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Ideal and real trajectory

Tire rolling, slip angle and tire-to-surface transmission of forces  

Massimo Clarke

The relation between tires and road surface is a rather complex one. Over the years, a number of production techniques have been adopted and standardized in order to guarantee better performances; as far as types and quantity of raw materials, though, it seems that every manufacturer follows its own path without sharing any information. And rightly so if they want to protect their technological edge. Different compounds are developed to allow tires to best perform their core functions, i.e. carry the load, guarantee a comfortable ride, provide good road handling and effectively transfer large forces to the ground. Clearly then, a good tire must possess an adequate elasticity. If perfectly rigid, as absurd as it may sound, it just would not work: the contact patch would be reduced to a line, and the load pressure exercised on the wheel (which, as known, is a force exerted on a surface per unit area) would not simply be tremendously high, it would be immeasurable!

No material could resist. This extreme situation is purely theoretical (perfectly rigid bodies do not exist!). Even when hard and compact materials are employed, such as certain types of steel or cast iron, where mechanical parts come into contact, a small elastic deformation takes place, which enables the type of force involved to be exerted over an area, however small it may be. That is precisely what happens, for example, in the case of metal wheels for railway use, or bearing rollers; the contact pressures are extremely high but bearable.

If wheels were too rigid, they would hardly contribute to a comfortable ride (no suspension, shock or vibration dumping….), but more importantly, they would not allow the driver to have any control of the vehicle at all. No wonder then, that vehicles equipped with metal wheels travel on rails.


It is clear at this point that the elastic deformations of tires must be considered seriously. And not just to ensure a comfortable ride, but most importantly to form the best possible contact patch. Which is precisely where the forces are transferred, both longitudinal and transversal, (the former while accelerating and braking, the latter acts when the vehicle is not travelling in a straight line). Grip, which is ultimately responsible for the effective transmission of such forces, is generated either by mechanical anchoring devices or by forms of adhesive phenomena. It relates to the tire’s ability to transfer forces to the ground. “Traction circles” are used to graphically indicate the total grip available, wherein the radius (the distance between the center of the tire and its circumference) indicates the maximum overall grip a tire can provide.

When braking, using all the grip that a tire is able to provide, lateral forces cannot be transmitted to the road. On the other hand, if all available grip is used to counter lateral forces, tires would not be in the position of dealing with longitudinal forces. That is why, when cornering at high speed, braking or accelerating becomes almost impossible. In intermediate conditions, the resultant of the two vectors that indicate the longitudinal and transversal forces involved must not be longer than the radius, i.e. they must remain within the traction circle (which, in fact, is almost always slightly elliptical). Otherwise, a loss of grip will occur, i.e. the grip required is greater than what is available.


Because of their elasticity, tires are deformed under the action of both the above-mentioned forces. At the contact patch, a tire must bear the load that weights on it, and therefore deforms elastically; this depends on, besides the load itself, the structural stiffness of the tire and (especially) the inflation pressure. This means that the actual rolling radius is lower than the geometrical radius of the wheel.

While rolling, a tire is compressed and stretched at the sidewalls, which, in turn, will affect the whole tire, while the contact patch or footprint changes continually. When standing still, the loads are exercised in a symmetrical manner, with the highest ratios in the mid-section; however, when turning freely, without longitudinal or transversal forces being transmitted, pressure distribution becomes asymmetrical, with a forward displacement of the maximum value. 

Travelling in a straight line, when longitudinal forces come into play (when accelerating or braking) no change occurs to the trajectory. Every element of the tire itself, and in particular the tread, deforms longitudinally starting from the contact patch. A slipping takes place and during acceleration the wheel rotates faster than the speed of the vehicle itself while, when braking, the opposite is true. Tire grip, therefore, also varies as the tire slips during acceleration or braking.

When cornering, transversal or lateral forces come into play. The rolling wheel’s actual direction of travel and the direction towards which it is pointing are different. Such behavior is caused by a lateral slip, caused by sideway forces greater than the tire’s friction resistance, which will produce an elastic transversal deformation of the contact patch. The front section of the contact patch will suffer a slight side shift compared with the rear section. The resulting slip angle is formed between the deflected and undeflected tread paths. 

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