Load carriage

What a foot march costs in energy carrying each firearm, its full magazines and its attachments on top of the rest of the kit, by two published load-carriage equations, and what those equations leave out.

What it computes

The load-carriage answer takes one or more firearms and a march: the marcher's body mass and body fat, the other kit they carry, the speed, the grade, the terrain and the distance. For each firearm it returns the metabolic rate in watts and the energy the march costs in kilojoules and kilocalories, carrying the firearm, its full magazines and any attachments named on top of that kit. When more than one firearm is compared, each is also set against the lightest.

It computes energy and nothing else. It does not predict fatigue, how well the marcher will shoot at the end of the march, or how long they can keep going before they stop. No validated closed-form equation exists for any of those, so the answer does not offer one.

Parameters, defaults and the response are in the reference: GET /v1/firearms/load-carriage

Where the masses come from

Every mass the march adds is one the catalogue records. The firearm is carried at its empty weight. A full magazine is the firearm's loaded weight less its empty weight: two figures the record states, subtracted, never estimated from a cartridge or a capacity. Attachments are carried at the weight their own records state.

L = m_{kit} + m_{empty} + n (m_{loaded} - m_{empty}) + m_{att}

where

$m_{kit}$: the other kit carried, in kilograms, as the request gives it
$m_{empty}, m_{loaded}$: the firearm's empty and loaded weights, as its record states them
$n$: the number of full magazines carried, the one in the firearm included
$m_{att}$: the summed weights of the attachments named

A firearm whose record has no empty weight, or no loaded weight when magazines are counted, is answered with an error in place of its cost, and the other firearms are still compared. An attachment with no recorded weight is refused outright, since it would otherwise be carried as nothing.

Loaded less empty weight is one magazine that comes out whole only when the firearm feeds from a detachable magazine. For a fixed magazine, a clip or a belt, the difference is the rounds alone, so when more than one reload is counted the answer carries a warning: each reload after the first is counted as its rounds, without the clip, belt or pouch that holds them.

The LCDA equation

The default model is the US Army's Load Carriage Decision Aid (LCDA) backpacking equation, fitted to modern military loads and more accurate on them than the older Pandolf equation. Its graded-walking term covers climbing and descending in one expression, and its resting rate is computed from lean mass. The answer is in watts for the whole marcher.

M = M_{rest} + W [0.19 + η (1.78 S^{0.58} + 0.27 S^{4} + M_{grade})] [1 + 1.96 {(\frac{L}{W})}^{1.36}]

M_{grade} = 34 S G (1 - {1.05}^{1 - {1.1}^{100 G + 32}})

M_{rest} = \frac{(500 + 22 W (1 - f)) \times 4184}{86400}

where

$M$: the metabolic rate, in watts
$W, L$: body mass and the load carried, in kilograms
$S$: walking speed, in metres per second
$G$: grade as rise over run: 0.05 is a 5% climb, −0.05 a 5% descent
$η$: the terrain factor, from the table below
$f$: the share of body mass that is fat, from 0 to 1

The graded-walking term is zero on the level and positive uphill. Downhill it is negative, because gravity does part of the work, and the saving shrinks again as the descent steepens and the marcher has to brake. The worked example shows it across grades.

The LCDA equation was fitted on speeds from 0.45 to 1.97 m/s and loads up to 66% of body mass. A march outside that range is still answered, because a caller exploring a range wants the whole curve, but the answer carries a warning naming the input that lies outside it.

The Pandolf equation

The Pandolf equation (1977) is the one most of the load-carriage literature reports against. It is offered as the alternative model so that a result can be set beside a published one. Here it answers level and uphill marches only, and it carries no fitted-range warning.

M = 1.5 W + 2.0 (W + L) {(\frac{L}{W})}^{2} + η (W + L) (1.5 V^{2} + 0.35 V G_{%})

where

$V$: walking speed, in metres per second
$G_{%}$: grade in percent: 5 is a 5% climb

Pandolf's equation has a published downhill correction (Santee et al. 2001). It is deliberately not implemented: its constants could not be verified against the paper, and a formula nobody has checked does not belong in a published answer. A downhill grade with the Pandolf model is refused with an error; the LCDA model answers it.

Terrain factors

Both models multiply the cost of moving, not of standing, by a terrain factor from Soule and Goldman (1972). A paved road or a treadmill is the baseline. The last column is the worked march below on each terrain, by the LCDA.

Terrain	In the request	η	The worked march
Paved road or treadmill	`paved`	1.0	1732 kcal
Dirt road	`dirt_road`	1.1	1862 kcal
Light brush	`light_brush`	1.2	1993 kcal
Heavy brush	`heavy_brush`	1.5	2383 kcal
Swampy bog	`swampy_bog`	1.8	2774 kcal
Loose sand	`loose_sand`	2.1	3164 kcal

The speed is the one the request gives. A factor raises the cost of moving at that speed; it does not slow the march down.

Worked example: a rifle on a road march

A marcher of 80 kg with 15% body fat carries 25 kg of other kit and a rifle of 3020 g with 7 full magazines of 400 g each, at 1.333 m/s over 20 km, level and on paved terrain. Every figure below is the answer's own functions, run as the page loads.

Load carried (1)	$L = 25 + 3.020 + 7 \times 0.400 = 30.82 kg, \frac{L}{W} = 0.385$
Resting rate (4)	$M_{rest} = \frac{(500 + 22 \times 68) \times 4184}{86400} = 96.7 W$
Metabolic rate, LCDA (2)	$M = 483.1 W = 6.04 W / kg$
Time on the march	$t = \frac{20000 m}{1.333 m / s} = 15004 s$
Energy, LCDA	$E = M t = 7248 kJ = 1732 kcal$
Pandolf, for comparison (5)	$M = 448.3 W, E = 1607 kcal$

The rifle and its magazines, 5.82 kg together, add 120 kcal to the march by the LCDA and 100 kcal by Pandolf, against the same march with the other kit alone.

The march lies inside the range the LCDA was fitted on, so the answer carries no warning.

The same march on grades from a descent to a climb, by the LCDA. Downhill the graded-walking term is negative and the march costs less; uphill it costs more, and quickly.

Grade	$M_{grade}$	Rate	The worked march
−10%	−1.333 W/kg	319.3 W	1145 kcal
−5%	−1.011 W/kg	358.9 W	1287 kcal
0%	0.000 W/kg	483.1 W	1732 kcal
5%	1.813 W/kg	705.8 W	2531 kcal
10%	4.203 W/kg	999.4 W	3584 kcal

Limitations

Both models treat every kilogram as carried on the body. A rifle held in both hands changes the gait, which neither equation models, so the cost of marching with it at the ready may differ from either answer.
Both give energy expenditure only. Neither predicts fatigue, marksmanship after the march, or time to exhaustion.
Both are population equations. One marcher's cost differs with fitness, stride, footwear and how the load is packed.
The masses are the catalogue's. A firearm whose record lacks one is answered with an error, never with an estimate.

Sources

The equations and every constant on this page are quoted from these papers.

Looney DP, Lavoie EM, Vangala SV, Holden LD, Figueiredo PS, Friedl KE, et al. (2022). Modeling the Metabolic Costs of Heavy Military Backpacking. Medicine & Science in Sports & Exercise 54(4):646-654.doi:10.1249/MSS.0000000000002833
Looney DP, Santee WR, Hansen EO, Bonventre PJ, Chalmers CR, Potter AW (2019). Estimating Energy Expenditure during Level, Uphill, and Downhill Walking. Medicine & Science in Sports & Exercise 51(9):1954-1960.doi:10.1249/MSS.0000000000002002
Pandolf KB, Givoni B, Goldman RF (1977). Predicting energy expenditure with loads while standing or walking very slowly. Journal of Applied Physiology 43(4):577-581.doi:10.1152/jappl.1977.43.4.577
Soule RG, Goldman RF (1972). Terrain coefficients for energy cost prediction. Journal of Applied Physiology 32(5):706-708.doi:10.1152/jappl.1972.32.5.706
Cunningham JJ (1991). Body composition as a determinant of energy expenditure: a synthetic review and a proposed general prediction equation. American Journal of Clinical Nutrition 54(6):963-969.doi:10.1093/ajcn/54.6.963