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Basics
Aug 3, 2019 21:59:04 GMT 5
Post by theropod on Aug 3, 2019 21:59:04 GMT 5
Good, then it was worth it.
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Basics
Aug 12, 2019 8:15:33 GMT 5
Post by Infinity Blade on Aug 12, 2019 8:15:33 GMT 5
@ theropodDo you have a source for the compressive strength of bone being <200 MPa? Sellers et al. (2017) claim it is not more than 100 MPa. 100 MPa is <200 MPa (duh), but I'm under the impression your original post implied it's just a little less than 200 MPa.
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Basics
Aug 12, 2019 12:55:40 GMT 5
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Post by theropod on Aug 12, 2019 12:55:40 GMT 5
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Basics
Aug 12, 2019 18:48:49 GMT 5
Post by Infinity Blade on Aug 12, 2019 18:48:49 GMT 5
Ah, that all makes sense. I wonder if it's possible that a decline in the compressive strength of bone with age occurs in all animals, and not just humans.
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Post by creature386 on Aug 12, 2019 20:13:23 GMT 5
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Basics
Aug 21, 2019 17:29:32 GMT 5
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Post by theropod on Aug 21, 2019 17:29:32 GMT 5
While most people here have a good grasp on this, let’s do this anyway in case it ever comes up: ScalingA common problem in (palaeo)biology is to scale up certain properties of one animal to another animal of different size. The important thing to keep in mind is what property is being scaled, and on how many dimensions it depends on. A length is a linear measurement, thus one dimension, and scales linearly (¹). Area, and associated muscle force, are two-dimensional, and scale with the square (²). Volume and mass are three-dimensional, and scale with the cube (³). Say we want to find body mass M of an animal of a known dimension (e.g. skull length, or total length) X from another animal of known equivalent dimension x and body mass m. Given the object has the same shape (called isometry), but is twice the length, it will be 2(longer)*2(taller)*2(wider)=8 times the volume. The formula for scaling volume is this: m*(X/x)³=M Of course it should be kept in mind that animals do in fact change proportions as they grow larger (this is called allometry), so the third power rule is merely an approximation, but it is often good enough and generally a reasonable default assumption. Analysis of the scaling of body mass with respect to dimensions in specific taxa may suggest a different relationship, e.g. 3.3rd or 2.7th power instead of 3rd power. If this has been found to be the case, then you might want to consider using that (it will usually come with a full equation). Other properties scale with different exponents. E.g. area scales to the second power, and with it force, because the force a muscle can produce depends on its cross-sectional area. So if a skull is 2 times longer, the area of the muscles, and the bite force, will be 2(wider)*2(thicker)=4 times higher. So if you are looking for bite (or other muscle) force F, and have known dimension X, and smaller animal with equivalent dimension x and force f, the formula you need is: f*(X/x)²=F You can also scale these with respect to each other. For that, simply take the root first, then the appropriate power. Say you know an animal with mass M, and another animal with mass m and dimension x. To find dimension X of the first animal, use: x*(M/m)^(1/3)=X If instead of a dimension, you are looking for a force F, and know force f, then you have to use the 2nd instead of the first power: f*(M/m)^(2/3)=F This is the same as: f*((M/m)^(1/3))²=F Some things to note: What this means in practice, is that a relatively small increase in length can mean a significant increase in force, and an even bigger increase in mass. Try to check, before scaling, if that length you are scaling on actually corresponds to increases in other dimensions as well.
E.g., if two animals seem to be essentially the same size, but one has a much longer tail, scaling body mass based on total length might not be a good idea. If two animals have similarly wide and tall skulls, but one is much longer, scaling bite force from skull length may not be a good idea. This also goes in reverse; a relatively large difference in mass can mean only a relatively small difference in dimensions, so if one animal is 200kg and has a 32cm skull, don’t mistake it as having a proportionately smaller skull than a 50kg animal with a 20cm skull–they are in fact the same, mass and length merely scale differently. This also has some biological implications. Because of the same mechanism, as animals grow larger, their mass grows bigger in relation to their surface area and the cross-sectional area of their limb bones. This can cause problems, such as dissipating body heat, or weight bearing. On the other hand, very small animals have proportionately very large surfaces, and can have the reverse problem, needing to prevent excessive heat loss, but their small size also means that they are relatively much stronger and more resilient than larger animals, exemplified by ants or beetles lifting tens or hundreds of times their own body weight, or insects falling from great height without injuring themselves. further reading: en.wikipedia.org/wiki/Square%E2%80%93cube_law
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Post by Infinity Blade on Aug 21, 2019 22:06:12 GMT 5
”Primitive", "superior", "more advanced”Quick note: I am not referring to the use of the words "primitive" or "advanced" under the context of them basically meaning "basal" or "derived".A perception that still exists among people today is that organisms are “superior” by being more “highly evolved” or “more advanced” or something along those lines, that the purpose of evolution is positive progress for organisms in being adapted to their environments. To put it another way, they see the evolution of organisms as akin to technological advancement, where two things might have the same function, but the more recent model is better at this function for being innovated upon specifically to be better than the earlier model. Animals living today are seen as somehow being the “best”. However, the evolution of organisms and the advancement of human technology are not comparable. Unlike technological advancement, evolution is not a progression from good to better. It is a progression from being well adapted to this environment to being well adapted to that environment. The present day is just another arbitrary point in time like any other; there is no reason to assume that organisms back then were any less adapted to surviving in their own environments than modern animals are to theirs. An organism may have evolved certain traits that allow it to prosper in one environment, but that doesn’t mean it would have fared well in an earlier, primordial environment with older, more “primitive”, and supposedly “inferior” species. A polar bear is well adapted to its icy habitat, but how would it have fared in the hot swamps of the Carboniferous? Or in the tropical forests of the Eocene? But to say a polar bear is somehow inferior to Carboniferous or Eocene-era animals is ridiculous and arbitrary. Similarly, to claim that an organism that went extinct thousands, if not millions of years ago is “inferior” in the context of any single present-day environment is completely arbitrary and doesn’t mean anything other than the fact that organisms are better suited to survive in certain environments than others. Contrast this with technological advancement. There was nothing about the context of the 18th century that would have made the use of single-shot muzzleloaders (and the manufacture thereof) better than the use of fully automatic assault rifles (and the manufacture thereof). There is, however, something about the context of an Eocene tropical rainforest that would make a polar bear ill-adapted to it. I'll let the reader read pp. 105-106 of Janis (1993)->, which is where I got part of what I've written down here. EDIT: I found a great explanation of this in my plant biology textbook. Botany: An Introduction to Plant Biology (Sixth Edition)->, p. 610. By James D. Mauseth.
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Aug 21, 2019 22:44:34 GMT 5
Post by creature386 on Aug 21, 2019 22:44:34 GMT 5
Related to the above: The correct way to read a cladogram is from the root (also called the node) to the tips (e.g. the taxa). Not from the lower taxon to the higher taxon or anything.
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Post by theropod on Aug 22, 2019 2:47:51 GMT 5
Also, a term some people use as a sort of more scientific equivalent for "primitive", but which actually has a very specific evolutionary meaning: Basal
You cannot just call a taxon "basal", because it can only be basal with respect to other taxa, and what constitutes "basal" depends on the perspective.
Sponges aren’t inherently "basal" or "primitive", their evolutionary history is just as long as that of all other lifeforms, they merely never developed a comparable complexity to some other animals. But with respect to taxa within Eumetazoa, sponges are basal animals, because they were the first to split from the lineage leading to Eumetazoans. But you can also turn this on its head, from a sponge perspective, Eumetazoans are basal animals, because they split from the lineage leading to sponges. So if you want to call sponges "primitive", or "basal", maybe think about whether what you want to say isn’t actually more properly described in term of more or less "complex", or better still, in terms of more specific features of the organisms you are talking about.
Similarly, one might call coelophysoids "basal neotheropods" or ceratosaurs "basal averostrans" or megalosaurs and carnosaurs "basal tetanurans", because usually we are looking at it from an extant bird perspective, and these taxa were the first within the respective clades to split from the lineage leading to birds. But while this would be unusual, you could also look at it from a ceratosaur perspective; from this viewpoint, tetanurans would be basal.
A helpful concept in this regard is the crown group. Strictly speaking, a crown group of a taxon is the group including all extant representatives of a given taxon, their last common ancestor, and all it’s descendants. When you have a crown group (e.g. extant birds with respect to other dinosaurs), you can refer to all the rest as (the so called stem lineage) as basal members, and it will usually be understood what you mean (e.g. basal physeteroids, basal cetaceans, basal avialans, basal synapsids...). However, take care, some people, very confusingly, use the term "crown group" more loosely to refer to major radiations within lineages, e.g. crown-group sauropods (=eusauropoda), even if they aren’t extant.
Great, I see that I now have to write a post on cladistics.
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Post by creature386 on Aug 23, 2019 19:38:35 GMT 5
An explanation of principal component analysis I sneaked into a different thread. I think it was good to explain it with a concrete example and what we can learn from it, although it made my post needlessly intimidating in the context of the debate. For those interested in context, go here: theworldofanimals.proboards.com/post/43954/threadFor the record, I don't understand PCA quite well myself. If theropod has a better explanation or if he has finds significant flaws/omissions, I'd be glad to hear his.
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Basics
Aug 23, 2019 22:48:25 GMT 5
Post by Verdugo on Aug 23, 2019 22:48:25 GMT 5
^ I log in just to like this post. Great job Creature! It's not exactly 'basics' like 'Force/Pressure' or 'square cube law' but oh well, i myself don't understand it fully either so reading your post does help me understand it a bit more.
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Post by Verdugo on Sept 17, 2019 13:23:56 GMT 5
Question about Second Moment of Area and Bending Strength?Recently, i came across these concepts why reading a paper on Canine strength of Felids by Per Christiansen. According to the paper, Bending Strength can be calculated as: I: is Second Moment of Area h: moment arm distance (distance from the Force application point to the point whose strength you're trying to measure) Second Moment of Area can be calculated as: B: pi x, y: linear dimensions (The formula is just the formula of Second Moment of Area of filled oval cross section because Felid Canines are oval in cross section. See wiki for details) Going by the formula, it's obvious that Second Moment would be to the Fourth power of linear dimensions (Unit is probably mm^4). Hence, the Bending Strength would be to the Third power of linear dimensions (mm^4 / mm = mm^3). This means that the Bending Strength will scale at a similar rate to Body Mass. Here is my question. For years, i have always thought that Bone Strength is only scaled to 2nd power while Mass is scaled to 3rd power. That's why larger animals are not simply just scaled up versions of smaller ones, they are usually also much more robust in their limb bone measurements (Tiger is more robust than House Cat for example). This makes sense because more robust bones would help compensate the relative decrease in Strength compared to Mass (as Strength is scaled as lower rate than Mass). It turns out that Bending Strength is scaled similarly to Mass. Obviously, this would beg questions. If Strength is scaled similarly to Mass than why larger Animals need more robust bones and shouldn't Animals grow to Godzilla size by now? What am i missing here? EDIT: No worries, i figured it out. Turn out the Second Moment Are Bending Strength is not the same as the Strength concept i usually think of (aka ability to withstand/generate Force). Reading Therrien (2005) (pg 183-184) help me figure that out.
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Post by theropod on Sept 17, 2019 15:57:06 GMT 5
Good post nevertheless!
The second moment of area is relevant for bending and torsional strength, but it is also an indicator of how much a structure will resist deflection (a beam will flex less for a given force applied to it if it is deeper in the direction of the force, but it will also take less deflection for it to fail because the tensional and compressional stress at a given deflection will be higher). In these loading cases, the area has moment arm around the bending axis, but the load has a moment arm too, and both grow larger as the structure grows larger. For example, second moment of area around the x-axis for an ellipse is 0.25*pi*r(x)*r(y)³, so the area times the (y axis radius)², but if it is loaded, the force also has a moment arm that also grows as the structure is scaled up. Deflection (in radians) for a beam supported on one end is FL^2/(2EI) (F being force, L being length, E module of elasticity and I the second moment of area)
The impact of the length is squared, so that brings us back down to the bending resistance scaling with the square, given that the force also has an isometrically scaling moment arm.
And of course resisting compressive loadings (such as in roughly columnar limb bones) would be completely unaffected by all of this, as it’s only the cross-sectional area that’s relevant here.
Similarly to how bite force also scales with the second power overall (barring allometry, of course). If we disregarded the fact that there are two moment arms, one might argue that bite force should scale with the cube, but this would only happen if only the in-lever of the force got larger, while the out-lever stayed the same.
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Post by Verdugo on Sept 17, 2019 20:26:02 GMT 5
Good post nevertheless! Deflection (in radians) for a beam supported on one end is FL^2/(2EI) (F being force, L being length, E module of elasticity and I the second moment of area) Don't really understand what you meant by this part. But i guess you're probably explaining the same thing that Therrien 2005 said. Personally, i think the explanation in Therrien to be perfectly understandable. The structure has its own Bending moment area and the Force loaded would also has its own Bending moment area. In order to work out the maximum Force the structure can handle, we'll just let the Force load till its Bending moment area is equal to the structure's Bending moment area. Thus, we'll get the equation according to Therrien 2005: Force = Structure's Bending Moment Area / Moment Arm Length This brings it back to 2nd power. Since Force and Strength are interchangeable. I think it would be most meaningful that Bending Moment Area be 'converted' to Force if you want to measure the Strength of something. However, i still have some questions. Is it meaningful to just compare the structure's Bending moment area on its own? Since it does not actually tell you how Strong the structure really is (how much Force it could be loaded till failure)? For example you have two identical Structures: Structures A and Structures B. They are identical in every ways, the only difference between them are their size as B is twice the size of A in linear dimensions. This means that the Bending moment area of B will be 8 times that of A. However, it's Strength (the amount of Force it can be loaded) is only 4 times that of A.
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Basics
Sept 17, 2019 21:01:11 GMT 5
Post by theropod on Sept 17, 2019 21:01:11 GMT 5
Well, almost. I may have overcomplicated it, Therrien et al. can certainly explain it better than I can, especially since I’ve always struggled with materials science myself.
I was talking about deflection, they are just talking about bending moments. The bending moment divided by the second moment of area I around the relevant axis gives you the maximum stress (that is one of its uses of I).
However, i still have some questions. Is it meaningful to just compare the structure's Bending moment area on its own? Since it does not actually tell you how Strong the structure really is (how much Force it could be loaded till failure)? For example you have two identical Structures: Structures A and Structures B. They are identical in every ways, the only difference between them are their size as B is twice the size of A in linear dimensions. This means that the Bending moment area of B will be 8 times that of A. However, it's Strength (the amount of Force it can be loaded) is only 4 times that of A. It is meaningful to compare second moments of area, they are related to strength, just not linearly.
Comparing bending moment itself isn’t very meaningful, because the bending moment is just force*length, so it depends on what force you apply, and where. But you can divide the known bending moment by the second moment of area, giving you the maximum stress caused by that bending moment in the structure, and then you can see whether that stress is below the yield stress of that particular material (presumably bone/dentine/enamel). Or you could transform that the product of second moment of area times the maximum stress of the material will give you the bending moment (F*L) it could withstand, and from that, you could figure out the force at a given length, or the length at a given force it could sustain.
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