A cantilever beam AB takes the form of a quadrant of a circle of radius, R, and is positioned on a horizontal plane. If the beam carries a vertically downward load, W, at its free end and its bending and torsional stiffnesses are EI and GJ, respectively, calculate the vertical component of the

Question

A cantilever beam AB takes the form of a quadrant of a circle of radius, R, and is positioned on a horizontal plane. If the beam carries a vertically downward load, W, at its free end and its bending and torsional stiffnesses are EI and GJ, respectively, calculate the vertical component of the deflection at its free end.

Accepted Answer

A plan view of the beam is shown in Fig. 4.17. To determine the vertical displacement of B, we apply a virtual unit load at B vertically downward (i.e., into the plane of the paper). At a section of the beam where the radius at the section makes an angle, [latex]\alpha[/latex], with the radius through B,

[latex]M_{ A }=W p=W R \sin \alpha, \quad M_{v}=1 R \sin \alpha[/latex]

[latex]T_{ A }=W(R-R \cos \alpha), \quad T_{v}=1(R-R \cos \alpha)[/latex]

The total internal virtual work done is given by the summation of Eqs. (4.20) and (4.22), that is,

[latex]w_{i, M}=\int_{L} \frac{M_{ A } M_{v}}{E I} d x[/latex] (4.20)

[latex]w_{i, T}=\int_{L} \frac{T_{ A } T_{\vee}}{G I_{ o }} d x[/latex] (4.22)

[latex]W_{i}=(1 / E I) \int_{0}^{\pi / 2} W R^{2} \sin ^{2} \alpha R d \alpha+(1 / G J) \int_{0}^{\pi / 2} W R^{2}(1-\cos \alpha)^{2} R d \alpha[/latex] (i)

Integrating and substituting the limits in Eq. (i) gives

[latex]W_{i}=W R^{3}\{(\pi / 4 E I)+(1 / G J)[(3 \pi / 4)-2]\}[/latex] (ii)

The external virtual work done by the unit load is [latex]1 \delta_{ B }[/latex], so that equating with Eq. (ii), we obtain

[latex]\delta_{ B }=W R^{3}\{(\pi / 4 E I)+(1 / G J)[(3 \pi / 4)-2]\}[/latex]

Question 4.11: A cantilever beam AB takes the form of a quadrant of a circl...

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