APPENDIX A

This Appendix explains how to derive an adjustment factor for non-preventable crashes in Chapter V Benefits. Chapter V uses the change in delta-V to estimate benefits from correcting the underinflated tire pressures. Change in delta-V at a given traveling distance is defined to be one half of the velocity difference between two scenarios: a vehicle with correct tire pressure and without. The total traveling distance of a vehicle after braking under the correctly inflated tire pressures is defined as the correct stopping distance. The incorrect stopping distance is the total traveling distance of a vehicle with underinflated tire pressures. Change in delta-V increases with the traveling distance after braking. Figure A-1 depicts a simplified curve relationship between change in delta-V and the traveling distance for illustration. The curvature varies with initial speed, deceleration, and traveling distance. For non-preventable crashes, the maximum change in delta-V occurs at the correct stopping distance. Therefore, applying the change in delta-V at this level to the total applicable baseline population would overestimate the benefits from correcting tire pressures for non-preventable crashes.

Figure A-1 Generalized Relationship Between Change in Delta-V and Traveling Distance

Ideally, the benefits would be estimated by applying the change in delta-V at any given traveling distance to the corresponding baseline population. However, the change in delta-V varies with the initial traveling speeds, deceleration, and traveling distance. There are too many initial traveling speeds and deceleration (or stopping distance) combinations to be exhaustively analyzed. Less ideally, the benefits would be estimated by applying the expected changes in delta-V to the applicable baseline population. This approach, too, encounters the same obstacles as in the ideal approach. In addition, the expected changes in delta-V might be fractions of 1 mile/hour, e.g., 0.1, 0.01 mile/hour. This would result in infinite ways to segment the measurement units for the delta-V based injury probability curves. Thus, this analysis only uses a weighted average initial speed, correct stopping distance, and incorrect stopping distance to estimate the expected change in delta-V with respect to the traveling distance. The expected change in delta-V can be considered as the mean for continuous variables (i.e., traveling distance). The adjustment factor is the ratio of the expected change in delta-V and change in delta-V at the correct stopping distance. The following two sections describe the process in detail. In the last section, a sensitivity analysis examines several scenarios to estimate the impact of different initial traveling speeds, decelerations, and stopping distances on the adjustment factors. Note, all the equations and functions were derived assuming constant decelerations.

Expected Change in Delta-V

The expected change in delta-V is an integral of the product of two functions: the probability density function of a non-preventable crash occurrence and the change in delta-V at any given traveling distance d. The change in delta-V function is one half of the velocity change function. This assumes a collision between two identical masses. This assumption is the best that can be done without performing a more detailed kinematic analysis of many different crash scenarios and applying weights to the results of each scenario. Insufficient information is known about the distribution of crash scenarios to perform such an analysis. The expected change in delta-V (EDV) is:

-------- (1)

Where, SD_c= the correct stopping distance

u(d) = the probability density function of a crash occurrence

DV(d) = the velocity change function.

The function _{[if !vml]>} is the change in delta-V. Note that the measurement unit among variables has to be consistent for Equation 1 and the rest of the equations. For example, if “feet-second” measurement is used, then the velocity, deceleration, and traveling distance have all been based on feet-second unit.

Probability Density Function

Assuming that the non-preventable crashes occurred uniformly at any traveling distance between the initial braking (d=0) and the correct distance, the probability density function u(d) has the property that. By solving this equation, u(d) is a constant function:

_---------(2)

where, SD_c = the correct stopping distance

Change in Delta-V Function

Change in delta-V function between two tire pressure conditions is half of the velocity change function at any traveling distance d. The velocity change function is

-------- (3)

Where, v_i = velocity with incorrect tire pressure

v_c = velocity with correct tire pressure

SD_c = the correct stopping distance

At any given traveling distance d, the velocity under a constant deceleration can be derived based on the following formula: , where v_o is the initial traveling speed and a the deceleration. Let variables a_i and a_c represent the deceleration under incorrect and correct tire pressure, respectively. Then,

-------- (4)

At the stopping distance, a braking vehicle has 0 velocity. Thus, for incorrect tire pressure:

and

--------(5)

where:

a_i = the deceleration with incorrect tire pressure

SD_i = the stopping distance with incorrect tire pressure.

Similarly, the deceleration formula for braking vehicles with the correct tire pressure is

-------- (6)

where:

a_c = deceleration with correct tire pressure

SD_c = the stopping distance with correct tire pressure

By substituting the right side of Equations 5 and 6 for a_i and a_c into Equation 4, the velocity change function can be rewritten as a function of initial traveling distance.

-------- (7)

Change in delta-V at any given traveling distance d would be

------- (8)

For passenger cars, the weighted average initial traveling speed, correct stopping distance, and incorrect stopping distances are:

V₀ = 45.078 mile/hour = 66.114 feet/second

SD_c = 132.080 feet

SD_i = 136.988 feet

At any given traveling distance, the change in delta-V is calculated by substituting these numbers into Equation 8. Figure A-2 shows the change in delta-V by traveling distance. At the correct stopping distance of 132.080 feet, for example, the change in delta-V DV(132.080) is:

Figure A-2. Change in Delta-V by Traveling Distance

Passenger Cars

For light trucks and vans, the weighted average initial traveling speed, correct stopping distance, and incorrect stopping distances are:

V₀ = 45.078 mile/hour = 66.114 feet/second

SD_c = 127.344 feet

SD_i = 131.530 feet

Figure A-3 shows the change in delta-V by traveling distance. At the correct stopping distance of 127.344 feet, the change in delta-V DV(127.344) is:

Figure A-3. Change in Delta-V by Traveling Distance

Light Trucks/Vans

After calculating the change in delta at the correct stopping distance, the expected change in delta-V must be derived to calculate the adjustment ratio.

Expected Change in Delta-V

The expected change in delta-V is an integral of the product of the probability density function of a non-preventable crash occurrence and the change in delta-V at any given traveling distance d. The crash probability density function (Equation 2) is a constant function as described in the previous section. With known correct and incorrect stopping distances and under a constant deceleration condition, the change in delta-V function is a function of the traveling distance (Equation 8). Substituting these two equations back to Equation 1, the expected change in delta-V function can be rewritten as:

Where, c₀ and c₁ are constants.

For passenger cars, the expected change in delta-V is:

For light trucks/vans, the expected change in delta-V is:

Adjustment Factors

The adjustment factor is the ratio of expected change in delta-V and change in delta-V at the correct stopping distance, i.e.

For passenger cars, under the following set of conditions:

the initial traveling speed V₀ = 45.078 mile/hour = feet/second,

the correct stopping distance SD_c = 132.080 feet, and

the incorrect stopping distance SD_i = 136.988 feet,

EDV = 0.453 mile/hour and DV(132.080)= 4.267 mile/hour.

The adjustment factor is 0.11 (= 0.453/4.267).

For light trucks and vans, under the following set of conditions:

the initial traveling speed V₀ = 45.078 mile/hour = feet/second,

the correct stopping distance SD_c = 127.344 feet, and

the incorrect stopping distance SD_i = 131.530 feet,

EDV = 0.406 mile/hour and DV(127.344) = 4.022 mile/hour.

The adjustment factor is 0.10 (= 0.406/4.022).

Sensitivity Study

The sensitivity study examines the variations of the adjustment factors under 12 different scenarios – combinations of three initial traveling speeds (35, 49, and 62 mph), two vehicle types (passenger cars, light trucks/vans), and two roadway conditions (dry, wet). Table A-1 lists the criteria of these 12 scenarios and the associated stopping distances and case weights. Readers can refer to Chapter V for detailed explanations on how the initial traveling speeds, stopping distances, and weights were derived for these 12 scenarios. Table A-1 also lists the calculated change in delta-V at the correct stopping distance, the expected change in delta-V, and the adjustment factor for each scenario. The adjustment factors range from 5 to 13 percent. It's not surprising that the adjustment factors are smaller for dry pavement roadways with a lower traveling speed. The overall weighted adjustment factor is about 9 percent which is slightly smaller than the overall 11 percent. Rounding errors contribute to the discrepancy.

Table A-1. Adjustment Factors and Related Statistics

	Initial Traveling Speed (mile/hour)	Correct Stopping Distance (feet)	Incorrect Stopping Distance (feet)	Change in Delta-V at the Correct Stopping Distance (mile/hour)	Expected Change in Delta-V (mile/hour)	Case Weights	Adjustment Factor
Passenger Cars, Dry Pavement
1	35	54.325	54.816	1.655	0.095	0.2920	0.06
2	49	100.869	103.072	3.581	0.305	0.3404	0.09
3	62	180.813	188.528	6.271	0.703	0.1110	0.11
Passenger Cars, Wet Pavement
4	35	117.356	121.753	3.327	0.354	0.1245	0.11
5	49	273.039	285.697	5.157	0.598	0.0963	0.12
6	62	582.906	621.712	7.745	1.032	0.0359	0.13
Light Trucks/Vans, Dry Pavement
7	35	53.936	54.284	1.402	0.069	0.2920	0.05
8	49	98.780	100.624	3.317	0.264	0.3404	0.08
9	62	173.642	180.026	5.836	0.617	0.1110	0.11
Light Trucks/Vans, Wet Pavement
10	35	112.792	116.778	3.235	0.336	0.1245	0.10
11	49	260.274	271.487	4.980	0.561	0.0963	0.11
12	62	546.161	578.827	7.362	0.942	0.0359	0.13