A cantilever beam of length L has a solid circular cross section of diameter D and carries a vertically downward load W at its free end. Calculate the distribution of shear stress in a cross section of the beam and hence the form factor β. What is the deflection due to shear at the free end of the cantilever? The shear modulus is G and note that $\int_{-\pi / 2}^{\pi / 2} \cos ^{6} \theta d \theta=5 \pi / 16$
Ans. $\beta=10 / 9,40 WL / 9 \pi G D^{2}$ .

Question

A cantilever beam of length L has a solid circular cross section of diameter D and carries a vertically downward load W at its free end. Calculate the distribution of shear stress in a cross section of the beam and hence the form factor β. What is the deflection due to shear at the free end of the cantilever? The shear modulus is G and note that [latex]\int_{-\pi / 2}^{\pi / 2} \cos ^{6} heta d heta=5 \pi / 16[/latex]
Ans. [latex]\beta=10 / 9,40 WL / 9 \pi G D^{2}[/latex].

Accepted Answer

In the solution of Ex. 10.5 the shear stress distribution is, at one stage, expressed in terms of an angular variable θ (see Fig.10.10)
i.e.

[latex] au=-\frac{16 S_{y}}{3 \pi D^{2}} \cos ^{2} heta[/latex]

Substituting this expression in Eq. (13.20)

[latex]\beta=\frac{A}{S_{y}^{2}} \int_{y_{1}}^{y_{2}} au^{2} b_{0} d y[/latex] (13.20)

gives

[latex]\beta=\frac{\left(\pi D^{2} / 4 ight)}{\left(S_{y}^{2} ight)} \int_{-\pi / 2}^{\pi / 2} \frac{\left(-16 S_{y} \cos ^{2} heta ight)}{\left(3 \pi D^{2} ight)}(D \cos heta)(D / 2) \cos heta d heta[/latex],

which simplifies to

[latex]\beta=\frac{32}{9 \pi} \int_{-\pi / 2}^{\pi / 2} \cos ^{6} heta d heta[/latex]

Using the given value of the integral

β = 10/9
The deflection of the free end of the cantilever is given by Eq. (13.22)

[latex]v_{s}=\frac{\beta}{G} \int_{L}\left(\frac{S_{y}}{A} ight) d x[/latex] (13.22)

[latex]v( ext { freeend })=\frac{10}{9 G} \int_{0}^{L} \frac{(W)}{\left(\pi D^{2} / 4 ight)} d x[/latex]

from which v(freeend) [latex]=\frac{40 W L}{9 \pi G D^{2}}[/latex]

Question 13.P.25: A cantilever beam of length L has a solid circular cross sec...

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