# 1.10: Strain-Displacement Relations for Circulate Plates

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The theory of circular plates is formulated in the cylindrical coordinate system \((r, \theta, z)\). The corresponding components of the displacement vector are \((u, v, w)\). In the remainder of the notes, the axi-symmetric deformation is assumed, which would require the loading to be axi-symmetric as well. This assumption brings four important implications

- The circumferential component of the displacement is zero, \(v \equiv 0\)
- There are no in-plane shear strains, \(\epsilon_{r\theta} = 0\)
- The radial and circumferential strains are principal strains
- The partial differential equations for plates reduces to the ordinary differential equation where the radius is the only space variable.

Many simple closed-form solutions can be obtained for circular and annular plates under different boundary and loading conditions. Therefore such plates are often treated as prototype structures on which certain physical principles could be easily explained.

The membrane strains on the middle surface are stated without derivation

\[\epsilon_{rr}^{\circ} = \frac{du}{dr} + \frac{1}{2} \left( \frac{dw}{dr} \right)^2\]

\[\epsilon_{\theta \theta}^{\circ} = \frac{u}{r}\]

The two principal curvatures are

\[\kappa_{rr} = - \frac{d^2w}{dr^2}\]

\[\kappa_{\theta \theta} = -\frac{1}{r}\frac{dw}{dr}\]

The sum of the bending and membrane strains is thus given by

\[\epsilon_{rr}(r,z) = \epsilon_{rr}^{\circ}(r) + z\kappa_{rr}\]

\[\epsilon_{\theta \theta}(r,z) = \epsilon_{\theta \theta}^{\circ}(r) + z\kappa_{\theta \theta}\]

It can be noticed that the expression for the radial strains and curvature are identical to those of the beam when \(r\) is replaced by \(x\). The expressions in the circumferential direction are quite different.