D&D 3E/3.5 A ropey issue

[Cleon]

While reading through the description of the Pathfinder magic item robe of infinite twine yesterday when my random thoughts led me to contemplate the humble hempen rope and a thought occurred to me.

How much weight can a standard hempen rope support? It'd be nice to have some idea how many adventurers can dangle from one before it runs the risk of snapping! For that matter, how thick is the rope?

Since there was nothing better to do this weekend I spent a pleasant few hours thinking about the question and rummaging through the internet on data about rope strength - which proved harder than I thought, since rope made out of hemp or other natural fibres is little used these days.

Anyhow, you might as well enjoy the results of this "research".

Basic Stats
Rope, Hempen, 50 ft. (3/3.5/Pathfinder): This rope has 2 hit points and can be burst with a DC 23 Strength check. Price 1 gp, Weight 10 lb.

First Approach - Break DCs vs Strength/Carrying Capacity
A Break DC of 23 means a character must have a Strength bonus of +3 or greater to snap the rope, requiring a Strength of 16 or higher. The Carrying Capacity rules say a character can lift twice their maximum load off the ground and stagger about with it, ergo they are applying a force equal to that weight. A Str 16 gives a max load of 230 lbs so their "max lift" is 460 lbs.

That seems too low, being roughly the weight of two medium-sized humanoid adventurers and their equipment. However, 460 pounds is the extreme edge case. The character is doing their utmost to break the rope and was extremely lucky (rolling a natural 20) to stretch the rope just right so they hit or create a weak spot on the rope causing it to fail. In normal usage that wouldn't be the case.

Instead let's consider a character who can "take 10" on a Strength check but will just fails to hit the rope's Break DC of 23. This requires a +12 Strength bonus so they would have have a Strength of 34 or 35, meaning their maximum load is 2800 or 3200 pounds and their maximum lift is 5,600 or 6,400 pounds.

Averaging that out suggests the standard hempen rope has a breaking load around 6,000 pounds.

That doesn't mean 5,000 pound Cloud Giants can go freely bungee-jumping with standard hemp rope however. That's the maximum load at which the rope is sure to break. The safe load of a rope is but a fraction of this, ranging from one-quarter to one-fifteenth of the breaking load depending on the likely use. For "your life depends on it" tasks good engineering practice recommends a factor around twelve, for a safe load of 500 lbs, especially if the rope is regularly exposed to strain and expected to last a while. However for the risk-casual adventurers a factor of six should be enough, for a safe load of 1,000 lbs beyond which there's a risk of the rope breaking.

Incidentally, merely tying a knot in a rope will reduce its breaking load by 50% or so, since knots concentrate stresses in the rope and make it more likely to break. So a 6,000 lb. breaking load rope will fairly definitely snap if you tie it round a 3,000 lb. rhinoceros and try to hoist it in the air. Going by some of the rope-related websites I read together the loss in breaking strength can be as little as 35% with the right knots, suggesting another task for a Use Rope skill check…

Second Approach - Real World Comparisons
Okay, we know 50 feet of this hempen rope weighs 10 pounds, therefore the rope weighs 0.2 pounds per foot (5 feet per pound) or, if you prefer S.I. units that comes to about 298 grammes/metre.

It took me way longer than I expected, but I eventually found enough sources on the weights of ropes made from actual hemp (Cannabis sativa) to determine that's the weight of a hemp rope 20mm in diameter (i.e. Buy Rope & Hemp Shop Rope), or 13/16th of an inch.

That Buy Rope datasheet gives an estimated breaking load of 2650 kg (5837 lb.) but the Hemp Shop Rope gives 1856 kg (4092 lb.). However the Hemp Shop's rope is 253g/m so is 5/6th the weight of the SRD hempen rope which therefore should be roughly 15-20% stronger, or 2183 kg (4816 lb.), which still is oddly low compared to the other figures I found which were all in the high-five-thousands to mid-six-thousand pound range. For example, this Smackdock Rope webpage (which turned out to be a good general reference source) lists 20 mm Italian hemp rope as having a 2,900 kg breaking load [6388 lb].

Overall, 6,000 pounds for the breaking load seems an acceptable compromise, especially as it matches the result of the first approach. What a remarkable coincidence!

For other "hempen" fibres, manila ropes made of abacá (Musa testilus) appear to be almost as strong as hemp, while sisal ropes made of sisal (Agave sisalana) are significantly weaker than hemp and manila.

Conclusion
The SRD standard hempen rope is a 13/16th inch diameter hemp rope that can support 1,000 pounds safely, or up to 3,000 lbs with an increasing risk of it breaking.

Note
Since this post is based on an evening of idle internet browsing through the internet, don't use the rope strength for real life applications!

 

[aco175]

The SRD standard hempen rope is a 13/16th inch diameter hemp rope that can support 1,000 pounds safely, or up to 3,000 lbs with an increasing risk of it breaking.

I can see this being about right. I see some websites saying the break DC is 17 for 5e games.

What kind of better rope is there? People talk about spider silk being 1/2 the weight, but can carry the same or more weight. I was thinking about thief's rope/ninja rope, but the spider silk is fine for most things.

 

[Cleon]

I can see this being about right. I see some websites saying the break DC is 17 for 5e games.

Yes, DC 17 seems way too low for a 5E hemp rope. That suggests an "average humanoid" with Strength 10 has a 20% chance of snapping the rope just by yanking on it with all their power.

Would you risk hanging from a cliff by a rope with a 20% chance of failure?

What kind of better rope is there? People talk about spider silk being 1/2 the weight, but can carry the same or more weight. I was thinking about thief's rope/ninja rope, but the spider silk is fine for most things.

Well while "researching" for rope data I wandered into checking on silk fibres since "rope, silk" is a standard SRD equipment item.

Here's the base stats:

Rope, Silk, 50 ft. (3/3.5/Pathfinder): This rope has 4 hit points and can be burst with a DC 24 Strength check. Price 10 gp, Weight 5 lb.

Rope, Spider's Silk (Pathfinder): This rope has 6 hit points and can be burst with a DC 25 Strength check. Price 100 gp, Weight 4 lb.

First Approach
Using the "Break DCs vs Strength/Carrying Capacity" method.

Silk Rope: Break DC 24 => safe Strength bonus +13 => Str 36-37 => max lift 3,680 or 4,160 lb. => max load 7,360 or 8,320 lb.
Spider's Silk Rope: Break DC 25=> safe Strength bonus +14 => Str 38-39 => max lift 4,800 or 5,600 lb. => max load 9,600 or 11,200 lb.

Rounding that to a convenient number, that suggests a regular silk rope has a breaking load of 7,800 pounds and a spider's silk rope 10,500 pounds. Assuming a "safe load" ratio of 1:6 that's 1,300 pounds for silk rope and 1,750 lb. for spider's silk rope.

A standard silk rope is probably made from regular insect silk (as in the stuff spun by silk moth larvae), while a spider's silk rope is specifically stated to be made from the silk of Monstrous Spiders. Possibly very big ones, since a Gargantuan Monstrous Spider's web only has a Break DC of 24. Then again, that Break DC may represent a single strand that "is strong enough to support the spider and one creature of the same size" and according to the Creature Size and Scale table that's a minimum of 32 tons or 64,000 pounds (since 3E uses US short tons).

More likely it's a silk strand from a Large Monstrous Spider as the standard weight range of a Large creature is 500-4,000 pounds, so a "safe load" of 1,750 lbs. falls within that range. A Large spider's web has a Break DC of 17 however. Possibly a Monstrous Spider web is easier to break than a single "support strand" because it's a collection of thin strands which can be snapped individually rather than a single large one.

A spider's silk rope could be made from hundreds of strands of silk from a Small or Tiny spider braided together like a normal hempen rope rather than a single strand from a Large one, if only because smaller Monstrous Spiders would be a lot easier and safer to keep "domesticated" than massive ones.

I guess it's possible the +7 difference between a Large Spider's Break DC 17 web and a Break DC 24 spider's silk rope is because the rope has been specially treated for strength and durability - some kind of secret drow alchemical mixture perchance? It'd explain why the stuff costs 100 gold pieces!

Also, a Large Monstrous Spider's web has 12 hit points per 5-foot section, twice the 6 hp of a spider's silk rope so maybe whatever treatment is used to turn their silk into rope makes it stronger but less durable? Or it just uses half the silk of a 5 ft. by 5 ft. stretch of Large spider webbing. A Medium Monstrous Spider's web has 6 hit points per 5 ft. section, for what it's worth.

Hmm, I seem to be wandering off the issue…

Second Approach
My best guess is spider silk rope should have approximately the same strength as high quality nylon rope, but the strongest known spider silks such as that from Darwin's bark spider (Caerostris darwini) are approximately twice as strong (up to 10 times stronger than kevlar!), so should produce a rope roughly twice as strong as nylon.

It's important to note that claims of a fibre being "X times stronger than steel/kevlar/whatever" are often deceptive, since they're usually for microscopic perfect fibres. An actual rope is only as strong as its weakest point, and will contain millions of individual fibres, many of which are weak, flawed or poorly bound to their neighbours, meaning the rope's breaking strength will be a lot less than the maximum theoretical strength of the material.

Regular silk from insect cocoons and the like is typically not as strong as spider silk, and can be much weaker depending on how the silk is harvested/spun and what particular insect it's from. I'll assume that regular silk ropes are made from the strongest varieties such as the standard silk of Mulberry silkworm cocoons from the silk moth (Bombyx mori) which is 50% to 60% spider strength.

Okay, so a regular silk rope is 10 feet long per pound (or 0.1 lbs. per foot). That's 148.9 grammes per metre. A spider's silk rope is 12.5 feet long per pound (or 0.08 lbs. per foot). That's 119.1 grammes per metre.

According to Engineering Toolbox data on Nylon Rope Strength:

A 16mm diameter nylon rope weighs 147 g/m and has a minimum breaking strength of 8,910 pounds.
A 14mm diameter nylon rope weighs 119 g/m and has a minimum breaking strength of 7,200 pounds.

So, if silkworm silk were 60% as strong as nylon, its minimum breaking strength ought to be 0.6 times the the breaking strength of 16mm nylon rope which'll be… 5,346 pounds, or approximately the same strength as the standard hempen rope.

If a spider's silk rope is as strong as nylon, its minimum breaking strength is 7,200 pounds.

Those figures seem a bit low. However, those are minimum breaking strength, and good engineering conduct is to include a healthy safety factor. The average breaking load could be noticeably higher.

According Smackdock Average Rope Breaking Loads:

A 16mm diameter three-strand nylon rope has an average breaking load of 6,640 kg (14,625 pounds).
A 14mm diameter three-strand nylon rope has an average breaking load of 5,100 kg (11,233 pounds).

Using the above:

60% of 5,100 kg would translate to a standard silk rope having a breaking load of 8,775 pounds.
50% of 5,100 kg would translate to a standard silk rope having a breaking load of 7,313 pounds.

And a standard spider's silk rope having a breaking load of 11,233 pounds.

Those figures are reasonably close to the First Approach's breaking loads of 7,800 lb. for silk ropes and 10,500 lb. for spider's silk ropes.

Silk's a bit denser than nylon (mulberry silk's typically 1.35 g/cm³, spider's dragline silk 1.31 g/cm³, nylon 6.6 is 1.15 g/cm³). That doesn't make a difference to our Strength calculation, but might affect the actual thickness of the rope.

All things being equal, the rope's diameter will be that of the equivalent nylon rope times the square root of the density ratio (nylon density divided by silk density).

Silk Rope: 16 mm × square root (1.15/1.35) = 14.77 mm => roughly 6/10th of an inch.
Spider's Silk Rope: 14 mm × square root (1.15/1.31) = 13.11 mm => roughly 1/2 of an inch.

However, a silk rope could be "fluffier" than a nylon rope with more airspace between its fibres. I don't really know, since I couldn't find any data on actual silk ropes.

This might be because a real-like silk rope would be much more vulnerable to abrasion damage than a nylon rope. Since individual silk fibres are so thin, it'll take very little to break one and a silk rope (in theory) might fray very easily when it rubs again something (including itself at points where it's knotted). A Monstrous Spider silk rope presumably would not suffer from this problem as its fibres would be much thicker than a regular Fine-size spider's.

As for a standard silk rope, maybe it's made from the cocoons of Giant Silkmoths the size of pigeons rather than regular sized insects? Considering the abundance of giant insects in D&D, Giant Silkworms are definitely a possibility and would be a simple answer to the abrasion problem.

Or silk rope is only used for a few days or weeks and chucked away or unwoven and used for regular fabric as soon as it becomes too worn to be safely used. That seems rather profligate, but adventurers tend to have money to burn.

Conclusion
Rounding to some convenient numbers:

The SRD standard silk rope is a 9/16th inch diameter silkworm silk rope (possibly from Giant Mulberrry Silkmoths) that can support 1,300 pounds safely, or up to 3,900 lbs with an increasing risk of it breaking.

The Pathfinder spider's silk rope is a 1/2 inch diameter Monstrous Spider silk rope that can support 1,800 pounds safely, or up to 5,400 lbs with an increasing risk of it breaking.

 

[Cleon]

Additional Notes

Stretchiness
Ropes stretch before they break. I couldn't find elasticity data for actual hemp with some casual internet browsing, but manila rope has a breaking strain of 20% according to smackdock and that fibre appears to have pretty similar properties to hemp. The percentage is how much longer the fibre gets before it snaps, meaning a 50 ft. length of rope will generally stretch to 60 ft. in length before it breaks under a load.

Regular silk from insect cocoons can have a wide range of breaking strains from 4% to 40%, with Mulberry silkworm silk ranging from 10-23.4% according to Wikipedia at least. Let's call it a nice round 20% on average, meaning a 50 ft. length of insect's silk will generally stretch to 60 ft. in length before it breaks under a load. It'd be more convenient if standard silk rope is made to be as "stretchy" as hempen rope.

Dragline spider silks (dragline silk, produced by the spider's Major Ampullate gland, being the strongest kind of silk spider's produce in terms of its maximum breaking load) appear to have breaking strains ranging from 18% to 33% according to Wikipedia. Let's call it 25% on average, so a 50 ft. length of spider's silk can stretch up to 62.5 ft. before it breaks.

However, Flag silk, produced by the spider's Flagelliform gland is MUCH stretchier, with breaking strains up to 400% (so they can stretch up to five times their length without breaking), although 270% is an average figure quoted by Ask an Entomologist. Unfortunately, flag silk is also far weaker, with typical breaking loads around half that of dragline silk, although its "toughness" (ability to absorb damage) is almost as good.

A really stretchy rope is of limited usefulness in nautical applications - a set of rope stays to keep a mast in position would be pretty useless it they're so elastic the mast flexes from side to side - but would be very handy for safety nets, climbing ropes, bungee cords and the like, since they can absorb damage by stretching under the impact of a blow or fall.

Come to think of it, that could justify the low Break DCs and Hardness 5 of a Monstrous Spider's webbing. The "capture portion" of a spider's web is made of flag silk, not the more load-resistant dragline silk. The hardness represents the web's ability to stretch without being damaged. So if you take a Large Monstrous Spider webbing's Break DC 17 and add its hardness of 5 to make 22, the 2-point difference between that and a spider silk's rope Break DC 24 is a sort-of-match to 2× increase in carrying capacity a 5-point increase in Strength produces (which'd increase a character's Strength modifier by +2 or +3).

'Course that's bending "reality" to match the numbers rather than vice-versa, but at least it's consistent.

Incidentally, nylon rope appears to have a breaking strain of 25 to 50%, which I believe/suspect varies according to the type of nylon. I haven't bothered trying to find out more since it didn't seem that germane to D&D rope use!

Notes on Falling Damage
Being secured by a rope does not make the impact from a fall disappear. It just (hopefully) shortens the distance.

Let's say you're an adventure climbing the Forbidden Cliff of Sheer Doom! with a hempen rope, one end of which is tied to you and the other end is secured to a piton in the Cliff of Doom. There's a 40 ft. length of rope between the piton and the rope harness you've rigged for yourself.

You've found a secret door 20 feet below the piton, but while trying to open it you fluff your Climb check and fall. The rope stops you falling once it stretches to 40 ft., so you would take damage from a 20 ft. fall (the rope's length minus the 20 ft. height the piton was above you).

That's why climber's keep on securing pitons just above them!

Conclusion
If you wish to add "really stretchy ropes" to your game, may I suggest:

Rope, Spider's Climbing, 50 ft. (Homebrew): An ingenious mix of different monstrous spider silks, this rope has 6 hit points and can be burst with a DC 24 Strength check. If you fall while secured with the rope it will stretch, reducing the falling damage by 5 points. The rope's tough but elastic silk can stretch up to one and a half times its normal length (75 ft.). A spider's climbing rope is about 1/2 inch thick and can support 1,300 pounds safely, or up to 3,900 lbs with an increasing risk of it breaking. Price 100 gp, Weight 5 lb.

Rope, Spider's Elastic, 50 ft. (Homebrew): This rope has 5 hit points and can be burst with a DC 23 Strength check. If you fall while secured with a spider's elastic rope the rope will stretch to absorb the falling damage. If there's enough height for you to fall four times the rope's normal length without hitting an obstacle (such as the ground) you take no falling damage, otherwise the effective falling distance is the rope's normal length reduced by 10 feet for every 30 feet of height the rope can stretch before you hit the obstacle. A spider's elastic rope is about 1/2 inch thick and can support 900 pounds safely, or up to 2,700 lbs with an increasing risk of it breaking. Price 100 gp, Weight 5 lb.

 

[Cleon]

Addendum
I forgot to mention that the listed diameters of ropes are for ropes under tension, since they are measured when the rope is being stretched during its manufacture (likely after it's spun together on a ropewalk in the case of D&D level technology).

If a silk or hempen rope were stretched to near its breaking point of 120-125% its normal length then, assuming its volume were the same, it would be roughly 110% that thickness when unstretched.

So so the "resting diameter" of a silk or hempen rope ought to be somewhere between 1% and 10% higher than its listed "under tension" diameter. Now I don't know how much professional D&D ropemakers test their products, but if they normally test a brand new rope to near its breaking load to ensure it doesn't have any flaws that would cause outraged adventurers to murderhobo them it could be that the actual thicknesses of the ropes listed above are 10% higher than the quoted diameters. However, they could just as easily only stretch the rope to, say, its maximum working load of 1/2 its breaking load, which'd make the resting diameter 5% or so higher.

A percent or ten doesn't see to be enough to make a big song-and-dance about.

The spider's elastic rope homebrew is really stretchy though, stretching up to 400% its normal length means its resting diameter should be 200% its fully stretched diameter. That might suggest I need to amend the "about 1/2 inch thick" to "about 1 inch thick?" but, upon reflection, I think the rope would be made under roughly the same stresses (up to 125%?) as a normal silk rope or spider's silk rope - a spider's capture web isn't under extreme tension after all, since the arachnid spins it to have a lot of "give" to absorb the impact of prey blundering into it.

Oh, and for the sake of completion the spider's climbing rope homebrew has a breaking stress of 150% (so a 50 ft. length may stretch to 75 ft. before snapping). That indicates it absorbs 1 point of falling damage per 5 foot of stretch, which is comparable to the spider's elastic rope's 1d6 per 30 foot (ten yards divided by max roll of one six-sided dice = 30/6 = 5 ft.).

Actually, I think I'll incorporate the stretch length in the writeup above.

 

[Cleon]

Well since I've got this far I might as well do the other item of rigging in the SRD:

Chain: Chain has hardness 10 and 5 hit points. It can be burst with a DC 26 Strength check. (10 ft., 30 gp, 2 lb).

First Approach
Using the "Break DCs vs Strength/Carrying Capacity" method.

Chain: Break DC 26 => safe Strength bonus +15 => Str 40-41 => max lift 6,400 or 7,360 lb. => max load 12,800 or 14,720 lb.

Okie-dokey, if we use the natural fibre rope's 1/6th safe load ratio that's a safe working load of 2,100 to 2,500 pounds and a max working load of 6,300 to 7,500 pounds.

Second Approach
Okay, the big problem is what is this blinking thing made of! It could be anything from pig iron to the best quality steel. Heck, I vaguely remember seeing pictures of a chain made out of wood.

The SRD chain is comparatively expensive (3 silver pieces per foot) and as heavy as hempen rope at 5 feet per pound (0.2 lb/ft. or 0.298 kg/m).

Come to think of it, the SRD chain costs four times as much per pound as a longsword (30 gp for 2 lb. versus 15 gp for 4 lb.). That suggests it's made from high quality metal at least as good as a typical sword.

Steel Rope
The closest match on engineering toolbox is wire rope - that weight lies between 5/16th inch (0.16 lb/ft., min breaking strength 8,520 lb.) and 3/8th inch (0.24 lb/ft., min breaking strength 12,200 lb.) diameters.

Averaging out the weight/length and min breaking strength suggests a 0.2 lb/ft. wire rope has a minimum breaking strength around 10,325 pounds, which is a bit lower than the first approach's 12,800 or 14,720 lb. breaking load. However, wire ropes and metal chains have a better safe load ratio than natural fibre (notice the table uses a 1:4 ratio rather than the 1:12 on the manila rope and sisal rope engineering toolbox pages), so the safe load of a 0.2 lb/ft. wire rope would be around 2,580 pounds - pretty much the safe working load derived from the First Approach, which seems close enough for casual D&D engineering maths.

However there's a slight problem - the stats are for wire rope, not a chain!

Steel Chain
A bit of internet rummaging turned up a typical rigging breaking loads for various types of metal rigging which tells me how strong they are, but unfortunately not how heavy they are.

Then I came across Chain Size – Break Strength and Safe Working Load on setsail.com which contained some useful factoids, namely:

• 3/8″/9.6mm Grade 70 – break strength 24,000 pounds/10,880 kg
• 5/8″/16mm Proof Coil – break strength 27,600 pounds/12.500 kg
  
  Assuming you carry 300 feet/90meters of chain, the 3/8″ will weight in at 408 pounds/185 kg. The 5/8″ weighs 1107 pounds/502kg.

The above appear reasonably in agreement to the rigging breaking loads data sheet - the 3/8" stainless rod rigging's breaking load is 22,500 pounds.

A 10 ft. length of chain like the SRD version will weigh one-thirtieth the above. So, putting that together:

3/8" grade 70 chain, break load 24,000 lb, 10 ft. length weighs 13.6 lb.
5/8" proof coil chain, break load 27,000 lb, 10 ft. length weighs 36.9 lb.

The grade 70 chain is 6.8 times heavier than the SRD chain, so all things being equal it'll be 6.8 times stronger. If we scale down the chain so the weigh becomes 2 pounds for a 10 foot length:

breaking strength = 24,000 / 6.8 = 3,529 pounds.
diameter = 0.375 / square root (6.8) = 0.144 inches (9/64 in. or 3.65 mm).

The metal the chain is made from, Grade 70 Carbon Steel, is a typical steel in terms of strength, with an ultimate tensile strength of 70,300-89,900 lb/in² and a yield strength of 37,700 lb/in².

With the recommended 1:4 safety ratio, that's a safe load of 882 pounds, which is WAY lower than the First Approach's result of 2,100-2,500 lbs.

That said, a safe load of 2,100 to 2,500 pounds is still possible since high quality steels (such as spring steel) can be much stronger than the typical Grade 70 used above. For example, 5160 spring steel has a breaking strength of 1,025 MPa and a yield strength of 650 MPa. Compared to Grade 70's 485-620 MPa breaking strength and 159–221 MPa yield strength that's roughly twice as strong overall, but it yields to loads three to four times heavier without permanent damage. The even mightier Eglin steel has breaking strength 1818 MPa and yield strength of 1,547 MPa, so is three times stronger than Grade 70 steel and its yield load is seven to nine times higher!

Wrought Iron Chain
I also found an applied science for metal workers' page on strength of chains that says it's better to make load-bearing chains out from wrought iron, since it's less brittle than carbon steel (so less prone to shatter under the shock of a sudden load) and easier to repair (since it's more amenable to welding).

According to Wikipedia, wrought iron has an ultimate tensile strength of 34,000-54,000 lb/in² and a yield strength of 23,000-32,000 lb/in².

That metal worker's webpage gives this formula for calculating the safe working strength of wrought iron chains:

Safe Load (pounds) = Bar Diameter (inches)² × 0.7854 × 40,000 [tensile strength] × 1.63 [chain link factor] × 0.5 [safety ratio]

Which boils down to:

Safe Load (pounds) = Bar Diameter (inches)² × 25,604.04

Which reverses to:

Bar Diameter (inches) = square root of [Safe Load (pounds) / 25,604.04]

Plugging a 2,200 pounds safe load into the above gives us a bar diameter of 0.29 inches (7.44 mm). So it's 78% as thick than the 0.375 inch diameter grade 70 steel chain above, which'd make it 0.78 × 0.78 = 61% as heavier*, or roughly 8 pounds for a 10-foot length instead of 2 pounds.

HOWEVER, note that the above formula uses a 2:1 safety factor rather than the 4:1 of the engineering references I came across. With a 0.25 safety ratio the chain would be twice as heavy or about 16 pounds, giving it a 0.41 inch bar diameter (10.5 mm).

*Fortunately all types of iron and steel have a pretty similar density (at least within a few percent) so there's no need to adjust the volume/thickness to account for the different metal.

Size & Diameter of Chains
Note that a chain's size measures the diameter of the bar used to make the chain, not the chain's overall dimensions. For example, a 1/2" chain would have links at least 1.5 inches across (a half inch for the bar on each side plus a gap in the middle wide enough for adjacent links to fit through). Thus the actual chain might be 1.6-1.75 inches across, or significantly wider if there's a lot of space within the link.

Unlike ropes, chains are not measured under tension, since they are forged rather than woven. Their listed diameter is their rest diameter. A two inch diameter Grade 70 steel rod will break when it stretches to around 121% its normal length. However, once it has stretched a certain point (I guesstimate about 110% if the material's Young's modulus is uniform under stress) the rod will have reaches its yield point and will begin to bend and distort at its weakest point(s), permanently damaging the chain.

Wrought iron has a very similar "stretchiness" to Grade 70 steel (its Young's Modulus is 193 GPa versus Grade 70's 200 GPa), but being roughly twice as weak (tensile strength 234-372 MPa vs. 485-620 MPa) it will typically break when it stretches by half as much as the steel (say 111%), though its decent yield strength (159–221 MPa vs. 260 MPa) mean it can stretch and recover a respectable amount (I'd guesstimate about 106-108.5%, sat 107.5 on average).

Notes on Chain Links
The links of a chain are generally made by bending short sections of metal bar in a loop. The cheapest of chains leave it at that, meaning any stress that's enough to unbend one of the links will cause the chain to fail.

Alternatively, up to half of a chain's links are cast as solid metal and bar links are made to string them together - this may be a bit cheaper and less labour intensive than bending every link. Being no expert on chain manufacturing I'm unable to say.

Proper chain has the ends of each bar link sealed shut. In thick modern chains the links are typically welded.

In Ye Olden Days it was a common practice to rivet the ends of each loop together. The rivets will generally be the weakest point of the chain. Riveted links was the standard method of making mail armour (aka chain armour, which is tautologically called chainmail in D&D), with each link looping around several of its neighbours before being riveted close.

Conclusion
The SRD chain is made from high-quality steel bars roughly 1/6 inch thick, resulting in a chain about 0.5 inches in diameter. It can support 2,400 pounds safely, or up to 7,200 lbs with an increasing risk of breakage.

Addition
The wrought iron chain inspired me to homebrew the following:

Chain, Iron: Made from 3/8" thick wrought iron bars, resulting in a chain about 1.25 inches in diameter. An iron chain has hardness 10 and 10 hit points, it can be burst with a DC 25 Strength check. The chain can support 1,600 pounds safely, or up to 4,800 lbs with an increasing risk of breakage. (10 ft., 3 gp, 4 lb).

Comparison to Spider's Silk
Spider silk is stronger than high quality steel per unit weight (while steel may have higher tensile strength in terms of MPa it is also roughly six times denser than silk).

The standard spider's silk rope is 40% as light as an SRD chain of the same length (4 pounds vs 10 pounds for a 50 foot length), so if it were as heavy as the chain it'd support 250% the load.

A spider's silk rope that's 250% heavier than normal should support 4,500 pounds safely, or up to 13,500 lbs with an increasing risk of it breaking. I estimate it has 15 hit points and Break DC 27.

The homebrew iron chain is five times heavier than a spider's silk rope (4 pounds vs 20 pounds for a 50 foot length).

A spider's silk rope that's 500% heavier than normal should support 9,000 pounds safely, or up to 27,000 lbs with an increasing risk of it breaking. I estimate it has 30 hit points and Break DC 29.

 

[Cleon]

Unlike ropes, chains are not measured under tension, since they are forged rather than woven. Their listed diameter is their rest diameter. A two inch diameter Grade 70 steel rod will break when it stretches to around 121% its normal length. However, once it has stretched a certain point (I guesstimate about 110% if the material's Young's modulus is uniform under stress) the rod will have reaches its yield point and will begin to bend and distort at its weakest point(s), permanently damaging the chain.

Wrought iron has a very similar "stretchiness" to Grade 70 steel (its Young's Modulus is 193 GPa versus Grade 70's 200 GPa), but being roughly twice as weak (tensile strength 234-372 MPa vs. 485-620 MPa) it will typically break when it stretches by half as much as the steel (say 111%), though its decent yield strength (159–221 MPa vs. 260 MPa) mean it can stretch and recover a respectable amount (I'd guesstimate about 106-108.5%, sat 107.5 on average).

Ropes have a yield point beyond which they stretch and weaken too, obviously, but unfortunately my casual internet browsing has failed to find any information on what that is for hemp or manila, so your guess is as good as mine as to when an SRD hempen rope is likely to permanently stretch.

Wikipedia's Yield (engineering) page quotes spider silk as yield strength 1150 MPa and breaking strength 1400 MPa. That indicates its resistance to plastic deformation is better than all but the most elastic steels, since it can take up to 82% its breaking load and bounce back to normal. Incidentally, that breaking strength seems a bit high - most sources I've seen quote 1000 to 1100 MPa for typical dragline silk, although a few species can do better than that (like the aforementioned Darwin's bark spider).

EDIT: Further on the above, I suspect Wikipedia's 1400 MPa spider's silk may be based on the silk of the giant golden orb weaver (Nephila pilipes), a species noted for particularly strong silk as per this paper. Note that paper also studied the silk of the tiger spider "Nephila plumipes" (now renamed Trichonephila plumipes) whose webs are of more typical strength - the Strain-Toughness/Stress Diagram shows the golden orb's silk reaching 1600 MPa while the tiger spider barely manages 900 MPa.

For comparison, Grade 70 steel can take somewhere between 32% to 36% (159/485 - 221/620); 5160 spring steel can take 63% (yield strength divided by tensile strength is 650/1,025), while Eglin steel can take 85% (1547/1818).

 

[Cleon]

After checking the sums and Carrying Capacity tables I've decided the Conclusion breaking loads for chains are a little too high, so I tweaked them down so the iron chain is slightly weaker than a spider's silk rope (which has the same Break DC) and the regular chain (which is +1 DC higher) was a nice round 50% stronger than its wrought iron cousin.

 

[Cleon]

Miscellaneous Cords
The Pathfinder SRD has a few additional items of general adventuring gear that seem pertinent:

String or Twine: Sold in balls or spools of 50 feet, string and twine are useful for rigging traps and alarms and are a vital component of grappling bolts and arrows. String or twine has hardness 0, 1 hit point, and a break DC of 14. Price 1 cp; Weight 1/2 lb.

Bloodvine Rope: This 50-foot length of tough, lightweight rope is made from alchemically treated bloodvine, a rare scarlet-colored vine that grows only in warm jungle environments. Though prized by climbers for its durability, bloodvine can also be used to bind creatures. Bloodvine rope has a hardness of 5 and 10 hit points, and can be broken with a DC 30 Strength check. A creature bound by bloodvine rope can escape with a DC 35 Escape Artist check or a DC 30 Strength check. The DC to create bloodvine rope with Craft (alchemy) is 30. Price 200 gp; Weight 5 lbs. Source Advanced Class Guide (2014).

Stretch cords: Sold in pairs, these cords have an elastic quality. Both cords measure 2 feet long but can be extended to 4 feet. Each end of a cord has a small metal hook. Employed together, these cords can be used to tie down all sorts of equipment and prevent damage from wind or other inhospitable weather. Price 5 sp; Weight 1/2 lb. Source Ultimate Wilderness (2017).

 

[Cleon]

Okay, I'll go through them in the above order:

String
Using the First Approach, Break DC 14 => +3 Strength bonus => Strength 16-17 max lift => 460 or 520 pounds => working load 76.66 or 86.66 pounds.

Using the Second Approach, 100 ft. of string weighs 1 pound, so one foot weighs 0.01 lb. Which is 67.18 m/kg, or 14.88 g/m, or 3.275 kg/220m.

That weight matches a cord somewhere around 4.5 mm to 5 mm in diameter, which to me sounds more like thin rope than thick string:

Engineering Toolbox 5mm sisal rope (0.01 lb/ft. or 0.02 kg/m, min break strength 290 lb or 1.29 kN)
Engineering Toolbox 5mm manila rope (0.014 lb/ft. or 0.02 kg/m, min break strength 405 lb or 1.8 kN)
Buy Rope 6mm hemp rope (6kg/220m [36.66 m/kg or 27.27g/m, break load 250 kg or 550 lb.])

The above breaking strengths translate to a working load of around 45 to 50 pounds assuming a 1:6 safety ration, which is distressingly low compared to the First Approach's Break DC derived working loads of 76 to 87 pounds. Those loads would require a natural fibre cord around 6 mm in diameter - that's definitely a rope!

The Pathfinder SRD lists "string or yarn" as costing 2 cp a pound, which is one-fifth the price-per-weight of hempen rope (1 gp for 10 lb => 1 1b. is 1 sp or 10 cp). This strongly suggests it's made from a cheaper and presumably inferior fibre than hemp. Maybe sisal or flax?

If standard Pathinder String is made from such a cheap material, then presumably better quality cord made of hemp or silk should be a thing too.

Conclusion
The Break DCs of Pathfinder string or yarn indicate it's a thin rope roughly one-quarter of an inch in diameter, able to support 70 pounds safely, or up to 210 lbs with an increasing risk of it breaking. It's too cheap to be made from hemp, so must be made from inexpensive but relatively weak plant fibres such as sisal or flax.

The listed weight is too low for the indicated Break DC; a plant fibre line that strong should be roughly twice as heavy, or about 50 feet to the pound.

A line one-quarter of an inch is way too thick to be realistically called "string or yarn", so I would recommend renaming it "Cord" and adding "Twine", "Line" and "Thread" for thinner yarns.

Twine tends to be 3 mm in thickness (about 1/8th of an inch), although I've seen it in 2 mm so could call it 1/10th of an inch on average (about 2.5 mm).

A fairly thick thread (size 69 of Tex 70 thread) is about 0.01 inches in diameter). So if 1,000 metres of Tex 70 weights 70 grams a tenth of a pound (45.4 grammes) of thread would be… 649 metres or 1,428 feet long. Silk thread is slightly denser than nylon, so a thread of that thickness would be heavier and shorter.

Putting that together could gives us stats for yarns along these lines:

String or Yarn​

Strings and yarns can range in thickness from cord the diameter of thin rope to fine threads thinner than elven hair. Yarn is sold in balls or spools. A spool of string is heavier than a ball of the same length due to the weight of the bobbin or spindle it's wound around.​

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Cord: Sold in lengths of 50 feet or 600 feet, cord has hardness 0, 1 hit point, and a break DC of 14. A cord can support 70 pounds safely, or up to 210 lbs with an increasing risk of it breaking. Price 2 cp for 50 ft.; or 2 sp for 600 ft.; Weight 1 lb. for 50 ft. (2 lb. in spool); or 10 lb. for 600 ft. (+5 lb. in spool)​

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Cord, Hempen: Sold in lengths of 50 feet or 600 feet, hempen cord has hardness 0, 1 hit point, and a break DC of 14. A hempen cord can support 80 pounds safely, or up to 240 pounds with an increasing risk of it breaking. Price 1 sp for 50 ft.; or 1 gp for 600 ft.; Weight 1 lb. for 50 ft. (2 lb. in spool); or 10 lb. for 600 ft. (+5 lb. in spool)​

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Cord, Silk: Sold in lengths of 120 feet or 600 feet, silk cord has hardness 0, 1 hit point, and a break DC of 15. A silk cord can support 100 pounds safely, or up to 300 pounds with an increasing risk of it breaking. Price 2 gp for 120 ft.; or 10 gp for 600 ft.; Weight 1 lb. for 120 ft. (2 lb. in spool); or 5 lb. for 600 ft. (+5 lb. in spool)​

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Cord, Spider's Silk: Sold in lengths of 150 feet or 600 feet, spider's silk cord has hardness 0, 1 hit point, and a break DC of 16. A spider's silk cord can support 150 pounds safely, or up to 450 pounds with an increasing risk of it breaking. Price 25 gp for 150 ft.; or 100 gp for 600 ft.; Weight 1 lb. for 150 ft. (2 lb. in spool); or 4 lb. for 600 ft. (+5 lb. in spool)​

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Twine: Sold in balls or spools of 200 feet, twine has hardness 0, 1 hit point, and a break DC of 9. A length of twine can support 17 pounds safely, or up to 50 pounds with an increasing risk of it breaking. Price 2 cp; Weight 1/2 lb. (1 lb. in spool)​

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Twine, Hempen: Sold in balls or spools of 200 feet, hempen twine has hardness 0, 1 hit point, and a break DC of 9. A length of hempen twine can support 20 pounds safely, or up to 60 pounds with an increasing risk of it breaking. Price 5 cp; Weight 1/2 lb. (1 lb. in spool)​

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Line, Silk: Sold in balls or spools of 400 feet, silk line has hardness 0, 1 hit point, and a break DC of 10. A length of silk line can support 25 pounds safely, or up to 75 pounds with an increasing risk of it breaking. Price 1 gp; Weight 1/2 lb. (1 lb. in spool)​

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Line, Spider's Silk: Sold in balls or spools of 500 feet, spider's silk line has hardness 0, 1 hit point, and a break DC of 11. A length of spider's silk line can support 35 pounds safely, or up to 100 pounds with an increasing risk of it breaking. Price 125 sp; Weight 1/2 lb. (1 lb. in spool)​

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Thread: Fine linen or cotton thread sold in spools of 800 feet, thread has hardness 0, 1 hit point, and a break DC of 2. A length of thread can support 2 pounds safely, or up to 6 pounds with an increasing risk of it breaking. Price 1 cp; Weight 1/4 lb. (1/10 lb. without bobbin)​

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Thread, Silk: Sold in spools of 1,600 feet, silk thread has hardness 0, 1 hit point, and a break DC of 3. A length of thread can support 3 pounds safely, or up to 9 pounds with an increasing risk of it breaking. Price 2 sp; Weight 1/4 lb. (1/10 lb. without bobbin)​

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Thread, Spider's Silk: Sold in spools of 2,000 feet, this thread has hardness 0, 1 hit point, and a break DC of 4. A length of spider's silk thread can support 4 pounds safely, or up to 12 pounds with an increasing risk of it breaking. Price 25 sp; Weight 1/4 lb. (1/10 lb. without bobbin)​