The beam is made from a plate that has a constant thickness b. If it is simply supported and carries a uniform load w, determine the variation of its depth as a function of x so that it maintains a constant maximum bending stress

Question

The beam is made from a plate that has a constant thickness b. If it is simply supported and carries a uniform load w, determine the variation of its depth as a function of x so that it maintains a constant maximum bending stress [latex]\sigma_{ ext {allow }}[/latex] throughout its length.

Accepted Answer

Moment Function: As shown on FBD (b).
Section Properties:

[latex]I=\frac{1}{12} b y^{3} \quad S=\frac{I}{c}=\frac{\frac{1}{12} b y^{3}}{\frac{y}{2}}=\frac{1}{6} b y^{2}[/latex]

Bending Stress: Applying the flexure formula.

[latex]\sigma_{ ext {allow }}=\frac{M}{S}=\frac{\frac{w}{8}\left(L^{2}-4 x^{2} ight)}{\frac{1}{6} b y^{2}}[/latex]

[latex]\sigma_{ ext {allow }}=\frac{3 w\left(L^{2}-4 x^{2} ight)}{4 b y^{2}}\quad[1][/latex]

At [latex]x=0, y=h_{0} [/latex]. From Eq. [1]

[latex]\sigma_{ ext {allow }}=\frac{3 w L^{2}}{4 b h_{0}^{2}}\quad[2][/latex]

Equating Eq. [1] and [2] yields

[latex]y^{2}=\frac{h_{0}^{2}}{L^{2}}\left(L^{2}-4 x^{2} ight)[/latex]

[latex]\frac{y^{2}}{h_{0}^{2}}+\frac{4 x^{2}}{L^{2}}=1[/latex]

The beam has a semi-elliptical shape.

Question 11.32: The beam is made from a plate that has a constant thickness ...

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