A thin-walled cantilever with walls of constant thickness t has the cross section shown in Fig. P.9.22. The cantilever is loaded by a vertical force P at the tip and a horizontal force 2P at the mid-section. Determine the direct stress at the points A and B in the cross section at the built-in

Question

A thin-walled cantilever with walls of constant thickness t has the cross section shown in Fig. P.9.22. The cantilever is loaded by a vertical force P at the tip and a horizontal force 2P at the mid-section. Determine the direct stress at the points A and B in the cross section at the built-in end.

Accepted Answer

The bending moments at the built in end are

[latex]M_{z}=P L, \quad M_{y}=2 P\left(\frac{L}{2} ight)=P L[/latex]

The centroid of area is at the centre of the enclosed part of the section and, using the approximations for thin-walled sections described in Section 9.6

[latex]\begin{aligned}&\bar{I}_{z}=2 t d\left(\frac{d}{2} ight)^{2}+\frac{2 t d^{3}}{12}=\frac{2 t d^{3}}{3} \&I_{y}=2 t d\left(\frac{d}{4} ight)^{2}+2\left[\frac{t d^{3}}{12}+t d\left(\frac{d}{4} ight)^{2} ight]=\frac{5 t d^{3}}{12} \&I_{z y}=t d\left(\frac{d}{4} ight)\left(\frac{d}{2} ight)+t d\left(\frac{-d}{4} ight)\left(\frac{-d}{2} ight)=\frac{t d^{3}}{4}\end{aligned}[/latex]

The denominator in Eq. (9.31) is equal to [latex]\left(t d^{3} ight)^{2}\left[(2 / 3)(5 / 12)-(1 / 4)^{2} ight]=0.215\left(t d^{3} ight)^{2}[/latex].

Substituting in Eq. (9.31) for [latex]M_{z}, M_{p}[/latex], etc.

[latex]\sigma_{x}=-\left(\frac{M_{y} I_{z}-M_{z} I_{z y}}{I_{z} I_{y}-I_{z y}^{2}} ight) z-\left(\frac{M_{z} I_{y}-M_{y} I_{z y}}{I_{z} I_{y}-I_{z y}^{2}} ight) y[/latex] (9.31)

[latex]\sigma_{x}=\frac{-P L(1.94 z+0.78 y)}{t d^{3}}[/latex]

Then at A where z=3d/4 and y=d/2

[latex]\sigma( A )=\frac{-1.85 PL }{t d^{2}}[/latex]

and at B where z=-d/4, y=d/2

[latex]\sigma( B )=\frac{0.1 PL }{t d^{2}}[/latex]

Question 9.22: A thin-walled cantilever with walls of constant thickness t ...

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