Introduction

The overall performance of a weapon is of great interest to handgun and rifle users. That performance is defined by the magnitude of specific variables such as muzzle velocity, recoil force, and the maximum pressure developed inside the barrel. These performance characteristics are related for the most part to bullet weight, propellant weight, type of propellant used, propellant burn rate, barrel length, barrel inside diameter, and frictional force between the bullet and the inside diameter of the barrel.

Calculation of the recoil force should be given special consideration by firearm designers and users because excessive recoil has an adverse effect on accuracy. It is therefore important to obtain an understanding of why excessive recoil occurs and how it might be mitigated. A clue is provided by the observation of what appears to be an anomaly in the case of the recoil force developed in each of two handguns that fire the same ammunition and are identical in all respects except for the material from which they are constructed and their weight. The recoil force developed by the lighter gun is significantly higher than that of the heavier gun, although the expectation is that they are equal.

Calculation of Recoil Force

Firearm recoil has been determined experimentally by using load cells and an apparatus with a special design¹, but the problem of determining recoil of guns that differ only in material of construction and weight has not been addressed.

Without apparatus to measure recoil, it was necessary to determine what the recoil might be from an analysis that made use of the mechanical properties of the gun materials, the barrel pressure, the coefficient of sliding friction between bullet and the barrel, and pertinent gun details such as the rifling and the barrel dimensions.

Since recoil is due to barrel pressure and barrel pressure should be the same for both the heavier gun and the lighter gun using the same ammunition, the hand gripping the gun should sense only the recoil force that is due to barrel pressure. Therefore, the recoil force should be the same for both guns because barrel pressure is the same for both.

Summary

Identical guns of different weight do not exhibit the same recoil force, probably as a result of the fact that the barrel of the lighter gun barrel is fabricated of an aluminum alloy that expands more when stressed because of its lower modulus of elasticity. Hence, the rifling is stressed less, and the frictional force between bullet and the rifling is reduced. Since the recoil force is equal to the recoil force without friction minus the frictional force, recoil force is higher for the lighter gun.

A possible means of avoiding excessive recoil and reduced accuracy of a firearm of lesser weight therefore might be to make use of high-strength alloy steel construction.

Thus, the question is as follows: Would alloy steel with a yield strength of perhaps 140,000 pounds per square inch (PSI) instead of 50,000 PSI reduce firearm weight very significantly yet retain the recoil of a heavier gun?

The following equations were used to calculate the difference in recoil between the lighter and heavier guns, and the calculated results are summarized in Table 1.

t = (Px)(Do)/((2Sd) + Px))                                                         (1)

where:
t = barrel wall thickness, inches
Px = maximum barrel pressure, pounds per square inch
Do = barrel outside diameter, inches
Sd = design stress of barrel material, pounds per square inch

The recoil force for the steel and aluminum alloy barreled guns, with the aluminum alloy gun having a thin-walled rifled steel liner, are given in Equations (2) and (3). Equation (2) is designed to ensure that a constant clearance is maintained between bullet and barrel and is based on a barrel pressure of zero at the muzzle end of the bullet. Solution of the equations also requires that a rifling stress be assumed for the steel barreled gun.  Thus,

((e)bs – (e)rs))steel barrel = ((e)ba – (e)rs))aluminum barrel (2)

where:
(e)bs = steel barrel deflection, inches
(e)rs = steel rifling deflection, inches
(e)ba = aluminum barrel deflection, inches

The subscripts “steel barrel” and “aluminum barrel” refer to the steel and aluminum barrel cases, respectively.

Favg = 3.14(Db)(Lb)(Cfric)(Arifling)(Savg)                              (3)

where
Favg = average frictional force, pounds, based on barrel pressure at breech end of bullet and muzzle end of bullet
Db = outside bullet diameter, inches
Lb = bullet length, inches
Cfric = coefficient of friction between bullet and rifling
Arifling = fraction of total bullet outside area contacted by rifling
Savg = average stress in rifling, PSI

(e/L) = (S/E)                                                     (4)

where:
e = deflection, inches
S = stress, PSI
E = modulus of elasticity, PSI
L = initial depth of the rifling or barrel inside radius, inches

The frictional force acts in a forward direction, opposite to the force caused by barrel pressure. The difference between the two is the recoil force:

(R)avg = (Ro)avg – (F)avg                                                       (5)

where:
(R)avg = average recoil force considering the effect of frictional force, pounds
(Ro)avg = average recoil force without considering the effect of frictional force, pounds

Table 1: Effect of Barrel Friction on Recoil of Two Identical Guns of Different Weight

	Heavyweight Gun	Lightweight Gun
Material of Construction	Steel	Aluminum Alloy
Weight, pounds	2.6	1.6
Barrel Design Stress, Sd, PSI	40,000	40,000
Rifling Stress, PSI*	20,000	5,000
Peak Barrel Pressure, Px, PSI	24,000	24,000
Modulus of Elasticity, million PSI	30	10
Calc. Barrel Wall Thickness, t, inches	0.18	0.18
Calc. Barrel Outside Diameter, Do, inches	0.8	0.8
Bullet-Rifling Friction Coefficient, Cfric†	0.4	0.4
Rifling Cross-Sectional Area, Arifling	0.2	0.2
(L)rs, Rifling Depth, Inches	0.01	0.01
(L)bs or (L)ba, Barrel Radius, Inches	0.22	0.22
Calc. Average Bullet Rifling Friction (F)avg. lbs	800	200
Max. Recoil Without Friction, Ro, pounds	3,714	3,714
Avg. Recoil Without Friction, Ro, pounds	1,857	1,857
Avg Recoil With Avg. Friction, (R)avg, pounds	1,057	1,657

*At muzzle end of bullet.

†E. A. Avalone, T. Baumeister, A. Sadegh. Marks Standard Handbook for Mechanical Engineers, 10th ed. McGraw-Hill, 1996, pp 5–3, 3–23.

Additional performance data for lightweight and heavyweight handguns firing the same ammunition were obtained. These data indicate that lightweight handguns with an aluminum alloy frame and a steel barrel have a much higher recoil than do identical handguns with a steel frame and a steel barrel. The data in Table 1 indicate that this was true of a handgun with an aluminum alloy frame and aluminum alloy barrel but did not address the case, as referred to above, of an aluminum alloy frame with steel barrel. Data for that case are provided in Table 2. These data were based on a somewhat revised calculation procedure and demonstrate that the new data properly reflect the observation that a steel barreled aluminum framed handgun also exhibits a much greater recoil than does a handgun fabricated entirely of steel. The analysis takes into account the fact that about a 1-inch-long threaded and rifled barrel projection extends inside the aluminum alloy frame. This results in an effective modulus of elasticity that may be calculated as follows:

E effective = ((Eal)(Lal) + (Estl)Lstl))/(Lal + Lstl) (5)

Table 2: Difference in Recoil between Three .38-Caliber Guns with 2-Inch Barrels of Different Weight and Material of Construction

	Lightweight Gun	Heavyweight Gun	Lightweight Gun
Barrel Material	Steel*	Steel	Aluminum Alloy†
Frame Material	Aluminum Alloy	Steel	Aluminum Alloy
E eff. Barrel, million PSI	23	30	10
Assumed Rifling Depth, inches	0.2	0.2	0.2
Barrel Wall Thickness, inches	0.38/0.38	0.38/0.38	0.38/0.38
Barrel Pressure on Rifling, PSI	64,200	83,100	27,700
Average Barrel Gas Pressure, PSI	30,000	30,000	30,000
Barre; Design Gas Pressure, PSI	100,000	100,000	100,000
Avg, Frictional Force, pounds‡	1,441	1,998	416
Recoil Force Without Friction, pounds	3,400	3,400	3,400
Recoil Force With Friction, pounds	1,959	1,402	2,984
Recoil Force, % higher than base case	39.7	base case	113

*The 1-inch rifled steel barrel section inside the aluminum alloy frame is assumed to be thin enough that the surrounding frame behaves as would an aluminum barrel section.

†The aluminum alloy surrounding the thin-walled rifle steel liner is assumed to be thick enough relative to the steel liner that the composite barrel behaves like an all-aluminum barrel.

‡Average of frictional forces at breech and muzzle ends of bullet

Reference

1. Matthew J. Hall. Measuring Felt Recoil of Sporting Arms. International Journal of Impact Engineering, vol. 35, no. 6, pp 540–549, 2008.