Membrane stresses in shells of revolution

The walls of thin vessels can be considered to be membranes supporting loads without significant bending or shear stresses similar to the walls of a balloon. 

The analysis of the membrane stresses induced in shells of revolution by internal pressure gives a basis for determining the minimum wall thickness required for vessel shells. The actual thickness required will also depend on the stresses arising from the other loads to which the vessel is subjected. 

02 = the circumferential or tangential stress, the stress acting along parallel circles (often called the hoop stress), r = the meridional radius of curvature, r2 = circumferential radius of curvature. 

Note the vessel has a double curvature the values of n and r2 are determined by the shape. 

Consider the forces acting on the element defined by the points a, b, c, d. Then the normal component (component acting at right angles to the surface) of the pressure force on the element 

---