2.9: Circular Plates
- Page ID
- 21699
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.