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

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10 S E C T I O N 2 This section summarizes the literature related to super- elevation criteria for sharp curves on steep grades and also summarizes current practice on this issue. The topics are orga- nized as follows: • Horizontal curve design • Heavy trucks • Driver comfort • Friction studies • Vehicle dynamics models • Current practice 2.1 Horizontal Curve Design Current AASHTO policy on horizontal curve design is based upon a point-mass model. From the basic laws of Newtonian physics, consider a point mass traveling in a curved roadway with a constant radius (R) and a constant velocity (V), as shown in Figure 1. The point mass undergoes a centripetal acceleration which acts toward the center of curvature. The centripetal acceleration is given as: a (1)r 2V R = Assume that the point mass is a vehicle. The acceleration is balanced by the side friction developed between the vehicle’s tires and the pavement surface, the component of vehicle’s weight acting parallel to the road due to superelevation, or a combination of both, as shown in Figure 2. Let the bank- ing angle of roadway be a (radians). The superelevation (e) is typically defined by the rise (change in elevation) in feet per 100 ft across the road (i.e., in the transverse direction). Hence, e/100 = tana. There are three forces acting on the point mass as shown in Figure 2: 1. Normal reaction from the road (N) 2. The tire–pavement friction cornering force acting at the road toward the center of the rotation (Fc) 3. Vehicle weight (W = mg; where m is the mass and g is the gravitational acceleration). Performing a force balance in the y-axis direction (refer- ring to the axis system shown in Figure 2), one obtains: i (2) 2 m a F m V R N W F r y y y cy ∑= ⇒ = + + And in the z-axis direction: 0 (3)F N W Fz z z cz∑= = + + The force components are given as: = − α = α = α = α = = cos , sin sin , cos , 0 (4) N N N N F F F F W mg W z y cz c cy c z y Expressions for the friction factor and superelevation are: 100 tan (5)f F N ec = = α Equation 3 can be solved for mass by substituting values from Equation 4 to obtain m = 1/g i (-Fc sin(a) + Ncos(a)). Sub- stituting this into Equation 2, and then simplifying the result by substituting expressions from Equation 5, one obtains: Literature Review

11 tan 1 tan 100 1 100 (6) 2V gR f f f e f e)(= + α − α = + − • • Rearranging terms, one gets the basic curve formula: 0.01 1 0.01 (7) 2V gR f e fe = + −     The product f i e/100 in the denominator is usually small and is generally ignored. The simplified formula can be used to solve for the curve radius allowable as a function of the maximum friction factor, the design speed, and the superelevation. R = +( )    • • V g f e 2 0 01 8 . ( ) The limiting factor for road design is the side friction fac- tor f. Also, the superelevation rate for a curve will not exceed a maximum value selected by the designer. Hence, for a given design speed of a roadway, practical lower limits on the radius of curvature, Rmin, are given by: 0.01 (9)min 2 max max R V g f e DS i i( )= + Here, fmax is the maximum demand side friction factor used in horizontal curve design, and emax is the maximum super- elevation rate for a given design speed, VDS. AASHTO uses Equation 9 for determining the minimum radius of curva- ture. This usage is generally justified since it provides a more conservative design than Equation 8. The basic side friction formula can be obtained by re- arranging terms in Equation 8 as follows: 0.01 (10) 2 f V g R eDS i i= − In AASHTO policy, f is called the “side friction factor” which represents the portion of lateral acceleration that is not balanced by superelevation. The term f represents a friction “demand” which must be resisted by the available “supply” of friction generated at the tire–pavement interface. In addition, the unbalanced lateral acceleration creates an overturning moment on the vehicle that must be resisted by the vehicle’s roll stability, which depends on vehicle design, loading, and suspension characteristics. The term “side friction factor,” as used in the Green Book, represents friction demand, not fric- tion supply. AASHTO design policy for horizontal curves is based on the assumption that the value of f can be determined as a function of vehicle speed, curve radius, and superelevation. An inher- ent assumption is that vehicles follow the curved path exactly. The tire–pavement interface can supply friction (ftire-pavement) to resist the tendency of the vehicle to skid due to lateral accel- eration as the vehicle traverses a curved path. The pavement friction generated at the tire–pavement interface is propor- tional to the normal load transmitted to the tire through the vehicle suspension which depends on tire and pavement properties. From the viewpoint of a point-mass model, the vehicle will skid if f > ftire-pavement, where ftire-pavement represents the maximum amount of friction that can be generated at the tire–pavement interface to counteract lateral acceleration and prevent skidding. Similarly, from the viewpoint of a point-mass model, the vehicle will overturn if f > frollover, where frollover represents the maximum lateral acceleration that a vehicle can expe- rience without overturning. frollover is referred to as the Figure 1. Point-mass model: vehicle traveling on a horizontal curve. Y Z Fc W = mg N R α Figure 2. Lateral forces acting on point mass during cornering.

12 “rollover threshold” of the vehicle. Rollover thresholds are a characteristic of vehicle design and loading that can be estimated from static tests, but are best determined from dynamic tests. The Green Book design criteria for horizontal curves are not based on any formal assumptions about the magnitudes of ftire-pavement and frollover. Rather, horizontal curve design is based on limiting the value of f to be less than or equal to a specified value, fmax, which has been selected based on driver comfort levels (i.e., driver tolerance for lateral acceleration). A further assumption, stated but not explicitly demonstrated in AASHTO policy, is that the values of fmax used in design have been selected such that fmax < ftire-pavement and fmax < frollover. The first criterion, fmax < ftire-pavement, is addressed in Green Book Figure 3-5, which shows that the values of fmax used in design are less than the values of ftire-pavement. The second criterion, fmax < frollover, is asserted but not demonstrated in the Green Book. Research by others, including Harwood et al. (1989) and Harwood et al. (2003), has shown that the assumptions of fmax < ftire-pavement and fmax < frollover do appear to be generally applicable to both passenger vehicles and trucks for horizontal curves designed in accordance with AASHTO policy. The point-mass model works reasonably well for the con- ceptual design of horizontal curves; however, there are sev- eral limitations to this simple approach to horizontal curve design (Easa and Abd El Halim, 2006). First, the model does not account for differences in vehicle dynamics between pas- senger vehicles and trucks, and the model ignores tire force differences between the front/rear or left/right tires of a vehi- cle (i.e., the forces acting on all tires are assumed to be the same). Second, the point-mass model ignores the combined characteristics of the highway alignment such that the hori- zontal alignment is designed in isolation without accounting for the overlapping vertical alignment. Third, the point-mass model assumes that vehicles traverse curves following a path of constant radius equal to the radius of the curve; however, it has been shown that at some points on a horizontal curve, some vehicles will over steer the curve, following a path less than the radius of the curve (Glennon and Weaver, 1972). Fourth, the point-mass model assumes vehicles traverse the curve at a constant speed and does not account for situations when vehicles may have to decelerate (i.e., apply the brakes) while traversing through the curve. Several research efforts have evaluated the adequacy of the current point-mass model approach to horizontal curve design. In the mid-1990s, Harwood and Mason (1994) evaluated the adequacy of the AASHTO geometric design policy to safely accommodate both passenger vehicles and trucks on hori- zontal curves. Harwood and Mason concluded there does not appear to be a need to modify existing high-speed criteria for determining the radius and superelevation of horizontal curves designed in accordance with current AASHTO policy. Existing design policies provide adequate margins of safety against skid- ding and rollover for both passenger vehicles and trucks as long as the design speed of the curve is selected realistically. Special care should be taken for curves with design speeds of 30 mph or less to assure that the selected design speed will not be exceeded, particularly by trucks. Design of superelevation transitions according to the ²⁄³-¹⁄³ rule provides an acceptable design, while spiral transitions would provide marginally lower lateral accel- erations. For minimum-radius horizontal curves designed in accordance with AASHTO low-speed criteria, AASHTO policy generally provides adequate margins of safety against skidding and rollover for passenger vehicles traveling at the design speed, but for design speeds of 10 to 20 mph, minimum-radius curves may not provide adequate margins of safety for trucks with poor tires on a poor, wet pavement or for trucks with low roll- over thresholds. Revision of the AASHTO low-speed horizon- tal design criteria should be considered, especially for locations with substantial truck volumes. In other research, Bonneson (1999) estimated statistical models of curve speed and side friction demand to develop limiting values of side friction demand for use in horizontal curve design. The relationship between maximum side fric- tion demand and horizontal curve approach speed derived for passenger vehicles is shown in Figure 3. The model illustrates that side friction demand decreases as the curve approach speed increases, while the side friction demand increases as the speed reduction between the curve approach speed and the speed at the mid-point of a horizontal curve (Va - Vc) increases. The side friction demand related to no speed reduction between the approach tangent and mid-point of a horizontal curve (Va - Vc = 0 mph) was proposed as the desirable upper limit on maximum design side friction factors. However, a maximum desirable speed reduction of Figure 3. Relationship between side friction demand and speed (Bonneson, 1999).

13 3 mph (5 km/h) was proposed to balance traffic flow and construction cost, thus allowable maximum side friction demands corresponding to the Va - Vc = 3 mph (5 km/h) trend line were recommended. To assess the margin of safety for the proposed side friction demand factors, Bonneson (2000b) compared side friction sup- ply for both slide and roll failure to the proposed side friction demand factors. A graphical representation of this assessment is shown in Figure 4 where the side friction demand for speed reductions of 0 to 1.86 mph (0 and 3 km/h) is plotted for both passenger vehicle and truck margins of safety against slide and roll failure. The results show that grades, particularly steep upgrades, reduce the margin of safety, particularly for trucks. Another trend observed was that roll failure is only observed in trucks on low-speed curves. Finally, Figure 4 shows that slide failure will occur prior to roll failure for passenger vehicles at any speed, and at higher speeds for trucks. Bonneson (2000a) also proposed limiting superelevation rates of 8.2%, 9.8%, 10.8%, 11.4%, and 11.8% proposed for design speeds of 18.6, 24.8, 31.0, 37.3, and 43.5 mph, respectively, and determined the optimal proportion of the superelevation runoff located prior to the point of curvature (PC) to be 80% at 18.6 mph and 70% at 74.6 mph for two-lane highways. A 10% increase in the proportion was proposed for each additional travel lane to be rotated on the transition curve. Later, Bonneson (2001) proposed a superelevation distribution method for horizontal curves based on established minimum and maximum super- elevation rate boundary conditions. Awadallah (2005) more recently proposed a method to determine design side friction factors based on side friction supply factors for skid and roll, and, about the same time, Tan (2005) replicated experiments conducted in the 1930s and 1940s to determine comfortable net lateral acceleration on horizontal curves. Tan concluded that AASHTO design side Figure 4. Margin of safety between side friction supply and demand (Bonneson, 2000b). Effect of speed reduction on passenger car margin of safety against slide failure. Effect of speed reduction on truck margin of safety against slide or roll failure.

14 friction values are conservative for contemporary passenger vehicles traveling at the design speed and recommended that the coefficient of friction at impending skid be revisited in the AASHTO Green Book to reflect current pavement design practices and performance. 2.2 Heavy Trucks Eck and French (2002) investigated problems faced by trucks on sharp curves on steep grades to determine appro- priate superelevation rates for trucks under these conditions. The primary findings and conclusions from this research included the following: • On downgrades, a portion of the available friction (side friction supply) is consumed in maintaining a steady speed. This leaves less than the maximum friction available for side friction demand. This is not a significant problem under normal steady-speed conditions, but the available side friction is severely reduced when braking. The down- grade also adds to the lateral acceleration. Two theoretical models support the use of additional superelevation on sharp curves on steep downgrades. • High superelevation rates (e.g., between 0.08 and 0.16) make horizontal curves on steep downgrades more for- giving. These high superelevation rates do not necessarily permit higher speeds but can better accommodate drivers making errors in safe speed selection for the curve and grade combination. • Reducing the superelevation of existing curves is not good highway geometric design practice, unless there is another safety issue that requires this reduction. Where the super- elevation rate has been reduced, significant increases in passenger vehicle crashes have been observed and are par- tially attributable to violation of driver expectancy. 2.3 Driver Comfort A key consideration in AASHTO’s policy in selecting maxi- mum side friction factors (fmax) for use in design is the level of centripetal or lateral acceleration sufficient to cause drivers to experience a feeling of discomfort and to react instinctively to avoid higher speeds. The general policy follows the assumption that at low speeds drivers are more tolerant to discomfort and hence higher values of side friction are sought, while at higher speeds a greater margin of safety should be sought; hence, the use of lower side friction factors at high speeds. This approach for selecting maximum side friction factors for design is based upon research from the 1930s and 1940s (Barnett et al., 1937; Moyer and Berry, 1940; Meyer, 1949; Stonex and Noble, 1940). More recent studies by Bonneson (2000b) and Tan (2005) reaf- firmed the appropriateness of the side friction factors currently recommended in AASHTO policy for horizontal curve design. 2.4 Friction Studies The basic side friction formula (Equation 10) gives an esti- mate of the side friction for a vehicle maneuver on a horizontal curve. One of the earliest studies on measuring the coefficient of friction at the point of impending skid on a roadway was done by Moyer (1934). Table 1 lists different coefficient of fric- tion values recorded by Moyer, and Figure 5 shows variation in friction levels (for different skid conditions) with respect to speed. In Figure 5, the side skid coefficients of friction reported are higher than straight skid coefficients of friction, which is usually not the case in modern measurements of tire behavior. The differences might be explained by Wong (2008) where he notes that modern passenger vehicles now use synthetic rub- ber which has significantly different properties from natural rubber, which is still sometimes used in truck tires. The differ- ence is that natural rubber has much better wear properties, Type of surface Type of skid Remarks Coefficient of friction Speed (mph) 5 10 15 20 25 30 Portland cement concrete, 19 × 4.75 tires, no chains Side Dry surface 1.01 1.01 0.97 0.95 0.92 0.89 Straight Dry surface 0.94 0.90 0.86 0.83 0.80 0.77 Side Wet surface 0.78 0.75 0.72 0.69 0.66 0.64 Straight Wet surface 0.67 0.63 0.59 0.55 0.51 0.46 Ice on pavement, no chains Side Smooth tread 0.20 0.19 0.19 0.20 – – Side New tread 0.19 0.19 0.22 0.19 – – Ice on pavement, 16 × 7.00 tires, no chains Straight New tread 0.18 0.15 0.17 0.21 – – Impending New tread 0.17 0.19 0.19 0.19 – – Side New tread 0.19 0.19 0.19 0.18 – – Table 1. Coefficient of friction vs. speed (Moyer, 1934).

15 ideal for trucks; but the coefficient of friction is much less for natural rubber tires, with the result that trucks have a stopping distance of 1.65 to 2.65 times farther than a passenger vehicle, assuming both are using high-grip tires of good condition. More recent work examining stopping distance includes that of Olson et al. (1984). Olson et al. proposed Equation 11 to calculate the skid number for a given velocity (V ): SN SN e (11)V 40 P V 40= ( )− where: SNV = Skid Number (= 100 × coefficient of friction) at given speed V = Speed in mph P = Normalized skid gradient (<0) Table 2 summarizes the formulae given by Olson et al. for sliding friction and maximum rolling friction for passen- ger vehicle tires and truck tires. The friction coefficients for truck tires are less than those for passenger vehicles. Olson et al.’s study also indicates a decrease in friction with the increasing speed. Table 3 lists the values of maximum and side friction coefficients of friction for different tires on dry as well as wet roads as determined by Fancher et al. (1986). Because of tire deformation characteristics, a wheel will exhibit different curves and different maximum friction val- ues depending on whether the force is in the lateral direc- tion or longitudinal direction, the condition of the tires, whether an anti-lock braking system (ABS) is employed, and the loading of the tires. The use of braking forces will reduce the available lateral friction, and the use of lateral force will reduce the available braking forces. This inter- relationship between lateral and longitudinal forces is called the friction ellipse. The sliding friction limit for a tire, regardless of direction, is determined by the coefficient of sliding friction times the load. The friction can be used for lateral force, brake force, or a combination of the two, in either the positive or nega- tive directions (Gillespie, 1992). However, the vector total of the two forces cannot exceed the friction limit. This leads to the friction ellipse (or circle) concept. As shown in Figure 6, utilization of friction in one direction decreases the friction reserve in the other direction. The friction ellipse equation represents the operating range of tire forces and is given by Equation 12 (Wong, 2008): 1 (12) ,max 2 ,max 2 2 F F F F n y y x x     +   = ≤ Here Fx is the tire’s longitudinal (braking) force, Fy is the tire’s lateral (cornering) force, and Fx,max and Fy,max are the maximum possible forces available in braking and corner- ing, respectively. The term n represents the total utilized friction and has a value of 1 when the tires are at the fric- tion limit. Values below 1 represent situations within the friction ellipse, whereas values above 1 are beyond the tire’s force capabilities. Thus, as long as the value of n is less than 1, the operating point (i.e., tire forces in x and y direction) lies inside the friction ellipse (i.e., the tire–pavement can generate required friction force). Equation 12 can be related Figure 5. Relation between static, side skid, and straight skid coefficients of friction on wet portland cement concrete (Moyer, 1934). Table 2. Formulae for forward friction coefficients (Olson et al., 1984). Passenger vehicle tire Truck tire Sliding friction (µs) 1.2 SNV 0.84 SNV Maximum rolling friction (µp) 0.2 + 1.12 µs µs

16 to pavement friction values in the lateral and longitudinal directions through a simple transformation since the friction factor is defined as force divided by vertical load. Specifically, the longitudinal and lateral friction demands are derived from the demanded tire forces as follows: Longitudinal Friction Factor (13)x x f F N = Lateral (i.e., Side) Friction Factor (14)f F N y y = Where Fx and Fy are braking and cornering forces on the tire, and N is the normal load the tire carries. Depending on the level of model complexity, this “tire” could be construed to represent either an individual tire or the sum of force effects on multiple tires. With these substitutions, the friction ellipse equation can be written in terms of the friction factors as: 1 (15) y y,max 2 x x,max 2 2 f f f f n     +     = ≤ Unless the tire is at an extreme angle to the road, the nor- mal force, Fz, in the tire’s coordinate system can be assumed to be the normal force, N, acting on the tire from the road as shown in Figure 2. The term n in Equations 12 and 15 can be 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). Again, one can usually infer that enough friction supply is available as long as n < 1. When n > 1, friction supply is exceeded. For the dry pavements, there is little to no significant change in the tire–road pavement friction with increasing speed, per- haps 10% to 20% at most, but there is a noticeable decrease in friction on wet surfaces with increasing speeds. The friction is found to be decreasing with increasing speeds as shown in Fig- ure 7 (Wong, 2008). This variation also depends on the type of road, condition of tire treads, etc. The shapes of these curves roughly match the driver comfort friction demand curves empirically determined for use in the design of horizontal curves. It thus seems likely that driver “comfort” may simply be a driver’s perception of inferred friction supply on wet roads. To summarize, the maximum lateral force acting on a tire or the maximum side friction factor depends on a range of main factors, including: • The normal force on the tire; • Longitudinal tire force; • Road surface condition (dry, wet, snow, ice, etc.); • Vertical load acting on the tire; • Speed (mainly for wet surfaces); • Tire condition (new, worn out); and • Tire composition. 2.5 Vehicle Dynamics Models Although the point-mass model serves as the basis for hori- zontal curve design, over the past few decades some research- ers have proposed two-axle models (i.e., the bicycle model) for horizontal curve design (Figure 8). The models in these studies represent modifications to the classical bicycle model used in vehicle stability analysis. This model is derived and discussed in detail in subsequent sections. The modifications include fac- tors such as inclusion of grade, braking/acceleration, consid- eration of the friction ellipse, etc. The advantage of the bicycle Tire type Tire construction Dry Wet µp µs µp µs Goodyear Super Hi Miler (Rib) Bias-ply 0.850 0.596 0.673 0.458 General GTX (Rib) Bias-ply 0.826 0.517 0.745 0.530 Firestone Transteel 1 (Rib) Radial-ply 0.809 0.536 0.655 0.477 Firestone Transport 1 (Rib) Bias-ply 0.804 0.557 0.825 0.579 Goodyear Unisteel R-1 (Rib) Radial-ply 0.802 0.506 0.700 0.445 Firestone Transteel Traction (Lug) Radial-ply 0.800 0.545 0.600 0.476 Goodyear Unisteel L-1 (Lug) Radial-ply 0.768 0.555 0.566 0.427 Michelin XZA (Rib) Radial-ply 0.768 0.524 0.573 0.443 Firestone Transport 200 (Lug) Bias-ply 0.748 0.538 0.625 0.476 Uniroyal Fleet Master Super Lug Bias-ply 0.739 0.553 0.513 0.376 Goodyear Custom Cross Rib Bias-ply 0.716 0.546 0.600 0.455 Michelin XZZ (Rib) Radial-ply 0.715 0.508 0.614 0.459 Average 0.756 0.540 0.641 0.467 Table 3. Coefficients of road adhesion for truck tires on dry and wet concrete pavement at 40 mph (Fancher et al., 1986).

17 model versus the point-mass model is that it examines not only force balance, but also moment balance. The moment balance in particular prevents the vehicle from “spinning out” on a roadway. Further, it is useful to examine whether individual axles will exhibit skidding prior to the entire vehicle exhibiting skidding. Figure 6. Friction ellipse diagram (right turn) (Milliken and Milliken, 1995). Using the bicycle model, Psarianos et al. (1998) studied the influence of vehicle parameters on horizontal curve design. Psarianos et al. indicate that the friction reserve might be exceeded for a passenger vehicle traveling 12 mph higher than the design speed of 50 mph on a minimum-radius curve (obtained from basic point-mass model) for downgrades

18 steeper than 5%. They pointed out that these maneuvers will be more critical for trucks since they have lower maximum side friction factors. Kontaratos et al. (1994) also developed an analytical two- axle vehicle model to determine the minimum horizontal curve radius as a function of vertical grade. In their bicycle- like model, Kontaratos et al. added the effects of the grade and superelevation, front-wheel versus rear-wheel drive, air resistance, etc. Their results suggest that the margins of safety against skidding are lower on steeper grades. Bonneson (2000b) developed a two-axle vehicle model in his analysis of horizontal curve design. In the analysis Bon- neson considered mild braking representative of the speed reduction upon entry to the curve. He developed slide (skid) failure and roll failure models separately to check if vehicle maneuvers are safe for given conditions. A decrease in the margin of safety (for the side friction factor) for trucks and passenger vehicles was reported on grades. None of the studies mentioned above consider a multi- axle vehicle model and thus omit all tractor semi-trailers. Further, few of these studies considered a tire model inclusive of the friction ellipse and representative combined braking/ turning situations. They also did not address load transfer, transient instabilities, and many steady-state instabilities as well. Also, except Bonneson (2000b), who used the Highway- Vehicle-Object Simulation Model (HVOSM) for a part of his study, there was no use of a multibody simulation model to comprehensively analyze vehicle stability while traversing a horizontal curve. In the vehicle dynamics literature, many papers and text- books (e.g., Dugoff, 1968; Ito, 1990; Milliken and Milliken, 1995; Wong, 2008; Gillespie, 1992) relevant to vehicle stability on a horizontal curve have been published, although none of these are clearly used at present in AASHTO policy. Of par- ticular interest, if a driver applies a steady steering input (e.g., during transition from a tangent to a horizontal curve) and maintains it, the vehicle will enter a curve of constant radius after a transition period. The behavior of the vehicle in this transition time period is called its “transient response charac- teristics.” Bundorf (1968) pointed out that such a behavior is quite important and the handling qualities of an automobile depend greatly upon its transient response. The bicycle model can predict curve onset transient behavior and other transient effects, for example, maneuvers such as a lane change where the radius of the curve is changing. 2.6 Current Practice The design policies/manuals of 40 state highway agencies were reviewed to understand their current practice concerning superelevation design criteria, specifically seeking to determine if state policies differed from AASHTO guidance on super- elevation criteria for sharp horizontal curves on grades. Of the 40 state design policies/manuals reviewed, most referred to the Green Book for detailed design procedures concerning superelevation. Only two state design policies/manuals pro- vided statements concerning superelevation design criteria on grades. The other state design policies/manuals are silent on this issue. Figure 7. Effect of speed on coefficient of road adhesion (Wong, 2008). Figure 8. Plan view of bicycle model.

19 The design manual for the Indiana Department of Trans- portation (INDOT) recommends the use of a higher speed in superelevation calculations than the design speed for the following conditions: • Transition area. Where a highway is transitioning from a predominantly rural environment to an urban environ- ment, travel speeds in the transition area within the urban environment may be higher than the urban design speed. • Downgrade. Where a horizontal curve is located at the bottom of a downgrade, travel speeds on the curve may be higher than the overall project design speed. As suggested adjustments, the design speed used for the horizontal curve may be 5 mph (grade of 3% to 5%) or 10 mph (grade > 5%) higher than the project design speed. This adjustment may be more appropriate for a divided facility than for a two- lane, two-way highway. • Long tangent. Where a horizontal curve is located at the end of a long tangent section, a design speed of up to 10 mph higher than the project design speed may be appropriate. The design manual for the Ohio Department of Transpor- tation (ODOT) provides the following guidance for design- ing superelevation on steep grades: On long and fairly steep grades, drivers tend to travel some- what slower in the upgrade direction and somewhat faster in the downgrade direction than on level roadways. In the case of divided highways, where each pavement can be superelevated independently, or on one-way roadways, such as ramps, this tendency should be recognized to see whether some adjust- ment in the superelevation rate would be desirable and/or feasible. On grades of 4% or greater with a length of 1000 ft (300 m) or more and a superelevation rate of 0.06 or more, the designer may adjust the superelevation rate by assuming a design speed which is 5 mph (10 km/h) less in the upgrade direction and 5 mph (10 km/h) higher in the downgrade direc- tion, providing that the assumed design speed is not less than the legal speed. On two-lane, two-way roadways and on other multilane, undivided roadways, such adjustments are less fea- sible, and should be disregarded. In summary, the guidance provided in the design policies/ manuals for INDOT and ODOT is very much consistent with AASHTO’s policy on superelevation criteria for curves on steep grades, but both provide more detail than AASHTO’s policy. Where AASHTO policy suggests assuming a higher design speed for the downgrade, the Indiana and Ohio policies/ manuals provide specific guidance on how much to increase the design speed. Also, Ohio’s manual indicates a specific length of grade for consideration.