A thin circular ring of radius r and uniform flexural stiffness carries two radial loads W applied along a diameter. Estimate the maximum bending moment in the ring.

Question

Accepted Answer

By symmetry the loading actions on a half-ring are [latex]\frac{1}{2}[/latex]W and M[latex]_0[/latex] . The bending moment at any angular position θ is

[latex]M = M_0 - \frac{1}{2}Wr \sin heta[/latex]

Then

[latex]C = \int_{0}^{\pi}{\left(M_0 - \frac{1}{2} Wr \sin heta ight)^2\frac{r d heta}{2EI}}[/latex]

But ∂C/∂M[latex]_0[/latex] = 0, so that

[latex]\int_{0}^{\pi}{M_0 d heta } = \frac{1}{2} Wr \int_{0}^{\pi}{\sin heta d heta} [/latex]

Then

[latex]M_0 = \frac{Wr}{\pi}[/latex]

Question 18.P.6: A thin circular ring of radius r and uniform flexural stiffn......

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