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...