Thank you TreeCo. The only reason I didn't do that was so the equation and numbers could be easily read, found that parts of letters got lost when reduced.
Trees and Physics
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Thank you TreeCo. The only reason I didn't do that was so the equation and numbers could be easily read, found that parts of letters got lost when reduced.
The article on rescue rope is very interesting, I plan on reading it in more depth later on. As for Hooke's Law, you are saying the rope abides by this in saying the amount it deforms is linearly proportional to the force required to cause deformation (approximately). Look up Hooke's Law on wikipedia, its always good to do some research before making a comment right? Also, saying that something is "spring like" is not synonymous with saying deformation and force have a linear relationship, there are plenty of springs in the world made specifically so this relationship is non-linear.
The ropes we use for rigging may very well exhibit a linear relationship but the only way to know this is through additional information. All I know about the Samson rope is the difference in elongation between 20% and 30% breaking strength is almost half of that between 10% and 20%. Using a straight line on this will result in more error then you imply, which is increased if the value at zero is included. I am not saying there could not be a linear relationship after some point, actually if the parabolic trend line I made is accurate there will be a point at which linearity can be assumed (until plastic deformation occurs but that would require loading close to breaking strength which is not good). I don’t think the two lines you drew on the graph would work for this as it will be further up on the parabola that linearity begins.
Why is it unreasonable for an arborist to load a rigging rope beyond 30% of its breaking strength? I am not saying they should bring it to 80 or 90%. A piece of oak 3’ long and 20” in diameter will can weigh around 420 lbs, think of how much force that translates to when coming to a stop in 3 ft.
You said we are only looking at the simplistic case of the piece falling vertical and no other variables but looking at the original post of this thread I disagree. The question included how falling in an arc will change loading. Pointing out all the variables is important as they can dramatically change the forces seen on these ropes. The simple calcs you are doing will not be true representations of what is seen on the job and I wanted to make sure it was stated, kind of like a little disclaimer. What I think you are trying to say is that we can still work on the ideal scenario and with that I fully agree.
As for the explanation of my background, again I am not trying to toot my own horn or anything. That was also not meant to be a job application, don’t really know where that came from. All I was trying to do is show a bit of my background to give my information some credibility. Figured my information would mean a bit more if you it was coming from a mechanical engineer, but I guess that’s not the case with you (moray). Why should anyone believe something without ensuring credibility of its source? The only thing I know about you (moray) is that you are a software designer and to me that does not inspire confidence in doing engineering analysis.
I am out of this pi$$ing contest now, further discussions on loads are more then welcome though.
The ropes we use for rigging may very well exhibit a linear relationship but the only way to know this is through additional information. All I know about the Samson rope is the difference in elongation between 20% and 30% breaking strength is almost half of that between 10% and 20%. Using a straight line on this will result in more error then you imply, which is increased if the value at zero is included. I am not saying there could not be a linear relationship after some point, actually if the parabolic trend line I made is accurate there will be a point at which linearity can be assumed (until plastic deformation occurs but that would require loading close to breaking strength which is not good). I don’t think the two lines you drew on the graph would work for this as it will be further up on the parabola that linearity begins.
Why is it unreasonable for an arborist to load a rigging rope beyond 30% of its breaking strength? I am not saying they should bring it to 80 or 90%. A piece of oak 3’ long and 20” in diameter will can weigh around 420 lbs, think of how much force that translates to when coming to a stop in 3 ft.
You said we are only looking at the simplistic case of the piece falling vertical and no other variables but looking at the original post of this thread I disagree. The question included how falling in an arc will change loading. Pointing out all the variables is important as they can dramatically change the forces seen on these ropes. The simple calcs you are doing will not be true representations of what is seen on the job and I wanted to make sure it was stated, kind of like a little disclaimer. What I think you are trying to say is that we can still work on the ideal scenario and with that I fully agree.
As for the explanation of my background, again I am not trying to toot my own horn or anything. That was also not meant to be a job application, don’t really know where that came from. All I was trying to do is show a bit of my background to give my information some credibility. Figured my information would mean a bit more if you it was coming from a mechanical engineer, but I guess that’s not the case with you (moray). Why should anyone believe something without ensuring credibility of its source? The only thing I know about you (moray) is that you are a software designer and to me that does not inspire confidence in doing engineering analysis.
I am out of this pi$$ing contest now, further discussions on loads are more then welcome though.
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Pilsnaman, thank you for your measured response. I don't like this kind of exchange at all, but I feel it is a price I have to pay to preserve good high-level discussions about ideas, including the one we were having until the offensive paragraph came along. I want to actually explain myself this time rather than simply express displeasure with the paragraph in question. I also want to leave you out of the equation entirely, and make a general philosophical case that is addressed to everyone, but first I need to identify the two particular items that I think are both illegitimate and harmful in any field of discourse. The first is the argument ad hominem in which you conclude, from two words in my Bio ("software developer") that anything I say in this thread must be discounted. The second is the related argument from Authority in which you explicitly claim anything you say, by contrast, has to be taken much more seriously. End of personal references.
Facts stand on their own feet. They are true or false or something in between regardless of who presents them. Arguments are the same--either spot on, a little off the mark, or irrelevant and worthless, no matter who the author may be. A mathematical relation is true or false or misapplied or whatever, again, irrespective of who writes it down. Part of the joy of a site like AS is the challenge of navigating a thicket of facts and pseudo-facts and good and bad arguments and good advice and not-so-good advice to reach your own understanding of something you didn't understand before. This is an experience that cannot simply be delivered by an "authority." I have learned a lot of good stuff here, and had hours of enjoyment in the bargain, and never once did I need to consider whether someone was an "authority" or not.
To shine a really bright light on the argument from Authority, I refer everyone to the "Monty Hall problem," which is well-described on Wikipedia. This was briefly quite famous when the Parade magazine column ran in 1990, and several hundred mathematicians, and thousands of normal folks, wrote the magazine to claim the published answer was wrong. What the Wikipedia article doesn't say, but I remember reading in some follow-up a few years later, was that some 100 or so of the mathematicians decided to indulge themselves by trotting out the argument from Authority: "I've been teaching college-level probability courses for 40 years and I am appalled that your columnist would actually get this elementary problem wrong." When it finally became clear that the mathematcians were wrong, and the columnist right, I am sorry to report that only 2 or 3 of the hundred who had enjoyed publicly waving their credentials actually had enough character to write back and apologize for their unworthy behavior.
I'm done with this, and ready to get back to rope calculations.
A Real Experiment: Background
Throughout this thread there has been a vigorous debate about the true nature of the stress-strain relationship for arborist rope, and whether that relationship is linear, or close enough to linear, that the rope can be treated like a spring for calculation purposes. Everyone seems to agree that the data published by the rope manufacturers, Samson in particular, are woefully abbreviated, and that much more extensive data sets would solve the problem. A parallel issue has been the nature of the underlying material from which the rope is made. Because the rope is a complex structure, the stress-strain relationship for the rope may not precisely mirror the stress-strain relationship of a single fiber of the rope. But it would be reasonable, if actual data for a rope are unavailable, to assume that the rope behavior won't be wildly different from that of a single fiber. The more we know about the actual behavior of the rope, the less we have to fill in the data gaps with assumptions and reasoning and calculations.
I did a bit of online searching to find some evidence for or against the categorical statements above. As to the contention that Hooke's law doesn't apply to synthetics, my Web search pretty much came up empty. Hard data on rope stress-strain relations weren't jumping out at me at every turn, either, but the following link is excellent: http://www.xmission.com/~tmoyer/testing/Qualifying_a_Rescue_Rope.pdf.
And here is the Samson link:
http://www.samsonrope.com/index.cfm
The first reference shows that for a number of ropes the stress-strain relationship is quite linear up to 1000 lbs, which is as far as the testing went. Data points from Samson, which I tabulated in an earlier post, are clearly not linear over the load range from zero to 30% of ultimate strength; it would be an approximation to treat them as if they were.
What I would really like to do is some experiments like those performed in the first reference above, but on arborist ropes. Naturally I don't have the big expensive machinery I would need for such a project, much as I would love to play with such stuff. But suddenly it occurred to me that I could perform a mini-experiment with simple materials ready at hand. I could measure the stress-strain relationship for a single yarn of an arborist rope!
The rope I chose was a length of 1/2 inch Samson Tree-Master. This is a 3-strand polyester rope, sometimes used for rigging, with a published break strength of 7000 lbs. There are 117 yarns in the rope, each of which is a mildly twisted bundle of dozens of individual parallel fibers. The yarns are bundled into mini 3-strand ropelets, twisted in the reverse direction. These ropelets are then bundled into groups of 13 and given a left-hand twist. These bundles of 13 comprise the major strands of the rope, which are given a right-hand twist. I mention all this to indicate how complicated even a "simple" 3-strand rope can be, and why we should hesitate to assume that the stretching behavior of a complex structure like this would necessarily be as simple as that of an individual fiber.
Since there are 117 yarns in the rope, each one should have a break strength of about 60 lbs. Since a single fiber of the rope, if you could see it all by itself, would take the form of a complicated set of spirals within spirals, it is clearly not lined up directly with tension in the rope, and it won't be as strong in that orientation as it would be if were simply a straight fiber under tension. This means one yarn from the rope will have to be a bit stronger than 60 lbs for the whole rope to have 7000 lbs of tensile strength. In any event, properly testing a yarn of about 60 lbs tensile strength was going to be well within my experimental means.
Throughout this thread there has been a vigorous debate about the true nature of the stress-strain relationship for arborist rope, and whether that relationship is linear, or close enough to linear, that the rope can be treated like a spring for calculation purposes. Everyone seems to agree that the data published by the rope manufacturers, Samson in particular, are woefully abbreviated, and that much more extensive data sets would solve the problem. A parallel issue has been the nature of the underlying material from which the rope is made. Because the rope is a complex structure, the stress-strain relationship for the rope may not precisely mirror the stress-strain relationship of a single fiber of the rope. But it would be reasonable, if actual data for a rope are unavailable, to assume that the rope behavior won't be wildly different from that of a single fiber. The more we know about the actual behavior of the rope, the less we have to fill in the data gaps with assumptions and reasoning and calculations.
I did a bit of online searching to find some evidence for or against the categorical statements above. As to the contention that Hooke's law doesn't apply to synthetics, my Web search pretty much came up empty. Hard data on rope stress-strain relations weren't jumping out at me at every turn, either, but the following link is excellent: http://www.xmission.com/~tmoyer/testing/Qualifying_a_Rescue_Rope.pdf.
And here is the Samson link:
http://www.samsonrope.com/index.cfm
The first reference shows that for a number of ropes the stress-strain relationship is quite linear up to 1000 lbs, which is as far as the testing went. Data points from Samson, which I tabulated in an earlier post, are clearly not linear over the load range from zero to 30% of ultimate strength; it would be an approximation to treat them as if they were.
What I would really like to do is some experiments like those performed in the first reference above, but on arborist ropes. Naturally I don't have the big expensive machinery I would need for such a project, much as I would love to play with such stuff. But suddenly it occurred to me that I could perform a mini-experiment with simple materials ready at hand. I could measure the stress-strain relationship for a single yarn of an arborist rope!
The rope I chose was a length of 1/2 inch Samson Tree-Master. This is a 3-strand polyester rope, sometimes used for rigging, with a published break strength of 7000 lbs. There are 117 yarns in the rope, each of which is a mildly twisted bundle of dozens of individual parallel fibers. The yarns are bundled into mini 3-strand ropelets, twisted in the reverse direction. These ropelets are then bundled into groups of 13 and given a left-hand twist. These bundles of 13 comprise the major strands of the rope, which are given a right-hand twist. I mention all this to indicate how complicated even a "simple" 3-strand rope can be, and why we should hesitate to assume that the stretching behavior of a complex structure like this would necessarily be as simple as that of an individual fiber.
Since there are 117 yarns in the rope, each one should have a break strength of about 60 lbs. Since a single fiber of the rope, if you could see it all by itself, would take the form of a complicated set of spirals within spirals, it is clearly not lined up directly with tension in the rope, and it won't be as strong in that orientation as it would be if were simply a straight fiber under tension. This means one yarn from the rope will have to be a bit stronger than 60 lbs for the whole rope to have 7000 lbs of tensile strength. In any event, properly testing a yarn of about 60 lbs tensile strength was going to be well within my experimental means.
A Real Experiment: Methods and Materials
I retrieved 3 yarns from an unused piece of rope, each about 5 feet long. The idea was to anchor the top of the yarn to a low tree limb and then add succesive units of weight to the bottom. Measuring the stretch of the yarn would be as simple as noting the position of a marked point on the rope against a background ruler. With each addition of weight, the rope would stretch a certain amount, the mark would descend a short distance, and the new location of the mark could be recorded.
Since I didn't have a whole box of identical weights, I decided use identical volumes of water for my weights. I suspended a 5-gallon plastic bucket from the bottom of the strand being tested. Each time I added water to the bucket, I would record the new position of the mark against the ruler. Since I wanted to be able to do this quickly and uniformly, I chose to fill a large tub with about 10 gallons of water, and scoop water from the tub with a pitcher. The pitcher, full to the brim, would then be slowly poured into the bucket to avoid applying a significant dynamic load to the tested strand. A full pitcher of water represented slightly more than one liter.
I decided to record the readings with a camera. This provided several advantanges. One, measurement-taking was very uniform. This meant no significant parallax effects, which would not have been the case if I had taken the readings by eye. Two, I could carefully study the photos later to locate the position of the index mark as accurately as I could. This could take several seconds, which I didn't want to waste while the experiment was in progress. Three, it very fast and easy.
I did a full experimental run on two separate strands. The first run was a practice run that allowed me to prove the method was viable and to eliminate a couple of small bugs in the method. The second run was the one from which I recorded the data.
The final setup was this: The empty bucket, which weighs about 2 lbs, was attached to the bottom of the yarn. The distance from the upper anchor to the bail of the bucket was a bit over 3 feet so there would be no interference between the ruler and the system under test. The ruler was also suspended from the same tree limb just behind the yarn anchor. Any sag or movement of the limb, or the whole tree for that matter, would be experienced equally by the yarn and the ruler, and should have no effect on the measurements. A marker pen was used to create an index mark about 30 inches below the upper anchor. Thus it was the stretching of this 30-inch yarn segment that was being measured by the experiment.
Once the empty bucket was suspended from the yarn, it was allowed to hang there for about 10 minutes before I started loading on weight. This allowed the yarn to slowly untwist and equilibrate, and gave me some time to finish preparations. The camera was set up on a tripod about 5 feet away. The tripod was adjusted so the camera lens would be about an inch below the index mark on the yarn. I knew from the first run that the index mark was going to move about 2 inches by the time the bucket was full, and I wanted the camera to be positioned about in the middle of that range. Then the camera zoom was set to give me good readability for the real measurements. With the camera locked in place, I was ready to begin. The photo below is the first photo of the run, taken when the bucket was empty and just before I started adding water. The index mark is visible at the 705-mm position.
It took about 20 seconds to go through one complete cycle, and I tried to keep this as uniform as possible: Scoop a full pitcher of water. Slowly pour it into the bucket over 3 or 4 seconds. Stabilize the bucket with one finger to dampen any swing. Put the pitcher on the ground. Take two steps over to the camera. Half-depress the shutter button and wait for the ready "beep." Take the shot. Return to the pitcher and water tub ready to do it all again.
The last data point was recorded when the bucket was full and on the verge of overflowing. I later measured the weight of an overflowing bucket at about 42 lbs. I guess it really does hold 5 gallons. Just to satisfy my own curiosity I went ahead and loaded the yarn to failure by pressing down on the full bucket with two hands. My subjective impression was that I pressed down with about 15 to 25 pounds. I am very sure I didn't use 40 lbs. I just wanted to get some sense of how far the experiment had pushed the yarn towards failure--probably something like 60 or 70 percent. If there were a compelling reason to know this more accurately, anyone could easily repeat the experiment using a 10-gallon bucket.
I retrieved 3 yarns from an unused piece of rope, each about 5 feet long. The idea was to anchor the top of the yarn to a low tree limb and then add succesive units of weight to the bottom. Measuring the stretch of the yarn would be as simple as noting the position of a marked point on the rope against a background ruler. With each addition of weight, the rope would stretch a certain amount, the mark would descend a short distance, and the new location of the mark could be recorded.
Since I didn't have a whole box of identical weights, I decided use identical volumes of water for my weights. I suspended a 5-gallon plastic bucket from the bottom of the strand being tested. Each time I added water to the bucket, I would record the new position of the mark against the ruler. Since I wanted to be able to do this quickly and uniformly, I chose to fill a large tub with about 10 gallons of water, and scoop water from the tub with a pitcher. The pitcher, full to the brim, would then be slowly poured into the bucket to avoid applying a significant dynamic load to the tested strand. A full pitcher of water represented slightly more than one liter.
I decided to record the readings with a camera. This provided several advantanges. One, measurement-taking was very uniform. This meant no significant parallax effects, which would not have been the case if I had taken the readings by eye. Two, I could carefully study the photos later to locate the position of the index mark as accurately as I could. This could take several seconds, which I didn't want to waste while the experiment was in progress. Three, it very fast and easy.
I did a full experimental run on two separate strands. The first run was a practice run that allowed me to prove the method was viable and to eliminate a couple of small bugs in the method. The second run was the one from which I recorded the data.
The final setup was this: The empty bucket, which weighs about 2 lbs, was attached to the bottom of the yarn. The distance from the upper anchor to the bail of the bucket was a bit over 3 feet so there would be no interference between the ruler and the system under test. The ruler was also suspended from the same tree limb just behind the yarn anchor. Any sag or movement of the limb, or the whole tree for that matter, would be experienced equally by the yarn and the ruler, and should have no effect on the measurements. A marker pen was used to create an index mark about 30 inches below the upper anchor. Thus it was the stretching of this 30-inch yarn segment that was being measured by the experiment.
Once the empty bucket was suspended from the yarn, it was allowed to hang there for about 10 minutes before I started loading on weight. This allowed the yarn to slowly untwist and equilibrate, and gave me some time to finish preparations. The camera was set up on a tripod about 5 feet away. The tripod was adjusted so the camera lens would be about an inch below the index mark on the yarn. I knew from the first run that the index mark was going to move about 2 inches by the time the bucket was full, and I wanted the camera to be positioned about in the middle of that range. Then the camera zoom was set to give me good readability for the real measurements. With the camera locked in place, I was ready to begin. The photo below is the first photo of the run, taken when the bucket was empty and just before I started adding water. The index mark is visible at the 705-mm position.
It took about 20 seconds to go through one complete cycle, and I tried to keep this as uniform as possible: Scoop a full pitcher of water. Slowly pour it into the bucket over 3 or 4 seconds. Stabilize the bucket with one finger to dampen any swing. Put the pitcher on the ground. Take two steps over to the camera. Half-depress the shutter button and wait for the ready "beep." Take the shot. Return to the pitcher and water tub ready to do it all again.
The last data point was recorded when the bucket was full and on the verge of overflowing. I later measured the weight of an overflowing bucket at about 42 lbs. I guess it really does hold 5 gallons. Just to satisfy my own curiosity I went ahead and loaded the yarn to failure by pressing down on the full bucket with two hands. My subjective impression was that I pressed down with about 15 to 25 pounds. I am very sure I didn't use 40 lbs. I just wanted to get some sense of how far the experiment had pushed the yarn towards failure--probably something like 60 or 70 percent. If there were a compelling reason to know this more accurately, anyone could easily repeat the experiment using a 10-gallon bucket.
A Real Experiment: Results
The photos were studied carefully to extract the readings, which were then entered into Excel. For readings that appeared to fall pretty squarely on a millimeter tick mark, I entered the reading as a whole number, as in 109. If the reading appeared to be about halfway between 109 and 110, I entered it as 109.5. So the resolution of the measurements was .5mm. What does this really mean? It means that if I were shown two photos, and in one of them the index mark was actually precisely .5mm higher than in the other, I could always detect the difference.
The only adjustment to the data was to subtract 705 from every reading. Since the empty bucket was the starting point for the experiment, and I wasn't interested in absolute numbers but in actual stretch, starting from zero at the starting point, after the subtraction step, the numbers along the vertical axis represent mm of stretch. For example, if one of the readings had been 725, it would now show up on the chart as 20, and that would indeed be the amount of stretch from the starting point.
The units along the horizontal axis are units of force--pitchers of water!! You won't find this in the physics books, I know. But we would have exactly the same curve if I had measured the weight of one pitcher to be, say, 2.18281 pounds, and then noted in the chart legend that 1 unit on the horizontal axis equals 2.18281 pounds. Same chart, same data, different legend.
Finally I let Excel draw in the best-fit straight line. This is hardly necessary; as anyone can see, the raw data show the stress-strain curve is almost perfectly linear.
The photos were studied carefully to extract the readings, which were then entered into Excel. For readings that appeared to fall pretty squarely on a millimeter tick mark, I entered the reading as a whole number, as in 109. If the reading appeared to be about halfway between 109 and 110, I entered it as 109.5. So the resolution of the measurements was .5mm. What does this really mean? It means that if I were shown two photos, and in one of them the index mark was actually precisely .5mm higher than in the other, I could always detect the difference.
The only adjustment to the data was to subtract 705 from every reading. Since the empty bucket was the starting point for the experiment, and I wasn't interested in absolute numbers but in actual stretch, starting from zero at the starting point, after the subtraction step, the numbers along the vertical axis represent mm of stretch. For example, if one of the readings had been 725, it would now show up on the chart as 20, and that would indeed be the amount of stretch from the starting point.
The units along the horizontal axis are units of force--pitchers of water!! You won't find this in the physics books, I know. But we would have exactly the same curve if I had measured the weight of one pitcher to be, say, 2.18281 pounds, and then noted in the chart legend that 1 unit on the horizontal axis equals 2.18281 pounds. Same chart, same data, different legend.
Finally I let Excel draw in the best-fit straight line. This is hardly necessary; as anyone can see, the raw data show the stress-strain curve is almost perfectly linear.
Moray, interesting test and the methods are well thought out. Other then my uncertainty in the use of a single strand vs the original rope configuration I would agree that the relationship is linear. I do not agree that braided line is lower in strength then straight line. It has always been my understanding that braiding rope increases strength over similar diameter mono-filament rope, I will see if this can be verified. Also, it is strange that your rope section didn't have the initial curve but was linear from the start, unlike the data on Samson's website. That does bring up a red flag but what do I know, according to you the people that study a topic are going to get it wrong and don't know what they are talking about. I guess we should ask a journalist to work this problem out for us. In the end a linear relationship is looking likely, I am the kind of person that doesn't believe things until they are truly tested. While I did contact Samson to try and obtain additional information they have not yet replied, not that I am expecting them to. If anyone has a scale that can read forces up to around 1500 lbs let me know so rope in its normal configuration can be tested, figure using a 3 ton floor jack would do the trick for adding weight.
I have enjoyed this bit of conversation and learned some stuff, always interesting, but don't feel much was gained from an arborist standpoint. What do we learn from examining an over simplified problem? Any force we calculate will be totally different with a groundy that lets it run and pieces that swing around instead of moving exactly vertical. In all this did the original question asked in post #1 ever get answered?
I have enjoyed this bit of conversation and learned some stuff, always interesting, but don't feel much was gained from an arborist standpoint. What do we learn from examining an over simplified problem? Any force we calculate will be totally different with a groundy that lets it run and pieces that swing around instead of moving exactly vertical. In all this did the original question asked in post #1 ever get answered?
Another Experiment: Nylon
Thanks.
I decided to do another test, this time on the nylon core from some Arbor-Master rope, left over from a splice I had done. The six yarns that make up the core are much larger than the polyester yarn in the experiment described above, but there are a number of smaller yarns that comprise the big one, and I was able to unwind and peel them out, one at a time.
I ran 3 tests. The small, slippery nylon yarn was much more difficult to handle than the polyester. I had a problem with slippage where I had wrapped it around the anchor limb. It also slipped on the bail of the bucket. By the final run I had solved both problems by creating and testing a loop of yarn. The loop ran through the same screw link from which the metal ruler was suspended, and at the lower end it ran through a keychain carabiner that was clipped to the bucket bail. The loop was formed by placing the ends of the original yarn together and tying a surgeon's knot in both at once. When it was pulled tight, it appeared to be rock solid.
Water was added in 1/2 liter increments using a glass measuring cup. The last data point was not of my choosing. The yarn held for several seconds, then broke while I was filling the cup for another go. In all three runs I noted the same behavior (and I got soaked every time)--the stress-strain behavior seemed smooth and uniform till the yarn broke without warning.
The results are plotted in the two charts below. As in the previous experiment, the starting point represents the empty bucket, or about 2 lbs.
Without question, the curve is not linear. In the upper figure, Excel has fitted a second degree polynomial (a parabola), and in the lower figure Excel has fitted two straight-line segments.
I show both of these to make the following point: In both figures the mathematical curves are approximations. We don't know what the "true" curve is, and cannot know. We can only take lots and lots of measurements and then try to connect the dots with a smooth curve. This is eminently reasonable and practical, but it is an approximation. For my money, both approximations I show look about equally good in terms of closeness of fit to the data points. That does not mean that both are equally close to describing the underlying physics. There is no way that the true relationship between force and stretch is linear in this case. It is certainly curvilinear and quite well matched with the parabolic curve I show.
If I want to use these data for the purpose of finding the answer to a stress-strain problem, I am not interested at that point in the underlying physics. I may just want my calculation to be easy to do and to give me an answer that is not far off the mark. For this purpose, using an approximation that does not mirror the underlying physics may work just fine.
Thanks.
I decided to do another test, this time on the nylon core from some Arbor-Master rope, left over from a splice I had done. The six yarns that make up the core are much larger than the polyester yarn in the experiment described above, but there are a number of smaller yarns that comprise the big one, and I was able to unwind and peel them out, one at a time.
I ran 3 tests. The small, slippery nylon yarn was much more difficult to handle than the polyester. I had a problem with slippage where I had wrapped it around the anchor limb. It also slipped on the bail of the bucket. By the final run I had solved both problems by creating and testing a loop of yarn. The loop ran through the same screw link from which the metal ruler was suspended, and at the lower end it ran through a keychain carabiner that was clipped to the bucket bail. The loop was formed by placing the ends of the original yarn together and tying a surgeon's knot in both at once. When it was pulled tight, it appeared to be rock solid.
Water was added in 1/2 liter increments using a glass measuring cup. The last data point was not of my choosing. The yarn held for several seconds, then broke while I was filling the cup for another go. In all three runs I noted the same behavior (and I got soaked every time)--the stress-strain behavior seemed smooth and uniform till the yarn broke without warning.
The results are plotted in the two charts below. As in the previous experiment, the starting point represents the empty bucket, or about 2 lbs.
Without question, the curve is not linear. In the upper figure, Excel has fitted a second degree polynomial (a parabola), and in the lower figure Excel has fitted two straight-line segments.
I show both of these to make the following point: In both figures the mathematical curves are approximations. We don't know what the "true" curve is, and cannot know. We can only take lots and lots of measurements and then try to connect the dots with a smooth curve. This is eminently reasonable and practical, but it is an approximation. For my money, both approximations I show look about equally good in terms of closeness of fit to the data points. That does not mean that both are equally close to describing the underlying physics. There is no way that the true relationship between force and stretch is linear in this case. It is certainly curvilinear and quite well matched with the parabolic curve I show.
If I want to use these data for the purpose of finding the answer to a stress-strain problem, I am not interested at that point in the underlying physics. I may just want my calculation to be easy to do and to give me an answer that is not far off the mark. For this purpose, using an approximation that does not mirror the underlying physics may work just fine.
Agreed, that would give an excellent fit.
After I plotted this, it occurred to me that the curvilinear characteristic of the Arbor-Master core might help explain something that I had always wondered about--why make the core out of nylon? One of the reasons I bought a bunch of Arbor-Master for my climbing line was that it seemed to be the only one out there where you didn't have to milk the slack out of the cover a few times before the rope would stabilize.
The curve in the plot looks a whole lot like the general curve for most of the Samson ropes--relatively soft in the beginning, but much stiffer at higher loads. The core in Arbor-Master is not braided or twisted--only the yarns have a mild twist. The curve I measured should be pretty similar to the curve for the core as a whole, because there is little in the way of constructional slack to complicate the equation. So the cover, which has constructional slack but is woven from a linear fiber, more or less matches the stretch curve for the core, which lacks constructional slack, but is built from non-linear fiber! Probably totally wrong, but maybe Samson will tell us...
Boy oh boy did you ever give me a scare, Pilsnaman! I was just settling down with a nice cup of coffee to enjoy some of your peevish remarks, when this one brought me up short. Holy ****, I thought, I know one! A journalist, that is.
I was sitting in the local coffee shop a couple of years ago, practicing tying knots with a bit of cord, when she approached me with a huge smile on her face. She said she couldn't help noticing all the knot tying, and just had to ask what it was for. When I explained, she allowed as how she knew a bit about knots, herself. Turns out she was part of the local search and rescue team, specializing in high angle rescue.
Over the next few weeks we became friends, and she taught me quite a lot of cool stuff about ropes and knots and rigging. The trunk of her car was full of carabiners and slings and quickdraws and belay plates and other stuff I had never seen before or even heard of. Even though I never got to watch her practice rescue rigging scenarios, she came over and climbed a big tree with me one time so she could see first hand what that was all about.
But what a close call, eh?!! If only your wise observation about the competence of journalists had been available to me a couple of years ago, I could have avoided this dangerous encounter! Whew! I just pray to God I haven't suffered any permanent damage.
But this has set me to thinking. I realize my upbringing has been sorely deficient in preparing me to deal with all the different kinds of people out there, and I need to grow up and learn a real system. The swiftness and certitude with which you can divine that a journalist can't do math just takes my breath away, and I want to see if I can learn your system. But as I am just a total beginner at this, and probably have no natural aptitude at all, I need your help. Could you maybe send me a list (you can PM me if you want) that shows all the categories you use (like homeowner, journalist, etc.) and what each one means? I think it would take me forever to figure this stuff out on my own, so I would much prefer to learn it from a pro like yourself.
Many thanks,
moray
Don't think you quite got the point there Moray. I was making a sarcastic remark on the following:
Also, the more I think about it the more I don't think testing single fibers or groups of fibers will result in data that represents a complete rope. Between the braiding of a rope and combination of a core and sheath there is no way we can assume the single strands will be good representations. Now I am not saying the data won't be a representation of the rope but there is no data or information to support this assumption.
I understand that you are trying to point out that because someone says they are a mathematician, engineer, etc doesn't mean they are always going to be right or knowledgeable on a topic within their field. That being said, would you rather have someone with an english or math degree work a math problem for you? While it would depend on the people involved, generally the mathematician will be a better choice.
Also, the more I think about it the more I don't think testing single fibers or groups of fibers will result in data that represents a complete rope. Between the braiding of a rope and combination of a core and sheath there is no way we can assume the single strands will be good representations. Now I am not saying the data won't be a representation of the rope but there is no data or information to support this assumption.
Well put; I totally agree. I wish the manufacturers would give us more data, or that I had some way of doing my own tests.
BobEMoto
ArboristSite Lurker
Just when I thought the movie 
opcorn: was over you guys get all mussy again.
There is more information out there. It's just hard to find. I've haven't kept track of all the links, but from what I recall: rope stretchiness has a little nonlinearity at the beginning indicated by some of the curve matching not hitting zero. This is probably caused by interfiber friction. Then it enters a near-linear region for the safe working load area and a little beyond. Then it starts to stretch less and less up to the breaking point. I would guess this has to do with the molecular characteristics of each fiber and that when it quits stretching, it breaks.
Treating the rope as a spring doesn't really tell us anything new. The force equations give the maximum and the smooth stretchiness is the spring constant in action.
What is more interesting is how to deal with the dynamic loads. I tried approximating the effects in the earlier chart I published. I now think my approximation wasn't very close. The issue I had was how to deal with the kinetic energy gained by the fall. This extra energy equates to an increased weight. The key seems to be that the potential energy initially gained is linear with distance. This is shown in Yale's somewhat cryptic chart:
http://www.yalecordage.com/html/pdf/industrial_marine/low/Pg8.pdf
So the forces previously published show more force then will really be achieved because the extra energy causes the rope to stretch more which causes the stopping force to be lower.
opcorn: was over you guys get all mussy again.There is more information out there. It's just hard to find. I've haven't kept track of all the links, but from what I recall: rope stretchiness has a little nonlinearity at the beginning indicated by some of the curve matching not hitting zero. This is probably caused by interfiber friction. Then it enters a near-linear region for the safe working load area and a little beyond. Then it starts to stretch less and less up to the breaking point. I would guess this has to do with the molecular characteristics of each fiber and that when it quits stretching, it breaks.
Treating the rope as a spring doesn't really tell us anything new. The force equations give the maximum and the smooth stretchiness is the spring constant in action.
What is more interesting is how to deal with the dynamic loads. I tried approximating the effects in the earlier chart I published. I now think my approximation wasn't very close. The issue I had was how to deal with the kinetic energy gained by the fall. This extra energy equates to an increased weight. The key seems to be that the potential energy initially gained is linear with distance. This is shown in Yale's somewhat cryptic chart:
http://www.yalecordage.com/html/pdf/industrial_marine/low/Pg8.pdf
So the forces previously published show more force then will really be achieved because the extra energy causes the rope to stretch more which causes the stopping force to be lower.
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If you need help with this just let me know what the weight is and length of fall. I can calculate the resulting velocity and post how the work was done. Also the energies can be calculated if you would like that too.
Just when I thought the movie
Yeah, sorry about that... Maybe we can get the Oscar in the Public Brawling category.
I would like to thank my wife and parents for all the practice throughout my life, I couldn't have done it without them.
What a great link--the Yale site is a gold mine! You can easily spend an hour or two with the tables and charts, the engineering descriptions, the discussion of working loads, and so on. I especially liked the charts in the Industrial section (why can't Samson do this?).
Comparing the charts for a number of different ropes several things become apparent. For one, the ropes are quite different. The stiffer the rope, in general, the less energy storage capacity it has. The range of values in energy storage per lb. of rope was more than 5:1. The static ropes arborists like to use for rigging are near the bad end of the scale in this regard. (Even though there is only one chart for each rope type, and energy storage per lb. is indicated on the chart, Yale notes that this quantity varies by size of rope for certain specific ropes.)
The working load for the rope, which Yale discusses at some length, also varies from rope to rope, but the energy storage at the working load is always just a small fraction of the rope's maximum.
The shape of the stress-strain curve varies a bit from rope to rope, from a nearly perfect straight line to somewhat curvy. Since the working load region of the curves is so short, in nearly every case a straight-line approximation for that section of the curve should work fine for rough calculation purposes.
In the second Engineering Index page on the site Yale goes through the dynamic load calculations for a scenario just like the one we have been discussing in this thread. It is based on the straightforward energy absorption method, but things are simplified a bit by the use of a quantity derived from the charts--energy absorption per lb. It occurred to me that for someone who uses the same rigging rope over and over again, this could be further simplified by converting this to energy absorption per foot of rope. I decided to do this for the rigging rope I have been using, 1/2 inch Samson Stable Braid.
Stable Braid
The following calculation applies to 1/2 inch Samson Stable Braid, a low-stretch double-braided polyester rope meant for rigging operations. The specs and construction of the rope seem quite similar to Yale's Double Esterlon, though the latter rope seems quite a bit stretchier.
The chart below shows the data points published by Samson; the trend line is Excel's best-fit straight line. If you ask Excel to fit a second-degree polynomial (a parabolic curve) instead, it looks almost exactly like the straight line--for much of the chart you can hardly tell there is more than one line. Why make life complicated? Choose the straight line.
The equation of the line is shown. You can actually figure a lot of things out using the chart directly, but it is easier to do the calculations numerically. Without the chart, though, it would be really really hard to understand this stuff.
The number I want to find is energy absorbed per foot of rope at the working load limit (WLL). Working load is a more or less arbitrary fraction of breaking load; For illustration purposes I chose 25%, which is the number used by Yale for Double Esterlon. If you choose a different number, obviously you will get a different value for absorbed energy.
The calculation is actually quite easy. The published break strength of Stable Braid is 10,400 lbs. Then the WLL is 1/4 of that, or 2600 lbs. The arrow on the chart points to the fake data point I installed at 2600 lbs. From the equation, or just by reading the chart, 2600 lbs. corresponds to a stretch of 2.25%. In a 100-ft piece of rope, that would be 2.25 feet. Just imagine the units along the X axis are feet instead of per cent; in this case the chart shows the stretch characteristic of 100 feet of 1/2 inch rope. The energy needed to stretch the rope 2.25 feet when the force increases linearly from 0 to 2600 lbs is 2927 ft. lbs. This is just the shaded area of the chart. The shaded area is the area of the triangle, which is given by the height times the base divided by two. If 100 feet of rope absorbs 2927 ft. lbs. of energy at the WLL, then 1 foot absorbs 1/100 as much, or 29 ft. lbs. This is the magic number I want to remember. (If you use Samson's recommendation of 20% break strength for WLL, the magic number is about 18.5 ft. lbs. per foot of rope.)
What is so cool about this number is that it allows you to solve fairly tough dynamic loading problems, the ones where you drop a load and the rope alone stops it, without any need for F = M x A, or velocity, or mass, or the gravitational constant. The three things needed for the calculation can be estimated by eye: how heavy is the load, how far will it drop before the rope catches it, and how much rope will do the catching.
The energy the rope must absorb is just the product of the load weight times the distance it drops, i.e., E = F x D. The length of rope doing the catching is the length of the rope from the load to the anchor. (If the rope passes over a pulley, we still measure all the way to the anchor. If there is a friction device in the line, or the load is allowed to run, then this calculation will not apply.) Now divide the energy by the rope length to get energy per foot. If it is less than 29, you have not exceeded the WLL; if it is more than 29 you have.
It is shockingly easy to exceed the WLL. Tie some Stable Braid to a low limb. A couple of feet below the limb, tie on a 30-lb barbell. Lift the weight up to the limb and drop it. You have just applied more than 2600 lbs of force to the rope and stressed it beyond the WLL.
The math is really easy. You are about to drop 100 lbs 5 feet. But the rope from the anchor through the block and down to the load will be 25 feet long. The 500 ft lbs of energy will be divided by 25 ft of rope, or 20 ft lbs per foot. This is well below the WLL at 29 ft lbs per foot of rope.
The following calculation applies to 1/2 inch Samson Stable Braid, a low-stretch double-braided polyester rope meant for rigging operations. The specs and construction of the rope seem quite similar to Yale's Double Esterlon, though the latter rope seems quite a bit stretchier.
The chart below shows the data points published by Samson; the trend line is Excel's best-fit straight line. If you ask Excel to fit a second-degree polynomial (a parabolic curve) instead, it looks almost exactly like the straight line--for much of the chart you can hardly tell there is more than one line. Why make life complicated? Choose the straight line.
The equation of the line is shown. You can actually figure a lot of things out using the chart directly, but it is easier to do the calculations numerically. Without the chart, though, it would be really really hard to understand this stuff.
The number I want to find is energy absorbed per foot of rope at the working load limit (WLL). Working load is a more or less arbitrary fraction of breaking load; For illustration purposes I chose 25%, which is the number used by Yale for Double Esterlon. If you choose a different number, obviously you will get a different value for absorbed energy.
The calculation is actually quite easy. The published break strength of Stable Braid is 10,400 lbs. Then the WLL is 1/4 of that, or 2600 lbs. The arrow on the chart points to the fake data point I installed at 2600 lbs. From the equation, or just by reading the chart, 2600 lbs. corresponds to a stretch of 2.25%. In a 100-ft piece of rope, that would be 2.25 feet. Just imagine the units along the X axis are feet instead of per cent; in this case the chart shows the stretch characteristic of 100 feet of 1/2 inch rope. The energy needed to stretch the rope 2.25 feet when the force increases linearly from 0 to 2600 lbs is 2927 ft. lbs. This is just the shaded area of the chart. The shaded area is the area of the triangle, which is given by the height times the base divided by two. If 100 feet of rope absorbs 2927 ft. lbs. of energy at the WLL, then 1 foot absorbs 1/100 as much, or 29 ft. lbs. This is the magic number I want to remember. (If you use Samson's recommendation of 20% break strength for WLL, the magic number is about 18.5 ft. lbs. per foot of rope.)
What is so cool about this number is that it allows you to solve fairly tough dynamic loading problems, the ones where you drop a load and the rope alone stops it, without any need for F = M x A, or velocity, or mass, or the gravitational constant. The three things needed for the calculation can be estimated by eye: how heavy is the load, how far will it drop before the rope catches it, and how much rope will do the catching.
The energy the rope must absorb is just the product of the load weight times the distance it drops, i.e., E = F x D. The length of rope doing the catching is the length of the rope from the load to the anchor. (If the rope passes over a pulley, we still measure all the way to the anchor. If there is a friction device in the line, or the load is allowed to run, then this calculation will not apply.) Now divide the energy by the rope length to get energy per foot. If it is less than 29, you have not exceeded the WLL; if it is more than 29 you have.
It is shockingly easy to exceed the WLL. Tie some Stable Braid to a low limb. A couple of feet below the limb, tie on a 30-lb barbell. Lift the weight up to the limb and drop it. You have just applied more than 2600 lbs of force to the rope and stressed it beyond the WLL.
The math is really easy. You are about to drop 100 lbs 5 feet. But the rope from the anchor through the block and down to the load will be 25 feet long. The 500 ft lbs of energy will be divided by 25 ft of rope, or 20 ft lbs per foot. This is well below the WLL at 29 ft lbs per foot of rope.
Stable Braid--a real event
This is the embarrassing report of the first piece I ever dropped on a rope, and it shows the folly of not knowing what you are doing. It was an experiment inasmuch as it wasn't necessary at all, but I wanted to see what it would be like, and it seemed like a good piece to practice on.
The piece was a 6-foot long vertical piece of a recently dead elm, and I guess it weighed about 60 lbs, more or less. The rope (1/2 in. Stable Braid) was hitched to the wood about 1 foot above the cut. A block was fastened about 6 in. below the cut. I figure the center of mass of the spar was about 3 feet above the cut as it was a bit bottom heavy. By the time the rope caught the falling wood, the center of mass had fallen about 7 feet. This amounts to 420 ft. lbs of energy the rope must absorb.
The length of rope catching the load was the 1.5 feet from wood to block, and roughly another 4 feet to where the rope took half a wrap around a limb, or 6.5 feet in all. Using the Samson recommended 20% maximum load for the WLL, we have, from the previous post, a figure of 18.5 ft. lbs. of energy absorbed per foot of rope, or 120.25. So the load actually delivered 420 ft. lbs. of energy, but only 120 was permitted. Oops. Note this does not mean the maximum force experienced by the rope was nearly 4 times the WLL. It would actually have been the square root of 420/120 times WLL, or a bit less than twice the WLL.
This was bad and I won't do it again. What about the end of the rope, you ask? I had a ground person holding the end who had never done this before, but had been instructed to let it run a bit. Assuming it did run at least a few inches, and accounting for a little slack in the sling holding the block, and some slack and slip where the rope was hitched to the falling wood, it seems likely that the rope got a significant amount of help from all these other elements and actually experienced somewhat less maximum force than the worst case I calculated. Nevertheless, a very simple calculation would have told me this was going to overload the rope.
Since the cut was about 25 feet above the ground, the same calculation would have shown me that the exact same setup described above, but with the rope anchored to the base of the tree, would have given me about 27 feet of rope to stretch. At 18.5 ft. lbs. per foot, this could have absorbed about 500 ft. lbs. of energy, safely within the WLL.
This is the embarrassing report of the first piece I ever dropped on a rope, and it shows the folly of not knowing what you are doing. It was an experiment inasmuch as it wasn't necessary at all, but I wanted to see what it would be like, and it seemed like a good piece to practice on.
The piece was a 6-foot long vertical piece of a recently dead elm, and I guess it weighed about 60 lbs, more or less. The rope (1/2 in. Stable Braid) was hitched to the wood about 1 foot above the cut. A block was fastened about 6 in. below the cut. I figure the center of mass of the spar was about 3 feet above the cut as it was a bit bottom heavy. By the time the rope caught the falling wood, the center of mass had fallen about 7 feet. This amounts to 420 ft. lbs of energy the rope must absorb.
The length of rope catching the load was the 1.5 feet from wood to block, and roughly another 4 feet to where the rope took half a wrap around a limb, or 6.5 feet in all. Using the Samson recommended 20% maximum load for the WLL, we have, from the previous post, a figure of 18.5 ft. lbs. of energy absorbed per foot of rope, or 120.25. So the load actually delivered 420 ft. lbs. of energy, but only 120 was permitted. Oops. Note this does not mean the maximum force experienced by the rope was nearly 4 times the WLL. It would actually have been the square root of 420/120 times WLL, or a bit less than twice the WLL.
This was bad and I won't do it again. What about the end of the rope, you ask? I had a ground person holding the end who had never done this before, but had been instructed to let it run a bit. Assuming it did run at least a few inches, and accounting for a little slack in the sling holding the block, and some slack and slip where the rope was hitched to the falling wood, it seems likely that the rope got a significant amount of help from all these other elements and actually experienced somewhat less maximum force than the worst case I calculated. Nevertheless, a very simple calculation would have told me this was going to overload the rope.
Since the cut was about 25 feet above the ground, the same calculation would have shown me that the exact same setup described above, but with the rope anchored to the base of the tree, would have given me about 27 feet of rope to stretch. At 18.5 ft. lbs. per foot, this could have absorbed about 500 ft. lbs. of energy, safely within the WLL.
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