This was something I posted on Battlefront.com sometime ago. while not directly related to the topic at hand, it is related to Ha$h's comments above, and why the simple cosine rule is not a very good way to estimate armors "effective" thickness....Of course effective plate thickness is not 140mm -- 140mm is the LOS thickness of the plate (Line of Sight Thickness -- simple cosine relationship). Effective thickness is much more complex than a simple cosine relationship and is a function of t/d. It can also be a function of projectile bending strength, projectile nose shape, plate hardness, plate quality...blah blah blah.
Projectile effectiveness within the game is very often driven (as it should be) by side angle or aspect of the target relative to the firer. So unless the shot is straight on, invariably obliquity will be compounded as a function of side angle. When you are talking about a base plate slope of 55-degrees, effective thickness will increase rather dramatically as the true attack angle is compounded as a result of vehicle cant, side angle, etc.. Moreover the difference between plate effectiveness at say 60-degrees is rather dramatic relative to say 55-degrees.
Invariably people keenly interested in this sort of thing want to discuss firing trials and how these should be interpreted -- such as those carried out at Kubinka or the like. But this presumably has been kicked about on other threads on this forum.
The “cosine rule” – while applicable under certain conditions of plate penetration – is really only valid under limited circumstances.
What’s the cosine rule? In the way it is typically put forth – ala this particular thread – is that plate ballistic resistance to armor piercing projectiles that rely predominately upon their kinetic energy to perforate armor is somehow directly proportional to the line of sight thickness of the armor plate. This isn't really true, at least under most conditions.
What is a plate’s Line Of Sight Thickness (LOSt)? Let’s say a plate has no inclination – it doesn’t slope. It’s actual thickness is the same as its LOSt thickness. If the plate is inclined at say 30-degrees, than it’s line of sight thickness (LOSt) is going to be greater than its actual thickness. The LOSt thickness is equal to the actual plate thickness divided by the cosine of the angle at which the plate is inclined. For example:
If a plates actual thickness is say 77.9mm and it is inclined at 30-degrees, than the LOSt is simply:
LOSt = 77.9mm/COS(30-deg)
Cosine(30-deg) = 0.86604
Therefore LOSt = 77.9 / 0.866 = 90mm
If a plates actual thickness is say 51.6mm and it is inclined at 55-degrees, than the LOSt is simply:
LOSt = 51.6mm/COS(55-deg)
Cosine(55-deg) = 0.57363
Therefore LOSt = 51.6 / 0.57363 = 90mm
IMAGE-1 Actual Plate Thickness vs. LOSt
As I was saying, folks often want to try and explain the superiority of an inclined plates ballistic protection capability by the plates line of sight thickness. The plate is thicker along the diagonal. Therefore it resists more. But there is more to inclined plate ballistic limit than simply the apparent thickness along the diagonal.
Now let’s consider how much umph it takes for a projectile to penetrate a given plate of roll hardened armor. The following example is for circa-WWII, US-Army 90mm M82 Armor Piercing capped projectile. This is a pretty common shell that was employed by the T-26 Pershing Tank and the M-36 Jackson Tank Destroyer.
The following figure is how much impact velocity the projectile requires to completely pass through a armor plate of various thicknesses and at various inclinations.
IMAGE-2: 90mm M82 APC Limit velocity vs. RHA as a function of t/d
How to read the above graph: The X-Axis or horizontal axis is presented in terms of t/d. t/d is simply a common means of presenting penetration data in terms of the projectile diameter or projectile caliber, divided by the thickness of the plate being attacked. For example:
At t/d = 1 the thickness of the plate is simply:
Projectile Diameter, d = 90mm
Plate Thickness, t = 90mm x 1 = 90mm.
The plate thickness at t/d = 1 is therefore 90mm.
At t/d = 1.5 the plate thickness is simply 1.5 x 90mm = 135mm
At t/d = 0.5 the plate thickness is simply 0.5 x 90mm = 45mm
As I am talking about the validity (or lack of validity) of the often quoted cosine-rule, the t/d values for the penetration data represents the line of sight thickness of the plate rather than the actual plate thickness. For example:
For 30-degrees and t/d=1 the LOSt – like that of the plate with zero inclination -- is also 90mm. However the actual plate thickness is only:
90mm x cosine(30) = 77.9mm.
The vertical axis of the graph is the amount of velocity the projectile requires at impact to completely pass through the plate. For example
At t/d = 1 and inclination = 0-degrees, the impact velocity required for the projectile to pass through the plate is about 1900-feet per second.
At t/d = 1 and inclination = 30-degrees, the impact velocity required for the projectile to pass through the plate is also about 1900-feet per second.
At t/d = 1 and inclination = 55-degrees, the impact velocity required for the projectile to pass through the plate is a smidge over 2200-feet per second.
Why the cosine-rule is not always wrong, but why it is not very accurate the vast majority of the time.
The following figure (Image-3) compares the ratios of velocity required for the 90mm M82 APC projectile to perforate a plate inclined at 0-degrees to that of a plate with the same line of sight thickness (LOSt) when the plate is inclined. So LOSt is being held constant between the zero degrees inclined plate and the plate inclined at either 30-degrees or 55-degrees. When this ratio of limit velocities is equal to 1 for a specific t/d value, than it can be said that the cosine rule accurately predicts the level of ballistic protection the plate provides against 90mm M82 APC. When the ratio is less than 1, or more than 1, than it can be said that the cosine rule does not accurately predict the level of protection provided by the plate at the given inclination.
As the figure shows, except for a couple of very discreate points, the cosine rule doesn’t accurately portray the level of ballistic protection provided by an inclined plate in the majority of situations considered. These discreate points are indicated by the two arrows on the graph. For all the t/d values considered for the plate inclined at 55-degrees, the plate is providing much more resistance than the simple cosine rule would imply. For 55-degrees inclination, you’re always better off with the inclined plate.
For the 30-degree inclination, when the LOSt t/d ratio is near 1, the cosine rule is pretty accurate. But for t/d less than 1, and 30-degrees inclination, you’re better off going with vertical armor. For t/d greater than 1, and 30-degrees inclination, you’re better off going with the inclined plate.
The cosine rule can be used as a sort of ballpark figure for plate inclinations of about 30-degrees or less (depending upon the projectile type). But for greater inclination – such as 55-degrees – it can be said that the plate is providing a much greater level of ballistic protection than the simple cosine rule would predict.
IMAGE-3: Ratio of Limit Velocity Required to Perforate LOSt at Inclination vs. Same Thickness of Plate at Zero Degrees Inclination. 90mm M82 APC vs. RHA.
And finally – just for grins – below is a sequence of images of a projectile perforating an inclined plate. As you can see the projectile does not follow a straight line path through the plate. The projectile is subjected to several direction changes as it passes through the plate. Each direction change is resulting in rather large amount of stress developing within the projectile. But as you can see, the projectile does not follow the straight LOSt path through the plate.
IMAGE-4: Projectile passage through inclined plate.