Equivalent Tyre Contact Radius

Mechanistic pavement analysis — the layered-elastic route, and the mechanistic-empirical methods built on top of it — needs a load it can integrate. The closed-form stress and strain solutions for a layered elastic half-space exist for a uniformly loaded circle. They do not exist for the shape a truck tyre actually leaves. So the first move in every such analysis is to replace the real footprint with the circle of equal area at equal pressure, and this is that circle's radius.

Two things about the square root are worth internalising. The radius grows as the square root of the load, so doubling the wheel load widens the patch by only 41%. The extra load therefore does not spread proportionally — it pushes the pressure bulb deeper. That is the mechanical picture behind why heavy axles reach down and stress the subgrade while light ones are absorbed by the surface layers, and it is worth carrying alongside the fourth-power arithmetic, which tells you the same thing without explaining it.

And the radius sets the depth scale for everything below. Stresses beneath a circular load are conventionally read in multiples of $a$: the vertical stress at a depth of one radius is roughly two-thirds of the contact pressure, at two radii about a quarter, at four radii under a tenth. A pavement structure is thin or thick relative to the contact radius, not in absolute inches, which is why the same 200 mm of asphalt behaves like a thick pavement under a car and a thin one under a loaded truck.

Where the circular idealisation is good and where it fails. Deep in the structure it is excellent, and the reason is Saint-Venant's principle: by two or three radii down, the stress field has forgotten the shape of the load and remembers only its resultant and its centroid. A rectangle, a circle and a pair of duals of the same total load are indistinguishable at subgrade level. Near the surface it is poor — and the surface is where the asphalt is. Tensile strain at the bottom of a thin asphalt layer, shear stress just outside the tyre edge, and top-down cracking, which initiates at the surface beside the tyre and propagates downward, all depend on the real footprint shape and the real non-uniform pressure. Top-down cracking is now recognised as a major distress mode on thick asphalt pavements, and the circular-load model cannot see it at all.

The dual-wheel case is the honest edge. Two tyres side by side with a gap between them are not one circle and are not two independent ones either: their stress bulbs merge somewhere below the surface, so near the top you must treat them separately and lower down you may treat them as a single larger load. Where the crossover happens depends on the spacing and the layer stiffnesses, which is precisely the sort of question a layered-elastic program is for and a hand calculation is not.