Question 11.38: Determine the variation in the width b as a function of x fo...

Mechanics of Materials – Instructor Solutions Manual [EXP-4667]

Determine the variation in the width b as a function of x for the cantilevered beam that supports a uniform distributed load along its centerline so that it has the same maximum bending stress $\sigma_{\text {allow }}$ throughout its length. The beam has a constant depth t.

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Section Properties:

I=\frac{1}{12} b t^{3} \quad S=\frac{I}{c}=\frac{\frac{1}{12} b t^{3}}{\frac{t}{2}}=\frac{t^{2}}{6} b

Bending Stress:

\sigma_{\text {allow }}=\frac{M}{S}=\frac{\frac{w x^{2}}{2}}{\frac{2}{6} b}=\frac{3 w x^{2}}{t^{2} b}\quad(1)

At $x = L , b = b_{0}$

\sigma_{\text {allow }}=\frac{3 w L^{2}}{t^{2} b_{0}}\quad(2)

Equating Eqs. (1) and (2) yields:

\frac{3 w x^{2}}{r^{2} b}=\frac{3 w L^{2}}{t^{2} b_{0}}

b=\frac{b_{0}}{L^{2}} x^{2}

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