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The goal is to find the optimum values of the diameter and the thickness of the circular cross-section of a cantilevered beam in bending, with the lowest weight (cross-sectional area). The main criterion is to make sure that the internal stress does not exceed a limit anywhere in the cross-section. The length of the beam has already been chosen. The approach so far has been to find the internal loading of the beam from the external loads, then use the following formula. I am using the thin-walled assumption for the tubular cross-section.

$\sigma = \frac{M_{y}}{I_{yy}} x$

However, it appears that I can increase the radius indefinitely, with the same area, and always achieve a better cross-section.

$\sigma = \frac{M_{y}}{\pi R^{3} t} x = \frac{2 M_{y}}{(2 \pi R^{} t) (R^{2})} x = \frac{2 M_{y}}{(const) (R^{2})} x$

Is the limit solely based on manufacturability, or is there something else I should consider? Also, if it is based on manufacturability, could you give some advice on what the limits would be for composites or metals? I am currently considering either an aluminum alloy or carbon fiber. This is for a beam for an eVTOL vehicle that is connected to a propeller, so if you have comments relating to my approach let me know. I am currently considering only the vertical force from the propeller since I think it would be the critical load.

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The equation for bending stress is $\sigma_b = \dfrac{My}{I}$

For a thin-walled circular shaft, $I = \dfrac{\pi d^3t}{8}$, $d = 2r$ (diameter)

Rewrite the equation, with $y = d/2$:

$\sigma_b = \dfrac{4M}{\pi d^2t} \le \sigma_{(a)llowable}$ for the composite or metals.

$d^2t = \dfrac{4M}{\pi \sigma_a}$

In order to satisfy the least weight criteria, we need to find the dimensional parameters "$d$" & "$t$", and optimize the $d/t$ ratio.

  • For stability and buckling concerns, the $d/t$ ratio shall be kept within $3300/f_y$.

Now, you can set "$t$" as a function of "$d$", and should be able to find the optimum section that satisfies all criteria.

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  • $\begingroup$ Thanks for your response. Could you tell me what 𝑓𝑦 is, and where that ratio comes from? $\endgroup$
    – fidafa123
    Jun 5 at 16:29
  • $\begingroup$ Fy is the yield strength of the material. The recommended b/t ratio was taken from the AISC steel manual. As it is mainly applicable to metals, you can set a b/t ratio in accordance with recommendations for other materials. $\endgroup$
    – r13
    Jun 5 at 17:28
  • $\begingroup$ Could you provide me with some sources on where to find these ratios? For aluminum, I was only able to find a paid copy of the ALUMINUM DESIGN MANUAL 2020. For composites, I found nothing. $\endgroup$
    – fidafa123
    Jun 5 at 19:26
  • $\begingroup$ Unfortunately, I don't have the data available, but you may consider setting the allowable stress equal to the critical buckling stress, or bending stress, whichever is lower. $\endgroup$
    – r13
    Jun 5 at 21:04

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