Skip to content
AxleGuard
Demo

Field notes

Seven ways the bridge formula gets implemented wrong

The formula is one line of arithmetic, which is why people are confident about it and why so many implementations are quietly incorrect. These seven errors turn up in spreadsheets, in printed tables and in shipped software, and every one of them returns a number that looks like an answer.

  • Seven errors
  • Every figure produced by the formula
  • US federal, August 2026

The rule being got wrong

The formula, once, for reference

Everything below is a deviation from this. If you want it worked through properly first, the full guide does that.

W = 500 × ((L × N) / (N − 1) + 12N + 36)

W is the maximum weight in pounds on the group. L is the distance in feet between its first and last axle. N is the number of axles in it. Round the result to the nearest 500 lb, cap it at 80,000 lb, and cap it again at 34,000 lb where the group is two axles spaced 8 ft or less. Apply it to every set of two or more consecutive axles on the combination.

Five instructions. The seven errors below are, in order, ignoring the last one, misapplying the third, misapplying the first, ignoring the second, and then three different ways of feeding it the wrong L or N.

Round to the nearest 500 lb. Cap at 80,000 lb. Cap again at 34,000 lb for a pair spaced eight feet or less. Apply all of it to every set of two or more consecutive axles.
Federal Bridge Gross Weight Formula, 23 CFR 658

The formula worked end to end, with all ten subsets enumerated, is inthe bridge formula guide.

One to four

Mistakes in the arithmetic

Four errors that live inside the calculation itself. Each of them produces a number, and the number looks like an answer.

01

Only the groups that have names get checked

The drive tandem, the trailer tandem and the gross weight are the three things everybody checks, because they are the three things that have names in a yard. The formula applies to every set of two or more consecutive axles, which on a five-axle combination is ten sets. Seven of them have no name, and it is usually one of those seven that binds.

Worked example

Worked example for Only the groups that have names get checked.
Subsets on a five-axle combination10
Subsets a typical check covers3
Interior four-axle group at 38 ft67,500 lb limit
Resulta load 200 lb over on a group nobody looked at

This is not an exotic failure. It is the ordinary one. A load can clear the steer axle, both tandems and the gross with hundreds of pounds to spare and still be over on the four-axle group that runs from the front drive axle to the rear trailer axle.

What to do about it

Enumerate. Take every run of consecutive axles of length two or more, from the tightest pair to the full outer spread, and evaluate all of them. It is a loop, not a judgement call, and the cost of running the seven extra checks is nothing.

02

The tandem exception is applied in the wrong direction

Two axles spaced 8 ft or less are capped at 34,000 lb. It is a cap. It can only ever lower a limit, never raise one. Implementations get this wrong in both directions, and the two errors have opposite consequences.

Worked example

Worked example for The tandem exception is applied in the wrong direction.
Two axles at 8 ft, formula result38,000 lb
Two axles at 8 ft, limit after the cap34,000 lb
Two axles at 10 ft, cap does not apply40,000 lb
Two axles at 3.5 ft, formula below the cap33,500 lb

Return the formula result for a pair at 8 ft and you have granted 4,000 lb that does not exist. Stamp 34,000 lb on a pair at 10 ft and you have denied 6,000 lb that does. And on a tightly coupled pair the formula itself returns 33,500 lb, which is below the cap, so the cap is not the answer there either.

What to do about it

Compute the formula first, then apply the cap as a minimum operation, and only where the spacing is 8 ft or less. Note also that a pair spaced under 40 in is not a tandem at all for weight purposes and is treated as a single axle group, which is a different rule again.

03

The rounding is done any way but to the nearest 500

The formula result is rounded to the nearest 500 lb. Implementations truncate instead, or round up, or carry the raw figure. It looks like a detail. It is worth up to 250 lb of legal capacity in either direction, on a check that is routinely decided by less than that.

Worked example

Worked example for The rounding is done any way but to the nearest 500.
Four axles at 50 ft, raw formula75,333 lb
Rounded to the nearest 50075,500 lb
Truncated downwards instead75,000 lb
Four axles at 38 ft, truncated67,000 lb, against a real limit of 67,500 lb

Truncating the 38 ft group gives 67,000 lb. A load carrying 67,200 lb on that group is legal by 300 lb and a truncating implementation calls it over by 200 lb. The load gets broken down and rebuilt for no reason, which costs a crew, a dock slot and a delivery window.

What to do about it

Round to the nearest 500, with the usual tie handling, and test it explicitly. Two test cases catch every version of this error: one group whose raw result rounds up and one whose raw result rounds down.

04

The 80,000 lb cap is left off

For long groups the formula returns a number larger than the federal gross limit. The cap is not optional and it is not implied by anything else in the calculation. Leave it out and the outer bridge check silently stops being a check.

Worked example

Worked example for The 80,000 lb cap is left off.
Five axles at 54 ft, formula result82,000 lb
Applied limit after the cap80,000 lb
Capacity granted by an uncapped implementation2,000 lb that does not exist

An uncapped check passes a load at 82,000 lb gross, which is 2,000 lb over the federal limit. The outer spread is exactly the group where the cap matters, and it is exactly the group people assume is safe because it is the longest.

What to do about it

Apply the cap to the result of every group, not only to the outer one. It costs one line and it converts a class of silent failure into no failure at all.

Five to seven

Mistakes in the inputs

Three errors where the formula is implemented perfectly and fed the wrong numbers. These are harder to find, because the arithmetic is not wrong.

05

The spacing is measured to the wrong point, or not rounded

L is the distance between the first and last axle of the group, and the regulation works in whole feet. Measure to the tire edge instead of the axle centre, take it from a drawing rather than the vehicle, or carry an unrounded figure through the formula, and the answer moves.

Worked example

Worked example for The spacing is measured to the wrong point, or not rounded.
Four-axle group measured at 38 ft67,500 lb
The same group measured at 37 ft66,500 lb
Cost of one foot of measurement error1,000 lb of limit

One foot is worth 1,000 lb on this group. That is four times the margin most failing loads fail by. A tape measure on the actual combination is a more reliable input than a specification sheet for a vehicle that has been rebuilt twice.

What to do about it

Measure axle centre to axle centre on the vehicle, record it against the unit rather than the model, and round to the nearest foot as the regulation does. Re-measure after any modification that moves an axle.

06

Lift axles are counted whether they are down or up

A raised lift axle carries nothing. It is not part of the group and it does not contribute to N. Implementations that read the axle count from the equipment record rather than from the axle's state will include it, and N is in the numerator and the denominator of the formula, so the error is large.

Worked example

Worked example for Lift axles are counted whether they are down or up.
Four axles across 38 ft, lift axle raised67,500 lb
Five axles across 38 ft, lift axle counted anyway72,000 lb
Imaginary capacity created4,500 lb

Counting one raised axle inflates the limit on that group by 4,500 lb. That is enough to pass a load that is thousands of pounds over, and the check will report a comfortable margin while doing it.

What to do about it

Treat the axle's deployed state as an input to the calculation, and re-solve when it changes. If the state is not known, the safe assumption is raised, because that is the assumption that cannot produce a false pass.

07

The federal table is assumed to apply everywhere

The formula above is the United States federal rule as it applies on the Interstate System. Off the Interstate, the governing weight table is the state's. Several states publish their own, and a number of routes carry grandfather provisions that predate the federal rule entirely.

Worked example

Worked example for The federal table is assumed to apply everywhere.
Where the federal formula governsthe Interstate System
Where it may notstate and local roads, and grandfathered routes
What a single-table implementation doesgives one confident answer for every road

A tool with one hard-coded table is right most of the time, which is the worst possible failure profile: it builds trust everywhere it is correct and spends that trust on the run where it is not.

What to do about it

Carry the applicable table as data attached to the jurisdiction and the road class, resolve it per leg, and have the verdict name which table it applied. A verdict that cannot say which rule bound is not auditable.

The arithmetic is not wrong. It was handed the wrong L, the wrong N, or the wrong table, and it answered confidently anyway.

Pounds of limit each error invents or throws away

lb
  • A raised lift axle counted in N4,500 lb

    capacity invented

  • The formula result used for a pair at 8 ft4,000 lb

    capacity invented

  • The 80,000 lb federal cap left off the outer spread2,000 lb

    capacity invented

  • One foot of spacing error on a four-axle group1,000 lb

    capacity thrown away

  • Truncating instead of rounding to the nearest 500500 lb

    capacity thrown away

Every figure is the gap between the correct limit and the one the error produces on the groups above. Two errors have no single number. Skipping interior subsets removes a check rather than changing a limit. Applying the federal table everywhere creates a gap equal to whatever the state's table says.

Testing

How to find out whether your own check is right

You do not need access to the code. Six loads will tell you what an implementation does, whether it is a spreadsheet, a printed table or somebody's software.

  1. 01

    Feed it a load that fails only on an interior group

    Build a case where the steer axle, both tandems and the gross are all comfortably legal and one four-axle group is a couple of hundred pounds over. If the tool passes it, it is checking three groups rather than ten, and nothing else you test matters until that is fixed.

  2. 02

    Ask it for a pair at exactly 8 ft

    The right answer is 34,000 lb. If it returns 38,000 lb, the tandem cap is not being applied. If it returns 34,000 lb for a pair at 10 ft, the cap is being applied where it should not be.

  3. 03

    Ask it for a group whose raw result ends in 333 or 666

    A four-axle group at 50 ft has a raw result of 75,333 lb. The right answer is 75,500 lb. An answer of 75,000 lb means it truncates, and it is costing you legal capacity on every load.

  4. 04

    Ask it for the outer spread on a long combination

    A five-axle unit at 54 ft has a raw result of 82,000 lb. The right answer is 80,000 lb. Anything higher means the federal gross cap is missing.

  5. 05

    Change one axle spacing by a foot

    Every group containing that axle should change. If only the pair changes and the three-axle and four-axle groups do not, the interior groups are not being recomputed, which means they are probably not being computed at all.

  6. 06

    Raise a lift axle and re-run

    Every limit should get tighter, because N falls. If nothing changes, the axle state is not an input, and any equipment with a lift axle is being checked against a limit that does not apply to it.

Six loads will tell you what an implementation does, whether it is a spreadsheet, a printed table or somebody's software.

A pass/fail badge is not a result

None of these tests are possible against a tool that returns only a verdict. To check an implementation you need to see the spacing, the axle count, the formula result and the applied limit for each group. If a product cannot show you its worksheet, you cannot audit it, and neither can anyone else in a dispute.

Next step

Put the bridge-formula mistakes against one real load.

Bring the route, configuration or scale ticket that made the bridge-formula mistakes matter. We will work through it in plain English and show where AxleGuard changes the decision.