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2.9: Circular Plates

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  • It is relatively easy to derive the equation of equilibrium of a circular plate from the principle of virtual work. Bending and in-plane responses is considered separately

    \[\int_{R_1}^{R_2} (M_r \delta \kappa_r + M_{\theta} \delta \kappa_{\theta}) dr + \int_{R_1}^{R_2} p \delta w r dr + r \bar{M}_{r} \delta w^{\prime} |_{R_1}^{R_2} + r \bar{V}_{r} \delta w |_{R_1}^{R_2} \label{2.9.1}\]

    where the radial and circumferential curvatures and their variations are defined (without proof) by

    \[\kappa_r = \frac{\partial^2 w}{\partial r^2}, \; \delta \kappa_r = \frac{\partial^2 (\delta w)}{\partial r^2}\]

    \[\kappa_{\theta} = \frac{1}{r} \frac{\partial w}{\partial r}, \; \delta \kappa_{\theta} = \frac{1}{r} \frac{\partial (\delta w)}{\partial r}\]

    Integrating the left hand side of Equation \ref{2.9.1} by parts and using similar arguments as in the case of a beam, one gets equilibrium:

    \[ \frac{d}{dr}\left( r \frac{dM_r}{dr}\right) + \frac{dM_r}{dr} - \frac{dM_{\theta}}{dr} = pr\]

    and boundary conditions

    \[ (M_r - \bar{M}_r) \delta w^{\prime} = 0 \]

    \[ (V_r - \bar{V}_r) \delta w = 0\]


    \[V_r = \frac{d}{dr}(rM_r)\]

    When \(R_1 = 0\), we have a circular plate. Otherwise the plate is annular with the inner and outer radius \(R_1\) and \(R_2\), respectively.

    When the circular plate is loaded in the in-plane direction only, it remains flat and the components of the mid-surface extensions and their variations are

    \[\epsilon_r^{\circ} = \frac{du_r}{dr}, \; \delta \epsilon_r^{\circ} = \frac{d}{dr} (\delta u_r)\]

    \[\epsilon_{\theta}^{\circ} = \frac{u_r}{r}, \; \delta\epsilon_{\theta}^{\circ} = \frac{\delta u_r}{r}\]

    The principle of virtual work can be easily established in the form

    \[\int_{R_1}^{R_2} (N_r \delta \epsilon_r^{\circ} + N_{\theta} \delta \epsilon_{\theta}^{\circ} ) r dr = rN_r\delta u_r |_{R_1}^{R_2} \]

    The equation of equilibrium in the in-plane direction are easily derived by integrating by parts

    \[\frac{d}{dr} (rN_r) − N_{\theta} = 0 \]

    subject to the boundary condition

    \[(N_r − \bar{N}_r) \delta u_r |^{R_2}_{R_1} = 0\]

    Note that \(\bar{N}_{\theta}\) is zero at the boundaries, ensuring that there will be no in-plane shearing force \(N_{r \theta}\) and the radial and hoop membrane forces are principal forces.