Pierścień Kołowy pod Działaniem Wewnętrznego i Zewnętrznego Ciśnienia Hydrostatycznego

The state of stress in a homogeneous circular ring under the action of internal and external hydrostatic pressure linearly variable along cne of the diameters is determined.
This problem corresponds to that of an empty tube (or a tube filled up with a light liquid) having a mass and immersed horizontally in a heavy liquid. It may be called the plane «bathyscaph» problem. Tubes so loaded are often met in practice. Suffice it to mention oil pipe lines in sand (in desert regions) or subaqueous cables in which the insulating oil is under hydrostatic pressure.
In Art. 1, the state of stress in a weightless circular ring loaded by an internal pressure p1 = cx and an external pressure P2 = (R1/R2)2p1: (the pressures are linearly variable in the x-direction) is determined.
R1 and R2 are the inner and the outer radius of the ring.
In Art. 2, the displacements are obtained for the same ring.
In Art. 3, the state of stress in the tube is found by completing the state of stress obtained in the Art. 1 by a normal, principal stress acting in the direction of the tube axis depending on the boundary conditions for the tube ends.
In Art. 4, the state of stress in a tube having a mass and immersed horizontally in a liquid is determined as a sum of the following:
(1) the state. of stress corresponding to the Lamé problem;
(2) the state of stress constituting a particular solution of the equations of the theory of elasticity taking into consideration mass forces (acting in the x-direction), (…) where Yo is the specific gravity of the tube; (3) the state of stress in a weightless tube loaded by an internal pressure pi cc and an external pressure p2 = (R1/R2)2p1, where R1 and R2 are the radii of the tube.