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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2$\begingroup$ Possible duplicate of How to calculate deflection of a cantilever beam subject to point loading $\endgroup$– Wasabi ♦Mar 12, 2017 at 1:59
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$\begingroup$ Not as far as my math skills go. $\endgroup$– Jim LewisMar 12, 2017 at 10:59
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$\begingroup$ Amount of deflection at end of beam. $\endgroup$– Jim LewisMar 13, 2017 at 13:58
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1 Answer
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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:
$$\dfrac{64WL^3}{3\pi Ed^4}$$
