barcode generator vb.net source code CYLINDRICAL SHELLS STRESS ANALYSIS in Software

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39.4 CYLINDRICAL SHELLS STRESS ANALYSIS
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Cylindrical shells are widely used in the manufacture of pressure vessels. They can be easily fabricated and have great structural strength. Depending on their function, they may be vertical or horizontal. Vertical pressure vessels are often preferred, especially for a thin-walled vessel operating under low internal pressure.
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PRESSURE CYLINDERS 39.5
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PRESSURE CYLINDERS
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The design of a vertical cylindrical vessel becomes simple because the additional bending stresses due to weight of the vessel itself and of the fluid can be eliminated. Cylindrical shells of malleable materials, such as steels, nonferrous metals, and their alloys, operating under internal pressures up to 10 MPa are fabricated mostly of rolled and welded sheets of corrosion-resistant material (see Ref. [39.1]). The minimum thickness of a shell rolled from low-carbon sheet metal and welded is 4.0 mm. The minimum thickness for an austenitic steel shell is 3.4 mm; for copper shells, 2.5 to 30 mm; and for cast shells, 20 to 25 mm.
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39.4.1 Cylindrical Storage Vessel Before the actual design calculation of a vertical cylindrical vessel is done, the relationship between the optimum height and the diameter should be determined. The volume of the sheet metal needed to make a vertical cylindrical storage tank (Fig. 39.1) is determined by the following formula: Vs = doHt1 + = doHt1 + Vessel capacity is V= The inside diameter of the vessel is di = 4V H (39.3) d2 i H 4 (39.2) d2 o (tk + ta) 4 d2 o t2 4 (39.1)
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Since thickness t1 is very small in comparison with do and di, we may consider di do. Substitution of Eq. (39.3) into Eq. (39.1) gives Vs = 2t1 VH + Vt2 H (39.4)
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The minimum volume of sheet metal is obtained by differentiating Eq. (39.4) with respect to H and equating the derivative to zero. Thus dVs = t1 dH V/H Vt2 =0 H2 (39.5)
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The optimum height and the optimum diameter of the tank are given by Hopt = and V t2 t1 Vt1 t2
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2 1/3
(39.6)
dopt = 2
(39.7)
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PRESSURE CYLINDERS 39.6
CLASSICAL STRESS AND DEFORMATION ANALYSIS
The shell thickness is given by t1 = Hopt dopt 2 a (39.8)
where = specific weight of the fluid in the tank, and a = allowable stress. Substituting the optimum values of height and outside diameter in Eq. (39.8), the tank capacity V is recovered as V = t2 1 3 a 3t2 (39.9)
Considering a minimum shell thickness of t1 = 4 mm and top and bottom thicknesses of 4 mm, we find t2 = 8 mm. The specific weight of the liquid can be obtained from appropriate property tables. The allowable stress for low-carbon steel is 137 MPa. Using this information in Eq. (39.9), one can easily find the corresponding tank volume. Pressure vessels subjected to internal and/or external pressures require application of the theoretical principles involved in shell analysis. The structural configurations of do relatively simple geometric shapes such as cyltk inders and spheres have been studied extensively in the theories of plates and shells (see Refs. [39.2], [39.4], and [39.5]). In fact, no single chapter or even an entire textbook can cover t1 all the advancements in the field, particularly H where internal pressure, external pressure, and other modes of loading are present. Therefore, attempts will be made here only to cover the materials which are pertinent for design of simple pressure vessels and piping.The majority of piping and vessel components are designed for ta internal pressure and have been analyzed to a FIGURE 39.1 Cylindrical storage vessel. great degree of sophistication. Numerous cases involve application of external pressure as well, where stresses, elastic stability, and possible structural failure must be analyzed and evaluated.
39.4.2 Membrane Theory The common geometric shapes of pressure vessels used in industrial processes are spheres, cylinders, and ellipsoids. Conical and toroidal configurations are also used. Membrane theory can be applied to determine the stresses and deformations of such vessels when they have small thicknesses compared with other dimensions and have limited and small bending resistance normal to their surface. The stresses are considered to be average tension or compression over the thickness of the pipe or vessel wall acting tangential to the surface subjected to normal pressure. The imaginary surface passing through the middle of the wall thickness, however, extends in two directions and calls for rather complicated mathematical analysis, particularly when more than one expression for the curvature is necessary to describe the displacement of a point in the shell wall. In fact, in the more general sense it is neces-
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