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The determination of the deflection, contact area, dimensions, and load carrying capacity for driven pneumatic tires operating on concrete pavement
Journal of Terramechanics, 1976, Vol. 13, No. 1, pp. 15 to 20. Pergamon Press Printed in Great Britain.
THE D E T E R M I N A T I O N OF THE DEFLECTION, CONTACT AREA, DIMENSIONS, AND LOAD CARRYING CAPACITY FOR DRIVEN PNEUMATIC TIRES OPERATING ON CONCRETE PAVEMENT G. KOMANDi* Summary--In order to predict the performance of pneumatic tires with respect to rolling dimensions and traction, it is necessary to determine the relationships between a tire's dimensions and its behaviour under load. In this paper, mathematical expressions are given describing tire deflection, contact area dimensions, and load carrying capacity. A means of determining ply rating when the required load capacity and dimensions are known is also presented. The relationships are all based on the results of tire tests. INTRODUCTION DRIVEN pneumatic tires have a dual role: they support the vehicle and generate the peripheral forces required to overcome the resistances. The peripheral force is generated by the adhesive forces acting between the tire and the pavement. Activities in the area of running gear research and development are aimed at the increase of the maximum tractive effort and at the decrease of the rolling resistance. Both depend on the soil and on the design of the running gear. Investigations of the mechanics of resistances and tractive forces are primarily concerned with their relationship with the soil. It is a fact that knowledge of the physical and mechanical properties of the soil is of great concern to us. But, it would be equally important to know what these properties are for tires which are in mechanical interaction with the soil. This is a difficult problem because a tire cannot be regarded as a homogeneous object, and because its surface cannot be developed (i.e., "unfolded into a plane"); it is manufactured in different sizes and types and its mechanical behavior depends on the manufacturing process and the properties of the material. THE ROLE OF THE PNEUMATIC TIRE IN THE GENERATION OF ROLLING RESISTANCE AND PERIPHERAL FORCE Rolling resistance is a function of the axle load and the strength of the soil. The strength of the soil is obtained by means of various penetration tests and is expressed by different corresponding relationships. These relationships, however, are only valid when the surface of the penetrating object is constant. But if the contact surface varies with the load, as in the case of tires, at least one more equation is needed for the prediction of the sinkage. Similarly, one has to be able to describe mathematically the tire soil interface *Department of Tractors and Automobiles, Universityof Agricultural Sciences, Godollo, Hungary. 15
i6
G. KOMANDI
surface as a function of the peripheral force in order to express the tractive force whiO~ can be exerted by a tire. The above considerations prompted the staff of the Department of Tractors and Automobiles of the University of Agricultural Sciences to perform a series of tests on pneumatic tires. The purpose of these tests was the determination of the behavior of tires under static load leading to information which would enhance the practical applicability of the equations established for the determination of the rolling resistance and the tractive force. DESCRIPTION OF TIRE TESTS
Ten conventional tires were tested on concrete pavement. Their sizes ranged from a 9-24 to a 15-30. The catalogue data for each tire were established and their geometric dimensions were checked. Next, the tires were loaded in increasing increments, the corresponding deflections were measured and the shape and size of the contact areas were determined. These tests were repeated for various inflation pressures from 0.4 kp/cm 2 to 1-6 kp/cm". The results of the tire tests were plotted for each tire. An attempt was made to find empirical equations to express the deflection, the width of contact area, the length of contact area as a function of the wheel load. The following relationships were derived Tire deflection."
Q
0"85
Ar = C1
BO.7 DO.43 p10"6
K (cm)
(1)
where: Ca is a parameter depending on tire design. Its value is 1.15 for conventional tires. It is assumed that C1 1-5 for radial tires. (Radial tires have not been tested.) K--15 X 103 X B+0"42 Q = wheel load (kp) B = width of tire (cm) D diameter of tire (cm) Pl :: inflation pressure (kp/cmZ). Width o f contact area: O ( 1 -- e 0"4omin .
b = C~ Bpl-°'x4
.
.
.
(cm)
(2)
where: C2 is 0.87 for a normal rim and it is 0.9 for a wide rim. B is the width of the tire (cm) Qmin is the tire load allowed at the lowest inflation pressure permitted (kp).
DRIVEN PNEUMATIC TIRES OPERA'IING ON CONCRETE PAVEMENT
17
Length o f contact area..
1' = I'7V'(D -- Ar) Ar
(cm)
(3)
where: D is the outer diameter of the tire (cm) r is the tire deflection (cm). Contact area..
F=(/'--b)
b + ~b 4
(cm 2)
(4)
Length o f equivalent rectangle: I~q.
= F
(cm).
(5)
b The total area of the tread imprints, which amounted to 22-24% of the total area, was also established. Therefore, an average value of F, = 0.23F is recommended. ESTABLISHMENT OF THE LOAD-CARRYING
CAPACITY
It is possible to calculate the load carrying capacity on the basis of the deflection/ load relationship. The maximum allowable deflection is a given percentage of the section height. (6)
r = k H = kBC.
Where: 0 < k < 1 ; H is the section height (cm) and C = H/B. So equating k B C with the right hand side of equation (1) gives: Ar = k C . B =
- C1 Q°'SSK BO.7 DO-43plo'6
(7)
(8)
-
or after rearrangement: k C B B°'7D°'4a pl 0'6 = C 1 Q°'SSK 1
1
(9) Q
\-C1
1
..........
So the load carrying capacity is:
c cr. (17o0410.)11. Tk = Q =
\~!
15.10-ZB + 0 . 4 2
18
G. KOMANDI
Tk ~ A B2°D°~tP~'"7 (15.1013B
(Io)
4 0.42) r~s(kp)
here B and D are given in cm-s and Pl is in kp/cm". For conventional pneumatic tires the nominal load causes 15% deflection, so k 0.15, C1 = 1.15 and C 0.87. And therefore A 766 ~ 10 4. DETERMINATION OF PLY RATING The relationship between lhe load carrying capacity of the tires, their dimensions and their ply rating has been examined. The so-called "specific load carrying capacity" which is the ratio between the maximum allowable load and the tire diameter, width and ply rating has been introduced• This, according to Fig. I, depends on ratio D/B.
7
0
PR=6 PR=4
7-
o . ...
•
.s
•.
; ..o . 'o o
6 -
E ac: E~
~g
TKmax 8D(PR)
2
-2 127
D/B
I0
t
0
I tO
I 20
I 30
I 40
I 50
I 60
I 7.0
D/B FIG, 1.
Relationship between "specific load carrying capacity" and ratio of tire diameter to width.
~ 1"27 D/B lo -2.
Tk ....
(it)
B D (er)
From here the ply rating is:
PR = 0.787 T,
max D2
7".
It) 2 ,.~ 8 0 ~'k max D2
(12)
The utility of the preceding relationships can be demonstrated through a few examples, as shown in Tables l, 2, and 3.
DRIVEN
TABLE 1.
PNEUMATIC
TIRES OPERATING
ON CONCRETE
PAVEMENT
COMPARISON OF CALCULATED TIRE DEFLECTION DATA TO THOSE FOUND IN D I N 7807 CATALOGUE Q kp
PI kp/cm 2
A r (m~ A r ~m) Cat. Comp.
A %
Type
D mm
B mm
15-34
1650
456
2565
1.4
65.0
60.5
-6.92
]5-30
1550
456
2415
1.4
59.0
61.2
+3.73
]2-36
1515
332
1530
].4
40.5
42.3
+4.44
]0-28
].205
272
1070
].7
33.5
32.2
-3.88
9-3~
1355
241
II00
2.0
28.5
29.8
+4.56
8-24
995
211
765
2.2
25.5
23.9
-6.27
TABLE 2.
19
_
COMPARISON OF CALCULATED TIRE DEFLECTIONS WITH DATA PUBLISHED BY M. I. DWYER [3] Type
D mm
B mm
].].-36
1453
321
]2-3R
]~55
347
Q kp
P1 kp/cm 2
A r (mm) A r (mm) Cat. Comp.
A~
]310
].0
49
48
-2.04
]730
].7
39
44
]2.8
1530
].0
56
54
-3.5
2000
].6
48
50
44.16
]4-30
1463
428
2120
].I
7]
66
-7.04
]3-34
]563
424
2270
].]
72
68
-5.55
]5-30
1~36
470
2540
1.1
86
74
-]3.0
20
G. KOMAND1 T A B L E 3.
E S T A B L I S H M E N T OF LOAD C A R R Y I N G C A P A C I T Y ]2-3f~ T I R E D 8
Pl
Tk
(inflation pressure)
: :
]h]. ~ cm %4. r cm 2
0.9
].0
i.I
1.2
|.~
].4
kp/cm
catalogue
1180
1255
1330
1400
]465
]530
kp
computed
1150
1240
1325
]392
1488
1568
kp
-2.54
-1.20
-0.38
-0.57
+1.57
+2.48
1.2
1.3
].4
kp/cm 2
1615
]686
kp
]676
1768
kp
+3.78
+4.86
(load c a r r v inq capacity):
~'°z
difference
RADIAL
Pl
Tk
cm
B =
34.2
cm
0.9
1.0
i.I
catalogue
1300
1385
1465
computed
1296
1397
1493
-0.31
0.87
+1.91
(inflation pressure) (load c a r r y i n q capacity):
rating PR
For
the PR
[I] [2] [3]
TIRE
D = 149.4
difference
PIy
(DIN 7807 ~,
- 80
for
L~%
]2-36
Tkmax D~
radial
1568
tire: - 80
1568 22952
~ 5
tire:
= 80 T k m a x D2
80 1 7 6 8 22320
~
6
REFERENCES Treibradreifen auf Breitfelgen fiir Ackerschlepper und Ackermaschinen. D I N 7807 (1967). Additional sheet to tyre handbook, Pirelli (1969). M . I . DWYER, D. R. COMELYand D. W. EVERNDEN, The field performance of some tractor tyres related to soil technical properties. J. Agric. Engng. Res. 19, 35-50 0974).
THE D E T E R M I N A T I O N OF THE DEFLECTION, CONTACT AREA, DIMENSIONS, AND LOAD CARRYING CAPACITY FOR DRIVEN PNEUMATIC TIRES OPERATING ON CONCRETE PAVEMENT G. KOMANDi* Summary--In order to predict the performance of pneumatic tires with respect to rolling dimensions and traction, it is necessary to determine the relationships between a tire's dimensions and its behaviour under load. In this paper, mathematical expressions are given describing tire deflection, contact area dimensions, and load carrying capacity. A means of determining ply rating when the required load capacity and dimensions are known is also presented. The relationships are all based on the results of tire tests. INTRODUCTION DRIVEN pneumatic tires have a dual role: they support the vehicle and generate the peripheral forces required to overcome the resistances. The peripheral force is generated by the adhesive forces acting between the tire and the pavement. Activities in the area of running gear research and development are aimed at the increase of the maximum tractive effort and at the decrease of the rolling resistance. Both depend on the soil and on the design of the running gear. Investigations of the mechanics of resistances and tractive forces are primarily concerned with their relationship with the soil. It is a fact that knowledge of the physical and mechanical properties of the soil is of great concern to us. But, it would be equally important to know what these properties are for tires which are in mechanical interaction with the soil. This is a difficult problem because a tire cannot be regarded as a homogeneous object, and because its surface cannot be developed (i.e., "unfolded into a plane"); it is manufactured in different sizes and types and its mechanical behavior depends on the manufacturing process and the properties of the material. THE ROLE OF THE PNEUMATIC TIRE IN THE GENERATION OF ROLLING RESISTANCE AND PERIPHERAL FORCE Rolling resistance is a function of the axle load and the strength of the soil. The strength of the soil is obtained by means of various penetration tests and is expressed by different corresponding relationships. These relationships, however, are only valid when the surface of the penetrating object is constant. But if the contact surface varies with the load, as in the case of tires, at least one more equation is needed for the prediction of the sinkage. Similarly, one has to be able to describe mathematically the tire soil interface *Department of Tractors and Automobiles, Universityof Agricultural Sciences, Godollo, Hungary. 15
i6
G. KOMANDI
surface as a function of the peripheral force in order to express the tractive force whiO~ can be exerted by a tire. The above considerations prompted the staff of the Department of Tractors and Automobiles of the University of Agricultural Sciences to perform a series of tests on pneumatic tires. The purpose of these tests was the determination of the behavior of tires under static load leading to information which would enhance the practical applicability of the equations established for the determination of the rolling resistance and the tractive force. DESCRIPTION OF TIRE TESTS
Ten conventional tires were tested on concrete pavement. Their sizes ranged from a 9-24 to a 15-30. The catalogue data for each tire were established and their geometric dimensions were checked. Next, the tires were loaded in increasing increments, the corresponding deflections were measured and the shape and size of the contact areas were determined. These tests were repeated for various inflation pressures from 0.4 kp/cm 2 to 1-6 kp/cm". The results of the tire tests were plotted for each tire. An attempt was made to find empirical equations to express the deflection, the width of contact area, the length of contact area as a function of the wheel load. The following relationships were derived Tire deflection."
Q
0"85
Ar = C1
BO.7 DO.43 p10"6
K (cm)
(1)
where: Ca is a parameter depending on tire design. Its value is 1.15 for conventional tires. It is assumed that C1 1-5 for radial tires. (Radial tires have not been tested.) K--15 X 103 X B+0"42 Q = wheel load (kp) B = width of tire (cm) D diameter of tire (cm) Pl :: inflation pressure (kp/cmZ). Width o f contact area: O ( 1 -- e 0"4omin .
b = C~ Bpl-°'x4
.
.
.
(cm)
(2)
where: C2 is 0.87 for a normal rim and it is 0.9 for a wide rim. B is the width of the tire (cm) Qmin is the tire load allowed at the lowest inflation pressure permitted (kp).
DRIVEN PNEUMATIC TIRES OPERA'IING ON CONCRETE PAVEMENT
17
Length o f contact area..
1' = I'7V'(D -- Ar) Ar
(cm)
(3)
where: D is the outer diameter of the tire (cm) r is the tire deflection (cm). Contact area..
F=(/'--b)
b + ~b 4
(cm 2)
(4)
Length o f equivalent rectangle: I~q.
= F
(cm).
(5)
b The total area of the tread imprints, which amounted to 22-24% of the total area, was also established. Therefore, an average value of F, = 0.23F is recommended. ESTABLISHMENT OF THE LOAD-CARRYING
CAPACITY
It is possible to calculate the load carrying capacity on the basis of the deflection/ load relationship. The maximum allowable deflection is a given percentage of the section height. (6)
r = k H = kBC.
Where: 0 < k < 1 ; H is the section height (cm) and C = H/B. So equating k B C with the right hand side of equation (1) gives: Ar = k C . B =
- C1 Q°'SSK BO.7 DO-43plo'6
(7)
(8)
-
or after rearrangement: k C B B°'7D°'4a pl 0'6 = C 1 Q°'SSK 1
1
(9) Q
\-C1
1
..........
So the load carrying capacity is:
c cr. (17o0410.)11. Tk = Q =
\~!
15.10-ZB + 0 . 4 2
18
G. KOMANDI
Tk ~ A B2°D°~tP~'"7 (15.1013B
(Io)
4 0.42) r~s(kp)
here B and D are given in cm-s and Pl is in kp/cm". For conventional pneumatic tires the nominal load causes 15% deflection, so k 0.15, C1 = 1.15 and C 0.87. And therefore A 766 ~ 10 4. DETERMINATION OF PLY RATING The relationship between lhe load carrying capacity of the tires, their dimensions and their ply rating has been examined. The so-called "specific load carrying capacity" which is the ratio between the maximum allowable load and the tire diameter, width and ply rating has been introduced• This, according to Fig. I, depends on ratio D/B.
7
0
PR=6 PR=4
7-
o . ...
•
.s
•.
; ..o . 'o o
6 -
E ac: E~
~g
TKmax 8D(PR)
2
-2 127
D/B
I0
t
0
I tO
I 20
I 30
I 40
I 50
I 60
I 7.0
D/B FIG, 1.
Relationship between "specific load carrying capacity" and ratio of tire diameter to width.
~ 1"27 D/B lo -2.
Tk ....
(it)
B D (er)
From here the ply rating is:
PR = 0.787 T,
max D2
7".
It) 2 ,.~ 8 0 ~'k max D2
(12)
The utility of the preceding relationships can be demonstrated through a few examples, as shown in Tables l, 2, and 3.
DRIVEN
TABLE 1.
PNEUMATIC
TIRES OPERATING
ON CONCRETE
PAVEMENT
COMPARISON OF CALCULATED TIRE DEFLECTION DATA TO THOSE FOUND IN D I N 7807 CATALOGUE Q kp
PI kp/cm 2
A r (m~ A r ~m) Cat. Comp.
A %
Type
D mm
B mm
15-34
1650
456
2565
1.4
65.0
60.5
-6.92
]5-30
1550
456
2415
1.4
59.0
61.2
+3.73
]2-36
1515
332
1530
].4
40.5
42.3
+4.44
]0-28
].205
272
1070
].7
33.5
32.2
-3.88
9-3~
1355
241
II00
2.0
28.5
29.8
+4.56
8-24
995
211
765
2.2
25.5
23.9
-6.27
TABLE 2.
19
_
COMPARISON OF CALCULATED TIRE DEFLECTIONS WITH DATA PUBLISHED BY M. I. DWYER [3] Type
D mm
B mm
].].-36
1453
321
]2-3R
]~55
347
Q kp
P1 kp/cm 2
A r (mm) A r (mm) Cat. Comp.
A~
]310
].0
49
48
-2.04
]730
].7
39
44
]2.8
1530
].0
56
54
-3.5
2000
].6
48
50
44.16
]4-30
1463
428
2120
].I
7]
66
-7.04
]3-34
]563
424
2270
].]
72
68
-5.55
]5-30
1~36
470
2540
1.1
86
74
-]3.0
20
G. KOMAND1 T A B L E 3.
E S T A B L I S H M E N T OF LOAD C A R R Y I N G C A P A C I T Y ]2-3f~ T I R E D 8
Pl
Tk
(inflation pressure)
: :
]h]. ~ cm %4. r cm 2
0.9
].0
i.I
1.2
|.~
].4
kp/cm
catalogue
1180
1255
1330
1400
]465
]530
kp
computed
1150
1240
1325
]392
1488
1568
kp
-2.54
-1.20
-0.38
-0.57
+1.57
+2.48
1.2
1.3
].4
kp/cm 2
1615
]686
kp
]676
1768
kp
+3.78
+4.86
(load c a r r v inq capacity):
~'°z
difference
RADIAL
Pl
Tk
cm
B =
34.2
cm
0.9
1.0
i.I
catalogue
1300
1385
1465
computed
1296
1397
1493
-0.31
0.87
+1.91
(inflation pressure) (load c a r r y i n q capacity):
rating PR
For
the PR
[I] [2] [3]
TIRE
D = 149.4
difference
PIy
(DIN 7807 ~,
- 80
for
L~%
]2-36
Tkmax D~
radial
1568
tire: - 80
1568 22952
~ 5
tire:
= 80 T k m a x D2
80 1 7 6 8 22320
~
6
REFERENCES Treibradreifen auf Breitfelgen fiir Ackerschlepper und Ackermaschinen. D I N 7807 (1967). Additional sheet to tyre handbook, Pirelli (1969). M . I . DWYER, D. R. COMELYand D. W. EVERNDEN, The field performance of some tractor tyres related to soil technical properties. J. Agric. Engng. Res. 19, 35-50 0974).