Determine the variation of the radius r of the cantilevered beam that supports the uniform distributed load so that it has a constant maximum bending stress \sigma_{\max } throughout its length.
Moment Function: As shown on FBD.
Question 11.36: Determine the variation of the radius r of the cantilevered ...
Section Properties:
I=\frac{\pi}{4} r^{4} \quad S=\frac{I}{c}=\frac{\frac{\pi}{4} r^{4}}{r}=\frac{\pi}{4} r^{3}Bending Stress: Applying the flexure formula.
\sigma_{\max }=\frac{M}{S}=\frac{\frac{w x^{2}}{2}}{\frac{\pi}{4} r^{3}}\sigma_{\max }=\frac{2 w x^{2}}{\pi r^{3}}\quad[1]
At x=L, r=r_{0}. From Eq. [1]
\sigma_{\max }=\frac{2 w L^{2}}{\pi r_{0}^{3}}\quad[2]Equating Eq. [1] and [2] yields
r^{3}=\frac{r_{0}^{3}}{L^{2}} x^{2}
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