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For a cantilever beam of circular cross section, loaded at the end, how does the deflection of the free end vary as a function of beam length and diameter?

I expect the answer will be something like:

$$\dfrac{L^2}{D^3}$$

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For a cantilever with a point load at the free end, the deflection at that free end is:

$$\dfrac{WL^3}{3EI}$$

Handbook of Structural Steelwork - 3rd edition

where

  • $W$ = point load
  • $L$ = beam length
  • $E$ = Young's modulus
  • $I$ = second moment of area

The Young's modulus depends on the material, see Wikipedia for various examples.

The second moment of area for a circle is:

$$\dfrac{\pi r^4}{4} = \dfrac{\pi d^4}{64}$$

So your overall equation is:

$$\dfrac{64WL^3}{3\pi Ed^4}$$

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