# 2.9: Circular Plates

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

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

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

where

$V_r = \frac{d}{dr}(rM_r)$

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.