Sauropods, Elephants, Weightlifters

Structural Issues

by Wayne Throop

Copyright © 1995-2003

[Text Last Updated: June 27, 1995]

[Links Updated: March 27, 2003]

• Limbs
• Necks

Limbs

Message-ID: <medved.778903263@access1>
Common sense says that there may be some difference in leverage
between Kazmayer and a big elephant or sauropod, but that there isn't
enough leverage in the world to let one of those guys get to 360,000
lbs, when Kaz, who is far more muscular than they, can only get to
21000 or so.

360000^(1/3) / ((1000+340)/340^(2/3)) ~= 2.6

Further note that the sauropod limbs don't need to be as heavily muscled as Kazmaier's, because the sauropod has four limbs and Kazmaier is employing only two. Thus the above is assuming limbs far less muscular than Kazmaier's. In fact, given that leverage factor, each limb would need only half the muscle cross section compared to Kazmaier's leg.

And finally, note that Ted's estimate of 360,000 lbs mass for ultrasaur is distinctly larger than any currently accepted figure. It was done via linear projection of a few vertebrae, not even a remotely complete skeleton, and in [tD] and [KoC], there are discussions of why ultrasaur's size and shape is uncertain at best. Most researchers put the maximum known sauropod at more like 80 to 100 tons, which would make the required leverage advantage

200000^(1/3) / ((1000+340)/340^(2/3)) ~= 2.1

So the question becomes, can a leverage advantage be discovered in sauropod limbs at perhaps 2-to-1 over human limbs? First, consider what Alexander has to say in [tHM]

The kneecap is not merely a device to allow the quadriceps tendon to slide over the femur like a rope on a pulley. The force in a rope is the same on both sides of a pulley but the quadriceps tendon pulls less hard on the tibia than the muscle pulls on it. This has been demonstrated by experiments on knees, like the one shown [here] in fig.4.6a. Force transducers were attached to the tendon above and below the kneecap. They were pulled on in appropriate directions, with the kneecap in the positions it would occupy at various knee angles. The force registered by transducer B equalled the force on transducer A when the knee was straight and the two tendons in line with each other, but when the knee was bent to a right angle, the force on B was only half that on A.

The point being, human knee joints have a 2-to-1 built-in leverage disadvantage. All a sauropod would have to do is have a slightly different arrangement of the tendon route over the knee to reap enough leverage to stand in 1g. And as can be seen when diagrams [seen below] of human and elephant knees are compared, there are many differences in structure, including size and placement of the patella, and in the structure of the load-bearing surfaces. Less is preserved in detail on sauropod joints in the references I've seen, but it is very plausible that sauropod joints could avoid the mechanical disadvantage, and be at a significant advantage when compared to human joints.

Note especially the exaggerated elbow and knee structures, which yield significant leverage, which can be seen in this diagram of an elephant skeleton from McGowan's book [DS&S].

Also note the slight differences in patella and socket structure in comparing a human and elephant knee.

Specifically, Alexander goes on to say

This reduction in force by the kneecap may seem like a disadvantage, but it is accompanied by a magnification of movement.

But that's only the beginning. Let's consider other ways of estimating limb leverage.

Message-Id: <medved.775504609@access1>
A look at a top powerlifter such as Kazmaier and an elephant or
sauropod scaled to the same weight should convince anybody that the
human lifter's legs are stouter, and that hence leverage would favor
the human;

Far less vulnerable to subjectivity is to use images of actual bones. For example, Tim Walters (twalters@intuit.com) posted a comparison image (seen here). From this image, which compares the actual sizes and shapes of a human and brachiosaur tibia scaled to equal lengths, it is clear that the brachiosaur has a 1.75 to 2 times leverage advantage, when leverage is estimated by joint width over limb segment length.

The argument might be made that we don't know how thick Kazmaier's tibia are. But exercise in humans generally thickens the shaft, not the joints and not the length. So it is unlikely to be extremely different.

Then consider a set of images (seen below) comparing skeletal members of a human leg from an anatomy book, a brachiosaur femur and tibia from the textbook [tD], the human skeleton "broadened" to 350 lbs, and finally "Jensen reconstruction" of ultrasaur's forelimb from Lessum's book [KoC].

In this case, the femur/humerus lengths are scaled equal, and subsequent measurements using the joint-width/limb-segment-length estimation also show a leverage advantage of about 1.75 to 1 in favor of the brachiosaur/ultrasaur.

We are still only estimating fairly crudely from images, but at least they are accurate representations of actual artifacts instead of an artistic interpretation such as Ted recommended. Better still would be to reconstruct the bones and deduce tendon attachments and routing, and make much more accurate measurements.

But even with these crude measures, it is plausible that tendon structure plus basic dimensional differences (which are independent) could combine to yield enough leverage difference to make sauropods possible in 1g, perhaps even as much as a factor of 3 or more, well over the factor of 2 that is needed. Certainly, Ted's burden of proof to show that his claim that "there isn't enough leverage in the world" to make sauropods possible in 1g has not been met.

Necks

You don't hang a 30,000 lb load 40' off into space even if it is made out of wood and structural materials, much less flesh and blood. No building inspector in America could be bribed sufficiently to let you build such a thing. A sauropod's neck, however, particularly in the case of the recent ultrasaur and seismosaur finds, weighed several times the weight of a large elephant and, if held outwards horizontally, would actually arch downwards (the wrong way).

::: medved@access4.digex.net (Ted Holden)
::: ultrasaur/supersaur/brachiosaur/seismosaur necks [...]
::: arch the wrong way when held horizontally.
:: gthomson@gpu.srv.ualberta.ca (Greg Thomson)
:: Oh really, ever see a suspension bridge?
: medved@access4.digex.net (Ted Holden)
: Ever see a suspension bridge supported only from one side?

This is the completed Knie Bridge, an image from "Cable Stayed Bridges: Theory and Design", Troitsky, Crosby Lockwood Staples, London, 1977, page 58. All of the support us supplied by tension in cables originating on the far shore.
This is an image from the same source, of the bridge under construction. It was built straight out over the river, with no double-ended support; as in a sauropod neck, it was supported by tension forces in the cables (nuchal ligament) and compression forces along the deck structure (spinal column).

Ted's assertion that an arch shape is necessary is wrong.

But the question remains: is the sauropod neck strong enough to have been supported when held out straight in 1g? This question is addressed in Alexander's book on Dinosaur dynamics [DD&EG].

Diplodocus and Apatosaurus have neck vertebrae with V-shaped neural spines. I suggest that the V was filled by an elastin ligament that ran the whole length of the neck and back into the trunk. This ligament would have helped to support the neck while allowing the dinosaur to raise and lower its head.

I made some experiments to find out whether the idea was feasible. [...] the mass of the real head and neck would have been 1,340*10 = 13,400 newtons. [...]

We can only guess how thick the ligament was, but it seems likely that it projected above the tops of the neural spines. If so, its center line, at the base of the neck, was about 0.42 meters above the center of the centrum. [...]

The weight acts 2.2 meters from the joint and the the ligament tension have been needed to balance the weight is 2.2 * 13,400/0.42 = 70,000 newtons (7 tons force). The third force shown in the diagram [seen here] is the force in the joint itself, where one centrum presses on the next.

The calculated tension may seem enormous, but the ligament was very thick. If it was as thick as in the diagram, its cross sectional area was 40,000 square millimeters and the stress in it, for a force of 70,000 newtons, would be 1.8 newtons per square millimeter. This is more than the stress in the ligamentum nuchae of a deer with its head down (about 0.6 newtons per square millimeter), and would be enough, or nearly enough, to break ligamentum nuchae. However, this stress would act only if the ligament supported the neck without any help from muscles. If neck muscles took some of the load (as they do in birds) the stress would have been less. The suggestion of an elastin ligament seems feasible.

Now, Ted has said that even if diplodocus is workable (borderline perhaps, but workable), the larger sauropods, especially ultrasaurus or seismosaurus, would be Right Out.

That turns out not to be the case. The actual neck proportions of other sauropods were different than those of Diplodocus. Even considering the seismosaur, which very much resembles a 1.6 times scaled up diplodocus, and therefore (if isometrically scaled) would have had 1.6 times the load per area in its nuchal ligament, also had (according to Gillette in [StES]) neural spines proportionally two times longer. The total force on the ligament goes up as the fourth power of length (mass times moment arm), and the loadbearing capability only goes up as the cube (ligament cross section times moment arm). Extending the ligament's moment arm as Gillette describes compensates for the increased load, and the seismosaur is thus not worse off than the diplodocus, overall.

It would be better still to actually measure in more detail, but the point is, seismosaur, ultrasaur, whateversaur you choose, plausibly had sufficient strength to move their neck around in 1g gravity. Even for the larger sauropods "the suggestion of an elastin ligament seems feasible".

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