Adopting the Performance Margin in Horizontal Curve Design

Abstract

A vehicle negotiating a curve experiences a centripetal force that is partially supplied by the tire–pavement friction. The American Association of State Highway and Transportation Officials’ design handbook, A Policy on Geometric Design of Highways and Streets (“Green Book”), dictates the design of horizontal curve segments according to the side friction (demand) factor, which is a fraction of the available friction used during a maneuver. However, the Green Book casts the side friction factor not as a measure of tire–pavement friction but as a measure of lateral acceleration. In this way, the Green Book conflates these independent concepts (tire–pavement friction versus lateral acceleration). It is shown in this work that keeping these curve design parameters independent has meaningful benefits. A rigorous mathematical clarification of the differences among the side friction (demand) factor, lateral acceleration, and coefficient of friction is developed. It is shown that changing the coefficient of friction affects the side friction factor, but the opposite is not necessarily true. An alternate metric is proposed, the performance margin $(PM)$ , that has several advantages for assessing friction margins. Currently, horizontal curves are designed only with a lateral friction reserve. Excluding longitudinal dynamics may not be realistic given the manner in which drivers react in limit handling maneuvers. The $PM$ accounts for both bidirectional dynamics and road geometry, and highlights regions where the Green Book’s standards are inconsistent with vehicle performance capabilities. The work concludes by posing recommendations aimed at implementing the $PM$ in future Green Book editions.

Keywords

infrastructure roadway design performance effects of geometric design curves horizontal alignment streets and highways geometric design super-elevation

Horizontal curves are commonplace on highways throughout the world, as they provide the transition between two straight road segments. They must be designed to accommodate the cornering capabilities of various vehicle types (ranging from agile passenger vehicles to heavy-duty trucks) and driver behaviors in all weather conditions. Curves are designed to subtend some portion of a circle so that physical laws are easily used to set bounds on design parameters. The American Association of State Highway and Transportation Officials (AASHTO) stipulates a specific design methodology in the policy, A Policy on Geometric Design of Highways and Streets (henceforth referred to as the “Green Book”) ( 1 ).

When cornering, there is a centripetal force directed toward the center of the curve. If the curve is unbanked, the frictional force between the road surface and the vehicle’s tires must generate the centripetal force. If the curve is super-elevated, a component of the normal force aids in supplying the centripetal force. The Green Book extensively employs the side friction factor (alternatively called the friction demand), $f_{Y_{V}}$ , to express the fraction of the available friction that is being used ( 2 ). The side friction factor, design speed, and super-elevation are vital design parameters. However, there are major theoretical and practical improvements that can be made with respect to the side friction factor.

The definition of the side friction factor conflates two independent concepts: tire–pavement friction and (lateral) acceleration. Thus, $f_{Y_{V}}$ cannot adequately convey the potential risk of undesirable vehicular maneuvers if the operating condition changes.

The Green Book assumes $f_{Y_{V}}$ will be reached before the tire–pavement friction is reached, which is not always true in practice (especially in adverse weather conditions).

The numeric values of $f_{Y_{V}}$ are derived from passenger comfort studies that may not reflect contemporary driving behavior, forecasted behaviors of autonomous vehicles (AVs), or both.

This work discusses each of these issues. The contributions are as follows:

a call for future research in driver behavior;

an equivalency between the Green Book and the Society of Automotive Engineers (SAE), which elucidates the difference between the coefficient of friction and the friction demand;

a comparison between the margin of safety used by the Green Book and the performance margin ( $PM$ ), a margin of safety used in vehicle dynamics.

Ultimately, the $PM$ is suggested as a replacement for AASHTO’s margin of safety for practical horizontal curve design purposes.

Furthermore, there are practical implementation issues. When designing horizontal curves, the Green Book bases the numeric values of the maximum allowable side friction factor on potentially outdated passenger comfort studies. In particular, Tan ( 3 ) demonstrated that modern-day motorists are “willing to accept higher levels of net lateral acceleration.” The current allowable side friction factors may also need to be revised if AVs become mainstream, as AVs may be programmed to drive more aggressively than the average human driver ( 4 ). Basing side friction factors on comfort studies that are unsubstantiated in the modern day, especially as AV implementation inches closer, is hazardous. The subjective nature of passenger comfort therefore makes the $f_{Y_{V}}$ values extremely sensitive. Adopting other comfort studies can change or eliminate AASHTO’s native safety margins. In addition, the Green Book assumes that the vehicle does not accelerate longitudinally through the curve, which is unlikely in practice. This invites the possibility of a vehicle having an inadequately large lateral safety margin.

Background

Performance Margin

Consider a vehicle model that responds to the road surface features: bank angle (super-elevation), slope (grade), and friction; and through driver commands: throttle, brake, and steering. The road geometry and friction provide a means by which the vehicle may generate forces and resulting accelerations; that is, the road surface determines the available acceleration. Similarly, the vehicle dynamics and driver commands result in the required acceleration to successfully navigate the path at a given speed. The performance envelope is defined by the locus of points for which the required acceleration equals the available acceleration in the ground plane. The longitudinal and lateral components of the available acceleration for a certain operating condition are a locus of points that define the performance envelope given by $A_{X}^{*}$ and $A_{Y}^{*}$ ; the longitudinal and lateral components of the required acceleration are given by $A_{X}$ and $A_{Y}$ (in units of gravitational acceleration). The equation defining the elliptical performance envelope is as follows:

\frac{{(A_{X}^{*} - \tan (θ_{s}))}^{2}}{μ_{X, E f f}^{2}} + \frac{{(A_{Y}^{*} - \tan (θ_{b}))}^{2}}{μ_{Y, E f f}^{2}} = 1

(1)

where $θ_{s}$ is the slope (so that $\tan (θ_{s})$ is the grade), $θ_{b}$ is the bank angle, and $μ_{X, Eff}$ and $μ_{Y, Eff}$ are the longitudinal and lateral effective friction coefficients. The effective friction coefficient is defined as the maximum fraction of the normal force that can be used to generate tractive force by the vehicle at the current operating condition ( 5 ). The effective coefficient of friction incorporates traditional road–tire interactions while simultaneously accounting for limitations imposed by vehicle dynamics, terrain roughness, and weather elements (surface contaminates, water films, etc.).

The PM is defined as the additional performance capability that can be drawn on beyond that demanded by the current operating condition ( 5 ). Figure 1 depicts the relationship among the $PM$ , the available acceleration (solid curved line in the first quadrant), and the required acceleration (indicated with an “X” inside the first quadrant).

Mathematically, the $PM$ is expressed as Equation 2 ( 5 ):

P M = \min (\sqrt{{(A_{X}^{*} - A_{X})}^{2} + {(A_{Y}^{*} - A_{Y})}^{2}}), \forall A_{X}^{*}, A_{Y}^{*}

(2)

The $PM$ is a function of the required and available accelerations, which themselves are functions of the vehicle dynamics and driver commands, the road geometry (cross-slope $θ_{b}$ and grade $θ_{s}$ ), and the effective friction coefficient. Values of the $PM$ typically range from zero to $μ$ for passenger vehicles. A $PM$ of zero means the vehicle is at its handling limit and the maximum available tractive force has been reached. Increasing values of $PM$ indicate that more traction is available for braking and cornering maneuvers and thereby additional maneuvering capability exists at this operating condition. Because the $PM$ incorporates traditional limit handling metrics, it can be used to quantitatively measure a vehicle’s performance during a maneuver. It is measured in $G s$ (units of gravitational acceleration), which is easy to understand and allows direct performance comparisons between vehicles with different system parameters ( 6 ).

Occupant’s Preference Metric

The $PM$ imposes a quantitative handling constraint on a vehicle. Another common metric is the ride quality, a subjective perception of the occupants’ comfort during a trip. Defining quantitative ride comfort metrics can be difficult because human preferences are subjective and varied. The occupant’s preference metric (OPM) sets preferable accelerations, decelerations, and jerks for various driver behavior types ranging from normal drivers to extremely aggressive drivers. These values are obtained by amalgamating prior passenger comfort research into one unified study with clearly defined comfort thresholds ( 7 – 15 ). The thresholds, defined with respect to both acceleration and jerk, are given in Table 1.

Table 1.

Comfortable Accelerations and Jerks ( 7 )

	Normal driver	Aggressive driver
$a_{X_{V}} (\frac{m}{s^{2}})$	1.5	3.1
$- a_{X_{V}} (\frac{m}{s^{2}})$	−2.0	−5.1
$\| a_{Y_{V}} \| (\frac{m}{s^{2}})$	3.0	5.6
$\| z_{X_{V}} \| (\frac{m}{s^{3}})$	0.6	2.0
$\| z_{Y_{V}} \| (\frac{m}{s^{3}})$	0.6	2.0

In Table 1, $a_{X_{V}}$ is the occupant’s longitudinal acceleration discomfort threshold, $- a_{X_{V}}$ is the occupant’s longitudinal deceleration discomfort threshold, $| a_{Y_{V}} |$ is the magnitude of the occupant’s lateral acceleration discomfort threshold, and $| z_{X_{V}} |$ and $| z_{Y_{V}} |$ are the magnitudes of the maximum allowable longitudinal and lateral jerks.

The OPM is used in this study because it not only confirms findings from numerous passenger comfort studies, but also recognizes and accommodates driver variations, offering comfort thresholds for varying driver aggressiveness levels.

Green Book Horizontal Curve Design Methodology

The Green Book thoroughly derives the equations of motion of a vehicle on a banked, curved road. Other literature ( 5 , 16 , 17 ) also contain extensive derivations. Thus, this section only highlights the final steps in a classical derivation. Consider a vehicle, modeled as a point mass, negotiating a curve banked at an angle $θ_{b}$ at a constant longitudinal speed $v_{X_{V}}$ . The free-body diagram is illustrated in Figure 2; the variables are as follows.

$R$ : curve radius (ft).

$θ_{b}$ : bank angle (°).

$e$ : super-elevation (feet of vertical rise per 100 ft of horizontal distance). The relationship between $e$ and $θ_{b}$ is $e = 100 \tan θ_{b}$ (%).

$F_{f}$ : side frictional force (lbf).

$W$ : vehicle weight (lbf), with normal and parallel components $W_{n}$ and $W_{p}$ .

$F_{c}$ : centripetal force (lbf), with normal and parallel components $F_{cn}$ and $F_{cp}$ .

Balancing the vertical and horizontal forces yields the following:

\frac{v_{X_{V}}^{2}}{gR} = \frac{\sin θ_{b} + f_{Y_{V}} \cos θ_{b}}{\cos θ_{b} - f_{Y_{V}} \sin θ_{b}}

(3)

where $f_{Y_{V}}$ is the side friction (demand) factor, the fraction of the available friction that is being used. The Green Book uses super-elevation instead of the bank angle. Applying the relationship $e = 100 \tan θ_{b}$ and the small-angle assumption to Equation 3 results in the following:

\frac{v_{X_{V}}^{2}}{gR} = \frac{0.01 e + f_{Y_{V}}}{1 - 0.01 f_{Y_{V}} e}

(4)

Equation 4 is referred to as the basic curve equation in the Green Book. The term $1 - 0.01 f_{Y_{V}} e$ is nearly unity; omitting it is common as a conservative measure ( 1 ). Doing so and solving for $f_{Y_{V}}$ (the side friction factor) yields the following:

f_{Y_{V}} = \frac{v_{X_{V}}^{2}}{gR} - 0.01 e

(5)

In other words, the side friction factor is the centripetal acceleration acting on the vehicle $(\frac{v_{X_{V}}^{2}}{gR})$ minus the portion of the centripetal acceleration sustained by the super-elevation $(0.01 e)$ ( 18 ). Thus, the side friction factor represents the lateral acceleration at an operating condition with units of Gs.

Friction Demand Versus Coefficient of Friction

Transportation engineers and automotive engineers utilize different terminology despite using the same first principles. In particular, the PM is written using SAE terminology. Translating between the transportation and automotive constituents creates consistency and comprehensibility. In this work, the transportation (AASHTO) terminology will be “translated” to SAE notation.

The left-hand side of Equation 3 represents the SAE definition of the centripetal acceleration:

A_{C} = \frac{v_{X_{V}}^{2}}{gR} = \frac{\sin θ_{b} + f_{Y_{V}} \cos θ_{b}}{\cos θ_{b} - f_{Y_{V}} \sin θ_{b}}

(6)

Solving for $f_{Y_{V}}$ :

f_{Y_{V}} = \frac{A_{C} \cos θ_{b} - \sin θ_{b}}{A_{C} \sin θ_{b} + \cos θ_{b}}

(7)

Kang and Ferris ( 5 ) derived expressions for the accelerations in the lateral and vertical directions in vehicle coordinates:

\begin{matrix} A_{Y_{V}} = A_{C} \cos θ_{b} - \sin θ_{b} \\ A_{Z_{V}} = A_{C} \sin θ_{b} + \cos θ_{b} \end{matrix}

(8)

These comprise the numerator and denominator of Equation 7:

f_{Y_{V}} = \frac{A_{Y_{V}}}{A_{Z_{V}}}

(9)

Similarly:

f_{X_{V}} = \frac{A_{X_{V}}}{A_{Z_{V}}}

(10)

The SAE definition of $f_{Y_{V}}$ (Equation 9) is identical to AASHTO’s definition of the side friction demand given in Equation 3.6 in the Green Book. However, Section 3.3.2.2 of the Green Book mislabels $f_{Y_{V}}$ as the coefficient of friction, which is defined as the “friction force divided by the component of the weight perpendicular to the pavement surface” ( 1 ). Clearly, the coefficient of friction, $μ_{Eff}$ , is defined as the maximum possible longitudinal or lateral force divided by the component of the weight perpendicular to the pavement surface. This mislabeling in Section 3.3.2.2 unnecessarily introduces an inconsistency between the AASHTO and SAE notation, and motivates a more precise distinction between the side friction demand ( $f_{X_{V}}$ or $f_{Y_{V}}$ ) and the effective coefficient of friction ( $μ_{X, Eff}$ or $μ_{Y, Eff}$ ). The definition of the coefficient of friction with respect to available acceleration ( $A_{X_{V}}^{*}$ or $A_{Y_{V}}^{*}$ ) is as follows:

\begin{matrix} μ_{Y, Eff} = \frac{F_{Y_{V}}^{*}}{F_{Z_{V}}} = \frac{A_{Y_{V}}^{*}}{A_{Z_{V}}} \\ μ_{X, Eff} = \frac{F_{X_{V}}^{*}}{F_{Z_{V}}} = \frac{A_{X_{V}}^{*}}{A_{Z_{V}}} \end{matrix}

(11)

because the acceleration is the mass-normalized force. Combining Equations 9–11 gives the following:

\begin{matrix} f_{Y_{V}} = (\frac{A_{Y_{V}}}{A_{Y_{V}}^{*}}) μ_{Y, Eff} \\ f_{X_{V}} = (\frac{A_{X_{V}}}{A_{X_{V}}^{*}}) μ_{X, Eff} \end{matrix}

(12)

Bonneson ( 19 ) recognizes that “the friction demand predicted by [Equation 5] is more of a conceptual convenience than it is a true measure of tire-pavement friction,” but goes on to say that “the predicted friction is approximately equal to the lateral acceleration (in Gs) experienced by the driver,” which is inconsistent with the derivation resulting in Equation 12. This misuse of $f_{Y_{V}}$ in Section 3.3.2.2 of the Green Book would only be correct if the vehicle were operating on the performance envelope $(A_{Y_{V}} = A_{Y_{V}}^{*})$ . When operating on the performance envelope, when the $PM$ is $0 G$ , no margin is left for additional maneuvering. Again, it should be clear that the side friction demand factor is a lateral acceleration; the coefficient of friction is a property of the tire–pavement interface. These are two distinct concepts that should not be conflated by using $f_{Y_{V}}$ to approximate the amount of tire–pavement friction at an operating condition.

Equation 12 expresses the longitudinal and lateral friction demands as the directional friction coefficient is scaled by the directional accelerations at the operating condition. This bears significant consequences: $f$ (either $f_{X_{V}}$ or $f_{Y_{V}}$ ) is a relative metric that depends on $μ$ (either $μ_{X, Eff}$ or $μ_{Y, Eff}$ ), an absolute metric. Another interpretation of $f$ versus $μ$ is that $f$ is dependent on the driver (by way of the vehicle’s speed) and the tire–pavement interaction, but $μ$ is only dependent on the tire–pavement interaction. Changing the vehicle’s velocity through the curve results in a change in the lateral acceleration (and lateral force), resulting in a change in $f$ . However, the coefficient of friction is unaffected by the change in vehicle speed on dry pavement because it is a property of the tire–pavement interface. On the other hand, Equation 12 states that changing the value of $μ$ will always change the value of $f$ for a constant operating condition.

Thus, changing $μ$ may precipitate a change in driver behavior (reflected in $f$ ) for safety, but an equivalent change in $f$ does not necessarily convey the potential dangers stemming from approaching the performance envelope. To further clarify the distinction between the side friction demand factor and the coefficient of friction, consider a vehicle negotiating a curve with no longitudinal acceleration. Two operating conditions and two coefficients of friction values are listed in Table 2.

Table 2.

Cases of $f_{Y_{V}}$ and $μ_{Y, Eff}$ for a Fictitious Scenario

	Case 1 $(Gs)$	Case 2 $(Gs)$
$f_{Y_{V}}$	0.2	0.2
$μ_{Y, Eff}$	0.8	0.2

In the first case, the required (lateral) acceleration is 25% of the available acceleration and there is a $0.6 G$ performance margin available to accommodate a worsening of circumstance. In Case 2, the side friction demand is unchanged from Case 1, but the lateral coefficient of friction is decreased such that $f_{Y_{V}} = μ_{Y, Eff}$ . The cause of the drop in $μ_{Y, Eff}$ may stem from a combination of deteriorated pavement and inclement weather (such as a prevalent water film buildup from rainfall or a layer of ice). In Case 2, the required acceleration is 100% of the available acceleration and there is no performance margin available because the vehicle is operating at the performance envelope. Because $f$ is a relative metric, only considering $f$ cannot capture the crucial difference in these operating conditions. The coefficient of friction must be provided in tandem with $f$ and then assessed to fully understand the ramifications of a variable operating condition.

Evaluating Margins of Safety

Horizontal curve design is treated in the Green Book to address the possibility of skidding and rollovers. The Green Book’s standards intend to provide an ample margin of safety against undesired vehicle maneuvers, but exactly how much is unclear ( 1 ). Furthermore, some studies used to inform the Green Book’s standards are nearly 100 years old and should be re-evaluated to ensure the standards reflect contemporary vehicles, driving styles, road construction and maintenance, and traffic management ( 16 ).

Side (Lateral) Friction Margin

Torbic et al. ( 16 ) developed super-elevation criteria for sharp horizontal curves on steep grades. As part of the study, the effective friction coefficients of numerous highways were measured to quantify the safety margins implanted in the Green Book’s policies. A formal definition of the lateral margin of safety begins with the friction ellipse equation ( 20 , 21 ):

{(\frac{F_{Y_{V}}}{F_{Y_{V}}^{*}})}^{2} + {(\frac{F_{X_{V}}}{F_{X_{V}}^{*}})}^{2} \leq 1

(13)

The friction ellipse can be written with respect to acceleration by normalizing each quantity in Equation 13 by the vehicle operating weight, $F_{Z_{V}} = N = mg$ , unless the road plane is extremely angled relative to the ground plane (significant slope and bank angle). This is equivalent to substituting Equations 9–11 into Equation 13. Doing so yields the following:

{(\frac{f_{Y_{V}}}{μ_{Y, Eff}})}^{2} + {(\frac{f_{X_{V}}}{μ_{X, Eff}})}^{2} \leq 1

(14)

Equation 14 expresses the friction ellipse with respect to acceleration (in Gs). From the definition of a margin, the lateral friction margin can be written as follows:

f_{Y, margin} = f_{Y, supply} - f_{Y_{V}}

(15)

where $f_{Y, supply}$ is equivalent to the available (lateral) acceleration at that operating condition. Here, $f_{Y, supply}$ is typically less than $μ_{Y, Eff}$ because of the presence of a nonzero longitudinal acceleration induced by braking. It is found by rearranging Equation 14 for the case $f_{Y_{V}} = f_{Y, supply}$ :

f_{Y, supply} = μ_{Y, Eff} \sqrt{1 - {(\frac{f_{X_{V}}}{μ_{X, Eff}})}^{2}}

(16)

Finally, the lateral friction margin is obtained by substituting Equation 16 into Equation 15:

f_{Y, margin} = [μ_{Y, Eff} \sqrt{1 - {(\frac{f_{X_{V}}}{μ_{X, Eff}})}^{2}}] - f_{Y_{V}}

(17)

These quantities are illustrated in Figure 3. The friction ellipse peaks laterally at $μ_{Y, Eff}$ and longitudinally at $μ_{X, Eff}$ . An operating condition $(f_{X_{V}}, f_{Y_{V}})$ is indicated by the red dot. The lateral friction supply is the available lateral acceleration at that operating condition. Because $f_{X_{V}} \neq 0, f_{Y, supply} < μ_{Y, Eff}$ . The lateral friction margin is the vertical line connecting the lateral friction supply to the operating condition.

Skidding will not occur if $f_{Y, margin} > 0 G$ . The vehicle is on the verge of skidding if $f_{Y, margin} = 0 G$ , and the vehicle may skid because of tire saturation if $f_{Y, margin} < 0 G$ . Torbic et al. ( 16 ) produced the following generalizations:

large safety margin: $f_{Y, margin} \geq 0.2 G$ ;

moderate safety margin: $0.1 G \leq f_{Y, margin} < 0.2 G$ ;

low safety margin: $0 G \leq f_{Y, margin} < 0.2 G$ ;

unacceptable safety margin: $f_{Y, margin} < 0 G$ .

It was found that the current AASHTO guidelines yield a sufficiently large safety margin ( 16 ). Other studies, such as the work by Morrall and Talarico ( 22 ) and Kordani and Molan ( 23 ), have reported similar findings. However, these findings are predicated on the usage of $f_{Y_{V}}$ as the friction coefficient rather than the fraction of available friction. The contributions developed in this work are extensions of these previous works in that concepts developed for lateral vehicle dynamics are also considered for longitudinal vehicle dynamics.

Determining the lateral safety margin is vital in road design. The Green Book sets maximum allowable $f_{Y_{V}}$ values as a function of design speed. These values are derived from a series of ball-bank readings representing passenger comfort from a study conducted in 1940 ( 24 ); clearly the vehicles and driving environment have changed in the last 80 years. The Green Book cites too few corroborative studies ( 19 , 25 ) to guide the official design practices, given the highly variable and subjective nature of passenger comfort. The choice of passenger comfort metrics and thresholds influences the size of various permissible operating regions, which in turn pinpoints areas where the AASHTO guidelines do not adequately capture the desired safety margins as vehicles approach their performance limits.

Performance Margin Versus Lateral Friction Margin

Consider the differences between the performance margin, $PM$ , and lateral friction margin, $f_{Y_{V}}$ , that represents the maximum allowable side friction factor as defined by AASHTO. Note that there are inconsistencies in the use of $f_{Y_{V}}$ , discussed in the third section.

On the GG diagram, the lateral friction margin is the difference between the lateral acceleration for this given operating condition and the lateral friction supply (portion of the performance envelope parallel to the lateral motion axis; see Figure 3). However, the lateral friction margin does not account for a margin in the longitudinal direction. In light of recent vehicle dynamics advancements such as robust vehicle control algorithms ( 6 ), improved torque vectoring strategies ( 26 , 27 ), and percipience of vehicle handling near the limits of tire adhesion ( 28 ), there exists a need to simultaneously measure the lateral and longitudinal friction margins since the longitudinal acceleration while cornering is increasingly likely to be significant. As noted by Torbic et al. ( 16 ), “if there is going to be an area of concern based on AASHTO’s current design policy, it will likely arise primarily from the interaction of braking and cornering forces.”

The performance margin is recommended as a replacement for the lateral friction margin for the following reasons.

It incorporates both the lateral and longitudinal margins of safety. The $PM$ is the minimum difference between an operating condition (represented as lateral and longitudinal acceleration and braking, the required acceleration) and the closest point of the performance envelope (the locus of points corresponding to the available acceleration). In this way it accounts for nonzero longitudinal accelerations. This makes the $PM$ more robust than the lateral friction margin. The lateral friction margin is a subset of the $PM$ whose equivalency only occurs during a pure braking and/or pure cornering maneuver (which is unlikely in practice).

Grade and super-elevation are accounted for in the $PM$ because they affect the available acceleration. The Green Book uses acceleration as a surrogate measure of tire–pavement friction at an operating condition; that is, the lateral friction margin represents the amount of tire–pavement friction remaining before the tire becomes fully saturated. The amount of friction alone cannot convey the effects of grade and super-elevation.

Operating condition changes can be assessed with the same performance envelope.

The $PM$ is rooted in acceleration and has the intuitive units of Gs.

Furthermore, a vehicle that is accelerating or braking around a curve can exceed the lateral safety margin even if the lateral acceleration conforms to the Green Book’s maximum allowable side friction factor.

Consider Figure 4, depicting the first quadrant of the GG diagram with coefficients of friction comprising the semi-major and semi-minor axes. The maximum side friction factor for a given design speed is superimposed; any operating condition within the orange region fulfills the Green Book’s design requirements. However, if the lateral safety margin is to be maintained, then the allowable side friction factors region cannot extend all the way to $μ_{X, Eff}$ because there comes a longitudinal acceleration at which the lateral safety margin is no longer sufficient. Thus, the allowable side friction factors region truly resembles the region in Figure 4b. This region illustrates the existence of a maximum allowable longitudinal acceleration, $A_{X_{Green Book}}$ . Any operating condition such that $A_{X} \leq A_{X_{Green Book}}$ and $A_{Y} \leq f_{Y_{V}}$ is guaranteed to have a “large” lateral safety margin.

Next consider a performance margin with the same 0.2G safety margin, shown in Figure 5 as a blue elliptical region. This region represents the operating conditions satisfying $PM \geq 0.2$ . There exists a maximum longitudinal acceleration $A_{X_{PM}}$ that may not necessarily equal $A_{X_{Green Book}}$ . Likewise, the lateral acceleration limit is bounded by $A_{Y_{PM}}$ , which may differ from $f_{Y_{V}}$ . In general, the $PM$ offers a different “safe” operating region than that prescribed by the lateral friction margin. Whether the $PM$ ’s operating range encompasses a larger area than the lateral friction margin’s range depends on the effective friction coefficients, as they determine the bounds of the performance envelope. However, because $f_{Y_{V}}$ is independent of the effective friction coefficients, the “safe” operating region is a constant size. This is especially problematic if $μ_{X, Eff}$ , $μ_{Y, Eff}$ , or both, are small.

Numerical Example

Consider a flat horizontal curve with a design speed of 70 mph. Because there is no bank angle or super-elevation, the performance envelope forms an ellipse defined by $μ_{X, Eff}$ and $μ_{Y, Eff}$ as the semi-major and semi-minor axes, respectively. A past study found that $μ_{X, Eff}$ and $μ_{Y, Eff}$ are normally distributed ( 29 ). The worst-case effective friction coefficient (in either direction) is defined as $μ_{Eff} = \bar{μ} - 2 σ$ , where $\bar{μ}$ is the mean friction coefficient and $σ$ is the corresponding standard deviation ( 16 ). Thus, the numerical values of $μ_{X, Eff} (= 0.65 G)$ and $μ_{Y, Eff} (= 0.51 G)$ are sourced from the data in Himes ( 29 ) and represent the worst-case effective friction coefficients.

Figure 6 is similar to Figure 5 with realistic accelerations (in $G s$ ) sourced from the Green Book and the normal driver thresholds of the OPM ( 7 ) to illustrate three points: the practicality of using the $PM$ over the lateral safety margin, the Green Book’s conflation of the side friction demand and the effective friction coefficient, and the sensitivity of the current $f_{Y_{V}}$ to comfort. Because the $f_{Y_{V}}$ values are rooted in comfort, a prior study (the OPM) was used in lieu of the studies employed by the Green Book. Figure 6 contains five sub-regions formed from the various overlapping regions; the labels N1–N5 describe the span of particular (sub)regions for clarity.

The operating region with an acceptable performance margin $(PM \geq 0.2 G)$ is indicated by the smaller ellipse nested inside the performance envelope and encompasses Regions N1–N4. To maintain a $PM$ of at least $0.2 G$ , $A_{X, PM}$ cannot exceed $0.45 G$ and $A_{Y, PM}$ cannot exceed $0.31 G$ . The acceptable $f_{Y_{V}}$ envelope (spanning Regions N2, N4, and N5) is sourced directly from the Green Book. Finally, the triangle spanning Regions N1 and N2 depict the “normal driver’s” comfort thresholds as defined by the OPM.

Each of the five regions represents operating conditions that satisfy certain metrics.

Region N1: these operating conditions satisfy the $PM$ and OPM limits but violate the Green Book’s $f_{Y_{V}}$ limit. A vehicle with no longitudinal acceleration can maintain a lateral acceleration of nearly three times the Green Book’s $f_{Y_{V}}$ limit before reaching the border of unacceptability (exceeding the $PM$ or OPM envelope). This suggests an overly conservative nature of the Green Book’s current maximum side friction demand limits in the case when the cornering requirements are small compared to the available friction $μ_{Y, Eff}$ .

Region N2: these operating conditions satisfy the $PM$ , OPM, and the Green Book’s thresholds. This region may be ideal for ultra-conservative motorists.

Region N3: these operating conditions satisfy the $PM$ limits but violate the OPM’s and the Green Book’s constraints. Motorists in this region will be uncomfortable but will retain a sufficient performance margin. This region illustrates that comfort (OPM and Green Book) and handling $(PM)$ , two inherently independent quantities, can be synthesized to form acceptable operating regimes. Furthermore, it illustrates that the $PM$ constitutes a “hard constraint” because violating the $PM$ may lead to vehicular instability. On the other hand, it may still be possible to maintain stability while outside the comfort regimes.

Region N4: these operating conditions satisfy the $PM$ and the Green Book’s limits but violate the OPM constraints. Motorists in this region will be stable and comfortable according to the Green Book’s comfort threshold $(f_{Y_{V}})$ , but uncomfortable according to the OPM’s comfort thresholds. Like Region N3, a motorist operating within this region will maintain a sufficient performance margin at the cost of comfort.

Region N5: these operating conditions satisfy the Green Book’s limits but violate the $PM$ and OPM limits. While a motorist will be comfortable according to the Green Book, they will not be according to the OPM. This region should be avoided because these operating conditions lie outside of the $PM$ envelope, thus compromising safety.

This numerical example illustrates the interplay between metrics. In particular, the size of Figure 6’s sub-regions are determined from each metric’s thresholds. Figure 7 reflects the changes in the shapes, locations, and existences of sub-regions when the OPM’s “aggressive” comfort envelope is used rather than the “normal” comfort thresholds.

Some sub-regions of interest include the following.

Region A1: these operating conditions satisfy the OPM limits but are beyond the performance envelope. Operating within this region may induce vehicular instability (tire saturation, etc.) but is comfortable. This sub-region does not exist in Figure 6.

Region A3: analogous to Region N1, but now the $PM$ provides the limiting acceleration rather than the OPM.

Region A5: analogous to Region N3, but there is a considerably smaller pool of operating conditions.

Region A6: these operating conditions satisfy the OPM and Green Book limits but violate the $PM$ limit. Like Region A1, this region does not exist in Figure 6.

Because the sub-regions are sensitive to the particular choice of comfort study, it is critical to synthesize a wide variety of comfort studies to formulate the most accurate comfort thresholds. Regularly updating the comfort studies and thresholds may also be important as AVs become mainstream and driving behaviors may evolve.

Hydroplaning Example

The discussion in the Numerical Example section illustrates the interplay between passenger comfort and handling. The Green Book assumes that the passenger comfort thresholds will be breached before the tire–pavement friction is exceeded. This is not always true, especially in adverse conditions. Wet pavement poses a significant threat to vehicle maneuverability. Wet pavement accounted for 15% of all vehicle crashes in the U.S.A. and killed 4050 people from 2007 to 2016 ( 30 ). NCHRP 300 ( 31 ) yielded the following equation defining the effective lateral friction coefficient as a function of the water film thickness (WFT) and vehicle operating speed:

\begin{matrix} μ_{Y} (v_{X_{V}}, W F T) = μ_{Y, d r y} \\ + (W F T) (\frac{\partial μ_{Y}}{\partial W F T}) \\ + (v_{X_{V}} - 25) (\frac{\partial μ_{Y}}{\partial v_{X_{V}}}) \end{matrix}

(18)

where $v_{X_{V}}$ is the operating speed (mph), $WFT$ is the water film thickness (mm), $μ_{Y, dry}$ is the effective lateral friction coefficient on dry pavement at 25 mph $(G)$ , $\frac{\partial μ_{y}}{\partial WFT} = - 0.036$ is the change in $μ_{Y, dry}$ per unit change in WFT $(\frac{G}{mm})$ , and $\frac{\partial μ_{y}}{\partial v_{X_{V}}} = - 0.0035$ is the change in $μ_{y, dry}$ per unit change in speed $(\frac{G}{mph})$ .

Because the lateral tire–pavement friction coefficient is a function of the water film thickness and operating speed, it is reasonable to assume that a combination of water buildup, high operating speeds, or both, can reduce $μ_{Y}$ such that the tire–pavement friction is reached before the Green Book’s suggested limits.

Figure 8 illustrates various friction and comfort thresholds. The Green Book’s side friction demand factors are represented by the blue asterisks; the “normal” and “aggressive” comfort thresholds (as defined by the OPM) are given as the black dashed/dotted horizontal lines. The two reddish horizontal lines represent the lateral friction coefficients of wet snow and ice ( 32 ). Finally, the two purple downward-sloping lines represent the lateral friction coefficients for two hydroplaning scenarios.

Assume a vehicle travels at the road’s design speed and does not accelerate longitudinally. If the road surface contains wet snow $(μ_{Y, snow} = 0.22 G)$ , the Green Book’s side friction demand factors will be exceeded before $μ_{Y, snow}$ is reached. However, the friction margin (a vertical line between two curves) between these quantities is slightly greater than $0.1 G$ at a speed of 70 mph, which is “low” according to Torbic et al. ( 16 ). Both of these curves are situated below the OPM’s thresholds, so passengers will remain comfortable.

If the road is icy $(μ_{Y, ice} = 0.14 G)$ , the Green Book’s side friction factors exceed $μ_{Y, ice}$ at speeds below 55 mph, which is problematic. Above 55 mph, the lateral friction margin is extremely small. Once again, both curves are underneath the OPM’s comfort limits. If the vehicle is being operated at a low speed, this may present a safety hazard, as occupants will feel comfortable despite exceeding the Green Book’s suggested acceleration.

The light purple downward-sloping line represents the lateral tire–pavement friction coefficient when the road has a 2 mm water film buildup after a rainfall event. A 2 mm WFT was selected based on the work in Torbic et al. ( 16 ). The vertical distance between the $μ_{Y, wet} (WFT = 2 mm)$ line and the $f_{Y_{V}}$ line is roughly $0.3 G$ for all speeds, which is a sufficiently large margin. Thus, the Green Book provides an adequately large margin against skidding failure for a water film thickness of 2 mm. On the other hand, the dark purple downward-slanting line represents the lateral friction coefficient for a WFT of 12.9 mm. Above 55 mph, $μ_{Y, wet} (WFT = 12.9 mm)$ will be exceeded before the side friction demand factors are reached. Although a 12.9 mm WFT seems implausibly large, rainy regions such as Tampa, Florida, experience this rainfall volume annually ( 31 , 33 ).

It is clear that some conditions may cause the tire–pavement friction coefficient to be less than the side friction demand factors. Thus, a realignment of the side friction demand factors to better account for variable road and/or weather conditions is suggested to ensure the updated side friction demand factors provide an ample margin against skidding failure for most conditions.

Updating Numeric Values of the Side Friction Demand

The numeric values of $f_{Y_{V}}$ in the Green Book are based on a series of passenger comfort studies conducted in the 20th century. Publishing a nationally adopted policy that promotes the use of values based on “differences in judgement as to what constitutes incipient instability or uncomfortable centrifugal sensation,” and not on something more concrete, is itself puzzling ( 34 ). Given the subjective nature of passenger comfort, it can be argued that comfort tolerances evolve over time, and that the currently used $f_{Y_{V}}$ values may not reflect modern comfort thresholds.

It should first be noted that passenger comfort may be influenced by the vehicle’s construction and characteristics. Erol et al. ( 35 ) demonstrated that comfort can be swayed solely by the appearance of a car seat. ISO Standard 2631 and British Standard 6841 define comfort as a function of the seat vibration, which may be influenced by a vehicle’s suspension, seat design, or both ( 36 – 38 ). Present-day vehicles are also constructed differently compared to the 20th century, when many of the passenger comfort studies used to inform the Green Book’s $f_{Y_{V}}$ values were underway. A 1931 Ford Model A Tudor sedan (a then-popular vehicle) has a lower center of gravity, weight, and driver position compared to a 1992 Ford Taurus (one of the top selling vehicles during the 1990s). According to a field demonstration, passengers in the 1931 Ford felt less comfortable than passengers in the 1992 Ford Taurus ( 4 ). It is therefore reasonable to think that modern vehicle designs account for passenger comfort to some degree. Because modern vehicle designs may be dissipating some of the previously noticeable discomforts, passengers may be more willing to tolerate greater accelerations.

Driver behavior can also play a role in determining passenger (dis)comfort. Tan ( 3 ) demonstrated that modern-day motorists are “willing to accept higher levels of net lateral acceleration.” Drivers may also be more aggressive today than in previous decades. For example, the number of registered vehicles on Australia’s road network is increasing, but the infrastructure and alternative transport options are not growing at the same pace. Congestion, a textbook trigger of driver aggression, is therefore increasing, leading to potentially increasingly aggressive drivers ( 39 ). The COVID-19 pandemic may also have increased driver aggression. Stephens et al. ( 39 ) surveyed 774 drivers, of which 33% self-reported heightened aggression in the past 5 years, and 47% noted that other drivers had become more dangerous during and after lockdown restrictions. Similar findings have been reported by Lopetrone and Biondi ( 40 ) and Meyer ( 41 ). A deeper study into post-lockdown driver behavior is needed. If drivers consistently exhibit more aggression, it may indicate a need to update the Green Book to reflect this behavior.

Similarly, the current allowable side friction factors may also need to be revised if AVs become mainstream, as AVs may be programmed to drive more aggressively than the average human driver ( 4 ).

This section is not meant to be an exhaustive literature review. Rather, the purpose of this section is to highlight a sample of relatively recent research endeavors supporting the assertion that currently used $f_{Y_{V}}$ values may not align with modern or future passenger comfort tolerances. An in-depth literature review is suggested for future research.

Conclusion

As this work highlights inconsistencies within the Green Book, this section is dedicated to providing major recommendations for future revision. There are three main issues with the Green Book. Firstly, the Green Book estimates the amount of tire–pavement friction at an operating condition as the lateral acceleration at that operating condition, which is only true in the limit $(A_{Y_{V}} = A_{Y_{V}}^{*})$ . This false equivalency conflates the concepts of the side friction demand factor (acceleration) and the coefficient of friction. Although these independent concepts are linked through friction (the coefficient of friction provides the limiting case for the acceleration), the equivalency is misleading and terminologically incorrect. The first major recommendation is to clarify the amount of tire–pavement friction at an operating condition that is approximated by the required acceleration and that $f_{Y_{V}}$ represents acceleration, not friction.

Secondly, the numerical values of $f_{Y_{V}}$ are contingent on the choice of comfort studies. Figure 6 includes the OPM and $f_{Y_{V}}$ , which both measure comfort but originate from different studies. The relative sizes of the five sub-regions within the figure are sensitive to the selected comfort studies; substituting another comfort study would change the geometry of each sub-region and could possibly reveal additional operating conditions that conform to one metric but not another. The second major recommendation is to update the numerical values of $f_{Y_{V}}$ based on the following.

The proper definition of $f_{Y_{V}}$ , as discussed in the third section.

A comprehensive, contemporary literature review on comfort studies. There are two main questions to be answered in future research. Firstly: do modern driving styles still align with the older studies that informed the Green Book (such as Stonex and Noble’s Pennsylvania Turnpike experiments [25])? However, comparing contemporary and prior studies may be difficult because of the earlier studies’ lack of experimental design details ( 3 ). Of particular interest is to what extent the COVID-19 pandemic shifted driver aggressiveness. Some studies assert that driver aggressiveness and speeding violations have increased post-pandemic ( 40 , 41 ), so future Green Book revisions could account for this heightened aggressiveness. Second: do drivers and passengers tolerate comfort differently? This distinction is critical, as AV occupants are passengers. Although full-scale AV adoption is years away ( 42 ), this foresight is consistent with the Green Book’s gradual policy revisions to accommodate AVs.

A coupling of vehicle dynamics and ride comfort. As illustrated in Figure 6, handling and comfort are two distinct concepts. A vehicle operating close to the performance envelope can maintain traction, but the ride may be uncomfortable (and vice versa). The current values of $f_{Y_{V}}$ cannot convey this distinction.

Finally, the performance margin is a more robust safety measure compared to the lateral safety margin. Whereas the lateral safety margin only accounts for a buffer in one direction, the $PM$ is bidirectional and can thus allow for more aggressive maneuvering, especially at small longitudinal accelerations. In addition, the $PM$ accounts for the road’s grade and roughness in both directions, but the lateral safety margin cannot. For these reasons, the third major recommendation is to adopt the $PM$ in lieu of the lateral safety margin.

In practice, implementing the $PM$ as a replacement for the side friction factor can lead to two opposing outcomes. Firstly, Figures 5–7 illustrate that the side friction factors may be too conservative at low longitudinal accelerations. A risk-averse practitioner may not be inclined to modify the current horizontal curve design process. However, if changes are to be made, the following is the recommended flowchart.

Determine the effective lateral tire–pavement friction coefficient, accounting for weather, contaminates, and other variables that reduce $μ_{Y}$ .

Determine the allowable $PM$ . There is not a unifying guideline for establishing the allowable $PM$ ; this may be at practitioners’ discretion based on the geography, terrain, driver aggression, and so forth.

Substitute $μ_{Y}$ , the allowable $PM$ , and an assumed value of the vehicles’ longitudinal accelerations $(A_{Y})$ into Equations 1 and 2 to obtain the maximum allowable lateral acceleration, $A_{Y}$ (referred to as $A_{Y_{PM}}$ in Figures 5–7.

If there is a sufficient (what constitutes “sufficient” is at the practitioner’s judgement) margin between $A_{Y_{PM}}$ and current values of $f_{Y_{V}}$ , the Green Book’s values are overly conservative. Thus, the curve’s design can be manipulated such that vehicles’ lateral accelerations reach $A_{Y_{PM}}$ .

However, if road and/or weather conditions reduce $μ_{Y}$ such that there is an insufficient (“insufficiency” is at the practitioner’s judgement) margin between $A_{Y_{PM}}$ and current values of $f_{Y_{V}}$ , the Green Book’s values inadequately account for the decreased $μ_{Y}$ value from the base conditions. The curve’s design can be manipulated to lower vehicles’ lateral accelerations.

The underlying goal of this process is to “tune” the highway’s parameters such that vehicles accelerate at or below $A_{Y_{PM}}$ . Suppose an existing horizontal curve is declared overly conservative (i.e., it falls under Item 4 above). To increase lateral accelerations on this curve, the curve radius can be decreased or the posted speed can be increased. Building tighter curves uses fewer materials, which implies monetary savings. Increasing the posted speed may facilitate greater throughput.

On the other hand, suppose an existing horizontal curve is deemed not conservative enough. Increasing the curve radius or decreasing the posted speed will reduce the lateral accelerations. In practice, implementing the $PM$ in lieu of the side friction factor may have monetary and/or capacity consequences, yielding more conservative or liberal designs depends on the circumstance.

Three avenues for future research are as follows:

analyzing crash data to better assess if highways facilitate (un)safe travel;

using more robust vehicle dynamics models;

special conditions that alter the tire–pavement friction coefficients (extreme weather, emerging asphalt technology, etc.).

Footnotes

Nomenclature

Every effort is made to use terminology and nomenclature as described by SAE J670 ( 43 ). Without loss of generalization, the concepts in this work are developed for a left turn.

$θ_{b}$ Cross-slope (or bank angle): the slope between the road plane and the ground plane projected onto the $(Y_{V}, Z_{V})$ plane, where the positive sense is such that the lower side of the road plane is closer to the center of the turn (a properly banked road).

$A_{X_{V}}$ Longitudinal acceleration: the scalar value of the component of the vehicle’s acceleration in the direction of the $X_{V}$ -axis.

$A_{Y_{V}}$ Lateral acceleration: the scalar value of the component of the vehicle’s acceleration in the direction of the $Y_{V}$ -axis.

$g$ Gravitational constant $(9.81 m / s^{2})$ .

$X_{V}, Y_{V}, Z_{V}$ Vehicle axis system: an axis system centered at the vehicle center of mass, with $X_{V}$ directed forward in the road plane, $Y_{V}$ directed laterally in the road plane, and $Z_{V}$ normal to the road plane.

Ground plane: a horizontal plane normal to the gravitational vector (no slope or cross-slope).

Road plane: a plane representing the road surface passing through the tire contact patches, supporting the tires and providing the friction necessary to generate tire shear forces.

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: T. Kim, J. Ferris; data collection: T. Kim, J. Ferris; analysis and interpretation of results: T. Kim, J. Ferris; draft manuscript preparation: T. Kim, J. Ferris. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Troy Jaisohn Kim

Data Accessibility Statement

The data that support the findings of this study are available from the corresponding author, T. Kim, on reasonable request.

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