How do UIAA falls add up over time?/When to replace a rope

Question

How do lead climbing falls add up over time based on their fall ratings?

For example, the rope I have is rated to 8 UIAA falls. A UIAA fall is a fall factor of 1.77. Lets say over the course of a year I take 28 falls of a factor or 0.5, that adds up to about 8 UIAA falls. Does this mean the rope is close to breaking? (28Falls * 0.5Factor = 8Falls * 1.77Factor)

Basically is this a linear relationship. Will the ropes always break around the same number of accumulated fall factors? (Num_FallsFall_Factor = Num_FallsFall_Factor)

My gut says that it probably isn't linear, but I would also be interested to know if any research has been done on this.

Answer 1

I am sure there is no simple formula to add everything up as the load of a high factor fall is not increasing linearly. There have been experiments conducted by Pit Schubert in which he tested a rope with repeated falls on factor 0.35-0.45 on a rope rated to 5 falls with an 80 kilo mass. (Basically the norm just with a low fall factor)

He reports that the rope lasted for more than 2 days of fall testing and broke after 220 falls. However, it was already stiff to unusability after 80 falls.
(Source: Pit Schubert, Sicherheit und Risiko in Fels und Eis, page 64)

What does this mean?

1. The fall factor is relatively high for sports climbing falls. A factor of 0.4 would be a 4m fall from 10m height into a quick-draw at 8m. Your typical fall (especially at the gym) will likely be much smaller.
2. The rope was fixed which means the fall was held completely static. In practice (belaying off the harness) there is always a dynamic catch, even with static devices like the GriGri. Loads in practice will be even lower.
3. With the rope being fixed, there is always the same part of the rope in the carabiner that holds the fall. In practice, this would be distributed over a bigger part of the rope.
4. As in all testing, this was performed using a metal weight to simulate the climber. A human climber's body will absorb some of the impact, therefore lowering the impact on the rope.

Conclusion
In practice, you will get a lot more than the 220 falls measured. Therefore, it should be impossible to break a rope simply by falling onto it, no matter how often you do. Check your equipment before climbing on it and if in doubt retire.
Retire the rope when it gets too stiff for comfortable handling, or if it reaches the maximum life span given by the manufacturer.

Answer 2

Manziel's answer is very good. A detail that was left out, however, is that ropes are just very big elastic bands. Instead of relying on polymer chains expanding, they work by deforming twisted nylon strands. These strands, after a fall, are elongated according to fall factor: the higher the factor, the more they lose their twist. This twist is partially or completely recovered after rest time, meaning that your rope is usually shorter the day after you used it.

A large set of falls is not an indicative of real life. No one falls 220 times in a day. I've seen people falling 15 times very near a quickdraw, a factor of maybe 0.1. I've never seen anyone taking a 0.5 factor fall and coming back 15 times. This doesn't happen.

In resume, add another item to the list why you shouldn't worry: ropes rest when you're not climbing and recover a large fraction of their elasticity. This renders them basically infinite. A UIAA fall is so dangerous just because it breaks this rule: even after rest, we expect some "scars" in the rope after such high-impact falls - which can accumulate and render the rope more static and very weak.