Interior and Terminal Ballistics for GURPS

by Douglas Hampton Cole

Ever had a player who wanted his Black Ops character to tote around some obscure or fictional piece of artillery that he'd read about in a back issue of a science fiction magazine? Or a player who wanted his Old West character to carry a lever action rifle and a revolver with the same ammunition? How about players who want to cut down a revolver or rifle to make it more easily concealable? In some campaigns, players ask for some very oddball pieces of equipment now and again.

Presented here is a method for taking some real-world information and converting it into GURPS damage. It also presents a slightly new way of looking at special ammunition types, and revisits the blow-through rule, giving it a nudge towards the realistic by making blow-through vary with a projectile's wounding potential. Finally, it provides a formula for estimating the kinetic energy (KE) imparted to a projectile given some real-world parameters.

Interior, Exterior, and Terminal Ballistics

Interior ballistics refers to everything that happens in a firearm from the moment the striker hits to the moment the bullet leaves the barrel. For TL5-7, this involves black powder or nitrocellulose powder expanding in a firing chamber, and then accelerating a projectile down a tube. For Ultra Tech projectiles, it might involve magnetic acceleration. We can even generalize the issue by claiming that "interior ballistics" can be stretched to include the force a bowstring exerts on an arrow, and cover TL3-4 missile weapons as well.

In all cases, the propellant is applying a force to the projectile over a certain distance (or draw length, for the case of bows and crossbows). Simple physics states that force applied over distance equals energy. If we turn that into kinetic energy by using the equation F.x = ½ mv2, we can determine the final velocity of the projectile, which is where interior ballistics ends, and exterior ballistics begins.

Once the projectile leaves the weapon, Sir Isaac takes over and we could calculate a trajectory. However, all of this is (usefully) abstracted into a skill roll in GURPS, which is far simpler. It is worthy of note, though, that with some knowledge of how the velocity of a projectile degrades with distance, you would not have to employ the mechanic of the ½ Damage range . . . you would simply solve for the projectile velocity at the target range, recalculate damage based on the final velocity, and roll some dice. That would require a computer to be used at the gaming table . . . some do it, but it's certainly not for everyone! Once the bullet arrives at its target, exterior ballistics ends.

While there is probably no more controversial subject in firearms literature than the effects that bullets have on people and animals, for gaming purposes the questions are fairly simple: how much armor can the projectile penetrate, and what does it do to the target once it gets through all that armor? That's terminal ballistics, and for gaming purposes, we're going to assume that projectiles do two things. First they penetrate defenses, and then they penetrate flesh, causing a wound as they do so.

Modeling the penetration of armor is actually not that hard. Penetration of solid objects tends to scale with the kinetic energy (KE) of the impactor divided by its cross sectional area (Xsect). In GURPS, DR scales linearly with thickness (in reality protection scales with the square of thickness), so at the very least our model will scale armor penetration ability with the square root of KE/Xsect. However, in order to maintain consistency with the vast majority of GURPS published material out there, we have to make some modifications to that scaling. We need to assume three things:

  1. The 585 Joule 124 grain 9mm NATO standard cartridge is fixed at 2d+2 damage, the value given in GURPS Basic and High Tech
  2. The .50 BMG is accurately recorded in Vehicles 2ed as 13d+1 (equivalent to an average penetration of about DR 46.5, or 17mm of steel armor)
  3. The penetration of a .30-06 should be roughly halfway between the 9mm and .50 BMG (hopefully matching the 7d+1 listed in GURPS High Tech).

Re-scaling the penetration figure to fit currently published GURPS stats may not be elegant, but in the end it will mean that fewer tweaks need to be made in existing campaigns. Chalk another one up to "playable abstraction!"

This article will deal with terminal ballistics first, proceed to ignore exterior ballistics, and then return to interior ballistics for those who just have to modify their .357 magnum to have an 18" barrel, or design an archer who can pull a 200lb longbow.

Icky Equations: Penetration Of Armor

The damage formula for armor penetration is

Damage (points) = sqrt(KE1.04/Xsect0.314)/13.3926

where KE is expressed in joules and Xsect is in square meters. Convert this value to dice by dividing by 3.5 and converting the remainder to "adds" by taking the decimal remainder, multiplying by 3.5, and dropping all fractions. This would convert damage of 13 points to 3d+2. Alternately, you can try to divide damage by either 3.5 or 7, and pick which is more satisfying, 13 being (approximately) 3d+2 or (exactly) 4d-1.

Example: We want to determine the penetration of a new 7mm (cross section equals π×(Bore2)/4 = π×(.007m)2/4 = 38.5×10-6 square meters) TL8 assault rifle. It fires a projectile with a KE of 3600 joules (a 10 gram bullet at 848.5m/s). The formula gives it a damage value of Sqrt[(36001.04)/(38.5x10-6)0.314]/13.3926 = 26.0 points. This converts to 7d+1 of damage with 0.5 points left over, which we drop. The base penetration of this round is 7d+1.

Armor Piercing Rounds

All bullets are not created equal. Some are designed to penetrate solid objects more effectively than others. This can be accomplished two ways: either with a very high energy and low cross section (such as using ammo with a discarding sabot) or by constructing the bullet out of special materials. The penetration value above assumes the bullet is constructed out of a copper or mild steel jacket wrapped around a lead core. Bullets which feature a hardened steel cap, or with tungsten or depleted uranium penetrators, will punch through armor more effectively given the same energy and cross section. Use the following armor divisors with this system:

    •  Steel cap, reinforced core, or semi-armor piercing:    

1.25

    •  Tungsten or tungsten carbide core:

1.9

    •  Depleted uranium (DU) core or penetrator:    

2.4

The rules are as usual; the armor divisor reduces DR, but also reduces damage by the same amount (see Using the Rules in Play for a modification). This represents a bullet that penetrates farther without destabilizing or fragmenting. Do not use an armor divisor of less than one for expanding bullets, such as soft lead (LRN), soft point (SP), or jacketed hollow point (JHP). These are treated below.

Example: Our TL8 7mm battle rifle can fire special AP bullets at the same energy but with a tungsten core (the bullet will be heavier and slower). The damage is still 7d+1, but with an armor divisor of 1.9, written 7d+1(1.9). It will lance through more than 49 points of armor on an average roll!

Special Rule: Armor piercing sabot ammunition no longer merely adds +1 damage per die. Rescale the penetration figure based on the actual diameter of the penetrator. As a rule of thumb, a tungsten round with 48% of the base bore diameter with equivalent KE to the original round will duplicate the performance of APS from p.UT18. Sabot rounds are commercially available today, with a .223 projectile fired from a .308 case at over 4000 feet per second (7d)!

Example: Our opponents, the Evil Grey Aliens have beefed up their Energy Cloth by some SuperScience, and are sporting DR of near 70! Since our Black Ops still need to kill them, we issue 7mm APS ammo with a DU penetrator that's still 3600 joules, but is only 3mm in diameter. The math now works out to about 34 points of damage, giving us 9d+2(2.4), capable of penetrating about DR80. The only good Grey is a dead Grey . . .

More Icky Equations: Wound Channel And Blow-Through

Taking the above formula to its logical extreme, we'd all be firing hypervelocity DU needles. These will have extremely high armor penetration, and yield very impressive numbers. A 1.5x30mm steel gauss needle with 1600 joules of KE will clock in with 7d+3 of penetration, much more effective than an M16 projectile of the same energy (5d).

We're still safe from the Munchkins, however. Once you've gotten through all that armor, you still have to leave your mark on the target. That mark is the wound channel.

Bullets cause damage by disrupting tissue, plain and simple. The old rule of thumb still holds: All other things being equal, make a big hole. While tiny projectiles might have the penetration advantage, they also don't cause large wounds (there are exceptions . . . see below!).

The wound channel modifier is what is used to scale the penetration ability with a projectile's ability to injure the target. The wound channel is given a figure of merit describing the average points of damage the round would do penetrating all the way through the torso. It is proportional to the momentum of the projectile multiplied by its effective cross-section, scaled to give our 9mm "reference bullet" a wound channel modifier of 1.0, preserving the core of the GURPS rules.

Wound Channel Damage (points) = MV × Xsect* x 26,220

where momentum is expressed in kg × m/s, and Xsect* is again expressed in square meters. To calculate the Wound Channel Multiplier, use the following equation:

WCM = sqrt[Wound Channel Damage/Penetration Damage].

Don't include the armor divisor in your penetration equations. Note that you'll need to know the mass and velocity of the projectile, and this might require some extra work to derive it for oddball projectiles. The wound channel modifier is exactly the same as the bullet size modifier in GURPS High Tech, and is used in place of that figure.

Example: We're back to the 7mm DU APS round we gave our Black Ops. After some math, we decide the penetrator is a 3mm DU rod with an 11.7:1 aspect ratio (so it's equal in length to the original full-size round) which has a mass of 3.17 grams and a velocity of 1505 m/sec (!). Entering this information into the WCM formula, we get Wound Channel Damage = (.00317kg × 1505m/sec × 7.1x10-6 m2 x 26,220) = 0.9 points, and the WCM = sqrt(.9/34) = 0.16, which we round to 0.2. The round that is so capable of punching holes through armor will average about 5.5 points of damage per hit. What's going on? It's reality, sort of. Small holes cause small wounds, all other things being equal.

Wound Ballistics: A Quick Primer

Note the asterisk in Xsect*, though. That's our first hint that all other things are not equal. When a bullet strikes a viscous medium like soft tissue, it will do two things. First, it carves out a "permanent wound channel" which is the volume of tissue physically disrupted by the path of the bullet. For low velocity bullets, this crush cavity is the primary mechanism of wounding. The second thing bullets do is create a temporary wound cavity, which usually doesn't contribute to wounding.

The WCM listed above assumes implicitly that the bullet travels straight through the target without changing its point-first orientation. For pistol bullets, this is mostly correct. For "spitzer or long ogive" shaped bullets (rifle bullets), this is not even close to true. The point forward orientation is metastable in air, and unstable in flesh. The bullet wants to reorient itself inside your body (ow!) to travel base first. This can vastly increase the effective cross section of the projectile, as the bullet is actually traveling sideways for a while. To what extent bullets "tumble" like this is primarily a function of velocity; bullets above about 600m/sec can be counted on to do this (eventually) with high reliability. To correct for this factor, we add a bit of complexity and take a weighted average of the head-on and tumbling cross section:

Xsect* = (1-V/600)×π(Bore/2)2 + V/600×Bore2×(Aspect Ratio)

Where aspect ratio is the length to bore ratio of the projectile, and if V is greater than 600m/sec, set V = 600.

Example: We reconsider the 7mm APS cartridge from above, replacing the smaller value of the projectile cross section with the "tumbling" cross section of (0.003m×0.003m x 11.7) = 105.3×10-6 m2. This increases the WCM to 0.6, making the round three times more effective as if we didn't include this correction.

Example: A .45ACP pistol fires a 230 grain FMJ bullet at 245m/sec, with an aspect ratio of about 1.5. The penetration equation gives us a base damage value of 7.6 points (2d), and the corrected Xsect* provides us with a WCM of 1.3. This is written in curly brackets, as 2d{1.3}.

Note that the WCM for the .45 is close to the generic bullet size modifier suggested in High Tech, and is used exactly the same, with one exception noted in the section on Using the Rules in Play.

Additional Complications: Expanding and Fragmenting Rifle Bullets

GURPS Compendium II (p. CII55) presented the option to players that allowed them to increase the effective damage of their weapons by loading hollow point or expanding ammunition (don't hold this against them . . . U.S. police agencies do this too, with good reason). Hollow point and other expanding ammunition can be handled in this rules expansion very simply: replace the cross-section in both the penetration and wound channel calculations with the expanded cross section of the bullet. Most well designed hollow points expand 1.5-2× on striking tissue (a rule of thumb of 1.66 is a good one, although soft pointed rifle bullets are designed to give about a 2× expansion). Do not apply the bullet tumbling correction. This will reduce the effective penetration of the round, but greatly increase the final wound. This is realistic; most JHP ammunition tends to be higher velocity than their FMJ cousins to offset this fact.

Example: We load our .45ACP with 230 grain jacketed hollow point bullets at the same velocity as the factory load: 245m/sec. The bullet expands to 1.66x its original diameter (0.747"). As a result, the Xsect value changes to 282.7×10-6 m2, changing both the penetration to 6.5 points (1d+2) and WCM to 2.0(!). If we used the higher velocity JHP at 259m/sec, we edge over the line and eke out 1d+3{2.1} from this round. 

The last form of complication is something particularly ugly that happens when a high velocity rifle bullet yaws in an elastic medium. Sometimes, during the time the bullet is traveling sideways, it cannot handle the stress and fragments . . . sometimes badly. The M16 (5.56×45mm) cartridge is noted for this effect below a range of 200m or so, as are the Yugoslav versions of the AK-47 round (7.62×39mm) and the West German 7.62x51mm (.308) round. These fragments lance through the wound area that is forming the temporary stretch cavity, literally creating perforations that turn what is typically a fairly harmless effect into a grotesque nightmare of a wound. This is primarily a function of bullet construction and velocity, and is allowed at GM discretion. Multiply the second half of the Xsect* equation by the velocity of the bullet (in m/sec) divided by 360, so the equation reads:

Xsect* = (1-V/600) ×π(Bore/2)2 + (V2)/216,000 ×Bore2 ×(Aspect Ratio)

Note that the destabilization which causes this yaw is not present in fin-stabilized projectiles, only spin-stabilized ones. If a projectile is described as APFSDS, don't allow this. This means that typically flechette shotgun rounds will probably not be as lethal as the marketing and sales folks would have you believe!

Example: Our fiendishly clever Tech Ops have modified our 7mm DU APS ammo to not only penetrate, but introduced a structure that fragments as it yaws. The wound cross section during yaw is increased by a factor of (1505m/s)/360 = 4.18, for a net Xsect* of 440.4x10-6 m2. The damage is now 9d+2(2.4){1.3}, increased from 9d+2(2.4){0.6}. 

Example: The M16 fires the M193 projectile, a 3.6 gram(55 grain) projectile at about 990m/sec with an aspect ratio of 3.95. This projectile is known to fragment badly upon striking soft tissue. The damage for this round is 5d+1{1.3}.

Example: That Ultra Tech gauss needler fires a 1.5mm steel projectile with 1600 joules of KE. We decide on an aspect ratio of 40 this time, and a mass of 1.35 grams (if you treat the needle as a cylinder you can derive the mass by setting the density of the needle to 7,860 kg/m3, the density of steel). The velocity of such a projectile is 2392 m/sec. If the needle doesn't fragment (or melt!), it will do damage equal to 7d+3{0.3}, yielding an average of 9.3 points of damage (equivalent to 3d-1) per needle assuming an unarmored target. The 1.5mm gatling needler (p. UT2 53) is listed as a 4d weapon; perhaps we have an lower-powered needle rifle here?

Interior Ballistics, Revisited

The critical data needed to calculate the terminal ballistics of a projectile are its diameter, mass, velocity, and aspect ratio. But what about how you accelerate that projectile to that velocity?

For firearms, a round of ammunition consists of a propellant and an accelerated mass (it could be a discarding sabot, a bean bag, or shot pellets as well as a single full-bore bullet). That propellant, when detonated, fills the barrel behind the bullet (beginning with the initial starting point for the round, called the chamber) with a certain pressure. For simplicity, we can say that the explosion instantly reaches its maximum pressure, and begins to accelerate the bullet down the barrel. The maximum pressure is maintained for a short period of time as the propellant completes burning, and then falls off (the system is now a piston being pushed by a pressurized gas). With a little work, you can calculate the energy imparted to the accelerated mass during its trip down the barrel, and convert that directly (albeit approximately) into kinetic energy. Without too much more explanation, the formula is:

KE (joules) = P ×{D ×A + V ln [(A ×L/V) +1]} ×6896

P is the max pressure in the chamber in psi,
D is the burn distance, the distance in meters the max pressure lasts,
A is the bore area, equal to the barrel cross-section (sq meters),
V is the fall-off volume, equal to the chamber volume plus the volume of the barrel when the powder completes its burn,
L is the acceleration length, typically equal to the length of the bore less the burn distance. The overall barrel length is usually quoted including the chamber, so the acceleration distance for most weapons is actually the barrel length minus the burn distance, minus the case length, plus the bore diameter, because bullets are usually seated in the cartridge by about that much.

Rules of Thumb

Max Pressure: Shotguns: 11,000psi; TL6 era autoloaders (like the .45ACP): 15-20,000 psi; Modern pistols: 30-40,000 psi; military rifles: 50-55,000 psi; machineguns and autocannon: up to about 60,000psi

Burn distance: about 7/24 the case length for handgun cartridges, 7/16 the case length for rifles.

DISCLAIMER: DON'T TRY THIS AT HOME! IF YOU USE THESE RULES OF THUMB TO RELOAD AMMO, YOU'LL GET YOURSELF KILLED.

Example: The .45ACP (11.43x23mm) operates at 17,500 psi from a 127mm barrel (acceleration distance 108.7mm). The bore area and chamber area are both the same, as the cartridge is not bottlenecked. The fall-off volume is calculated at 1.581x10-6 m3. The burn distance is estimated at 7/24*23 = 6.7mm. The final KE for the round is 466.2 joules, which will propel the standard 230 grain cartridge just shy of 250m/sec. That's right in the ballpark! The damage for this round is 2d{1.3}

Example: We decide to take that same cartridge and launch it out of a 304.8mm (12") submachine gun barrel. How much damage will it do? We only need increase the acceleration distance by 304.8-127 = 177.8mm to 286.5mm. The predicted energy is 630.6 joules, yielding 291m/s and 2d+2{1.4} damage.

Example: Let's bend the system. We take a 23 gram wooden arrow, and apply a "pressure" of 1000psi. We use an 8mm diameter, an aspect ratio of 95:1, and a "barrel length" and "burn length" of 480mm (for a 760mm arrow); the case length we set at 1mm (it's irrelevant). This converts to about a 78lb draw weight, and propels the arrow at 121 m/sec, inflicting 1d+1 damage. Not bad! A 120 lb bow (1560psi) with the same properties would inflict 1d+2, firing the arrow at 151m/sec. Medieval bows were quite strong; it's not unrealistic to assume a 120lb bow, and the damage matches up very well for a longbow that requires ST15 to pull (a Thrust of 1d+1). The predicted velocities are a bit high compared to reality, but the predicted damages work well with GURPS canon material.

Wound Channel Modifier and Blow-Through

The wound channel modifier can be used to augment or eliminate the "blow-through" rule that states that damage done after HT to the torso is lost. This allows more damaging bullets to cause more severe wounds.

Optional Rule 1: The wound channel modifier also defines the blow-through modifier for that round. Multiply the wound channel modifier by the subject's HP to get the blow-through value; this is tripled as normal for shots to the vitals.

Optional Rule 2: As above, but either multiply by HT (excluding extra hit points, to give that advantage more "pull" in high tech games), or by a flat value of 10, which also increases the value of high HT in high tech games.

Using these rules on an average HP individual, the .45ACP cannot ever blow through (max damage 13 pts; blow through at 13 points), while the 9mm (at 2d+2{1} by definition) will blow through more frequently (max penetration 14 points, blow through at 10). This tends to model fairly well the arguments for and against these rounds. The 9mm is frequently accused of "overpenetrating," while the .45 is frequently defended as The Worlds Greatest Pistol Cartridge because it avoids this, among other things.

Optional Rule 3: Bullets that don't kill by central nervous system damage kill by causing massive blood loss. Don't discard all that damage before it blows through. For each 7 points of damage inflicted before blow-through is applied, a wound will bleed for one HP per minute. Reduce the time interval for bleeding for large wounds, not the damage per full minute. Do include the multipliers for hits to the vitals. So a .223 round that hits the torso for 5d+1{1.3} and rolls 21 points of damage will blow through after 13, but the wound will bleed for 1 HP every 20 seconds. Thus a HT10 character will have a 50% chance of being "instantly" unconscious (less than zero HT), and will be in grave danger of bleeding out (-5×HT) after about 16 minutes. The GM as always has discretion as to whether bleeding stops by itself. Damage to the veins and arteries location bleeds three times as fast; the heart bleeds out at a rate of hits per second, rather than per minute.

Optional Rule 4: Fragmenting rifle wounds are ugly. First aid might slow bleeding, but it won't restore any HP. To do that typically requires a roll against the Surgery skill.

Optional Rule 5: Armor piercing rounds tend to pass through a body without tumbling. As an option, do not reduce the damage rolled by the armor divisor. Instead, re-calculate the wound channel modifier using the unmodified bullet frontal cross-section, rather than increasing it for yaw. This will drastically reduce the wounding potential of AP and APS rounds, as they pay for their increased penetration with much decreased wounding.

Using these Rules with GURPS Vehicles

To use these rules with the Vehicles 2ed design system, first design your weapon normally. You will end up with stats such as bore size, approximate barrel length, and weight per shot. Use about 2/3 of the WPS to get the projectile weight. Back-calculate the kinetic energy requirement from the bore size, and from there you can determine the projectile's velocity. Make some assumptions for chamber pressure and case length by comparison to existing cartridges, and find a barrel length that is in the length class of the weapon you designed. Note the parameters you've settled on for future reference, in case you want to design short barreled carbines or heavy barreled support weapons using the same ammo.

A Final Word

All of these calculations seem cumbersome, and they are, if you use a calculator. These rules come into their own when you enter the formulae into your favorite spreadsheet program, and can use its power to solve for damage, velocity, or other unknown parameters, or play with designs to achieve the desired results. One can duplicate the performance of nearly any cannon system, from small arms to the largest I've tried yet, the 16" guns of the Iowa class battleships (6d ×137{2050}, if you're curious!).

Bibliography and Suggested Reading

Books

Websites

For cartridge information:

Wound ballistics:

* * *

My thanks to Shadowman, Hans-Christian Vortisch, and Nigel McCarty-Eigenmann for helpful suggestions and comments. Also thanks to Sean Punch and David Pulver, who gave me real encouragement to keep at this task. And to my wonderful wife, who's had to put up with years of me working on my spreadsheet and ignoring her. At least she still games with me.



Article publication date: May 24, 2002

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