Membrane Stress Equations for a General Openended Shell
For a general axisymmetric shell with internal pressure, P, the membrane stress resultants in the meridian and circumferential (or hoop) can be calculated from the equilibrium equations as

(1) 

(2) 
where subscripts and refer to the meridian and circumferential directions, respectively. The angle is the colatitude angle, as shown in Fig. 1. The shear stress resultant is zero since the shell geometry, loading, and material properties are all axisymmetric. At the open dome, the meridian stress resultant, must reduce to zero to match the boundary condition at . The meridian and circumferential radius of curvatures and , respectively, are

(3) 
Using the relationships and , Eq. 1 can be evaluated as

(4) 

(5) 
These two equations are plotted in Fig. 2 for a series of openellipsoidal shells including a spheroid shell where a = b = 1. Since the material properties were not used in deriving these stress resultants, they are valid for fiber reinforced composite materials as well as for isotropic materials. However, for filament wound structures it is important to know the stresses in the material property coordinate system, i.e., the fiber direction and transverse direction. To calculate these stresses, as well as the shell displacements, the material properties of the shell and the shell thickness are required. Before developing the elastic relationships, some preliminary relationships concerning the filament winding of pressure vessels and composite materials will be presented.
© 1997 Kurt Gramoll, The University of Oklahoma, All Rights Reserved.