Thin-walled cylindrical shells

The cylindrical shell is frequently used in pressure vessel design. For initial designs, it is useful to calculate the stresses in a thin-walled cylindrical shell that is uniformly loaded with internal pressure. For thin-walled pressure vessel calculations to be valid, the radial stresses in the shell need to be negligible. This is usually taken to be a valid assumption when the ratio of the vessel inner radius to the wall thickness (R/t) is greater than 10. "... 

Procedure 2-15 Buckling of Thin Wall Cylindrical Shells [21]... 

E.L. Paradis, Fabrication of thin wall cylindrical shells by sputtering, Thin Solid Films 72 (1980) 327. 

Thin-walled cylindrical vessels imder axial loads can fail in two manners by column action as in the case of Euler s buckling or by wrinkling between tray supports. The stiffness provided by the internal structures of columns containing trays, tray supports, downcomers, etc., provide additional rigidity. As a result, Euler s buckling, which is produced by bending of the shell as a whole, is seldom a controlling factor in tall vertical vessels. 

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

CVD of tungsten and tungsten-rhenium (co-deposition of WFg and ReFe) is of commercial importance for the production of X-ray targets. W-Re layers of up to 1 mm are deposited onto graphite disks with diameters between 100 and 150mm [5.78]. Fiuther-more, CVD can be used to produce cylindrical or conical shells, cones (such as shaped charge liners), crucibles, and thin-walled tubes. 

From basic theoretical considerations for thin-walled vessels, the minimum thickness of a closure for a cylindrical shell is obtained when the closure has a hemispherical shape. In this case, the membrane theory predicts that, using the same construction materials and allowable stresses, a hemispherical closure needs to be only half as thick as the cylindrical shell to which it is attached. 

The 1962 ASME Code Section VIII, Division 1 gives equation 4-4 for thin-walled (f <0.356r) spherical shells, which also applies to hemispherical heads. If the thick-wall correction factor (-0.6p and -0.2p for cylindrical shell and hemispheric head, respectively) is omitted, the thicknesses are 1 2, respectively. 

The ellipsoidal dished head with a major to minor axis ratio of 2 1 is popular for economic reasons, even though the theory for thin-walled vessels predicts that the head of this shape should have twice the thickness of a hemispherical head where the major and minor axes are equal. Such an ellipsoidal head used for vessels under internal pressure has the same thickness as the cylindrical shell if the same allowable stresses and joint efficiencies are applied to both parts. The 1962 ASME Code Section VIII, Division 1 gives the following equation for the thin-walled ellipsoidal dished heads with a 2 1 major to minor axis ratio ... 

Consider a long, thin cylindrical shell of mean diameter D and wall thickness t subjected to an external pressure P. The cylinder is in a stable configuration as long as it remains circular in shape. If there is an initial ellipticity, the cylinder will be in an unstable condition and will eventually buckle. ... 

Since the thin walls of the prismatic beams under discussion resemble the cylindrical thin shells of Section 6.2.2, the associated formulation of strains may be adopted. Thereby, also the respective assumptions are inherited. A comparison of Remarks 6.4 and 7.6 reveals that the ratio of thickness and radius of curvature is additionally confined and Remark 6.5 indicates a hnear strain displacement relation. These shell strains are given by Eqs. (6.12) and (6.15). The insertion of Eqs. (7.26) and (7.24) leads to rather complicated expressions. Through tedious manipulations with the aid of Eqs. (7.17), (7.22), and (7.23), a remarkably compact formulation may be found. For the sake of correlation to the beam displacements and rotations again in anticipation of the upcoming considerations of torsional warping, the warping function 0 s) is employed and also appears in the abbreviation... 

The key observation for a general treatment of thin-walled beams is the inevitability of an adequate strain formulation for cylindrical thin shells as presented here on the basis of the theory of Sanders [158] and Koiter [114]. Therefore, all terms have to be retained and distinction through imderline-ment is not necessary any more. With these considerations and the resulting initiation of the beam shear strains, Eq. (7.28) reduces significantly. 

Formula (1) is based on the static analysis of the shell of a cylindrical tank with constant wall thickness. Let us consider a thin-walled circular cylindrical tank of height //, internal radius r (distance from the vertical axis of symmetry to the wall surface) and wall thickness t, see Figure 3. 

Schneider W, Biede A (2005) Consistent equivalent geometric imperfections for the numerical buckling strength verification of cylindrical shells under uniform external pressure. Thin-Walled Struct 43(2) 175-188 Schneider W, Timmel I, Hdhn K (2005) The conception of quasi-collapse-affine imperfections a new approach to unfavourable imperfections of thin-walled shell stmc-tures. Thin-Walled Struct 43(8) 1202-1224 Stricklin JA, Haisler WE, von Riesemann WA (1973) Evaluation of solution procedures for material and/or geometrically nonlinear structural analysis. AIAAJll(3) 292-299... 

Terekhova LP, Skoryi lA (1973) Stresses in bonded joints of thin cylindrical shells. Strength Mater 4 1271 Tong L, Sun X (2003) Nonlinear stress analysis for bonded patch to curved thin-walled structures. Int J Adhes Adhes 23 349... 

Thin-wallcd and thick-walled cylinders arc forms of pipeline primarily used in the offshore industry to transport gas and hquid under pressure. Cylindrical pipes are also used as a stmctural load bearing member to restrain the motion of a platform, for example, in response to wind, waves and currents (to within a specified limit). When a pipe is exposed to a combination of internal and external pressure, the pipe material will be subjected to pressure loading and, as a result, will develop stresses in all directions. The resulting normal stresses induced by the pressure loading are functions of the pipe diameter, wall thickness and cylindrical shell boundary conditions (open-ended cylinder or closed-ended cylinder) (Kashani and Young, 2008). 

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