The document details the evolution of the Tire and Rim Association (TRA) load formula for passenger car tires, outlining its historical adjustments and the relationship between tire load, deflection, and stiffness. It introduces a new approach that correlates tire stiffness with inflation pressure, footprint width, and outside diameter, presenting various equations for evaluating these parameters. Additionally, it discusses the implications of deflection on tire durability and includes tables of standard dimensions for rim codes and tire sizes.

1. TIRE LOAD CAPACITY
𝟏𝟓𝑻𝑯
𝑺𝒆𝒑𝒕𝒆𝒎𝒃𝒆𝒓 𝟐𝟎𝟐𝟐

2. Content
• Objectives
• History: TRA load formula
• A new approach (load, deflection & stiffness)
• TRA load formula vs. deflection
• Deflection Analysis
• Proposed load formula

3. Objectives
The Tire and Rim Association, Inc. (TRA)
• This chapter presents the evolution of the TRA load formula
for passenger car tires from the early years to its current
application.
• The Tire and Rim Association, Inc. (TRA) for over 100 years
has been the establishment of interchangeability standards
for tires
• The load formula is then compared to one based upon
constant relative deflection.

4. History
TRA load formula
Prior to 1928:
• Loads were
not
dependent
on rim
diameters;
• Loads varied
in relation to
tire section
width;
• Loads varied
in direct ratio
to the air
pressure.
• For lesser loads the ratings were determined by taking a direct proportion of the
maximum inflation pressure as illustrated by the formula below:
𝐿 = 𝐿0 𝑃/𝑃0
Where:
L = load limit at pressure P
𝐿0= maximum load limit at 𝑃0
• The first formula adopted by TRA was developed in the mid 1930’s by C. G. Hoover, a
mathematician who later served as the staff director of TRA.
𝐿 = 6.65 × 𝑃0.585
× 𝑆1.702
× (𝐷𝑅 + 𝑆)/(19 + 𝑆)
Where:
𝐿 = tire load carrying capacity at pressure P
P = tire inflation pressure
S = tire section width (on rim width = 62.5% of tire
section width)
𝐷𝑅=nominal rim diameter
• The above formula was revised in 1936 to the following:
𝐿 = 𝐾 × 0.425 × 𝑃0.585
× 𝑆1.39
× (𝐷𝑅 + 𝑆) …………………………(1)

5. Basic formula:
The origins of the load formula are not well documented. However, based on the available information, Hoover related the tire load carrying
capacity to the tire volume in developing the formula.
• The direct proportionality, 𝐿 = 𝐿0 𝑃/𝑃0 , was adjusted to 𝐿 = 𝐿0 𝑃/𝑃0
𝑛
, with n = 0.585, since it was thought that the load-pressure
relationship would not really be a linear one.
• The new relationship indicates that for each tire design, a constant value exists for the ratio 𝐿/𝑃𝑛
=𝐿0/𝑃0
𝑛
.
• The tire load carrying capacity was assumed to be directly proportional to the air volume 𝑉. Thus, 𝐿/𝑃𝑛
would depend linearly on air
volume 𝑉. 𝐿/𝑃𝑛
=const. 𝑉.
As the cross-section of a tire was approximately circular in the 1930’s, the volume 𝑉 was given by :
𝑉= 𝑐𝑜𝑛𝑠𝑡. 𝑆2
(𝐷𝑅 + 𝑆)
Where 𝐷𝑅 is rim diameter, 𝑆 is section diameter. Combining the above equations yields:
𝐿= 𝑐𝑜𝑛𝑠𝑡. 𝑃𝑛
× 𝑆2
(𝐷𝑅 + 𝑆)
However the volume of a circular annulus is not exactly proportional to 𝑆2
. The exponent of 2 for 𝑆 was reduced to 1.39 – probably based on
field experience of tire performance at that time – so that the basic tire load formula became:
𝐿= 𝑐𝑜𝑛𝑠𝑡. 𝑃𝑛
× 𝑆1.39
(𝐷𝑅 + 𝑆)

6. A new approach
Tire load/deflection and stiffness
Fig.- load-deflection curves at various operational inflation pressures
• The tire stiffness at a given pressure is derived from the
slope (the tangent vertical stiffness) of the individual
curves, which appear to be quite linear over normal
ranges of operating load.

7. A new approach
Tire load/deflection and stiffness
• As a result of Rhyne’s work, we know that the tangent stiffness 𝐾𝑍 , is a function of tire pressure,
footprint width and outside diameter and may be expressed as follows:
𝐾𝑍 = 0.00028 × 𝑃 𝑊 × 𝑂𝐷 + 3.45
…………………. (2)
Where,
𝐾𝑍= tangent stiffness(kg/mm)
P= tire inflation pressure(kPa)
W= tire footprint width(mm)
OD= outside diameter(mm)

8. A new approach
Tire load/deflection and stiffness
Relationship between footprint width and nominal section
width:-
𝑊 ≈ −0.004 𝐴𝑅 + 1.03 𝑆𝑁 ≈ 𝑎 × 𝑆𝑁
………..(3)
Where:
W = Footprint width(mm)
AR = Aspect Ratio
𝑆𝑁= Nominal Section Width(mm)
a= factor from table 1
• The effect of equation (3) is that as the Aspect Ratio
decreases, the factor ‘a’ by which the nominal section
width, 𝑆𝑁, is multiplied, increases, as shown in Table 1.
• Thus, for lower aspect ratio tires, the footprint width as a
percentage of the width of the tire section, is inversely
proportional to the aspect ratio.
Aspect ratio “a”
25 .93
30 .91
35 .89
40 .87
45 .85
50 .83
55 .81
60 .97
65 .77
70 .75
75 .73
80 .71
Table:-1

9. A new approach
Tire load/deflection and stiffness
By using equation (2) & equation (3),
𝐾𝑍 = 0.00028 × 𝑃 −0.004 𝐴𝑅 + 1.03 𝑆𝑁 × 𝑂𝐷 + 3.45 … … … … (4)
We know, 𝑂𝐷 = 2𝐻 + 𝐷𝑅 ……………(5)
Where,
H = Design Section Height (mm)
𝐷𝑅= Rim Dia Code (mm)
And 𝐻 =
𝑆𝑁
100
× 𝐴𝑅 ………………….(6)
Where,
𝑆𝑁= nominal section width(mm)
𝐴𝑅= aspect ratio
Thus 𝑂𝐷 may be expressed in terms of section width, aspect ratio and rim code as follows:
𝑂𝐷 =
𝑆𝑁×𝐴𝑅
50
+ 𝐷𝑅 ………………..(7)

10. A new approach
Tire load/deflection and stiffness
Equation (4) can be expressed by which is used by
engineers and standardized units:-
𝐾𝑍
= 0.00028
× 𝑃 −0.004 𝐴𝑅 + 1.03 𝑆𝑁 ×
𝑆𝑁 × 𝐴𝑅
50
+ 𝐷𝑅
+ 3.45 … … … … . (8)
Using equation (8) we can now compare predicted
versus measured values of 𝐾𝑍 for a large sample of tires.
Table 2 lists values of 𝐷𝑅 for current rim codes.
Rim diameter
code
𝐷𝑅(mm)
12 305
13 330
14 356
15 381
16 406
17 432
18 457
19 483
20 508
21 533
22 559
23 584
24 610
Table:-2 – Values for 𝐷𝑅
(rim code x 25.4)

11. TRA load formula vs. deflection
Having shown that the tangential vertical stiffness is correlated for a wide range of tires. We know look at the
TRA load equation and calculate the relative deflection of tires at maximum load and under normal load.
• As the deflection increases, the tire is strained more severely and therefore more heat is generated.
Consequently the operating temperature increases. The energy expended in rolling also increases. Thus
any review of load capacity should consider the corresponding deflection.
Fig:- schematic of a tire mounted on a rim

12. TRA load formula vs. deflection
Equivalent static deflection formula can be written as:-
𝑑 =
𝐿
𝐾𝑍
…………………….(9)
Where,
d = deflection(mm)
L= load(in kg)
𝐾𝑍= tangential stiffness(kg/mm)
• The deflection under load is considered to be the main determinant of tire durability and for different types of
tire the relative deflection is the appropriate measure.
• Thus, to compare a variety of tire diameters, aspect ratios and rim diameters, it is desirable to express the
deflection as a percent of the section height (SH).
%𝑑 =
𝑑
𝑆𝐻
× 100 =
2𝐷
𝑂𝐷−𝐹𝐷
× 100 …………….(10)
where: FD = rim flange diameter (mm)
SH = section height (above rim flange) (mm)
OD = outside diameter (mm)

13. TRA load formula vs. deflection
A listing of standard dimensions for today’s rims is
presented in Table .3 Using equations (8) through (10),
deflections for the entire range of tire sizes may easily be
calculated.
Rim
diameter
code
Rim
diameter
(D) in
mm
Flange
diameter
(FD) mm
14 354.8 389.8
15 380.2 415.2
16 405.6 440.2
17 436.6 471.6
18 462.0 497.0
19 487.4 522.4
20 512.8 547.8
21 538.2 573.2
22 563.6 598.6
23 589.0 524.0
24 614.4 649.4
Table .3 – Basic rim dimensions

14. Deflection Analysis:-
Fig :- 75 series – standard load – 180kPa
Fig :- 75 series – standard load – 240kPa
• Both figures show that as the section width of these tires is increased the relative deflection decreases
slightly, but as the rim code increases there is a more significant increase in deflection.

15. Deflection Analysis:-
Figure : P225/arRrc - TRA - light load
Figure : P225/arRrc - TRA - light load
• for a given aspect ratio, as the rim diameters are increased there is a significant increase in
relative deflection, and the rate of increase is similar for all aspect ratios.
• The above graphs clearly show that the existing TRA formula penalizes larger rim diameters,
that is, it requires the tire to deflect more as the rim diameter increases.

16. From equation (9), for the linear case, we see that load is equal to the product of deflection and stiffness, and we
have already developed equation (8) to calculate stiffness.
Transposing equation (10) to give the deflection in terms of the nominal section width and aspect ratio results in
the following equation:
𝑑 =
%𝑑×(𝐻−17.5)
100
=
%𝑑×(
𝑆𝑁×𝐴𝑅
100
−17.5)
100
……………..(11)
From equation (8) and (11)-
𝐿
=
%𝑑 × (
𝑆𝑁 × 𝐴𝑅
100
− 17.5)
100
× [0.00028 × 𝑃 −0.004 𝐴𝑅 + 1.03 𝑆𝑁 ×
𝑆𝑁 × 𝐴𝑅
50
+ 𝐷𝑅
+ 3.45 ] … … … … . (12)
Proposed load formula:-