I've seen a number of mentions in the tensegrity literature asserting (without proof or reference) that, in the context of a tensegrity rod under pure axial compression, hollow columns perform better than solid columns for the same weight.
Is this true? There must surely be papers comparing the two in a definitive and controlled way, can anyone point me towards one or more of these?
In calculating buckling loads Euler’s formula is used. $F = \frac{n π^2 E I}{L^2}$ from
where
F = allowable load (lb, N)
n = factor accounting for the end conditions
E = modulus of elastisity (lb/in2, Pa (N/m2))
L = length of column (in, m)
I = Moment of inertia (in4, m4) (technically, the second moment)
As Euler was one of the greatest mathematicians ever, it’s doubtful that either you or I could find fault with his derivation. It does however make the important assumption that axial strain is small compared to buckling strain. Also it generally overestimates the load that may be safely carried by a column as it assumes perfect materials. It does not account for imperfections in the material or faults incurred during manufacturing, quality control or installation.
The maximum safe height of the column is given by the slenderness ratio $\frac{L}{r}$ where $r=\sqrt{\frac{I}{A}}$
The slenderness ratio r for a solid column one unit in diameter is $\frac{1}{4}$ and rises to a maximum of $\frac{1}{\sqrt{8}}$ for a hollow column of the same mass.
r value vs column diameter
However, here is a problem; you could increase the diameter of the column until its wall was only one molecule thick and you would have that same value for r, yet this obviously would buckle! There has to be some diameter of column that gives a maximum r value that can also maintain its integrity, but I don’t see a formula that gives that.
That said, when the outer radius of the column has reached twice its original diameter you are approaching that maximum r value (0.33 vs 0.354max) Solid column r=0.25
There may be also considerations of the amount of space each column takes up if that space is to be used. (For example, building a Greek Temple)
Here are some very useful websites:
https://pkel015.connect.amazon.auckland.ac.nz/SolidMechanicsBooks/Part_I/index.html
(part 7.5 buckling)
https://www.engineeringtoolbox.com/euler-column-formula-d_1813.html
http://www.engineeringcorecourses.com/solidmechanics2/C5-buckling/C5.1-eulers-buckling-formula/theory/
In terms of buckling, having a larger structure is advantageous. Things are rarely ever pure so even under "pure" axial loading, the failure mode will tend to have some lateral component.
Impurities in tangential directions reduce displacements via allowing some deformations inward (think Poisson's ratio and how materials try to expand in the perpendicular plane when compressed) for the hollow entity. Hollow will also have material spread further apart, reducing shear (angular) stresses.
This however may not always be better. In order to withstand high load, one may need a high displacement to be acceptable. Resisting displacement is however often the primary purpose of structure.
Rather than seeking papers, try to model your cylinder and tube, but instead of making them straight, perhaps put the same small lateral offset to their midsections.
