[Andrew Gill]

Hi Everyone

Tod's videos and the recent discussion stirred up about the tiny so-called assasin's crossbow got me doing a lot of overdue research into crossbows. Being a mechanical engineer but lacking anything more interesting than a homemade hickory flatbow to test, I wanted to try my hand at some numerical modelling (disclaimer, I specialise in fluid dynamics and structural analysis is very much a second string to my lyre, so to speak). Sean Manning pointed out a very interesting thesis on using FEA methods to model a bow, which I'm trying to emulate in my spare time. These results I will also put up here if there is interest, when I'm fairly certain of them.

However, while reading Payne-Gallwey's Book of the crossbow, I realised that it was worth taking a look at the bolt as well.

I wrote a simple computer program to numerically calculate the flight path of an arrow, javelin or crossbow bolt, given its mass, "muzzle velocity", firing angle, approximate frontal area and drag coefficient (more on these two later). I can get various bits of information out - the velocity of the arrow at an arbitrary point in its flight, how far it will travel before hitting the ground (assuming level ground!) and how much energy it loses due to air resistance.

There's some uncertainty about the exact drag coefficent of an arrow in the published literature. A commonly accepted value is a little over 2, but have seen values as high as 4 and as low as about 1.5. The ugly truth is that drag-coefficients aren't actually constant, so its naive to assume that there is a single value to rule them all.

I checked it by looking at olympic javelin data, and got a range of about 85 m for the official test conditions (the actual range from both javelin gun and selected athletes was about 83m iirc).
Then I discovered Tod's video with the longbow arrow speeds at ranges of 25, 50, 75 and 100m. He discovered that an arrow loses 15-20% of its speed after 100m of flight.
I had to make a couple more assumptions, such as the angle that the arrow began its flight at, but my results were again fairly close for 100m - I got a value of between 20-24% decrease in speed, which I found pretty encoraging, given the uncertainty of the drag coefficient of the arrow, and the guesses that I had to make. So I have some confidence that I've done this as correctly as such a simple model can be done.

Now, having probably put a lot of people to sleep, the (hopefully) more exciting bit.
P-G's famous shot across the Strait of Menai is claimed to have been 440 or 450 yards, which is about 420m (google confirms this, by the way.)
My calculations suggest that for a drag coefficient of around 2 in still air, he would have had to shoot his arrow at nearly 100m/s (!) at 45 degrees (he was aware that this would give best range, as he comments on it later). Its initial kinetic energy would have had to be about 425 J, and it would have lost roughly two-thirds of that to air resistance by the time it landed.
On the face of it, these numbers are so far above what Tod is reporting that they cast his account in serious doubt in my mind unless there were some other factors at play here, or somehow renaissance crossbow makers really could somehow overcome the inherent limitations on steel prods; these numbers look a lot like those given by Andreas Bichler for his giant 1200lb horn crossbow, at least in terms of energy. Unless there is something else going on here...

Remember that I mentioned that the drag coefficient of arrows was quite difficult to pin down?
Well, another part of the problem is simply that some arrows simply have more air-resistance than others.
I have a suspicion (no more than a suspicion, mind you!) that this is part of what bit Tod with the long headed (type 7?) bodkin arrow which lost speed faster than the shorter headed one despite its greater mass. Contrary to popular belief, a long spike is actually more poorly streamlined for subsonic flight than a shorter, more rounded one (the reverse is true for supersonic flight, which is why rifle bullets are often pointy and subsonic pistol bullets rounded). We don't know what sort of bolt P-G was using, beyond that it probably weighed around 30z, or (80g for us metric types, I believe.) My little program suggests that a fairly small difference in drag coefficient can easily make a difference of 50m or more to an arrow's range in still air. We also know from Tod's experiments that head shape can have a noticeable aerodynamic effect on the arrows (the giant swallowtail on the crossbow effectively ruined the arrow's stability, for instance). People in the Renaissance had to use arrowheads that were effective at piercing their targets. But PG was trying to win a friendly bet with some friends, and had no such restrictions on head design (and also did a lot of trial-and-error experiments with different bolt designs).

Which brings me to my second point.
PG completely fails to mention the weather conditions at the time of his shot, though he mentions that it was in autumn.
Well, according to modern online average weather data, it seems that there is often a wind blowing from the south or east during the autumn months and it can be blowing quite strongly - 14 miles/hour is not uncommon, and 20 mi/h is by no means unknown. Conveniently, this would be at P-G's back when he made his shot. I made a small modification to my program to take into account the effect of a head- or tail-wind, to get best-case and worst-case scenarios. Again, it can make a fairly big difference to range - again more than 50 meters, if I've not botched the programming. I've attached a graph showing several trajectories for likely best and worst cases for

I also investigated the effect it would have had if he were standing on the 6m high battery when he fired his shot, but this seems to make less than 10m difference to the range either way, so is probably not relevant.

So, if P-G (who seems to have had a friendly bet on the matter) chose a particularly aerodynamic arrow design(he could have determined this by trial-and-error experimentation, which he seems to have done a lot of), and made his shots with a decently strong tailwind, the whole thing starts looking slightly more plausible than before. Using high-drag arrows in still air seems to give a maximum range of about 270m, while super-streamlined arrows with a strong but not unreasonable tailwind will just about do 420m. Given the amount of educated guesswork involved here, this is not even remotely conclusive, but it is suggestive...
And, if not totally convincing, it at least seems a little more plausible to me than that either Tod's magnificent bows are terrible, or that PG, who was apparently a noted professional ballistics expert of his time, was lying through his teeth.

And perhaps it shows that we should be cautious of ascribing some reputed performance to the superiority of one particular part of these quite complex weapons, such as the prod.

I don't know if this is of interest to anyone besides me, but if it is, I'll post whatever I find when my current investigation of the crossbow lath is complete (which will possibly take a bit longer). If any of this does interest anyone, or someone thinks I've bungled my modelling attempt (which would be embarrassing, but is certainly not impossible dispite my checks), please let me know.

Andrew

Edit: for some reason I can't upload the graph.

[Pieter B.]

Thanks for the write up!

What would your model say if you kept the same drag coefficient but raised the bolt weight to around 120-140 grams?

[Andrew Gill]

Hi Pieter

Thanks! That's a good question; I actually tested that but forgot to include it in my (already long) writeup. Increasing mass while keeping the drag coefficient constant makes the arrow fly quite a bit further (at least 40m), if you fire both light and heavy bolts at the same speed.

That's the problem, though, because the bolt's initial speed is also controlled to some extent by its mass (heavier bolts will accelerate more slowly from the force of the string, and thus attain a lower speed) and to some extent by the dynamic properties of the bow limbs. Once I do a finite-element model of the lath's behaviour, I'll be able to predict that with some accuracy, but it's pretty difficult to do with any certainty using simpler methods.

These numbers are for 80m/s, by the way, which is the speed I determined that P-G would probably have needed to shoot his bolt at if my hypothetical tailwind was blowing.

Andrew

[Pieter B.]

[Duncan Hill]

Bravo Andrew on you program and calculations, and to you both for working on these problems - I'm very interested to learn more!

[Jean Henri Chandler]

[Andrew Gill]

Hi everyone

Pieter, you make a very good point about heavier bolts being more efficient. If the bolt mass is above some value, the acceleration of the bolt will become the limiting factor in the combined lath-string-and-bolt system - ie. the bow is no longer fighting with itself more than it is fighting with the bolt to try to get the latter to move faster - which is of course a desirable situation that just gets better with heavier bolts, as long as the bolt still leaves the bow at a useful speed. (Tod illustrated this in one experiment by firing a very light modern bolt form a 960 lb crossbow - the speed gain is negligible because the lath's and string's response becomes the limiting factor) It's probably also why Andreas Bicher often fires much heavier bolts (200g or more iirc?) from his giant composite crossbows. And of course once it leaves the crossbow, the bolt is less susceptible to aerodynamic drag, assuming this stays roughly equal, so you get less of a range drop-off there as well.
And the bolt is less likely to outpace the bowstring at any point (I'm not convinced this can actually occur but if it did, it would certainly be more likely with a lighter bolt). And I'm sure there are other factors that I've neglected here. So heavier is definitely better!

Jean, I was hoping you'd join in here.
I've watched Andreas Bicher's videos and they are incredibly impressive. I would go so far as to say that if P-G had used a large composite crossbow rather than the steel one he mentioned, I don't think many would doubt his claims, or at least not to such a great degree.
The issue for me (and part of the reason why I'm investigating) is the apparently huge discrepancy between his (P-G's) claims for a steel bow, and the work of Tod, the french chap (Lebel?) and also my own rusty knowledge of the mechanics of leaf springs and cantilever beams and also the properties of steel (which I've devoted a decent effort to understanding because it is integral to my interest in medieval and renaissance weapons and armour, as well as being of some importance in my profession). I'm not at all convinced by the argument that there were lost methods either metallurgical of making such vastly superior steel crossbow laths, but I would love to prove myself wrong (even aside from the fascinating insights we'd gain into medieval technology, the benefits to making modern leaf springs alone would be worth the effort). A good scientist should never be sure of what he knows (and engineers are, or should be, applied scientists, after all).

By the way, on the subject of metallurgy:
I've examined the published properties of steels from modern maraging steel, high carbon steels and mild steel, to Japanese tamahagane and wootz and bulat (there's an excellent paper in which the researchers compare historical samples with "synthetic wootz" samples that they made themselves - so apparently making it is possible!) In all cases the modulus of elasticity was within a fairly narrow band of between 180 and 220 Gpa - there was one single old wootz sample which reached 250 GPa, but that's only a 13% "improvement" on the common value of 220, and this was unusual even among the historical samples investigated. (in fact, I think a higher modulus would actually probably make worse bows, though I'm aware you weren't arguing for using wootz specifically). The yield strength does vary quite a bit between alloys, which is relevant since this represents the limit to which a given bow can be loaded before it snaps or takes a permanent set, but in all the measured data I've seen, the better modern alloys always come out on top. If I've missed something I'd love to know, but based on published experimental data including old metal samples, I just don't see the answer being in the metallurgy.

If I were to look for a way to win a bit of extra performance from a steel bow lath, I'd look at fine-tuning the shape. As an engineer, my reaction to how to design a good crossbow would be to treat it like any other cantilever beam, and make sure that the stresses are within it are roughly constant along its length and slightly below the yield stress by some safety factor when drawn. From my rusty knowledge of structural mechanics, this results in a linear taper. But this is considering a bow or lath as a static system, and it is not - it is decidedly dynamic when it is doing its job. Iif one designs for efficiency, with the different speeds at which different points along the bow move when it is released, it might be possible to tune the profile and taper slightly differently, while still keeping yield stresses in reign. This is where the FEM model will come in.
It's a pretty obvious point, so I'm probably not the first to think of it (I'd be surprised if modern-style crossbow designers didn't do this as a matter of course), but it will be instructive to me and hopefully help me bows better. How much can we gain in this way? I've no idea! I share Tod's suspicion that there aren't huge improvements to be made, but until I crank the numbers, its no more than a suspicion.

There's another interesting point that I discovered: when you use a linen bowstring on a wooden bow, you have a bowstring material that has a modulus of elasticity about five times higher than the bow wood. When you use it on a steel crossbow lath or prod, it is about five times lower than the bow steel (these numbers are very rough; I need to check them). Now this is not as exciting as it may at first sound because the elasticity or stiffness of the bow string under tension is also dependent on the sectional area of the string, not just the modulus of elasticity, and crossbow strings are much thicker proportionally than longbow strings, so I don't think that the elasticity of the bowstring is actually a major factor. But its another thing to check.

So, to finish another marathon post: If anyone has data on the cross-sectional measurements and taper at several points along the length of an extant steel crossbow lath, that would be a huge help.

Also, Jean, if you can pass on any data from the range of shots, mass of bolts, etc during the schuzenfests in the mid to late 1400s (when steel crossbows were probably most common?) I'd be very much obliged. More data would help a lot!

Andrew

edited for clarity and spelling errors

[Michael Zimmermann]

Dear Andrew,

since I am partially to blame for suggesting FEM modelling might be an interesting avenue to pursue here, let me offer a couple of suggestions as to people, who might be worthwhile contacting. Do keep in mind that crossbows are not at all part of my research, so...

There appears to be an equivalent to Andreas Bichler (who is Austrian?) in Germany, who has devoted himself to the historical (steel) crossbow since 1990; he is an engineer, like you, and renowned for his bowsmithing over here, I gather. His name is Jens Sensfelder and he built a crossbow for this exhibition in Berlin last year:

https://www.dhm.de/ausstellungen/archiv/2019/die-armbrust-schrecken-und-schoenheit/

He publishes a periodical, which occasionally has articles in English, called 'Jahrblatt der Interessengemeinschaft Historische Armbrust'.

Jens also supplies this workshop with bows:

https://www.armbrust-manufaktur.de/index.html

Their 'biggest' specimen is this guy:

https://www.armbrust-manufaktur.de/schwere_armbrust4.html

Sadly, no further data is supplied here, but might be available on request.

Best,

[Andrew Gill]

Hmm I said my structural mechanics were rusty, didn't I? I was right.
The taper on an end-loaded cantilever beam should not be linear (the internal shear-forces vary linearly, of course); the maximum stress is roughly proportional to one over the square of of the thickness of the beam in the bending direction, though they are linear to the width of the beam. So I think keep constant maximum stresses, you ideally want the thickness to vary in some way proprotionally to the square root of the distance from the bow tiller (assuming constant width in the other direction - varying the relationship between both tapers is one of the ways we can play with the bow's profile). I'll check this before going on.
Tod has mentioned that steel bows should taper down to roughly 60% of the center dimensions in both thickness and width. I get the impression that is based on both measurement of historical examples and some sort of structural calculation (I'd love to know more, if he reads this).

Just a small but important correction. Duh!

Edit: this doesn't in any way affect the calculations I did on the projectile flight, luckily!

[Andrew Gill]

Thanks Michael, that's a very good starting point!

Andrew

[Jean Henri Chandler]

A couple of comments

[Andrew Gill]

Hi Jean

The principle paper from which I got my info on wootz was "Nanoindentation behavior and microstructure of several wootz-type steels", by Paetzke et al of the technical university of Dresden. They used electron microscopes and some other rather specialised equipment to study five different samples, including an old indian one. It seemed pretty rigorous and was nicely written (I could understand most of it based on my engineering metallurgy courses). It is on researchgate, but I'm not sure if I'm allowed to post the link here - you can find it easily enough via google or I can PM the link. If you have a paper which refutes or qualifies their findings please send the details so I can read it. In fact, send me whatever literature you have on wootz. I disagree with some of your conclusions about it, but its a fascinating material whichever way you look at it.

As for whether I'm working backwards on the crossbow question: Yes I am, consciously. I want to see whether the claims of historical sources are plausible within the laws of physics. Can I independently design a steel bow lath that "on paper" matches these claims? But I also want to work forwards, by examining the historical data and artifacts to see what solution they arrived at. If the two converge, well, then we know! Otherwise I'll at least have a better inderstanding of the physics of bows.

In the previous thread, you spoke about faith. For me personally, faith is anaethma to research. I need to question everything, no matter how dearly I hold the belief, to arrive at the truth . I want to examine the historical records and modern experiments, but I will not decide beforehand which (if either) is correct. And I'm not even convinced that this is the right question to ask, honestly. This is my personal view, it shapes my approach to research, but I can't expect others to follow it.

So if you have data on renaissance crossbow performance, please share it with me. If we discover an ancient way to design excellent springs for crossbow laths, it will be incredibly exciting for everyone here. If we don't, we'll still learn a huge amount about the technology of this fascinating period, as well as how to at least make better modern steel crossbows. Which is also exciting.

Andrew
PS. The synthetic wootz was not pattern-welded. The account of its preparation is pretty fascinating.

[Jean Henri Chandler]

The only thing I question from that article (which i have not yet read though I might if I have time) is the notion that they have in fact created true synthetic wootz. I do know that there have been many attempts to create modern or synthetic wootz, (two examples are mentioned in the article I linked about Lawrence Livermore) and there have been some interesting results producing metals with unusual properties, but so far as I am aware nobody has yet succeeded in emulating the original, or even come very close to it.

As for faith - there is more than one type. Basically there is true, inherent spiritual faith, and there is something more like 'earned' or acqujred faith. I have an 'earned' faith that when I turn the knob on my sink water will come out of the tap, because it almost always has in the past. I have an earned faith that if I drink a whole liter of whiskey tonight, I'm going to end up with a headache tomorrow. Because that is what usually happens (though I can get away with a bit more of it if I stick to the single-malt )

I have an acquired faith in the late medieval sources in aggregate. Time and again these people have proven smarter than we give them credit for. Time and again, with regard to weapons specifically, I have seen confident so called experts saying things like 'medieval swords were like (barely) sharpened crowbars' or that they 'weighed 10 or 15 lbs' or that 'medieval armor was so heavy that if the rider fell off of his horse, or fell down in the mud, he couldn't get back up again' and so on. From History channel or BBC / PBS documentaries to University courses. Then we (collectively, as a community in places like this and within the HEMA revival and so on) put it to the test, Ewart Oakeshotts work bubbled up. Alan Williams work became famous. Now we know better.

I should note though, it took more than 3 or 4 people making replicas, and a lot of research, and a lot of very close examination of antiques.

I personally also do have an acquired faith in empirical research and modern engineering. I do fly in commercial aircraft for example. But I know from my own professional life that as powerful as our modern approach is, it is only as good as the data that goes into it. Working backward from replicas without looking closely at antiques wouldn't qualify to me.

I have posted some raw data from schützenfest and some other records to this forum pretty recently, but as I said in the other thread, I'm not in a position right now where I can spend the (hours and hours of) time wading into detailed arguments. When I do have time I will write a paper on that specific subject (schützenfest records). The problem with posting some of that raw data here is that there is enough wiggle room to debate it, and inevitably that will just go on and on. Distances for example are recorded in units of measure like ells and cubits which could and did vary depending on the specific place. Each town and principality had their own units of measure then and they changed over time. Fortunately on the shooting contest invitations we have the length of the unit of measure they refer to physically drawn on the sheet, but we don't have that kind of verification in all the other records.

That said, here is some data to consider.

One invitation from Strasbourg to Rottweill in 1494 said that targets at 90 ells distance had to be hit 4 times out of six shots to qualify to enter the tournament. That was a 20 cm target. By our estimate it worked out to 102 meters distance. Other invitations from Strasbourg earlier in the 15th Century worked out to targets at 70-90 meters.

Another invitation (1549 from Landshut) said that the target (shown as a circle on the invitation as being about 18 cm) had to be hit 8 times out of 12 shots at a distance of 96 Landshuter cubits, which works out to 43 meters. Once qualified the shooters would have 24 shots at the target. The same invitation specified 260 cubits distance for the arquebus target.

That invitation can be viewed here https://digital.staatsbibliothek-berlin.de/werkansicht/?PPN=PPN1022206435

Keep in mind these are small targets not human or horse-sized.

I have 6 other ones that we haven't fully translated yet. But there are dozens out there.

In a letter in 1408 the Teutonic knights Komptur of Elbing mentioned a 'safe distance' of 300 ells from a defended castle against crossbows, with different ranges listed for culverin and hand-büschen.

In an earlier manuscript (Chronicon terrae Prussiae) from 1326 they listed a safe distance of 250 ells. So apparently the weapons got a bit more dangerous. Or possibly the ell meant a different unit of measure then. The Chronicler doesn't specify.

In a document from 1427 (just before attack by Hussite armies), the city of Wroclaw specified that certain defensive outworks be built at the limit of crossbow range from the wall, which they listed as 600 cubits, which works out (possibly) to about 270 meters.

I think Christine de Pisan actually may have something on this too in her war book but I can't remember, I'll need to scan that book again.

[Sean Manning]

There have been Europeans and settlers making horn bows since at least 1960 (Edward McEwan in London said "25 years" in 1983). Not as many work on the short-powerstroke European horn crossbows, which have an odd method of construction, but the basic idea of horn bows is not a lost art. But yes, the very high draw-weight, particular construction, and short draw might give different performance than other kinds of horn bows!

One of the Paston letters describes a group of 3 or 4 steel crossbows which need to be repaired because they are shedding splinters of steel and won't shoot well any more. The citation is in Hardy and Strickland's Great Warbow.

Edit: John Paston III to John Paston II (year 1469), 05 in Paston letters and papers of the fifteenth century, Part I p. 543 in Michigan and other places online

[Jean Henri Chandler]

The crossbows are a different thing, they use different materials. Nobody makes a 1200 lb draw recurve self bow either.

[Sean Manning]

[Jonathan Dean]

[Andrew Gill]

Thanks Jean and Sean. Some good information!

Jean, I take your point about faith - I'd perhaps use the word "trust" instead, but I understand what you are saying.
For what its worth, I don't believe that people of the renaissance were any less reliable observers or recorders in this regard than modern people - I mistrust both equally. No measurement is ever perfectly accurate, and no observer is perfectly reliable. We always need to cross-check from different sources and different approaches. I tell my students that if they are to be good researchers, they should question and check everything, including what I tell them.

The range data which you did post now is already rather useful. If I understand you correctly, we can say with reasonable certainty that between 270 and 300m was considered outside crossbow range in the early 1400s? I suppose the question then becomes what they considered safe from crossbow bolts, ie. no crossbow bolts land beyond this point, or they might still reach here, but have lost so much energy that they're not much of a threat. Do any sources that you know of comment on this? I can now check various scenarios and see how much energy the crossbow bolts retain at these distances - maybe effective range and maximum range are not very different, maybe they are. This won't give us certainty, of course, but hopefully it gives some more insight. Naively, though, it actually seems pretty close to modern range measurements? Tod shot about 250 or 260m, Lebel estimated about 270m, and Andreas Lebel got about that much with his 1200lb horn bow, though with a much heavier bolt, I admit (which does make a big difference). Or am I missing something?

The accuracy data is quite impressive and pretty conclusive - but I didn't really have any quibble on that score for military or hunting crossbows at least.

Sean, thanks for the link to the Paston letters. Those look very interesting indeed! I had a look at your website and found lots of great stuff. The information on the italian crossbow-guild rules was fascinating. It gives a little window into the sort of maintenance that horn crossbows required, and the sort of sharp practise that was presumably common enough to need to banned. It sounds like they were particularly fiddly and temperamental pieces of equipment, even by the standards of the day. It was also interesting that some kinds of damage seem to have been considered irreparable. For these reasons alone, I'd probably have considered getting a steel bow if I could (or at least I thought so until you mentioned them shedding steel splinters -eek!).

Andrew

[Jean Henri Chandler]

[Bartek Strojek]

From what I've seen, Bichler famous 190+J, 69.8 m/s shots were made with pretty hefty, 81 g bolts.

So, quite possibly, he could have came close, or surpass it with lighter ones, only he (?) knows obviously.

Adam Karpowicz needed arrows below 40g to surpass 80 m/s with very powerful and fast Turkish bows, while he was going at almost 55 m/s with 100g arrow.

https://www.atarn.org/islamic/akarpowicz/turkish_bow_tests.htm

So doing very rough math, if reducing arrow weigh to just ~30% (34g) drops the efficiency by 15%, then reducing bolt weight to this ~34 g would drop efficiency by about 12.5%.

We would then still be looking at about 170J with 85 m/s velocity. Lots of assumptions though, particularly assuming that this crossbow would be as phenomenally efficient with lighter arrows as this turkish bow.

Given the weight and bulk of 1200 pounds prod it may not be the case, hard to say.

As mentioned, Tod's steel bows so far seem to have very limited efficiency with smaller bolts, barely any difference in velocity between 80 and 125 gram projectile.