Break testing of the Alpine Butterfly Knot

Practical Knots
1

I recently posted a method of tying the Alpine Butterfly Knot, which lead to some interesting discussion of the breaking strength of the ABK. Specifically, whether or not there is a stronger and weaker side of the knot due to the slight internal asymmetry of the knot. I did some break testing and have concluded that the strength of the ABK is symmetrical: there is not a weaker and stronger side.

I’ve attached a report on my results.

I have a larger file size report, containing pictures for clarity, that I am happy to provide to anyone interested. Or, if someone can explain to me a convenient way of posting a 400kb file to this forum, I’ll provide the full pdf file of the report.

[modified 4/28/15]

I have overcome my technological challenges :slight_smile:

check out this link:

http://tinyurl.com/ohlsd43

you can view and download the report, with the pictures. there is also a folder of all the pictures I took, and a video.

2

“strength is symmetric(al)” :: This sounds quite peculiar, IMO,
vs. “is equal” and so on.

Can’t you post photos of your knots, in <100kb-size files,
w/o much to-do? --or cite ones that are already posted
in the forum somewhere. Just “butterfly knot” doesn’t
denote a specific geometry, and certainly not the particular
one(s) that met your testing. It can be edifying to see how
knots transform when push comes to shove.

Thanks,
–dl*

3

kr your document claims to disprove the notion that they are unequal. In technical factual sense not only does it not do this, but this is impossible to do.

An asymmetry can never be measured to be exactly zero. It can only be measured to be consistent with zero with the precision of the experiment.

Actually your two results agree too well, to the point that a suspicious person would have a right to be suspicious.

With 14 trials you can expect the range to represent about 1.5 standard deviations (an estimate which itself has a larger error). But if I go with that, then that gives you a standard error of about 88 pounds which produces an error in the mean of about 24 lbs. In other words if you repeat the exact same experiment, you could it expect it to be off by about 24 lbs more or less. Of course you COULD get the same result within 1/4 of a pound but the odds on that are probably about 1 in 50.

Anyway, the accuracy of your measurement is probably not better than about 23 lbs so a little over 2%. All you can really say is that they are the same, at the level of a couple of percent (give or take a little).

HOWEVER

The accuracy of this experiment is already far better than the certainly you can have with any one rope you are hanging from at a given moment, and it may even turn out that the asymmetry is smaller than you could ever practically measure with ANY number of breaks, because all that statistical precision can only be trusted so far. Statistics alone cannot reach .00001 standard deviation measurements in most any practical applications even if you do have enough trials , because except in extreme laboratory conditions studying fundamental results, you can never trust the systematics that well. So there is a point where I say if the difference can NEVER be measured then it has no observable meaning, ie the difference is meaningless, which I think in standard parlance means there is no difference. I am perfectly fine with that.

You are NOT at that level but you are at the level where you can already say the difference can have no bearing on how you decide to use the knot. That’s good enough for me. So yes, I think as far as anything that matters goes, these are the same. Use a proper safety factor and don’t worry about which way you point the knot. If you have even the slightest reason to think it matter, you should stop and do something safer.

4

DL:

Can't you post photos of your knots, in <100kb-size files, w/o much to-do? --or cite ones that are already posted in the forum somewhere. Just "butterfly knot" doesn't denote a specific geometry, and certainly not the particular one(s) that met your testing. It can be edifying to see how knots transform when push comes to shove.

I seemed to have been suffering from technological tunnel vision yesterday. :slight_smile:

check out this link:

http://tinyurl.com/ohlsd43

The report as a PDF as well as all the pics I took, and even a video. There is a whole sequence of pictures of a severely loaded, but unbroken ABK if you’re curious how the knot behaves under heavy loading.
I updated my original post, and put the link there as well.

TeX:

Actually your two results agree too well, to the point that a suspicious person would have a right to be suspicious.

To a person more suspicious than I, that statement could be taken as an accusation of dis-honorable behaviour :wink: You’ll just have to take my data at face value, and trust me that I didn’t doctor it.

Overall, I completely agree with what you say Tex. Perhaps I shouldn’t have used the word “disproved”, or clarified “disproved within the accuracy of this experiment”

...the (tiny) difference can have no bearing on how you decide to use the knot.

That’s pretty much exactly what I was looking to answer.

5
To a person more suspicious than I, that statement could be taken as an accusation of dis-honorable behaviour ;) You'll just have to take my data at face value, and trust me that I didn't doctor it.

Well I decided not to qualify it and let you guess, but you guessed right. It certainly was not an accusation, just to raise awareness that a passerby should not be too impressed by that incredible agreement. You CANNOT conclude that they are within a fraction of a pound of the same from that data. You can only conclude that they are within 25 lbs OR SO (could be less, could be 75 too though with low probability) of the same.

6

Anyway this is great data and great work. If you want to take your report to pro level, produce a mostly unassailable conclusion, and avoid much unneeded criticism (xarax is coming) then do the following little extra bit of work that will take 5 minutes. If you ignore it, obviously that’s fine too.

  1. put all the numbers in excel and ask it for the standard deviation for each test, s1 and s2

  2. divide each by the square root of 14 to get the standard error e1 and e2

  3. Find the mutual error em=square root of ( e1 squared + e2 squared), or just choose the biggest one and multiply by 1.4 (close enough and could have just used 28 in step 2 above to get the same result edit: scratch that, got carried away)

  4. multiply em by 2 (2 sigma is two sided 95% confidence interval) and write the result, f, as a percent of the average value.

  5. Instead of stating you proved they are not different, state “With 95% confidence level I find that the strengths are within f% of identical”

I did not make the 2 sigma correction nor the 1.4 correction in my quick estimate earlier so it will get worse than what I said. At 95% maybe it’s a 5% measurement, but you have to do the math and see.

Xarax will stipulate that he has always said that they differ by less than 1% and this will be far less your sensitivity. When you state things this way he can be happy to obsess over his 1% and you can be happy that you made a now document conclusion at the level everyone else cares about. Someone else will argue about the 5% confidence and that’s fine too. People can take the statement for what they want but the statement will be valid and you can shrug it off.

7

Worrying about exactly what distribution you have (I assumed gaussian of course) is by far second fiddle to first making an estimation of the general scale of the spread of the distribution and converting that into some error scale, as I described.

8
(xarax is coming)

:smiley: LOL ;D

9

The first priority is to understand what knot is tested.
Here, it is the “legs-abutting” version of the butterfly.
In this case, I hope that my prior remarks about
the orientations, “the geometries” of the “butterfly knot”
have been understood; alas, though, for there to be
several comments on the report without any mention
of this, I’m afraid not.
(Please refer to that image shown by Alan Lee, e.g. :
that is of a different, “legs-crossed” geometry.)

TO ME, I must see the relatively good results of this
tested orientation to challenge my notions of better
curvature and so on which I see for the “legs-crossed”
version (by either end loaded).

To Xarax, I still don’t understand what HE thinks is
the difference, of the 1-dia vs. 2-dia curvatures, for
the shown-here orientation --both seem more of the
1-diameter U-turn, IMO.

To Knot_Rigger, thank you for quite some work!!
(–as I’ve done similar or not even, and know it to
be a chore). You’ve “outdone yourself”, and raised
the bar! That said, the black cord makes discerning
the details of the material difficult if not impossible.
We might need to ask you to report what your eyeballs
up close & personal see of the Real McCoy(s), if you
can abide such query.

I note that the fig.8 eye knots seem to be loaded
in what I have called “the strong form” of what Xarax
in another thread posted a clear image of (well, w/o
indication of loading; but of the “perfect form” geometry)
in response to my remarking on two other images.
IMO, this orientation will benefit from hard loading
of the tails, in setting --something that might be not
practical in many cases, given strength of material
(i.e., that one cannot manually do “hard” for it).
My thought is that, in this orientation, the S.Part
bears against its twin part (i.e., the tail) as it
flows into the knot and goes then to its U-turn
around the eye legs (which is where the break
appears, in your one case --or maybe we should
see that in-knot-broken-end as needing to be
strettttttched farther along, so nearer the entry!?).

There is more that we can wonder about, in the spirit
of pure testing, at least --vs. practical significance-- :
would it matter were the mid-line eye-knot first loaded
end-2-end / “through”, and then loaded qua eye knot?!
Essentially, what a hard setting by loading ends would
do!? --and vice versa : how the knot would perform
end-2-end after taking a hard eye load and whatever
distortions this might impart. We’d be concerned about
such things mostly (most reasonably) if there were some
practical circumstance in which these are loadings.

–dl*

10

You do not have to understand it, you have to see it - if you still do not, you need a new pair of glasses, I am afraid.
I have taken those pictures, and I have marked them drawing the diameters of the two first curves with red ( for the wider ) and yellow ( for the narrower) circles - I can do no more !

http://igkt.net/sm/index.php?topic=3204.msg34668#msg34668
http://igkt.net/sm/index.php?topic=5269.msg34592#msg34592

P.S. Perhaps this difference will remain in place, only if the knot is first loaded end-to-end, so the shown geometrical difference will be “fixed”, and only afterwards end-to-eye, as you notice. I can not tell, because I have never loaded a Butterfly loop ( or any other knot ) near its strength limits…

11

Hi again

Tex, thanks for your encouragement, and for the brief lesson in Statistics. My appologies, but some of that is over my head. I get it in theory, but the i’m not great with the math. Perhaps you could recommend a resource to me, so that I can re-figure the data I have and understand the steps I’m taking to do it. And/or perhaps you could run you’re analysis on the data in my report… maybe that would help me understand the details. I added the raw data in a google spread sheet file to the shared folder in my google drive if anyone is interested in looking at it closer. The link is in the original post.

DL: Sorry that the pictures suck! You’re right that the black line is problematic, but it’s what I had for free :wink: I’ll try and give you some qualitative analysis from the breaking.

So there were two, very similar modes of breaking: 1) the line broke where the standing part entered the knot. breaking right at the collar, leaving the body of the knot fully intact. 2) the breaking point was under the collar. the point of rupture visible stretched out and after the break you could see a little daylight through the collar where the standing part used to be. These two modes of breaking are VERY similar, but I did make the distinction in recording the data. In none of the tests did the line break inside the ABK.

The breaking point in the figure 8 knot was inside the body of the knot. Where the stading part takes its sharpest bend, at the loop side of the body of the knot. I did tie each fig 8 in it’s “strong” form intentionally. You can see in the video how the fig 8 reacted to the high load, the standing part slips down and under it’s twin (the tail) at the loop end of the body of the knot. I consistently saw this behaviour during the test but didn’t record it as data, as I was concentrating on the ABK. I’ve also seen this deformation on heavily loaded fig 8 knots in larger rope, “in the field”. As an aside, I’m not convinced that there is a stronger and weaker way to tie the figure 8 loop knot. I’ve seen it said in one source that there is, and I’ve seen it said in another source that they are equal. Most sources don’t mention a distinction at all. Perhaps that will be my next break testing experiment (tex i may need your help with the math)

It’s hard to see the pictures of the heavily loaded ABK in the shared files (sorry, black line). I looked closedly at each of the ABK that survived the testing, the ones where the break happened at the fig 8. Both are heavily deformed, but maintain the same basic shape as a set and dressed ABK (the version I tied, without the loop legs crossing, and dressed with the X on the “back side”… ) Specifically I was looking to see if the X was still there, or if maybe it slips out under heavy loading, it was there on both. I found the same thing with all the broken ABK in the test as well, I did not see any capsizing of the form of the knot under heavy loading.

As you can (sorta) see in the video, as the tension was applied these things happened in approximately this order: 1) the ABK tightens up, yielding line to the standing part 2) the fig 8 tightens up, yielding line to the standing part 3) the ABK tightens some more, yielding line to the loop of the ABK 4) rupture.

With most samples there were two “pops”. The first pop was at peak force, and after it the tension was lower, then a little more pulling yielded a final rupture, but not at the full force. This is due to first, the load bearing core of the line rupturing, but leaving the jacket, then the jacket rupturing. I’ve seen this behavior before in other break testing of similar construction of rope, where the core provides most of the strength (ie kernmantle)

hope this info is interesting and helpful

12

References, ok, type error analysis in amazon.com. Choose the first book that comes up. It’s actually a great one (older editions are cheaper though). The next 10 or so all look good too though. Wikipedia is a is too dense but it will do too once you know what to look for.

You need basically three concepts.

  1. standard deviation, you know that one (excel knows how to calculate this, and wikipedia too)
  2. standard error in the mean. = SD/sqrt(N), simple. (let’s call this e1 and e2)

This is what you would expect the standard deviation of the AVERAGE itself to be if you repeated the whole experiment 10 times. (and for this reason it still gets written with the same symbol, sigma, which can confuse people) It also thus tells you how close you can expect your average to be to the true value. This number is smaller than SD because if you draw a big gaussian curve, even a randomly bumpy one, you can tell where the middle is (even by eye) with much better acurracy than 1 SD. An SD is sort of half the width of the whole distribution, loosely speaking. You know where the middle is much better than that, hence the division by something. Obviously that doesn’t prove the formula, just the idea.

Now you can calculate the ratio of the two values (knot up this way or that way) or the difference. I went with difference because the error calculation is SLIGHTLY more basic.

  1. The error in the sum or difference of two numbers is sqaure_root_of (e1^2+e2^2). e1 SHOULD be about equal to e2 since the situations were clearly pretty equal. So this is also just e1square_root_of two =e11.4 I said choose the bigger of e1 or e2, just to be generous.

You can find a nice proof of this in any of those books, but that won’t help you do the calculation. You just need the formula. If you want it in writing from a less respectable source:

http://en.wikipedia.org/wiki/Propagation_of_uncertainty#Simplification

Scroll down to the big example formulas table (as usual wikipedia over-generalizes the formalism until it’s useless for learning purposes. That’s too bad.)
You want the one f=aA+bB. The lower-case letters are meant to be constants, 1 (and -1) in our case. you can ignore sigma_ab, the last term in the answer. That’s for correlated results. Now you see the formula I gave you.

Long story short simply square the numbers from step 2, add them, and take the square root.

putting it all together, 14 trials, using root 2 method, what we have so far amounts simply to SD*0.38 .

That’s it. That is your one sigma error in the difference. 3 easy steps reduced to 1.

Divide by the average if you want to write it as a percent and call it a 1-sigma fractional uncertainty. If you were publishing in a real journal this would be enough. You could stop there. Congratulations.

But usually if you’re seeing nothing, you want to set a limit on the range of likely values so you google a z table and how to use it. You find that 2.5% of the area of a gaussian is above mean+2sigma, so 5% is is above or below mean plus or minus 2sigma. So you multiply your answer by two and say that with a 95% confidence level the values are at least that close together.

This does all assume Gaussian distributions, especially the z table bit. The other parts work reasonably well with any reasonable distribution though. With this little data it would be hard to prove you didn’t have Gaussian distributions . The process is an estimate of what you don’t know so more if and buts really aren’t useful. These assumptions are so standard they often go unstated as do the steps in the calculations.

13

P.S. if you want to just make a big improvement and aren’t comfortable making statements to back up such numbers, you can just state your standard deviations in place of the range. The standard deviation won’t get better as you take more data (the standard error will), but it least it won’t get worse. The problem with the range is, it actually gets worse with more data and doesn’t really describe the distribution. Someone who knows how can at least quickly use the SD to calculate what I wrote.

14

Yes, 2s should be enough, for practical knots, IMHO. However, I think that when you examine data coming from destructive tests of elastic materials, you do not get a Gaussian distribution - you probably get a Weibull distribution ( but that is what I had only read somewhere - I have no real knowledge or experience with those kinds of things…). Also, I think that one should have a method to find out which data he should ignore, the “outliers” - and I do not know if different distributions are related to different methods of determining the outliers, and different numbers of outliers, and, if they do, how they are related…

15

Tex

Thanks for the statistics lesson. I’ll work on the data analysis some more, and let you know how it goes.

16

xarax. Yes one should probably throw out outliers. I would certainly hope that in climbing rope (which maybe this isn’t?), there are no outliers. If there are any it should be from mis-tied knots or such, but I haven’t looked

You are right that nothing with a strict limit of zero can have a true gaussian distribution.
Gaussian is much easier to work with, and is probably close enough. It is also commonly used as an approximation for poisson distributions. There are a few cases where the real distribution can be know at least out to many sigma, but for most things like this, it cannot. kr mentioned two breaking mechanisms. This could even very easily produce a lopsided or even bimodal distribution, which has a much bigger impact that the Weibull vs Gaussian issue.

For finding the “true” mean, it doesn’t even really matter if your estimate is biased though, because the true mean is not important for its own sake here (there is no true rope). Any well defined measure of central tendency will probably do just as well.

As for the 2S you should realize that this has less to do with the distribution than you think. The “S” at that point is not the standard deviation of the breaking distribution. It is the standard deviation of repeated whole experiments, which is why 2 of these , let’s call it 2e instead, equals 0.75 of the distribution SD. The distribution of the means is not the same distribution) does not have the same shape) as that of the data itself. This 95% confidence estimate has nothing to do with how confident we are about a particular rope breaking of course. It only has to do with how confident we are that the distribution of breaking points is not affected by this knot orientation. Your particular rope still has a 95% chance (well, let’s just say high) to break within 2S, not 2e, but for this estimate, we CANNOT remove outliers. Outliers will kill you. For digging into what we can learn about the underlying details, we can remove them.

Nobody can know the exact distribution. In truth standard methods are used because they are standard, and they give pretty useful results that tend to work quite well (and I didn’t say 100% confidence). This result will be MUCH more reasonable than just taking the difference in the means for instance, and that’s the main point.

17

Right ! :slight_smile: I had not thought of that …

18

But a bimodal distribution does not produce a bimodal distribution of means. The mean is still the mean and is and fluctautions around it from one data set to the next are still random, with no preference (spikes, modalities) for any other false mean. A different single-data distribution will have some impact on the width of the distribution of means, but the effect is probably not as strong as one might think (but the distribution of what people might think tends to have many outliers).

19

By the way, I don’t have much experience with weibull distributions except for one special case of them, but I’m not that impressed with them for this. In the most general definition(not the one on wikipedia) it seems they can describe darn near any(maybe actually any, requires more thought) unimodal distribution that hits the origin, but I think the philosophical point behind them that makes them attractive is deceptive. We are just measuring the distribution of strengths of particular knots tied on particular ropes. Unlike time till failure the chance that a particular knot might die early doesn’t exactly limit its chance to die late. That configuration at that moment is as strong as it is and it will die where it dies. I can speculate on many reasons for distributions of strength within the ropes and some may produce slightly different curves than others, some more weibully inspired than others (and maybe others know more about the physical defects and how they collude), but it’s still speculation.

I do think at a given near-failure tension, there should be a very simple weibul distrubtion of survival probability vs rope length. This is about probability of defects vs legth, and that’s very 0-power weibullish, which has its own special name; it’s called exponentinal.

20

My (rudimental ) understanding is that, since Weibull distribution is the proper tool in situations involving a weakest link, it will be able to describe the rupture of ropes under tension, which is local. Of course, if ropes not only break, but also melt, and the heat generated by friction in one area can be transferred, along the fibres, to another, and act there, the whole argument in favour of its utilization becomes less convincing.