Axle Group Load Against the Legal Cap

This is the simplest equation in the shard — a load divided by a limit — and it is here because the interesting part is not the arithmetic but the gap between two percentages.

A tandem weighed at 36,000 lb against a 34,000 lb cap is 5.9% overweight. At a load exponent of 4, it is doing 26% more damage. Ten per cent over is 46% more damage. Twenty per cent over is 107% more — that axle group is doing the work of two legal ones. That divergence is the entire justification for weight enforcement, and it is why an overweight fine looks disproportionate to the tonnage until you do the arithmetic. The haulier saved 6% on a load; the road paid a quarter again.

Which cap applies is a real question, which is why the cap is an input here. On the US interstate system the flat figures are 20,000 lb single, 34,000 lb tandem, 80,000 lb gross — and the bridge formula sets a further, spacing-dependent limit on every group of two or more consecutive axles. These are not alternatives to choose between: the binding limit is the lowest of all of them. Off the interstate system, states set their own. Canadian provinces work from a different framework entirely. Permit loads negotiate their own numbers. Hard-coding 34,000 into a calculator would make it wrong for most of the world's roads.

And then there is the spring thaw, which is the sharpest application of everything on this page. Through a northern winter the subgrade freezes from the top down. In spring it thaws from the top down as well, which means the melting layer sits on top of ground that is still frozen and therefore impermeable. The meltwater has nowhere to go. For a few weeks the upper subgrade is effectively saturated and its stiffness collapses — a road that carried legal loads all winter can lose a large fraction of its bearing capacity — and the pavement above it is riding on something close to soup. Agencies respond with seasonal load restrictions, cutting allowable axle loads by anywhere from 20% to 50% for six to eight weeks.

Put that through the exponent and you can see why it is worth the disruption. Halving the axle load cuts the damage to $(0.5)^4 = 6\%$ of what it was. The restriction is not a modest precaution; it is a sixteen-fold reduction in the damage each pass does, applied for exactly the window during which the pavement is least able to take it. A load that is routine in August is genuinely destructive in April, and the same fourth power that makes overloading expensive makes the restriction cheap.

Two practical notes on using the check. A scale ticket is a snapshot — fuel burns off the drive axles as a trip goes on, bulk loads settle and shift under braking, and rain on an open trailer of aggregate is worth real weight. Most operators load to 95-98% of the cap for that reason. And when two parties disagree about a percentage, the usual cause is that they are quoting against different caps: one against the flat statutory figure, the other against the bridge-formula result for the actual spacing.