The short version: the bridge formula is not the whole rule. Three caps sit on top of it — 20,000 lb per axle, 34,000 lb for any group spanning 8 ft or less, and the vehicle's own gross weight — and omitting any one of them over-permits a real load. The published federal table also disagrees with the federal formula in 5 cells; we follow the formula and record the disagreement. Run it on your own axle layout →
The formula
23 U.S.C. §127(a) / 23 CFR Part 658, as published in Bridge Formula Weights, FHWA-HOP-19-028, August 2019. Public domain.verbatim
W = 500 × ( (L × N) / (N − 1) + 12N + 36 )
W = the maximum weight in pounds that can be carried on a group of two or more axles to the nearest 500 pounds
L = the distance in feet between the outer axles of any two or more consecutive axles
N = the number of axles being considered
The formula exists to keep a heavy load from concentrating its mass over a short span of bridge deck. Spread 80,000 lb over 51 ft and the bridge is fine; bunch it into 20 ft and it is not.
The three caps the formula alone does not tell you about
The published table is not the bare formula. Three limits sit on top of it, and omitting any one over-permits a real load.
- 20,000 lb × N. No group may exceed its own axles' individual limits. This is why the two-axle column flattens at 40,000 lb: the raw formula at 11 ft with two axles gives 41,000, and two axles can never legally carry more than 40,000.
- 34,000 lb for any group spanning 8 ft or less — the statutory tandem limit, applied by span rather than by axle count, which is how the published table shows 34,000 in both the two- and three-axle columns at short spans.
- The vehicle's own gross weight. A group is a subset of the vehicle's axles, so it cannot carry more than the whole vehicle carries. That is arithmetic rather than statute, which is exactly why it is safe at any weight — including on a permitted load well above the federal gross limit.
That third cap is the one that is easy to get wrong. Hard-coding it at the federal 80,000 lb gross limit fabricates bridge-formula overages of up to 25,500 lb on permitted loads whose axle groups are in fact compliant — on precisely the heavy-haul moves a permit calculator exists for. The federal gross limit is a limit on the vehicle, and it is enforced once, in its own place.
The one span where the table contradicts interpolation
The federal table prints whole-foot rows plus exactly one interval row: "over 8 but less than 9 feet", reading 38,000 lb at two axles and 42,000 lb at three. Those are precisely the formula's values at 8 ft — so across that whole open interval the published table holds the 8-foot value flat rather than letting the formula climb.
Interpolating there over-permits: at 8.99 ft with two axles the bare formula gives 38,990, rounding to 39,000 — a full 1,000 lb more than the 38,000 the table prints for that exact span. Over-permitting is the dangerous direction, so the published flat value wins. This is not generalised to every fractional span: nowhere else does the table say anything about fractional feet, and the statute defines the weight as a continuous function of the span.
The bridge formula table
Maximum gross weight in pounds on a group of consecutive axles, by the distance in feet between the outermost axles of the group and the number of axles in it. Computed from 23 U.S.C. §127(a) with the three caps above applied, rounded to the nearest 500 lb with ties down.
| Span (L) | 2 axles | 3 axles | 4 axles | 5 axles | 6 axles | 7 axles | 8 axles | 9 axles |
|---|---|---|---|---|---|---|---|---|
| 4 ft | 34,000 | — | — | — | — | — | — | — |
| 5 ft | 34,000 | — | — | — | — | — | — | — |
| 6 ft | 34,000 | — | — | — | — | — | — | — |
| 7 ft | 34,000 | — | — | — | — | — | — | — |
| 8 ft | 34,000 | 34,000 | — | — | — | — | — | — |
| 9 ft | 39,000 | 42,500 | — | — | — | — | — | — |
| 10 ft | 40,000 | 43,500 | — | — | — | — | — | — |
| 11 ft | 40,000 | 44,000 | — | — | — | — | — | — |
| 12 ft | 40,000 | 45,000 | 50,000 | — | — | — | — | — |
| 13 ft | 40,000 | 45,500 | 50,500 | — | — | — | — | — |
| 14 ft | 40,000 | 46,500 | 51,500 | — | — | — | — | — |
| 15 ft | 40,000 | 47,000 | 52,000 | — | — | — | — | — |
| 16 ft | 40,000 | 48,000 | 52,500 | 58,000 | — | — | — | — |
| 17 ft | 40,000 | 48,500 | 53,500 | 58,500 | — | — | — | — |
| 18 ft | 40,000 | 49,500 | 54,000 | 59,000 | — | — | — | — |
| 19 ft | 40,000 | 50,000 | 54,500 | 60,000 | — | — | — | — |
| 20 ft | 40,000 | 51,000 | 55,500 | 60,500 | 66,000 | — | — | — |
| 21 ft | 40,000 | 51,500 | 56,000 | 61,000 | 66,500 | — | — | — |
| 22 ft | 40,000 | 52,500 | 56,500 | 61,500 | 67,000 | — | — | — |
| 23 ft | 40,000 | 53,000 | 57,500 | 62,500 | 68,000 | — | — | — |
| 24 ft | 40,000 | 54,000 | 58,000 | 63,000 | 68,500 | 74,000 | — | — |
| 25 ft | 40,000 | 54,500 | 58,500 | 63,500 | 69,000 | 74,500 | — | — |
| 26 ft | 40,000 | 55,500 | 59,500 | 64,000 | 69,500 | 75,000 | — | — |
| 27 ft | 40,000 | 56,000 | 60,000 | 65,000 | 70,000 | 75,500 | — | — |
| 28 ft | 40,000 | 57,000 | 60,500 | 65,500 | 71,000 | 76,500 | 82,000 | — |
| 29 ft | 40,000 | 57,500 | 61,500 | 66,000 | 71,500 | 77,000 | 82,500 | — |
| 30 ft | 40,000 | 58,500 | 62,000 | 66,500 | 72,000 | 77,500 | 83,000 | — |
| 31 ft | 40,000 | 59,000 | 62,500 | 67,500 | 72,500 | 78,000 | 83,500 | — |
| 32 ft | 40,000 | 60,000 | 63,500 | 68,000 | 73,000 | 78,500 | 84,500 | 90,000 |
| 33 ft | 40,000 | 60,000 | 64,000 | 68,500 | 74,000 | 79,000 | 85,000 | 90,500 |
| 34 ft | 40,000 | 60,000 | 64,500 | 69,000 | 74,500 | 80,000 | 85,500 | 91,000 |
| 35 ft | 40,000 | 60,000 | 65,500 | 70,000 | 75,000 | 80,500 | 86,000 | 91,500 |
| 36 ft | 40,000 | 60,000 | 66,000 | 70,500 | 75,500 | 81,000 | 86,500 | 92,000 |
| 37 ft | 40,000 | 60,000 | 66,500 | 71,000 | 76,000 | 81,500 | 87,000 | 93,000 |
| 38 ft | 40,000 | 60,000 | 67,500 | 71,500 | 77,000 | 82,000 | 87,500 | 93,500 |
| 39 ft | 40,000 | 60,000 | 68,000 | 72,500 | 77,500 | 82,500 | 88,500 | 94,000 |
| 40 ft | 40,000 | 60,000 | 68,500 | 73,000 | 78,000 | 83,500 | 89,000 | 94,500 |
| 41 ft | 40,000 | 60,000 | 69,500 | 73,500 | 78,500 | 84,000 | 89,500 | 95,000 |
| 42 ft | 40,000 | 60,000 | 70,000 | 74,000 | 79,000 | 84,500 | 90,000 | 95,500 |
| 43 ft | 40,000 | 60,000 | 70,500 | 75,000 | 80,000 | 85,000 | 90,500 | 96,000 |
| 44 ft | 40,000 | 60,000 | 71,500 | 75,500 | 80,500 | 85,500 | 91,000 | 96,500 |
| 45 ft | 40,000 | 60,000 | 72,000 | 76,000 | 81,000 | 86,000 | 91,500 | 97,500 |
| 46 ft | 40,000 | 60,000 | 72,500 | 76,500 | 81,500 | 87,000 | 92,500 | 98,000 |
| 47 ft | 40,000 | 60,000 | 73,500 | 77,500 | 82,000 | 87,500 | 93,000 | 98,500 |
| 48 ft | 40,000 | 60,000 | 74,000 | 78,000 | 83,000 | 88,000 | 93,500 | 99,000 |
| 49 ft | 40,000 | 60,000 | 74,500 | 78,500 | 83,500 | 88,500 | 94,000 | 99,500 |
| 50 ft | 40,000 | 60,000 | 75,500 | 79,000 | 84,000 | 89,000 | 94,500 | 100,000 |
| 51 ft | 40,000 | 60,000 | 76,000 | 80,000 | 84,500 | 89,500 | 95,000 | 100,500 |
| 52 ft | 40,000 | 60,000 | 76,500 | 80,500 | 85,000 | 90,500 | 95,500 | 101,000 |
| 53 ft | 40,000 | 60,000 | 77,500 | 81,000 | 86,000 | 91,000 | 96,500 | 102,000 |
| 54 ft | 40,000 | 60,000 | 78,000 | 81,500 | 86,500 | 91,500 | 97,000 | 102,500 |
| 55 ft | 40,000 | 60,000 | 78,500 | 82,500 | 87,000 | 92,000 | 97,500 | 103,000 |
| 56 ft | 40,000 | 60,000 | 79,500 | 83,000 | 87,500 | 92,500 | 98,000 | 103,500 |
| 57 ft | 40,000 | 60,000 | 80,000 | 83,500 | 88,000 | 93,000 | 98,500 | 104,000 |
| 58 ft | 40,000 | 60,000 | 80,000 | 84,000 | 89,000 | 94,000 | 99,000 | 104,500 |
| 59 ft | 40,000 | 60,000 | 80,000 | 85,000 | 89,500 | 94,500 | 99,500 | 105,000 |
| 60 ft | 40,000 | 60,000 | 80,000 | 85,500 | 90,000 | 95,000 | 100,500 | 105,500 |
A dash means no cell: a group of that many axles cannot physically span that short a distance, and the federal table prints nothing there either.
5 cells where the federal table disagrees with the federal formula
The statute defines the weight by the formula, "to the nearest 500 pounds". The printed table is a reader's aid derived from it, and in 5 cells the derivation is simply wrong — each prints exactly 500 lb below the nearest-500 value while every neighbouring cell in the same column is correct.
| Cell | Raw formula | Nearest 500 (ours) | Published |
|---|---|---|---|
| 39 ft, 5 axles | 72,375 | 72,500 | 72,000 |
| 53 ft, 9 axles | 101,812.5 | 102,000 | 101,500 |
| 54 ft, 9 axles | 102,375 | 102,500 | 102,000 |
| 55 ft, 9 axles | 102,937.5 | 103,000 | 102,500 |
| 56 ft, 9 axles | 103,500 | 103,500 | 103,000 |
The 56 ft, 9-axle cell settles it: 500 × (63 + 108 + 36) is 103,500 on the nose, an exact multiple of 500 with no rounding decision to make at all, and the table still prints 103,000. No rounding rule produces that. None of the five is an exact tie either, so ties-down does not explain them — and the 52 ft, 9-axle cell is an exact tie, and the table rounds it down correctly. These are typos in a table, not a rule anyone has failed to find.
We follow the formula. The disagreement is recorded in our code as a named constant, and a test walks all 265 published cells and expects exactly these five to differ by exactly +500 — so if the table is ever reprinted, the test says which way it moved.
Questions
Which axle groups does the bridge formula apply to?
Every group of two or more consecutive axles — all N(N−1)/2 of them. A five-axle tractor-semitrailer has ten such groups. The classic failure is a rig whose steer, drives and trailer tandems each pass while an interior group spanning axles two through five does not; checking only the obvious three misses exactly the group a bridge cares about.
How is the result rounded?
To the nearest 500 pounds, with exact ties going DOWN. The statute says "to the nearest 500 pounds", and every exact tie in the published table rounds down — a raw 42,750 prints as 42,500. Ordinary rounding, which sends halves up, would permit 500 lb that the law does not.
Why is a five-axle van with 34,000 lb on each tandem legal?
Because of a statutory carve-out: two consecutive sets of tandem axles may each carry 68,000 lb between them provided the outer axles of the two tandems are at least 36 ft apart. The bare formula at that geometry yields 66,000 lb, so implementing the formula alone would flag every legal 5-axle van in the country as overweight.
Does passing the bridge formula mean I need no permit?
Not on its own. A vehicle also has to clear the flat statutory limits — 20,000 lb on a single axle, 34,000 lb on a tandem, 80,000 lb gross — and it has to be legal on size. Above any of them you are in permit territory.