If I'm climbing down a rope, and get near the bottom, how much would I have to be carrying to break the rope?
What is the strength of a standard 50 foot rope from the equipment list?
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So, given @Szega's very, very valid "5E is not a physic simulator" point but running with the suggestion of extrapolating from the DC 17 strength check anyway...
The 5E lifting and carrying rule is quite coarse:
You can push, drag, or lift a weight in pounds up to twice your carrying capacity (or 30 times your Strength score).
If we assume that a +7 to Strength will hit a DC 17 reliably, that means 24 Strength (only monsters can break ropes every time, it seems) ... and 30 × 17 gives 720 lbs.
The answer from @chaoticgeek looks at real-world ropes and gives 675lbs as the "safe load factor" — ¹⁄₁₂th the tensile strength. The number based on the DC is right in that ballpark, and if we use 12× ~700lbs as maximum load with stress, we get around 8500 lbs. So this all seems reasonable.
Or, one could argue that someone with Str 4 — a -3 bonus — can hit a DC 17 some of the time, and that the safe load should cover that, because that's what "safe" means. That gives a safe load for standard hemp or silk rope at only 68 pounds, with max weight at a little over 800 lbs. This one seems low.
Also, note that the uncommon magical item Rope of Climbing says:
This 60-foot length of silk rope weighs 3 pounds and can hold up to 3,000 pounds
But it doesn't have a given strength DC to break. Instead, it has AC 20, 20 hit points, and (slow) regeneration. So it's probably safe to just say that its limit doesn't necessarily correspond to that of regular rope.
However much your DM says.
DnD 5e is not a physics simulator and does not bother with that level of detail. The only guidance the PHB offers is that "it can be burst with a DC 17 Strength check" (p.153). If you are so inclined you could try to convert that DC to a weight limit, but the system is not built like that.
This page shows various pound (force), expressed as lbf, for a hemp rope of given diameter to break. If we assume that hemp rope in D&D is constructed in similar ways then according to the chart a 1" diameter rope has a minimum breaking strength of 8100lbf, and a safe load factor (1/12 the breaking strength) of 675lbf. And through some searches 1lbf = 1lb of weight.
So in general your 1" rope could stand up to 8100lbs before it would snap, and safely 675lbs to not stress beyond a safe load. The safe load would take into account any sort of large movements, swinging or falling and grabbing the rope, that could cause more force beyond the weight on it or general wear and tear.
Just use common sense: assuming you are playing in a kind of medieval technology world like Forgotten Realms, and this is not a special rope made of some rare material, hemp rope of about 8–12mm should do, which would yield a breaking strength of from 900 to 2385 pounds. The length of the rope actually has nothing to do with it (other than it adds to the weight)
Remember most climbing ropes break at the knot or at some edge. So depending an care and knowledge, it could break significantly sooner.
I also found a 1E reference that gives 1500 lbs as a figure, which fits with the real-world range above.
The best real-world equivalent is Manila (Jute) rope. It supplanted the use of hemp rope in ship rigging, and is the strongest natural-fiber rope easily available. It's much more durable than cotton or hemp, stronger, more flexible, resists salt water damage, weathers well, and gets more "tooth" (friction) over time and use, making knots grip stronger. It also floats on water.
Rope strength depends on thickness and material. A 50ft spool that weights ten pounds corresponds to a thickness of one inch. The break strength on a one-inch Manila rope is 8,100 lbs. Cotton weighs slightly less, but it's break strength at that weight/50ft is only 3,000 lbs.
I actually couldn't find the tensile strength of silk rope, because it's now most often used for... private things.
First let's look at how strong the game indicates that the rope is (PHB page 153):
Rope, whether made of hemp or silk, has 2 hit points and can be burst with a DC 17 Strength check.
How can we relate a DC17 strength check to a given amount of force?
A humanoid with a strength score of 24 can consistently lift 720lbs using the following rule that doesn't involve any die rolls (PHB page 176):
Push, Drag, or Lift. You can push, drag, or lift a weight in pounds up to twice your carrying capacity (or 30 times your Strength score).
A humanoid with a strength score of 24 can consistently reach a DC17 strength check using the rules for Passive Ability check (PHB page 175):
A passive check is a special kind of ability check that doesn't involve any die rolls.
Here's how to determine a character's total for a passive check:
10 + all modifiers that normally apply to the check
In this case, a Strength Score 24 humanoid would have a +7 strength modifier so their passive strength check is 17.
This infers that the force that will break the rope is 720lbs as a humanoid who can consistently lift 720 pounds must also be able to consistently apply 720 pounds of force to a rope to break it.
You cannot infer anything from the weight of the rope per foot as the rope could be of vastly different quality for the same weight.
But you can infer that this rope is very weak for its weight as hempen rope that break around 720 pounds of force applied to it should weight less than 2 pounds for a 50ft length of it.
Safe Load Factor has nothing to do with this. That is a reference to many considerations of modern safety standards and accounting for factors that this game does not account for and aren't relevant to the question asked which is simply the matter of a static suspended weight.
As Szega says there are no specific rules for this but the characters are assumed to be competent and climbers are never caught without a sufficiently strong rope so the same should hold true for adventurers.
I would have this as a non-issue personally.
If you as a GM want the rope to snap consider it rubbing on a sharp piece of rock rather than a weight check.