**New Alpha Factor Determination as a Function of**

**Number of Wheels and Number of Coverages**

**Letter Report**

**Prepared**

**by**

**Gordon F. Hayhoe**

**Airport Technology Research and Development Branch**

**AJP 6310 (AAR-410)**

**Submitted**

**to**

**Associate Administrator for Airports**

**Office of the Director, Airport Safety and Standards**

**AAS-1**

**November 16, 2006**

[1.  INTRODUCTION]{.underline}.

The International Civil Aviation Organization (ICAO) has proposed that
the Alpha Factors for four- and six-wheel gears at 10,000 Coverages be
redefined for use in calculating the Aircraft Classification Number
(ACN) of airplanes operating on flexible pavements. The proposed Alpha
Factors are 0.800 and 0.720 and the existing values are 0.825 and 0.788,
with the new values being based on an analysis of a recently expanded
set of full-scale test data for four- and six-wheel traffic. With these
changes, Alpha Factors at other numbers of wheels, and over the full
range of coverages for design, also require redefinition to maintain
consistency in the CBR (California Bearing Ratio) procedure for flexible
pavement thickness design as required for ACN calculations, even though
additional full-scale test data is not available for other than four-
and six-wheel gears.

A new set of Alpha Factors at 10,000 Coverages is defined first based on
recommendations contained in ICAO AOSWG/4 Discussion Paper No. 21. A new
set of functions is then defined to allow the calculation of Alpha
Factor over the full range of Coverages expected in flexible pavement
thickness design for commercial airplane operations. The new set of
functions defining Alpha Factor versus Coverages is compatible with the
new set of Alpha Factors at 10,000 Coverages. The functions are also
compatible, to the extent possible, with existing definitions of Alpha
Factor versus Coverages.

[2.  ALPHA FACTOR AT 10,000 COVERAGES VERSUS NUMBER OF
WHEELS]{.underline}.

The Alpha Factors embedded in ICAO Computer Program No. 2 for
calculating flexible pavement ACN values for 1, 2, 4, 6, 8, 12, 18, and
24 wheels (reference 1) are listed in table 1 and shown graphically in
figure 1.

Table 1.  Alpha factors embedded in ICAO Computer Program No. 2,
Flexible Pavement ACN.

  ---------------------- ------------------------------------------------
  Number of Wheels       Alpha Factor from ICAO Computer Program No. 2

  1                      0.995

  2                      0.900

  4                      0.825

  6                      0.788

  8                      0.755

  12                     0.722

  18                     0.700

  24                     0.689
  ---------------------- ------------------------------------------------

![](media/image1.wmf){width="4.5368055555555555in"
height="3.567361111111111in"}

Figure 1.  Alpha factors embedded in ICAO Computer Program No. 2,
Flexible Pavement ACN, with recommended changes for 4- and 6-wheel
airplanes.

ICAO AOSWG/4 Discussion Paper No. 21 (DP21) recommends that the
following change be made to the Alpha Factors for the computation of ACN
values:

-   Change the Alpha Factors for all 4-wheel and 6-wheel aircraft (0.80
    and ~~0.71/~~0.72), and alter the Alpha Factors for other wheel
    numbers so that the ranking of damaging effect is consistent. ACNs
    for all aircraft would change, and the Alpha Factors would be
    technically correct.

Figure 1 also shows the new Alpha Factors for 4- and 6-wheel airplanes
as well as the original Alpha Factors.

The Alpha Factors in table 1 were originally derived from curves of
Alpha Factor versus Coverages published in references 2 and 3 for gears
with 1, 2, 4, 6, 8, 12 18, and 24 wheels. Reference 2 published the 1-,
4-, and 12-wheel curves based on full-scale test data for single, dual
tandem, and two C-5 six-wheel gears in tandem (for the 12-wheel gear
curve). Reference 3 added the additional curves by interpolation (for
2-, 6-, and 8-wheel gears) and extrapolation (for 18- and 24-wheel
gears). Extrapolation outside the range of the existing full-scale test
data was important because the CBR design procedure as defined in
reference 4 requires that pavement thickness be computed for all
combinations of wheels in the main landing gear, and that the largest of
these thicknesses be used for design. Quoting the relevant passage from
reference 4 (pages 70 and 71):

"The *α~i~* curves have been developed using the load on one landing
gear assembly of an aircraft. In actual practice, designs are to be
based upon the load on all main tires. Therefore, the *α~i~* versus
passes curves (figure 82) show the total number of aircraft main gear
tires represented by the gear type. Use of the criteria, therefore, is
accomplished by determining the ESWL and the *α~i~* for all main gear
tires, which generally results in maximum thickness requirements for a
specific aircraft. However, where it is shown that some combination or
grouping of tires other than all main gear tires will produce a greater
thickness requirement, then the other combination or group will be
used."

Reference 3 (page 27) contains a similar statement:

"The number of tires used to compute the ESWL is that combination of
tires which gives the greatest ESWL ..."

These two statements are not necessarily consistent, but the intent is
clear. In particular, the pavement thickness design charts for the B747
airplane published in Federal Aviation Administration (FAA) AC
150/5320-6C and -6D are based on equivalent single wheel loads (ESWLs)
computed using 16 wheels and Alpha Factor curves for 16 wheels.

Looking at figure 1, it is clear that, in order to integrate the new
Alpha Factors (0.8 and 0.72) into the Aircraft Classification Number --
Pavement Classification Number (ACN-PCN) system, adjustments need to be
made to the Alpha Factors for wheel numbers other than 4 and 6. The
1-wheel Alpha Factor is based on a reasonably large set of full-scale
tests (see reference 2) and there does not appear to be a compelling
reason to change it. The 2-wheel Alpha Factor was based on interpolation
between the 1-wheel and 4-wheel Alpha Factor curves and it seems
reasonable to also not change its value, although a reduction in its
value proportionate to the reduction of the 4-wheel value from 0.825 to
0.80 would also be reasonable. That is, a new Alpha Factor for 2 wheels
which has the same proportions relative to the new 1- and 4-wheel Alpha
Factors as the old 2-wheel Alpha Factor has to the old 1- and 4-wheel
Alpha Factors could be defined by:

$\begin{array}{r}
\alpha_{2\text{new}} = 0\text{.}\text{995} - \frac{(0\text{.}\text{995} - 0\text{.}\text{900}) \times (0\text{.}\text{995} - 0\text{.}\text{800})}{(0\text{.}\text{995} - 0\text{.}\text{825})} \\
0\text{.}\text{886}
\end{array}$ (1)

The remaining Alpha Factors, for 8 wheels and above, have no basis in
full-scale test data except that the C5 test data had originally been
treated as a 12-wheel gear. This formed the basis for the old 12-wheel
Alpha Factor with the old 6-wheel Alpha Factor being an interpolation
between the 4- and 12-wheel values. But after test data had been
collected with a triple-dual-tandem (3D) gear at higher coverage levels
than for the C5 tests, the C5 was re-categorized as two 6-wheel gears in
tandem so that the two data sets could be combined (see reference 5 for
more details). As stated above, ICAO AOSWG/4 DP 21 recommends that
undefined new Alpha Factors be altered "so that the ranking of damaging
effect is consistent." The Alpha Factors for 8 wheels and above have
therefore been adjusted so that the ratio of ACN relative to the ACN for
a 6-wheel gear is, to a very good approximation, the same in the new
system as in the old. The equation used to calculate the new Alpha
Factors is:

$\alpha_{\text{Inew}} \approx \left( \frac{\alpha_{\text{Iold}}}{\alpha_{6\text{old}}} \right) \times \alpha_{6\text{new}}$
(2)

where:

  ---------- ----------------------------------------------------------------
  *~Iold~* = old Alpha Factor for *I* number of wheels as listed in table 1
             for *I* = 8, 12, 18, and 24

  *~6old~* = old Alpha Factor for 6 wheels = 0.788 as listed in table 1

  *~6new~* = new Alpha Factor for 6 wheels = 0.72
  ---------- ----------------------------------------------------------------

Equation 2 is derived in a separate section below.

Table 2 shows the new alpha factors as defined by full-scale test data
(1, 2, 4, and 6 wheels) and from equation 2 (8, 12, 18, and 24 wheels).

Table 2.  New Alpha factors at 10,000 Coverages.

  ---------------------- ------------------------------------------------
  Number of Wheels       New Alpha Factor

  1                      0.995

  2                      0.900

  4                      0.800

  6                      0.720

  8                      0.690

  12                     0.660

  18                     0.640

  24                     0.630
  ---------------------- ------------------------------------------------

In order to determine values of Alpha Factor for numbers of wheels other
than those in table 2, a curve fitting or interpolation scheme is
needed. A curve fitting computer program (CurveExpert 1.3) was used to
find equations describing the relationship between Alpha Factor (at
10,000 Coverages) versus number of wheels. The best fit equation for the
old Alpha Factors had the form

$\alpha = a\text{.}b^{I}\text{.}I^{c}$ (3)

where:

  --------------------- -------------------------------------------------
  =                     Alpha Factor

  *I* =                 number of wheels

  *a*, *b*, and *c* =   curve fitting coefficients
  --------------------- -------------------------------------------------

Figure 2 shows the old and new Alpha Factors with curve fits for
equation 3 also shown. The curve fit for the old Alpha Factors is very
good, but that for the new Alpha Factors is poor, particularly over the
critical range of 2 to 12 wheels. This curve fit cannot therefore be
used to compute Alpha Factor without either giving inaccurate values for
the defined points (when the defined points are computed with the curve
fit equation) or giving inaccurate relative values between undefined and
defined points (when the undefined points are computed with the curve
fit equation and the defined values are used for the defined points).
Other equations give better correlation coefficients for the new Alpha
Factors, but none satisfactorily solve the problem described in the
previous sentence. A simple linear interpolation is therefore proposed
for computing the Alpha Factor values for numbers of wheels between
those defined in table 2.

![](media/image2.wmf){width="4.5368055555555555in"
height="3.5708333333333333in"}

Figure 2.  Old and new Alpha Factors with curve fits.

[2.1  DERIVATION OF EQUATION 3]{.underline}.  The CBR equation used for
thickness design in the CBR method for airport pavements is:

$t^{2} = \alpha^{2}\left( \frac{P}{8\text{.}1\text{CBR}} - \frac{A}{\pi} \right) = \alpha^{2}\left( \frac{P\text{.}\pi - 8\text{.}1\text{CBR}\text{.}}{8\text{.}1\text{CBR}\text{.}\pi} \right)$
(4)

Where:

$$\begin{array}{r}
t = \text{ total pavement thickness, in} \\
\alpha = \text{ load repetition factor }(\text{alpha factor}) \\
\text{CBR} = \text{ subgrade strength} \\
P = \text{ equivalent single wheel load }(\text{ESWL}) \\
A = \text{ contact area of one tire on the gear, in}^{2}
\end{array}$$

To keep the ratio of ACN for *I* wheels to ACN for 6 wheels in the new
system to be the same as in the old system:

$\frac{\text{ACN}_{\text{Inew}}}{\text{ACN}_{6\text{new}}} = \frac{\text{ACN}_{\text{Iold}}}{\text{ACN}_{6\text{old}}}$
(5)

The definition of ACN (reference 1) is:

$\text{ACN} = \frac{t_{\text{ac}}^{2}}{t_{\text{ref}}^{2}}$ (6)

Where:

$$\begin{array}{r}
t_{\text{ac}} = \text{ total pavement design thickness for the aircraft in question} \\
t_{\text{ref}} = \text{ total pavement design thickness for a reference single wheel load}
\end{array}$$

and therefore:

$\frac{\text{ACN}_{I}}{\text{ACN}_{6}} = \frac{t_{I}^{2}}{t_{\text{ref}}^{2}}\frac{t_{\text{ref}}^{2}}{t_{6}^{2}} = \frac{t_{I}^{2}}{t_{6}^{2}}$
(7)

Substituting for *t* from equation 4 (the CBR equation):

$\begin{array}{r}
\frac{\text{ACN}_{I}}{\text{ACN}_{6}} = \frac{\alpha_{I}^{2}}{\alpha_{6}^{2}}\frac{\left( P_{I}\text{.}\pi - 8\text{.}1\text{CBR}\text{.}A_{I} \right)}{8\text{.}1\text{CBR}\text{.}\pi}\frac{8\text{.}1\text{CBR}\text{.}\pi}{\left( P_{6}\text{.}\pi - 8\text{.}1\text{CBR}\text{.}A_{6} \right)} \\
\frac{\alpha_{I}^{2}}{\alpha_{6}^{2}}\frac{\left( P_{I}\text{.}\pi - 8\text{.}1\text{CBR}\text{.}A_{I} \right)}{\left( P_{6}\text{.}\pi - 8\text{.}1\text{CBR}\text{.}A_{6} \right)}
\end{array}$ (8)

The second terms in the parentheses are small compared to the first
terms, and:

$\frac{\text{ACN}_{I}}{\text{ACN}_{6}} \approx \frac{\alpha_{I}^{2}}{\alpha_{6}^{2}}\frac{P_{I}}{P_{6}}$
(9)

Now, from equation 5:

$\begin{array}{r}
\frac{\alpha_{\text{Inew}}^{2}}{\alpha_{6\text{new}}^{2}}\frac{P_{\text{Inew}}}{P_{6\text{new}}} \approx \frac{\alpha_{\text{Iold}}^{2}}{\alpha_{6\text{old}}^{2}}\frac{P_{\text{Iold}}}{P_{6\text{old}}} \\
\therefore\ \alpha_{\text{Inew}}^{2} \approx \frac{\alpha_{\text{Iold}}^{2}}{\alpha_{6\text{old}}^{2}}\frac{P_{\text{Iold}}}{P_{6\text{old}}}\frac{P_{6\text{new}}}{P_{\text{Inew}}}\alpha_{6\text{new}}^{2}
\end{array}$ (10)

This equation does not have an explicit solution because *P* changes
with pavement thickness and pavement thickness changes with Alpha
Factor. However, as is demonstrated below in a numerical example, the
term
$\frac{P_{\text{Iold}}}{P_{6\text{old}}}\frac{P_{6\text{new}}}{P_{\text{Inew}}}$
has a value close to one and, to a first approximation, equation 10 can
be simplified to

$\alpha_{\text{Inew}} \approx \frac{\alpha_{\text{Iold}}}{\alpha_{6\text{old}}}\alpha_{6\text{new}}$

Which is the relationship used above (equation 2) to generate the new
Alpha Factors listed in table 2 for 8, 12, 18, and 24 wheels.

[2.2  NUMERICAL EXAMPLES]{.underline}.  Two numerical examples are
worked through to establish how well the new Alpha Factors for 8, 12,
18, and 24 wheels maintain the same relative ACN ratios as in the old
system. The first example compares two C5 airplane gear configurations.
The first configuration consists of one six-wheel gear and the second
configuration consists of two six-wheel gears in tandem. The two-gear
configuration is treated as a twelve-wheel gear for ACN computation. The
second example compares a triple-dual-tandem (3D) six-wheel
configuration with a 5D ten-wheel configuration. Both configurations
have the same dual and tandem wheel spacing and the 3D configuration is
close to B777-300 gear geometry and wheel loading. These two examples
represent two extremes of wheel configuration and wheel loading.

[2.2.1  Example 1 -- C5 Landing Gear]{.underline}.  Figures 3 and 4 show
the wheel configurations for the six- and twelve-wheel cases. Gear
geometry is standard and wheel loads are 30,000 lbs (133 kN). COMFAA was
used to calculate ACNs with both the old and the new Alpha Factors.
Tables 3 and 4 list the results for 3 CBR and 15 CBR, together with
pavement thicknesses and equivalent single-wheel loads (ESWLs). The ACN
ratios for old and new Alpha Factors are summarized in table 5.

![](media/image3.png){width="5.458333333333333in" height="4.0875in"}

Figure 3.  COMFAA screen shot showing C5 single-gear six-wheel
configuration.

![](media/image4.png){width="5.458333333333333in" height="4.09375in"}

Figure 4.  COMFAA screen shot showing C5 double-gear twelve-wheel
configuration.

Table 3.  C5 ACN results for old Alpha Factors

+-------------+--------+----------+------------+-----------+---------+
| Gear        | CBR    | ACN      | Pavement   | Pavement  | ESWL,   |
| Co          |        |          | Thickness, | Thickness |         |
| nfiguration |        |          | in         | Squared,  | lbs     |
|             |        |          |            | in^2^     |         |
+-------------+--------+----------+------------+-----------+---------+
| 12-Wheel    | 3      | 61.7     | 51.75      | 2,678     | 127,111 |
+-------------+--------+----------+------------+-----------+---------+
| 6-Wheel     | 3      | 58.7     | 50.47      | 2,547     | 101,903 |
+-------------+--------+----------+------------+-----------+---------+
| Ratio       |        | 1.051    | 1.025      | 1.051     | 1.247   |
| 12Whl:6Whl  |        |          |            |           |         |
+-------------+--------+----------+------------+-----------+---------+
|             |        |          |            |           |         |
+-------------+--------+----------+------------+-----------+---------+
| 12-Wheel    | 15     | 30.9     | 14.84      | 220.2     | 61,702  |
+-------------+--------+----------+------------+-----------+---------+
| 6-Wheel     | 15     | 31.3     | 14.96      | 223.8     | 54,280  |
+-------------+--------+----------+------------+-----------+---------+
| Ratio       |        | 0.987    | 0.992      | 0.984     | 1.137   |
| 12Whl:6Whl  |        |          |            |           |         |
+-------------+--------+----------+------------+-----------+---------+

Table 4.  C5 ACN results for new Alpha Factors

+-------------+--------+----------+------------+-----------+---------+
| Gear        | CBR    | ACN      | Pavement   | Pavement  | ESWL,   |
| Co          |        |          | Thickness, | Thickness |         |
| nfiguration |        |          | in         | Squared,  | lbs     |
|             |        |          |            | in^2^     |         |
+-------------+--------+----------+------------+-----------+---------+
| 12-Wheel    | 3      | 46.7     | 45.04      | 2,029     | 115,349 |
+-------------+--------+----------+------------+-----------+---------+
| 6-Wheel     | 3      | 44.7     | 44.06      | 1,941     | 93,257  |
+-------------+--------+----------+------------+-----------+---------+
| Ratio       |        | 1.045    | 1.022      | 1.045     | 1.237   |
| 12Whl:6Whl  |        |          |            |           |         |
+-------------+--------+----------+------------+-----------+---------+
|             |        |          |            |           |         |
+-------------+--------+----------+------------+-----------+---------+
| 12-Wheel    | 15     | 24.5     | 13.23      | 175.0     | 59,126  |
+-------------+--------+----------+------------+-----------+---------+
| 6-Wheel     | 15     | 25.2     | 13.42      | 180.1     | 52,403  |
+-------------+--------+----------+------------+-----------+---------+
| Ratio       |        | 0.972    | 0.986      | 0.972     | 1.128   |
| 12Whl:6Whl  |        |          |            |           |         |
+-------------+--------+----------+------------+-----------+---------+

Table 5.  ACN and ESWL ratios (12Whl:6Whl) for C5 gears (from tables 3
and 4)

+-----------+----------------+----------------+-----------+----------+
| CBR       | ACN Ratio with | ACN Ratio with | Percent   | $        |
|           |                |                |           | $\sqrt{\ |
|           | Old Alpha      | New Alpha      | D         | frac{P_{ |
|           | Factors        | Factors        | ifference | \text{Io |
|           |                |                |           | ld}}}{P_ |
|           |                |                |           | {6\text{ |
|           |                |                |           | old}}}\f |
|           |                |                |           | rac{P_{6 |
|           |                |                |           | \text{ne |
|           |                |                |           | w}}}{P_{ |
|           |                |                |           | \text{In |
|           |                |                |           | ew}}}}$$ |
+-----------+----------------+----------------+-----------+----------+
| 3         | 1.051          | 1.045          | 0.6       | 1.0042   |
+-----------+----------------+----------------+-----------+----------+
| 15        | 0.987          | 0.972          | 1.5       | 1.0037   |
+-----------+----------------+----------------+-----------+----------+

[2.2.2  Example 2 -- 3D and 5D Landing Gear]{.underline}.  Figures 5 and
6 show the wheel configurations for the six- and ten-wheel 3D and 5D
gears. Gear geometry is 54 inches dual spacing and 57 inches tandem
spacing. Wheel loads are 55,000 lbs (245 kN). Tables 6 and 7 list the
results for 3 CBR and 15 CBR, together with pavement thicknesses and
equivalent single-wheel loads (ESWLs). The ACN ratios for old and new
Alpha Factors are summarized in table 8.

![](media/image5.png){width="5.458333333333333in" height="4.0875in"}

Figure 5.  COMFAA screen shot showing 3D six-wheel gear.

![](media/image6.png){width="5.466666666666667in" height="4.10625in"}

Figure 6.  COMFAA screen shot showing 5D ten-wheel gear.

Table 6.  3D and 5D ACN results for old Alpha Factors

+-------------+--------+----------+------------+-----------+---------+
| Gear        | CBR    | ACN      | Pavement   | Pavement  | ESWL,   |
| Co          |        |          | Thickness, | Thickness |         |
| nfiguration |        |          | in         | Squared,  | lbs     |
|             |        |          |            | in^2^     |         |
+-------------+--------+----------+------------+-----------+---------+
| 10-Wheel    | 3      | 174.1    | 86.92      | 7,555     | 338,278 |
+-------------+--------+----------+------------+-----------+---------+
| 6-Wheel     | 3      | 137.8    | 77.35      | 5,983     | 235,986 |
+-------------+--------+----------+------------+-----------+---------+
| Ratio       |        | 1.263    | 1.124      | 1.263     | 1.433   |
| 10Whl:6Whl  |        |          |            |           |         |
+-------------+--------+----------+------------+-----------+---------+
|             |        |          |            |           |         |
+-------------+--------+----------+------------+-----------+---------+
| 10-Wheel    | 15     | 81.1     | 24.06      | 578.9     | 139,310 |
+-------------+--------+----------+------------+-----------+---------+
| 6-Wheel     | 15     | 72.5     | 22.75      | 517.6     | 111,887 |
+-------------+--------+----------+------------+-----------+---------+
| Ratio       |        | 1.119    | 1.0576     | 1.118     | 1.245   |
| 10Whl:6Whl  |        |          |            |           |         |
+-------------+--------+----------+------------+-----------+---------+

Table 7.  3D and 5D ACN results for new Alpha Factors

+-------------+--------+----------+------------+-----------+---------+
| Gear        | CBR    | ACN      | Pavement   | Pavement  | ESWL,   |
| Co          |        |          | Thickness, | Thickness |         |
| nfiguration |        |          | in         | Squared,  | lbs     |
|             |        |          |            | in^2^     |         |
+-------------+--------+----------+------------+-----------+---------+
| 10-Wheel    | 3      | 134.4    | 76.40      | 5,837     | 313,265 |
+-------------+--------+----------+------------+-----------+---------+
| 6-Wheel     | 3      | 108.0    | 68.46      | 4,687     | 221,748 |
+-------------+--------+----------+------------+-----------+---------+
| Ratio       |        | 1.244    | 1.116      | 1.245     | 1.413   |
| 10Whl:6Whl  |        |          |            |           |         |
+-------------+--------+----------+------------+-----------+---------+
|             |        |          |            |           |         |
+-------------+--------+----------+------------+-----------+---------+
| 10-Wheel    | 15     | 62.8     | 21.18      | 448.6     | 129,780 |
+-------------+--------+----------+------------+-----------+---------+
| 6-Wheel     | 15     | 56.8     | 20.12      | 404.8     | 105,625 |
+-------------+--------+----------+------------+-----------+---------+
| Ratio       |        | 1.106    | 1.0527     | 1.108     | 1.229   |
| 10Whl:6Whl  |        |          |            |           |         |
+-------------+--------+----------+------------+-----------+---------+

Table 8.  ACN and ESWL ratios for 3D and 5D Gears (from tables 6 and 7)

+-----------+----------------+----------------+-----------+----------+
| CBR       | ACN Ratio with | ACN Ratio with | Percent   | $        |
|           |                |                |           | $\sqrt{\ |
|           | Old Alpha      | New Alpha      | D         | frac{P_{ |
|           | Factors        | Factors        | ifference | \text{Io |
|           |                |                |           | ld}}}{P_ |
|           |                |                |           | {6\text{ |
|           |                |                |           | old}}}\f |
|           |                |                |           | rac{P_{6 |
|           |                |                |           | \text{ne |
|           |                |                |           | w}}}{P_{ |
|           |                |                |           | \text{In |
|           |                |                |           | ew}}}}$$ |
+-----------+----------------+----------------+-----------+----------+
| 3         | 1.263          | 1.244          | 1.5       | 1.0073   |
+-----------+----------------+----------------+-----------+----------+
| 15        | 1.119          | 1.106          | 1.2       | 1.0067   |
+-----------+----------------+----------------+-----------+----------+

In both examples, the ACN ratios for the new Alpha Factors are
consistently higher than those for the old Alpha Factors, but by less
than 2 percent. The discrepancies are presumably due to the
approximations made in deriving equation 2 and, for the second example,
the linear interpolation between eight- and twelve-wheel values. To
determine the order of the errors in the new Alpha Factors, the new
Alpha Factor for twelve wheels in example 1 was adjusted until the ACN
ratio for the new Alpha Factors at 3 CBR was the same as the ACN ratio
for the old Alpha Factors at 3 CBR. This resulted in a corrected
twelve-wheel Alpha Factor value of 0.661 instead of the value of 0.660
from table 2. Performing the same process for example 2 resulted in a
corrected ten-wheel Alpha Factor value of 0.678 instead of the value of
0.675 found by interpolating between the eight- and twelve-wheel Alpha
Factors in table 2. These differences are negligible and it is
recommended that the values in table 2 be used for the new Alpha Factors
at the indicated number of wheels and that intermediate values be found
by linear interpolation.

Also of note is that the six-wheel gear has a lower ACN than the gear
with more than six wheels in example 1 at 3 CBR and in example 2 at both
3 and 15 CBR. But the ACN of the six-wheel gear is higher than the ACN
of the twelve-wheel gear in example 1. A major difference in the
examples is that the twelve-wheel C5 configuration has two groups of six
wheels spaced a considerable distance apart whereas the wheels in the
ten-wheel gear of example 2 all have the same tandem spacing as the
wheels in the six-wheel gear. In order to get a clearer picture of the
variation of ACN with number of wheels, a B747 configuration was set up
with sixteen wheels in the main landing gear (see figure 7). ACN was
computed for the initial configuration and repeatedly after removing
single wheels in the group until only one wheel remained. Wheels were
removed from right to left in one case and from the left and right sides
towards the center in another case. In a third case, the 5D gear of
figure 6 was extended to a 6D twelve-wheel gear and ACN computed as
pairs of wheels were removed from one end of the gear to the other.
Figures 8, 9, and 10 show the results.

![](media/image7.png){width="5.458333333333333in" height="4.09375in"}

Figure 7.  COMFAA screen shot showing B747 16-wheel landing gear.

![](media/image8.wmf){width="5.385416666666667in"
height="3.9791666666666665in"}

Figure 8.  ACNs for B747 configuration. Wheels removed from right to
left.

![](media/image9.wmf){width="5.375in" height="3.9791666666666665in"}

Figure 9.  ACNs for B747 configuration. Wheels removed from the outside
to the center.

![](media/image10.wmf){width="5.395833333333333in"
height="3.9791666666666665in"}

Figure 10.  ACNs for a 6D configuration. Wheels removed in pairs from
one end to the other.

In figure 8, with the 747 wheels removed from right to left, the old
system gives ACN values which decrease almost continuously from 16
wheels to 1 wheel (the curve is almost flat from eight to five wheels).
But with the new system, the ACN values decrease continuously from
sixteen to six wheels and then increase significantly from six to four
wheels before decreasing from four to one wheel (with almost the same
values as with the old system, as is to be expected). This local minimum
does not make physical sense unless the procedure is considered to be
exhibiting the same kind of behavior noted in reference 6, which was
caused by reversals in vertical strain damage computations with distance
from the point of load application. The ACN values in figure 9, with the
747 wheels removed from the outside inward, show similar trends, but
less extreme.

In figure 10, with dual wheels removed from the 6D gear from one end to
the other, the old and new systems both show continuously decreasing ACN
values. The old system shows a smoother curve than the new system at 3
CBR, but the reverse is the case at 15 CBR, indicating yet again that
the Alpha Factor system, as implemented in the CBR design procedure,
cannot account for the effects of varying gear and pavement parameters
much outside the range of the full-scale test parameters. Therefore,
when applying the CBR design procedure to the computation of ACN values
using the new Alpha Factor system, the wheel configurations used to
compute ESWLs should be as close to the wheel configurations from which
the Alpha Factors were derived in full-scale tests. This means that the
wheel configuration used to compute the ACN for any particular airplane
should be selected on the basis of the wheels forming a recognizable
group, with the spacing between adjacent wheels being about the same for
all wheels in the group. With the exception of one or two unusual and
low-damage airplanes, this is the scheme which has been followed by ICAO
for computing ACNs of multiple-gear airplanes, in particular the Boeing
747 where a single group of four wheels on the same strut is used to
calculate ACN. It is recommended that the same scheme be followed when
computing ACNs for other large multiple-gear airplanes such as the A380.

An alternative procedure could be adopted, such as the one investigated
in reference 4, in which the vertical deflection computed by elastic
theory for a single-wheel load is attenuated by a factor which decreases
in value with distance from the point of application of the load (the
computed deflection is multiplied by the attenuation factor, which has a
value of one directly under the wheel load and a value of zero at some
specified number of tire radii from the point of load application).
However, such empirically derived modifications of theoretical
computations are subject to the same kind of modeling constraints
discussed above for Alpha Factors, except that the empirical
modification is shifted from the traffic model (in the case of the Alpha
Factor model) to the response model (in the case of the attenuation
model). The attenuation model was abandoned in reference 4 because:
"Analysis of traffic test data with the two gear configurations
\[four-wheel and twelve-wheel\] did not yield values of ESWL consistent
with the elastic theory or any one modification thereof. The elastic
theory without modification was selected as best representing the
relationship of ESWL for all multiple-wheel assemblies" (reference 4,
page 55). The Alpha Factor model was then adopted to account for the
apparent overestimation of ESWL for multiple-wheel gears.

[3.  ALPHA FACTOR VERSUS COVERAGES FOR THICKNESS DESIGN]{.underline}.

Computation of ACN values requires Alpha Factors defined at only 10,000
Coverages. However, to determine the airplane for which ACN is to be
found typically requires that pavement thickness designs be made in
order to determine the "design" airplane for a specified mix operating
on the pavement of interest (references 1 and 7). This requires the
definition of Alpha Factors over the full range of Coverages expected at
an airport. The computer program F806FAA, published by the FAA in 1989
for flexible pavement thickness design by the CBR method (reference 8),
contains the following definitions of Alpha Factor as a function of
number of wheels and log~10~(Coverages) (logCov in the code below).

If LogCov \< 1 Then \' Less than 10 Coverages.

Alpha(1) = 0.405 + 0.2 \* (LogCov - 1) \' Single.

Alpha(2) = 0.400 + 0.2 \* (LogCov - 1) \' Dual.

Alpha(3) = 0.395 + 0.2 \* (LogCov - 1) \' Dual tandem.

Alpha(4) = 0.390 + 0.2 \* (LogCov - 1) \' Six wheels.

Alpha(5) = 0.380 + 0.2 \* (LogCov - 1) \' Eight wheels.

Alpha(6) = 0.370 + 0.2 \* (LogCov - 1) \' Twelve wheels.

Alpha(7) = 0.360 + 0.2 \* (LogCov - 1) \' Sixteen wheels.

Alpha(8) = 0.359 + 0.2 \* (LogCov - 1) \' Twenty four wheels.

ElseIf 1 \<= LogCov And LogCov \<= 4 Then \' Between 10 and 10,000
Coverages.

Alpha(1) = 0.405 + (LogCov - 1) \* \_

(0.205 + (LogCov - 2) \* (-0.0025 - 0.0025000 \* (LogCov - 3)))

Alpha(2) = 0.400 + (LogCov - 1) \* \_

(0.190 + (LogCov - 2) \* (-0.0125 - 0.0008330 \* (LogCov - 3)))

Alpha(3) = 0.395 + (LogCov - 1) \* \_

(0.180 + (LogCov - 2) \* (-0.0225 + 0.0041667 \* (LogCov - 3)))

Alpha(4) = 0.390 + (LogCov - 1) \* \_

(0.175 + (LogCov - 2) \* (-0.0260 + 0.0048330 \* (LogCov - 3)))

Alpha(5) = 0.380 + (LogCov - 1) \* \_

(0.175 + (LogCov - 2) \* (-0.0375 + 0.0141667 \* (LogCov - 3)))

Alpha(6) = 0.370 + (LogCov - 1) \* \_

(0.175 + (LogCov - 2) \* (-0.0350 + 0.0066670 \* (LogCov - 3)))

Alpha(7) = 0.360 + (LogCov - 1) \* \_

(0.175 + (LogCov - 2) \* (-0.0350 + 0.0050000 \* (LogCov - 3)))

Alpha(8) = 0.359 + (LogCov - 1) \* \_

(0.171 + (LogCov - 2) \* (-0.0355 + 0.0035000 \* (LogCov - 3)))

ElseIf LogCov \> 4 Then \' Greater than 10,000 Coverages.

Alpha(1) = 0.990 + 0.1700 \* (LogCov - 4)

Alpha(2) = 0.890 + 0.1250 \* (LogCov - 4)

Alpha(3) = 0.825 + 0.0875 \* (LogCov - 4)

Alpha(4) = 0.788 + 0.0800 \* (LogCov - 4)

Alpha(5) = 0.765 + 0.0700 \* (LogCov - 4)

Alpha(6) = 0.725 + 0.0500 \* (LogCov - 4)

Alpha(7) = 0.705 + 0.0375 \* (LogCov - 4)

Alpha(8) = 0.680 + 0.0200 \* (LogCov - 4)

End If

Each curve of Alpha Factor versus Coverages is divided into three
separate functions which are continuous at the changeover points (10 and
10,000 Coverages). Figure 11 shows the curves plotted in the range 1 to
1,000,000 Coverages. The curve marked "Base" is the linear equation,
 = 0.23 Coverages + 0.15, in use before the introduction of Alpha
Factors.

![](media/image11.wmf){width="5.377083333333333in"
height="4.071527777777778in"}

Figure 11.  ACN versus log~10~(Coverages) as implemented in Computer
Program F806FAA.

The same equations were implemented in the computer program COMFAA
except that the curve for eight wheels was deleted because it does not
have the same trend as the other curves and does not fit uniformly
between the six- and twelve-wheel curves. To find the Alpha Factor for a
number of wheels other than those in the F806FAA set, COMFAA does a
cubic spline fit through the defined Alpha Factor curve values at the
log~10~(Coverages) of interest and then interpolates at the desired
number of wheels. COMFAA is therefore compatible with the previous FAA
methods for flexible pavement design by the CBR method but is applicable
in a consistent manner for pavement design with any number of wheels up
to a maximum of twenty four. Now that the defined Alpha Factors at
10,000 Coverages have changed for ACN computation, the thickness design
portion of COMFAA has to be modified to be compatible with the ACN
portion.

As described in references 9 and 10, and from ICAO panel deliberations,
the new Alpha Factors at 10,000 Coverages for four- and six-wheel gears
were derived from quadratic curve fits to the full-scale test data.
These curve fits are only valid within the range of the test data and an
alternative procedure has been developed to extrapolate outside the
range of the test data in a simple and consistent manner.

Tables 3 and 4 show the full-scale test data from references 4 and 10
used to determine the Alpha Factor curves for an assumed equivalency of
1.4 between standard quality base and subbase materials (data for SQS =
1.4 CA in reference 10).

Table 3.  Alpha Factor versus Coverages from four-wheel full-scale test
data.

  ------------------- ---------------- -------------------- ----------------
  Test Series         Coverages to     log~10~(Coverages)   Alpha Factor
                      Failure                               

  MWHGL, 4-Whl        40               1.602                0.524

  MWHGL, 4-Whl        40               1.602                0.538

  NAPTF, 4-Whl        55.9             1.747                0.634

  MWHGL, 4-Whl        280              2.447                0.621

  NAPTF, 4-Whl        1,258            3.100                0.745

  NAPTF, 4-Whl        5,825            3.765                0.818

  NAPTF, 4-Whl        9,223            3.965                0.774

  NAPTF, 4-Whl        16,949           4.229                0.803
  ------------------- ---------------- -------------------- ----------------

Table 4.  Alpha Factor versus Coverages from six-wheel full-scale test
data.

  ------------------- ---------------- -------------------- ----------------
  Test Series         Coverages to     log~10~(Coverages)   Alpha Factor
                      Failure                               

  MWHGL, 6-Whl        8                0.903                0.362

  NAPTF, 6-Whl        57.3             1.758                0.517

  MWHGL, 6-Whl        104              2.017                0.572

  NAPTF, 6-Whl        1,009            3.004                0.645

  MWHGL, 6-Whl        1,500            3.176                0.667

  MWHGL, 6-Whl        1,500            3.176                0.684

  NAPTF, 6-Whl        8,280            3.918                0.736

  NAPTF, 6-Whl        12,739           4.105                0.693
  ------------------- ---------------- -------------------- ----------------

The curve fitting computer program CurveExpert 1.3 was used to select
the best fitting function in terms of goodness of fit and applicability
to representing Alpha Factor as a function of log~10~(Coverages) for
pavement design. The function selected is called Exponential Association
(3) in the CurveExpert program and is defined by equation 11.

$\alpha = a\left( b - e^{- c\text{.}\text{log}(\text{Coverages})} \right)$
(11)

Figures 12 and 13 show screen shots of the output from the program. The
coefficients a, b, and c corresponding to figures 12 and 13 are listed
in table 5.

![](media/image12.png){width="4.009027777777778in"
height="2.7493055555555554in"}

Figure 12.  Four-wheel Alpha Factor versus log~10~(Coverages) from curve
fit program for the exponential association (3) function.

![](media/image13.png){width="4.009027777777778in"
height="2.7618055555555556in"}

Figure 13.  Six-wheel Alpha Factor versus log~10~(Coverages) from curve
fit program for the exponential association (3) function.

Table 5.  Coefficients for the exponential association (3) function
curve fits in figures 12 and 13.

  --------------------- ------------------------ ------------------------
  Coefficient           Four-Wheel Data          Six-Wheel Data

  a                     0.759554                 0.714537

  b                     1.269902                 1.093644

  c                     0.378882                 0.586352
  --------------------- ------------------------ ------------------------

Figure 14 shows the two curves plotted on the same graph together with
the base curve from figure 11 and the quadratic curves from reference
10. It can be seen that the exponential association curves agree very
well with the quadratic curves over the range of the full-scale test
data (about 10 to 15,000 Coverages, or 1 to 4.2 log~10~(Coverages)). But
outside the range of the test data the two sets of curves diverge
significantly. Coverage levels below 10 are of relatively little
interest, particularly for commercial operations. Coverage levels from
15,000 to 100,000, and higher, are, however, well within normal
commercial operations at large airports and figure 14 demonstrates
clearly that the quadratic curves are not suitable for extrapolation
outside the range of the test data. The design program F806FAA
extrapolates outside the range of the test data with linear functions of
log~10~(Coverages) and it is considered that this is a reasonable
strategy at low coverage levels. But at high coverage levels it seems
more reasonable to continue the curved characteristic, as long as the
slope does not become negative as can happen with the quadratic curves.
The exponential association curve has a very simple and well defined
characteristic at all coverage levels beyond 10,000 when constrained to
pass through a defined Alpha Factor at 10,000 Coverages. On this basis,
and considering that it appears to be a physically reasonable
continuation of the existing test data, the exponential association
curve was selected for extrapolation at high coverage levels.

![](media/image14.wmf){width="5.367361111111111in"
height="3.953472222222222in"}

Figure 14.  Four- and six-wheel Alpha Factor versus log~10~(Coverages)
showing raw data and quadratic and exponential association (3) curves.

In order to make the COMFAA thickness designs compatible with the Alpha
Factor values at 10,000 Coverages of table 2, the exponential
association curve for any number of wheels is constrained to pass
through the point defined by the Alpha Factor value in the table (or
linearly interpolated from the values in the table) and
log~10~(Coverages) = 4.0. At the low end, the curves for all numbers of
wheels are constrained to pass through the points log~10~(Coverages) =
0.0, Alpha Factor = 0.1, and log~10~(Coverages) = 1.0, Alpha Factor =
0.38.

[3.1  DERIVATION OF THE CONSTRAINT EQUATIONS]{.underline}.  The Alpha
Factor versus Coverages curve is defined by:

$\alpha = a\left( b - e^{- c\text{.}\text{LogCov}} \right)$ (11)

Where:

  --------------------- -------------------------------------------------
  =                     Alpha Factor

  *a*, *b*, and *c* =   curve fitting coefficients

  *LogCov* =            log~10~(Coverages)
  --------------------- -------------------------------------------------

When Coverages = 1.0, *LogCov* = 0.0, ~0~ = *a*(*b* -- 1), and

$a\text{.}b = \alpha_{0} + a$ (12)

When Coverages = 10, *LogCov* = 1.0, and

$\alpha_{1} = a\left( b - e^{- c} \right)$ (13)

When Coverages = 10,000, *LogCov* = 4.0, and

$\alpha_{4} = a\left( b - e^{- 4c} \right)$ (14)

where ~4~ is found from table 2 for the number of wheels of interest.

Rearranging equation 13 and substituting for *a.b* from equation 12
gives:

$e^{- c} = \frac{\alpha_{0} + a - \alpha_{1}}{a}$

and
$- c = \text{ln}\left( \frac{\alpha_{0} + a - \alpha_{1}}{a} \right)$
(15)

Rearranging equation 14 and substituting for *a.b* from equation 12
gives:

$e^{- 4c} = \frac{\alpha_{0} + a - \alpha_{4}}{a}$

and
$- c = \frac{1}{4}\text{ln}\left( \frac{\alpha_{0} + a - \alpha_{4}}{a} \right)$
(16)

Subtracting equation 16 from equation 15 gives:

$\text{ln}\left( \frac{\alpha_{0} + a - \alpha_{4}}{a} \right) - \frac{1}{4}\text{ln}\left( \frac{\alpha_{0} + a - \alpha_{4}}{a} \right) = 0$
(17)

Equation 17 can be solved iteratively for the coefficient *a*.
Coefficient *b* is then found from equation 12, *c* is found from
equation 15 or 16, and the Alpha Factor for the number of wheels of
interest at *LogCov* log~10~(Coverages) is found from equation 11. A
Visual Basic function routine implementing this procedure is listed in
Appendix A.

Figure 15 shows the new Alpha Factor curves versus log~10~(Coverages)
for four-, and six-wheel gears. The raw data for the single-wheel test
data from reference 2 has also been included in the figure (see table 5)
and extrapolation below log~10~(Coverages) = 1.0 is along the Base
curve. In comparison with the best fit curves of figure 14, the
six-wheel curve in figure 15 is almost identical with that of figure 14.
The four-wheel curves in figures 14 and 15 correspond closely above
1,000 Coverages, but diverge below that value. The four-wheel curve in
figure 15 could be made to match better with that in figure 14 below
1,000 Coverages by making ~1~ a function of the number of wheels to
provide a spread in the curves similar to that shown in figure 11.
However, this would probably be an unnecessary complication. Quoting
again from reference 4, page 71: "The unique limiting curve shown \[in
figure 82\] from 1 to 100 passes represents a composite of the single,
twin-tandem, and 12-wheel curves shown in figure 80. Actually, the
composite curve for the very low operational level is used for
convenience because there is such a small difference in repetitions
effect and because it is difficult to differentiate failure at a low
operational level." The wide scatter of the single-wheel raw data seen
in figure 15 at low Coverages supports these comments.

![](media/image15.wmf){width="5.372222222222222in"
height="3.9555555555555557in"}

Figure 15.  Four- and six-wheel Alpha Factor versus log~10~(Coverages)
showing raw data and constrained exponential association (3) curves.

Table 5.  Alpha Factors versus Coverages from single-wheel full-scale
test data (reference 2).

  -------------- -------------- -------------- -------------------- --------------
  Wheel Load,    Subgrade CBR   Coverages to   log~10~(Coverages)   Alpha Factor
  kips                          Failure                             

  200            6.0            150            2.176                0.646

  200            9.0            1,700          3.230                0.924

  200            16.0           10             1.000                0.550

  200            18.0           60             1.778                0.686

  200            15.5           360            2.556                0.704

  200            17.5           1,500          3.176                0.930

  200            8.0            1,300          3.114                0.959

  15             8.0            3,760          3.575                0.813

  15             9.0            3,760          3.575                0.893

  50             3.7            6              0.778                0.378

  50             4.4            200            2.301                0.662

  30             3.7            120            2.079                0.497

  30             14.0           216            2.334                0.816

  30             7.0            178            2.250                0.548

  30             6.0            203            2.307                0.502

  10             6.0            40             1.602                0.376
  -------------- -------------- -------------- -------------------- --------------

Figure 16 shows Alpha Factor versus log~10~(Coverages) for constrained
exponential association (3) curves and the corresponding curves from the
F806 program (see figure 11). Curves are shown only for the single- and
four-wheel gears because the new and old curves for six wheels and
higher diverge too much at 10,000 coverages for the shape comparison to
have any real meaning. The single-wheel curves are almost identical
below 10,000 Coverages and show minimal divergence at higher coverage
levels, with the new curve being less conservative than the old curve
for thickness design. The four-wheel curves show a similar
correspondence except that the divergence above 10,000 Coverages is
exaggerated by the enforced difference of 0.25 at 10,000 Coverages.

![](media/image16.wmf){width="5.377083333333333in"
height="3.9659722222222222in"}

Figure 16.  Single- and four-wheel Alpha Factor versus
log~10~(Coverages) showing constrained exponential association (3)
curves and the corresponding curves from the F806 program.

Finally, figure 17 shows the family of new Alpha Factor constrained
exponential association curves corresponding to the old F806FAA curves
shown in figure 11. Curves for any other number of wheels can be found
by linear interpolation between the values of Alpha Factor given in
table 2 and the application of equation 11 as described above.

For thickness design of flexible pavements, it is recommended that the
design be based on that number of wheels which gives the thickest
pavement out of all combinations of wheels from the complete main
landing gear group. This will maintain compatibility with the CBR
flexible pavement thickness design procedure as specified in reference
4.

![](media/image17.wmf){width="5.377083333333333in"
height="3.8715277777777777in"}

Figure 17.  Family of constrained exponential association (3) Alpha
Factor versus log~10~(Coverages) curves for use in COMFAA.

[4.  SUMMARY]{.underline}.

ICAO has proposed that the Alpha Factors for four- and six-wheel gears
at 10,000 Coverages be redefined for use in calculating the ACN of
airplanes operating on flexible pavements. The proposed Alpha Factors
are 0.800 and 0.720 compared to the existing values of 0.825 and 0.788,
with the new values being based on an analysis of an expanded set of
full-scale test data for four- and six-wheel traffic. With these
changes, Alpha Factors at other numbers of wheels, and over the full
range of coverages for design, required redefinition to maintain
consistency in the CBR procedure for flexible pavement thickness design
as required for ACN calculations, even though additional full-scale test
data is not available for other than four- and six-wheel gears. In fact,
full-scale test data is not available at all for gears with more than
six wheels. The Alpha Factors have been redefined as follows.

1.  The standard set of Alpha Factors at 10,000 Coverages for 1, 2, 4,
    6, 8, 12, 18, and 24 wheels have the new values: 0.995, 0.900,
    0.800, 0.720, 0.690, 0.660, 0.640, and 0.630. The values for one and
    two wheels are unchanged from the old set because the old values
    were based on existing full-scale test data. The values for more
    than six wheels were derived based on maintaining a consistent
    ranking of damaging effect, as recommended in ICAO AOSWG/4
    Discussion Paper No. 21. This was done by making the ratio of ACN
    for the higher numbers of wheels to ACN for six wheels in the new
    set of values approximately the same as in the old set of values.

2.  Alpha Factors at 10,000 Coverages for numbers of wheels other than
    those in the standard set (in item 1 above) should be found by
    linear interpolation between the two closest numbers of wheels in
    the standard set.

3.  It is recommended that the wheel configuration used to compute the
    ACN for any particular airplane should be selected on the basis of
    the wheels used in the computation forming a recognizable group,
    with the spacing between adjacent wheels being about the same for
    all wheels in the group. This is compatible with current ICAO ACN
    calculations for multiple-gear airplanes.

4.  A new set of functions has been established for describing the
    variation of Alpha Factor with Coverages over the full range of 1 to
    24 wheels. The functions are defined by an exponential equation
    constrained to pass through the points:

    a.  Alpha Factor as defined in items 1 and 2 above and Coverages =
        10,000.

    b.  Alpha Factor = 0.38 and Coverages = 10.

    c.  Alpha Factor = 0.1 and Coverages = 0.

5.  If the new set of functions is to be used for flexible pavement
    thickness design, it is recommended that the design be based on that
    number of wheels which gives the thickest pavement out of all
    combinations of wheels from the complete main landing gear group.
    This is compatible with the CBR flexible pavement thickness design
    procedure as recommended by the U.S. Army Corps of Engineers and as
    implemented by the FAA for multiple-gear airplanes.

[5.  REFERENCES]{.underline}.

1.  International Civil Aviation Organization, Doc 9157-AN/901,
    "Aerodrome Design Manual, Part 3, Pavements," Montreal Quebec,
    Canada, Second Edition, 1983.

2.  Cooksey, D. L., and Ladd, D. M., "Pavement Design for Various Levels
    of Traffic Volume, AFWL-TR-70-133, prepared by U. S. Army Engineer
    Waterways Experiment Station, CE, for Air Force Weapons Laboratory,
    Kirtland AFB, NM, July 1970.

3.  Pereira, A. Taboza, Instruction Report S-77-1, "Procedures for
    Development of CBR Design Curves," U. S. Army Engineer Waterways
    Experiment Station, CE, June 1977.

4.  Ahlvin, R. G., Ulery, H. H., Hutchinson, R. L., and Rice, J. L.,
    Technical Report S-71-17, "Multiple-Wheel Heavy Gear Load Pavement
    Tests, Vol. 1, Basic Report," U. S. Army Engineer Waterways
    Experiment Station, CE, June 1971.

5.  Hayhoe, Gordon F., "Alpha Factor Determination Using Data Collected
    At The National Airport Pavement Test Facility," Report
    DOT/FAA/AR-06/7, March, 2006.

6.  Rodway, Bruce, Wardle, Leigh, J, and Wickham, Gary, "Interaction
    Between Wheels and Wheel Groups of New Large Aircraft," 1999 FAA
    Airport Technology Transfer Conference, Atlantic City, New Jersey,
    April, 1999.

7.  FAA AC 150/5335-5A, "Standardized Method of Reporting Airport
    Pavement Strength -- PCN."

8.  FAA Engineering Brief No. 43, "Computerized Pavement Design," FAA
    Engineering Specifications Division, AAS-200, April, 1989.

9.  Hayhoe, Gordon F., "Alpha Factor Determination Using Data Collected
    at the National Airport Pavement Test Facility," FAA Report
    DOT/FAA/AR-06/7, March, 2006.

10. Hayhoe, Gordon F., "Subgrade CBR Values for Alpha Factor
    Determination Using Data Collected at the National Airport Pavement
    Test Facility," FAA Technical Note DOT/FAA/TN-07/-, in publication.

APPENDIX A -- VISUAL BASIC FUNCTION ROUTINE FOR COMPUTING ALPHA FACTOR

Function New06AlphaFactorFromCurve(NW As Integer, Coverages As Double)
As Double

\' NW = number of wheels.

Dim I As Long

Dim A As Double, B As Double, C As Double

Dim A0 As Double, A1 As Double, A4 As Double

Dim LogCovA0 As Double, LogCovA1 As Double, LogCovA4 As Double

Dim A4define() As Double, NWA4define() As Long, NA4define As Long

Dim Resid As Double, ResidM1 As Double, DelA As Double, Slope As Double

\' Fixed points to define alpha versus log(coverages) curves.

\' log(coverage) values.

LogCovA0 = 0: LogCovA1 = 1: LogCovA4 = 4

\' Alphas at log(coverages)

A0 = 0.1: A1 = 0.38 \' A4 = derived from new set of alpha at 10,000
coverages.

\' Incremental damage relative to six-wheels the same as old alphas for
higher than 6-wheels.

NA4define = 8 \' Number of defined values of alpha at 10,000 coverages.

\' NWA4define() = number of wheels. A4define = alpha for NWA4define
wheels.

ReDim NWA4define(NA4define), A4define(NA4define)

NWA4define(1) = 1: NWA4define(2) = 2: NWA4define(3) = 4

NWA4define(4) = 6: NWA4define(5) = 8: NWA4define(6) = 12

NWA4define(7) = 18: NWA4define(8) = 24

A4define(1) = 0.995: A4define(2) = 0.9: A4define(3) = 0.8

A4define(4) = 0.72: A4define(5) = 0.69: A4define(6) = 0.66

A4define(7) = 0.64: A4define(8) = 0.63

If NW \< NWA4define(1) Then \' Should never happen. Would be an error.

New06AlphaFactorFromCurve = A4define(1)

Exit Function

End If

If Coverages \< 10 Then \' Lower end of curves is independent of number
of wheels.

New06AlphaFactorFromCurve = 0.15 + 0.23 \* Log10(Coverages)

Exit Function

End If

\' Linear interpolation for undefined alpha factors at 10,000 coverages.

For I = 1 To NA4define - 1

If NW \< NWA4define(I + 1) Then

Slope = (A4define(I + 1) - A4define(I)) / \_

CDbl(NWA4define(I + 1) - NWA4define(I))

A4 = A4define(I) + Slope \* CDbl(NW - NWA4define(I))

Exit For

Else

A4 = A4define(NA4define)

End If

Next I

\' Find the coefficients of the defining exponential association (3)
curve

\' Alpha = A \* (B - e\^(-C \* log10(Coverages))), (see CurveExpert
program).

\' Solve for Resid function (below) by Newton\'s method.

A = A4 - A0 + 0.1 \' The second log argument is -ve if A \< A4 - A0.

DelA = A / 1000

ResidM1 = Log((A0 + A - A1) / A) / LogCovA1 - Log((A0 + A - A4) / A) /
LogCovA4

Do

A = A + DelA

Resid = Log((A0 + A - A1) / A) / LogCovA1 - Log((A0 + A - A4) / A) /
LogCovA4

A = A - Resid \* DelA / (Resid - ResidM1)

If A \< A4 - A0 + 0.001 Then

A = A4 - A0 + 0.001 \' Jumped too far because of nonlinearity.

End If

ResidM1 = Log((A0 + A - A1) / A) / LogCovA1 - Log((A0 + A - A4) / A) /
LogCovA4

DoEvents \' In case the iteration does not terminate. Allows
interruption.

Loop Until Abs(ResidM1) \< 0.000001

B = A0 / A + 1

C = -Log((A0 + A - A4) / A) / LogCovA4

New06AlphaFactorFromCurve = A \* (B - Exp(-C \* Log10(Coverages)))

End Function