Tensions and stresses of ellipsoidal chambers.

Equations for calculating wall tensions in an ellipsoidal chamber might be useful in analyses of elongated chambers whose transverse sections are not round, and they should be useful for examining the tension distribution associated with such shapes. Considering the forces changing a prolate spheroid (semiaxes a > b = c) into a general ellipsoid (semiaxes a > b > c) led to an equation for tensions at the poles of an ellipsoid. Considering the thickness distribution of a chamber of uniform average stress led to an equation for the average of orthogonal tensions at any point on an ellipsoidal chamber. Applying these equations with Laplace's law to points along an axis plane showed that tension normal to that plane is a weighted average of tensions normal to that plane at the intersections of the ellipsoid with the other two axis planes. It was postulated that this rule also would apply to the tension component normal to a plane coincident with any hoop (line of constant distance from one axis plane), and this postulate led to an equation for tension orthogonal to a hoop at any point. These three equations (pole tensions, local average tension, local hoop-orthogonal tension) allowed calculation of the tension tensor at any point. The equations and their algorithm were validated by four tests: the surface integral of the average of orthogonal tensions is as necessary for tensile work to equal hydraulic work in a symmetrical displacement (satisfying chamber equilibrium), the line integral of the component of tension normal to any hoop-coincident plane is equal to the product of pressure and area in the hoop (satisfying force balance), at any point the tensions predicted from the tensor for the directions of greatest and least curvature are compatible with Laplace's law (satisfying local equilibrium), and the calculated principal-tension lines relate properly to the nodes where tension is the same in all surface directions. These tests could be used to validate finite-element analyses of complex chambers.