Friction in the rope system can increase fall forces significantly when you are further out on the pitch, right?

Fundamentally, it's not a question of where you are on the pitch, but rather, what the "nominal" fall factor is. What friction does is to increase the fall factor, producing an "effective" fall factor. But the effective fall factor still can't exceed the maximum theoretical value of two.¹ In the UIAA test, the fall factor (1.78) is close to 2, so there's not much "room" for friction to have an effect.

However, if the nominal fall factor is lower—as it normally would be further from the belay, with more running pro placed—then friction can cause a larger effect. In principle, with enough friction, you could turn a harmless 0.15-factor fall into a horrendous fall approaching fall factor 2. But, qualitatively, the results are the same, regardless of the nominal fall factor: friction will increase the force on the climber, and decrease it on the belayer and the top anchor.

The methodology for these calculations is explained here.

However the thorny problem still arises in that one is assuming that the rope is moving from the T2 to the T1 side at a constant rate (or a least with a constant decceleration) but this is not what occurs. The frictional constants used in most papers are a simple fudge factor which gives reasonable results on a larger time scale but when we look at higher speed traces of the forces it is apparent that friction and the T1/T2 issue are not following the rules.
Depending on the fall rate and the way the rope is braked it is clear that at some times the rope will actually go backwards through the karabiner which is going to make a unified theory of everything a bit difficult!
It´s also apparent that µ (if we extract it from T2/T1) varies between 0.35 and 0 with negative values when T1 becomes greater than T2 (and in some applications we are getting up to 0.6 but these are not concerned with falls).

For most practical purposes jt´s model gives a good match to an experimental force curve when it has been considerably smoothed out but the short term peaks are massively higher than one predicts (and the corresponding troughs naturally). The damping of the rope smooths a lot of these out but you would still be suprised to see some of the forces that are appearing, though they don´t seem to be breaking things so perhaps the make it unbreakable concept is still the way to go for gear designers!

Thanks for posting that article.

Attaway seems to make an incorrect assumption, but maybe it is correct in a way that is too subtle for me. I'll use your notation to describe it.

If we say that T2=k*y2/L2, we assume that the length L2 of rope, once stretched, is under tension T2. However, the distance between the belay end of the rope and the top anchor is fixed (L2), and whatever rope moves to the climber's side of the biner is actually under tension T1.

The way I've seen similar problems modeled is to use the length of unstretched rope that moves to the climber's side of the biner as basic variable--let's call it s2 and then say that T2=k*s2/(L2-s2). That is, a rope of length L2-s2 stretches until its length is L2. On the other side, we have now L1+s2 unstretched rope, which stretches according to T1=k*y/(L1+s2), where y is the total amount by which the entire rope lengthens.

y is the total amount by which the entire rope lengthens.

19/04/2010
Thought I'd share a recent experience of a "Factor 2" fall with the hope that anyone interested may learn from it:

My gf and I were on "Lamplighter" at araps. It was her lead on the second pitch – her first on a multi pitch route, having not led much on trad before. As she left the belay I encouraged her to put in a piece early, or clip the top piece of the anchor on the way past. I was a bit surprised when she charged up a couple of metres without slotting anything but wasn’t concerned because she had been completely solid on lead for the whole trip and it looked like there were good slots for gear just above. It was a bit of an awkward stance – slabby, but without a customary grade 14 bucket to hang off – and after trying to place a nut or two, it was clear that those slots weren’t so good after all. She had skipped the fixed pin with my encouragement (“not really trustworthy”).

At this point she started getting a little flustered and poked in a small cam (a yellow WC Zero). It wasn’t aligned with the direction of the fall so I got her to move it, but apparently that made the cam lobes go crooked and offset and it looked dodgey so she moved it back. After all this frigging around she was getting pumped and stressed and suddenly the delicate moves in an increasingly airy position got a bit hard. She tried the next move but half fell, half retreated to rest on the cam. The cam ripped out and she went for the big plummet. Thankfully she sailed right out over the belay ledge, over the slab below and the ropes pulled her up on the steeper section below that.

She came up pretty much without a scratch, very fortunate given that she’d just taken a factor two fall which probably ended up being greater than six metres from top to bottom. It was by far the nastiest fall I’ve seen. She heroically got back up, and after a brief pause to check all limbs were intact, she charged on and led the pitch (one of the best pitches ever, though some of the gloss was probably taken off for both of us). This time she headed off with a bomber nut placed just above the anchors and also clipping the fixed pin above that.

Unfortunately I didn’t come off quite so well. I was sitting on the edge of the big comfy belay ledge, and as she sailed past the ropes ran over my leg. As the ropes came up tight, my leg got squished between the ropes and lip of the ledge. At first I thought my leg just had a deep, juicy bruise but it turns out that I had torn my hamstring as well. My guide hand (on the climber’s side of the belay device) got smashed into the rock. Just as predicted in the manuals, some rope slipped through the device and gave me some minor rope burn. I may also have damaged my sciatic nerve, possibly from the impact being transferred through my harness and into my lower back. My physio and I are still working out the exact nature of the damage to my leg and back, but whatever, it’s bloody painful and I may be out of action for a while.

Interestingly, it seemed that the anchor had taken very little of the impact. All the pieces popped out easily (two nuts and a hex) when I cleaned the anchor. The knot in my cordelette may have tightened up, but only a fraction. I was clove hitched into the ‘power point’ of the anchor with both strands of our double ropes and there wasn’t any slack. It appeared that most of the energy from the fall was transferred elsewhere in the system, a lot of it into my body – possibly an example of just how effective the old-school method of hip belaying from a good stance could be?

Anyway, the message is: Don’t be a muppet! Learn from our mistakes and remember...

- Factor two falls are bad for your health and bad for your belayer too.

- On multi pitch, place that gear as soon as possible off the belay, even if it is while you are standing right there next to your belayer on that cosy ledge. If the anchor is bomber (we could have towed away the Pharos with ours), consider clipping one of the pieces.

- Small cams need to be placed with care and with and good understanding of their limitations if you are to trust them. They may not be the best tool for beginner trad leaders.

- If climbing with an inexperienced partner make sure they understand the concept of fall factors before setting off on that multi pitch classic, even if you expect it to be a cruise.

- Learn all you can about building good anchors, and make sure they are up to taking a factor 2. You never know when you might need it so save the bacon of you and your climbing buddy.