SRT Base Tied TIP Forces

I don't have any X rigging rings, but based on a perhaps 1" diameter for the rope to slide over I'm guesstimating a tension ratio of 1.6 to 1.8 range. So that means 100 lbs hand side for lowering a 160/180 lb log which means a 60 or 80 lb difference up at the rigging tip. 260/280 lbs total at the tip instead of theoretical 320/360 lbs at the tip. Theoretical because a pulley runs about 1.1 tension ratio or 145/164 lbs hand side for the same 160/180 lb log, not right on double - for lowering - for raising it makes the total more than double.

Bet that's the first time anyone's quantified the tip friction load change for a rigging ring.

Edit - Crow eating time. Dr. Kane measured the tension ratio on X Rings and did a writeup:
Nov 2019 http://digimag.tcia.org/publication/?m=54984&i=629325&p=34&pp=1

He measured the forces using the Donzelli 1999 method of hanging weights on both ends of the rope and then measuring how much more force started the rope sliding. He got 1.5:1 and 0.7:1 which inverts to 1.43:1 (he slid the rope in the opposite direction) so pretty repeatable. I believe 1/2" line on large rings.

He also measured during a log drop and got a ratio of 1.2, but I question the influence of dynamics i.e. is it during a net acceleration or a net deceleration etc which is very hard to pin down on an instantaneous basis even from high speed video, which I think wasn't done.

So punchline my estimate looks high at 1.6 to 1.8 but I don't have slippery hard anodise coating data to work from.
 
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The I'm bored Biner Double Whip Corollary

Sling on log with rope in biner on sling.

Was using double whip the other day and pondered what friction was at play as I lowered the pieces. A rope doing a 180 around a biner has an about 1.8 tension ratio, so the stationary overhead anchored side of the rope has 1.8 times the tension than in the rope you're spooling down to the biner. So (1.0 + 1.8) x spooling rope tension = log weight. Or spooling tension = log weight / 2.8. Rope wrench territory. So if you bare hand the rope, you feel log weight/ 2.8.

If you throw the moving rope 180 degrees u turn over a natural crotch and then hold it, the natural crotch does it's magic. Let's say it's not super rough at 2.5 tension ratio, but average at 2.0 tension ratio. Now the (log weight/ 2.8) you would have bare handed gets reduced to (log weight/ 2.8)/2.0 = log weight/ 5.6. This is BMS Belay spool 1 1/2 wrap or Morgan block territory. On a rough bark tree this would be log weight/ 7.0. And if you want more advantage throw in another 1/4 to 1/2 wrap natural redirect down where you are situated cutting the base of the log, where it's convenient to tie off the rope before/while you cut off the log (assume rigging tip above, not -ve rigging - don't shock load biners).

The eureka for me was how beneficial the seemingly poor choice of biner and sling was friction wise, for such a simple setup. And climber releasable lowered pieces from aloft until you run out of biners and slings.
 
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The I'm Still Up Pulley System Mechanical Advantage Corollary

Browsing some threads about n:1 pulley systems and recalled how once I tried to partially right a root failed cherry tree - some success, pruned it weird, had a 45 degree trunk on a prop and produced nice eating cherries that were super easy to pick - it was a bitch moving it despite a many to one pulley system. Future me lightbulb on - friction did me in all those years ago.

Take a 4 or 5:1 system with 4 rotating pulleys, 2 at each end where the rope does a 180 turn.
Now add: W47 big blue rollgliss kit pulley, bronze bushing, 1.97" groove dia x 0.58" wide groove x 0.19" deep groove, tension ratio 1.20 equivalent mu 0.06 - typical of such used pulley.

Each pass around the pulley ups the effort by 1.20:1. 1.20x1.20x1.20x1.20 = 2.07

So you set up 4:1 but get screwed by 2.07 from pulley friction and end up at 2:1 in reality. Don't even think about small pulleys. Maybe real good ball bearing pulleys will get you an effective 3:1.

DOH!!!!!!!

Edit - I found the Sheehan 2004 reference from the HSE chapter 8 - about pulley friction as above -
 
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Donzelli 1999 arborist block friction: (I can't get the link to copy, it automatically downloads the paper)

Donzelli was prompted into some analysis perhaps by:

"If the block were frictionless, and both parts of line were parallel and directed vertically, a balance of forces predicts the reaction force at the block to be twice the line pull measured at the base of the tree. In 3 trials, the reaction force exceeded twice the line pull by 7.5%, 10.5%, and 9.3%."

So I did some ciphering to convert this to a tension ratio:

logT > baseT because the friction through the block drops the basal/porty rope tension

TipT = (1.075 to 1.105)x 2xbaseT
TipT = baseT + logT they're both hanging at 180 degrees around the block

baseT + logT = (1.075 to 1.105)x 2xbaseT call it (1.1) x 2 x baseT i.e. pick a nominal value

logT = (1.1) x 2 x baseT - baseT = (1.1x2 - 1) x baseT

logT/baseT = (1.1x2 - 1) = 2.2 - 1 = 1.2 !!!!!!!!


Holy crap he got the same tension ratio as my W47 blue pulley. At least it's a vote for nominal consistency amongst typical pulleys. I'd take it to the bank as a rule of thumb. His impending motion hanging weights tests corroborated the about 20% difference in tensions except for one block that was probably better that had about 10% or a 1.10 ratio.
 
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The SRT TIP Redirect Corollary

There was another thread in which a chart of rope passing around a redirect resolved perfectly into equal vectors and a symmetric centered/bisected angle resultant force on on the stub rope holding the redirect. I Neil Degrasse Tyson'ed it due to the effects of friction. I was waiting for someone to apply the bollard equation to an SRT redirect and maybe come up with an angle chart. The difference here is I'm not doing a sling plus carabiner redirect, but instead a natural crotch redirect.

Simplest intuitive case, redirect is level with your tip. Your rope comes up, bends 90 degrees around the tip, goes horizontally out to the tip and bends 90 degrees again and goes down to you.
Tclimber/Thorizontal = exp(mu x 90 degrees)
90 degrees is pi/2 and mu for typical bark is 0.2 and the ratio calculates to 1.37

Thorizontal/Tbasaltie = exp(mu x 90 degrees)
90 degrees is pi/2 and mu for typical bark is 0.2 and the ratio calculates to 1.37
Looks familiar? There's an equal tension drop at each rope contact.


Now let's try a redirect angled up at 45 degrees from the TIP. Your rope comes up, bends 45 degrees around the tip, goes 45 degrees up angle out to the tip and bends 135 degrees again and goes down to you.
Tclimber/T45up = exp(mu x 135 degrees)
135 degrees is 3pi/4 and mu for typical bark is 0.2 and the ratio calculates to 1.601

T45up/Tbasaltie = exp(mu x 45 degrees)
45 degrees is pi/4 and mu for typical bark is 0.2 and the ratio calculates to 1.170
Here's where it gets interesting. 1.37x1.37 = 1.873859 1.601x1.170 = 1.8738
The ratio of climber weight to Basal tie tension is the same in both cases, with the intermediary rope tension being different.



Now let's try a redirect angled down at 45 degrees from the TIP. Your rope comes up, bends 135 degrees around the tip, goes 45 degrees down angled out to the tip and bends 45 degrees again and goes down to you.
Tclimber/T45down = exp(mu x 45 degrees)
45 degrees is pi/4 and mu for typical bark is 0.2 and the ratio calculates to 1.170

T45down/Tbasaltie = exp(mu x 135 degrees)
135 degrees is 3pi/4 and mu for typical bark is 0.2 and the ratio calculates to 1.601
Wait, we've seen these numbers before. 1.601x1.170 = 1.8738

Dollars to donuts you can pick any pair of angles like 80 degree+10 degrees etc and it works out.

The ratio of climber weight to Basal tie tension is the same in all 3 cases, with the intermediary rope tension being different. This is the corollary.


The astute mathematicians in the crowd may spot that the secret lies in the cascading of exponential equations. The curious student can ponder the effect of the redirect branch bending. The preceding math applies most simplistically to something like an oak tree.
 
The You can't Always Get What You Want, But If You Try Sometimes, You Get What You Need Mechanical Advantage Corollary

An earlier post worked through the losses of a typical aborist rope doing 180 degree U turns through typical pulleys in a mechanical advantage rope and pulleys system. The simple tension ratio gives the rope tension loss per bend but adding up the losses for each configuration lead to the question of whether there was a limit of gains by using yet more pulleys.

The addition of tensions is done with reduced rope tension each time, so the tension ratio of 1.2 is used inverted 1/1.2=0.83

1 + 0.83 + 0.83 squared + 0.83 cubed + 0.83 to the fourth power + ... is the list of tensions at each leg of rope. This is actually the proportion of the initial rope pull, e.g. 100 lbs, 83 lbs etc if you initially pulled with 100 lbs.

such a sum of ratio numbers in a series is called the sum of a geometric series with ratio r between its terms. There's a formula to add up n terms.

ok here goes. 1/ 1.2 = 0.83 the ratio r

sum of the series = term1 x (1-r to the n) / (1-r)

n = 6 pulleys term1 is hand pull on rope so
(1 - .83 to the 6th) / (1 - .83) = (1 - .327) / (.17) = .573 / .17 = 3.37

so the hand pull diminished by 6 pulleys but adding up all 6 attempted tension gains gives 3.37:1 actual mechanical advantage. Moral of the story - use good bigger pulleys on your fat rigging line!

try r = 1 / 1.1 = 0.91
(1 - .91 to the 6th) / (1 - .91) = (1 - .568) / (.09) = .432 / .09 = 4.8 significantly closer to 6:1

real rope on a real pulley unfortunately is the 1.2 tension ratio. doh! maybe smaller amsteel on a 4" pulley would give 1.1 tension ratio. For extra points do it with rope on biner friction. fail.

So the geometric sum formula predicts the limit as you add infinite pulleys. Just try the formula with a very big n value. (1-r to the n) becomes 1.0 with a big enough n exponent and makes the 1/(1-r) formula.

1/(1-r) gives 5.88:1 max advantage on 1.2 tension ratio rope/pulleys. 11.11:1 for 1.1 tension ratio rope/pulleys. That's even if you set up a 100:1 or more system!!! It's all because the rope tension diminishes each time it goes around yet another pulley and makes smaller and smaller contributions.

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The Shadowscape SRT base Tie Fall Arrest Shock Absorber Corollary

A Yates brand screamer is a folded up runner with the folds stitched together with thread intended to tear out and in doing so absorb fall energy. There are other brands on the market as well. They generally trigger (tear out) at 2 kN (560 lbs). In rock climbing they are used at the "tip" a piece of climbing hardware in the ice or rock and the climber's rope can either do a 180 degree bend if pure vertical ascent or a 90 degree bend if doing a traverse. The point of bringing that up is that the screamer sees the vector result of the "climber" side and the "basal" side ropes.

In SRT you could use a screamer either at the climber's saddle, or in series with the down rope to the basal tie. If at the climber the climber would experience the forcexstroke limit of 560 lbs directly.

The corollary is that if e.g. the tension ratio over average tree bark was 1.5, a climber would have to generate (experience) 1.5 x 560 lbs = 810 lbs on the climb line to create 560 lbs at the basal tie to trigger the screamer. The rougher the tree bark, the higher the tension ratio and the rougher the stop ride will be. That being said, in a rock climber fall test video 170 lb climbers were experiencing 2 kN casually and the threshold for bodily damage is substantially higher as per UIAA.

There is an opportunity for testing in that rope skidding was seen to grind off the roughness of bark and possibly high tension ratio bark would act with a lower effective tension ratio in the course of rope stretch/travel/skidding over the crotch. This has been partially assumed in using 1.5 in the example calculation.

A recommended possible safety addition to a basal SRT anchor is a screamer in series. Or similar absorption device perhaps paralleled onto the down rope with a prussic, leaving appropriate travel/slack in the intact basal tie rope.

The current challenge is the specific parameters of the absorption characteristic to balance G loading vs stopping distance in the face of variable freefall distance of the climber. This application is for redundant catch tip SRT configurations i.e. if you break out your tip your rope lands on a lower tip catching you before you greet Mother Earth.
 
I will add this to your program. Whether I am operating DRT or SRT, I never run my climb line over a crotch. Never. I run an anchor rope over the crotch with which I pull up my climb line, stopping just before the crotch is reached. I do this for a number of reasons. 1) my climb lines do not get the abrasion of being dragged over a barky crotch. 2) If it is a pine or spruce my climb lines don't get dragged through pitch. 3) Because I run mostly 200' ropes, whether it is my climb line or my anchor line, there is always enough anchor rope available to lower me down. 4) Once I am up there, I can unhook my climb line and reposition it or advance if I choose to do so, or I can use it as a cinch while working my way down while blocking down the stem.
That said, the logical placement of a screamer is on the climb side of the crotch between the anchor rope and the climb line system, whether it is SRT or DRT. In the DRT it would be between the anchor line and the DRT pulley. In SRT between the two ropes.
The Yates shorty screamer is fairly compact and doesn't take up that much space. About 7". And it would not need to be dragged over the crotch, just pulled up to it.

I had mine on the base anchor system, between a Petzl Rig and a 5/8" Tenex Tec ultra sling around the stem. Not the best placement but was made after the climb line was in place, and I was too lazy to bring it back down and do it properly. But something is better than nothing, as they say.

A side note: I tie a slip knot in the anchor line after the Rig in case the Rig were to slip, and a carabiner through the knot to the ultra sling. The Rig is only rated for max 11.5mm ropes, and most of mine are 11.7/11.8 variety. I expected to find that the Rig had slipped after my fall and came to rest on the knot, but I was surprised to find it had not slipped at all. That was one of the good things that came out of this incident. Let's me have a little more confidence in that piece of equipment for the future.

Update on condition: Been sleeping in the recliner. A lot easier than trying to get myself horizontal on the bed. Typing is not so hot with a good portion of my left hand bandaged up, but doable. Jaw seems to be fine. Right shoulder has some torn stuff that will probably be the last thing to heal up, if it does. Oh, and I broke a DMM PerfectO carabiner that was just hanging on my harness, not being used. Must have gotten between me and the tree. That might also explain the black and blue spot on my hip.
 
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Climbers/workers can control cutting climb ropes and/or lanyards, but have no control over freak failures of a tip. SRT tip distribution and redundancy has been discussed in other threads. I'm trying to keep this thread technical as it's been an eye opening journey for me. First post even temporarily threw me for a loop till I analyzed the graphs carefully.

edit - Climbers/workers can control cutting climb ropes, lanyards, bridges, people or trees that when cut will cause damage/injury, but have no control over freak failures of a tip. More inclusive list.
 
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DSMc said:
But an obligate doubling of exposure, is no more true than is the doubling of load at the tip.
Click to expand...
If a climber has a canopy anchor at 50' they are able to cut their rope anywhere along that 50'. If they use the same union but anchor the rope basally, they now have 100' of rope that could be cut. Now, not only can the climber sever their own line, but a ground worker has access to it as well. Twice as much rope. Twice as many people.
 
If you are interested in this subject, consider starting a new thread on it, saving the original purpose and content of this one.
 
Shadowscape said:
I will add this to your program. Whether I am operating DRT or SRT, I never run my climb line over a crotch. Never. I run an anchor rope over the crotch with which I pull up my climb line, stopping just before the crotch is reached. I do this for a number of reasons. 1) my climb lines do not get the abrasion of being dragged over a barky crotch. 2) If it is a pine or spruce my climb lines don't get dragged through pitch. 3) Because I run mostly 200' ropes, whether it is my climb line or my anchor line, there is always enough anchor rope available to lower me down. 4) Once I am up there, I can unhook my climb line and reposition it or advance if I choose to do so, or I can use it as a cinch while working my way down while blocking down the stem.
That said, the logical placement of a screamer is on the climb side of the crotch between the anchor rope and the climb line system, whether it is SRT or DRT. In the DRT it would be between the anchor line and the DRT pulley. In SRT between the two ropes.
The Yates shorty screamer is fairly compact and doesn't take up that much space. About 7". And it would not need to be dragged over the crotch, just pulled up to it.

I had mine on the base anchor system, between a Petzl Rig and a 5/8" Tenex Tec ultra sling around the stem. Not the best placement but was made after the climb line was in place, and I was too lazy to bring it back down and do it properly. But something is better than nothing, as they say.

A side note: I tie a slip knot in the anchor line after the Rig in case the Rig were to slip, and a carabiner through the knot to the ultra sling. The Rig is only rated for max 11.5mm ropes, and most of mine are 11.7/11.8 variety. I expected to find that the Rig had slipped after my fall and came to rest on the knot, but I was surprised to find it had not slipped at all. That was one of the good things that came out of this incident. Let's me have a little more confidence in that piece of equipment for the future.

Update on condition: Been sleeping in the recliner. A lot easier than trying to get myself horizontal on the bed. Typing is not so hot with a good portion of my left hand bandaged up, but doable. Jaw seems to be fine. Right shoulder has some torn stuff that will probably be the last thing to heal up, if it does. Oh, and I broke a DMM PerfectO carabiner that was just hanging on my harness, not being used. Must have gotten between me and the tree. That might also explain the black and blue spot on my hip.
Click to expand...
Heal up quick buddy, and I am sending you heartfelt thoughts and prayers over the loss of your beloved DMM biner..
 
Shadowscape, despite the circumstances your experience brought to light something that could help other people and contributed to the knowledge of the topic. Consider it making the tech personally relatable. Be well;)
 
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