Update (5/14): This post was based on assuming yield strength values of 50 ksi and maximum bending stress of 24 ksi, based on the values for structural steel. It’s come to my attention that the values for rail steel may be higher, at least for modern steel. One company claims a yield strength of 67 ksi for their rail steel, and claims that 65 ksi is standard. I have not found data on maximum bending stress for rail steel, but if it is the same fraction of the yield strength that it is for structural steel, and if the added yield strength is still present at high temperatures (which is unclear – comments from anyone with more knowledge on this point would be welcome), the values here will need suitable revision. The post below, however, has been left unaltered; the (perhaps) necessary modifications are left as an (easy) exercise for the reader. (Note also that this issue does not apply to the deflection based analysis, as young’s modulus is not significantly affected by the type and treatment of the steel.)
In my previous discussion of the bending of rails at the Reinhardt camps, I focused entirely on deflection as a mode of failure of the rails. There are three main things that could cause a beam can fail: deflection, yield strength, and shear. (There is also buckling, which I will say something about later; it will make things even worse for the exterminationists.) This post will look at the Reinhardt cremation facilities from the standpoint of yield strength.
I should add that I have thus far considered and am considering only total, catastrophic failure. Even if the facilities were designed to avoid this, there would be additional problems with accumulated bending over several cremations, requiring frequent replacement of the rains, which is not part of the story (a late testimony from Blatt on Sobibor and a statement by Stangl on the failure of preliminary facilities with trolley rails notwithstanding).
The yield strength of steel varies with the grade of the steel, but many modern steels have yield strength 50 ksi (thousand pounds per square inch) in tension, while for older steels a more typical number was 36 ksi. However, we are not interested in pure tension but resistance to bending. The maximum bending stress for modern steels might typically be 24 ksi, and for older steels 18 ksi. This data book from Carnegie steel gives a value for maximum bending stress of 18 ksi.
Now, bending stress is not just load divided by area. Rather,
bending stress = My/I, where M = bending moment, y = distance from center of beam to top of beam, I = moment of inertia
(This is explained here: AECT360-Lecture 8)
How will this work out for the Treblinka cremation facilities? That depends on how we describe them of course. Our previous deflection based analysis clearly showed that they could not have worked as generally described. If they were to work, the spans would need to be shorter. For simplicity’s sake we’ll assume a simple span. Multiple spans will be a little different – for instance, the maximum bending moment of a quadruple span is only 85.7% of that of a single span – but that’s easy enough to work into the analysis later.
(Our decision to use a single span is also supported by Samuel Rajzman’s previously cited statement that the rails for the cremation facility were carried by 6 to 8 Jews, who were severely weakened by the camp regime. Given that rails of the strength we’re assuming will weigh around 60 kg per meter, it’s obvious that these rails could not have been anywhere near 30 meters long.)
A reminder on the effect of temperature:
We’re interested in a single span, and the bending stress at the center point. The formula for the bending moment of a single span with uniform load is wL^2/8, where L = length, w = load per unit of length. Just to annoy everyone, I’ll use imperial units. I’ll assume rail height 6 inches (as rails are asymmetrical this is really more complicated, but I’ll push ahead anyway, since the asymmetry is going to make them worse beams, not better). I will take the moment of inertia of rails to be 70 in^4
The Treblinka cremation facility is generally described as (somewhat less than) 100 feet long and capable of carrying 2000-3000 bodies at once. We’ll assume 2000 bodies, and use the rather low figure of 100 pounds per body. Thus,
load = 20 bodies per foot = 2000 pounds per foot
Now for some sample calculations.
span length = 20 feet
moment at center = (166 2/3)(240^2)/8 = 1200 kip-in
if number of rails = 4, then moment = 300 kip-in for each rail
so bending stress = 300*3/70 = 12.86 ksi
The allowable bending stress for steel beams varies somewhat, but for rails from the second world war it can probably be taken to be 18 ksi – today it might be 24 ksi. So the rails are within their yield strength limit – if they are not heated. But at 600 C, the minimum temperature for cremation, yield strength is reduced by ~60%, and the rails would fail due to yield strength.
if number of rails = 6, then bending stress = 8.57 ksi, and even if we use 24 ksi as our max bending stress, the rails would still exceed their yield strength by 650 C.
Thus with 20 foot ~= 6 meter spans, the Treblinka cremation facilities are totally unworkable due to the yield strength of steel, even if the temperature is kept remarkably low (which would be impossible due to the design of the facilities, with all the fuel below the rails and all the bodies above it, so that the rails have to be subject to the most intense fire if the cremation is to work.
Could the spans be even shorter?
if span length = 13 feet (less than 4 meters)
number of rails = 4
moment = 126.75 kip-in
bending stress = 5.43 ksi
so in this case if we take max bending stress = 24 ksi, the yield strength would not be exceeded until it had been reduced by heat to 22.6% of its room temperature value, which will happen at around 700 C. This temperature will be reached for certain.
if span length = 13 feet (less than 4 meters)
number of rails = 6
moment = 84.5 kip-in
bending stress = 3.62 ksi
So again taking max bending stress = 24 ksi, the rails would exceed their yield strength at around 15.1% of their room temperature strength, which will happen at around 750 C.
if span length = 10 feet
number of rails = 4
moment = 75 kip-in
bending stress = 3.21 ksi
This is 13.4% of 24 ksi; steel’s yield strength would drop to this at ~750-800 C
number of rails = 6, bending stress = 2.14 ksi, which is 8.9% of 24 ksi; steel’s yield strength would drop to this at ~850 C
All of these temperatures will be reached, as was seen in section 6 here.
Using multiple spans rather than single spans will improve things a little – as previously mentioned, the maximum bending moment in a quadruple span is only 85.7% of what it is in a single span – but nowhere near enough to save the story.
Using less accommodating assumptions
One very generous assumption was the moment of inertia of 70 in^4 and height of 6 inches. Rail with that high a moment of inertia will likely have a somewhat greater height, but more importantly, many witnesses describe the use of worn rail. This book gives 1440 cm^4 = 34.6 in^4 as the moment of inertia of worn rail. This is less than half what we assumed. The head height loss of worn rail can be estimated at 0.5 inches, so even if we assume rail height for new rail of 6 inches, hence height of worn rail of 5.5 inches, the bending stress will be increased by a factor of 1.855. Hence where with a 13 foot span and 6 rails we found that the yield strength would be exceeded at 15.1% of room temperature strength, at ~750 C, when assuming worn rail we find that this occurs at 28% of their room temperature strength, which occurs between 650 and 700 C.
We could have used 18 ksi rather than 24 ksi for our maximum bending stress.
We’ve used 4 rails (following Wiernik and Leleko) and 6 rails (following Rajchman). One could also do the calculation for 2 rails, following Rosenberg; obviously this would make for a weaker structure.
Our load of 20 bodies per foot means less than the 2000-3000 bodies assumed to have been piled on 25-30 meter long rails, so it’s on the low end. Using Rajchman’s story about 2500 (he claims they counted them, so it should be accurate) on a 30 meter long grid we get 25.4 bodies per foot.
Our figure of 100 pounds per corpse is rather low. Since there are supposed to have been loads of fresh corpses as well as loads of decomposed corpses, at least some of the time the load per corpse would have been considerably greater than 100 pounds (especially with all those fat women who kindled the fire).