The beam is made from a plate that has a constant thickness b. If it is simply supported and carries the distributed loading shown, 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 the distributed loading shown, 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

Support Reactions: As shown on the free-body diagram of the entire beam, Fig. a.
Moment Function: The distributed load as a function of x is

[latex]\frac{w}{x}=\frac{w_{0}}{L / 2} \quad w=\frac{2 w_{0}}{L} x[/latex]

The free-body diagram of the beam's left cut segment is shown in Fig. b. Considering the moment equilibrium of this free-body diagram,

[latex]\curvearrowleft +\Sigma M_{o}=0 ; \quad M+\frac{1}{2}\left[\frac{2 w_{0}}{L} x\right]{x}\left(\frac{x}{3}\right)-\frac{1}{4} w_{0} L x=0[/latex]

[latex]M=\frac{w_{0}}{12 L}\left(3 L^{2} x-4 x^{3}\right)[/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{w_{0}}{12 L}\left(3 L^{2} x-4 x^{3}\right)}{\frac{1}{6} b h^{2}}=\frac{w_{0}}{2 b h^{2} L}\left(3 L^{2} x-4 x^{3}\right)\quad(1)[/latex]

At [latex]x=\frac{L}{2}, h=h_{0}[/latex]. From Eq. (1),

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

Equating Eqs. (1) and (2),

[latex]\frac{w_{0}}{2 b h^{2} L}\left(3 L^{2} x-4 x^{3}\right)=\frac{w_{0} L^{2}}{2 b h_{0}^{2}}[/latex]

[latex]h=\frac{h_{0}}{L^{3 / 2}}\left(3 L^{2} x-4 x^{3}\right)^{1 / 2}[/latex]

Question 11.34: The beam is made from a plate that has a constant thickness ...

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