The Chieftain's Hatch: Turret Maths

In last week’s Hatch, the report governing the likelihood of encounter of a tank-penetrating gun at any particular range was covered. It’s a two-part report, though, and the second part of it covers the likelihood that a tank will be struck from any particular aspect. i.e., is the tank even facing the gun which hit it?

So I turn the keyboard back over to the Ballistics Research Lab and whoever was doing the typing in late 1951.

 Angular distribution.

The percentages of hits* on the various surface of knocked-out tanks appear in a British report, “A survey of Tank Casualties”. Extracts from this report giving percentages of hits on the front and rear of the hull and turret appear in Table II. The hull and turret are treated as separate units. Thus the percentages of hits on the front, sides and rear of the hull add up to 100%. The same holds for the turret. This break-down must be made since the hull and turret have two separate hit distributions. This is due to the fact that the turret is not always facing forward when hit.

Table II: Sites of hits on Allied tanks knocked out by A.P. shot in NW Europe during WWII

Source  Type of Tank    # of tanks            Percent of Hull Hits                         Percent of Turret hits

                                                               Hull front             Hull Rear              Turret Front       Turret Rear

1st Army    Sherman              239         43%                        5%                          61.8%                    2.7%

                       Stuart                  31         52%                        6%                          68.7%                    12.5%

British Units        All                135         49%                        2%                         31.1%                    4.9%

Normandy (Brit) All                227         30%                        0%                         37.3%                    12.4%

F.V.D.D. Files     All                  88          19%                        15%                      35.3%                    4.6%

Totals                                       720         40.3%                    4.8%                      43.5%                    7.8%


*Although ranges were given for tank casualties, angular distributions of hits were given, without reference to whether or not a hit produced a casualty.[Chieftain’s note: In other words, it would be possible for multiple hits to be accounted for on a single tank, some of which may have failed to penetrate, or some tanks may have suffered multiple penetrations before the opposition stopped shooting at them.]

The function chosen to express the angular frequency distribution of hits on the hull is:

Where θ = angle of attack in radians measured from the front of the hull.

 [Chieftain: Bad news, guys. More maths follows]

 The corresponding function for the turret is


Where θ = angle of attack in radians measured from the front of the turret.

 These two functions can be written in the common form:



 a= constant = 1 when considering hits on the hull, ¾ when considering hits on the turret.

 [End Extract]. I’m sure this means something to someone. There followed about 8 more pages of data and formulae, but the bottom line from it all was that about half of the hits on the hull came from the sector between the angles of 47° on each side of the front. The corresponding sector for the turret is between plus and minus 55°. See Figure 12 for a graphical depiction.

What is particularly interesting, and perhaps may seem counter-intuitive, is that it appears that turrets were more likely to be hit in the rear than hulls were. Something must have been attracting their attention, and there were overlapping sectors of fire involved. At least, that's how I would explain it.

In any case, the paper then attempted to make sense of the data. The “So what”, as it is perhaps known:

Range distribution and Terrain:

In the preceding sections, empirical equations were fitted to the range distribution of casualties and the angle of distribution of hits by German anti-tank guns and tanks on US and British tanks in NorthWest Europe. It was noted that the distributions of casualties in range for the three armies were about the same, although they represented operations in slightly different territory. This suggests that the range distribution of casualties is determined by some factor that influences all of the engagements in the same way. The following observations and deductions are offered as an explanation. Operational data shows the following things:

1. Four out of five engagements between single tanks were won by the tank that fired first.

2. The “Survey of Tank Casualties” showed that half of all casualties were caused by a single hit, and the average number of hit per casualty was less than two.

3. German weapons could perforate Allied armour in most cases out to 2,000 yards, whereas Allied guns could perforate German armor only out to 800 yards.

From these facts the following deductions can be made:

1. Few rounds were fired by a tank in an engagement on the average. If many rounds were fired by each of two combatants on the average, the importance of firing the first round would not be as high as indicated under 1 above.

2. If few rounds were fired to determine an engagement the single shot probability of hitting and killing must have been high. This corresponds to short ranges.

3. As noted earlier, the average range at which casualties were obtained was also short, and this is consistent with the above prediction that the battles took place at short ranges, that range change during a battle was small, and that the distribution of battle ranges and distribution of casualties should be about the same.

4. Because the Germans outgunned the Allied armor they would have preferred long battle ranges at which their armor would have protected them from Allied fire while their guns would still perforate Allied armor. Since such long battle ranges were not in fact observed, it is deduced that the Germans were prevented from engaging in long range combat either by Allied maneuver or by terrain. It will be shown in subsequent paragraphs that the distribution of battle ranges can be attributed to characteristics of the terrain.

[Chieftain’s note: This then diverged into seven pages of mathematical formulae showing exactly what they said it would. I understood so little, I didn’t spend time scanning them. Sorry. It then moved to Turret Theory.

The angular distribution of attack for the turret is different from that of the hull. Although the turret is faced with the same distribution of guns with respect to the forward direction of the hull, it is not always facing in that same direction. Thus the turret distribution depends on the direction in which the turret is facing when attacked.

A tank can assume three different attitudes towards an attacking gun. These attitudes describe the manner in which the turret is used.

A tank can assume a passive attitude. That is, the tank does not know the direction from which to expect an attack and it carries its gun in a forward position. Under these conditions, the angular distribution of attack is equal to the corresponding distribution for the hull.

A tank can be retaliating or engaging the attacking gun when hit. Only the front of the turret can be hit when this attitude is assumed.

Finally, the tank can assume an active attitude by engaging or expecting to engage a target other than the attacking gun. There is a distribution in the direction toward which the turret is facing when the tank is attacked.

The data from the Second British Army, previously quoted for the distribution of ranges also includes, where the information was available, the direction towards which the turret was facing and the conditions when the tank was knocked out. In only 3 of the 85 cases cited were the tanks actually engaging the gun that knocked them out. These 3.5% of retaliatory cases will be neglected. The direction towards which the turret was facing in the other 82 cases is shown in Fig 17. The ordinate represents the estimated turret azimuth in both the clock system of measuring angles as was used in the data, and in degrees. The abscissa gives the number of cases having this estimated azimuth. It can be interpreted as the number of cases within the 15° sector including that angle. The great number of cases at 0° shows that the turret was facing forward a good part of the time when the tank was hit. The 40 cases at 0° include all the passive as well as some of the active cases.

Figure 17 shows that, apart from what the situation is in this forward sector, the directions toward which the turret faced were spread around the tank with a tendency to cluster around the forward direction. It is reasonable to assume that this is a smooth distribution and that the number of active cases in the forward sector is roughly equal to the number in the adjoining sectors. Since there were 7 active cases at 11:00, 4 at 11:30, 3 at 12:30 and 4 at 1:00, it might be inferred that there were about 3-6 active cases at 12:00 and the remaining 34-37 passive.


[End extract]

Many more pages of incomprehensible mathematics followed.


So what does all this from this and last week’s article tell us?

Perhaps firstly, the importance of terrain. It would be highly instructive to see some sort of similar assessment of a similar time period on the Soviet front. Possibly as much by luck, the Western Allies went into battle with tanks quite well suited for the terrain in which it fought, where reaction time counted for more than long-range accuracy and hard survivability. Look out the nearest window, and try to figure out what’s within 350m, then imagine defenses set up in that band. Again, the average range of engagement is not the most likely range. Those occasional 2,000m shots tend to skew the average up a tad. That’s quite a wide angle to be covered, and it likely has a bearing on the second problem: Just how many times a tank was not shot at from the front.

Go back up to the table near the beginning of this article. More allied tanks were killed by going through the side or rear armor than front. An initial impression may well be “Well, duh. The front armour is thicker, it bounced more shots. If the front armour worked as advertised, the impacted tank wouldn’t be part of this survey of knocked-out tanks!” Perhaps, but there are several other points to note on this.

Firstly, the somewhat astonishing figure that only 3% of the killed vehicles of which their activities were known were engaging the weapon that killed them, which certainly brings into question the idea of a ‘tank duel’ as being common in Western Europe. There are a couple of different reasons which may be explain this.

Primarily, it is really hard to spot a threat, especially if you’re on the move. I would go as far as to suggest that the destruction of the tank was the first clue for a large portion of tankers that the enemy had them in their sights. Mechanised combat can be brutal and fast. The related figure was that about half the vehicles were in a ‘passive’ position, gun forwards.

Then, given the short nature of the typical engagement range, two enemies only 100m apart from each other, which is unusually close, will still provide quite an angle differential for the target vehicle. If the Allied tank is facing one enemy, the other enemy is already getting a pretty good look at the side of the allied tank. If in the ‘most likely’ engagement range of 330m, the Germans place a tank 300m from an anti-tank gun, which is not unreasonable, and the allied tank turns to face the German tank, that’s about 60 degree offset angle and a very definite opportunity for a shot into the side armor from the AT gun.

This is well and good for the Allies, but how does it affect the Germans? What conclusions can we draw from this data which is purely Allied-related? There are two very significant differences. Firstly, that the Allies were usually the ones advancing into the German engagement areas. Secondly, that some of the German vehicles were pretty tough nuts to crack from the front.

Perhaps the most important thing to consider is the likelihood that Allied tanks were not rolling around on their own. The fact that most of the killed allied tanks seem to have not known that their killers were out there leads to the conclusion that once battle was joined, far fewer allied tanks were lost. If it were a sustained fight, they would obviously be aware of the enemy. Again, several factors can account for this effect. By the way, for this whole section, I’m theorizing and extrapolating here, feel free to engage me on the forum and we’ll see if we can hammer something out.

Firstly, the fact that a late-war AT gun is a Die-in-Place piece of equipment. One is not going to be able to pick up and withdraw a PaK-40, or worse, a PaK-43 after firing a round or two. Shoot-and-scoot is not a possibility. Any engagement for a PaK is going to end one of two ways: Everything with sight to it is dead, or it goes down in a blaze of glory. Now, add to this the fact that any Allied tank is going to have friends, and those friends such as Shermans are going to have good broad fields of vision and fast target lay rates, and things don’t look good for the AT gun.

The above commentary on how only a few hundred meters separation can create a side shot also applies, of course, to the ambushing vehicles. A Jagdpanther facing a target vehicle and providing a very tough armoured target may well be providing a flank shot to that target’s platoon-mates... Who can then slew and deal with the target. This sort of effect is doubtless part of the reason that the US were so successful at Arracourt, where many German vehicles were presumably killed by side impact. It is not unreasonable to presume that in instances when the Germans were attacking, the ratio of hits on each aspect of hull and turret would be more or less similar to that of the Allied tanks above. That’s a lot of hits in the sides and rear, and the US had a lot of guns to make them.

In any case, if so many Allied tanks were destroyed without knowledge of being attacked, and so few were destroyed whilst dueling, the logical conclusion appear to me that the first kill on an allied tank in any particular engagement was also often the last kill on an allied tank in any particular engagement. What could result in this?

Again, there aren’t many ways for an engagement to end (All of the below can have Germans and Allies reversed):

First choice, the first volley by the Germans kills off all the Allied tanks and there is nothing left to either kill in a duel. Secondly, the first volley by the Germans kills off a number of Allied tanks, and their friends kill off the Germans in response. The next option is that the Germans or allies withdraw. In a few rare cases, an extended duel may occur. That’s about it.

The problem with the first circumstance is that, by and large and as mentioned above, the Allied tanks didn’t wander around on their own. After the first salvo, the remaining allied forces would have been well aware that there was something out there, and ‘passive’ attitudes would be over. This leads logically to the majority of incidents being either that the Allies destroy the Germans, drive them off, or fall back. Well, in the long term, the Allies didn’t do all that much falling back. Which of course leads us to destruction or withdrawal as the two most likely outcomes. For AT guns, withdrawal isn’t an option. I’d hate to be a PaK gunner. For vehicles, either is a possibility, but attrition from all those off-angle shots, or simply mobility kills, will eventually result in destroyed AFVs. And, besides, the infantry the tanks or TDs were supporting would not be happy, so they would likely withdraw as well. The Allied advance towards Germany would continue. And that was in circumstances where the Germans fired first. In the event that accompanying forces such as recon troops or infantry notified the Allies of German AT gun or AFV presence, the Allies could get that first shot off, or simply suppress the opposition by fires in order to maneuver.

Again, this is speculation on my part, but it is not unfounded. Basically, where I’m going with this is that an assessment of the data can lead to interesting lines of thought. One can imagine how, say, a British or French tank designer can use this information to design his tanks to best survive on a future battlefield on the same terrain, or perhaps a planner can use such data to better form his tactics. One can also use this data to do the reverse: Look at the designs of vehicles, and try to understand why the results were as they were. For example, why were Shermans considered to be more effective vehicles than Panthers, when normally the figures would say that they weren’t?

Anyway, Bob will bring you back to the forum thread. We’ll do a little less abstract work next time.

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