Axisymmetric clamped-plate deflection
= Axisymmetric clamped-plate deflection
{title2=$h=p(R^2-r^2)^2/(64B)$}
A circular <elastic plate> with <bending stiffness> $B$, constant excess pressure $p$ and <clamped boundary conditions> at radius $R$ obeys $B\nabla_r^4h=p$, where $\nabla_r^2=r^{-1}\partial_r(r\partial_r)$. Its regular solution is $h=p(R^2-r^2)^2/(64B)$. Consequently its enclosed volume is $V=\pi pR^6/(192B)$, and its edge <curvature> is $h_{rr}(R)=pR^2/(8B)=24V/(\pi R^4)$.