Superelevation Criteria for Sharp Horizontal Curves on Steep Grades (2014)

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A-1 A P P E N D I X A This appendix presents the definition of terms/notation used throughout the report. Nomenclature Notation Definition Location first used ar Centripetal acceleration Eq. 1 R Radius of curve Eq. 1 Rmin Minimum radius of curve, which is a function of the maximum rate of superelevation and the maximum demand side friction used in horizontal curve design Eq. 9 ftire-pavement Side friction supply, which represents the available friction that can be developed between pavement and vehicle tires to prevent skidding along a horizontal curve, also referred to as the coefficient of friction Page 11 F Side friction factor, which represents the unbalanced portion of lateral acceleration or the portion of lateral acceleration that is not balanced by superelevation. The side friction factor represents demand side friction. Eq. 5 fmax Maximum side friction, which represents the maximum side friction demand set forth in the AASHTO Green Book for use in horizontal curve design. The values are based on driver comfort levels (i.e., tolerance for lateral acceleration). It is also referred to as the limiting side friction factor. Eq. 9 e Superelevation, typically defined by the rise (change in elevation) in feet per 100 ft across the road (i.e., in the transverse direction) Eq. 5 emax Maximum rate of superelevation, which represents the maximum banking or cross slope of the roadway cross section within a horizontal curve. This value ranges from 4% to 12%, depending on climatic conditions, area type, terrain, and the frequency of very slow-moving vehicles in the traffic stream. Eq. 9 Banking angle of road (in radians) Eq. 4 VDS Design speed, which represents the selected speed used to determine the various geometric design features of the roadway Eq. 9 V Constant velocity Eq. 1 N Normal reaction from the road Figure 2 Fc The tire–pavement cornering force acting at the road toward the center of the rotation Figure 2 W Vehicle weight (W = mg) Figure 2 m Mass of the vehicle Eq. 2 g Gravitational acceleration Eq. 4 Fy Sum of forces acting in the y-axis direction Eq. 2 Ny Normal force acting in the y-axis direction Eq. 2 Wy Vehicle weight acting in the y-axis direction Eq. 2 Fcy Tire–pavement friction force acting in the y-axis direction Eq. 2

A-2 Notation Definition Location first used Fz Sum of forces acting in the z-axis direction, point-mass model Eq. 3 Nz Normal force acting in the z-axis direction Eq. 3 Wz Vehicle weight acting in the z-axis direction Eq. 3 Fcz Tire-pavement friction force acting in the z-axis direction Eq. 3 Va Curve approach speed Figure 3 Vc Speed at the mid-point of a horizontal curve Figure 3 PC Point of curvature Eq. 76 SNV Skid number at a given speed Eq. 11 P Normalized skid gradient Eq. 11 µs Sliding friction Table 2 µp Maximum rolling (peak) friction Table 2 Fx Tire longitudinal force, or braking force Eq. 12 (Figure 6) Fy Tire lateral force, or cornering force Eq. 12 (Figure 6) Fx,max Maximum longitudinal force, e.g., the maximum braking force that a tire can generate Eq. 12 Fy,max Maximum lateral force, e.g., the maximum cornering force that a tire can generate Eq. 12 N Referred to as the utilized amount of tire–pavement friction or the measure of friction supplied (often referred to as friction reserve by vehicle dynamicists) Eq. 12 fx Friction factor in the x-direction, defined as the tire’s lateral force divided by the normal force on the tire. (fx = Fx/Fz) Eq. 13 fy Friction factor in the y-direction, defined as the tire’s lateral force divided by the normal force on the tire. (fy = Fy/Fz) Eq. 13 fx, max The maximum extent of the tire’s friction ellipse for longitudinal (braking) forces Eq. 15 fy, max The maximum extent of the tire’s friction ellipse for lateral (cornering) forces Eq. 15 CG Center of gravity Eq. 27 tskid Time duration (in seconds) that a tire or axle is skidding Eq. 88 yLat Dev The lateral distance (in feet) that the vehicle will deviate from its normal lane position due to skidding Eq. 88 z The local deflection of a tire contact patch, in the LuGre tire model Eq. 19 vr The relative velocity of a tire contact patch to the pavement surface below, as used in the LuGre tire model Eq. 19 Fxi The braking force of an element of the tire contact patch, in the LuGre tire model Eq. 19 Fzi The normal force on an element of the tire contact patch, in the LuGre tire model Eq. 19 Constants in the LuGre tire model Eq. 19 ax Acceleration in the x-axis (braking) Eq. 21 G Grade, in percentage Eq. 21 The modified braking acceleration, used for calculating stopping sight distances in AASHTO guidelines Eq. 26 T Track width of the vehicle Eq. 27 h Height of the vehicle’s CG above the road surface Eq. 27 RMay Rollover margin based on lateral acceleration, which represents the difference between current lateral acceleration and the maximum lateral acceleration that a vehicle can experience without overturning. A value of 0 indicates the onset of wheel lift. Eq. 32 Fzi, Fzo Normal load on the inside, outside tires of the vehicle Figure 40 Fyi, Fyo Lateral force (cornering force) from the inside, outside tires Figure 40 Roll angle of the vehicle relative to the road surface Eq. 28 hr Height of the vehicle’s roll axis above road surface Eq. 28 ay Acceleration in the y-axis (cornering) Eq. 29 R Roll gain, e.g., constant that gives the angle of body roll produced by suspension per unit of lateral acceleration, in rad/g Eq. 31 Fbf,Fbr Braking force from the front and rear axles Eq. 35

A-3 Notation Definition Location first used Fcf,Fcr Cornering force from the front and rear axles Eq. 36 Nf, Nr Normal force from the front and rear axles on pavement Eq. 37 a, b Distance from CG to front and rear axles Eq. 41/40 fyf, fyr Lateral friction factors on front, rear axle (cornering) Eq. 44 Rtire Rolling radius of the tire Eq. 46 Gf, Gr Brake gain (front and rear axles), converts applied brake pressure to wheel torques Eq. 46 Pf, Pr Brake pressure applied to front and rear axles Eq. 46 Pa Brake pressure applied to proportioning valve Eq. 47 Pa’ Brake pressure at which proportioning valve engages Eq. 47 Fb’ Brake force at which proportioning valve engages Eq. 48 ax,p Deceleration level at which brake proportioning valve engages Eq. 49 fxf, fxr Longitudinal friction factors on front, rear axle (braking) Eq. 56 fmargin,f, fmargin,r Front and rear axles margin of safety (cornering) Eq. 59, 60 fyf,supply, fyr,supply Lateral friction supply per axle on front and rear axles Eq. 59, 60 r The spin rate of the vehicle (rad/s) measured around the z-axis Eq. 62 Vy The lateral velocity of the vehicle, e.g., the sideways sliding velocity, measured at the CG Eq. 63 Izz The moment of inertia of the vehicle about the z-axis. In the case of articulated vehicles, this is the inertia of the tractor only. Eq. 64 The slip angle of the tire relative to the road Figure 48 C , C The front and rear cornering stiffness which predicts how much force a tire produces per radian of the tire’s rotation to the road Eq. 65 CC, CCoffset The cornering coefficient and cornering coefficient offset for a tire that is used to calculate cornering stiffnesses from vertical load Eq. 66 f, r The front and rear tire slip angles Eq. 67 The steering angle of the front tire (radians), measured at the interface of the tire and the road surface. It is the angle between the longitudinal axis of the vehicle and the tire’s centerline. Eq. 67 R' The rotation radius, as measured in the vehicle’s body-fixed coordinate system Eq. 73 fyf,tangent, fyr,tangent The front and rear friction factors on the tangent, immediately prior to curve entry Eq. 76 LTR Load-transfer ratio of a vehicle, a measure of the percentage of load on a particular axle carried by the inside tire. A value of 1 means all load is on the inside tire; 0 is equally balanced inside/outside loading; and a value of 1 is when all load is on the outside tire. Values of 1 or 1 represent the onset of wheel lift. Eq. 89 Ni, No Normal force on inside, outside tires of an axle Eq. 89 RMLTR Rollover margin defined by the proximity of the load-transfer ratio to an absolute value of unity, e.g., how close an axle is to experiencing wheel lift. A value of 0 indicates the onset of wheel lift Eq. 90 TRACTOR TRAILER MODEL m1,m2 Mass of tractor, trailer Table 25 Fcf, Fcr, Fct Cornering force on tractor front axle, tractor rear axles, trailer axles Table 25 Fbf, Fbr, Fbt Braking force on tractor front axle, tractor rear axles, trailer axles Table 25 Nf, Nr, Nt, Nh Normal force on tractor front axle, tractor rear axles, trailer axles, and on hitch Table 25 ft Distance from hitch to trailer CG Table 25 gt Distance from hitch to trailer axle Table 25 dh Tractor CG to hitch distance Table 25 h1, h2, hh Height of tractor CG, trailer CG, and hitch point Table 25 L The vehicle wheelbase, e.g., the distance from the front to the rear axle as measured from the center of the contact patch of each tire Table 25