Shardplate Math!

[+hwiles]

I'm trying to get some actual numbers for the characteristics of shardplate from in-book descriptions.  Things like: How much strength does shardplate lend its user, how much impact can shardplate absorb without visible cracking, how much does shardplate weigh, and are there any serious discrepancies between these values for the various suits that have been used onscreen.  I'll try to be as brief and non-technical as possible while still conveying what I think I've found so far, criticisms and suggestions welcome!

I'll try to clearly identify any assumptions and estimates appropriately.  I'm not doing citations right now though.

It's been stated that a large horse can safely carry a full-grown man in shardplate so long as they mount the animal "carefully."  I take this to mean that the weight of the rider, plate, and saddle is close to the maximum weight the horse can carry around for a more-or-less extended period of time without injury, about 20-30% of its body weight.  A horse can weigh up to about 2,000 lbs.  0.3 x 2,000 lbs = 600 lbs of total carrying capacity, which would make the armor weigh at most about 400 lbs, and more likely closer to 250-300 lbs (since horses that large typically aren't suitable for riding).  This is also happens to be just heavy enough that most people would have extreme difficulty moving if the suit were to lose power, which also agrees with in-book references.

I'm going to assume Dalinar weighs 200 lbs, is 2 m tall, his armor weighs 300 lbs, and a parshendi weighs ~250 lbs

Dalinar demonstrates a maximum vertical jump of 8 ft (a tremendous feat even considering the reduced gravity on Roshar).  Neglecting air-resistance and assuming gravity on Roshar is 0.7g (I think there's a WoB on this?), this would require an initial vertical velocity of about 5.7 m/s.  Assuming he crouched down before the jump, resulting in a lowering of his center of gravity (which I'll assume is exactly at his belly-button for simplicity) by about 6 inches (I'm not a professional athlete, but this number seemed reasonable to me) his average force from both legs would come out to about 25,000 Newtons, or approximately 2,300 lbs of force from each leg.

In another scene, Dalinar kicks a parshendi corpse about 20ft.  This is far enough that air resistance starts to matter, however we don't have a great way of accounting for it with the information in the scene, so just know that the force calculated here is a lower-bound.  Without air resistance, assuming he kicks the body at somewhere between a 30 and 45 degree angle, and his foot contacts the body for about 1 foot of distance (anyone who plays soccer can feel free to jump in with a better number here if you have one) he would need to exert a force of about 1,700 lbs (According to my research, this slightly exceeds the amount of force required to dismember a human body so we'll assume parshendi bodies are a little stronger than human's, and/or he hit the corpse in the center of the torso).  What's interesting about this figure is that it suggests that Dalinar could have hit the body much harder, but chose not to (probably instinctively) because doing so would have caused it to explode as his foot passed through it rather than go flying (gross, but neat!).

 

I'm planning to review some more scenes, specifically those between multiple shardbearers and see if I can come up with some more refined and rigorous calculations, but wanted to share what I had so far.  If you think this sort of thing is interesting let me know what you think and I'll try to pick up the pace; challenges to my numbers or your own calculations are also welcome!

 

TLDR;

A person in plate could easily tear a person's limbs off (and dunk like no one's business)

[Windrunner]

Very cool, I've never seen anyone try to quantify Shardplate's strength before. Can't wait to see what else you come up with!

[Erunion]

Yay math! 

Looks good so far. Except that imperial units make me cringe... (Stick to Metric man! Makes life so much easier!)

[+hwiles]

5 minutes ago, Erunion said:

Yay math! 

Looks good so far. Except that imperial units make me cringe... (Stick to Metric man! Makes life so much easier!)

As to that...I was trained as a physicist, so I do all my math in metric however, since getting employed as an engineer (in the US), I've been browbeaten into reporting my final answers in imperial...

[CaptainRyan]

@hwiles The scene where Dalinar digs a latrine trench in the Way of Kings might be instructive. It has two decent "comparisons" - the first is two men struggling to lift the warhammer that Dalinar easily lifts with one hand and the other is when Dalinar removes a chunk of rock that would take 4-5 men to move and he tosses it aside one handed. It is in Chapter 28 if that helps.

Btw, I think this idea is AWESOME! I applaud your efforts!

[+hwiles]

1 minute ago, CaptainRyan said:

@hwiles The scene where Dalinar digs a latrine trench in the Way of Kings might be instructive. It has two decent "comparisons" - the first is two men struggling to lift the warhammer that Dalinar easily lifts with one hand and the other is when Dalinar removes a chunk of rock that would take 4-5 men to move and he tosses it aside one handed. It is in Chapter 28 if that helps.

Btw, I think this idea is AWESOME! I applaud your efforts!

Thanks, I'll definitely take a look!  The absolute most valuable scenes will be those involving the plate, or objects it's interacting with, being anecdotally pushed just to their limit.  If you recall any scenes where plate that hasn't already taken significant damage is struck hard but doesn't visibly crack, or scenes where a person in shardplate hits something as hard as they can without completely destroying it, or struggles to just barely lift or stop something, that would help fantastically.  Scenes where the plate is only very slightly damaged by an action and those where numbers are mentioned or hinted at or scenes where direct comparisons are made are definitely valuable too though.

 

Does anyone know if the book ever mention what the draw weight is on those shardbows?

[CaptainRyan]

The scene where Kaladin fights the Shardbearer while protecting Amaram has a moment where Kaladin lands a solid, full strength hit with the butt of his spear (shatters the spear with no damage to the spear), where his squad lands multiple hits with spears (no damage) and another moment where Kaladin lands a solid, full strength hit with the tip of his spear and it merely scratches the paint of the Shardplate. 

Another scene, in the early middle section of the book, where Adolin and Dalinar go on a plateau run has Adolin straight punching Parshendi in the face hard enough to kill them, shatter their bones etc. but there is no mention of the gauntlets taking any damage from smashing into the hardened armor/bones of the Warform Parshendi.

I dunno if any of these help but they are what came to mind when I read your reply.

48 minutes ago, hwiles said:

Does anyone know if the book ever mention what the draw weight is on those shardbows?

It does not mention the draw weight of the Grandbows (sorry, Adolin dislikes the name Shardbows) but it does mention that the draw weight is so high that the steel itself would be warped and unusable without the gemstome fabrial enhancing the strength of the steel. So, the draw weight literally exceeds the strength of the steel.

[Emerald101]

While I highly applaud your efforts, I feel I have to note that Brandon was probably not calculating how much force the plate exerted on things as he writes the story. I suspect that even given perfect descriptions and hard numbers to work with we would still end up with inconsistent numbers.

Don't let my cynicism discourage you though. You go and do your math on these things, because it's wicked cool. just don't expect to get fully consistent answers.

[AndrolGenhald]

11 minutes ago, Emerald101 said:

While I highly applaud your efforts, I feel I have to note that Brandon was probably not calculating how much force the plate exerted on things as he writes the story. I suspect that even given perfect descriptions and hard numbers to work with we would still end up with inconsistent numbers.

Don't let my cynicism discourage you though. You go and do your math on these things, because it's wicked cool. just don't expect to get fully consistent answers.

You might be surprised by how thorough Brandon and Peter can be.

[CaptainRyan]

Another battle scene where we see the same Parshendi hit Dalinar twice with full strength and only one of the blows seems to cause damage.

page 780 of 1008 of the kindle version of the Way of Kings (emphasis mine):

Quote

[...] swinging his hammer. [...]
    His blow cracked into Dalinar's helm. It pushed his head to the side but the Plate absorbed most of the hit, a few tiny weblike lines cracking along its length. [...]
    Dalinar raised an armored arm and blocked the next hammer blow.

Note that the hammer causes a few tiny cracks in the helm but not on the arm. Same hammer swung by the same angry, determined male Parshendi Warform. I do not know if this will help you quantify anything but I thought it worthy of note.

[CaptainRyan]

Sorry for the double post but page 822 of 1008 of the Kindle version of the Way of Kings gives the weight of Adolin's Shardplate:

Quote

Adolin jumped back out of the way, Shardplate-enhanced legs giving him a nimbleness that defied the fact that he was wearing over a hundred stoneweights of thick armor.

 

[Oudeis]

On Earth a "stone" is 14 pounds... do we assume a "stoneweight" is the same?

[AndrolGenhald]

1 hour ago, Oudeis said:

On Earth a "stone" is 14 pounds... do we assume a "stoneweight" is the same?

"Over a hundred stoneweights" sounds like it's between 100 and 200 stoneweights, so between about 140 and 280 pounds, which matches previous conclusions fairly well, although if it were closer to 200 I'd expect him to say "almost 200 stoneweights". Has the .7g been factored into this?

[Dunkum]

12 hours ago, AndrolGenhald said:

"Over a hundred stoneweights" sounds like it's between 100 and 200 stoneweights, so between about 140 and 280 pounds, which matches previous conclusions fairly well, although if it were closer to 200 I'd expect him to say "almost 200 stoneweights". Has the .7g been factored into this?

you're off by an order of magnitude there.  100 stone (using the british unit) would come out to 1400lbs not 140.  that is definitely unreasonable for a horse to carry on its back, so I don't think we can assume that those measures are close.

I haven't reread any of the books in a little bit, but the scene where Kaladin and Adolin fight the duel might be informative.  Kaladin's multiple lashing kick was enough to damage a (already somewhat damaged if I recall) shardplate, so you might be able to get force measures for that if the scene gives the number of lashings that Kaladin used.

[AndrolGenhald]

4 minutes ago, Dunkum said:

you're off by an order of magnitude there.  100 stone (using the british unit) would come out to 1400lbs not 140.  that is definitely unreasonable for a horse to carry on its back, so I don't think we can assume that those measures are close.

I haven't reread any of the books in a little bit, but the scene where Kaladin and Adolin fight the duel might be informative.  Kaladin's multiple lashing kick was enough to damage a (already somewhat damaged if I recall) shardplate, so you might be able to get force measures for that if the scene gives the number of lashings that Kaladin used.

Quite right, stoneweight must be a Rosharian specific measurement.

[+hwiles]

On 6/17/2016 at 5:49 PM, Emerald101 said:

While I highly applaud your efforts, I feel I have to note that Brandon was probably not calculating how much force the plate exerted on things as he writes the story. I suspect that even given perfect descriptions and hard numbers to work with we would still end up with inconsistent numbers.

Don't let my cynicism discourage you though. You go and do your math on these things, because it's wicked cool. just don't expect to get fully consistent answers.

I absolutely, positively, 100% believe you are correct in this assumption as he would, IMO, be absolutely mad to do so.  However, I would speculate that the editors and pre-publishing readers usually (though definitely not all the time) have the presence of mind to check most major scenes for internal consistency, particularly when they make mention of specific numbers.

As I continue this work I intend to (if possible...) find a way to flesh out the finer details of that internal consistency.  Where potential conflicts exists I hope to identify the most plausible assumptions that can explain them; for example, I expect that at some point the weight/carrying capacity of the species of horse that lives on Roshar might come into question, since there's no technical reason it should be the same as the species we're familiar with.  As fans of the series the responsibility falls on us to fill in any logical gaps and explain/analyze how and why things happen.  If you need any convincing about what makes a story truly great, enduring, and life-altering, just google "Wookiepedia."

 

TLDR;

Emerald101 is right, Sanderson's time is better spent writing books than doing math, Starwars is totally boss, Ima keep working on my calcs and post when I think I have enough to be interesting again.

 

Thanks for all the support!  Glad I'm not the only one who has an easier time visualizing things when some numbers are attached (even if they might be a bit rough).

[Erunion]

On 6/17/2016 at 10:42 AM, hwiles said:

As to that...I was trained as a physicist, so I do all my math in metric however, since getting employed as an engineer (in the US), I've been browbeaten into reporting my final answers in imperial...

Aiyah! I feel for you. I'm being trained as an Engineer north of the border (in Canada, if that's not obvious) - they occasionally have us do stuff in Imperial in case we're hired by or work with people from down South (the USA). I'll send you a commiserative hug. 

On 6/17/2016 at 11:51 AM, CaptainRyan said:

It does not mention the draw weight of the Grandbows (sorry, Adolin dislikes the name Shardbows) but it does mention that the draw weight is so high that the steel itself would be warped and unusable without the gemstome fabrial enhancing the strength of the steel. So, the draw weight literally exceeds the strength of the steel.

For a bow, they'd likely use different types of spring steels (like those used in some crossbows). These have a yield strength of anywhere from 450 MPa to 1000 MPa (although that's more accurately the yield strength of a specific steel alloy, with more than just carbon and iron).

To really figure it out though, you have to figure out the thickness of the bow (is it hollow? Or solid? Thick or thin?). And then deal with the tension/compression characteristics, which I'm not up to handling right now (have an exam in an unrelated topic tomorrow...)

[+hwiles]

7 hours ago, Erunion said:

Aiyah! I feel for you. I'm being trained as an Engineer north of the border (in Canada, if that's not obvious) - they occasionally have us do stuff in Imperial in case we're hired by or work with people from down South (the USA). I'll send you a commiserative hug. 

For a bow, they'd likely use different types of spring steels (like those used in some crossbows). These have a yield strength of anywhere from 450 MPa to 1000 MPa (although that's more accurately the yield strength of a specific steel alloy, with more than just carbon and iron).

To really figure it out though, you have to figure out the thickness of the bow (is it hollow? Or solid? Thick or thin?). And then deal with the tension/compression characteristics, which I'm not up to handling right now (have an exam in an unrelated topic tomorrow...)

The shafts they fire are "three fingers thick."  That sounds like between 1-1/2 and 2 inches to me.  I don't work with aerodynamics much, but it seems to me that it wouldn't make sense to have an arrow that thick unless it was solid (not hollow) for a number of design and physics reasons (I think this is a fairly rock-solid assumption).  I need to do some more research, but I'm confident that the length of the shafts can be estimated from the height of the shardbearers, which will give us a good figure for the weight of the arrow.  I believe there are some more-or-less standard formulas for estimating the ideal draw-weight from shaft-weight.  Probably won't have time to flesh out the numbers for a day or two, but wanted to throw this out there.

[CaptainRyan]

2 hours ago, hwiles said:

The shafts they fire are "three fingers thick."  That sounds like between 1-1/2 and 2 inches to me.  I don't work with aerodynamics much, but it seems to me that it wouldn't make sense to have an arrow that thick unless it was solid (not hollow) for a number of design and physics reasons (I think this is a fairly rock-solid assumption).  I need to do some more research, but I'm confident that the length of the shafts can be estimated from the height of the shardbearers, which will give us a good figure for the weight of the arrow.  I believe there are some more-or-less standard formulas for estimating the ideal draw-weight from shaft-weight.  Probably won't have time to flesh out the numbers for a day or two, but wanted to throw this out there.

If Anticipation spren existed in our world I would literally be surrounded by them!

[Physicist of the Cosmere]

On 6/17/2016 at 10:42 AM, hwiles said:

As to that...I was trained as a physicist, so I do all my math in metric however, since getting employed as an engineer (in the US), I've been browbeaten into reporting my final answers in imperial...

My condolences as well. But now that I think about it, I'm going to have to do that too, won't I  

[Erunion]

8 hours ago, hwiles said:

The shafts they fire are "three fingers thick."  That sounds like between 1-1/2 and 2 inches to me.  I don't work with aerodynamics much, but it seems to me that it wouldn't make sense to have an arrow that thick unless it was solid (not hollow) for a number of design and physics reasons (I think this is a fairly rock-solid assumption).  I need to do some more research, but I'm confident that the length of the shafts can be estimated from the height of the shardbearers, which will give us a good figure for the weight of the arrow.  I believe there are some more-or-less standard formulas for estimating the ideal draw-weight from shaft-weight.  Probably won't have time to flesh out the numbers for a day or two, but wanted to throw this out there.

That's a good way to go about it! As an archer, I'm quite sure those tables exist (recently bought some arrows and the shopkeeper went through them to confirm which arrows I should have for my bow). 
I wasn't thinking hollow arrows, but a hollow bow - so a hollow tube of springsteel to serve as the bow itself (with the built in fabrials). The risk of buckling under compressive loads might not be worth the reduced weight though :/

[+hwiles]

Okay, here we go again:

On average, a person's arm-span is very close to their height.  Let's look at Adolin this time, who IIRC is described a handful of times as being relatively tall; I'm going to take this to mean about 6ft, giving him a 6ft (183cm) arm-span.  As it happens, a good estimate for a person's ideal draw-length is calculated as arm-span / 2.5 or 73cm in Adolin's case.  That would make the arrows about 75cm long.  The volume of an arrow fired by a Grandbow is therefore between 850 and 1,500 cc's depending on what exactly you consider 3 fingers thick to be; let's call it 1,000cc's to make things easier.  Flipping through the book I saw that the arrows for the Grandbows are described as "black" which doesn't tell us what they're made of...however, we can reasonably assume they're either steel, or some kind of hardwood.  We'll examine both possibilities: if they're made of steel, they'll have a mass of about 8 kg, if they're wood they'll have a mass of about 0.7 kg.

Lighter arrows tend to fly straighter and faster for longer, and speed is more important than mass in terms of damaging a target, however, shortening, shaving-down, or otherwise reducing the weight of an arrow below a certain point compared to the draw-weight of a bow introduces a number of technical and safety concerns.  Modern standards recommend an arrow weigh at least 5 grains (which is a crazy archaic imperial unit for measuring barley...) per 1 lb of draw-weight.  8kg = 123,459 grains, giving a maximum draw-weight of 617,000 lbs, which sounds unreasonably high.  0.7kg = 10,802 grains, giving a maximum draw weight of 54,000 lbs...Unfortunately, this still sounds at least an order of magnitude too high to me, luckily, it's technically just an upper-bound.  BUT WAIT!  Recall that we estimated that during a vertical jump a shardbearer's legs can each exert about 2,300 lbs of force on the ground, and I believe Sadeas is loosely described as firing the bolts off in relatively quick succession without exerting himself.  Giving us a really rough estimate of Grandbow draw-weight of between 1,000 and 2,000 lbs.  The scientifically accepted methods for predicting an arrow's velocity don't apply well to arrows that are this massive...however...by assuming a Grandbows provides a velocity similar to that of a normal soldier's recurve bow (about 225 ft/s) I came up with a required draw-weight of 1,200 lbs to 1,500 lbs.  I think this is pretty much as good a figure as is obtainable without actually trying to build and test a Grandbow.

Conclusions:

Grandbow arrows are around 3.8cm thick, 75cm long, and probably made of wood, since metal arrows of this size wouldn't fly very fast or accurately without an impossibly high draw-weight (even for a shardbearer).  The Grandbow has a draw-weight of around 1,200 lbs to 1,500 lbs, which is about the weight of a smart car and a little bit more than the weight of a particularly large polar bear (and more than an order of magnitude higher than a normal bow).

 

TLDR;

Adolin could easily dead-lift a smart car and probably throw a polar bear pretty far (depending on how he gripped it).

 

EDIT: Sorry that the units are all over the place, that's sort of just how we do it in the States (and after so long I've just gotten used to it)...Not an excuse, just an explanation. 

Edited June 22, 2016 by hwiles

[Oudeis]

What I find especially fascinating is that in a recent signing line he specifically said that he didn't imagine that a pewterarm could lift a car. Interesting that this is the comparison you make.

[Erunion]

hrmm. Nice analysis hwiles!

Just some considerations: If it's that thick and black, the arrow might well be hollow metal. We use hollow metal arrows nowadays. 
But these are practically handheld ballistae.

Good analysis on the strength of the bearers!

[+hwiles]

15 hours ago, Erunion said:

hrmm. Nice analysis hwiles!

Just some considerations: If it's that thick and black, the arrow might well be hollow metal. We use hollow metal arrows nowadays. 
But these are practically handheld ballistae.

Good analysis on the strength of the bearers!

True, I can't prove that the arrows aren't hollow, however, for projectiles as large as these, the amount of air-resistance would be tremendous.  Without access to both an aviation and well-stocked material science lab it's difficult to say, but I'm pretty confident that, if they wanted to reduce the weight of the shafts, shaving the outside down, rather than hollowing them out would result in much better accuracy and range, and a solid shaft of that size probably wouldn't rip apart at those speeds and forces.  I feel like I may have glazed over these points before, so thanks for the opportunity to clarify!