Sense of Krupp Armor Penetration Equation

[marty1]

Making Sense of the Krupp All-Purpose Armor Penetration Equation

I am interested in comparing the Milne version of the DeMarre Armor Penetration equation with results from the Krupp All-Purpose Armor Penetration Equation. While I am reasonably well versed in extracting penetration data from the Milne Equation, I am a bit at a loss with Krupp.

My first question: Is the Krupp equation specific to AP, or can it be reliably employed with APCBC as well?

My second question: What penetration criteria is entailed with the Krupp Equation. Is this a 50% probability of complete penetration? Or is penetration criteria defined as something else?

My third question regards the Krupp constant. What are reasonable ranges for this constant? From studying Nathun Okun’s web site he provides Krupp Equation Constants relative to projectile type and armor target type. For Wotan Hart Armor – which appears to equate loosely with roll hardened armor – the range of “C” values for the Krupp equation is from about 640 to 690. However I can’t discern any specific pattern to the “C” values. Perhaps I am trying to read too much into projectile caliber in this regard in trying to corrolate caliber with magnitude of "C"? In any case should one expect that all projectiles will fall within the range of “C” values described on N.Okun’s web page – i.e. the Krupp Constant "C" will vary within a range of approximately 640 to 690 for AP vs. Wotan Hart?

My last equation regards penetration as a function of obliquity. Unlike the Milne Penetration Formula, the Krupp Equation has no direct input for angle of attack. I am interested in comparing Milne Predicted penetration at 0-degrees, 30-degrees and 60-degrees but have no way of directly comparing Milne results at obliquity with Krupp results at obliquity. How did Krupp modify penetration to account for obliquity?

Thank you for any replies in advance.

Sam H.

[Nathan Okun]

This formula was, to my knowledge, developed using both uncapped and capped shells against both homogeneous, ductile armor (Wh, for the most part) and face hardened, brittle armor (KC n/A). The use of a single numerical constant and kinetic energy power exponent for the entire range of plate thickness values is roughly correct with KC n/A, which always fails the same way (brittle fracture of the rigid face followed by punching out an armor plug from the soft back as much of the face is pushed through the plate and out the back by the projectile), but homogeneous armor fails in several different ways depending on the nose shape of the shell, the angle of impact, and the thickness of the plate in multiples of the projectile diameter -- at right angles, thin plates act like trampolines (stretch to absorb energy for a wide area around the impact point before tearing in the center), intermediate plates act like cold taffy being pushed through by somebody's finger (bulge back only in the area near the finger and fold back thick pointed "petals" of steel ringing the hole), and thick plates act like a block of clay punched through by an ice pick (almost no bulging back and only a small region at the back forming petals, with most of the metal in the hole being pushed sideways out of the way, bulging the armor plate both at the front and the back to make room). Thus, the formula is very much a crude average of a lot of different effects.

The effects of AP caps (make the nose much blunter, at least until the cap breaks up against thicker plates, and protecting the nose from damage) are also kind of blurred out, since sometimes they help and sometimes they hurt, but this is not separated out, either.

The formula has a rather large scaling effect (large projectiles penetrate much more easily than small projectiles against scaled plates of identical composition), which is rather at odds with my homogeneous armor data, but agrees very closely with my face-hardened armor data. However, this formula does not correct for that (using the 'fudge factor' "C") for homogeneous armor when hit with projectiles over 8" (20.3cm), though it does compensate for smaller shells. Since I assume the results were matched to actual ballistic firing trials, this implies that the rather over-hard Wh material (Percent Elongation in tests of only 18-20% compared to most contemporary foreign armor values of 23-25% or more) was experiencing a scaling effect rather similar to KC n/A, which hurt its protective ability against battleship-size shells, though no loss in strength against cruiser-size shells or smaller is evident.

The effects of obliquity were not called out in the formula and I tried to match the penetration curves with my own formula, with some success. The projectiles used are again a mix of capped and uncapped shells, blurring the difference. I now have the actual formula used, but again it turns out to be a general-purpose equation that has its constants matched to a given set of test results and then used for all plates from then-on (my approximation is thus probably just as "good" as the original one!). It had a definite thickness factor that made thicker plates increase their resistance faster than thinner plates as obliquity increased. When projectile damage occurs or against very thick plates, this is indeed true since the initial oval-shaped hole or, at extreme obliquities, long slot being formed in the plate may not reach the plate back before the projectile is deflected to glance off (in pieces, if broken). However, for thinner plates and no major projectile damage, U.S. and British test results show that this doesn't happen and from a thin plate through a medium-thickness plate, the plotted percentage increase in resistance (required increase in striking energy to penetrate) with increasing obliquity more-or-less is the same. Again, this increase in resistance using the German curves with thickness may be due to more damage occurring against thicker plates, so the "effective" resistance goes up when all impacts are mixed up together, broken or intact, capped and uncapped. Not sure.

If you look at my FACEHARD program's Source Files (which are regular text files with BAS instead of TXT), you will find my basic obliquity formula and tables of constants for various projectiles that colculate the effects of obliquity based on these German curves, though I apply this only to face-hardened armor, not all armors, as Krupp did, and not to all projectiles, even there (some projectiles are stronger than others!).

Nathan Okun

[marty1]

Thanks Nathan. Interesting observation regarding scale effects. I had not really considered this aspect of the equation.

I will have a look through your FACEHARD data files for slope effects. Will slope effects differ dependent upon whether the target is Face Hardened Armor or RHA?

Do you know off hand how penetration was being defined for the Krupp Equation. I presume that the equation is based upon the projectile completely through the target plate, and that the bursting charge is effective – but I don’t know this for sure.

In addition, is there a probability of success associated with the formula. Moreover can I assume that the limit velocity predicted by the Krupp Equation for a given projectile and plate combination represents a 50% success probability?

[marty1]

I had the chance to play around with two of Nathan’s penetration programs this morning:

M79apclc.exe
&
M79lprnt.exe

Very nice programs.

I wanted to compare the M79*.exe results with the Krupp Universal Equation. I stuck with using M79 AP vs armor inclined at 0-degree obliquity on each of my runs. Input for the M79*.exe programs were as follows:

Projectile = 3-inch M79 AP-T
Projectile Dia = 3-inch
Projectile Weight = 15-lbs

Plate Quality = 1
Plate Elongation = 25%

The naval ballistic limits (NBL) for 3-inch plate = 1583-fps
The NBL for 4-inch plate = 1902-fps.

I than plugged the above into the Krupp Universal – i.e. the same inputs for diameter and weight, and the above striking velocities corresponding to the NBL limits described above for 3-inch RHA and 4-inch RHA (i.e. 1583-fps and 1902-fps). The trick is the Krupp constant. If I am interpreting the article on Nathan’s Web page correctly, and using Wotan Hart armor, it looks like the range of Krupp constants is about C=655 to C=690. Going off of w/d^3 values M79 works out to be about 0.555. If I plot w/d^3 vs Krupp C I would reckon that my Krupp C for M79 would be about 660 to 670. However the relationship between w/d^3 vs Krupp C is rather erratic. I therefore tried a range of possible Krupp C values from 655 to 690. The output is as follows:

Using C=655 and Impact Velocity=1583-fps; penetration = only 2.5-inches
Using C=690 and Impact Velocity=1583-fps; penetration = only 2.35-inches

Using C=655 and Impact Velocity=1902-fps; penetration = only 3.15-inches
Using C=690 and Impact Velocity=1902-fps; penetration = only 2.95-inches

There is obviously a fair amount of contrast between the M79*.exe program output and the Universal Krupp equation output. Presumably the range of Krupp APC Constants do not translate well into possible Krupp-C values for plain old AP.

Back calculating the Krupp-C values using M79*.exe as the basis for predicating ballistic limit results in a Krupp-C value of M79 AP of about 540 to 560. I have no idea if these are reasonable Krupp-C values for AP vs Wotan Hart or not. They are certainly well below the 655 to 690 range for APC vs Wotan Hart.

I can already see that this will result in problems in trying to use slope effects directly pulled from M79*.exe and plugging them into the universal krupp equation.

[George Elder]

In many cases, trail data is collapsed and penetration curves are greatly simplified, with most countries being guilty of this. In GK 100 and other documents we see that the Germans often used quick and dirty penetration formulations and ofen eschewed calculating nuances such as the influence of decapping, yaw, etc.. They were certainly aware of these effects, as ADM 213/951 shows -- and they devoted considerable study to measuring these effects. However, for general purposes they employed reductionist methodologies that gave somewhat crude approximations. I am not at all sure why this was done, other than to note that the Germans were not alone in this tendency. Then again, they didn't have the sophisticated software and computers.

George Elder

[marty1]

In all cases that I know of in which a regression equation -- ala the Krupp Universal Equation -- has been developed, the function was developed by "collapsing" -- as you call it -- experimental data. It’s the nature of the beast. Some of us might refer to this as developing a best fit curve and a function that will predict points along that curve.

But my questions are specific to derivation of the Krupp equations constant, as well as how the Germans accounted for the effect of obliquity.