Ah I have clearly misled you. When I said tightly strung, I did not mean so tight as to be in the range you envisage (but I will come back to this). So let?s start with a description of the situation which I had in mind. Basically you have a caver wearing a set of Cows Tails which roughly are waist level on a ledge. Now let there be say four anchors, also at waist height, the first two of which are linked by the SRT rope for protection on the lead out to the last two from which the pitch is rigged in a Y hang. I inappropriately used the phrase ?tightly strung? for the rope strung between the first two anchors, but what I wanted to convey is that it is not so loose as to have a large loop between the two anchors.
First the peak loading comes about because the need to arrest a fall of a certain height by absorbing the kinetic energy created from the release of the potential energy equivalent to the distance of the fall. As I said previously, what the French report makes clear, is the concept of Fall Factor has to be closely analysed when you are dealing with short lengths of rope since the knot and crabs start to dominate matters. A pitch head situation is quite complex. So let us get into gritty detail.
Case A - If the caver has clipped their Cows Tails into an anchor and then falls, then the height of the fall before it is started to be arrested is the length of the Cows Tails plus crabs etc, call it CT. And the shock loading from the fall would only be absorbed by the rope in the Cows Tails (I will ignore the body and other smaller contributors), call it R.
Case B - If a caver clipped onto the rope between the first two anchors, then the height of the fall is the small distance equal to the distance the SRT rope hangs below the anchors (call it x) plus the length of Cows Tails before the rope and Cows Tails start to arrest the fall. But now the shock absorption would be undertaken by not only the Cows Tails but also the length of rope strung between the first two anchors. This length of rope is not the distance x but something much larger.
Case C - If a caver clipped onto the rope between the third and forth anchors (ie the rope Y hang for the pitch), then height of the fall is equal to the distance the drop of the Y hang (call it y) plus the length of Cows Tails before the rope and Cows Tails start to arrest the fall. I hope it is clear that in this case y is much larger than x. Again the shock absorption would be undertaken by the Cows Tails but also the length of rope involved in this Y hang (plus the extra knot).
Case D - And although this is not realistic, consider the caver clips onto the rope below the Y hang and then falls (call it z) but is arrested by a knot some located above the floor. In this case the shock loading is taken by the long length of rope z plus the very small bit by comparison in the Cows Tails.
The point I was trying to make is that in Case A, the energy to be absorbed is that of a fall of CT and that absorption of energy is undertaken by R, the length of rope in the Cows Tails. Where as in Case B, the fall is CR plus x; ie a bit more. However in this case, the absorption of energy is undertaken by the length of rope in the Cows Tails plus the length of the SRT rope strung between the first two anchors. What I am embarrassed to admit is that I overlooked the point you made about increasing loads with angles. However in practice I would suggest ropes are not so tightly strung that one gets up into the range where there is a potential for breaking the rope. I made the stupid assumption that one would be gaining by the extension of rope length (which is strung between the two anchors) in absorbing the energy of the fall without taking into account the increase in load. Thinking a bit further, I predict there is a cross over between these two competing components, but I do not have the time to work out where it occurs.
In Case C, the fall is Y plus y; ie much more than in Case B, much depending upon how much of Y hang one puts in the rope. The absorption of energy in this case is undertaken by the length of rope in the Cows Tails plus the length of rope in the Y hang. So there is more energy to be absorbed and a little more SRT rope plus Cows Tail rope to do it in. In Case D, one has a fall of z and energy absorption by the length z of SRT rope plus Cows Tails.
My claim is that this means the peak loading the caver is subjected to, varies according to the height fallen and the amount of SRT rope and Cows Tails rope involved. My guess was that Case B would give the least peak force, with Case A being larger and Case C being larger still. I predict Case D will be yet larger (because there is only one rope involved below the Y hang) but it will flatten out when the length of rope becomes sufficient to make the contribution of the rope in the Cows Tails and the rope in the Y hang insignificant so that the concept of Fall Factor comes into play. I now have to amend that claim since as you point out, for very tightly strung rope between the first two anchors, the Case B situation will magnify the loads and thus lead to rope break which whilst may be reducing peak force seen from the action of the rope, will unfortunately lead to another peak force seen when the caver hits the floor below.
I know this is highly theoretical and I have probably lost many readers a long time ago. But I hope I have got the point across that predicting peak forces is not that simple in real world situations and in some cases will come out to your benefit rather than to your detriment.
Apologies for the delay in posting this, but I had other things I had to do. I will now retire in embarrassment for forgetting the point about tightly strung ropes loadings. :-[
