Laplace Equilibrium Equation for a Membrane Shell

For an element of a thin shell in a membrane state under normal pressure $p$, local equilibrium in the surface-normal direction gives:

$$\frac{N_1}{R_1}+\frac{N_2}{R_2}=p.$$

• $N_1$, $N_2$ — membrane normal force resultants per unit length in the two principal directions of the midsurface;
• $R_1$, $R_2$ — corresponding principal radii of curvature;
• $p$ — resultant normal load per unit area of the midsurface.

For a shell of constant thickness $t$, the corresponding average normal stresses are $\sigma_1=N_1/t$ and $\sigma_2=N_2/t$. Signs must follow the adopted convention for curvature, pressure, and membrane forces.