The elasticity and resistance of the materials of engineering . Thick, Hollow Cylinders and Spheres, and Torsion. Art. 9.—Thick, Hollow Cylinders. In Fig. I is represented a section, taken normal to its axis,of a circular cylinder whose walls are of the appreciable thick-ness /. Let/ and/i represent the interior and exterior inten-sities of pressures, respectively. The material will not bestressed with uniform intensity throughout the thickness /. Yet if that thickness, comparatively speak-ing, is small, the variation will also besmall; or, in other words, the intensityof stress throughout the
The elasticity and resistance of the materials of engineering . Thick, Hollow Cylinders and Spheres, and Torsion. Art. 9.—Thick, Hollow Cylinders. In Fig. I is represented a section, taken normal to its axis,of a circular cylinder whose walls are of the appreciable thick-ness /. Let/ and/i represent the interior and exterior inten-sities of pressures, respectively. The material will not bestressed with uniform intensity throughout the thickness /. Yet if that thickness, comparatively speak-ing, is small, the variation will also besmall; or, in other words, the intensityof stress throughout the thickness tmay be considered constant. Thisapproximate case will first be con-sidered. The interior intensity / will be considered greater than the exterior /i, consequently the tendency will ^ be toward rupture along a diametral plane. If, at the same time, the ends of the cylinder are taken as closed, as will be done, a tendency to rupture through the section shown in the Fig. will exist. The force tending to produce rupture of the latter kindwill be :. F = n{pr^ - p,r^) (I) Art. 9-] THICK, HOLLOW CYLINDERS. 37 If iVj represents the intensity of stress developed by thisforce, N = . ^ = />r^-Ar. .... (2) If the exterior pressure is zero, and if r is nearly equal to ^ - 2{r- r)~ 2t ^^^ In this same approximate case, the tendency to split thecylinder along a diametral plane, for unit of length, will be: F = pr - /,r,. If N is the intensity of stress developed by F: ^^^^Pr-P^r, (4) N is thus seen to be twice as great as Nj_ when/^ = o. If,therefore, the material has the same ultimate resistance in bothdirections the cylinder will fail longitudinally when the interiorintensity is only half great enough to produce transverse rup-ture ; the tJiickness being assumed to be very small and the ex-terior pressure zero, iVj and N are tensile stresses, because the interior pressurewas assumed to be large compared with the exterior. If theopposite assumption were made, they would be f
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