 Axisymmetric clamped-plate deflection

A circular elastic plate with bending stiffness $B$, constant excess pressure $p$ and clamped boundary conditions at radius $R$ obeys $B{\nabla }_r^{4}h=p$, 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 $V=\pi p{R}^6/(192B)$, and its edge curvature is $h_{rr}(R)=p{R}^2/(8B)=24V/(\pi R^4)$.