## Sunday, 22 May 2011

### How many legs are best for megamonsters?

About a year ago I wrote two posts about what happens to legs when an animal is scaled up (here and here). In a nutshell, if you make an animal's body twice as big, the new body will weigh eight times as much as the old one and not twice as much. If you make the legs twice as big, they will not be strong enough to carry the new weight, so the only solution is to make the legs more than twice as thick. The result of all this is that legs have to make up a larger proportion of a very large animal than of a small one. There is a limit to how big you can make an animal: at some point the legs need more food than the body can deliver, or something equally silly.

Click to enlarge; copyright Gert van Dijk

Last week Jan asked a question on the bulletin board of the Furaha site asking which body plan would be best for really big animals. That question made me think: the more legs there are, the smaller each leg can be to carry the body. What does that do to the mass of all legs together? If the total mass of six slender legs would be less than that of two thick legs while doing the same job, than having six legs would be a better design for very large animals than having two. That would be a nice outlandish and unearthly solution! It is shown above in a rather silly image (the human figure came with the program and is there for scale only).

Whether it would work or not was not intuitive to me, so I did some homework and came up with the work of Robert McNeill Alexander (if you are interested in biomechanics you will encounter his work many times). In this case, part of the answer was described in this book Optima for Animals.

Click to enlarge; copyright Gert van Dijk

The reasoning starts with tubes. There are excellent reasons why vertebrate leg bones and insect legs are tubes, and why tubes are such important structural elements in technology. They are about as strong as solid rods of the same diameter, but weigh a lot less, and doing the same job with less bone is a good idea. The three tubes above all have the same outer diameter, but the hole down their lengths differs in diameter. In papers on the subject you will find an very important parameter 'k': it describes the width of the inner hole as a fraction of he outer diameter. In the left one k is 1, so the hole is tiny. In the middle bone k is 0.5, and in the right one k is 0.9, meaning there is just a thin shell of bone. A value of 0 means no hole at all, and a value of 1 would mean the bone is infinitesimally thin (in simple words: there is no bone!).
Are all these bones equally strong? No, they are not. If you just take bending forces, there is a nice formula which contains three items of interest: the 'bending moment' M (the force that the bone needs to withstand), our friend 'k', and the radius r of the outer side of the bone (there is only one thing else and that is a constant K for the material - ignore it-).

Here it is: r=[M/K(1-k^4)]^0.33

If you keep the force M constant you can calculate what the radius is for any given value of k. Let's do so.

Click to enlarge; copyright Gert van Dijk

The image above shows what happens for four values of k: 0, 0.3, 0.6 and 0.9. As you can see, the value of the radius increases as well: the thinner bones have to be wider to withstand the same pressure. Note that that is hardly the case when the holes are fairly small. In fact, the effect is only really noticeable when k increased form 0.6 to 0.9. Does it matter? yes: the cubes in front of the bones represent the mass of the bone itself, and that nicely shows why tubes are good. They are all equally strong, but the thin-walled ones weigh a lot less.
On Earth, things are more complicated for mammals because the holes in the bones contain marrow. Marrow, while less heavy than bone itself, still makes the bone as a whole heavier. A bone with a very thin shell will be wide and will contain lots of marrow, defeating the purpose to make bones light. For mammals with marrow in their bones there is an optimal value for k, at which the bone as a whole weighs the least; that value turns about to be about 0.63. If, however, you manage to put air in the hole instead of marrow like birds, than the story becomes different and you can increase k. Above around 0.9 the bones the become too susceptible to buckling, so there is another optimum value for air-filled bones: a value of 0.9 is excellent. Our hypothetical megamonster shall therefore have air-filled tubular bones with a value of k of 0.9!

We still are not there yet. The real question was what happens if we give the animal more legs. Let's assume that the forces are simply divided among the legs, so with four legs each leg has to carry exactly one fourth of the burden. Remember that there were three parameters of interest in the formula: The bending moment M, the outer bone radius r and k. Set k to 0.9, and then we can calculate r for four bending values. The values per leg are 1 (the animal has one leg), 0.5 (two legs), 0.25 (four legs) and 0.125 (eight legs).

Click to enlarge; copyright Gert van Dijk

This picture show the resulting bones, along with the number of bones. As you can see, the bone for an animal with eight legs is a lot less thick than for the one with just one leg. So far so good. The smaller bone must weigh a lot less than the bone for the one-legged animal, which is what we wanted. Then again, there are now eight such bones, so the question is what their combined weight is.
Each bone of the two-legged megamonster weighs 63% of the one-legged one, so the two bones together weigh 126% of the one bone. That is not what we wanted, as the two legs weigh more than the one leg. Does it get better if we add more legs? Well, for four legs each one weighs about 40% of the one bone, and together they weigh 159%. For eight legs, each one weighs 25% of the one bone, so the total weighs 200% of the one bone.

How disappointing... I had hoped it would be the other way around. Now it seems that fewer legs is the better way to save weight if you need a mega-monster. Obviously, giving it just one leg is not practical; there would be a big risk of falling, and the only way to move would be to jump in a series of bone-shattering hops. Two legs is quite feasible; just think of carnivorous dinosaurs. Four is also good. In a last-ditch attempt to save the concept of multi-legged megamonsters I could say that having six or eight legs provides safety as a possible advantage. A two-legged monster with a broken leg is doomed with certainty, and a four-legged one probably is. But a six-legged one could deal with one broken leg and hobble away.
You might expect animals with a multi-legged body plan to lose some limbs as they grow bigger and bigger as a measure to save weight. Such limbs might be given another purpose than locomotion, so they could develop into, well, just about anything. They could develop clavigerism or centaurisation, also interesting.

I'm still disappointed though...

j. w. bjerk said...

I think though that you haven't put the final nail in the coffin of plausible many-legged giants. I don't question your math, but i suspect there are additional factors in biological systems that you haven't taken into account.

Your calculations imply that 2 hollow bone legs are optimal for carrying weight, but the really large ground birds --like ostriches-- tend to have much less air pocketing (if any) in their leg bones than flying birds. Presumably normal highly air-filled leg bones don't work for the big ground-dwelling birds.

I'm not much of an engineer, but bending isn't the only force a bone needs to endure. Perhaps the very high values for k the bone is much more vulnerable to shattering, or some other failure from sudden forces?

I wonder if the results would be different assuming a much lower value for k....

Zerraspace said...

Out of curiosity I decided to plug in some more numbers to see how other numbers of legs would fare. For 3 legs the mass is 144%, for five it is 169%, for six it is 181% and for seven it is 191%. Just for fun I decided to see how a rusp would fare and it turned out with 24 legs the result was 288% - not all that far off from 200% for 8 given that it has triple the number of legs. Past a certain point perhaps you could just keep adding them with minimal weight gain. Maybe if the creatures' body were long enough adding extra legs would be preferable to relieve the stress on the spine.

Personally I wonder why the legs of such giants are as long as they are. Why not shorten them till the body is held only a few inches off the ground? Past a certain size it is unlikely the creature will be hunted, and they can't take much damage from falling or tripping with such minimal distance to the ground. For that matter they could lay on their stomachs when not moving, reducing the strain on their limbs. A gastropod's system of locomotion might be best suited to such lifestyle - if only one could foresee how this creature could reach such a prodigious size in the first place...

I'm not sure I'm completely convinced that hollow bones are the best way to go - wouldn't it be better to at least add a crisscrossing matrix or a honeycomb on the inside to increase structural integrity with minimal weight gain? For that matter, what would be the effect of filling these bones with a fluid? I know it would increase weight tremendously, but surely that too could increase the outward strength of the bone due to its incompressibility.

j. w. bjerk said...

Zerraspace:

legs that held the body only a few inches off the ground, would be very lousy for moving -- i.e. one stride would only move it a few inches. It might as well sit on it's belly.

Bird's hollow bones have criss-crossing struts for support, no doubt they are structurally useful. I don't think he's arguing that bones are better without the struts, nor that bones should be perfect cylinders -- it's just the math is much easier if you simplify the geometry.

Zerraspace said...

I didn't consider that about the bones when looking at the pictures and I apologize. I was aware that such short legs wouldn't be much good for moving, but I reasoned that such a large creature with little to no predators wouldn't need to move quickly anyway.

Jan said...

Very interesting, but this concerns only stationary animals, right? When in motion, multilegged animals can move only a few of its legs, unlike animals with just a few legs. I hope there are some way of saving giant millipedes...

Luke said...

For a very long animal there's no choice but to have several legs, to support the back. The animals are long presumably to accommodate an extra long gut.

Sigmund Nastrazzurro said...

JW: There are indeed other factors at play, but an analysis like this isolates one factor and looks at what it does on its own. As for bending, that is not the only force bones have to withstand. McNeill Alexander's book went into that, and also took buckling into consideration. Thin bones with a large radius are very prone to buckling, and in them failure s more likely to be due to buckling than to bending. For thick bones with a small radius the opposite is true. This is why the value of k=0.9 was optimal: it minimises the risk for both. I mentioned that in one sentence, and felt it would take matters too far to discuss all this as well.

Zerraspace: if you look at the rusp, you will see that its legs are in fact on the short side, meaning that total leg weight is manageable. The design has other consequences: for one, rusps never fall, and breaking one leg doesn't harm them much. Then again, their top speed is sedate. They are almost impervious to attack regardless, as you guessed. Jumping is out of the question. When they have to cross gulleys they suspend the front end of the body, which works well (better than in elephants; elephants cannot jump at all).
As for having some kind of connecting slender rods inside the tubes, definitely. These probably help to resist buckling, according to McNeil Alexander. I simply omitted them to limit the post to one factor only.

Jan: No, the text concerns animals that do move a lot. I assume you re thinking about legs knocking into one another. That is a risk for mammals with relatively long legs, and there are ways to solve it. The risk also exists for hexapods, whose body is longer than a comparable mammal's, but not twice as long. I am keeping some of the solutions to myself (but you will probably think of them yourself).

Luke: Tyrannosaurs come to mind though: the big heavy front end and the long tail stick out a long way in either direction without legs of their own. But at some increase in length you are right.

Evan Black said...

I agree that this isn't necessarily the final nail in the coffin for multilegged megamonsters, as other biological factors may fall under consideration. For example, in the guestbook Luke brought up the concept of ballonts returning to a generally landbased life. While I think that such a lineage would eventually lose lighter-than-air adaptations, the process of atrophy could potentially make some very interesting megafauna. Multiple legs could serve to spread out what weight does need support, but given time these traits too will give way to biological equilibrium.

If nothing else, this blog entry reminds me how biology isn't always intuitive.

Jan said...

What I had in mind is that in motion, one leg of two legged animals must withstand twice of normal weight, so the situation is a bit different. I do not know if it is important.
If multiple legs are so disadvantageous, why relatively large invertebrates with soft bodies (caterpillars, velvet worms) have them? Even the largest terrestrial arthropod ever, Arthropleura, had multiple legs.

Besides, this clearly demonstrates not only the advantages of bipedal motion, but also of simple tree trunk. Still, I would prefer some other solutions.

Sigmund Nastrazzurro said...

My friends,

It seems as if I've given you the impression that multi-legged giant animals are out of the question. Not so!

What I did was to use a common scientific trick: to isolate one factor, scrutinise its consequences, and stop right there. If leg bone mass were the only important factor, yes, then I would remain disappointed, while now I think there are enough arguments to consider multilegged giants feasible. There are other arguments to consider, such as safety factors, the weight of a body plan that may not be readily susceptible to evolutionary change, etc. So, while my studies convinced me that leg mass considerations favour fewer legs, rusps are going to stay.

Sigmund Nastrazzurro said...

This is from an email by Metalraptor Maximus:

My first comment is in response to your "multi-legged megamonsters" post, and some of the comments I was able to see before my computer acted up. I found the allometric scaling of the legs interesting, especially how having six legs to support the same mass requires more leg mass than having one or two legs.

However, as some people have said (including yourself), I can see reasons for super-sized alien organisms having more than four legs. One that was mentioned quite a bit was the idea that multiple legs allow more support of the digestive tract, and could allow multi-limbed animals to evolve a longer body or a more massive digestive tract. Indeed, several times animals have shifted around their internal anatomy in order to accommodate a larger digestive tract. For example, it is thought that ornithischian dinosaurs originally evolved their distinctive pubis in order to have a larger digestive tract but still be able to balance their weight in a way to maintain their ancestral bipedal posture.

Interestingly, most gigantic vertebrates have a hindgut fermentation system, which would benefit from a multi-legged body. I have oftentimes heard that a hindgut fermentation system benefits from a larger body because most hindgut fermenters feed on low-quality plant matter. Larger bodies mean that more food can be consumed per unit of time in hindgut fermenters than in foregut fermenters, because food moves through the digestive system of the former faster. In addition, larger bodies would mean food spends more time in the digestive tract breaking down.

Another reason why multi-limbed megamonsters could be feasible has to do with safety factors. As you said, if a quadruped or biped breaks a leg, they are pretty much doomed. This is especially the case for hollow-boned giants, because from what I have read the lack of large marrow deposits in the bone make it difficult for the bone to mend. Walking on only one/three legs can be a problem, because then an animal is more likely to trip and fall over. And falls for megacritters can be lethal. Heck, it doesn't even have to be due to an injury. More legs means an animal can distribute its weight out more, and therefore be less likely to fall over and die.
Finally, I wanted to point out that there is a possibility for multi-limbed megamonsters that has not already been considered: agility. Yes, the limbs of the one-legged megamonster were less massive than those of eight-legged one, but the one limb was also much more massive. Large animals with fewer limbs would soon reach the size where these kinds of legs are too fat to walk on. But a multi-limbed animal would have thinner legs, and therefore greater mobility at this size. Heck, thinner limbs could mean that multi-limbed megamonsters could be more agile (relatively speaking, and only up to a point) than quadrupeds.

Evan Black said...

One thing that I've never really been able to grasp with these blogs about leg width and supporting body weight: do the muscles, tendons, and other indegument play a role in supporting weight too, or does it all depend on the bone?

Sigmund Nastrazzurro said...

Metalraptor: I like the argument about agility, and you may have a point there.

Evan: tendons and muscles do play a role in supporting weight, but a minor one. after all, these tissues are not stiff and cannot resist compressive forces, so by themselves they are not useful for this purpose. What they can do is to help direct the directions of force. If a bone is subject to bending forces, and if that bone is bent a bit in the other direction by tendons and/or muscles, this will diminish the bending forces on the bone at the price of a bit more compressive force.

Sigmund Nastrazzurro said...

I contacted professor McNeill Alexander to acknowledge my gratitude for the many fine books he has written over the years, and to draw his attention to this particular post. He replied with something I had not foreseen, but that makes perfect sense. here is his answer:

"Thank you for sending this. I am not surprised by your conclusion. I discussed a related problem in my textbook The Chordates (Cambridge, 1975). Primitive mammals have five digits on each foot, but some fast runners have only two (e.g. antelopes) or one (horses). I showed by an argument similar to yours that a reduction in the number of toes allowed a sufficiently strong foot to be made lighter.
Best wishes, Neill Alexander"

Fascinating, isn't it? Fast moving animals must accelerate and decelerate their legs much more than slow-moving ones, making a reduction in leg mass worthwhile. This is also the reason why the muscles moving the legs in such animals are not in the legs themselves but higher up. I had not realised that this reason, together with the mechanics of tubes, acts as a drive to reduce the number of toes!

Petr said...

@SN and yet the fastest terrestrial runner has four digits in contact with the ground when running. :P

Anyway I am not entirely convinced that you used "correct" numbers in your calculations, you have written that the two legged animal has to carry half of the animals weight, but that's only when it's standing still, there are times where it only stands on one leg and the other one is in the air, so the two legged creature should also have been calculated with the value of 1 (I now realize that it was here where you touched upon monopods for the first time, but I still really appreciate your recent post on them, but I think this post renders monopods as absolutely unfeasable, especially because they have no leg to stand on when their leg is making a "step" =)) But back to the point. if we asume a gait in which half of the legs are in the air and half of the legs are on the ground, the four-legged beast's leg would have to carry 0.5 of the total weight, not 0.25 and the eight-legged one would have four legs touching the ground at any time when walking, so each leg would have to carry 0.25 of the weight and not 0.125. (for a hexapod, it would be 0.33 of total weight per leg. I hope you understand what I mean, this is not to be taken as a harsh criticism, just an observation of what I have noticed and sharing my thoughts here, I apologize for commenting so long after this has been posted =)
I am just interested in what the effects may be that the bones would have to be heavier than you anticipated and the final numbers would be higher than yours ;)

Sigmund Nastrazzurro said...

Petr: the other comments and my replies stress that the post is about one isolated factor only: standing still on n legs.
You are of course right that walking involves times where the weight is on fewer legs, and that indeed influences the results. An even more important consideration would be the stresses due to active movements. Dynamic stresses not only require different calculations, but also need assumptions regarding speed, stride length, etc., etc.
By isolating one factor people may understand its importance, and may then build on the reasoning for themselves. It seems to work for you ;=)

Petr said...

I am an idiot, I'm sorry, yeas I've read the comments above but my brain made a leap. =/

Sigmund Nastrazzurro said...

Supposing brains have gaits, I prefer a leaping over a plodding one...

Petr said...

You're wonderful, thank you. :D