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In my NAVWEAPS Web Site "Nathan Okun Armor Data" pages, there is a document entitled MISCELLANEOUS ARMOR PENETRATION FORMULAE. The US THOMPSON "F" FORMULA with the 1931-derived -- by Dr. Thompson himself at the US Naval Proving Ground at Dahlgren, Virginia -- Empirical Formula for "F" Values (given in this document and plotted as a set of graphs in US Navy BuOrd Ordnance Sketch #78841) was developed from many tests against STS/Class "B" homogeneous armor of WWI vintage by the now-obsolete 8" Mark 11 Mod 1 soft-capped AP projectile (the first of the "Midvale Unbreakable" AP projectile designs that was introduced in 1911 and used until shortly after WWI, when uncapped and capped 8" "Special" Common projectiles replaced it). The Thompson F Formula had no scaling factor whatsoever (unlike most other formulae including the widely-used DeMarre Nickel-Steel Armor Penetration Formula), but the Empirical Formula for F Values was developed from tests up to 75 degrees obliquity, which was far, far higher than the DeMarre formula was good for (a mere 30 degrees obliquity was pushing the DeMarre formula's accuracy to its limit). Even though the basic F formula was very simple and totally kinetic-energy-driven, the ability to use the built-in "fudge factor" F to adjust the curves to match the actual test results allowed this formula to be empirically matched to all armors and all projectiles, with the brunt of the work in trying to explain why a certain value of F worked in any given case -- this allowed one to "focus" on the problem more concisely. It also could be used as a baseline onto which could be mapped actual test results as adjustments once that 1931 data set had been analyzed and the average standard formula for F derived. This "standard" F value for any given projectile, plate, and obliquity of impact was called the "Standard Navy Ballistic Limit" (NBL) and was adjusted to the actual test results by computing the "Percentage NBL" of that test or test result -- for example, if a plate impact at 2500 ft/sec gave a barely completely penetrating projectile at the given obliquity (the "true" NBL for that plate) and a computed F from the 1931 formula resulted in an NBL of, say, 2000 ft/sec when that F was plugged into the generic Thompson F formula, then the plate's true NBL was 100 x 2500/2000 = 125% of the Standard NBL. This does NOT mean that the plate is 25% better than other plates; it merely means that if the old 8" Mk 11 MOD 1 AP projectile had been used against a scaled up or down plate of the same type (assuming the shell being tested was not 8") and this good result still had occurred, then and only then would the plate be considered 25% better than the standard WWI-era STS/Class "B" homogeneous armor. Since we are not using that old projectile in most tests and the projectile size and armor thickness may be larger or smaller than used with a scaled 8" test projectile and we are using later armors and projectile designs (with hard caps, for example), the fact that the result was not exactly what that old projectile would have given (on the average) is not surprising. By carefully recording these % NBL results (offsets from the Standard F Formula baseline), one can track how armor penetration ability changes with scale using scale-model projectiles against thicker or thinner plates, how striking velocity can change results if projectile damage occurs on impact, how changing obliquity can change the plate resistance to a given projectile design, how changing nose shape or AP cap design causes changes to the plate resistance at a given obliquity, how changing plate metallurgical properties changes plate resistance when everything else is kept constant, and so on and on, each set of % NBL values being developed for a given characteristic of the plate and projectile and then used to calibrate other tests when more than one change was made simultaneously (as with testing a foreign projectile or plate with several simultaneous characteristics that are diiferent from the US standards, which was virtually certain to be true). This technique even allowed the % NBL system to be used with face-hardened Class "A" armor where a rather huge scale factor revealed itself (% NBL for a small cruiser AP shell might be in the circa 125% range against a caliber-thickness Class "A" plate while identical plates and projectiles scaled up to battleship shell size gave only about 90% under identical test conditions). This allowed scaling, as with this example, to be nailed down tight from the test results when one was careful as to picking tests with similar shell designs and plate characteristics of different total sizes. This is not so easily done with other formulae, which mix up various parameters in non-user-friendly ways. Once could systematically track all such parameters and work out equations to handle them, one by one, which is how I developed FACEHARD.

Even if the enemy armor was known precisely ("Agent X" got their entire test results library and specs for making and testing all of their armor, as well as a few projectiles and plates to calibrate this against your own weapons) and your own projectiles are spot-on from test to test (neglible quality problems), in real life things will still vary quite a bit from the nominal values you tested them with. For example, you fire a rather flat-trajectory salvo at an enemy ship and the shells, all fired at precisely the same elevation at the same time, spread out in range by say 200 yards short to 200 yards over. If the shell impact patteren was centered precisely on the target and it took 10 seconds for the shell that hit the target to get there, the shells that hit shortest went, on the average, 20 yards/second slower than the range-table figure, while the shells that went over the target furthest went 20 yards/second faster, again on the average. Assuming that the drag was fixed for the entire flight (good enough for this estimate, I would think), then the final impact velocity is actually twice this difference, so the range of impact velocities in the salvo was from 40 yards/second (= 120 ft/sec) low to 40 yards/second (120 ft/sec) high, an 80 yards/second (240 ft/sec) spread in striking velocities. If this was typical for the guns of that ship, then this kind of spread means that you have to know not only the nominal aim point of the spread center, but where in range in that spread the target actually was when it got hit to estimate the actual striking velocity deviation from the range table value for the given range and time of flight. On top of this is the pitch and roll angles of the target, its EXACT course relative to the line of fire (horizontal Target Angle), how fast the target ship and own ship was going (the former when the gun was fired, adding or subtracting a part of this speed from the muzzle velocity, similarly at the impact time against the target's motion), etc., etc. As you can see, things a kind of blurred and you really have to "box" in the results by changing the important things (angles, velocities, etc.) by reasonable amounts up and down (or right and left or in and out, as the case may be) about the expected tabled values, recalculating this again and again with various combinations of values and seeing if that changes the results much -- if not, then the result is "hard"; if so, then you have to "roll the dice" to see what was the chance of a given result actually happening. This is called the "Monte Carlo" approach and is used in actual weapon design testing today. Messy!!!