## 2 Theoretical Background
### The Pressure Head of the Tyre  
Vehicles stand on the pavement on their wheels. The wheel load is transmitted to the road pavement structure by the tyres. A plausible guess is that the \(F\) pressing force of the vehicle load is evenly distributed at the \(A\) contact surface area of the tyre:  
\[p=\tfrac{F}{A}, \tag{1}\]  
where \(p\) is the inflation pressure of the tyre, which is approximately equal to \(q\), the specific contact pressure transmitted to the pavement by the tyre. Strictly speaking, the equality \(q\approx p\) is true only in the case of an absolutely flexible tyre with no inner stiffness (Figure 1a). In case of a tyre that has an inner stiffness, the contact pressure can be higher or lower, even by 10-30%, than the inflation pressure of the tyre, according to the former literature (see, e.g., [8; 9; 10]).  
These contradictory results have been explained by [11], who verified, by a physical consideration, that the specific contact pressure \(q\) of a balloon is, by necessity, lower than its inner air pressure \(p\) (Figure 1b):  
\[q=p-\tfrac{4T}{D}\sin\theta+\tfrac{F^{\prime}}{A}, \tag{2}\]  
where \(p\) is the inflation pressure, \(q\) is the contact pressure, \(T\) is the tensile force/unit of length (at the rim of the contact surface), \(\theta\) is the angle between horizontal plane and direction of force \(T\), \(A\) is the contact circular surface, \(D\) is the diameter of the contact circular surface, and \(F^{\prime}\) is the load transmitted by the wall of the balloon.  
Since the tensile force \(T\) depends on both the inner air pressure \(p\) and the external loading force \(F\), the unknown average contact pressure \(q\) also depends on \(p\) and \(F\); therefore, generally, the following equation can be defined:  
\[q=f_{1}(p)+f_{2}(F,p), \tag{3}\]  
where \(f_{1}(p)\) is the function of inflation pressure and \(f_{2}(F,p)\) is the combined function of inflation pressure and load.  
[PERSON] extrapolated Equation (3), concerning balloons, to rubber tyres, where the physical structure of the tyre, its size, its tread type, the rubber toughness, and the wall thickness are introduced by a \(C^{\prime}\) factor [11] as follows:  
\[q=f_{1}(p)+f_{2}(F,p)+C^{\prime}. \tag{4}\]  
This theoretical function (4) was verified by [11] via laboratory measurements on an 8.25-20 \(\times\) 10-ply rubber tyre:  
\[q=(0.013p+10.5)F+0.119p+125.9 \tag{5}\]  
in a generalised form,  
\[q=(a_{1}p+a_{2})F+b_{1}p+C^{\prime}, \tag{6}\]  
where \(q\) and \(p\) are in kPa and the wheel load \(F\) is in kN. According to the empirical Equation (6), the contact pressure depends on the wheel load, inflation pressure of the tyre and, mainly, the \(C^{\prime}\) constant characterising the \"flexibility\" of the tyre. Similar relationships were provided by [11] for further tyre types and tested in the laboratory; their parameters can be found in Table 1.  
The theoretical correlations can be verified by a finite element analysis. Among the most widely presently used finite element software is ABAQUS/CAE (Complete Abaqus Environment), which provides an efficient and complete solution for a wide range of industrial tasks. Leading tyre manufacturers, i.e., Hankook, started to utilise ABAQUS software at the beginning of the 2000s for developing modelling techniques that are in use today [12]; therefore, this software was chosen as suitable for our current research work.  
Figure 1: The power play of an absolutely flexible balloon (**a**) and a balloon with wall stiffness (**b**) contacting a smooth horizontal plane.