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The moment-free theory of thin-walled shells: Pressure vessels, whose cylinders and heads are for the most part thin-walled rotational shells, are characterized by a uniform internal pressure that acts vertically on the inner surface of the vessel walls; this force is borne by the vessel walls. The pressure on the head will cause the cylindrical section to crack in its cross-section; the stress distributed along this cross-section is a1, and its direction of action is along the axis of the container, and it is referred to as \"longitudinal stress\" or \"axial stress\". The pressure on the cylinder wall will cause it to crack along the circumferential direction; this stress that acts in the tangent direction of the circle is known as \"circumferential stress\" or \"hoop stress\". To analyze and determine these two stress values in thin-walled shells, two theories can be used, namely the moment-free theory and the moment-bearing theory. Image: The moment-free theory is also known as the membrane theory. It assumes that the wall thickness is very small compared to the diameter, considering the wall to be so thin as to be almost like a film; it is subjected only to tensile or compressive stresses, not to bending moments. It also assumes that the stresses within the shell are uniformly distributed across the thickness, and such stresses in the vessel wall are referred to as \"film stresses\". According to the design specifications, a ratio of wall thickness to inner diameter of less than 1/10, that is, a situation where the ratio K of the outer diameter to the inner diameter of the shell is ≤1.2, corresponds to a rotating thin shell. According to the moment theory, not only tensile or compressive stresses exist within the shell, but also bending stresses; however, analysis shows that these bending stresses are a very small value compared to the circumferential stresses. To simplify the calculations, the assumptions of the moment-free theory are adopted, and the accuracy of the resulting calculations is also sufficient to meet engineering requirements. Assumptions of the moment-free theory: (1) The shell should have a continuous surface. Deformation discontinuities will directly lead to local bending, disrupting the stress state free of moment. (2) The external loads on the housing should be continuous. As a moment-free shell, it cannot actually withstand concentrated forces and moments perpendicular to the shell walls. Under the action of a concentrated force, the stress condition of the shell will be momentous. (3) The fixing mode of the shell boundary should be free support. In the objective world, the only thing that seems to meet all the above requirements is probably a \"balloon\". In design, it is often necessary to use film theory to discuss various solutions; in such cases, introducing the \"balloon\" model helps to understand the key points and grasp the essence.