,*) A RETURN END C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) 1  1  C s u b r o u t i n e t o c a l c u l a t e t h e f a i l u r e f u n c t i o n f o r THE MAXIMUM C COMFORTABLE SPEED AROUND A HORIZONTAL CURVE FOR A CAR ON WET OR C DRY PAVEMENT IMPLICIT R E A L * 8 ( A - H , O - Z ) DIMENSION X(N) COMMON/B/ A FMAX=((X(l)*1000/3600)**2)/(9.81*A) -.06 IF (IMODE.EQ.l) GXP=X(3)-FMAX IF (IMODE.EQ.2) GXP=X(2)-FMAX C RETURN END  132  Template C  f o r performance  function  subroutines  SUBROUTINE DETERM(IMODE) C subroutine for reading deterministic variables IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/B/ A , C , D C I F ( I M O D E . G T . l ) GO TO 5 0 WRITE(*,100) 10 0 F O R M A T ( / ' ENTER DESIGN SSD I N m ' / ) READ(*,*)•A WRITE(*,101) 101 F O R M A T ( / ' ENTER RADIUS I N m ' / ) READ(*,*) C WRITE(*,102) 102 F O R M A T ( / • ENTER SUPERELEVATION I N m/m ' / ) READ(*,*) D C 50 CONTINUE C WRITE(*,103) C103 F O R M A T ( / ' ENTER DESIGN SSD I N m 7) C READ(*,*) A C WRITE ( * , 1 0 4 ) C104 F O R M A T ( / ' ENTER RADIUS I N m ' / ) C READ(*,*) C C WRITE(*,105) C105 F O R M A T ( / ' ENTER SUPERELEVATION I N m/m ' / ) C READ(*,*) D RETURN END C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) C s u b r o u t i n e t o c a l c u l a t e t h e f a i l u r e f u n c t i o n f o r SSD ON HORIZONTAL C CURVE IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION X(N) COMMON/B/ A , C , D FMAX=(((X(l)*1000/3600)**2)/(9.81*C))-D RMAX=FMAX/X(3) F T = ( S O R T ( 1 - R M A X * * 2 ) ) * X (3) S D I S T = X ( 1 ) * 1 0 0 0/3 6 0 0 * X ( 2 ) BDIST=((X(l)*1000/3600)**2)/(9.81*FT) IF ( I M O D E . E Q . l ) GXP=X(3)-FMAX I F (IMODE.EQ.2) GXP=A-(SDIST+BDIST) C RETURN END  133  C Template f o r performance f u n c t i o n subroutines C SUBROUTINE DETERM(IMODE) C subroutine for reading deterministic, variables IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/B/ A , C , D C I F ( I M O D E . G T . l ) GO TO 50 WRITE(*,100) 10 0 F O R M A T ( / ' ENTER LENGTH OF VC I N m ' / ) READ(*,*) A WRITE(*,101) 101 F O R M A T ( / ' ENTER GRADE INTO VC I N PERCENT '/) READ (*,*) C WRITE(*,102) 102 F O R M A T ( / ' ENTER GRADE OUT OF VC I N PERCENT '/) READ (*,*) D C C50 CONTINUE C WRITE(*,102) C102 F O R M A T ( / ' ENTER * * * * * ' / ) C READ(*,*) ***** RETURN END C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) C s u b r o u t i n e t o c a l c u l a t e t h e f a i l u r e f u n c t i o n f o r SAG VC FOR CAR SSD C no PDO c o l l i s i o n IMPLICIT REAL*8(A-H,O-Z) DIMENSION X(N) COMMON/B/ A , C , D SDIST=X(1)*1000/3600*X(2) BDIST=(X(l)*1000/3600)**2/(2*9.81*X(3)) DDIST=SDIST+BDIST ADIF=ABS(C-D) IF (DDIST.GT.A) SAG=DDIST**2/(200*(X(4)+DDIST*TAN(1*3.14/180))) I F ( D D I S T . L E . A ) THEN T1=2*DDIST/ADIF T2=200*(X(4)+DDIST*TAN(1*3.14/180))/(ADIF**2) SAG=T1-T2 ENDIF VCL=SAG*ADIF GXP=A-VCL RETURN END  134  C Template f o r performance f u n c t i o n subroutines C SUBROUTINE DETERM(IMODE) C subroutine for reading deterministic variables IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON/B/ A , C , D C I F ( I M O D E . G T . l ) GO TO 50 W R I T E ( * , 1 0 0) 100 F O R M A T ( / ' ENTER LENGTH OF VC I N m ' / ) READ(*,*) A WRITE(*,101) 101 F O R M A T ( / ' ENTER GRADE INTO VC IN PERCENT •/) READ(*,*) C WRITE(*,102) 102 F O R M A T ( / ' ENTER GRADE OUT OF VC I N PERCENT '/) READ(*,*) D C C50 CONTINUE C WRITE(*,102) C102 F O R M A T ( / ' ENTER * * * * * ' / ) C READ(*,*) ***** RETURN END C SUBROUTINE GFUN ( X , N , I M O D E , G X P ) . C s u b r o u t i n e t o c a l c u l a t e t h e f a i l u r e f u n c t i o n f o r SAG VC FOR CAR SSD C no PDO c o l l i s i o n IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION X(N) COMMON/B/ A , C , D SDIST=X(1)*1000/3600*X(2) BDIST=(X(l)*1000/3600)**2/(2*9.81*X(3)*X(4)) DDIST=SDIST+BDIST ADIF=ABS(C-D) IF (DDIST.GT.A) SAG=DDIST**2/(200*(X(5)+DDIST*TAN(1*3.14/180))) I F ( D D I S T . L E . A ) THEN T1=2*DDIST/ADIF T2=2 0 0 * ( X ( 5 ) + D D I S T * T A N ( 1 * 3 . 1 4 / 1 8 0 ) ) / ( A D I F * * 2 ) SAG=T1-T2 ENDIF VCL=SAG*ADIF GXP=A-VCL RETURN END  135  APPENDIX B CHI-SQUARED RESULTS  136  Table B I : Chi Squared Test for Truck Braking Efficiency N N Value Category  Actual  Normal Distribution  Predicted  0.3 0.4 0.5 0.6 0.7 0.8 0.9  0 1 13 23 26 8 2  0.002 0.025 0.165 0.503 0.839 0.976 0.998  0.119 1.711 10.18 24.69 24.54 9.995 1.659  73 4 Degrees of freedom  Sum Max. chi sq  Chi Squared 0.1191922 0.2952741 0.7810448 0.1151372 0.0873861 0.3982477 0.0700941 1.8663763 9.49  Table B2: Chi Squared Test for Non-serious Pedestrian Injury Impact Speed Category  Cumulative Actual  Cumulative Predicted  Chi Squared  8.433382138 12.88433382 16.39824305 18.74084919 21.08345534 24.12884334 26.7057101 29.7510981 34.5534407 50.36603221  10 20 30 40 50 60 70 80 90 100  9.381966436 17.15430542 25.5794069 32.20809751 39.45773418 49.41812331 57.90019609 67.44244368 80.25724854 98.47517094  0.040713 0.472067 0.76396 1.885046 2.816669 2.265892 2.52858 2.338175 1.182712 0.023611  7  Degrees of Freedom  Sum Max. Chi Sq.  14.31742 15.987  137  Table B 3 : C h i Squared Test for Coefficient of Friction on Wet Pavement at 80 km/h Coefficient of Friction Category  Cumulative Actual  Cumulative Predicted  Chi Squared  0 0.145985401 0.23649635 0.299270073 0.318248175 0.335766423 0.367883212 0.401459854 0.437956204 0.583941606 0.59  0 10 20 30 40 50 60 70 80 90 100  0 6.345137 21.23074 38.45531 44.39739 50 60.19196 70.13821 79.444 97.70386 97.95642  1.335803 0.075736 2.383077 0.483427 9.53E-18 0.000614 0.000273 0.003864 0.659439 0.041762  Sum  4.983994  Max. Chi Sq.  15.987  Degrees of 7  Freedom  Table B 4 : C h i Squared Test for Coefficient of Friction on Wet Pavement at 50 km/h Coefficient of Friction Cumulative Cumulative Chi Squared Category Actual Predicted  0 0.233577 0.379562 0.426277 0.448175 0.480292 0.49635 0.525547 0.544526 0.671533 0.68  7  0 10 20 30 40 50 60 70 80 90 100  0.003435252 2.043396315 20.18908502 32.71875627 39.50443494 49.99999998 55.29420132 64.62045839 70.27724734 94.35252039 95.10633786  Degrees of Freedom  138  Sum Max. Chi Sq.  6.330754 0.001788 0.246388 0.00614 9.53E-18 0.369076 0.413421 1.181649 0.210494 0.239479 8.999188 15.987  Table B5: C h i Squared Test for Coefficient of Friction on Dry Pavement Coefficient of Friction Category  Cumulative Actual  Cumulative Predicted  Chi Squared  .85 .9 .95 1.0  2 4 7 9  0.716566 0.27929 ' 0.007331 0.115329 1.118516  2  Degrees of Freedom  3.197135 5.056958 6.773461 7.981197 Sum Max. Chi Sq.  5.991  139  APPENDIX C F O R M / SORM RESULTS (No  RESULTS FOR SERIOUS INJURY SCENARIOS)  140  Table C - l : FORM/SORM Results Probability of Being Unable to Stop Within the Specified SSD for a Car on Dry Pavement Vehicle  Perception  SSD  Probability  FORM  Probability  SORM  Speed  Reaction  Coefficient  (m)  of F a i l u r e  Beta  of F a i l u r e  Beta  (km/h)  T i m e (s)  of F r i c t i o n  50 65 80 100 113 139 150 200  0.659174 0.360827 0.152782 0.035711 0.01209 0.001119 0.000386 2.55E-06  -0.41 0.356 1.025 1.803 2.254 3.057 3.363 4.56  0.672139 0.37616 0.163511 0.039678 0.013766 0.001337 0.00047 3.33E-06  -0.446 0.316 0.98 1.754 2.204 3.003 3.308 4.504  70.972 82.13 91.463 101.79 107.48 116.97 120.35 132.11  1.389 1.5085 1.6385 1.8274 1.9597  0.87587 0.86734 0.85741 0.84318 0.83381 0.81594 0.80889 0.78345  FORM  SORM  2.2486  2.3804 3.0628  Table C-2: FORM/SORM Results Probability of Being Unable to Stop Within the Specified SSD for a Car on Wet Pavement FORM  SORM  Vehicle  Perception  SSD  Probability  FORM  Probability  SORM  Speed  Reaction  Coefficient  (m)  of F a i l u r e  Beta  of F a i l u r e  Beta  (km/h)  T i m e (s)  of F r i c t i o n  65 80 100 113 125 139 150 175 200 250  0.831458 0.694346 0.502964 0.398168 0.312301 0.23687 0.191311 0.121137 0.08087 0.042324  -0.96 -0.508 -0.007 0.267 0.489 0.716 0.873 1.169 1.399 1.724  0.846666 0.718103 0.532539 6.424263 0.339682 0.260601 0.211889 0.135087 0.090061 0.046349  -1.022 -0.577 -0.082 0.191 0.413 0.641 0.8 1.103 1.34 1.681  63.361 70.05 76.904 80.298 82.794 85.034 86.36 88.247 89.063 89.11  1.3757 1.4113 1.4488 1.4666 1.4785 1.4877 1.4919 1.4944 1.4918 1.4824  0.38709 0.36731 0.33685 0.31583 0.29625 0.2738 0.25681 0.22134 0.19127 0.1464  141  Table C-3: F O R M / S O R M Results Probability of Being Unable to Stop Within the Specified SSD for a Truck on Dry Pavement  SSD (m)  FORM Probability of Failure  FORM Beta  SORM Probability of Failure  SORM Beta  Vehicle Speed (km/h)  Perception Reaction Time (s)  Coeff. of Friction  N  50 65 80 100 113 125 139 150 175 200  0.8504326 0.6555937 0.4392727 0.2141224 0.1230383 0.0705754 0.0354956 0.0202751 0.0055016 0.0014955  -1.038 -0.4 0.153 0.792 1.16 1.472 1.806 2.048 2.543 2.969  0.8614487 0.6761834 0.4641341 0.2349737 0.1386459 0.0815496 0.0422486 0.0246752 0.0070394 0.0019988  -1.087 -0.457 0.09 0.723 1.086 1.395 1.725 1.966 2.455 2.878  61.386 71.085 79.218 88.237 93.209 97.278 101.47 104.38 109.91 114.04  1.3444 1.4064 1.4664 1.5403 1.5832 1.6185 1.6542 1.6771 1.7148 1.7304  0.88202 0.87648 0.86959 0.85908 0.85179 0.84488 0.83678 0.83043 0.81661 0.80422  0.621 0.609 0.595 0.573 0.557 0.542 0.524 0.509 0.474 0.438  Table C-4: F O R M / S O R M Results Probability of Being Unable to Stop Within the Specified SSD for a Truck on Wet Pavement SSD (m)  FORM Probability of Failure  FORM Beta  SORM Probability of Failure  SORM Beta  Vehicle Speed (km/h)  Perception Reaction Time (s)  Coeff. of Friction  N  50 65 80 100 113 125 139 150 175 200 250 300  0.978967 0.943719 0.88591 0.779989 0.701268 0.627173 0.543598 0.482369 0.362642 0.270983 0.15486 0.094307  -2.033 -1.587 -1.205 -0.772 -0.528 -0.324 -0.11 0.044 0.351 0.61 1.016 1.315  0.98025 0.949471 0.897654 0.800242 0.726253 0.655498 0.574414 0.514129 0.393911 0.29946 0.175754 0.108648  -2.059 -1.64 -1.268 -0.842 -0.602 -0.4 -0.188 -0.035 0.269 0.526 0,932 1.234  46.739 54.052 60.008 66.593 70.073 72.852 75.641 77.536 81.027 83.602 86.789 88.277  1.3398 1.3634 1.3852 1.4102 1.4237 1.4343 1.4446 1.4512 1.4624 1.4692 1.4744 1.474  0.41559 0.40907 0.39882 0.38183 0.36964 0.35794 0.34405 0.33308 0.30845 0.28476 0.24161 0.20512  0.634 0.6308 0.6258 0.6179 0.6125 0.6076 0.6020 0.5978 0.5891 0.5819 0.5717 0.5663  142  Table C - 5 : F O R M / S O R M Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 5 % Downgrade on Dry Pavement  SSD (m)  FORM Probability of Failure  FORM Beta  SORM Probability of Failure  SORM Beta  Vehicle Speed (km/h)  50 75 100 113 125 139 150 175 200  0.72649705 0.28487879 0.063320292 0.024587038 0.009536533 0.002944848 0.00112226 0.000113902 1.06621E-05  -0.602 0.568 1.527 1.967 2.344 2.754 3.056 3.686 4.251  0.7383453 -0.638 0.29998683 0.524 0.069713971 - 1.47.8 0.027744071 1.915 0.011012812 2.29 0.003494214 2.697 0.001360035 2.998 0.000145376 3.623 1.4338E-05 4.184  67.998 85.274 98.671 104.54 109.43 114.57 118.24 125.53 131.59  Perception Reaction Coefficient Time (s) . of Friction  1.37 1.5385 1.726 1.8305 1.9321 2.0537 2.1539 2.3978 2.6642  0.8779 0.86387 0.84633 0.83653 0.8273 0.81648 0.80803 0.78975 0.77299  Table C-6: F O R M / S O R M Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 5 % Downgrade on Wet Pavement  SSD (m)  FORM Probability of Failure  FORM Beta  SORM Probability of Failure  SORM Beta  Vehicle Speed (km/h)  Perception Reaction Time (s)  Coefficient of Friction  50 75 100 113 125 139 150 175 200 250 300  0.959865 0.832687 0.638461 0.535188 0.447776 0.359527 0.301352 0.202333 0.13887 0.072603 0.043621  -1.749 -0.965 -0.354 -0.088 0.131 0.36 0.521 0.833 1.085 1.457 1.71  0.963071 0.846866 0.662504 0.56166 0.474715 0.385315 0.325386 0.221275 0.152324 0.07981 0.047379  -1.787 -1.023 -0.419 -0.155 0.063 0.292 0.453 0.768 1.024 1.406 1.671  50.912 63.307 72.252 75.856 78.646 81.326 83.047 85.909 87.626 88.965 88.95  1.3356 1.3864 1.4276 1.4442 1.4566 1.4677 1.4741 1.4827 1.4852 1.4819 1.4759  0.40963 0.38927 0.35973 0.34263 0.32635 0.30722 0.29233 0.25989 0.2304 0.18237 0.14761  143  Table C-7: F O R M / S O R M Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 10% Downgrade on Dry Pavement  SSD (m)  FORM Probability of Failure  FORM Beta  SORM Probability of Failure  SORM Beta  Vehicle Speed (km/h)  50 75 100 113 125 139 150 175 200  0.7799175 0.3628176 0.10090452 0.044319493 0.01926014 0.006774226 0.002850819 0.000357619 4.04717E-05  -0.772 0.351 1.276 1.703 2.069 2.469 2.764 3.384 3.942  0.79035022 0.37930883 0.1098687 0.049376146 0.021932549 0.007919716 0.003404261 0.00044882 5.37304E-05  -0.808 0.307 1.227 1.651 2.015 2.413 2.706 3.321 3.873  65.316 82.199 95.52 101.44 106.42 111.71 115.52 123.25 129.85  Perception Reaction Coefficient of Friction Time (s)  1.3571 1.4984 1.6521 1.7363 1.8165 1.9128 1.9906 2.1739 2.3678  0.87964 0.86682 0.85055 0.84036 0.83104 0.81984 0.81088 0.79026 0.7699  Table C-8: F O R M / S O R M Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 10% Downgrade on Wet Pavement  SSD (m)  FORM Probability of Failure  50 75 100 113 125 139 150 175 250 300  0.975223 0.89115 0.743586 0.654894 0.573348 0.483929 0.420359 0.30155 0.117182 0.069978  SORM FORM Probability SORM Beta of Failure Beta  Vehicle Speed (km/h)  -1.964 -1.233 -0.654 -0.399 -0.185 0.04 0.201 0.52 1.189 1.476  47.393 59.186 67.984 71.657 74.586 77.509 79.47 82.984 88.018 88.849  0.976749 0.900384 0.761865 0.676685 0.597188 0.508696 0.444899 0.323483 0.128061 0.076303  -1.991 -1.284 -0.712 -0.458 -0.246 -0.022 0.139 0.458 1.136 1.43  144  Perception Reaction Coefficient of Friction Time(s)  0.13388 1.3793 1.4137 1.4285 1.4401 1.4513 1.4584 1.4698 1.4786 1.4756  0.41473 0.4002 0.3769 0.36295 0.34945 0.33328 0.32047 0.2917 0.21656 0.17903  Table C-9: F O R M / S O R M Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 15% Downgrade on Dry Pavement  SSD (m)  FORM Probability of Failure  FORM Beta  SORM Probability of Failure  SORM Beta  Vehicle Speed (km/h)  Perception Reaction Time (s)  Coefficient of Friction  50 75 100 113 125 139 150 175 200  0.82357554 0.44133055 0.14940837 0.073401368 0.035451215 0.014109843 0.006538131 0.001014677 0.00014041  -0.929 0.148 1.039 1.451 1.806 2.194 2.482 3.086 3.632  0.83250872 0.45825574 0.16085809 0.080673128 0.03974033 0.01619779 0.007651863 0.001243523 0.000180909  -0.964 0.105 0.991 1.401 1.754 2.139 2.425 3.025 3.566  62.797 79.217 92.336 98.228 103.21 108.55 112.42 120.34 127.21  1.3482 1.4679 0.15955 1.6643 1.729 1.806 1.8673 2.0093 2.1536  0.88119 0.86965 0.854 0.84476 0.83572 0.82469 0.81573 0.79467 0.77293  Table C-10: F O R M / S O R M Results Probability of Being Unable to Stop Within the Specified SSD for a Car on a 15% Downgrade on Wet Pavement  SSD (m)  FORM Probability of Failure  FORM Beta  SORM Probability of Failure  SORM Beta  Vehicle Speed (km/h)  Perception Reaction Time (s)  Coefficient of Friction  50 75 100 113 125 139 150 175 200 250 300  0.983909 0.927366 0.818011 0.746129 0.67571 0.593271 0.53096 0.404769 0.304672 0.174524 0.105888  -2.142 -1.456 -0.908 -0.662 -0.456 -0.236 -0.078 0.241 0.511 0.936 1.249  0.984513 0.93338 0.831513 0.763248 0.695528 0.61524 0.553833 0.427487 0.325187 0.188894 0.115097  -2.157 -1.501 -0.96 -0.717 -0.512 -0.293 -0.135 0.183 0.453 0.882 1.2  44.441 55.663 64.232 67.895 70.873 73.913 76.006 79.922 82.85 86.49 88.165  1.3434 1.3762 1.4051 1.418 1.4285 1.439 1.4461 1.4585 1.4665 1.4735 1.4737  0.41812 0.40801 0.38954 0.37808 0.36681 0.3531 0.34209 0.31685 0.2921 0.2466 0.20837  145  v  Table C - l l : F O R M / S O R M Results Probability of Skidding Out for a C a r on Horizontal Curve on Dry Pavement at 50 km/h  Radius (m)  FORM Probability of Failure  FORM Beta  SORM Probability of Failure  SORM Beta  Vehicle Speed (km/h)  Coefficient of Friction  10 20 30 40 50 60 70 80 90  0.85082026 0.57472421 0.32315959 0.15867952 0.070364527 0.028745061 0.010865825 0.003722811 0.001086543  -1.04 -0.188 0.459 1 1.473 1.9 2.295 2.676 3.065  0.85279658 0.57911893 0.32767759 0.16185657 0.072112556 0.029511569 0.011075661 0.003653571 0.000941695  -1.049 -0.2 0.446 0.987 1.46 1.888 2.288 2.683 3.108  33.333 46.991 57.294 65.804 73.124 79.55 85.207 90.031 93.661  0.87991 0.87395 0.86555 0.85564 0.84449 0.83197 0.81708 0.79668 0.76396  Table C-12: F O R M / S O R M Results Probability of Skidding Out for a C a r on Horizontal Curve on Wet Pavement at 50 km/h  Radius (m)  FORM Probability of Failure  FORM Beta  SORM Probability of Failure  SORM Beta  Vehicle Speed (km/h)  Coefficient of Friction  10 20 30 40 50 60 70 80 90  0.93491967 0.80412448 0.64259006 0.48501662 0.35103216 0.24663978 0.16987921 0.11558477 0.078154925  -1.513 -0.856 -0.365 0.038 0.383 0.685 0.955 1.197 1.418  0.93602271 0.80735672 0.64786435 0.49150897 0.35779757 0.2529642 0.17536381 0.12009062 0.081710504  -1.522 -0.868 -0.38 0.021 0.364 0.665 0.933 1.175 1.394  26.025 36.595 44.36 50.571 55.73 60.098 63.835 67.047 69.808  0.51105 0.50452 0.49288 0.4788 0.46333 0.447 0.43014 0.41294 0.39555  146  Table C-13: F O R M / S O R M Results Probability of Feeling Uncomfortable on a Horizontal Curve in a C a r on Wet or D r y Pavement at 50 km/h  Radius (m)  FORM Prob. of Failure  FORM Beta  SORM Prob. of SORM Failure Beta  Vehicle Speed (km/h)  Coefficient of Friction  Comfortable Coeff. of Lateral Friction  0.37361 0.947787 -1.624 0.94783 -1.624 23.479 0.87187 10 0.87187 0.3735 0.847629 -1.026 0.84774 -1.027 33.201 20 0.37323 0.716032 -0.571 0.71623 -0.572 40.65 0.87187 30 0.37287 0.87187 40 0.574659 -0.188 0.57492 -0.189 46.919 0.87187 0.37245 0.440933 0.149 0.44123 0.148 52.431 50 0.372 0.325308 0.453 0.32560 0.452 57.405 0.87187 60 0.37151 61.97 0.87187 70 0.231887 0.733 0.23215 0.732 0.371 66.209 0.87187 80 0.160333 0.993 0.16056 0.992 70.182 0.87187 0.37046 0.107863 1.238 0.10804 1.237 90 Note: Probability of failure is for the mode of failure in which passengers feel uncomfortable. At this speed, the probability of feeling uncomfortable was the same for both wet and dry conditions. Table C-14: F O R M / S O R M Results Probability of Skidding Out for a Car on Horizontal Curve on Dry Pavement at 80 km/h  Radius (m)  FORM Probability of Failure  FORM Beta  SORM Probability of Failure  SORM Beta  Vehicle Speed (km/h)  Coefficient of Friction  50 75 100 125 150 175 200 225 250  0.56587769 0.20634559 0.05164426 0.01008947 0.00168538 0.000256126 3.68939E-05 5.15977E-06 7.06915E-07  -0.166 0.819 1.629 2.323 2.932 3.474 3.964 4.41 4.823  0.57251869 0.21114252 0.05329528 0.01046577 0.001750609 0.000264493 3.72799E-05 4.95011E-06 6.21855E-07  -0.183 0.802 1.614 2.309 2.92 3.466 3.961 4.419 4.849  74.327 90.038 102.59 112.98 122.2 129.28 136.95 143.03 148.19  0.87467 0.854227 0.83006 0.80345 0.78304 0.74722 0.73915 0.71978 0.69098  147  Table C-15: F O R M / S O R M Results Probability of Skidding Out for a C a r on Horizontal Curve on Wet Pavement at 80 km/h  Radius (m)  FORM Probability of Failure  FORM Beta  SORM Probability of Failure  SORM Beta  Vehicle Speed (km/h)  Coefficient of Friction  50 75 100 125 150 175 200 225 250  0.9435196 0.82563097 0.66533151 0.5040834 0.36776951 0.26359202 0.18833248 0.1354696 0.098708122  -1.585 -0.937 -0.427 -0.01 0.338 0.632 0.884 1.101 1.289  0.94575637 0.83171816 0.6750116 0.51561362 0.37933688 0.27402824 0.19713996 0.1425949 0.10428836  -1.605 -0.961 -0.454 -0.039 0.307 0.601 0.852 1.069 1.257  52.967 63.354 71.044 76.864 81.301 84.675 87.405 89.55 91.312  0.41225 0.3902 0,36432 0.33703 0.30983 0.28352 0.26006 0.23873 0.22039  Table C-17: F O R M / S O R M Results Probability of Feeling Uncomfortable on a Horizontal Curve in a C a r on Dry Pavement at 80 km/h  Radius (m)  FORM Prob.of Failure  FORM Beta  SORM Prob. of Failure  SORM Beta  Vehicle Speed (km/h)  Coefficient of Friction  Comfortable Coeff. of Lateral Friction  50 0.953176 -1.676 0.95425 -1.688 52.561 0.33629 0.37459 • 75 0.84974 0.846398 -1.021 -1.035 64.321 0.87187 0.37388 100 0.687257 -0.488 0.693369 -0.505 74.193 0.87187 0.37296 125 0.515923 -0.04 0.524105 -0.06 82.849 0.87187 0.37191 150 0.36632 0.342 0.375258 0.318 90.636 0.87187 0.37076 175 0.252423 0.667 0.260933 0.64 0.87187 0.36954 97.759 200 0.172832 0.943 0.180214 0.915 104.35 0.87187 0.36827 225 0.119728 1.176 0.125738 1.147 110.5 0.87187 0.36687 250 0.084841 1.373 0.089531 1.344 116.28 0.87187 0.36545 Note: Probability of failure is a system probability because there are two failure modes. In this instance the system probability of failure and beta are the same as the probability of failure and beta for the passenger discomfort mode. The design point is for the passenger comfort failure mode.  148  Table C-18: F O R M / S O R M Results Probability of Feeling Uncomfortable on a Horizontal Curve in a C a r on Wet Pavement at 80 km/h  Radius (m)  FORM Prob. of Failure  FORM Beta  SORM Prob. of Failure  SORM Beta  Vehicle Speed (km/h)  Coefficient of Friction  Comfortable Coeff. of Lateral Friction  50 0.953176 -1.676 0.95425 0.37265 -1.688 54.711 0.41086 75 0.846398 -1.021 0.84974 -1.035 0.38572 0.37265 . 65.193 100 0.687257 -0.488 0.693369 -0.505 72.817 0.35705 0.37265 125 0.515923 -0.04 0.524105 0.37265 -0.06 78.495 0.3277 0.36632 150 0.342 0.375258 0.318 0.29888 0.37265 82.729 175 0.252423 0.667 0.260933 0.64 0.37265 85.867 0.27139 200 0.172832 0.943 0.180214 0.915 88.162 0.24567 0.37265 225 0.119728 1.176 0.125738 1.147 89.812 0.37265 0.22198 250 0.084841 1.373 0.089531 1.344 90.974 0.37265 0.20039 Note: Probability of failure is a system probability because there are two failure modes. The design point is for the skidding out failure mode. When the horizontal curve radius is about 250 m or greater, the influence of the passenger discomfort failure mode has on the system probability of failure and associated beta is minimal. In contrast the  Table C-19: F O R M / S O R M Results Probability of Being Unable to Stop on a Crest Vertical Curve for a C a r on Dry Pavement  VC (m)  FORM Probability of Failure  FORM Beta  SORM Probability of Failure  SORM Beta  Vehicle Speed (km/h)  Perception Reaction Time (s)  Coeff. of Friction  Height of Eye (m)  20  0.87195172  -1.136  0.878107  -1.166  60.031  1.3008  0.88106  1.1513  40  0.58957485  -0.226  0.6041215  -0.264  73.691  1.4151  0.87411  1.1495  60  0.35408614  0.374  0.36963754  0.333  82.381  1.5116  0.8671  1.148  80  0.20114159  0.838  0.21366867  0.794  88.879  1.5993  0.86043  1.1468  100  0.10489192  1.254  0.11348619  1.208  94.558  1.6894  0.85351  1.146  125  0.04152097  1.733  0.04599341  1.685  100.88  1.8083  0.84458  1.1452  150  0.01487797  2.173  0.01687399  2.123  106.46  1.9341  0.83557  1.1446  175  0.00493951  2.58  0.00573245  2.528  111.42  2.0669  0.82667  1.1442  200  0.00154864  2.958  0.00184083  2.904  115.83  2.2082  0.81823  1.1439  250  0.00013591  3.641  0.00013842  3.585  123.29  2.5122  0.80254  1.1434  300  1.10139E-05  4.243  1.41799E-05  4.186  129.22  2.8519  0.78975  1.1433  I ]  149  Table C-20: F O R M / S O R M Results Probability of Being Unable to Stop on a Crest Vertical Curve for a C a r on Wet Pavement  vc  (m)  FORM Probability of Failure  FORM Beta  SORM Probability of Failure  SORM Beta  Vehicle Speed (km/h)  Perception Reaction Time (s)  Coeff. of Friction  Height of Eye (m)  40  0.9138968  -1.365  0.9220427  -1.419  57.441  1.3396  0.40172  1.1514  80  0.73632385  -0.632  0.75795082  -0.7  68.261  1.4016  0.37339  1.1501  100  0.63924974  -0.356  0.66551424  -0.428  72.202  1.4231  0.35906  1.1496  125  0.51996491  -0.05  0.54941781  -0.124  76.35  1.4459  0.3398  1.1491  150  0.41321128  0.219  0.44295791  0.143  79.732  1.4637  0.31968  1.1487  175  0.32413277  0.456  0.35199491  0.38  82.437  1.4769  0.29928  1.1485  200  0.2532185  0.664  0.2779519  0.589  84.549  1.4858  0.27912  1.1484  250  0.15672378  1.008  0.17430224  0.937  87.321  1.4938  0.24109  1.1483  300  0.10139382  1.274  0.11311384  1.21  88.688  1.4937  0.20786  1.1483  400  0.05002664  1.645  0.05508867  1.597  89.206  1.485  0.15741  1.1484  500  0.15486  1.016  0.03209570  0.932  88.686  1.4765  0.1241  1.1486  Table C-21: F O R M / S O R M Results Probability of Being Unable to Stop on a Crest Vertical Curve for a Truck on Dry Pavement  VC (m)  FORM Prob. of Failure  FORM Beta  SORM Prob. of Failure  SORM Beta  Vehicle Speed (km/h)  Percep. Reaction Time (s)  Coeff. of Friction  Height of Eye (m)  N  20  0.89746  -1.267  0.90659  -1.32  57.661  1.3347  0.76267  2.4544  0.62  40  0.67620  -0.457  0.69963  -0.523  70.159  1.4047  0.76089  2.452  0.61  60  0.47079  0.073  0.50036  -0.001  78.081  1.4568  0.75903  2.4499  0.60  80  0.31768  0.474  0.34678  0.394  83.902  1.4984  0.75718  2.4481  0.58  100  0.21233  0.798  0.23780  0.713  88.488  1.5328  0.75538  2.4466  0.57  125  0.12838  1.134  0.14812  1.045  93.107  1.5684  0.75318  2.445  0.56  150  0.07568  1.435  0.09000  1.341  97.114  1.5995  0.7509  2.4441  0.54  175  0.04327  1.714  0.05306  1.616  100.71  1.6269  0.74848  2.4434  0.53  200  0.02421  1.974  0.03058  1.872  103.91  1.6499  0.74594  2.443  0.51  250  0.00731  2.442  0.00978  2.335  109.28  1.6839  0.74071  2.4424  0.48  300  0.00219  2.849  0.00308  2.739  113.36  1.7  0.73555  2.4423  0.44  400  0.00022  3.512  0.00034  3.399  118.13  1.6904  0.72725  2.4428  037 I I  150  Table C-22: F O R M / S O R M Results Probability of Being Unable to Stop on a Crest Vertical Curve for a Truck on Wet Pavement  VC (m)  FORM Prob. of Failure  FORM Beta  SORM Prob. of Failure  SORM Beta  Vehicle Speed (km/h)  Percep. Reaction Time (s)  Coeff. of Friction  Height of Eye (m)  N  80  0.79382  -0.82  0.81312  -0.889  65.903  1.4075  0.38397  2.4527  0.62  100  0.72503  -0.598  0.74873  -0.671  69.098  1.4199  0.37329  2.452  0.61  125  0.64529  -0.373  0.55208  -0.131  72.207  1.4318  0.36081  2.4513  0.61  150  0.56959  -0.175  0.59973  -0.253  74.804  1.4415  0.34845  2.4507  0.60  175  0.49833  0.004  0.52986  -0.075  77.051  1.4495  33599  2.4502  0.60  0.087  78.993  1.4561  0.32356  2.4498  0.59  200  0.43328  0.168  0.46517  250  0.32443  0.455  0.35477  0.372  82.105  1.4654  0.29922  2.4493  0.59  300  0.24265  0.698  0.26972  0.614  84.388  1.4709  0.27599  2.449  0.58  400  0.14007  1.08  0.15954  0.996  87.175  1.4746  0.23403  2.4489  0.57  500  0.08650  1.363  0.09980  1.283  88.43  1.4736  0.19883  2.449  0.57  750  0.03503  1.811  0.04003  1.75  88.61  1.4665  0.1372  2.4493  0.56  Table C-23: F O R M / S O R M Results Probability of Being Unable to Stop on a Sag Vertical Curve for a C a r on Dry Pavement  VC (m)  FORM Prob. of Failure  FORM Beta  SORM Prob. of Failure  SORM Beta  Vehicle Speed (km/h)  Perception Reaction Time (s)  Coeff. of Friction  Headlight Height (m)  20  0.9861784  -2.201  0.9867029  -2.217  43.346  1.2116  0.88357  0.65305  40  0.8946909  -1.252  0.8998415  -1.281  58.25  1.2887  0.88162  0.65121  50  0.8024642  -0.85  0.8113294  -0.883  64.378  1.3328  0.87935  0.65038  60  0.6828731  -0.48  0.6967027  -0.515  69.942  1.3796  0.87649  0.64973  80  0.4251455  0.189  0.4407637  0.149  79.729  1.4799  0.86947  0.666488  100  0.2174077  0.781  0.2303179  0.738  88.103  1.588  0.86129  0.64809  125  0.0750518  1.439  0.0818794  1.393  97.037  1.7334  0.85017  0.64751  150  0.0210670  2.026  0.0240700  1.976  104.62  1.8902  0.83866  0.64712  175  0.0053218  2.554  0.0061636  2.503  111.13  2.0583  0.82723  0.64685  200  0.0012022  3.035  0.0014350  2.981  116.72  2.2395  0.81644  0.64649  250  5.186E-05  3.882  6.537E-05  3.825  125.73  2.64008  0.79741  0.64646  300  2.045E-06  4.607  2.685E-06  4.55  132.48  3.0982  0.78294  0.6464  400  3.390E-09  5.796  3.980E-09  5.769  142.37  4.0883  0.76319  0.64642  151  Table C-24: F O R M / S O R M Results Probability of Being Unable to Stop on a Sag Vertical Curve for a C a r on Wet Pavement  (m)  FORM Prob. of Failure  FORM Beta  SORM Prob. of Failure  SORM Beta  Vehicle Speed (km/h)  Perception Reaction Time (s)  Coeff. of Friction  Headlight Height (m)  20  0.9973248  -2.785  0.9954370  -2.607  33.771  1.2663  0.40524  0.65274  40  0.9819903  -2.097  0.9824002  -2.106  45.272  1.2968  0.40942  0.65167  50  0.9647211  -1.808  0.9473403  -1.843  49.999  1.3142  0.40722  0.6511  60  0.9387259  -1.544  0.9442923  -1.592  54.243  1.3319  0.40318  0.65065  80  0.8588662  -1.075  0.8871904  -1.1335  61.603  1.3667  0.3911  0.64999  100  0.7487515  -0.671  0.7695472  -0.737  67.695  1.3986  0.37523  0.64954  125  0.5937461  -0.237  0.6214777  -0.309  73.849  1.4322  0.35201  0.64917  150  0.4483744  0.13  0.4782358  0.055  78.641  1.4581  0.32672  0.64893  175  0.3295790  0.441  0.3575584  0.365  82.274  1.4761  0.30067  0.64879  200  0.2403902  0.705  0.2643391  0.63  84.928  1.4873  0.27493  0.64872  250  0.1317288  1.118  0.1468097  1.05  87.981  1.4944  0.22767  64868  300  0.0783050  1.417  0.0871757  1.358  89.095  1.4914  0.18886  0.6487  400  0.0357397  1.802  0.0389093  1.763  88.94  1.4796  0.1355  0.64879  500  0.0208411  2.037  0.0221647  2.011  88.078  1.471  0.10373  0.64885  750  0.0091756  2.358  0.0094161  2.349  86.395  1.46009  0.06453  0.64892  vc  Table C-25: F O R M / S O R M Results Probability of Being Unable to Stop on a Sag Vertical Curve for a Truck on Dry Pavement VC (m)  FORM Prob. of Failure  FORM Beta  SORM Prob. of Failure  SORM Beta  Vehicle Speed (km/h)  Percep. Reaction Time (s)  Coeff. of Friction  N  20  0.99534  -2.6  0.99536  -2.601  35.112  1.7734  0.76269  .625  1.1133  48.294  1.8089  0.76285  .626  1.11309  Headlight Height (m)  40  0.96777  -1.849  0.99701  -1.882  50  0.94024  -1.557  0.94471  -1.596  52.958  1.8279  0.76268  .625  1.1297  60  0.90159  -1.291  0.90892  -1.334  57.167  1.847  0.76241  .623  1.128  80  0.78853  -0.801  0.80302  -0.852  64.807  1.8867  0.7616  .616  1.1251  100  0.64022  -0.359  0.66152  -0.417  71.591  1.9266  0.7605  .608  1.123  125  0.44406  0.141  0.46975  0.076  79.092  1.9751  0.75878  .595  1.1211  150  0.27695  0.592  0.30134  0.521  85.686  2.0209  0.75673  .581  1.1196  175  0.15805  1.003  0.17745  0.925  91.508  2.063  0.75437  .564  1.1186  200  0.08407  1.378  0.09761  1.295  96.651  2.1004  0.75174  .547  1.1178  250  0.02065  2.041  0.02570  1.948  105.16  2.1592  0.74581  .509  1.1168  300  0.00463  2.602  0.00616  2.503  111.56  2.1948  0.7394  .467  1.1164  400  0.00025  3.482  0.00037  3.375  118.79  2.1989  0.72811  .379  1.1167  500  2.07E-5  4.099  3.24E-5  3.994  129.81  2.1497  0.72414  .293  1.1176  6.69E-7  4.834  109.79  2.0332  0.73698  .147  1.1166  750  4.62E-7  4.907  152  i  Table C-26: F O R M / S O R M Results Probability of Being Unable to Stop on a Sag Vertical Curve for a Truck on Wet Pavement VC (m)  FORM Prob. of Failure  FORM Beta  SORM Prob. of Failure  SORM Beta  Vehicle Speed (km/h)  Percep. Reaction Time (s)  Coeff. of Friction  Height of Eye (m)  N  20  0.99876  -3.026  0.99720  -2.77  29.572  1.8196  0.40275  0.628  1.13  40  0.99300  -2.457  0.99229  -2.422  39.328  1.8344  0.4109  0.632  1.13  50  0.98738  -2.238  0.98758  -2.244  43.043  1.8431  0.41151  0.632  1.13  -2.065  46.374  1.8519  0.411  0.632  1.13  60  0.97928  -2.039  0.98052  80  0.95331  -1.678  0.95774  -1.725  52.328  1.8701  0.40703  0.630  1.13  100  0.91257  -1.357  0.92132  -1.414  57.489  1.8879  0.40037  0.627  1.13  125  0.84171  -1.001  0.85677  -1.066  63.009  1.9083  0.38938  0.622  1.12  150  0.75454  -0.689  0.77580  -0.758  67.664  1.9262  0.37645  0.616  1.12  175  0.65978  -0.412  0.68609  -0.485  71.591  1.9413  0.36231  0.610  1.12  200  0.56564  -0.165  0.59533  -0.241  74.898  1.9536  0.34748  0.604  1.12  250  0.40028  0.253  0.43172  0.172  80.002  1.9711  0.31714  0.592  1.12  300  0.27753  0.59  0.30619  0.507  83.534  1.9811  0.2874  0.583  1.12  400  0.13751  1.092  0.15690  1.007  87.424  1.9874  0.23366  0.571  1.12  500  0.07560  1.435  0.08749  1.356  88.806  1.9855  0.19006  0.565  1.12  750  0.02677  1.931  0.03036  1.876  88.443  1.9765  0.12115  0.566  1.12  153  APPENDIX D MONTE CARLO SIMULATION RESULTS  154  Table D - l : Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a Car on Dry Pavement Available SSD (m)  Probability of Failure  Beta  40 65 80 100 113 139 175 200  0.846 0.375 0.165 0.0403 0.0137 0.00152 0.00006 0.000008  -1.014 0.318 0.975 1.747 2.205 2.964 3.846 4.314  Table D-2: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a Car on Wet Pavement Available SSD (m)  Probability of Failure  Beta  40 65 80 100 113 139 150 175 200 250 300  0.976 0.845 0.717 0.532 0.425 0.261 0.212 0.137 0.0912 0.0476 0.029  -1.975 -1.017 -0.574 -0.081 0.19 0.64 0.798 1.098 1.334 1.669 1.896  155  Table D-3: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a Truck on Dry Pavement Available SSD (m)  Probability of Failure  Beta  50 65 80 100 113 125 139 150 175 200  0.92578 0.79351 0.60599 0.35567 0.22735 0.14343 0.07962 0.04859 0.01524 0.00454  -1.445 -0.819 -0.269 0.37 0.748 1.065 1.408 1.659 2.164 2.609  Table D-4: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a Truck on Wet Pavement Available SSD (m)  Probability of Failure  Beta  50 65 80 100 113 125 139 150 175 200 250 300  0.988997 0.96685 0.92863 0.84683 0.78093 0.71412 0.63348 0.57185 0.44444 0.34028 0.20001 0.12274  -2.289 -1.836 -1.466 -1.023 -0.775 -0.565 -0.341 -0.181 0.14 0.412 0.842 1.161  156  Table D-5: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a Truck to C a r on Dry Pavement Available SSD (m)  Probability of Failure  Beta  20 30 40 50 60 65 70 80 90 100 125 150 175 200  0.83674 0.69351 0.51605 0.35111 0.22098 0.17123 0.13046 0.07382 0.03861 0.01959 0.00313 0.00041 0.00003 0.000007  -0.981 -0.506 0.04 0.382 0.769 0.949 1.124 1.448 1.767 2.062 2.734 3.346 4.013 4.344  Table D-6: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a Truck to C a r on Wet Pavement Available SSD (m)  Probability of Failure  Beta  20 30 40 50 60 65 70 80 90 100 125 150 175 200  0.8538 0.76323 0.64889 0.52976 0.41697 0.3654 0.31918 0.23955 0.17687 0.12868 0.05691 0.02596 0.01153 0.0058  -1.053 -0.717 -0.382 -0.075 0.21 0.344 0.47 0.708 0.927 1.133 1.581 1.952 2.272 2.524 I  157  Table D-7: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a C a r to Pedestrian on Dry Pavement Available SSD (m)  Probability of Failure  Beta  20 30 40 50 60 65 70 80 90 100 125  0.7166 0.45527 0.23638 0.10461 0.0397 0.02366 0.01399 0.00438 0.00105 0.00023 0.000016  -0.573 0.112 0.718 1.256 1.754 1.983 2.198 2.621 3.076 3.47 4.159  Table D-8: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a Car to Pedestrian on Wet Pavement Available SSD (m)  Probability of Failure  Beta  20 30 40 50 60 65 70 80 90 100 125 150 175 200 250  0.7775 0.59833 0.41839 0.26852 0.16456 0.12571 0.09536 0.05383 0.03065 0.01713 0.00436 0.00149 0.00073 0.00047 0.00021  -0.764 -0.249 0.206 0.617 0.976 1.147 1.308 1.609 1.871 2.117 2.623 2.97 3.187 3.308 3.527  158  Table D-9: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a C a r to C a r on Dry Pavement Available SSD (m)  Probability of Failure  Beta  20 40 50 60 75 90 100 113 125 139 150 175  0.96618 0.7609 0.57491 0.38341 0.16849 0.05835 0.02546 0.00815 0.0025 0.00063 0.00016 0.000007  -1.827 -0.709 -0.189 0.297 0.96 1.569 1.952 2.402 2.807 3.225 3.599 4.344  Table D-10: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a C a r to C a r on Wet Pavement Available SSD (m)  Probability of Failure  Beta  20 40 50 60 75 90 100 113 125 139 150 175  0.97055 0.82943 0.68848 0.52387 0.29426 0.13829 0.07543 0.03057 0.01252 0.00397 0.00153 0.00011  -1.889 -0.952 -0.492 -0.06 0.541 1.088 1.437 1.872 2.241 2.663 2.962 3.695  159  Table D - l l : Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a Reckless C a r to Pedestrian on Dry Pavement Available SSD (m)  Probability of Failure  Beta  50 60 65 70 80 90 100 125  0.98744 0.83791 0.67165 0.47712 0.17948 0.04933 0.01219 0.00028  -2.24 -0.986 -0.444 0.057 0.917 1.651 2.251 3.45  Table D-12: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a Reckless C a r to Pedestrian on Wet Pavement Available SSD (m)  50 75 90 100 110 120 130 140 • 150 160 170 180 200 250 300  Probability of Failure  Beta  0.99992 0.98304 0.9166 0.83597 0.72964 0.6102 0.49457 0.38894 0.29729 0.2236 0.16413 0.1209 0.06262 0.01125 0.00205  -3.775 -2.121 -1.383 -0.978 -0.612 -0.28 0.014 0.282 0.532 0.76 0.978 1.17 1.533 2.283 2.87  160  Table D-13: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 5% Downgrade on Dry Pavement Available SSD (m)  Probability of Failure  Beta  50 75 100 113 125 139 150 175 200  0.73611 0.29784 0.06975 0.02746 0.01122 0.00392 0.00166 0.00014 0.00002  -0.631 0.531 1.478 1.92 2.283 2.659 2.936 3.633 4.107  Table D-14: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 5% Downgrade on Wet Pavement Available SSD (m)  Probability of Failure  Beta  50 75 100 113 125 139 150 175 200 250 300  0.96377 0.84438 0.66014 0.55873 0.47206 0.38386 0.32332 0.22127 0.15404 0.081 0.04806  -1.796 -1.013 -0.413 -0.148 0.07 0.295 0.458 0.768 1.019 1.404 1.664  161  Table D-15: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 10% Downgrade on Dry Pavement Available SSD (m)  Probability of Failure  50 75 100 113 125 139 150 175 200  0.78753 0.37718 0.10963 0.04951 0.02169 0.00843 0.00374 0.00049 0.00006  Beta  -0.798 0.313 1.228 . 1.65 2.02 2.39 2.675 3.296 3.846  Table D-16: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 10% Downgrade on Wet Pavement Available SSD (m)  Probability of Failure  Beta  50 75 100 113 125 139 150 175 200 250 300  0.97798 0.89922 0.7601 0.67417 0.59517 0.50643 0.44305 0.32175 0.23434 0.12806 0.07652  -2.014 -1.277 -0.707 -0.451 -0.241 -0.016 0.143 0.463 0.725 1.136 1.429  162  Table D-17: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 15% Downgrade on Dry Pavement Available SSD (m)  Probability of Failure  Beta  50 75 100 113 125 139 150 175 200  0.8289 0.45663 0.16091 0.08044 0.0396 0.01644 0.00813 0.00149 0.00021  -0.95 0.109 0.991 1.402 1.755 2.134 2.403 2.97 3.527  Table D-18: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 15% Downgrade on Wet Pavement Available SSD (m)  Probability of Failure  Beta  50 75 100 113 125 139 150 175 200 250 300  0.98577 0.93308 0.8291 0.76103 0.69301 0.61318 0.55157 0.4259 0.32363 0.18896 0.11567  -2.191 -1.499 -0.951 -0.71 -0.504 -0.288 -0.13 0.187 0.458 0.882 1.197  163  Table D-19: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 225 m Radius Horizontal Curve on Dry Pavement Available SSD (m)  Probability of Failure  Beta  50 75 100 113 125 139 150 175 200  0.90223 0.64265 0.34825 0.231 0.15236 0.09061 0.06014 0.02795 0.02063  -1.294 -0.366 0.39 0.736 1.025 1.337 1.554 1.912 2.041  Note: The probability of failure and beta are based on system failure as there are two modes of failure. Table D-20: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a Car on a 250 m Radius Horizontal Curve on Dry Pavement Available SSD (m)  Probability of Failure  Beta  50 75 100 113 125 139 150 175 200 250  0.8183 0.64105 0.34424 0.22571 0.14723 0.0881 0.0552 0.0176 0.0095 0.0073  -0.909 -0.361 0.401 0.753 1.048 1.353 1.596 2.106 2.346 2.442  Note: The probability of failure and beta are based on system failure as there are two modes of failure.  164  Table D-21: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 300 m Radius Horizontal Curve on Dry Pavement Available SSD (m)  Probability of Failure  Beta  50 75 100 113 125 139 150 175 200 250  0.9019 0.63919 0.33947 0.22018 0.14135 0.087 0.04997 0.01595 0.00509 0.00509  -1.292 -0.356 0.414 0.772 1.074 1.399 1:645 2.146 2.57 2.57  Note: The probability of failure and beta are based on system failure as there are two modes of failure. Table D-22: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 500 m Radius Horizontal Curve on Dry Pavement Available SSD (m)  Probability of Failure  Beta  50 75 100 113 125 139 150 175 200 250  0.90184 0.63731 0.33332 0.21275 0.13367 0.07406 0.04386 0.01263 0.00328 0.00022  -1.292 -0.351 0.431 0.797 1.109 1.446 1.708 2.237 2.718 3.515  Note: The probability of failure and beta are based on system failure as there are two modes of failure.  165  Table D-23: Monte Carlo Simulation Results Probability of Being Unable to Stop Within the Specified SSD for a C a r on a 500 m Radius Horizontal Curve on Damp Pavement Available SSD (m)  Probability of Failure  Beta  100 150 200 250  0.55826 0.18466 0.04678 0.01209  -0.147 0.898 1.677 2.254  Note: The probability of failure and beta are based on system failure as there are two modes of failure. Table D-24: Monte Carlo Simulation Results Probability of Skidding Out for a C a r on Dry Pavement at 50 km/h Design Speed Radius (m)  Probability of Failure  Beta  10 20 30 40 50 60 70 80 90  0.85166 0.57998 0.33042 0.1641 0.07306 0.02297 0.01102 0.00366 0.00078  -1.044 -0.202 0.439 0.978 1.453 1.885 2.29 2.682 3.163  Table D-25: Monte Carlo Simulation Results Probability of Skidding Out for a C a r on Wet Pavement at 50 km/h Design Speed Radius (m)  Probability of Failure  Beta  10 20 30 40 50 60 70 80 90  0.93477 0.8068 0.64939 0.49206 0.35971 0.25469 0.17734 0.12111 0.08216  -1.512 -0.866 -0.384 0.02 0.359 0.66 0.926 1.169 1.391  166  Table D-26: Monte Carlo Simulation Results Probability of Feeling Uncomfortable for a Car on Wet or Dry Pavement at 50 km/h Design Speed Radius (m)  Probability of Failure  Beta  10 20 30 40 50 60 70 80 90  0.94651 0.84544 0.71416 0.5736 0.43856 0.32243 0.23118 0.15892 0.10633  -1.612 -1.017 -0.566 -0.186 0.154 0.461 0.735 0.999 1.246  Note: The probability of failure and beta are based on system failure as there are two modes of failure. Table D-27: Monte Carlo Simulation Results Probability of Skidding Out for a Car on Dry Pavement at 80 km/h Design Speed Radius (m)  Probability of Failure  Beta  50 75 100 125 150 175 200 225  0.57404 0.21484 0.05331 0.01043 0.00177 0.00022 0.00003 0.000024  -0.187 0.79 1.6414 2.311 2.916 3.515 4.013 4.065  167  Table D-28: Monte Carlo Simulation Results Probability of Skidding Out for a Car on Wet Pavement at 80 km/h Design Speed Radius (m)  Probability of Failure  Beta  50 75 100 125 150 175 200 225 250  0.94483 0.83191 0.6751 0.51697 0.38151 0.27519 0.19762 0.14314 0.10438  -1.597 -0.962 -0.454 -0.043 0.302 0.597 0.85 1.066 1.257  Table D-29: Monte Carlo Simulation Results System Probability of Feeling Uncomfortable for a Car on Dry Pavement at 80 km/h Design Speed Radius (m)  Probability of Failure  Beta  50 75 100 125 150 175 200 225 250  0.93063 0.77751 0.56683 0.35795 0.20076 0.10043 0.04613 0.01993 0.00806  -1.48 -0.764 -0.168 0.364 0.839 1.279 1.684 2.055 2.406  Note: The probability of failure and beta are based on system failure as there are two modes of failure.  168  Table D-30: Monte Carlo Simulation Results System Probability of Feeling Uncomfortable for a Car on Wet Pavement at 80 km/h Design Speed Radius (m)  Probability of Failure  Beta  50 75 100 125 150 175 200 225 250  0.95462 0.84867 0.6933 0.5218 0.37082 0.25602 0.17787 0.12307 0.08743  -1.688 -1.031 -0.505 -0.055 0.33 0.656 0.924 1.16 1.357  Note: The probability of failure and beta are based on system failure as there are two modes of failure. Table D-31: Monte Carlo Simulation Results Probability of Being Unable to Stop on a Crest Vertical Curve for a Car on Dry Pavement Length of Vertical Curve (m)  Probability of Failure  Beta  20 40 60 80 100 125 150 175 200 250 300  0.87651 0.60398 0.36781 0.21327 0.11329 0.04551 0.01666 0.00562 0.00179 0.00016 0.000017  -1.158 -0.264 0.338 0.795 1.209 1.69 2.128 2.535 2.913 3.599 4.145  169  Table D-32: Monte Carlo Simulation Results Probability of Being Unable to Stop on a Crest Vertical Curve for a C a r on Wet Pavement Length of Vertical Curve (m)  Probability of Failure  Beta  40 80 100 125 150 175 200 250 300 400 500  0.92162 0.75706 0.66375 0.54912 0.44198 0.35213 0.27746 0.17477 0.11412 0.05606 0.03259  -1.416 -0.697 -0.423 -0.123 0.146 0.38 0.59 0.935 1.205 1.589 1.844  Table D-33: Monte Carlo Simulation Results Probability of Being Unable to Stop on a Crest Vertical Curve for a Truck on Dry Pavement Length of Vertical Curve (m)  Probability of Failure  Beta  20 40 60 80 100 125 150 175 200 250 300 400  0.94875 0.8007 0.626 0.46942 0.34271 0.23 0.14274 0.0935 0.05627 0.01904 0.00642 0.00066  -1.683 -0.844 -0.321 0.077 0.405 0.739 1.042 1.32 1.587 2.74 2.488 3.212  170  Table D-34: Monte Carlo Simulation Results Probability of Being Unable to Stop on a Crest Vertical Curve for a Truck on Wet Pavement Length of Vertical Curve (m)  Probability of Failure  Beta  80 100 125 150 175 200 250 300 400 500 750  0.85569 0.79859 0.727709 0.65489 0.58488 0.51804 0.39951 0.30554 0.18023 0.11163 0:04347  -1.061 -0.837 -0.604 -0.399 -0.214 -0.045 0.255 0.509 0.914 1.218 1.712  Table D-35: Monte Carlo Simulation Results Probability of Being Unable to Stop on a Sag Vertical Curve for a Car on Dry Pavement Length of Vertical Curve (m)  Probability of Failure  Beta  20 40 50 60 80 100 125 150 175 200 250  0.98697 0.89789 0.80928 0.69522 0.4396 0.22901 0.08048 0.02326 0.00607 0.00136 0.00005  -2.225 -1.27 -0.875 -0.511 0.152 0.742 1.402 1.991 2.508 2.998 3.891  171  Table D-36: Monte Carlo Simulation Results Probability of Being Unable to Stop on a Sag Vertical Curve for a C a r on Wet Pavement Length of Vertical Curve (m)  Probability of Failure  Beta  20 40 50 60 80 100 125 150 175 200 250 300 400 500 750  0.99764 0.9845 0.96842 0.94487 0.87063 0.76915 0.62054 0.47789 0.35708 0.26434 0.14709 0.08852 0.03987 0.02259 0.00938  -2.826 -2.157 -1.858 -1.597 -1.129 -0.736 -0.307 0.055 0.366 0.63 1.049 1.35 1.752 2.003 2.35  Table D-37: Monte Carlo Simulation Results Probability of Being Unable to Stop on a Sag Vertical Curve for a Truck on Dry Pavement Length of Vertical Curve (m)  Probability of Failure  Beta  20 40 50 60 80 100 125 150 175 200 250 300 400 500  0.99548 0.96938 0.94283 0.9063 0.7996 0.65728 0.46646 0.29869 0.17684 0.09799 0.02523 0.0059 0.00038 0.00003  -2.611 -1.872 -1.572 -1.318 -0.84 -0.405 0.084 0.528 0.927 1.293 1.956 2.518 3.367 4.013  172  Table D-38: Monte Carlo Simulation Results Probability of Being Unable to Stop on a Sag Vertical Curve for a Truck on Wet Pavement Length of Vertical Curve (m)  Probability of Failure  Beta  20 40 50 60 80 100 125 150 175 200 250 300 400 500 750  0.99876 0.99381 0.98882' 0.98138 0.95719 0.92073 0.85339 0.77304 0.68303 0.59297 0.43097 0.30685 0.15612 0.08698 0.03106  -3.026 -2.501 -2.284 -2.083 -1.719 -1.41 -1.051 -0.749 -0.476 -0.235 0.176 0.505 1.011 1.36 1.865  173  APPENDIX E ADAPTIVE SIMULATION RESULTS (FOR SERIOUS INJURY PREVENTION SCENARIOS O N L Y )  174  Table E - l : Adaptive Sampling Results Probability of Being Unable to Stop Within the Specified SSD for Reckless C a r to Pedestrian on Dry Pavement  SSD (m)  Probability of Non-Compliance  Beta  Perception Reaction Time (s)  Coefficient of Friction  Injury Speed (km/h)  50 60 65 70 80 90 100  0.990677 0.845829 0.671515 0.484826 0.181582 0.05022 0.011702  -2.353 -1.019 -0.444 0.038 0.909 1.643 2.267  1.4522 1.5465 1.6391 1.7333 2.03 2.3945 2.7283  0.87679 0.87499 0.8724 0.86626 0.86207 0.85873 0.84873  23.997 23.314 23.92 23.321 22.884 22.728 22.364  Table E-2: Adaptive Sampling Results Probability of Being Unable to Stop Within the Specified SSD for Reckless C a r to Pedestrian on Wet Pavement  SSD  Cm)  50 60 65 75 90 100 110 120 130 140 150 170 180 200 250 300  Probability of Non-Compliance  1 1 1 0.98322 0.921541 0.840882 0.728579 0.606406 0.498208 0.405059 0.22109 0.172451 0.11829 0.060358 0.011378 0.002054  . Beta  Perception Reaction Time (s)  Coefficient of Friction  Injury Speed (km/h)  1.4391 1.4582 1.4589 1.4513 1.4652 1.4787 1.4852 1.5115 1.4849 1.5167 1.4946 1.4928 1.5051 1.4988 1.5145 1.4972  0.31286 0.31387 0.31567 0.31014 0.30258 0.29078 0.27328 0.25795 0.24378 0.23456 0.20375 0.19689 0.18079 1.6592 0.12952 0.1054  24.581 24.384 24.052 24.486 24.478 23.523 23.159 24.112 23.564 22.361 22.994 22.346 21.394 22.2 20.062 20.944  -8.21 -8.21 -8.21 -2.125 -1.416 -0.998 -0.609 -0.27 0.004 0.24 0.769 0.945 1.184 1.552 2.278 2.87  175  Table E - 3 : Adaptive Sampling Results Probability of Being Unable to Stop Within the Specified SSD for C a r to Pedestrian on D r y Pavement  SSD (m)  Probability of Non-Compliance  20 30 40 50 60 65 70 80  0.713278 0.456735 0.227386 0.10731 0.039458 0.023226 0.01298 0.004164  Beta  Vehicle Speed (km/h)  Perception Reaction Time (s)  Coefficient of Friction  Injury Speed (km/h)  -0.563 0.109 0.747 1.241 1.757 1.991 2.227 2.638  56.719 62.267 68.485 72.712 77.771 79.149 81.851 86.2  1.4913 1.4904 1.573 1.6335 .1.6612 1.7748 1.8323 1.9526  0.8717 0.86762 0.86978 0.86741 0.85768 0.86776 0.86268 0.8532  24.163 23.538 24.213 23.893 23.244 24.169 23.148 22.999  Table E-4: Adaptive Sampling Results Probability of Being Unable to Stop Within the Specified SSD for C a r to Pedestrian on Wet Pavement  SSD (m)  Probability of Non-Compliance  Beta  Vehicle Speed (km/h)  20 30 40 50 60 65 70 80 90 100 125  0.769044 0.612079 0.39984 0.265544 0.166611 0.124303 0.093841 0.052506 0.031229 0.017037 0.004277  -0.736 -0.285 0.254 0.626 0.968 1.154 1.317 1.621 1.863 2.119 2.629  55.667 59.913 62.87 67.586 7.218 72.548 72.644 75.634 77.255 77.503 79.685  176  Perception Reaction Time (s)  Coefficient of Friction  Injury Speed (km/h)  1.4444 1.4718 1.5005 1.4976 1.5388 1.5239 1.5483 1.5324 1.591 1.6041 1.5605  0.4799 0.47472 0.473 0.46188 0.45261 0.44221 0.4339 0.42311 0.41208 0.3695 0.3002  23.954 22.834 22.978 23.371 22.88 22.633 23.225 22.658 22.437 22.568 23.167  Table E-5: Adaptive Sampling Results Probability of Being Unable to Stop Within the Specified SSD for C a r to C a r on Dry Pavement  P r o b a b i l i t y of  Vehicle  Perception  Speed  Reaction  Coefficient  Injury Speed  S S D (m)  Non-Compliance  Beta  (km/h)  T i m e (s)  of F r i c t i o n  (km/h)  20 40 50 60 75 90 100 113  0.962705 0.765071 0.567135 0.37218 0.165479 0.058375 0.026608 0.008229  -1.783 -0.723 -0.169 0.326 0.972 1.569 1.933 2.399  78.612 82.832 86.829 91.373 97.759 103.75 108.17 114.14  1.5006 1.5312 1.5707 1.5944 1.6382 1.7211 1.761 1.799  0.87422 0.87299 0.87125 0.86884 0.8643 0.85842 0.85294 0.84487  28.277 28.108 26.583 25.922 25.585 25.213 25.073 25.271  Table E - 6 : Adaptive Sampling Results Probability of Being Unable to Stop Within the Specified SSD for C a r to C a r on Wet Pavement  Probability of  Vehicle  Perception  Speed  Reaction  Coefficient  Injury Speed  S S D (m)  Non-Compliance  Beta  (km/h)  T i m e (s)  of F r i c t i o n  (km/h)  20 40 50 60 75 90 100 113 125 139  0.968452 0.838487 0.686234 0.519708 0.285465 0.142086 0.076118 0.030821 0.012141 0.003792  -1.859 -0.988 -0.485 -0.049 0.567 1.071 1.432 1.869 2.253 2.67  78.455 81.329 83.392 0.88313 92.754 98.415 103.41 108.92 112.66 118.12  1.4856 1.5293 1.5323 1.5579 1.6252 1.6399 1.6673 1.7003 1.7845 1.7911  0.63363 0.6335 0.63442 0.633 0.63201 0.63065 0.63169 0.62742 0.62859 0.62495  27.18 26.472 26.176 26.23 25.256 24.866 24.854 25 24.247 24.238  177  Table E-7: Adaptive Sampling Results Probability of Being Unable to Stop Within the Specified SSD for Truck to C a r on Dry Pavement  SSD (m)  Probability of NonCompliance  20 30 40 50 60 65 70 80 90 100 125 140  0.848149 0.716244 0.513694 0.358027 0.22436 0.171483 0.133318 0.071998 0.037288 0.019785 0.003194 0.000913  (  Beta  Vehicle Speed (km/h)  Perception Reaction Time (s)  Coefficient of Friction  Injury Speed (km/h)  N  -1.029 -0.572 -0.034 0.364 0.758 0.948 1.111 1.461 1.783 2.058 2.727 3.117  53.819 56.633 60.827 65.167 68.882 71.236 72.681 76.531 79.215 81.238 87.151 87.519  1.9857 1.9762 1.9814 2.027 2.0389 2.0561 2.59 2.0639 2.148 2.1035 2.1809 2.2044  0.87051 0.87114 0.87062 0.86606 0.86639 0.86747 0.86571 0.86568 0.86253 0.86013 0.84707 0.83662  24.481 23.59 23.739 24.016 23.443 23.426 23.382 22.414 22.814 21.893 21.009 20.957  0.59848 0.60181 0.59247 0.6 0.59013 0.58559 0.58448 0.58083 0.56838 0.55442 0.53711 0.51053  Table E-8: Adaptive Sampling Results Probability of Being Unable to Stop Within the Specified SSD for Truck to C a r on Wet Pavement  SSD (m)  Probability of NonCompliance  20 30 40 50 60 65 70 80 90 100 125 150 175 200  0.837946 0.763335 0.651146 0.512653 0.401327 0.363843 0.324864 0.240476 0.17481 0.12201 0.056375 0.024278 0.010501 0.002072  Beta  Vehicle Speed (km/h)  Perception Reaction Time (s)  Coefficient of Friction  Injury Speed (km/h)  N  -0.986 -0.717 -0.388 -0.032 0.25 0.348 0.454 0.705 0.935 1.165 1.586 1.972 2.308 2.571  53.201 56.208 57.834 60.871 63.158 64.783 65.471 67.296 68.086 71.278 76.128 75.007 78.275 78.039  1.9702 1.9587 1.9547 1.9818 1.9994 2.0081 1.9941 2.0017 2.0061 1.991 2.0115 2.04 2.0072 2.0161  0.4895 0.47836 0.47296 0.47951 0.47874 0.46175 0.45135 0.46042 0.4662 0.43525 0.42486 0.3802 0.3562 0.29749  23.606 23.514 23.472 23.212 22.953 23.811 22.748 22.552 21.968 22.949 21.65 21.212 21.588 19.676  0.59622 0.59917 0.59866 0.59669 0.58582 0.58911 0.58919 0.59014 0.5799 0.57984 0.56724 0.5513 0.56031 0.53803  178  APPENDIX F COMPARISON OF RESULTS FROM DIFFERENT COMPUTATIONAL METHODOLOGIES  179  a.  <  WD  o  o  o  o  Ajnfui snouas juaAajj o) dots o» ^ICjBun 6ujag jo Aiujqeqojd  CU >.  a CS <<  S o CU  V  >>  is 'a  •w a cs &  3  a  O  «  cs  U  no  cn  #  c  *£ CU "S  CS o U CU  JS  H  isio an  a -a  ck  ox o a U "cu  £  s C5 M H o a U  CU  JS  «M O  fl o c«  CJ a £  o U  CU u SS  on to  —I O o  1  O  O o  1  ^  C  o  O o  1  I  O o  1  T  J o  1  - 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