= Elastic plate
{wiki}
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 stiffness>
$$
B=\frac{Eh^3}{12(1-\nu^2)}.
$$
With constant $B$, no applied in-plane tension and transverse load per <area> $p$, static deflection $w$ satisfies $B\nabla^4w=p$, 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.
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