Determine the height h of the rectangular cantilever beam of constant width b in terms of h0, L, and x so that the maximum normal stress in the beam is constant throughout its length.

Question

Determine the height h of the rectangular cantilever beam of constant width b in terms of [latex]h_{0}[/latex], L, and x so that the maximum normal stress in the beam is constant throughout its length.

Accepted Answer

Moment Functions: Considering the moment equilibrium of the free-body diagram of the beam's right cut segment, Fig. a,

[latex]\curvearrowleft +\Sigma M_{O}=0\quad M-w x\left(\frac{x}{2}\right)=0 \quad M=\frac{1}{2} w x^{2}[/latex]

Section Properties: At position x, the height of the beam's cross section is h. Thus

[latex]I=\frac{1}{12} b h^{3}[/latex]

Then

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

Bending Stress: The maximum bending stress [latex]\sigma_{\max }[/latex] as a function of x can be obtained by applying the flexure formula.

[latex]\sigma_{\max }=\frac{M}{S}=\frac{\frac{1}{2} w x^{2}}{\frac{1}{6} b h^{2}}=\frac{3 w}{b h^{2}} x^{2} \quad(1)[/latex]

At x=L, [latex]h=h_{0}[/latex] . From Eq. (1),

[latex]\sigma_{\max }=\frac{3 w L^{2}}{b h_{0}^{2}}\quad(2)[/latex]

Equating Eqs. (1) and (2),

[latex]\frac{3 w}{b h^{2}} x^{2}=\frac{3 w L^{2}}{b h_{0}^{2}}[/latex]

[latex]h=\frac{h_{0}}{L} x[/latex]

Question 11.37: Determine the height h of the rectangular cantilever beam of...

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