How big can a tentacle be before it can no longer support its own weight?

Question

A tentacle is a muscular hydrostat; that is, it is a boneless, coreless appendage that's essentially all-muscle. Instead of pushing and pulling off of a solid core of bone or a fluid, the muscles in a tentacle push and pull off of other muscles.

This means that, while a tentacle is really good at exerting compressive force, as well as pushing and pulling, it's not very good at handling lateral loads at all. As such, a long tentacle that's extended too far horizontally will be unable to support its own weight against gravity.

I recognize that the minimum ratio of width to length here is the dimensions of an elephant's trunk (a muscular hydrostat); that is, about 15 centimeters wide by 182-244 centimeters long, resulting in a width-to-length ratio of ~1:12 to ~1:16. Elephants can lift hundreds of kilograms with a trunk 12-16 times longer than it is wide.

However, that's what evolution dictated for the elephant, not necessarily how big a trunk/tentacle can actually get - evolution optimizes for reproductive fitness, not necessarily the coolest possible bodily structures. With that in mind, and the question of evolution defenestrated: how long is it physically capable for a tentacle or a muscular hydrostat to be relative to its width before it cannot be extended parallel to the ground without collapsing?

Assume:

• that this tentacle is made out of human muscle tissue

• that this tentacle is being extended on land, rather than underwater

• that this tentacle is operating under Earth-standard sea-level conditions - i.e. 9.8 m/s^2 gravity, 1 atmosphere pressure, etc.

Note that I am not interested in determining what evolutionary pressures might lead to this, nor am I interested in how this might affect the biology of the creature it's attached to - i.e. things like blood flow issues or increased nutritional needs. Additionally, I am not referring to this previous question of mine; I am uninterested in how long and heavy such a thing can get relative to the animal it's attached to. All I am interested in is how long a tentacle can get relative to its width while still remaining capable of standing up under gravity.

Answer 1

Depends how large the base is

Imagine two tentacles B carrying another one A. And two-each C carrying B (plus another two in C to handle the weight of A). You can chain that as far as you like.

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As an estimate to feed other answers...

https://pubmed.ncbi.nlm.nih.gov/25831858/

Breaking strain of goat muscle is about 0.44 MPa when it's parallel to the fibres, and half that when at 45 degrees. 45 degrees is probably the optimal angle for making the tentacle flexible, though I haven't done the calculations.

So, assuming that there is a core (water or always-contracted muscle) which is wrapped in muscle fibres both lengthwise and reinforced by circumferential bands, your core can reach 0.44 MPa before rupture. That's about 4kg/cm^2. (This also works with a core of muscle-connected bones, since it's the muscle that's being strained to breaking.)

For an upper-bound, we'll assume the core has perfect pressure redistribution and your tentacle moves "slowly". (Resting strain is orders of magnitude lower.)

Now we come up against the problem that rockets do: Being a little bigger makes you a lot heavier, because you need more structure to support the other structure. That turns into an exponential growth. (Rocket height, tentacle length.)

An elephant's truck is about 45kg (based on the density of water) and has a cross-section of 225 cm^2, which works out to about 900kg breaking strain if the load is applied lengthwise. So this back-of-the-envelope calculation suggests that a trunk can withstand 20g in ideal conditions - call it 16 for nice numbers.

For a tentacle twice that length, you'd be limited to 4g, and at four times the length, it can (in laboratory-ideal conditions) support itself.

At this point, we can make it longer by adding more trunks at the back end. I'm no longer enjoying writing this answer, so I'll leave it to someone else to crunch the numbers for $x$, but you'll reach a point where $R_{base} = x^{length}$.

Lower bound

Elephants are probably doing well, with 1/4 of the length which would make it near-immobile and 16-20g of strain that they can handle.