Bending of plates

Definition

Let a thin isotropic plate have mid-plane $\Omega \subset \R^2$, thickness $h$, Young's modulus $E$, Poisson's ratio $\nu$ and bending stiffness $D = Eh^3/(12(1 - \nu^2))$. Let $q$ be a transverse load per unit area, and let each part of $\partial\Omega$ be clamped, simply supported or free. The plate bending problem in Kirchhoff–Love theory is to find the deflection $w$ such that:

1. $D \Delta^2 w = q$ in $\Omega$, the biharmonic plate equation.
2. On a clamped edge, $w = 0$ and $\partial_n w = 0$.
3. On a simply supported edge, $w = 0$ and the normal bending moment $M_{nn}$ vanishes.
4. On a free edge, $M_{nn} = 0$ and the Kirchhoff effective shear force $Q_n + \partial_t M_{nt}$ vanishes, where $t$ is the tangent direction.

The bending stresses then are $\sigma_{\alpha\beta} = 12 z M_{\alpha\beta}/h^3$ for $\abs{z} \le h/2$, with $M_{\alpha\beta}$ the bending moments of the plate.