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Introduction
Circumferential stress is a type of mechanical stress on cylindrical or spherical elements, as a result of internal or external pressure.
A classic example of hoop stress is the tension applied to the iron bands, or wooden hoops, of a barrel. In a straight, closed pipe, any force applied to the wall of the cylindrical tube by a differential pressure ultimately results in hoop stresses. Similarly, if the pipe has flat end plugs, any force applied to them by static pressure will induce a perpendicular axial stress in the wall of the pipe itself. Thin sections often have negligibly small radial stresses, but accurate models of thicker-walled, cylindrical shells must take such stresses into account.
Definitions
One of the types of mechanical stress is hoop stress, which arises in rotationally symmetric objects. This force is contained in the plane perpendicular to the axis of symmetry and is perpendicular to the radius of the object. Every particle of the cylinder wall suffers it in both directions. In general it does not have to be the same throughout the thickness but can vary. It is represented by σ.
If we consider a tube of internal radius r of wall thickness e and length l filled with a fluid at a pressure P, as shown in fig. 1. To simplify, we are going to consider the unit length l = 1. We decompose the pressure that is equal by Pascal's principle into a horizontal and a vertical component. The vertical components are symmetrical and cancel. And only the horizontal value of pressure remains. multiplying P by the projection of the surface 2r·l, and since we consider l=1, we have a force per unit length of 2·r·P. Since the tube is in equilibrium, the sum of forces must be zero, and due to the symmetry of the problem we consider that each wall of the tube exerts half the force r·P. If we consider that the stress is distributed uniformly within the wall we have a stress σ =r·P/e. From this expression it can be seen that although the pressure is maintained, the stress to which the tube wall is subjected depends on the radius; the larger the radius, the greater the stress. To consider that the stress is distributed uniformly, the ratio between the radius and the thickness must be greater than 10, according to other authors it is 20. If the radius-to-thickness ratio is lower, although the total force to be supported is the calculated one, a uniform distribution of the stress cannot be assumed within the thickness of the wall, there are areas where the stress is greater than the average. And shear stresses cannot be ignored.