Home Forums General Game Design Adding to the penetration formula collection — or not.

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    Avatar photoJohn D Salt

    As you will know if I have bored you on the topic before, I collect (simple, one-line) armour penetration formulae, having some time ago decided that this was a less harmfully stupid activity than collecting armour penetration data.

    Excitement is something that doesn’t visit my part of Swindon very often, so it was a bit of a red letter day when I found a method of estimating armour penetration I hadn’t seen before. It was in a paper published by the Institute for Defense Analyses: Gillingham and Patel, “Method of Estimating the Principal Characteristics of an Infantry Fighting Vehicle from Basic Performance Requirements”, IDA Paper P-5032, August 2013. IDA, like several other American defence organisations, generously makes it findings available to the public, so you can get your own copy from


    If anyone has played the game “The President’s New Tank”, where competing teams allocate resources to firepower, mobility and protection to try to produce the best overall design for a tank, this is like a much more complicated version of that, for infantry fighting vehicles. The paper gives handy rules of thumb for estimating things like how much internal space to allow per crew member, how much weight the engine and transmission will take up for a given automotive performance, how much fuel will be needed for a specified range, and so on. One of the key things to estimate is the weight of armour needed to achieve a specified level of protection. Appendix D is entitled “Derivation of Scaling Law for the Minimum Metallic Armor Thickness to Defeat Armor-Piercing Ammunition”. Lurking in there, right at the back of the report, is a formula to find the ballistic limit thickness (the ballistic limit, V50, being defined as on that gives 50% shot wins and 50% plate wins). It is:

    t = sqrt(mv^2/(0.7*UTS*pi*d))


    t is the ballistic limit thickness of armour, in mm
    m is the mass of the projectile, in Kg
    v is the striking velocity, in m/s
    UTS is the ultimate tensile strength of the plate, in MPa
    d is the calibre of the projectile, in m

    Obviously it’s more than I can resist to thwack this immediately into a spreadsheet, and check it against the “book” armour penetration figures they give for various projectiles of the kind IFVs will be interested in keeping out — small arms AP bullets, HMGs, 20mm to 30mm cannon, a few strange long rods, and 90mm APC.

    Gillingham and Patel seem quite pleased with their formula, and say “the data over a wide range of parameters fit the dependence on mass, diameter, and velocity very well, with an r-squared of 97 percent”, which sounds great. The data I plotted got the same r-squared value. However, the average absolute error between “book” penetration and formula prediction was about 16%, with the worst estimate being 63% out. Put in those terms, it didn’t sound so great, so I decided to test a couple more formulae from the collection to see how they performed.

    Morin’s formula, the distinguished old campaigner from 1833, which appeals to common sense by making penetration proprtional to kinetic energy over impact area, came off rather badly, with an r-squared of only 0.74 and worst case of 149%. Data points on the plot against “book” values were all over the place like a madwoman’s breakfast. The worst case was for a 7.8 x 78mm tungsten alloy rod arriving at 1101 m/s, which is rather outside the area of expected validity for Morin. Thanks to a simple typo in the report, the calibre of this projectile is entered in the table as 78mm, which would make it more of a coaster than a rod; before I corrected this to 7.8mm, it produced alarmingly way out results for any formula. Mr. Picky always wonders how many data points suffer from similar minor transcription errors that don’t produce such obviously daft results.

    Still, onwards and upwards. Gillingham and Patel say the model describes penetration by plugging, so the Krupp formula might make an interesting comparison. The traditional fudge-factor used, notably by the Russians, is K=2400, but we can tune this to match the numbers we want. Incidentally there seems to be a bizarre habit, especially from Russian sources, of referring to this formula as the Jacob de Marre formula. I have seen a fair variety of things described as “de Marre” formulae, many bearing little resemblance to his orignal formulation; but this one is very definitely one of Krupp’s formulae, so I’m at a loss to know why people so often misname it.

    The form I used was

    t = (v * sqrt(m))/(k * sqrt(d))


    t is the thickness of armour penetrated, in mm
    m is the mass of the projectile, in Kg
    v is the striking velocity, in m/s
    d is the calibre of the projectile, in m
    k is the plate quality fudge factor, 2400 by tradition

    Now to the twiddling of the K value. After a bit of plotting and some careful entwiddlement, it became horribly obvious that Gillingham and Patel’s formula was, in fact, the Krupp formula with a false beard and stuck-on nose. Choosing the right K value for a given target strength produced precisely the same results as in Gillingham and Patel’s formulation. Theirs is, I will admit, more useful in that it makes the armour plate strength an explicit part of the formula, so my time has not been completely wasted, but I am disappointed to find something not as new as I was expecting. It seems odd to me that they do not reference Krupp anywhere in the report; it is possible that they may have independently re-invented it.

    I also found it interesting (low boredom thresholds run in the family) that the K values needed to align the results were so low, in the region of 1600 for 1000 MPa plate. The difference from the Russian traditional K=2400 aligns with the observation that published Russian figures are consistently less optimistic in the depth of penetration achieved than those produced by Western nations, thanks to their more demanding penetration criteria.

    My last experiment to date was to fling in Tierberg’s penetration formula, to see how that compared. I have a sneaking fondness for this one, one of the more recent additions to my collection. I found it a few years ago in TNA document ADM 213/951, “German Steel Armour Piercing Projectiles and Theory of Penetration”, a 1945 report by a couple of Royal Artillery officers on prisoner interrogations. This report says:

    “Tierberg, the head Krupp designer until recently, was clearly a man of the old school who hated innovations. He was very autocratic and simply could not conceive of any design being better than his own. The only formula he ever used was

    t^0.8 = V/C sqrt(W/D)

    for normal attack.”

    With the fudge-factor (here C rather than K) appropriately tuned this gives about the same average error as Krupp, but the maximum error on the data given is 42%, rather than 63%. I would tend to prefer Tierberg were it not for the fact that his biggest errors occur with the projectiles I am most likely to be interested in as a WW2 wargamer, the 90mm APC. A certain percentage error in penetration for piddling little rifle-calibre bullets will not matter to the rules-writer as much as the same percentage error for a big tank-killing beast like the 90mm.

    There the matter rests for the moment. I may make the effort to do a comparison with Dehn’s formula, but that will need me to guess the density of projectiles in the data table, as it is not stated. If anyone is still awake, I may post the results.

    All the best,


    Avatar photoWhirlwind

    I am awake, so you need to post the results.

    And an e-mail sent, on a related subject!

    Avatar photoShaun Travers

    I love a good unearthing story, especially one I am interested in as well.  Keen to here the result with Dehn’s formula if you can find the time for guessing.

    Avatar photoJohn D Salt

    So, some findings, and some maunderings. I warn you that I have, like Pascal, not had the time to write a short letter.

    I tried out the Morin, Krupp (as formulated by Gillingham and Patel) and Dehn formulae on two sets of test data. The first was the one from Gillingham and Patel. The second was a collection of WW2 data that I had collected from my most-trusted sources in support of a Facebook argument I had some weeks ago.

    Morin’s formula assumes penetration by ductile hole enlargement, so penetration is proportional to kinetic energy loading. Penetration therefore increases more than proportionally with velocity.

    Krupp’s formula assumes penetration by driving out a cylindrical plug, so penetration is less than proportional to kinetic energy loading, and penetration increases proportionally with velocity.

    Dehn’s formula is a bit more complicated, somewhere between the two. The formulations I used of Morin and Krupp both allow for the ultimate tensile strength of the plate; Dehn’s formula also allows for the density of the plate. I would have expected this to tell in its favour with the Gillingham and Patel figures, as they include quite a lot of data points using aluminium armour.

    On the Gillingham and Patel data, Dehn did rather disappointingly. Both Morin and Dehn gave about 30% average error from the “book” value, and Krupp gave 18% (a bit different from the last figure I quoted, because last time I framped the book thicknesses to be corresponding thicknesses of 1000 MPa strength armour, whereas this time I explicitly included the strength in the calculations). Morin’s worst estimate was 123% out, and Dehn’s 171%, both on that troublesome tungsten rod. Krupp’s worst estimate, off by 91%, was, weirdly, for 7.62x54R API (B32), which Dehn got within 2%. In terms of r-squared number against the book data, Morin got 0.537, Dehn 0.498, and Krupp a very creditable 0.932. This is worse than Gillingham and Patel’s own figure, I suspect because they were forcing a zero intercept, on the face of it not unreasonable. If one does this, the r-squared numbers improve to Morin 0.750, Dehn 0.717, and Krupp 0.973. Either way Krupp is the clear winner on that data set.

    On the WW2 “trustworthy sources” data, it’s a different story. Whereas the Gillingham and Patel numbers are for the sort of thing that you would throw at an APC, IFV or LAV, and therefore tend to be small calibres by tanky standards, the WW2 set includes “proper” tank-punching guns, up to the mighty 122mm, 32-pounder and 12.8cm. Here the order of merit is pretty much reversed. Dehn’s average error is 17%, worst case 70%. Morin’s average is 34%, worst case 160%. Krupp, in contrast to its previous magnificent performance, has an average error of 45% and a worst case of 178%. Krupp’s worst effort is for the Russian L/16 76mm gun firing BR350, which Dehn puts 19% over, and Morin gets spot-on.

    The WW2 data included both full-calibre and sub-calibre projectiles (APCR, APCNR and APDS). Breaking out the results by these classes gave the average and worst-case percentage errors tabulated here:

    				Morin   Krupp   Dehn
    Full Calibre	Average error	23%	63%	16%
    		Worst case	66%	178%	57%
    Sub-calibre	Average error	52%	16%	19%
    		Worst case	160%	32%	70%

    It is noticeable that Dehn does pretty well on both cases; Morin is much better for full-calibre projectiles; and Krupp, while miserable overall and for full-calibre projectsiles, does very well for sub-calibre projectiles, even beating Dehn.

    All this suggests to me that Krupp really comes into its own for screamingly fast, tiny, very pointy projectiles. A table I stumbled across in “Impact and Explosion” gave some estimates of shatter velocity according to angle of impact and nose shape, and showed that very pointy (3 crh) projectiles had higher shatter velocities at normal impact than blunter ones, but the position was reversed by the time you got to 30 degrees from normal. Given that all Gillingham and Patel’s figures were for normal impact, I wonder how much worse some of the pointier things might have done at 30 degrees.

    In terms of r-squared values on the WW2 data, Dehn came out best with 0.876, Morin second with 0.815, and Krupp trailing badly on 0.480. Again, forcing a zero intercept makes everyone look better, Dehn getting 0.978, Morin 0.966, and Krupp 0.907.

    Even though I am working with the best data I can get, it still leaves much to be desired. None of the WW2 data includes any information about the target plate, so for the sake of a round number I simply assumed everything had an ultimate tensile strength of 1000 MPa (which is bloody good for WW2). I have found it very difficult to find good data on the dimensions and masses of sub-calibre penetrating cores, and for a lot of the small-arms projectiles in Gillingham and Patel’s data there should probably be an allowance for jacketed penetrators, which none of the formulae include. There’s also going to be a good deal of wonkiness in the striking velocities, as I have used penetration figures at 500m against penetration calculations using the muzzle velocity. It would not be surprising if all three formulae produced better estimates when fed better data.

    Of course I could have tried to estimate the velocity loss over 500m, but decided not to. Apart from laziness, my reasons were the lack of firing table data for most of the guns, and also a growing mistrust of such tables after re-reading Jochen Peelen’s “A Look at Drag Models in Old Small Arms Firing Tables” (a proper gun-nerd title for a proper gun-nerd book), in which he concludes that “it seems in order to be quite sceptical about the data in old firing tables”.

    Indeed, since most published penetration tables were based on desk caluclations to expand a minimal number of real test shots, I think that there is often no particular reason for the believing that the “book” figure is any more representative of reality than the calculated one. I have raised enough complications not to mention the additional one that armour plate generally decreases in strength as it increases in thickness. Ooops, mentioned it.

    I said that the WW2 data had been collected originally for an argument on Facebook. This arose when a bloke in a wargaming group posted a gussied-up version of a graph from the late Stuart Asquith’s book on WW2 wargaming (which I’ve not read). This seemed most remarkable to me, as he was apparently proposing that armour penetration should be determined with reference to gun calibre. Weirder yet, it was not a simple function of calibre, but instead used a line plotted on the graph, which zigged and zagged with no rhyme or reason that I could detect. One of the indigestible consequences of this scheme was that there were guns that could successfully defeat a heavy Churchill (Mk VII or VIII) from the frontal aspect, but could not defeat its much thinner side plates. This, let’s face it, is barking bloody mad. I said so with my customary tact and diplomacy, thus causing tremendous offence to one Facebooker, who delivered a finger-wagging phillipic admonishing me for various crimes, including the mortal sin of not realising that it was a game and supposed to be simple and fun. As a demonstration that, if you were going to pick a single attribute of a gun system to assess its armour penetrating capability, there were better choices than calibre (unless we are talking about HEAT rounds), I plotted the correlation between armour penetration at 500m and barrel length (a very easy number to collect if you know that a gun is a 50/60, or a 75/48). Given the expected roughness of using such an estimator, I could not be bothered to distinguish between nations that measure calibre length to the breech face and those that just measure the barrel, and anyway I don’t know offhand which countries do which. I did however sling together a collection of “Most trusted” penetration data from 40 different guns; 6 American, 7 British, 13 Russian and 14 German. At first I used just the data for APCBC (or the nearest thing to APCBC the weapon fired); later I added sub-calibre shot for another 29 entries.

    Now, it seems fair to assume that barrel length is going to be correlated with muzzle velocity, and so with penetration performance. I was, however, surprised at the goodness of the fit, with an r-squared number of 0.888 for APCBC rounds (more if one forced a zero intercept), and an average error of only 14% in estimating “book” penetration. Since I had projectile mass and velocity data to hand, I wondered how much better the correlation would be with muzzle energy density. For the sake of something different, I also calculated momentum density. In order to be able to score sub-calibre projectiles using barrel length, I produced plots for each ammunition nature, and applied a relevant fitted fudge-factor to each to produce a penetration estimate. To my surprise, all three measures produce similary good estimates. Since I am fitting fudge-factors to get the measures to produce estimates close to the “book” figures it shouldn’t be surprising that they bear some resemblance to the data, but the overall average error for all three estimators was about 16%, a bit better for full-calibre and a bit worse for sub-calibre in each case. The r-squared numbers for energy density, momentum density, and barrel length were 0.837, 0.862, and 0.832 respectively, and much better if a zero intercept was forced. There is not much to choose between any of the three, and I was really surprised that momentum density did so well.

    For comparison with the obviously-bad idea of basing penetration on calibre, I plotted that for APCBC rounds only, and it produced an r-squared number of 0.441. Looking at the points on the graph makes the very obvious, ummm, point that there is an awful lot of variation in the penetration of 75mm and 76mm guns, from flowerpot guns like the Russian L/16 and the German L/24 to screaming beasts like the 75/70 and 17-pounder. Whatever the mysterious reasoning behind Stuart Asquith’s line on his penetration graph, it did appear to have some positive effect, as the r-squared number for the correlation between his results and the “book” numbers was rather better, at 0.556. And, as a reminder that r-squared for these problems always seems to be made better by forcing a zero intercept, these would have been 0.896 for “raw” calibre and 0.905 for the Asquith line. A reminder not to take r-squared number as an indicator of model quality without looking at other things as well, because Asquith’s suggestion is still daft.

    What I take away from this is that it may well be possible, with a plentiful helping of Dr. Fudge’s all-purpose cavity filler, to produce quite good estimates of penetration performance for WW2 guns about which you know nothing but the barrel length (rather a difficult attribute to conceal). While you may as well take advantage of knowledge about muzzle veocities and projectile masses if you have it, the additional (really quite minimal) complexity of using one of the respected penetration formulae might not add as much fidelity as you would imagine. And, remember, in the relative absence of data from trial shoots giving striking velocity and penetration achieved together with full information on plate and projectile and criterion used, a lot of the data we have is not all that trustworthy in the first place.

    Here, then, are Dr. John’s Patented Fudge Factors for producing not-entirely-unbelievable estimates for armour penetration at 500m, using only barrel length:

    For APCBC type projectiles, score             27mm of penetration per metre of barrel length
    For German or Russian style APCR, score       38mm of penetration per metre of barrel length
    For Brtish APDS or American HVAP, score       48mm of penetration per metre of barrel length
    For APCNR, score                              44mm of penetration per metre of barrel length

    And, because Stuart Asquith’s idea of basing penetration on calibre isn’t daft for HEAT,

    For spun HEAT, score                         1mm of penetration per 1mm of calibre
    For unspun HEAT, score                       1.32mm of penetration per 1mm of calibre

    The APCNR and unspun HEAT estimates will probably be a bit squinty, but the others are likely to be little worse than you would get from a more sophisticated calculation using a proper armour penetration formula, for which you will at least need to know the mass, calibre and striking velocity of the projectile or its penetrating core, and have the extra problem of picking the best penetration formula.

    Obviously there is no real need to estimate penetration data from a single system feature unless one has a fetish about it, but it may prove useful if you dig up a reference to, say, the little-known Nether Crashbanian 65/37 gun-howitzer, about which no other data is available.

    All the best,


    Avatar photoWhirlwind

    Thanks very much for posting that John, very useful for getting an estimate into the right ballpark.

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