 Elastic plate

An elastic plate is a body whose thickness is small compared with its in-plane dimensions, allowing a three-dimensional elasticity problem to be reduced to its middle surface. It can carry both in-plane stretching loads and transverse bending loads. A sheet supported around its edges and pressed sideways is a simple bending experiment; its response depends strongly on thickness and on the boundary conditions.

For an isotropic plate of thickness $h$, Young's modulus $E$ and Poisson's ratio $\nu$, small-slope bending has bending stiffnessWith constant $B$, no applied in-plane tension and transverse load per area $p$, static deflection $w$ satisfies $B{\nabla }^{4}w=p$, using a consistent sign for load and deflection. The fourth-order biharmonic equation needs both displacement-related and moment-related edge conditions. Doubling thickness multiplies $B$ by eight, illustrating why a geometrical change can greatly stiffen a sheet without changing its material.