Road Vehicle Performance
EXAMPLE 2.10 THEORETICAL MINIMUM STOPPING DISTANCE WITH AND WITHOUT ANTILOCK BRAKES
A car is traveling up a 3% grade on a road that has good, wet pavement. The engine is running at 2500 revolutions per minute. The radius of the wheels is 15 inches, the driveline slippage is 3%, and the overall gear reduction ratio is 2.5 to 1. A deer jumps out onto the road and the driver applies the brakes 291 ft from it. The driver hits the deer at a speed of 20mi/h. If the driver did not have antilock brakes, and the wheels were locked the entire distance, would a deer-impact speed of 20 mi/h be possible?
SOLUTION
Next, using Eq. 2.5 the coefficient of rolling resistance is computed using the average speed, (V1 + V2)/2 as an approximation of V (with a V2 of 20 mi/h): value on good, wet pavement is a coefficient of road adhesion of 0.6 (ȝ = 0.6) from Table 2.4. Applying Eq. 2.43 with ȝ = 0.6, a 3% grade (so sinθg § 0.03), and γb = 1.04,
2.9 Principles of Braking
37
Because a braking efficiency of 1.33 is not possible, the driver would hit the deer at a higher speed if the wheels were locked the entire distance. Note that to achieve a deer-impact speed of 20 mi/h or less, ηbȝ in Eq. 2.43 must be 0.8 (1.33×0.6) or greater. The maximum coefficient of road adhesion for good, wet pavements is 0.9 (from Table 2.4). So, to achieve a deer-impact speed of 20 mi/h or less (with ȝ = 0.9),
Thus a braking efficiency of at least 89% is needed (if the vehicle has a functional antilock braking system) in order to achieve a deer-impact speed of 20 mi/h or less.
2.9.5 15B15B15BPractical Stopping Distance
As mentioned earlier, one of the most critical concerns in the design of a highway is the provision of adequate driver sight distance to permit a safe stop. The theoretical assessment of vehicle stopping distance presented in the previous section provided the principles of braking for an individual vehicle under specified roadway surface conditions. However, highway engineers face a more complex problem because they must design for a variety of driver skill levels (which can affect whether or not the brakes lock and reduce the coefficient of road adhesion to slide values), vehicle types (with varying aerodynamics, weight distributions, and brake efficiencies), and weather conditions (which change the roadway’s coefficient of adhesion). As a result of the wide variability inherent in the determination of braking distance, an equation is required that provides an estimate of typical observed braking distances and is more simplistic and usable than Eq. 2.42.
The basic physics equation on rectilinear motion, assuming constant deceleration, is chosen as the basis of a practical equation for stopping distance:
2 2
a = acceleration (negative for deceleration) in ft/s2, and d = deceleration distance (practical stopping distance) in ft.
Rearranging Eq. 2.44 and assuming a is negative for deceleration gives
2 2
To make this equation generally applicable for design purposes, a deceleration rate, a, must be chosen that is representative of appropriately conservative braking behavior. AASHTO [2011] recommends a deceleration rate of 11.2 ft/s2. Empirical studies [Fambro et al. 1997] have shown that approximately 90% of drivers decelerate at rates greater than this, and that this deceleration rate is well within a driver’s capability to maintain steering control during a braking maneuver on wet surfaces. Additionally, empirical studies [Fambro et al. 1997] have confirmed that most vehicle braking systems and tire-pavement friction levels are capable of supporting this deceleration rate, even under wet conditions.
To account for the effect of grade, Eq. 2.46 is modified as follows:
12
2 d V
g a G
g
= §§ · ·
¨¨ ¸± ¸
©© ¹ ¹
(2.47)
where
g = gravitational constant, 32.2 ft/s2,
G = roadway grade (+ for uphill, − for downhill) in percent/100, and Other terms are as defined previously.
It is important to note that Eq. 2.47 is consistent with Eq. 2.43 (the theoretical stopping distance ignoring aerodynamic resistance). Rewriting Eq. 2.43 with the assumption that the vehicle comes to a stop (V2 = 0), that sin θg = tan θg = G (for small grades), and that γb and frl can be ignored due to their small and essentially offsetting effects, we have
( )
12
2 b
S V
g η µ G
= ± (2.48)
Recall that ηbµ = gmax (Eq. 2.33). However, rather than determining the maximum deceleration rate (in g’s) for a specific vehicle braking efficiency and specific coefficient of road adhesion, the AASHTO-recommended maximum deceleration rate (again, an appropriately conservative value for the overall driver and vehicle population) is used. Thus, a maximum deceleration of 0.35 g’s (11.2/32.2) is used for Eq. 2.47.
The recommended deceleration rate as determined empirically already accounts for the effects of aerodynamic resistance, braking efficiency, coefficient of road adhesion, and inertia during braking (the braking mass factor). This value reflects current vehicle technologies and driving behavior. It is important to recognize that as vehicle braking technology and other vehicle characteristics change, as well as possibly driver behavior, the recommended value of a should be reviewed to determine if it is still applicable for highway design purposes. The relationship between changing vehicle characteristics and changing highway design guidelines is one that must always be kept in the design engineer’s mind.
2.9 Principles of Braking
39
EXAMPLE 2.11 BRAKING EFFICIENCY AND STOPPING DISTANCEA car [W = 2200 lb, CD = 0.25, Af = 21.5 ft2] has an antilock braking system that gives it a braking efficiency of 100%. The car’s stopping distance is tested on a level roadway with poor, wet pavement (with tires at the point of impending skid), and ρ = 0.00238 slugs/ft3. How inaccurate will the stopping distance predicted by the practical-stopping-distance equation be compared with the theoretical stopping distance, assuming the car is initially traveling at 60 mi/h? How inaccurate will the practical-stopping-distance equation be if the same car has a braking efficiency of 85%?
SOLUTION
First, to calculate the theoretical minimum stopping distance, Eq. 2.42 is applied with γb = 1.04, θg = 0°, µ = 0.60 (maximum for poor, wet pavement, from Table 2.4), and
2 32.2 0.0064 1.0 0.60 2200 0.013 2200 0
For the same conditions but with a vehicle braking efficiency of 85%, Eq. 2.42 gives
( )
( )( ) ( )
( )( )( ) ( )( )
1.04 2200 0.0064 60 5280 / 3600 2
ln 1
2 32.2 0.0064 0.85 0.60 2200 0.013 2200 0 234.11ft
Now applying Eq. 2.46 (since G = 0) for the practical stopping distance, we find
( )
In the first case, the error is 145.36 ft. In the case of 85% braking efficiency, the error is 111.60 ft. Rearranging Eq. 2.46 to solve for a, we find that stopping distances of 200.35 ft and 234.11 ft correspond to deceleration rates of 19.33 ft/s2 and 16.54 ft/s2, respectively.
Studies [Fambro et al. 1997] have shown that most drivers decelerate at rates of 18.4 ft/s2 or greater in emergency stopping situations. Thus, this range of theoretical values is consistent with observed distances for situations in which minimum stopping distances are being attempted. Comparing these theoretical values to the AASHTO-recommended deceleration rate of 11.2 ft/s2, it is readily apparent that a considerable level of conservatism is built into the deceleration rate for practical stopping distance.
2.9.6 16B16B16BDistance Traveled During Driver Perception/Reaction
Until now the focus has been directed toward the distance required to stop the vehicle from the point of brake application. However, in providing sufficient sight distance for a driver to stop safely, it is also necessary to consider the distance traveled during the time the driver is perceiving and reacting to the need to stop. The distance traveled during perception/reaction (dr) is given by
1
r r
d V t= × (2.49)
where
V1 = initial vehicle speed in ft/s, and
tr = time required to perceive and react to the need to stop, in s.
The perception/reaction time of a driver is a function of a number of factors, including the driver’s age, physical condition, and emotional state, as well as the complexity of the situation and the strength of the stimuli requiring a stopping action.
For highway design, a conservative perception/reaction time has been determined to be 2.5 seconds [AASHTO 2011]. For comparison, average drivers have perception/reaction times of approximately 1.0 to 1.5 seconds.
Thus, the total required stopping distance is a combination of the braking distance, either theoretical (Eq. 2.42 or 2.43) or practical (Eq. 2.47), and the distance traveled during perception/reaction (Eq. 2.49), as shown in Eq. 2.50:
s r
d d d= + (2.50)
where
ds = total stopping distance (including perception/reaction) in ft, d = distance traveled during braking in ft, and
dr = distance traveled during perception/reaction in ft.
The combination of practical stopping distance and the distance traveled during perception/reaction is a primary consideration in highway design, as will be discussed in detail in Chapter 3.
EXAMPLE 2.12 PRACTICAL STOPPING DISTANCE AND PERCEPTION/REACTION