Our purpose is to explain why highways in California seem to be damaged by the heavy trucks more than routes in Brittany by les camions

For the solution to the Five Axles puzzle, we shall refer to sketches and make comparisons of semi trailer-trucks.  The most common configurations plying the respective roadways are the C-rig, an 18-wheeler typically observed in California and the B-rig, which is a 12-wheeler typically observed in Brittany.  Both rigs have five axles. Normalized dimensions will be used, wherein the length of the load-carrying compartments of both rigs are postulated as the same, setting xC = 50 feet (15.2 meters ).  By coincidence, the length of the wheel-base for the B-rig xW = 50 feet, and for the C-rig the wheel-base xW = 63 feet, or about 25% longer.  Complying with regulations, the ‘roadway footprint’ of the B-rig is designed to be 13% shorter than the C-rig (62 feet bumper-to-bumper vs 70 feet).

The center of gravity (CG) for the uniformly loaded trailer in each rig is shown in the sketch below.  Its location relative to the Kingpin is represented by the symbol xLOne expects that for both rigs, the loaded weight of trailers wL must be much greater than the weights of tractors wT.  Let wL/wT ≈ 8, say.  The CG for the B-rig tractor relative to its Kingpin xT is shifted slightly forward (3 feet) from that in the C-rig, which is explainable by the cabine avancé (cab-over-engine) configuration on the B-rig versus cabine conventionelle (conventional cab) on the C-rig. The location of the Kingpin on the B-rig trailer is shown to be shifted rearward by 5 feet compared to that in the C-rig trailer.  Dimensions are summarized in the table below the sketches.

 Table of  Dimensions Symbol Description C-rig B-rig (alpha) (relative to Kingpin*) feet meters feet meters xB Highway Footprint 70.0 21.3 62.0 18.8 xC Load Capacity 50.0 15.2 50.0 15.2 xF Front Axle* 19.0 5.7 19.0 5.7 xL Load CG* 22.0 6.8 15.0 4.6 xR Rear Axles* 41.0 12.6 26.0 7.8 xS Axle Separation 6.0 1.7 6.0 1.7 xT Tractor CG* 8.0 2.2 10.0 3.1 xW Wheelbase 63.0 19.2 50.0 15.2

Weight regulations for all kinds of roadway vehicles are administered based on the distribution of static loads among axles.  Accordingly, for the solution to the Five Axles puzzle, we must estimate the individual axle-loadings (forces f1 to f5), and that takes some algebra (algebra lovers are invited to request the derivations here).

For the sketch below, consider yourself to be a stationary observer positioned at the far left of the horizontal line taking measurements while either the C-rig truck and the B-rig truck rolls over your observation point from right to left.  For comparing static loads, we will consider fully loaded (80,000 lbs) trucks to be moving ever so slowly over a weighing station.

The forces supported by the roadway
have been depicted in the sketch below as bell-shaped curves superimposed upon the axle locations.  These forces are used for regulatingEach force curve reaches its peak as the respective axle passes directly over the observation point.  Our reasoning is elementary.  The force is 'felt' in the pavement structure below the observation point in pavement layers both ahead of and behind the axle.  The dashed curves represent estimates for the axle-loads for an unloaded trailer.
The most pronounced outcome shown in the sketch below, is that for the B-rig the peak axle-load  fB2 = 34,240 lbs which  is greater than any of the axle loads for the C-rig -- by a factor of about 1.75-to-one.

The C-rig axle-loadings on fC4 and fC5 are carried by eight trailer wheels, while the B-rig loadings on fB3, fB4 and fB5 are carried on only six wheels.  For load-per-wheel, that represents a ratio of more than 1.4-to-one in favor of the C-rig.  Clearly, our solution to the Five Axles puzzle will allow us to invoke neither axle nor wheel loadings for the B-rig.

Static vs Dynamic Loads

In common with regulations of truck loading, our attempted solution to the Five Axles puzzle has made a tacit assumption that static loads will provide the explanation.

Accordingly, we have not considered dynamic forces attributable to truck wheels in contact with pavements at highway speeds.
Of course, even a perfect roadway experiences dynamic forces imposed by wheels as vehicles of all kinds come rolling along.  A typical segment of flat and level pavement is called upon to support the rather sudden application of a truck's static weight under pressure from its wheels.  After a period of time, which depends on the speed of the truck, that weight is just as suddenly relieved.  Until the next axle-load arrives at that segment.  For a given segment along the path of a truck wheel, pavement impacts take place five times for either the C-rig or the B-rig.
Solvers may want to review how axle-load, which is a force, creates stress in the roadway materials under the wheels and necessarily results in strainTwo types of stress are especially important: compression and shear. There are a half-dozen conventional failure theories, including stress (normal or shear), energy (strain or distortion), fracture or creep.

Roadways are seldom perfect.  Imperfections take many forms, and our analytic efforts must figure out how they can be affected differently between two truck types -- more specifically explaining why more damage seems to be caused by C-rigs than B-rigsOne supposes that a small imperfection will be made worse by vehicle wheels, trucks more than cars.  Indeed, one authoritative U.S. reference estimates that...

 One 80,000-pound (36,000 kg) truck does as much damage to roads as 750 3,800-pound (1,700 kg) cars

...which is hard to reconcile on a per-wheel basis.  Using those numbers, a car imposes 3,800 / 4 = 950 lbs on the pavement under each of its four wheels, while an 18-wheel truck carries 80,000 / 18 = 4,444 lbs per wheel.  That's a big difference, but 4,444 / 950 is only 4.68 times heavier not 750 times.  Tire pressures might be a factor.  Car-tire inflation may be as low as 32 psi, while truck-tires may be pressurized as high as 120 psi.  That's 120 / 32 = 3.75 times higher.  Even so, the product of 4.68 and 3.75 gives us a ratio of 17.5 not 750.  It is clear that differences in static wheel loadings will not provide an explanation and therefore a solution to the Five Axles puzzle.

What are we missing here?

Solvers who have tried to come up with the reason for why one of the axles on the B-rig is retracted may have a clue.  It's called a 'dead axle' or 'lazy axle' in one reference.   Fully loaded, the trailer on the B-rig has three axles (fB3, fB4, fB5) instead of just two axles on the C-rig (fC4, fC5).  WfC4, (fC4, fCheels B-rig B-rig B-rigon the B-rig are dragged onto circles that have incompatible radii.  Then too, retracting the axle saves tire wear when the truck is lightly loaded.

The B-rigB in the photograph is shown with its lazy axle retracted while apparently operating on the open highway.  The truck may be lightly loaded or even  unloaded.  Also, with an unloaded trailer, the retraction of fB3 some amount of normal force is preserved on fB2.  Come to think of it, trucks do operate unloaded much of the time, 'dead heading' between deliveries.

A popular subject in assuring roadability of vehicle designs is unsprung mass.  The main concerns are handling by drivers and riding comfort for passengers.
An empty trailer does get bounced around a lot without a payload on board.
Solvers will discover the explanation for why C-rigs do more damage to roadways than B-rigs in the solution to...

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