THE PENNSYLVANIA STATE UNIVERSITY SCHREYER HONORS COLLEGE DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING A COMPARISON OF FRICTION SUPPLY, FRICTION DEMAND, AND MAXIMUM DESIGN FRICTION ON SHARP HORIZONTAL CURVES WITH STEEP GRADES REBECCA LUTZ SPRING 2013 A thesis submitted in partial fulfillment of the requirements for a baccalaureate degree in Civil Engineering with honors in Civil Engineering Reviewed and approved* by the following: Eric Donnell Associate Professor of Civil Engineering Thesis Supervisor and Honors Advisor Patrick Reed Associate Professor of Civil Engineering Faculty Reader * Signatures are on file in the Schreyer Honors College.
i ABSTRACT The American Association of State Highway and Transportation Officials (AASHTO) have a set of recommended parameters to use for horizontal curve design. In their publication A Policy on Geometric Design of Highways and Streets (commonly referred to as the Green Book), design guidelines are outlined based upon a roadway s design speed, maximum superelevation, the radius of the curve, and the maximum design side friction. These values are considered to be very conservative as they are based on driver comfort studies and not the physics of the vehicle. It is important that these criteria do in fact have a large factor of safety. Otherwise the probability of vehicle skidding or rollover will increase, leading to an increased probability of crashes upon the roadway. The calculations that are commonly used to find friction supply and friction demand are simplified. Friction supply calculations ignore the effects of grade, rolling resistance, aerodynamic resistance (drag), and longitudinal acceleration/deceleration. The friction demand calculations assume some terms are small enough to be considered zero. In this thesis, these calculations are compared to their modified calculations which do not ignore any of these components. This is done to ensure that these modified friction calculations do not exceed those used by AASHTO. AASHTO has also never differentiated the maximum design side friction values for twolane or multi-lane highways. Intuitively the pavement characteristics and driver tendencies vary between these two facility types. These variations could contribute to a difference in friction supply or friction demand that could merit different design criteria for the different facility types.
ii The data analyzed for this thesis were acquired from two different reports: FHWA Information Report to Achieve Target Speeds on Rural and Suburban Highways, and NCHRP Report 15-39: Superelvation Criteria for Sharp Horizontal Curves on Steep Grades. These reports included curve radii, superelevation and vehicle speed for passenger cars and trucks, as well as friction readings as generated by field testing equipment. These data were utilized in the calculation of simple and modified friction demand and supply. The analysis of the data showed that simple and modified friction demand were nearly identical. At the upper end of the speed range, simple friction demand was slightly higher than modified friction demand. At 15 out of 24 sites, simple friction demand was greater than simple friction supply for some portion of the speed range. In all cases, modified friction supply was less than simple friction supply, but it was always higher than the AASHTO recommended maximum design side friction values. Five of the 7 multi-lane facilities had comparable simple friction supply values as the two-lane facilities, but the other 2 multi-lane facilities had higher values. The friction demand values at the two-lane sites were consistently higher than at the multi-lane sites. Based on these results, it can be concluded that the AASHTO maximum recommended design side friction values incorporate some factor of safety across the range of feasible vehicle speeds. AASHTO is also justified in using its simplified friction demand model since it is slightly more conservative than the modified model at higher speeds. At many of the sites, a small proportion of drivers are choosing speeds which require a higher friction demand than the roadway can supply. The comparison of friction supply values on multi-lane and two-lane facilities was inconclusive, but the comparison of the friction demand values showed that the
iii two-lane facilities consistently demand more friction than the multi-lane facilities. The collection of more data from a large variety of sites would lend more credence to this conclusion.
iv TABLE OF CONTENTS Abstract... i List of Figures... vi List of Tables... viii Chapter 1 Introduction... 1 1.1 Overview... 1 1.2 Research Objectives... 2 1.3 Organization of Thesis... 3 Chapter 2 Literature Review... 4 2.1 Introduction... 4 2.2 Side Friction and Superelevation... 4 2.2.1 The Friction Ellipse... 7 2.3 Current Horizontal Curve Design... 11 2.3.1 Horizontal Curve Transition... 12 2.3.2 Shortcomings of Existing Factors of Safety... 15 2.4 Safety Concerns Specific to Trucks... 16 2.4.1 Offtracking... 18 2.4.2 Braking Requirements... 20 2.4.3 Current Design Methodology to Accommodate Trucks... 21 Chapter 3 Derivation of the Point Mass and the Modified Point Mass... 23 3.1 Basic Vehicle Mechanics on a Tangent... 23 3.1.1 Friction... 24 3.1.2 Aerodynamic Resistance... 24 3.1.3 Rolling Resistance... 25 3.1.4 Grade Resistance... 27 3.1.5 Application to Roadway Design... 27 3.2 Basic Vehicle Mechanics during Cornering... 29 3.3 The Modified Point Mass Model... 31 Chapter 4 Methodology... 33 4.1 Lateral Friction Demand from the Point-Mass Equation... 33 4.2 Lateral Friction from the Modified Point-Mass Equation... 33 4.3 Finding Friction Supply... 34 4.4 Finding Friction Supply Using Modified Point Mass Equation... 36 4.5 Analysis Method... 38
v Chapter 5 Field Site Descriptions... 39 5.1 Multi-Lane Highway Field Site Descriptions... 39 5.2 Two-Lane Field Site Descriptions... 40 5.3 Friction Data from All Sites... 40 Chapter 6 Data Analysis and Results... 42 6.1 Comparison of Friction Demand, Friction Supply, and AASHTO Recommended Friction... 42 6.2 Comparison of Modified Friction Supply to AASHTO Maximum Design Side Friction 45 6.3 Comparison of Modified Friction Demand for Passenger Cars and Trucks... 47 6.4 Comparison of Simple Friction Demand for Multi-Lane and Two-Lane Facilities.. 49 Chapter 7 Conclusion... 53 Appendix A Friction Calculations... 55 1. Simple Friction Supply from All Sample Sites at 40 mph... 55 2. Speed Data... 61 3. Passenger Car Friction Calculations... 62 3.1 Simple Friction Demand... 62 3.2 Modified Friction Demand... 64 3.3 Simple Friction Supply... 65 3.4 Modified Friction Supply... 67 4. Truck Friction Calculations... 69 4.1 Simple Friction Demand... 69 4.2 Modified Friction Demand... 70 4.3 Simple Friction Supply... 72 4.4 Modified Friction Supply... 73 Appendix B Friction Graphs... 75 1. Passenger Car Friction Graphs... 75 2. Truck Friction Graphs... 81 REFERENCES... 87
vi LIST OF FIGURES Figure 2.1 A free-body diagram of a vehicle going around a curve. The only forces acting on the vehicle are gravity (G), the normal force (N), and the force of side friction (F s ).... 5 Figure 2.2 Diagram of the side friction and normal forces which combine to create centripetal acceleration on a superelevated roadway. Since gravity does not contribute to the centripetal acceleration, it has been omitted for clarity.... 6 Figure 2.3 Top view of the friction forces action on the tire of a vehicle cornering to the left 8 Figure 2.4 Sample of the friction ellipse. Source: Milliken and Milliken, 1995... 9 Figure 2.5 Differences in efficiency of cornering and braking friction at different percent slips. Source: NCHRP Report 505... 10 Figure 2.6 Diagram of a simple circular curve. The tangent section ends and the curve beings at the PC. The vehicle will go through the curve of constant radius (R) until the curve ends at the PT. Source: civilengineering.com... 12 Figure 2.7 Diagram of a spiral curve. Notice how the radius starts large (R1) and ends small (R5). Source: mccombsurveying.com... 13 Figure 2.8 Comparison of a circular (top) and spiral (bottom) curve. Source: eng2.uconn.edu 14 Figure 2.9 Truck offtracking through a horizontal curve. Source: (Harwood et al, 2003)... 19 Figure 2.10 Illustrations of truck behavior during locked-wheel braking. Source: (Harwood et al, 2003)... 20 Figure 3.1 Diagram of the forces acting on a vehicle while traversing a tangent section of roadway. Source: The Principles of Highway Engineering and Traffic Analysis.... 23 Figure 3.2 Diagram of forces acting on a vehicle while cornering on a superelevated roadway. 29 Figure 6.1 A comparison of the different friction values experienced by passenger cars at site MD 3.... 42 Figure 6.2. Friction Values for passenger cars at the WV Route 32 Comparison site. Notice that the friction demand exceeds simple friction supply, but only at the highest end of the speed range.... 45 Figure 6.3. Friction values for passenger cars at the WV1 site. Here, friction demand values exceed friction supply values near the mid point of the speed range.... 45 Figure 6.4. Simple friction supply values for the high speed facility sites.... 50 Figure 6.5. Simple friction supply values for the two-lane rural facility sites.... 50
vii Figure 6.6. Simple friction demand for all high speed facility sites.... 51 Figure 6.7. Simple friction demand values for all two-lane rural facility sites.... 52
viii LIST OF TABLES Table 2.1 A comparison of the factors of safety for passenger cars and trucks under a variety of conditions.... 17 Table 5.1 Geometric data for the 8 field sites in Maryland and West Virginia... 39 Table 5.2 Geometric and Speed Data for 7 field sites in West Virginia... 40 Table 5.3 Friction Supply values from all 15 field sites... 41 Table 6.1 Simple and Modified Friction Demand at site MD 3.... 43 Table 6.2. Comparison of factors of safety for passenger cars.... 46 Table 6.3. Comparison of factors of safety for trucks.... 47 Table 6.4. Modified friction supply values for passenger cars and trucks on a diverse sample of roadways.... 48
1 Chapter 1 Introduction 1.1 Overview The American Association of State Highway and Transportation Officials (AASHTO) A Policy on Geometric Design of Highways and Streets (herein referred to as the Green Book) contains roadway design controls and criteria for new and major reconstruction projects. Horizontal and vertical alignment, as well as cross-section, design elements are included in the policy. The horizontal curve design criteria are based on the point-mass model, which is shown below in equation (1.1). 2 V e max + f max = (1.1) 15R min where: e max = maximum rate of superelevation (decimal); f max = maximum design side friction (unitless); V = design speed (mph); R min = minimum radius of horizontal curve (feet). Geometric designers select a design speed (V) based on the anticipated operating speed of the highway, along with the topography and land use in which the alignment traverses. The maximum rate of superelevation (e max ) is limited to 12 percent in regions that do not experience
2 snow or ice accumulation, but is limited to 8 percent in areas that do amass snow or ice. The maximum design side friction (f max ) is based on driver comfort thresholds, resulting from research in the 1940 s (Stonex and Noble, 1940; Moyer 1934; Moyer and Berry 1940).The minimum radius of curve (R min ) results from the design speed, superelevation, and friction input, and is the criterion by which horizontal curve design is assessed. The point-mass model is a simplification of the forces and resistances that act on a vehicle when traversing a horizontal curve. The friction that is available at the tire-pavement interface (f supply ) is not used in horizontal curve design criteria. Further, vehicle operating speeds differ from the design speed. Recent research (Donnell, Himes, Mahoney, and Porter; 2009) shows that operating speeds exceed geometric design speeds. The side friction that drivers demand (f demand ), based on their speed choice, will differ from the maximum design side friction (f max ) and the friction supplied at the tire-pavement interface (f supply ). This suggests that there is a margin of safety integrated into geometric design criteria, because the maximum design side friction values used in the Green Book are substantially lower than both the driver demand side friction (f demand ) and the tire-pavement friction (f supply ). 1.2 Research Objectives The objectives of this research are to:
3 1. Derive the point-mass model used in horizontal curve design, and assess the forces and resistances that act on a vehicle when traversing a curve (i.e., modified point-mass model). 2. Determine the margin of safety used in horizontal curve design by considering the relationship between maximum design side friction (f max ), friction supplied by the tirepavement interaction (f supply ), and driver demand friction (f demand ). 3. Compare the margins of safety across multi-lane and two-lane highways, vehicle types, and varying vertical grades. 1.3 Organization of Thesis This thesis is organized into 7 chapters. Chapter 2 contains a review of the extant literature related to horizontal curve design criteria, friction, and factors that can contribute to vehicle crashes. Chapter 3 derives the point mass and modified point mass equations, both of which are used later in the analysis of the data collected in the field. Chapter 4 contains an explanation of the methods used to obtain the different friction values and the anticipated results of their analysis. Chapter 5 outlines the characteristic of the different study sites, and Chapter 6 contains the actual analysis and comparison of the friction demand and friction supply (obtained through both the regular and modified point mass equations) to current maximum design side friction as recommended by AASHTO. The final chapter is a summary of the findings and conclusions drawn from them.
4 Chapter 2 Literature Review 2.1 Introduction For this study, a review of the current literature available on the mechanics of vehicle cornering and design of horizontal curves was conducted. Of particular interest was the difference in the vehicle demands between passenger cars and trucks. The results of the literature review are presented in the following sections. 2.2 Side Friction and Superelevation The most basic model of a moving vehicle is that of the point-mass model, which treats the vehicle as a single point whose mass is located at the vehicle s center of gravity. When traversing a circular curve, the vehicle experiences centripetal acceleration (a c ), which is computed as shown in Equation (2.1): a c = V2 R (2.1) a c = Centripetal acceleration V = Speed of the vehicle R=Distance from the center of the curve to the location of the point mass
5 On a flat road, this centripetal acceleration is provided by only the side friction (F s ) acting between the vehicle s tires and the roadway, as seen in Figure 2.1. Any surface has some frictional characteristics. The most important of these is the friction factor (f), which is shown below in Equation (2.2): f = F f,max N (2.2) F f, max = Maximum available friction N= Normal force N F c F s G Figure 2.1 A free-body diagram of a vehicle going around a curve. The only forces acting on the vehicle are gravity (G), the normal force (N), and the force of side friction (F s ). A high friction factor means that there is a lot of friction available, while a low friction factor means that there is little available. At the roadway surface-tire interface, the friction factor is dependent on the type of roadway (asphalt or cement), the level of wear the roadway has experienced (new to very poor), and the condition of the road (dry, wet, icy). For asphalt roadways, these values range from >1.00 (good, dry) to 0.25 (packed snow or ice). If the tires lock and slide across the pavement, the friction factor decreases to a range of 0.8 to 0.1 (Mannering, 2009). Horizontal curve design, however, is based on driver comfort thresholds, called design side friction.
6 In some cases, the amount of centripetal acceleration needed for a vehicle to track a curve of a fixed radius cannot be provided in full by the side friction. This happens when the curve is particularly sharp (has a small radius), the design speed is high, or a combination of both. When N N y a c N x a c F s F s,y F s,x Figure 2.2 Diagram of the side friction and normal forces which combine to create centripetal acceleration on a superelevated roadway. Since gravity does not contribute to the centripetal acceleration, it has been omitted for clarity. this occurs, the road can be banked, or superelevated, to provide additional centripetal acceleration force. On a banked road, the normal force is no longer strictly vertical; a portion of it points inwards towards the center of the curve (see Figure 2.2). This horizontal component of the normal force contributes to the centripetal acceleration. The amount of normal force acting in the horizontal direction is directly related to the amount of superelevation; the greater the superelevation, the greater the horizontal component. In highway design, superelevation is denoted by the letter e and is the number of vertical feet of rise per 100 feet of horizontal distance. It is almost always written as a percent. Values for e range from 0% to 12%, but most states set an e max of 8%, particularly where snow and ice are accumulation occur. At present, the American Association of State Highway and Transportation Officials (AASHTO) A Policy on Geometric Design of Highways and Streets recommends that superelevation be found using Equation (2.3):
7 V= V D = Design speed (mph) e=e max = maximum superelevation, percent f=f max = maximum allowable side friction factor R=R min =minimum radius (feet) e = V2 f (2.3) 15R The derivation of this equation will be explained further in Chapter 3. 2.2.1 The Friction Ellipse As previously discussed, the friction factor represents the amount of friction available between the roadway and the vehicle s tires. However, this friction is not just the side friction (or lateral force), but also the braking or traction friction (see Figure 2.3). As Gillespie explains in Fundamentals of Vehicle Dynamics (1992), the friction factor is essentially the friction limit. Friction can be used for lateral force, braking/traction forces, or a combination of the two; but the vector sum of the two forces cannot exceed the friction limit. This gives way to the idea of a Friction Ellipse or Friction Circle, the radius of which is equal to the friction factor. The shape of the relationship is determined by the type of tire on the vehicle. Some tires have been specialized for optimized braking or lateral traction, resulting in the elliptical shape; otherwise, the relationship is circular.
8 Y Top View of a tire X Lateral Friction Path of Vehicle Movement Braking (or Traction) Friction Figure 2.3 Top view of the friction forces action on the tire of a vehicle cornering to the left Due to this relationship, a vehicle may be able to brake or corner and draw on the whole friction factor, but it cannot do these things simultaneously. In the event that the vehicle is both braking and cornering, only a fraction of the entire friction factor will be available for either of the two actions. This is important because, when entering a curve, most drivers will be braking since the centripetal acceleration increases driver discomfort. By lowering their speed, the driver lowers the centripetal acceleration (2.1) and thus decreases the discomfort caused by the centripetal acceleration. Thus it is unrealistic to assume that a driver will ever have the maximum friction factor available for the lateral acceleration needed to track the curve since a portion of that friction factor will almost always be used to brake. This problem is exacerbated when the vehicle is travelling downhill. As a vehicle proceeds downhill, it tends to pick up speed as its potential energy is converted into kinetic energy. The faster a vehicle is going while trying to transverse a horizontal curve, the more side friction is required to keep it from leaving the road. Logically then, the driver would brake to reduce their speed and thus reduce the amount of side friction required to keep the vehicle on the
9 road. However, this means that the driver will be braking even harder than he would on a flat grade and using up even more of the available friction in the friction ellipse. As a result, there Figure 2.4 Sample of the friction ellipse. Source: Milliken and Milliken, 1995 may not be enough available friction to provide the necessary side friction to track the curve. In effect, if the driver enters the curve too quickly and the curve demands a large amount of side
10 friction (i.e. a sharp curve), it may be impossible for the driver to compensate while in the curve without crashing the vehicle (Eck and French, 2002). Figure 2.5 Differences in efficiency of cornering and braking friction at different percent slips. Source: NCHRP Report 505 Moreover, the lateral force (also referred to as the cornering force) and the braking force functions differ depending on the percentage of slip which the tires are experiencing. Percent slip is a way of describing how well the tires are rolling as the vehicle moves. A freely rotating wheel
11 exhibits zero percent slip, while a fully locked wheel exhibits 100 percent slip. As shown in Figure 2.5, the lateral force is maximized when there is no slip, while the braking force is maximized when there is 10 to 15 percent slip. Thus, not only are the two forces drawing from the same total possible friction reserve, but they will also never simultaneously be in their condition for maximum efficiency. 2.3 Current Horizontal Curve Design AASHTO has created policies which limit the amount of unbalanced acceleration (mathematically equivalent to the friction factor, f ) based on the design speed of the road. f max ranges from 0.17g for a road with a design speed of 20 mph to 0.10g for a design speed of 70 mph. It is worth noting that these values come from studies completed between 1936 and 1949 on driver comfort (Harwood and Mason, 1994). Thus these values reflect the curve geometries which drivers find comfortable and are not a product of an analysis of the physics at work on the vehicle. Based on the design speed of the roadway, the minimum radius of curvature is found using: R min = 2 V D 15(f max +e max ) (2.4) If the design radius is greater than R min, then a superelevation which is less than e max is chosen using tables 3-8 to 3-13 in the AASHTO Green Book (Harwood and Mason, 1994).
12 2.3.1 Horizontal Curve Transition When designing a horizontal curve, there are two ways of transitioning between the straight, tangent section and the horizontal curve. Today, the most common of these is the simple circular curve, where the straight section runs immediately into the curve such that the two are tangent to one another where they meet. Figure 2.6 shows the basic elements of a simple circular curve, with two tangent sections being connected by a curve of a constant radius, R. This means that the roadway is instantaneously transitioning from having no curvature to having the radius of the horizontal curve. Without some pavement cross-slope transitions, the friction demands on the vehicle will instantaneously change. This could be accommodated if the road was designed such that it either did not require superelevation to safely transverse the curve or if the road was fully superelevated at the beginning of the curve. However, it is common practice today to use the two-thirds, one-thirds rule where the roadway is only at 67% of its Figure 2.6 Diagram of a simple circular curve. The tangent section ends and the curve beings at the PC. The vehicle will go through the curve of constant radius (R) until the curve ends at the PT. Source: civilengineering.com
13 full superelevation at the beginning of the curve. For the first part of the curve, the cornering forces (f side +e) are lower than the values which were assumed when the roadway was being designed (Eck and French, 2002). The other, less common way of designing the transition from a tangent to a curve is to use a spiral curve. A spiral curve is a curve of non-constant radius which starts out very large and gradually decreases until it reaches the radius of the simple circular curve. Figure 2.7 Diagram of a spiral curve. Notice how the radius starts large (R1) and ends small (R5). Source: mccombsurveying.com By designing the curve in this way, the centripetal acceleration is introduced gradually, as the radius of the curve decreases. In this case, implementing the two-thirds rule would produce a safer road because the force needed to keep the car on the road while the radius of the curve is still large would be sufficiently supplied by the friction between the road and the tires. By the time the radius of the curve is small enough to require superelevation, that superelevation will have been fully developed. While spiral curves offer advantages over simple circular curves, they are difficult to construct properly in the field. It is mainly for this reason that simple circular curves are used in geometric design of highways.
14 Figure 2.8 Comparison of a circular (top) and spiral (bottom) curve. Source: eng2.uconn.edu Interestingly, drivers inherently follow a spiral path when traversing a curve, whether the roadway is designed as such or not. Typically, one will tend to the outside part of the lane when entering the curve, and the inside of the lane when halfway through the curve. In other words, the driver gradually decreases the radius of the curve while traversing the curve (Eck and French, 2004). While some may say that this proves that the construction of spiral curves is unnecessary, Harwood and Mason (1994) have suggested that drivers steering a spiral path on a simple circular curve have the dangerous tendency to oversteer at some point in the curve. Since they
15 are not following the designed path while traversing the curve, drivers frequently continue to tighten their radius to a point where they are driving a smaller radius curve than the design radius. When this happens, accidents could occur because: 1) the driver feels uncomfortable and may compensate too quickly or 2) the increased lateral acceleration forces cause the vehicle to slide or roll over. If the road is designed as a spiral, drivers will simply follow the designed path and avoid oversteering. 2.3.2 Shortcomings of Existing Factors of Safety As discussed previously, the AASHTO policy regarding f max was derived from studies on driver comfort levels, not the physical limitations of the vehicle. It was assumed that drivers become uncomfortable before they are in serious danger of skidding or rolling over, so by making these the maximum allowable friction factors, a factor of safety has been included into the point mass model. However, these studies are over 60 years old. The vehicle fleet has changed completely since then, and it is unknown how that could affect the design policy. Equations 2.3 and 2.4, the basis of horizontal curve design, are based upon the point mass model. This model assumes that the forces acting on all tires of the vehicle are approximately equal. For passenger cars this may be true, but the forces acting on a semi tractor trailer s wheels are not equally distributed. The result is that, on any given curve, trucks will demand approximately 10% more side friction than a passenger car. Harwood and Mason (1994) termed this higher side friction demand effective side friction demand of trucks. Passenger cars are much more likely to skid off the road instead of rolling over since rolling over requires a lateral acceleration of about 1.2g. For trucks however, rollover crashes are a much greater concern. This is due to the truck s higher center of gravity as well as the elaborate
16 suspension system which prevents the truck from acting like a rigid body. Some trucks will roll over when they experience a lateral acceleration as low as 0.3g. Since AASHTO allows for the design of roadways with lateral accelerations of up to 0.17g on lower speed roads, there is a much lower factor of safety for trucks on these facilities than there is for passenger vehicles. Harwood and Mason (1994) took these details into consideration and calculated the factors of safety for both passenger cars and trucks at different design speeds and superelevations. The results are shown in Table 2.1. Clearly, trucks driving on wet pavement have a factor of safety which is half as great as the factor of safety for passenger cars driving under the same conditions. When one considers that vehicles frequently travel above design speeds and oversteer on curves which do not have spiral transitions, this factor of safety becomes even smaller. These issues are most prevalent on off-ramps at interchanges. This is because these ramps tend to have sharp curves and unrealistically low design speeds (Harwood and Mason, 1994). 2.4 Safety Concerns Specific to Trucks As mentioned previously, the physical mechanics of a truck differ significantly from those of a passenger car. This is due to their weight distribution, size and general geometry, braking mechanism, and articulated nature. Because of these factors, there are certain driving scenarios which may be particularly dangerous for a truck.
Passenger Cars Trucks 17 Design Speed (mph) Maximum superelevation (percent) Maximum comfortable lateral acceleration (g) Maximum demand (f) (in g) Minimum radius (feet) Available f (wet pavement) (in g) Margin of Safety (wet) (in g) Margin of Safety (Dry) (in g) Maximum comfortable lateral acceleration (g) Maximum demand (f) (in g) Minimum radius (feet) Available f (wet pavement) (in g) Margin of Safety (wet) (in g) 20 4 0.17 0.17 127 0.58 0.41 0.77 0.17 0.19 127 0.41 0.22 0.47 30 4 0.16 0.16 302 0.51 0.35 0.76 0.16 0.18 302 0.36 0.18 0.48 40 4 0.15 0.15 573 0.46 0.31 0.79 0.15 0.17 573 0.32 0.15 0.49 50 4 0.14 0.14 955 0.44 0.30 0.80 0.14 0.15 955 0.30 0.15 0.51 60 4 0.12 0.12 1626 0.42 0.30 0.82 0.12 0.13 1626 0.29 0.16 0.53 Margin of Safety (Dry) (in g) 20 6 0.17 0.17 116 0.58 0.41 0.77 0.17 0.19 116 0.41 0.22 0.47 30 6 0.16 0.16 273 0.51 0.35 0.76 0.16 0.18 273 0.36 0.18 0.48 40 6 0.15 0.15 609 0.46 0.31 0.79 0.15 0.17 609 0.32 0.15 0.49 50 6 0.14 0.14 849 0.44 0.30 0.80 0.14 0.15 849 0.30 0.15 0.51 60 6 0.12 0.12 1348 0.42 0.30 0.82 0.12 0.13 1348 0.29 0.16 0.53 70 6 0.10 0.10 2083 0.41 0.31 0.84 0.10 0.11 2083 0.28 0.17 0.55 20 8 0.17 0.17 107 0.58 0.41 0.77 0.17 0.19 107 0.41 0.22 0.47 30 8 0.16 0.16 252 0.51 0.35 0.76 0.16 0.18 252 0.36 0.18 0.48 40 8 0.15 0.15 466 0.46 0.31 0.79 0.15 0.17 466 0.32 0.15 0.49 50 8 0.14 0.14 764 0.44 0.30 0.80 0.14 0.15 764 0.30 0.15 0.51 60 8 0.12 0.12 1206 0.42 0.30 0.82 0.12 0.13 1206 0.29 0.16 0.53 70 8 0.10 0.10 1910 0.41 0.31 0.84 0.10 0.11 1910 0.28 0.17 0.55 20 10 0.17 0.17 99 0.58 0.41 0.77 0.17 0.19 99 0.41 0.22 0.47 30 10 0.16 0.16 231 0.51 0.35 0.76 0.16 0.18 231 0.36 0.18 0.48 40 10 0.15 0.15 432 0.46 0.31 0.79 0.15 0.17 432 0.32 0.15 0.49 50 10 0.14 0.14 694 0.44 0.30 0.80 0.14 0.15 694 0.30 0.15 0.51 60 10 0.12 0.12 1091 0.42 0.30 0.82 0.12 0.13 1091 0.29 0.16 0.53 70 10 0.10 0.10 1837 0.41 0.31 0.84 0.10 0.11 1837 0.28 0.17 0.55 Table 2.1 A comparison of the factors of safety for passenger cars and trucks under a variety of conditions.
18 2.4.1 Offtracking For smaller vehicles (bicycles and automobiles), the rear wheels of a vehicle very nearly follow the path of the front, driving wheels. In other words, offtracking is negligible. However, for trucks, which are much longer and articulated, offtracking is a very serious concern and must be taken into account when designing intersections and sharp curves. There are two different varieties of offtracking, low speed and high speed. In low speed offtracking, the front wheels of the vehicle try to drag the rear wheels towards them, but fail to do so completely, leaving the rear wheels to the inside of the curve. This phenomenon is depicted in Figure 2.9. If the curve is long enough, the offtracking will eventually reach a constant value, referred to as the fully developed offtracking. For common, two-axle trucks, this constant value can be found using a variation of the Pythagorean Theorem: OT = R + R 2 + l 2 OT= Offtracking (feet) R= Radius of the curve (feet) l= Distance between axles (feet) Once this fully developed offtracking value has been determined, the road can be designed to accommodate the truck. However, if the curve is too short or the radius of the curve is too small, the offtracking will never be fully developed. This situation is called partial offtracking and cannot be accurately predicted with an equation. Most engineers use a software
package called AutoTURN to model different turning vehicles and ensure that the trucks will be able to make the turn successfully. 19 Figure 2.9 Truck offtracking through a horizontal curve. Source: (Harwood et al, 2003) High speed offtracking results in the opposite phenomenon the rear wheels try to swing to the outside of the path of the steer axle instead of to the inside. Thus, the faster a truck is going, the smaller the offtracking, until a speed is reached where the effects of low- and highspeed offtracking cancel each other out. If the truck travels any faster than this, the wheels will begin to swing to the outside of the driven path. (Harwood et al, 2003)
20 2.4.2 Braking Requirements When a driver wishes to stop, he applies the brakes to his vehicle. The faster he wishes to stop, the harder the brakes are applied. At a certain point, the brakes will lock the wheels, preventing them from rolling while the vehicle decelerates. During locked-wheel braking, the vehicle is in a 100 percent slip condition. As shown in Figure 2.5, it is clear that the coefficient of braking while in 100 percent slip is slightly lower than the peak braking coefficient, making this form of braking less efficient for any vehicle. For trucks, however, locked-wheel braking presents a much more significant danger. All of the axles will not lock simultaneously. Depending on which one does lock first, a plow out, jackknife, or trailer swing will develop, as shown below in Figure 2.10. It is almost impossible to recover from any of these. Figure 2.10 Illustrations of truck behavior during locked-wheel braking. Source: (Harwood et al, 2003) In order to avoid locked-wheel braking, the driver must modulate or pump the brakes. This will result in a slower deceleration rate, which must be accounted for when finding the stopping sight distance during roadway design. (Harwood et al, 2003)
21 2.4.3 Current Design Methodology to Accommodate Trucks When a passenger car is travelling downhill, the speed which it might accrue from its descent is mostly countered by the rolling friction and aerodynamic drag passenger cars require minimal braking. Large trucks and trailers, however, have too much mass for these forces to sufficiently slow its speed significant amounts of braking are required. With so much braking, there is the possibility for the braking mechanism on these trucks to overheat, decreasing its efficiency and possibly causing the driver to lose control of the vehicle. Because of this, the Federal Highway Administration (FHWA) has created 4 different speed criteria which must be taken into account when designing downhill sections of roadway which will be traversed by tractor trailers (Harwood et al, 2003). Each criterion is specific to the particular model of truck and the grade of the roadway. They are: 1. The maximum speed at which the specified truck can descend the specified grade without the loss of braking ability 2. The maximum speed at which the specified truck can descend the specified grade without rolling over when traversing a horizontal curve 3. The maximum speed at which the specified truck can descend the specified grade without losing the ability to brake at a rate of 3.4 m/s 2 or more. This is the stopping sight distance which AASHTO assumes when calculating stopping sight distance. 4. The maximum speed at which the specified truck can descend the specified grade without losing the ability to slow to the desired speed for any horizontal curves on the downhill.
22 The first two criteria represent only a bare minimum of safety, allowing the vehicle to approach its physical limitations where an accident may be expected. The second two criteria, however, represent a much higher level of safety. The 3 rd criterion ensures that the behavior of the vehicle during braking will follow the assumptions used when designing the roadway. The 4 th allows for drivers to traverse the curves not only without rolling over, but also at a speed which they find comfortable. FHWA recommends the use of the lesser of criteria 3 and 4 as the design speed for the road. (Harwood et al, 2003)
23 Chapter 3 Derivation of the Point Mass and the Modified Point Mass 3.1 Basic Vehicle Mechanics on a Tangent A vehicle traveling along a tangent section of roadway has a number of forces acting upon it. A diagram of these forces and how they act on the car is shown in Figure 3.1. Figure 3.1 Diagram of the forces acting on a vehicle while traversing a tangent section of roadway. Source: The Principles of Highway Engineering and Traffic Analysis.
vehicle. It is import to understand each of these forces, what causes them, and how they act on the 24 3.1.1 Friction Friction is the force which is most responsible for propelling the vehicle along its chosen path or stopping it during braking. In Figure 3.1, the vehicle is braking, so the friction force is represented by F b, the braking force. When the vehicle is trying to accelerate, friction is represented as F, or the available tractive effort. F would be acting in the opposite direction than F b currently is. The magnitude of the friction force is equal to the product of the friction factor (f) and the normal force (N), as shown in Equation (2.2). Technically, F b is the sum of F bf and F br, the braking force acting upon the front and rear tires, respectively. These forces should act in the same direction (i.e. the direction of braking), so it is accepted practice to treat them as a single force. (Mannering, 2009): 3.1.2 Aerodynamic Resistance Aerodynamic Resistance (R a ) has a number of sources. The most significant of these is the turbulent flow of air around the vehicle body while the vehicle is in motion. This accounts for approximately 85% of the aerodynamic drag, and it is mostly a function of the vehicle s shape and speed. The other components are the friction of the air passing over the body of the vehicle (12% of R a ) and air flow through vehicle components such as radiators and air vents (3%) (Mannering, 2009).
25 Aerodynamic Resistance can be calculated using Equation (3.1): R a = ρ C 2 DA f V 2 (3.1) ρ = air density (slugs/ft 3 ) C D = coefficient of drag (unitless) A f = frontal area of the vehicle (projected area of the vehicle in direction of travel) in ft 2 V= speed of the vehicle relative to the prevailing wind speed (ft/sec) Air density is a function of elevation and temperature, with larger values at higher temperatures and low elevations. The drag coefficient is specific to each vehicle and implicitly accounts for the three different previously discussed causes of aerodynamic drag. It can only be found empirically. Typical values for C D range from 0.25-0.55 for passenger cars, 0.5-0.7 for buses, 0.6-1.3 for tractor-trailers, and 0.27-1.8 for motorcycles. All of these values assume that the vehicle is operating with its windows up and with the top of a convertible up. If the windows are down or the top is down, R a can increase by 5 or 7%, respectively. If both are down, it can increase by 23% (Mannering, 2009). 3.1.3 Rolling Resistance Rolling Resistance (R rl ) incorporates the resistance caused by the internal mechanics of the vehicle and the interaction of the air-filled tires with the roadway surface. As the tire rotates, the portion of the tire which touches the roadway flattens somewhat, increasing the surface area of the tire which is in contact with the road. The majority (about 90%) of rolling resistance
26 comes from this deformation of the tire when it interacts with the roadway surface. The tire is also not completely rigid, allowing for the tire to compress and penetrate the surface of the roadway on the microscopic level. This constitutes about 4% of the R rl value. The final 6% is due to things like the slippage of the tire on the roadway surface and the air circulation around the tire and wheel during rotation. There are factors which will affect the magnitude of the rolling resistance. The mass of the vehicle is very important; the heavier the vehicle, the more weight is pushing the tire into the road, the higher the resistance. Roadway characteristics play a role as well. A hard, smooth, and dry roadway offers less rolling resistance. High tire pressure also decreases the deformation of the tire which in turn decreases the resistance. Counter intuitively, higher temperature also decreases resistance. This is because the higher temperature makes the tire more flexible, so less energy is expended when the tire is compressing and decompressing while rotating. Finally, the vehicle speed affects the magnitude of the resistance because higher vehicle speed leads to more vibration and flexing of the tire. While all of these factors are important, only vehicle speed and weight are directly accounted for when engineers calculate rolling resistance using Equation (3.2) (Mannering, 2009): R rl = Wcosθ g 0.01 1 + V (3.2) 147 W = Weight of the vehicle (lbs) V = Vehicle speed (in ft/sec) θ g = Angle of the roadway with respect to the vertical. cosθ g is typically assumed to be 1
27 3.1.4 Grade Resistance Grade resistance (Rg) is caused by the gravitational force which acts parallel to the roadway instead of perpendicular to it. This force is only in effect when the vehicle is on a grade. Rg can be calculated using Equation (3.3) below: R g = W sin θ g (3.3) Grade resistance is positive for uphill grades and negative for downhill grades. In most cases, sin θ g tan θ g = G Where G is the grade of the roadway, in decimal value. Thus, (3.3) can be rewritten as follows (Mannering, 2009): R g = WG (3.4) 3.1.5 Application to Roadway Design From a safety perspective, the time when all of these forces acting on a vehicle are most important is when the vehicle is trying to slow or stop from V 1 to V 2. ΣF = γ b ma = F b + ΣR Of course, many of these forces are a function of the speed of the vehicle which is changing over time, so integration is necessary. a ds = F b + ΣR ds γ b m = V dv
28 V 2 S = γ b m V 1 V 2 S = γ b m V 1 V dv F b + ΣR V dv F b + ρ 2 C DA f V 2 + Wcosθ g 0.01 1 + V 147 ± W sin θ g γ b = Mass factor accounting for moments of inertia during braking, which is given as 1.04 for automobiles. Letting m = W/g and F b = fw S = γ bw 2g ρ ln fw+ 2 C DA f ρ 2 C DA f V 2 1 +Wcosθ g 0.01 1+ V 147 ±W sin θ g (3.5) fw+ ρ 2 C DA f V 2 2 +Wcosθ g 0.01 1+ V 147 ±W sin θ g S= the stopping distance in feet However Equation (3.5) is cumbersome and thus is often simplified in highway engineering. Equation (3.6) is the simplified braking or stopping distance equation used in geometric design: S = V 2 2 V 1 2 2g a g ±G (3.6) a = the vehicle deceleration rate. According to AASHTO policy, this should be assumed to be 11.2 ft/sec 2 Equation (3.6) only directly accounts for grade resistance. The other resistances are implicitly accounted for using a deceleration rate of 11.2 ft/sec 2 since this rate is well within a driver s capability to maintain steering control during a braking maneuver. (Mannering, 2009)
29 3.2 Basic Vehicle Mechanics during Cornering Any object moving on a circular path experiences a centripetal force (F c ) which always acts towards the center of the circle. Any force acting on an object creates a change in that object s velocity; the centripetal force generates the vehicle s change in direction along the circular path, not its change in speed. As was demonstrated in Chapter 2, when a vehicle is cornering on a flat surface, the force responsible for generating the necessary centripetal force is the friction force acting between the roadway and each of the vehicle s tires. Modeling a vehicle in this way is very complicated, so engineers have simplified the process by considering the vehicle to be a single point, The Point Mass Model. Figure 3.2 Diagram of forces acting on a vehicle while cornering on a superelevated roadway.
30 The friction available at the roadway-tire interface is often not capable of supplying the required amount of friction necessary to maintain driver comfort while tracking the curve. For this reason, the roadway is often superelevated through the horizontal curve. During cornering maneuvers where the roadway is superelevated, the friction force (F) and the weight (W) of the vehicle act together to generate the centripetal force, as shown in Figure 3.2. The portion of the centripetal force which is most responsible for keeping the vehicle in its desired lane of travel is the portion which acts parallel to the roadway, F cp. As shown in Figure 3.2, F cp = W p + F f (3.7) Where, F f = f(w n + F cn ) Equation (3.7) can be re-written as: WV 2 gr v cos α = W sin α + f(w cos α + WV2 gr v sin α) (3.8) W= Vehicle weight in lbs V= Vehicle velocity in ft/sec Dividing both sides of Equation (3.8) by Wcos α and rearranging yields: V 2 gr v (1 f tan α) = tan α + f (3.9)
31 Since values of f and α are typically quite small, the term f tan α is conservatively assumed to be zero. With e = 100 tan α, and converting from ft/sec to mph, we obtain Equation (3.10), which is what AASHTO advises engineers use when designing horizontal curves (Mannering, 2009): 0.01e + f = V2 15R (3.10) 3.3 The Modified Point Mass Model While Equation (3.10) accurately accounts for the forces acting on a vehicle in the y- and z- directions, it does not account for those forces which act in the x-direction, or direction of the vehicle s motion: braking, aerodynamic resistance, rolling resistance, and grade resistance. When a vehicle is travelling downgrade, the forces acting on it in the x-direction are: F x = ma x = Grade Resistance (Braking + Aerodynic Resistance + Rolling Resistance) = mg sin θ g f x mg cos θ + ρ 2 C DA f V 2 + mgcosθ g 0.01 1 + V 147 (3.11) Where f x is the amount of friction available in the longitudinal direction for braking, and forces are considered positive if they act in the direction of vehicle motion. As discussed in Section 2.2.1, the amount of friction being utilized in the x-direction affects the amount of friction available in the y-direction due to the friction ellipse. For this reason, it is important to find f x as a function of the forces acting on the vehicle. a x = g sin θ g f x g cos θ + ρ 2m C DA f V 2 + gcosθ g 0.01 1 + V 147 a x g sin θ g + ρ 2m C DA f V 2 + gcosθ g 0.01 1 + V 147 = f xg cos θ
32 f x = a x + tan θ ρ g cos θ g C 2mg cos θ DA f V 2 + 0.01 1 + V (3.12) 147 The friction ellipse is defined as: f 2 x + f 2 y = n 2 1 (3.13) f x,max f y,max Thus, for any known longitudinal friction demand,f x, the amount of friction remaining for use in the lateral direction, f y,supply, can be found using: f y,supply = f y,max 1 f 2 x f x,max (3.14) f y is the lateral friction demand. Rearranging of Equation (3.9) gives us an expression for this friction term: V 2 gr v tan α = f y 1 + tan α V2 gr v f y = V 2 gr v tan α 1 + V2 gr v tan α f y = V 2 gr v e 100 1 + V2 e gr v 100 (3.15) As long as f y f y,supply, the roadway offers sufficient friction for simultaneous braking and cornering. If f y > f y,supply, the vehicle will not be able to brake while tracking the curve, which could possibly result in vehicle skidding or rollover. In order to derive these equations, a method very similar to that used by Varunjikar (2011) was implemented.
33 Chapter 4 Methodology 4.1 Lateral Friction Demand from the Point-Mass Equation Friction, superelvation, speed, and radius have already been related using Equation 3.9 from Chapter 3. Using simple algebraic concepts, this equation may be rearranged to solve for the friction as a function of the vehicle s velocity and the radius and superelevation of the roadway. This relationship is: f d = V2 15R e 100 (4.1) f d = friction demand V= vehicle speed (mph) This friction, f d, is known as the friction demand because it is the friction demanded or needed by the vehicle in order to complete its turn at a given speed, radius, and superelevation. It is different from the friction supply, f supply, which is the maximum amount of friction available at the tire-road interface. If the vehicle is able to corner without skidding, that indicates that f d <f supply. 4.2 Lateral Friction from the Modified Point-Mass Equation With the modified point-mass equation, the relationship between friction demand and the other parameters is more complicated. In this derivation, no terms were ignored even though
34 their values would be close to zero. This equation was already derived in Section 3.3. In order to differentiate the friction demand derived from the modified point-mass equation from the friction demand found using Equation 4.1, the friction demand here will be denoted as f d,modified : f d,modified = V 2 gr v e 100 1 + V2 e gr v 100 (3.14) V= vehicle speed (ft/sec) 4.3 Finding Friction Supply Friction supply is a function of the micro- and macro-textures of the roadway. Microtexture refers to the surface roughness of the individual aggregate particles in the roadway. Macro-texture is a surface roughness quality defined by the mixture properties of an asphalt roadway (size, shape, gradation) or the finishing/texturing method of a concrete roadway. In order to test the micro-texture of a roadway, a Dynamic Friction (DF) Tester device is used in accordance with the ASTM E1911-09a standard. The DF Tester consists of a horizontal spinning disk with three spring-loaded rubber pads, or sliders, that contact the roadway surface as the horizontal disk spins. The rotational speed of the disk decreases due to the friction supplied between the sliders and the paved surface. A water supply unit delivers water to the surface being tested in order to simulate wet roadway conditions. The DF Tester is able to measure the amount of torque required to turn the three rubber pads on the surface in question at a variety of speeds. These torque measurements are then used to calculate the friction supply as a function of speed. Typical speeds range from 55 to 3 mph. All measurements from the DF