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(39.12)
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When the deformation reaches plastic state, a definite amount of deformation energy is assumed (see Ref. [39.1]), and the value of the Poisson s coefficient v is taken as equal to 0.5. Substituting Eqs. (39.16) and (39.12) into Eq. (39.46) and assuming the value of Poisson s coefficient v = 0.5, the design stress can be expressed as d = 0.87 piro t (39.47)
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where ro = external radius of the shell. The external radius ro = do/2, where do = external diameter of the shell. It is observed that according to the distortion-energy theory, considering the combined effect of the longitudinal and tangential stresses, the design stress for plastic material is 13.0 percent less compared with the maximum value of the main stress. The allowable stress a is therefore given by a = pido 2.3t (39.48)
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And the thickness of the shell t is given by t= pdo 2.3 a (39.49)
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Introducing the joint efficiency factor j and allowance factor tc for corrosion, erosion, and negative tolerance of the shell thickness, the following relation for design thickness is obtained: td = pdo + tc 2.3 a j (39.50)
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Azbel and Cheremisinoff [39.1], on the basis of similar analysis, give the following design formula for the determination of shell thickness: td = pidi + tc 2.3 j a pi (39.51)
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where do/di 1.5. The proper selection of allowable stress to provide safe operation of a vessel is an important design consideration. The allowable stress value is influenced by a number of factors, such as (1) the strength and ductility of the material, (2) variations in load over time, (3) variations in temperature and their influence on ductility and strength of the material, and (4) the effects of local stress concentration, impact loading, fatigue, and corrosion. In shell design, the criterion of determining the allowable stresses in an environmentally moderate temperature is the ultimate tensile strength. The pressure-vessel codes [39.3] use a safety factor of 4 based on the yield strength Sy for determining the allowable stresses for pressure vessels. The safety factor y is defined as
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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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PRESSURE CYLINDERS 39.21
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PRESSURE CYLINDERS
y =
Sy a
(39.52)
It is known that for ductile metals, an increase in temperature results in an increase in ductility and a decrease in the yield-strength value. The allowable stress at higher temperatures can be estimated by using the following relation: a = ST y y (39.53)
where ST = yield strength of the material at a given operating temperature. y Implied in the preceding analysis is an assumption that no creeping of the shell material is involved. This is generally a valid assumption for ferrous metals under load at temperatures less than 360 C. However, at higher operating temperatures, the material creeps under load, and an increase in stress takes place with time. The resulting stresses do not exceed the yield-strength values. It is to be noted that the yield point at high temperatures cannot generally be used as a criterion for allowance of the stressed state because creep of the material may induce failures as a result of the increased deformation over a long period of time.The basic design criterion for shells operating at moderate temperatures is the stability of size and dimensional integrity of the loaded elements. For a shell operating at higher temperatures, increase in size, which ensures that creep does not exceed a tolerable limit, should be considered. When a loaded material is under creep, there is an associated relaxation of stress. The stress decreases over time as a result of plastic deformation. The creep rate is dependent on temperature and state of stress in the metal. Azbel and Cheremisinoff [39.1] discuss the effects of creep and stress relaxation on loaded shells and give a formula for determining the allowable stresses and safety factors on the basis of fatigue stress.
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