National Physical Laboratory Equation

[Thoddy]

with regard to the NPL-formula 

excerpt from

"Summary Technical Report of Division 2, NDRC, Volume 1 - Effects of Impact and Explosion (1946)"

See number 2 and also note  the restrictions with regard to plate quality of thick plates

 

that may also mean "the greater the thickness to be penetrated in the case of angled plates the higher the (potential) optimal hardness.

One problem I see. The british experiments were done only at normal incidence.

I dont know if its really applicable to angled plates especially for plates angled at 45 degrees and more.

Edited May 15, 2025 by Thoddy

[Peasant]

Hi, @Thoddy, glad to see you again.

Firstly, you are mistaken about 

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The british experiments were done only at normal incidence.

as described here, a second series of experiments were conducted later on where the targets were shot at obliquity and the basic formula was slightly modified and extended to include those results:

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which is what I used here.

Secondly, yes, I am aware of the issues with thicker plates. AFAIK the best crystalline structure for steel armour is tempered martensite, which offers the best combination of hardness and toughness, but in sections above roughly 4.5in. thick the center cannot be cooled fast enough to achieve this.

Which is why thicker sections are processed to obtain bainitic microstructure. Its impact toughness is lower that that of tempered martensite, given equal hardness, which is why its limited in maximum hardness to about 260 BHN to still have acceptable toughness for armor applications. Therefore extrapolating the NPL formula above this thickness would not produce accurate results and would overestimate the optimal hardness level.

In the example earlier I used a 75mm thick plate, which is comfortably below this limit.

Edited May 15, 2025 by Peasant

[Thoddy]

Should have looked into SUPP 6-910

;&

 

[Peasant]

I would like to take this moment to reiterate that the NPL formula is only accurate for conditions that do not lead to a significant deformation and/or shatter of the attacking AP shell (see this data).

Source.

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Edited October 22, 2025 by Peasant

[Peasant]

While analyzing records of testing of 20mm scale model of the US 90mm T33 AP shot, it came to my mind to compare it's experimental performance with that predicted by the NPL equation. While doing so I found some interesting tidbits that I'd like to share now.

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The results at 0° and 20° obliquity are, overall, unremarkable, so I'd like to first examine those at 30°:

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Here AP shot with higher nose hardness(56 - 63 Rc) performs better, defeating the armour at lower velocities than the one with softer nose (49 Rc). Although once the armour becomes thick enough that all examined here types of shot shatter on impact, the difference in performance becomes much smaller.

NPL provides good estimate here, for conditions where the attacking shot remains mostly un-deformed/intact. The unusually low experimental ballistic limit against a 3/4in. plate (1800fps vs expected ~1921fps) is explained by the fact that the Protection criteria used in this series of experiments does not require the whole projectile to pass through the plate, only to make a hole large enough to push some fragments through, while the NPL model was calibrated on the british OB's W/R limit criterion.

Now, lets compare this behaviors to the one at 45°:

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Already against the 3/8in. plate we see that the negative effect of soft nose(49 Rc) is greatly diminished, closing the gap in velocities between the hard and the soft-nosed shot.

Against the 5/8in. plate the situation is reversed: unlike what we saw for 30° obliquity, here the immediate breaking off of the soft projectile's nose upon impact actually helps it defeat the armour by inhibiting its tendency of ricochet, while the more robust projectiles require higher striking velocities to defeat the armour. Clearly the british AP projectiles are more akin to the latter, as the NPL predictions show a higher ballistic limit here, in line of those obtain with the 56-63 Rc AP shot.

Finally, lets note that against the 3/4in. plate here, the ballistic limit of a hard (63 Rc) projectile is substantially unchanged from the one obtained against a significantly thinner(5/8in.) plate (both ~2400fps). This is because this projectile has finally also reached its integrity limit, as the soft-er projectile did previously, and is promptly breaking on impact, increasing it's efficiency.

 

The behavior of full caliber AP against RHA at obliquities around 45° is much more complex that those towards the either end of the spectrum, and successfully predicting their behavior under these circumstances requires us to be mindful of how things that we take for granted to be constant for a given obliquity, are in fact changing here depending on variables.

[Peasant]

Perhaps you've seen US ballisticians reporting exact measurements of a specific projectile's diameter before and after (if it remains in one piece) it perforated the armour.

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This is not a mere academic curiosity as this information can be used to account for the effect of increase in projectile's nominal diameter during perforation of the armour that forces it to make a larger hole than it normally would, thus reducing it's measured residual velocity and skewing the calculation of exact Ballistic Limit of the armour plate.

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Here I've used NPL formula to estimate this effect for a 1,33 calibers thick, 280 BHN RHA plate at 0° inclination.

Edited November 7, 2025 by Peasant

[Thoddy]

limit energy for M79

[Peasant]

Comparing the graph from here with NPL equation predictions:

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As you can see the match is excellent, except for the 20mm projectile. It has an unusual rounded conical nose which might be the cause of this discrepancy.

20mm M75 AP.

[Peasant]

The exact shape of the curve "Critical Velocity" vs Hardness of the plate baked into the NPL formula is not the only correct one. A more richly alloyed steel and/or with less concentration of embrittling impurities, mainly Sulfur and Phosphorus, can be tempered to a higher hardness level without negative effects on its ballistic performance. But this increase in ballistic resistance does not apply to normal/low hardness levels, where it will exhibit almost identical performance of a less performing chemical mix.

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Edited November 28, 2025 by Peasant

[Peasant]

Be advised: the formula reported in this source differs from one given in other sources in that the central constant has a value of 929 instead of 916. From my experience the lower value here fits better the experimental results for normal impact. Cant say whether the opposite is true at this point.

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Edited December 1, 2025 by Peasant

[Thoddy]

 

DEFE 15-452 A Fundamental Investigation into the Optimum Hardness for a Capped Armour Piercing Shot

https://drive.google.com/file/d/0B12aaMD-x_k_eDBMS1VFWTBWRGs/view?usp=sharing&resourcekey=0-J-gT2aXt_me7Kwio92kC4w

 

Capped shots werent used in this investigatio

Edited 13 hours ago by Thoddy