I'm thinking of a single rope that it's safe to take long lead falls, but also light to carry around. I weight 75 kg and I might climb with heavier partners (80-90 kg)
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Failure by cutting is a primary concern
In terms of safety (rather than e.g. rope life) laboratory (UIAA) fall testing may not be the most important concern. Even a new, thick rope can be cut in a single fall across a sharp edge. This also applies to so-called "edge resistant" (defunct UIAA 108 standard) ropes. See this Yellow Spur fatality report.
Generally a thicker rope is more cut resistant than a thin one of the same construction, though of course construction is very important. Here is a test report primarily of static rope, but also an unnamed 11mm dynamic rope, 8mm cord, and 1" tubular webbing: Qualifying a Rescue Rope by Tom Moyer. (Of note is that the 1" webbing is downright dangerous in its poor abrasion resistance, contrary to what has been stated elsewhere.)
Double or Twin rope technique is believed to reduce the likelihood of catastrophic failure by cutting.
Rope design is a compromise between fall and abrasion durability
Using the manufacturer's stated sheath proportion for the Genesis and Serenity ropes described above the estimated cross-sections look like this (1:1 scale on a 100ppi display):
Climber's size is very important
From Jean-Franck Charlet UIAA SAFCOM President as quoted by Francis Baker on SuperTopo.com:
From the empirically determined formula we see that for survivability of the rope (and climber) in a fall weight matters in the extreme; a fourth degree exponential relationship. Of course not every rope can be expected to rigidly follow this exact relationship but it was determined to be reliable enough to be the basis for a very important safety standard. (See addendum.)
Applying the formula: a rope that will survive five 80kg UIAA drop tests, the minimum single-rope standard, will probably only survive a single 105kg UIAA-type drop (
This plot includes the minimum spec for each UIAA standard, as well two more robust samples each for the Single and Half types. While the UIAA drop test includes a margin of safety via a rigid drop mass that exerts considerably greater impact force than a human body, old ropes lose a considerable portion of their strength.
It should be apparent that rope size should be matched to climber size. This is not usually "the way it's done" but anything else goes against both empirical data and simple engineering principles. Observe that the calculated margin of safety for a 100 kilogram climber on a burly 11mm rope with a 15-fall rating is about the same as that for a 60 kilogram climber leading on a single strand of mid-range (9 fall) "Half" rope. This should not actually be a surprise but it probably comes as one to many anyway. The strength of a given construction of rope is proportional to its cross-sectional area, and the area (or linear mass) of an 8.5mm Half rope is 60% that of an 11mm rope.
Skinny single ropes which manage to pass or exceed the UIAA fall rating do so at the expense of abrasion and likely cut resistance, so "there is no free lunch" in this regard.
Of course there are not reports of above-80kg-climbers routinely falling to their death as a result of rope breakage. Surely this is partly because the UIAA single rope standard is practically quite high, but it is probably also because ropes are rarely tested with extreme falls. For the paper Hang 'Em High: How Far Can You Trust Your Belay Device? the authors conducted a survey of climber's weights and their biggest lifetime falls (results on pages 8 and 10). According to it falls approaching factor 2 are rare, as are climbers above about 100kg. As ropes continue to be manufactured thinner and thinner, if you or your partner is on the heavy side do you really want to be the test case?
On the other end of the scale lightweight climbers are arguably climbing on greatly over-specified ropes much of the time. Since Half ropes pass the same drop test as Single ropes, but with a 55kg rather than 80kg mass, such ropes appear to provide as much safety for a 55kg climber (including gear) as does a Single rope for an 80kg climber. As stated at the beginning of this post fall rating is not the only concern, but it is unlikely that a Half rope is cut more easily by a 55kg climber than a Single is by an 80kg climber as the PSI (weight over area) is the same. Further, as described by the Mammut representative (above) a robust Half rope like the Genesis has a much more burly sheath than a skinny single like the Serenity.
Further examination of the relationship between mass and drops held
In the comments DavidR questioned the formula used to approximate the relationship between test mass and drops held. Therefore I went searching for empirical data. Unfortunately I do not have access to raw test data so I compiled a table of manufacturer ratings for every multi-certified dynamic rope that I could find. Some ropes or data points were omitted because they were rated e.g. "20+" rather than a specific value. Please be aware that to the best of my knowledge manufacturers are under no legal obligation to provide accurate drop test numbers; they are only required to meet the minimum standard, and the UIAA only provides a "PASS" rating to the public. Here are the results:
On the left is the name of the rope; the next two numbers are the drop mass (80: Single, 55: Half, 40: Twin), and the number on the right is the derived exponent required match the drop test numbers. Triple-certified ropes have three pairings on the right.
Two things should be quickly apparent: there is wide variation in the figures, and none of them are as high as four. Some of the variation may come from the manufacturers rounding figures e.g. a specification of "Falls: 6 Half, 20 Twin" instead of "21 Twin" etc., but I suspect a lot of the variation is real as well. Since none of the derived exponents exceed four, I conjecture that 4 in the formula given by Jean-Franck Charlet was arrived at as an upper bound; indeed this makes sense given the way it is used in the safety standard. As a histogram the data looks like this:
The mean is 2.97 with a SD of 0.55.
Using an exponent of three the simulated drops chart looks like this:
Note that on this chart the UIAA Twin and Half standards are very close. While one cannot assume that a Twin could pass the Half test, given the large extreme spread if nothing else, there is evidence that the two standards are in fact quite close: the skinny double-certified ropes themselves:
What Rory Alsop posted is great info that is "internet safe" and a great rule of thumb. Many companies, however, will rate their ropes down to 9mm as single rope safe. In areas where rope protectors are often used, climbers safely go down to 8mm.
I regularly climb with a Mammut Infinity 9.5mm. Compare that to Petzl Xion 10.1mm
Now with good reason the 60m length Mammut costs about 40 USD more, but you get the picture. For 20 USD more than the Mammut Infinity you can get the Mammut Serenity 8.9mm which has similar stats with even less weight (it brags to be the "lightest single rope in the world") and diameter. As a rule of thumb I would stick to 10mm and above but there are definitely exceptions, and anything rated by the manufacturer as a single rope should be fine.
The abc of rock climbing website has a useful article when choosing ropes.
I have a 10mm 70m rope and a 10.5mm 50m rope - the longer one is heavier, but the slightly thinner diameter means it isn't too unwieldy.