Guide

The bridge formula, worked end to end

The formula is three lines of arithmetic and almost nobody applies it to every group it covers. This page runs it across all ten consecutive axle subsets on an ordinary five-axle combination, and shows a load that passes every check a planner normally makes while failing one of them.

Ten subsets, all enumerated
Two worked examples
US federal, August 2026

The rule

What the formula actually says

It is not a weight limit. It is a limit on how much weight you may put on a given length of road at once, and it applies to every group of consecutive axles on the vehicle.

The Federal Bridge Gross Weight Formula exists because a bridge does not care what your truck weighs. It cares how much of that weight is sitting on it at the same moment. Eighty thousand pounds spread across fifty-four feet is a manageable load for a span. The same eighty thousand pounds crammed into twenty feet is not. So the limit is written as a function of length rather than as a single number.

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

W is the maximum legal weight in pounds carried on the group. L is the distance in feet between the first and last axle of the group. N is the number of axles in the group. The result is rounded to the nearest 500 lb, and it is capped at 80,000 lb, because 80,000 lb is the gross limit on the Interstate System and no arithmetic gets you past it.

The word that does the damage is "every"

The formula applies to every set of two or more consecutive axles on the combination. Not the drive tandem. Not the trailer tandem. Not the outer bridge. All of them, including the interior groups that do not correspond to anything a driver would name out loud.

On a standard five-axle tractor-semitrailer that is ten separate checks: four adjacent pairs, three three-axle groups, two four-axle groups and the full outer spread. Most people check three of the ten, and the three they check are the three that almost never fail.

The exception that catches people out

Two axles spaced 8 ft or less apart are capped at 34,000 lb, regardless of what the formula returns for them. This is the tandem exception, and it is why a drive tandem is a 34,000 lb group rather than whatever the arithmetic would suggest. Apply the raw formula to a close-coupled tandem and you will produce a number that looks legal on paper and is wrong in the field.

Note which direction the exception runs. It is a cap, not a floor. It can only make the limit lower, never higher.

The formula applies to every set of two or more consecutive axles. On a five-axle truck that is ten checks, and seven of them have no name in a yard.

What the formula allows a four-axle group as it lengthens

Each point is W = 500 × ((L×4)/3 + 84) rounded to the nearest 500 lb, which is the formula with N set to four. It rises in a straight line and then stops: from 57 ft onward the arithmetic returns 80,000 lb or more, and the federal gross cap is what applies instead.

The formula

W = 500 × ((L×N)/(N−1) + 12N + 36)

L in feet, N axles, W in pounds

Rounding

Nearest 500 lb

applied to the formula result

Hard cap

80,000 lb

no group may exceed the federal gross

Tandem cap

34,000 lb

two axles spaced 8 ft or less

The combination

A five-axle truck, measured

Every number on the rest of this page comes from these five axle positions. They are a worked example of an ordinary tractor and a 53 ft van, not a measurement of any particular vehicle.

Axle positions and weights for the two five-axle Bridge Formula examples
Axle	Role	Position from steer	Case A load	Case B load
A1	Steer	0 ft	11,900 lb	11,600 lb
A2	Drive 1	16 ft	16,850 lb	17,000 lb
A3	Drive 2	20 ft	16,850 lb	17,000 lb
A4	Trailer 1	50 ft	16,850 lb	16,850 lb
A5	Trailer 2	54 ft	16,850 lb	16,850 lb
Gross			79,300 lb	79,300 lb

Both cases weigh the same. Case B is Case A after the planner slid the fifth wheel back by a notch to take 300 lb off a steer axle that was reading a little high. Every obvious check still passes. One of them does not.

Two axles spaced eight feet or less are held to 34,000 lb, whatever the arithmetic returns for them. The exception can only lower a limit, never raise one.

Federal Bridge Gross Weight Formula, 23 CFR 658

The enumeration

All ten subsets, with the arithmetic printed

This is the whole check. The formula column shows the unrounded result so you can follow where the rounding and the caps take effect.

Every consecutive axle subset on a five-axle combination, with its spacing, axle count, formula result, applied limit and the margin for two load cases.
Group	L	N	Formula	Limit	Case A	Case B
A1–A2	16 ft	2	46,000	46,000	28,750(−17,250)	28,600(−17,400)
A2–A3tandem cap applied	4 ft	2	34,000	34,000	33,700(−300)	34,000(−0)
A3–A4	30 ft	2	60,000	60,000	33,700(−26,300)	33,850(−26,150)
A4–A5tandem cap applied	4 ft	2	34,000	34,000	33,700(−300)	33,700(−300)
A1–A3	20 ft	3	51,000	51,000	45,600(−5,400)	45,600(−5,400)
A2–A4	34 ft	3	61,500	61,500	50,550(−10,950)	50,850(−10,650)
A3–A5	34 ft	3	61,500	61,500	50,550(−10,950)	50,700(−10,800)
A1–A4	50 ft	4	75,333	75,500	62,450(−13,050)	62,450(−13,050)
A2–A5	38 ft	4	67,333	67,500	67,400(−100)	67,700(+200)
A1–A5capped at the federal gross	54 ft	5	81,750	80,000	79,300(−700)	79,300(−700)

Formula column is the unrounded result of W = 500 × ((L×N)/(N−1) + 12N + 36). The limit column is that result rounded to the nearest 500 lb, capped at 80,000 lb, with the 34,000 lb tandem cap applied where the spacing is 8 ft or less. Bracketed figures are the margin to the limit.

Reading the table

What the two cases tell you

The interesting column is not the limit. It is the margin.

Case A is legal, and it is legal by 100 lb

Run the checks a planner normally runs. Gross is 79,300 lb, which is 700 lb under the cap. The steer axle is 11,900 lb against a 12,000 lb limit. The drive tandem is 33,700 lb and the trailer tandem is 33,700 lb, both against 34,000 lb. Nothing looks tight. Every one of those checks has at least 300 lb of room.

Now look at A2–A5: the four-axle group running from the front drive axle to the rear trailer axle, 38 ft end to end. Its limit is 67,500 lb and it is carrying 67,400 lb. The margin is 100 lb. That group is the one that decides whether this truck is legal, and it is not a group anyone names in a yard.

Case B is illegal, and nothing you would look at says so

Case B is the same load after a reasonable-looking correction. The steer was reading 11,900 lb, the planner slid the fifth wheel back a notch, and 300 lb moved off the steer and onto the drive tandem. Now the steer is 11,600 lb, which is better. The drive tandem is exactly 34,000 lb, which is exactly legal. Gross is unchanged. A dispatcher signing this off has no reason to hesitate.

And A2–A5 is now 67,700 lb against a limit of 67,500 lb. That is 200 lb over, on a group that does not appear on a scale ticket, in a spreadsheet, or in most software.

This is the shape of the problem. The bridge formula does not usually fail loudly. It fails on an interior group, by a few hundred pounds, on a truck that is comfortably under gross.

Case B, every subset against its own limit

A1–A228,600 lb
2 axles across 16 ft, limit 46,000 lb
A2–A334,000 lb
2 axles across 4 ft, limit 34,000 lb
A3–A433,850 lb
2 axles across 30 ft, limit 60,000 lb
A4–A533,700 lb
2 axles across 4 ft, limit 34,000 lb
A1–A345,600 lb
3 axles across 20 ft, limit 51,000 lb
A2–A450,850 lb
3 axles across 34 ft, limit 61,500 lb
A3–A550,700 lb
3 axles across 34 ft, limit 61,500 lb
A1–A462,450 lb
4 axles across 50 ft, limit 75,500 lb
A2–A567,700 lb
4 axles across 38 ft, limit 67,500 lb
A1–A579,300 lb
5 axles across 54 ft, limit 80,000 lb

The marker on each bar is that group's own limit, not one number applied to all of them. 9 of the 10 groups clear it, and A2–A5 misses by 200 lb. Nothing on a scale ticket would tell you which.

The load that gets a ticket is rarely the load that looked heavy. It is the load that looked fine.

Why the correction made it worse

Moving weight from the steer axle to the drive tandem moves itinto the A2–A5 group. The steer axle sits outside that group, so weight parked there costs nothing against it. Relieving the steer is the one adjustment that loads the interior group without changing gross at all.

The fix

How to get Case B legal again

Three moves work. One of them is the obvious one and it makes things worse.

01
Put the weight back on the steer axle
Slide the fifth wheel forward again. The steer axle has headroom to 12,000 lb, and every pound that lands there is a pound that leaves the failing four-axle group. Moving 200 lb forward clears the violation; moving 300 lb forward restores Case A. The trade is that the steer axle now carries more, so check it against the tire rating as well as the statute — the tire rating on a standard steer tire is often the lower of the two.
02
Lengthen the group by sliding the trailer tandem back
L is the lever the formula turns on. Taking the trailer tandem back 2 ft moves the group from 38 ft to 40 ft and lifts its limit from 67,500 lb to 68,500 lb, which clears the problem outright. The catch is that sliding the tandem rearward transfers load from the trailer axles onto the kingpin, and from the kingpin onto the drive tandem, which in Case B is already at exactly 34,000 lb. This move only works if the drive tandem has room, and here it does not.
03
Take the weight off the truck
Two hundred pounds of freight removed from anywhere between the drive tandem and the rear of the trailer fixes it. On a full trailer of uniform pallets this is one pallet short, which is a conversation with the shipper rather than an adjustment on the yard. It is the last resort for a reason, but on a non-divisible load it may be the only option other than a permit.
04
Re-run all ten subsets, not the one you fixed
Every one of these moves changes more than one group. Sliding the fifth wheel forward loads the steer and the A1 groups. Sliding the tandem back loads the drives and shortens nothing else. A fix that clears one subset and breaks another is worse than the original problem, because now you believe it is solved.

Edges

Where the formula behaves unexpectedly

Four cases worth knowing before you trust an answer, including one where the arithmetic gives a number that is legal and useless.

Rounding decides real cases

The A1–A4 group in the table above returns 75,333 lb from the formula and a limit of 75,500 lb after rounding. Rounding to the nearest 500 lb moved that limit up. It moves other limits down. On a load sitting within a few hundred pounds of a group limit, the rounding rule is the difference between a ticket and a clean pass, so it is not a detail to implement casually.

Spread axles gain capacity as they separate

Two axles more than 8 ft apart escape the 34,000 lb tandem cap and get the formula result instead. At 10 ft apart the formula gives 40,000 lb for the pair. This is why spread-axle trailers exist, and it is also why a spread-axle trailer produces a completely different set of subset limits from a close-coupled tandem on an otherwise identical truck. Software that assumes 34,000 lb for any two trailer axles will call a legal spread illegal, and will call an illegal one legal the moment the slider moves inside 8 ft.

The formula can return more than the axles are rated for

A 30 ft gap between two axles returns 60,000 lb from the formula. No pair of single axles is going to carry that, because each of them is capped at 20,000 lb by statute long before the group limit is reached. The bridge formula is one of several limits, not a licence. Run it alongside the per-axle statutory limits, the tire rating and the manufacturer's axle rating, and take the lowest.

State bridge tables are not always the federal one

The formula above is the United States federal rule as applied on the Interstate System. A handful of states publish their own bridge tables for non-Interstate routes, and a few carry grandfather provisions that predate the federal rule entirely. If a route leaves the Interstate, the applicable table is a question about that state, not a question about arithmetic.

Dated, and honest about it

Assessed from publicly published documentation as of August 2026. The formula and the federal limits quoted here have been stable for a long time, but they are law rather than physics, and law changes. Check the current text before you rely on it in a dispute.

Put the bridge-formula walkthrough against one real load.

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

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