DISCUSSION THREAD
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Note that since the tank is not closed with an internal gas pressure, there is no axial component of stress from the internal fluid. The weight of the tank walls do contribute the axial stress, however. The hydrostatic pressure due to the fluid produces a hoop stress that is dependent on position.
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I am confused on finding the area for the axial stress. Does the force act on the cross-section of the tank? If so, I would then subtract the outside area from the inside, correct?
The axial stress acts on the area of the tank cross-section (make a cut perpendicular to the longitudinal axis of the tank, and the area of the cross-section is seen). That area is basically the area of the outer section subtracted away the area of the inner section, as I think that you are suggesting here.