Evaluation of Agriculture Tires Deformation Using Innovative 3D Scanning Method

Abstract

This study presents the results of research related to agriculture tire deformation under variable vertical load and inflation pressure. The research objects were two tires of the same size and different internal structures. Three levels of inflation pressure and five levels of vertical load were used. The loaded tire with each inflation pressure was scanned using the 3D scanner—the effect of this operation was a three-dimensional image of a tire part (near the place of contact with the surface). The next step was the creation of vertical and horizontal cross-sections of the tire profile, which allowed the analysis of tested parameters: profile height, location of the point of maximum tire deflection, the width of the tire profile, and the area of horizontal cross-sections. Finally, the mathematical model was formulated, describing contact areas of horizontal cross-sections as a function of the factors. Based on the conducted research, it was stated that an increase in vertical load caused reductions in both types of heights. Moreover, the width of tire profiles and the area of horizontal cross-sections increased due to the increase in vertical load (for bias-ply, increases were smaller than for radial tires). Similar changes were observed after the reduction of inflation pressure.

1. Introduction

The development of modern agriculture and intensification of production has caused a need for new machines with higher operational width [1,2]. These make it possible to achieve high efficiency in field operations, but they can cause higher vertical load (often, machines with greater width have higher masses) [3]. The use of these machines can lead to higher contact pressure, and the soil can be exposed to damage related to high compaction [4,5]. The results of previous research emphasize serious changes in the physical and mechanical properties of the soil compacted by the wheels of agricultural vehicles [6,7]. Compaction of the soil is a major problem influencing (in a destructive way) natural environment functioning [8,9,10]. The ability to uptake nutrients by plants is disturbed [11], which can lead to insufficient contents of carbon dioxide and rainwater. These disruptions often make field operations difficult and lower the yield of the plants [12,13]. On the other hand, high energy losses of agricultural machinery can occur [14].

Environmental protection requirements create new demands to reduce the negative consequences of the compaction of the soil by tractor wheels [15]. One agronomical way to improve the management of the compaction is the cultivation of deep-rooted plants, which can have a positive effect by loosening the soil structure [16]. However, attention to the technical and operational parameters of the machines is needed (especially in the aspect of the chassis parameters) [17].

Nowadays in agriculture, wheeled and tracked chassis are used. The most popular type is a wheeled system with the tire as the main element—it has direct contact with the ground [18]. Agricultural tires are divided into two types depending on their internal structure. The first is bias-ply tires, which have the same quantity of material in each part of the profile. Bias-ply tires are characterized by relatively high mechanical strength, but they are relatively stiff. The second type of agricultural tire has a radial structure. In this case, the material near the tread is thicker than on the side walls. These can cause better grip and lower soil compaction, but on the other hand, radial tires are more prone to mechanical damage [19,20].

The tire, as an element of the chassis, plays an important role in generating pressure on the soil. Its values are dependent on the contact area, which is determined by the type of the tire, the size of the wheel, inflation pressure, and vertical load [21,22,23]. The problem of excessive compaction has been the subject of intensive scientific research. One of the frequently described factors is the internal design of the tire. Originally, in agricultural vehicles, bias-ply tires were used. They are characterized by higher mechanical strength, but their traction abilities are worse than radial tires [24]. Moreover, due to their higher stiffness, bias-ply tires cause higher compactness than radial tires. It was reported that the use of radial tires in agriculture made it possible to obtain lower unit pressure and, as a consequence, less intensive compactness [25,26]. Other significant factors in the research concerning the tire–surface system are inflation pressure and the vertical load of the wheel [27,28]. Renčin et al. [29] concluded that even a small increase in inflation pressure (from 90 kPa to 120 kPa) led to large increases in the values of unit pressure on the soil. In the same research, a non-linear increase in the contact area of a tire (as a consequence of vertical load increases) was reported. Filipovic et al. [30] also analyzed the inflation pressure and stated that this factor is crucial for unit pressure and has an impact on the risk of excessive compactness.

In the literature, there are many publications regarding the contact area between the tire and the ground, but this problem is complex and hard to describe [31,32]. Most of the research is based on measuring the tire’s footprint on the soil [33,34]. In research conducted by Lamande et al. [35], loaded tires were placed on the soil, and the surface around them was sprinkled with gypsum. After lifting a tire, the footprint was drawn on foil, and then it was measured. Another method was used by Kumar et al. [18]. In this case, two sheets of carbon paper were placed between the tire and soil, and then the footprint generated on the paper was measured. The development of digital technology gave new possibilities to measure the tire–surface contact area [36]. One is a photometric method in which photographs of the footprints are analyzed in special software [37]. In research conducted by Farhadi et al. [14], a 3D scanner was used to measure a previously created gypsum imprint.

Those methods of assessing the tire’s impact on the soil are based on measurements in the place of the contact, and deformable surfaces are often needed. Due to the differences between the tire’s designs, their deformability will differ, so their impact on the soil will also differ. This confirms the need to conduct further research aimed at the assessment of the unit pressure generated by tires on the soil. For this reason, the aim of this study was the evaluation of changes related to tire deformation using a three-dimensional scanning method. In the first stage, the relationship between parameters (inflation pressure and vertical load of the tire) and dimensions of the vertical cross-sections of the tire will be analyzed. Then, the area of horizontal cross-sections at different exploitation parameters will be analyzed. Finally, the mathematical models will be formulated (separately for each of the tires).

2. Materials and Methods

The research was conducted in laboratory conditions on a non-deformable surface. Two types of tires (bias-ply and radial) of the same size were tested, with a profile width: 500 mm, profile height: 250 mm, rim diameter: 17 inches. Variable parameters used in the experiment were inflation pressure (3 levels: 0.8 bar, 1.6 bar, and 2.4 bar) and vertical load acting on the tires (5 levels of forces corresponding to the following masses: 800, 1200, 1600, 2000, and 2400 kg). The method of research was based on the scheme proposed by Ptak et al. [38], but some modifications related to the scanning process were used.

2.1. Test Bench

The test bench made it possible to study non-driven tires in static conditions; the scheme of the bench is presented in Figure 1. It was built from steel elements. The basis of the design was an external frame (3) connected to an internal frame (4). A hydraulic jack (6) was mounted between both frames because it was necessary to implement smooth changes in the vertical load on the tires. Vertical load values were measured using a TecSisforce sensor (5) with an accuracy of 50 N and a measurement range of 0–100 kN. The tire was placed on the shaft, which was mounted (using bearings) on the internal frame. Screw mechanisms (7) were intended to lock the internal frame (i.e., in the case of the accidental reduction of oil pressure in the hydraulic jack). The surface used in the experiment (1) was a flat steel bar 4 mm thick; this element was the reference basis in the optical analysis of the tire profile. The changes in inflation pressure were obtained using an HLO 215-25 compressor.

2.2. Scanning Process

A SMARTTECH3D scanner was used for the scanning process, the main parameters of which are presented in

Table 1. Technical specification of 3D scanner.

Parameter	Description
Scanning technology	white structural light-LED
Measuring volume (x, y, z) [mm]	400 × 300 × 240
Distance between points [mm]	0.156
Accuracy [mm]	0.08
Power consumption during measurement [W]	200
Weight [kg]	4.40
Working temperature [°C]	20 ± 0.5

. The scanner was connected to a notebook fitted with special SMARTTECH3D measuring software. Before the start of the scanning process, the tested tire was covered with a white matte layer, which made it possible to obtain higher contrast. After mounting the tire, the test bench was placed on a rotary table connected to the software of the scanner, and then the main parameters, such as vertical load and inflation pressure, were entered. The rotary table made it possible to determine the basic parameters of the scanning process, i.e., the angular range of the scanning, angles of each rotation, and the number of scanning steps. In the experiment, the angular range of 360° was used, and a single step corresponded to the rotation of 20°. The effects of this operation were 18 individual scans with the ground created by clouds of points. As an effect of rotational scanning, a spatial picture of the tire was obtained. The process was applied in each factor combination (tire—inflation pressure—vertical load).

2.3. Management of Obtained Data

The effects of the scanning process were represented by a cloud of points which reflected the geometry and shape of the tire in real conditions. Performing further analysis was possible after the transformation of the cloud into a triangular mesh. In turn, the mesh was used to draw a vertical and horizontal cross-section of the tire profile. The cross-section was exported to the AutoCad software, which made it possible to determine specific dimensions of the tire profile. The plane used to perform the vertical cross-section was perpendicular to the surface, and it covered the center line of the tire. Figure 2 shows an example of a vertical cross-section with typical dimensions (total height of the profile and height measured at the maximum transverse deformation of the profile). In Figure 2, red lines show the part of the tire tread that could not be scanned (due to the lack of access to laser beams). However, this part was not the subject of the research—the main attention was paid to the edges of the profile and their deformations.

At the height corresponding to the maximum deformation, the horizontal cross-section was made, which was further used to read the width of the profile. Then, in AutoCad software, the area of the cross-section at the maximum deflection was read; it was assumed that it would be the area of the tire’s contact with the surface (at the maximum deformation). The example of the horizontal cross-section is presented in Figure 3.

2.4. Statistical Analysis of Results

The results were examined using statistical analysis. The first part included the evaluation of the factor’s impact on tire dimensions and the area of horizontal cross-section. First, the test of distribution normality was done (using the Shapiro–Wilk test at the significance level α = 0.05). Then, the test of variance homogeneity was conducted (Levene test at the significance level α = 0.05). The statistical procedures had to determine the possibility of ANOVA use to evaluate the influence of the factors on the parameters. When the criterion was fulfilled (distributions in accordance with normal distribution, homogeneous variances), the two-factor ANOVA test at the significance level α = 0.05 was used. In other cases, the evaluation was conducted using the non-parametrical Kruskal–Wallis test at the significance level α = 0.05. Both in the first and other cases, the statistical procedure included post-hoc tests—they had to show significant differences between each of the levels of the factors.

The statistical analysis also included formulating a mathematical model separately for each of the tires. These models had to describe the relationship between the factors (vertical load, inflation pressure) and the horizontal cross-section area. Mathematical models were created using TableCurve 3D software.

3. Results

The experiment led to the results regarding vertical and horizontal deformation of two tires with the same external dimensions and different internal structures. The variable factors were inflation pressure and the vertical load of the wheel.

Figure 4 shows the visualization of vertical cross-sections of both tires with all values of vertical load and inflation pressure. The highest deflection of the tire was observed at the vertical load of 2400 kg (gray color). However, at the lowest inflation pressure (0.8 bar) in the case of the radial tire, differences in deflections were quite low. It can be stated that in these cases, the deflection was not dependent on vertical load. The bias-ply tire with the inflation pressure of 1.6 bar had greater differences in deflection; the two lowest levels of vertical load (800 kg and 1200 kg) caused a significant difference compared to other levels of the load (1600 kg, 2000 kg, and 2400 kg).

Parameters read from vertical cross-sections were height of the tire profile (h_p) and height corresponded to maximum deflection (h_u). Figure 5 shows the values of h_p and h_u at all factor combinations (inflation pressure, vertical load) for both tires. In each case, decreases in heights due to an increase in vertical load were different. The highest value of total profile height of the bias-ply tire was observed at the inflation pressure of 2.4 bar, and the lowest vertical load, 205 mm. For the radial tire at the same vertical load and inflation pressure, the height of the profile was 197 mm. In the case of the radial tire at 0.8 bar inflation pressure, an increase in vertical load from 1200 to 2400 kg caused a reduction in deflection by 6.4 mm. It was a relatively small difference, and it could be assumed that there is some limit to deflection; a further increase in vertical load, especially at low inflation pressure, will not cause high changes in deflection. These changes do not have to be related to the changes in the height of maximum deflection (h_u). The comparison of both tires made it possible to conclude that the h_u parameter for the radial tire was more differentiated than for the bias-ply tire. In the latter case, the height of maximum deflection decreased due to an increase in vertical load; this tendency was observed at all levels of inflation pressure. The lowest value of the h_u parameter was observed at the inflation pressure of 0.8 bar. In the case of the radial tire, it was observed at the vertical load of 1600 kg (the h_u value was 50.4 mm), while for the bias-ply tire, this situation occurred at the vertical load of 2400 kg (h_u = 37.2 mm). At the highest vertical load and highest inflation pressure, the heights of maximum deflection were 68.2 mm and 74.4 mm for the radial tire and bias-ply tire, respectively. Similar values were observed for the radial tire at the inflation pressure of 2.4 bar and vertical load of 800 kg. For the bias-ply tire, this tendency occurred at the inflation pressure of 0.8 bar and vertical load of 1200 kg.

The next step in the research description was the preparation of horizontal cross-sections at the height of maximum deflection, shown in Figure 6. The cross-sections correspond to vertical cross-sections and confirm the previously described tendency concerning tire deformation. Moreover, Figure 6 shows the difference in the thread-parts of the tire with their edges at different distances from each other. It shows that the points of maximum deflection were located at different heights.

The parameters read from horizontal cross-sections were their total area (A_hu) and the width (b_hu). Figure 7 shows the values of the area of horizontal cross-sections. The highest values of the parameter were observed for both tires at the highest vertical load (2400 kg) and the lowest inflation pressure (0.8 bar). The areas were 0.325 m² and 0.316 m² for the radial and bias-ply tire, respectively. At the highest (2.4 bar) and lowest (0.8 bar) inflation pressures for each vertical load, higher cross-section areas were observed for the radial tire. Only in the case of the inflation pressure of 1.6 bar and vertical load of 800 kg was the area of cross-section higher for the bias-ply tire.

Figure 8 shows the values of cross-section width (b_hu parameter). The highest values of this parameter were obtained at the inflation pressure of 0.8 bar, which was observed for both tires. In the case of the radial tire, similar values of areas were observed at the vertical loads of 1200 kg, 2000 kg, and 2400 kg; in each of these three cases, they were about 540 mm (0.54 m). For the radial tire, the values higher than 540 mm were obtained only at the vertical load of 2400 kg (546 mm). The proportional increase in the cross-section area due to a load increase was observed in each inflation pressure and vertical load combination for both tires; the exception was the radial tire at the inflation pressure of 0.8 bar. Comparison of both tires made it possible to conclude that higher values of the cross-section area were obtained for the radial tire. It can be the basis for the statement that the radial tire is more elastic and more exposed to deformations.

The results were confirmed using statistical analysis. The first step was the evaluation of the possibility of using ANOVA tests to describe the factor impact on the analyzed parameters. Then, two-factor ANOVA tests were conducted separately for each of the tires.

Table 2. Results of statistical analysis for radial tire, significance level α = 0.05, SD—standard deviation.

Analyzed Parameter	Factor	Factor Level	Arithmetic Mean	± SD	p-Value
Height of tire profile (hp), mm	Vertical load	800 kg	175.1 A	24.03	<0.00001
		1200 kg	164.7 B	26.41
		1600 kg	155.8 C	22.01
		2000 kg	149.8 CD	18.21
		2400 kg	143.9 D	17.25
	Inflation pressure	0.08 MPa	132.6 A	9.89	<0.00001
		0.16 MPa	163.3 B	15.71
		0.24 MPa	177.9 C	16.35
Height of maximum deformation of the tire (hu), mm	Vertical load	800 kg	72.2 A	7.92	0.00008
		1200 kg	74.4 A	12.58
		1600 kg	64.1 B	13.02
		2000 kg	64.9 B	6.82
		2400 kg	63.1 B	5.42
	Inflation pressure	0.08 MPa	58.9 A	5.58	<0.00001
		0.16 MPa	66.8 B	7.71
		0.24 MPa	77.5 C	7.70
Width of the tire (bhu), mm	Vertical load	800 kg	522.5 A	17.87	0.00002
		1200 kg	527.4 A	16.57
		1600 kg	529.2 AB	10.50
		2000 kg	536.5 B	9.35
		2400 kg	541.5 C	7.08
	Inflation pressure	0.08 MPa	544.1 A	5.68
		0.16 MPa	530.8 B	9.60
		0.24 MPa	519.3 C	13.07
Area of cross-section (Ahu) m2	Vertical load	800 kg	0.290 A	0.019	<0.00001
		1200 kg	0.300 B	0.015
		1600 kg	0.303 B	0.008
		2000 kg	0.313 C	0.009
		2400 kg	0.317 C	0.007
	Inflation pressure	0.08 MPa	0.319 A	0.007	<0.00001
		0.16 MPa	0.298 B	0.015
		0.24 MPa	0.297 B	0.011

The letters at arithmetic means (A, B, C, D) denote separate homogenous groups.

shows the results of the statistical analysis for the radial tire. The p-value describes the probability of acceptance of the hypothesis about the lack of significant impact of the factor. When the p-value is greater than the significance level (in this case, α = 0.05), the factor is not significant for the parameter. Homogenous groups are denoted by letters A–D, and they mean the levels of factors which did not cause significant differences in the analyzed parameters.

Based on Table 2, it can be stated that both vertical load and inflation pressure had a significant impact on all parameters (in all cases, p-values were much lower than the significance level). Post-hoc tests conducted for the first factor (vertical load) showed that significant differences in h_p values were observed between each of the first four levels. Only the two highest levels of vertical load (2000 kg and 2400 kg) were classified into one homogeneous group, which means that there were no significant differences in h_p values between these levels. In the case of h_u values, homogenous groups were observed for the two first levels (800, 1200 kg) and the other three levels (1600, 2000, 2400 kg). Analysis of changes in tire width showed that the first homogeneous group was observed for the first three levels of vertical loads. The next homogeneous group was created by levels 1600 and 2000 kg (1600 kg was classified to the first homogeneous group, while the last homogeneous group was characterized by the highest vertical load (2400 kg)). Significant differences for the last parameter (the area of cross-section) were observed between the lowest level, two medium levels (1200, 1600 kg) and two highest levels (2000, 2400 kg). A post-hoc test for the second factor (inflation pressure) showed the differences in h_p values between each of the levels. The same situation was observed for h_u values and the values of tire width (b_hu). Significant differences in the cross-section area were observed between the lowest level (0.8 bar) and two other levels (1.6, 2.4 bar).

The statistical analysis for the bias-ply tire is presented in

Table 3. Results of statistical analysis for bias-ply tire, the significance level α = 0.05, SD—standard deviation.

Analyzed Parameter	Factor	Factor Level	Arithmetic Mean	± SD	p-Value
Height of tire profile (hp), mm	Vertical load	800 kg	195.3 A	12.34	<0.00001
		1200 kg	188.9 A	11.32
		1600 kg	170.7 B	17.38
		2000 kg	159.7 B	19.72
		2400 kg	145.7 C	24.88
	Inflation pressure	0.08 MPa	159.2 A	25.94	<0.00001
		0.16 MPa	166.8 A	23.33
		0.24 MPa	190.2 B	14.40
Height of maximum deformation of the tire (hu), mm	Vertical load	800 kg	89.8 A	6.54	<0.00001
		1200 kg	85.6 A	6.15
		1600 kg	71.3 B	11.63
		2000 kg	65.7 B	12.92
		2400 kg	51.9 C	17.33
	Inflation pressure	0.08 MPa	65.5 A	18.40	<0.00001
		0.16 MPa	67.0 A	17.33
		0.24 MPa	86.0 B	8.94
Width of the tire (bhu), mm	Vertical load	800 kg	497.5 A	7.38	<0.00001
		1200 kg	501.8 A	10.05
		1600 kg	512.8 B	14.14
		2000 kg	521.0 B	15.38
		2400 kg	528.9 C	19.04
	Inflation pressure	0.08 MPa	524.2 A	16.56
		0.16 MPa	515.1 B	16.55
		0.24 MPa	497.9 C	7.27
Area of cross-section (Ahu), m2	Vertical load	800 kg	0.280 A	0.005	<0.00001
		1200 kg	0.284 AB	0.005
		1600 kg	0.288 B	0.008
		2000 kg	0.297 C	0.005
		2400 kg	0.302 C	0.012
	Inflation pressure	0.08 MPa	0.297 A	0.012	<0.00001
		0.16 MPa	0.290 B	0.010
		0.24 MPa	0.283 C	0.007

The letters at arithmetic means (A, B, C) denote separate homogenous groups.

. Explanations of symbols are the same as in Table 2.

Statistical analysis of the results obtained for the bias-ply tire showed that all factors had a significant impact on all analyzed parameters (p-values were smaller than the significance level α). Post-hoc tests for h_p values showed that the two lowest levels created a separate group, then the levels 1600 kg and 2000 kg created the next homogeneous group; finally, the higher vertical load level created a separate group. Moreover, significant differences were observed between the two groups—the first was created by the lowest inflation pressure while the second by the highest pressure. Post-hoc tests for h_u values made it possible to conclude that non-significant differences in this parameter were observed at the two first levels of vertical load (they created one homogeneous group). A similar situation was observed for the two next levels (1600 kg and 2000 kg); the highest level of the load created a separate homogenous group. For the second factor (inflation pressure), significant differences were observed between the highest level (2.4 bar) and two other levels. In the case of tire width, the influence of vertical load was the same as in the case of the h_p parameter (homogenous groups were created by the same levels of vertical load). A post-hoc test for the second factor showed significant differences between each of the levels. Significant differences in values of areas of cross-sections were observed between three groups: the first was created by two lowest levels, the second group included the levels of 1200 kg and 1600 kg, while the third group was created by two highest levels. The second factor (inflation pressure) caused significant differences between all three levels (at each inflation pressure value, there were significant differences between the cross-section areas).

Statistical analysis included both analysis of variance and mathematical model formulation. After verification of factor significance, equations describing the cross-section area as a function of vertical load and inflation pressure were created. The models were formulated separately for the radial and bias-ply tire.

3.1. Model for Radial Tire

The overall form of the equation (Equation (1)) is:

$$A_{hu}=\mathrm{a}+\mathrm{b}·\mathrm{lnG}+\mathrm{c}·{\left(\mathrm{lnG}\right)}^2+\frac{\mathrm{d}}{p}$$

(1)

$${\mathrm{R}}^2=0.881$$

where:

• A_hu—area of horizontal cross-section (m²),
• G—vertical load (kg),
• p—inflation pressure (bar),
• a, b, c, d—parameters (constants of equation).

The final form of the equation (Equation (2)) is:

$$A_{hu}=0.491-0.081·\mathrm{lnG}+0.007·{\left(\mathrm{lnG}\right)}^2+\frac{0.028}{p}$$

(2)

In Table 4 the values of parameters of Equation (2) are presented while Table 5 shows the results of the verification of the model concerning the radial tire. It includes a comparison of measured values of the cross-section area (the arithmetic mean from the replications) and values calculated from the mathematical model. Moreover, the values of an estimation error were presented; the error was calculated from the following equation (Equation (3)):

$$\mathrm{Err}=100·\left(\frac{{\mathrm{A}}_{\mathrm{m}}-{\mathrm{A}}_{\mathrm{c}}}{{\mathrm{A}}_{\mathrm{m}}}\right),\%$$

(3)

where:

• Err—error of estimation, %,
• A_m—measured area of cross-section, m²,
• A_c—calculated area of cross-section, m².
• Analysis of obtained results made it possible to conclude that the highest error was 4.68%, the lowest error was 0.17%, and the mean error was at the level of 1.62%.

Table 4. Values of equation parameters for radial tire.

Parameter	Value
a	0.491
b	−0.081
c	0.007
d	0.028

Table 4. Values of equation parameters for radial tire.

Parameter	Value
a	0.491
b	−0.081
c	0.007
d	0.028

Table 5. Verification of mathematical model for radial tire.

Vertical Load, kg	Inflation Pressure, bar	Area of Cross-Section (Measured)	Area of Cross-Section (Calculated from Model)	Error, %
800	0.8	0.315	0.305	3.22
1200	0.8	0.322	0.313	2.93
1600	0.8	0.309	0.319	3.21
2000	0.8	0.324	0.325	0.29
2400	0.8	0.325	0.331	1.71
800	1.6	0.275	0.288	4.68
1200	1.6	0.297	0.295	0.73
1600	1.6	0.294	0.302	2.47
2000	1.6	0.312	0.308	1.32
2400	1.6	0.314	0.313	0.34
800	2.4	0.281	0.282	0.17
1200	2.4	0.291	0.289	0.59
1600	2.4	0.296	0.296	0.22
2000	2.4	0.304	0.302	0.72
2400	2.4	0.313	0.307	1.76

Table 5. Verification of mathematical model for radial tire.

Vertical Load, kg	Inflation Pressure, bar	Area of Cross-Section (Measured)	Area of Cross-Section (Calculated from Model)	Error, %
800	0.8	0.315	0.305	3.22
1200	0.8	0.322	0.313	2.93
1600	0.8	0.309	0.319	3.21
2000	0.8	0.324	0.325	0.29
2400	0.8	0.325	0.331	1.71
800	1.6	0.275	0.288	4.68
1200	1.6	0.297	0.295	0.73
1600	1.6	0.294	0.302	2.47
2000	1.6	0.312	0.308	1.32
2400	1.6	0.314	0.313	0.34
800	2.4	0.281	0.282	0.17
1200	2.4	0.291	0.289	0.59
1600	2.4	0.296	0.296	0.22
2000	2.4	0.304	0.302	0.72
2400	2.4	0.313	0.307	1.76

3.2. Model for Bias-Ply Tire

The overall form of the equation (Equation (4)) is:

$$A_{hu}=\mathrm{a}+\mathrm{b}·\mathrm{lnG}+\mathrm{c}·{\left(\mathrm{lnG}\right)}^2+\mathrm{d}·{\left(\mathrm{lnG}\right)}^3+\mathrm{e}·p$$

(4)

$${\mathrm{R}}^2=0.895$$

where:

• A_hu—area of horizontal cross-section (m²),
• G—vertical load (kg),
• p—inflation pressure (bar),
• a, b, c, d, e—parameters (constants of equation).

The final form of the equation (Equation (5)) is:

$$A_{hu}=3.523-1.250·\mathrm{G}+0.158·{\left(\ln \mathrm{G}\right)}^2-0.006·{\left(\ln \mathrm{G}\right)}^3-0.008·p$$

(5)

Table 6. Values of equation parameters for bias-ply tire.

Parameter	Value
a	3.523
b	−1.250
c	0.158
d	−0.006
e	−0.008

presents the values of parameters of Equation (4).

Table 7. Verification of mathematical model for bias-ply tire.

Vertical Load, kg	Inflation Pressure, bar	Area of Cross-Section (Measured)	Area of Cross-Section (Calculated from Model)	Error, %
800	0.8	0.285	0.287	0.65
1200	0.8	0.288	0.290	0.66
1600	0.8	0.295	0.296	0.17
2000	0.8	0.301	0.303	0.78
2400	0.8	0.316	0.310	2.14
800	1.6	0.278	0.280	0.95
1200	1.6	0.284	0.283	0.29
1600	1.6	0.287	0.290	1.03
2000	1.6	0.301	0.296	1.39
2400	1.6	0.301	0.303	0.54
800	2.4	0.278	0.274	1.52
1200	2.4	0.280	0.277	1.10
1600	2.4	0.281	0.283	0.71
2000	2.4	0.292	0.290	0.84
2400	2.4	0.290	0.296	2.27

shows the results of the verification of the model for the bias-ply tire. The error of estimation was calculated using the same formula as in the case of the radial tire.

Based on the results (Table 7), it can be concluded that the highest value of the error was 2.27%, while the lowest error was 0.17%. The mean error calculated from all cases was 1.01%.

4. Discussion

The results made it possible to determine the differences in the deformation of radial and bias-ply tires with the same dimensions at different levels of vertical load and inflation pressure. The experiment was conducted on a non-deformable surface. Based on obtained results, two mathematical models were created. Similar experiments were conducted by other researchers. Sharma and Pandey [39] and Grečenko [40] created models and formulas describing the deformation of agricultural tires. However, these authors found that mathematical models needed large amounts of data obtained from experiments conducted on many tires under different conditions. In addition, there are studies describing the deformation of tires and the contact area on deformable surfaces (often on agricultural soil). The main advantage of the present models is related to their simple form. Unlike other models, they used just two initial parameters (inflation pressure and vertical load, while other models also need parameters of the surface).

It was concluded that an increase in vertical load caused an increase in the cross-section area (it can be equated with the contact area). In turn, an increase in inflation pressure caused a reduction in the cross-section area. This tendency is also confirmed by Teimourlou and Taghavifar [41]. In an experiment conducted by Renčin et al. [29], the reduction in inflation pressure of a radial tire (from 1.6 bar to 0.8 bar) caused an increase in the tire imprint area by 21%. In our research, a similar change in inflation pressure caused increases in the cross-section area by 6–13% (it was dependent on vertical load). Different results were obtained by Raper et al. [28]. In this case, the changes in contact area after the change in inflation pressure were significantly smaller than in our research. However, the impact of a vertical load increase (reported by Raper et al. [28]) was different from the results of our research (an increase in vertical load by 90% caused an increase in the area by more than 30%, while in our research an increase in vertical load by 100% caused an increase in the area by 1–4%, dependent on inflation pressure).

Some studies described the accuracy of methods; according to one of them, using only the tire size to determine the contact area [42] can produce an error of 70% [43]. Schjønning et al. [44] compared two radial tires with different dimensions (650/65R30.5 and 800/50R34) from different manufacturers. Based on an elliptical model of the tire contact area, they concluded that a tire with a lower size caused a much higher footprint than a tire with larger dimensions. In turn, Way and Kishimoto [45] tested an 18.4R38 tire, and they observed very similar values of the contact area in each factor combination (described by different values of vertical load and inflation pressure).

In summary, comparing changes in tire performance under variable vertical load and inflation pressure is necessary to obtain information about tire deformation and the area of its contact with the surface. Knowledge about the relationship between tire elasticity and the contact area is crucial to obtaining better soil protection.

5. Conclusions

The analysis of the relationship between exploitation factors and deformability of the tires made it possible to conclude that both inflation pressure and vertical load were significant factors affecting the analyzed parameters.

1. An increase in vertical load of the wheel caused reductions in analyzed heights (both height of tire profile and height of maximum deflection of the tire). This tendency was observed for both tires, but in the case of the radial tire, the values of the heights were lower than for the bias-ply tire.

2. A reduction in inflation pressure led to reductions in values of the heights (height of tire profile and height of maximum deflection) both for the radial and bias-ply tire. For the radial tire, the impact of inflation pressure on the heights was lower than in the case of the bias-ply tire.

3. The values of width of the tire profile were comparable for the radial and bias-ply tire. An increase in vertical load of the wheel with the radial tire caused an increase in the width of the tire profile—higher differences were observed at high values of inflation pressure. However, an increase in vertical load of the bias-ply tire caused an increase in profile width only at two lower levels of inflation pressure (0.8 bar and 1.6 bar).

4. In the case of the radial tire, the area of the horizontal cross-section was higher than for the bias-ply tire. A reduction in inflation pressure of the radial tire caused significant increases in the area of horizontal cross-sections. The highest differences were observed after the reduction of inflation pressure from 1.6 bar to 0.8 bar. In the case of the bias-ply tire, an inflation pressure reduction caused lower increases in the area of cross-section. This tire was more exposed to the changes in the area of cross-section after the changes in vertical load. At each inflation pressure, proportional increases in the cross-section area due to the increases in vertical load were observed.