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PRESSURE CYLINDERS 39.16
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CLASSICAL STRESS AND DEFORMATION ANALYSIS
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39.6.2 Thermal Stresses in Thick Cylindrical Vessels When a thick-walled cylindrical vessel is subjected to a thermal gradient, nonuniform deformations are induced, and consequently, thermal stresses are developed. Fluids under high pressure and temperature are generally transported through structures such as boilers, piping, heat exchangers, and other pressure vessels. L Owing to the presence of a large temperature gradient between the inner and outer walls, thermal stresses are produced in these structures. Radial, hoop, and longitudinal stresses in thick, hollow cylinders with a thermal gradient across the wall may be estimated analytically, but generally the computar ro tions are lengthy, tedious, and timeconsuming. Bhaduri [39.9] developed a ri set of dimensionless graphs from comT1 puter solutions, and they can be used to find the thermal-stress components with a few simple calculations. T In this technique, the temperature of the shell s inner surface at radius ri and FIGURE 39.8 Hollow cylinder subjected to a temperature gradient. outer surface at radius ro are considered to be Ti and To, respectively. The ends of the cylindrical shell are considered unrestrained. The longitudinal strain developed as a result of the thermal stresses is assumed to be uniform and constant. The temperature distribution in the shell wall is given by T = Ti ln ro/r ln ro/ri
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when the outer surface temperature To = 0, that is, when the temperature differences are measured relative to the outer surface temperature. The dimensionless hoopstress function Fh, radial stress function Fr, and longitudinal stress function Fz are obtained analytically for cylindrical shells of various thickness ratios. The stress functions are given by the following equations: Fh = Fr = Fz = where h r z = = = = 1 2(1 ) h (Ro/R)2 + 1 R = 1 ln o ln Ro R ETi ln Ro R2 1 o 2(1 ) r 1 = ETi ln Ro ln (Ro/R)2 1 R + ln Ro Ro R2 1 o (39.45)
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1 R 2(1 ) z 2 = 1 2 ln o 2 ln Ro ETi ln Ro R Ro 1 hoop stress radial stress longitudinal stress Poisson s ratio
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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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coefficient of thermal expansion Young s modulus r/ri ro/ri
The stress functions are shown graphically for the radii ratios ro/ri = 2.0, 2.5, and 3.00, respectively, in Fig. 39.9. These curves are general enough to compute the hoop, radial, and shear stresses produced by temperature gradients encountered in most cylindrical-shell designs. The procedure is simple. The value of Ro must be known to select the appropriate curve. For a particular value of R, the corresponding values of the stress functions can be read from the ordinate.The stresses at any radial location of the shell wall can be easily calculated from the known values of the stress function, property values of the shell material, and temperature of the inside surface of the shell.
39.7 FABRICATION METHODS AND MATERIALS
Welding is the most common method of fabrication of pressure-vessel shells. Structural parts such as stiffening rings, lifting lugs, support clips for piping, internal trays, and other parts are also attached to the vessel wall by welding. Welded joints are used for pipe-to-vessel connections to ensure optimum leak-proof design, particularly when the vessel contains hazardous fluid at a very high temperature. A structure whose parts are joined by welding is called a weldment. The most widely used industrial welding method is arc welding. It is any of several fusion welding processes wherein the heat of fusion is generated by electric arc. Residual stresses in a weld and in the adjoining areas cannot be avoided. They are quite complex. If the weld residual stress is superposed on the stress due to external loads and the resultant stress exceeds the yield point of the material, local plastic yielding will result in redistribution of the stress in ductile materials. A good weld requires a highly ductile material. In order to prevent loss of ductility in the welded region, low-carbon steels with less than 0.35 percent carbon content are used as construction materials. Carbon itself is a steel hardener. However, in the presence of a manganese content of 0.30 to 0.80 percent, carbon does not cause difficulties when present in steel up to 0.30 percent. Welding is a highly specialized manufacturing process. In pressurevessel fabrication, the designer has to follow the code developed by the American Society of Mechanical Engineers. Chuse [39.3] provides simplified guidelines for designing welding joints for pressure vessels. For design purposes, welding is classified into three basic types: groove, fillet, and plug welds. Welding joints are described by the position of the pieces to be joined and are divided into five basic types: butt, tee, lap, corner, and edge. Bednar [39.2] describes various types of welds and outlines methods for stress calculation.
39.7.1 Construction Materials A designer of a cylindrical pressure vessel should be familiar with commonly used construction materials to be able to specify them correctly in the design and in material specification. The selection of materials for code pressure vessels has to be made from the code-approved material specifications.
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