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PRESSURE CYLINDERS 39.7
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PRESSURE CYLINDERS
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sary to define a normal force, two transverse shear forces, two bending moments, and a torque in order to evaluate the state of stress at a point. Membrane theory simplifies this analysis to a great extent and permits one to ignore bending and twisting moments when shell thickness is small. In many practical cases, consideration of equilibrium of the forces allows us to develop necessary relations for stresses and displacements in terms of the shell parameters for adequate design.
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39.4.3 Thin Cylindrical Shells under Internal Pressure When a thin cylinder is subjected to an internal pressure, three mutually perpendicular principal stresses hoop stress, longitudinal stress, and radial stress are developed in the cylinder material. If the ratio of thickness t and the inside diameter of the cylinder di is less than 1:20, membrane theory may be applied and we may assume that the hoop and longitudinal stresses are approximately constant across the wall thickness. The magnitude of radial stress is negligibly small and can be ignored. It is to be understood that this simplified approximation is used extensively for the design of thin cylindrical pressure vessels. However, in reality, radial stress varies from zero at the outside surface to a value equal to the internal pressure at the inside surface. The ends of the cylinder are assumed closed. Hoop stress is set up in resisting the bursting effect of the applied pressure and is treated by taking the equilibrium of half of the cylindrical vessel, as shown in Fig. 39.2. Total force acting on the half cylinder is Fh = pidiL (39.10)
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where di = inside diameter of cylinder, and L = length of cylinder. The resisting force due to hoop stress h acting on the cylinder wall, for equilibrium, must equal the force Fh. Thus Fh = 2 htT (39.11)
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Substituting for Fh from Eq. (39.10) into Eq. (39.11), one obtains the following relation: h = pidi 2t or h = piri t (39.12)
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Despite its simplicity, Eq. (39.12) has wide practical applications involving boiler drums, accumulators, piping, casing chemical processing vessels, and nuclear pressure vessels. Equation (39.12) gives the maximum tangential stress in the vessel wall on the assumption that the end closures do not provide any support, as is the case with long cylinders and pipes. Hoop stress can also be expressed in terms of the radius of the circle passing through the midpoint of the thickness. Then we can write h = pi(ri + 0.5t) t (39.13)
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The shell thickness is then expressed as t= piri h 0.5pi (39.14)
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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 39.8
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CLASSICAL STRESS AND DEFORMATION ANALYSIS
r t di Pi h t L Pi ri h
FIGURE 39.2 Thin cylindrical shell under internal pressure.
The code stress and shell thickness formulas based on inside radius approximate the more accurate thick-wall formula of Lam , which is t= piri Se 0.6pi (39.15)
where e = code weld-joint efficiency, and S = allowable code stress. Consideration of the equilibrium forces in the axial direction gives the longitudinal stress as = or = pidi 4t piri 2t (39.16)
Equations (39.12) and Eqs. (39.16) reveal that the efficiency of the circumferential joint needs only be one-half that of the longitudinal joint. The preceding relations are good for elastic deformation only. The consequent changes in length, diameter, and intervolume of the cylindrical vessel subjected to inside fluid pressure can be determined easily.
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PRESSURE CYLINDERS 39.9
PRESSURE CYLINDERS
The change in length is determined from the longitudinal strain given by = 1 ( h) E (39.17)
where E = modulus of elasticity, and = Poisson s ratio. The change in length L is then given by L = or L = L ( h) E pidi (1 2 )L 4Et (39.18)
Radial growth or dilatation under internal pressure is an important criterion in pipe and vessel analysis. For a long cylindrical vessel, the change in diameter is determined from consideration of the change in the circumference due to hoop stress.The change in circumference is obtained by multiplying hoop strain h by the original circumference. The changed, or new, circumference is found to be equal ( di + di h). This is the circumference of the circle of diameter di(1 + h). It can be shown easily that the diametral strain equals the hoop or circumferential strain. The change in diameter di is given by di = or di = di ( h ) E pid2 i (2 ) 4Et (39.19)
The change in the internal volume V of the cylindrical vessel is obtained by multiplying the volume strain by the original volume of the vessel and is given by V = pidi (5 4 )Vo 4Et (39.20)
where V = change in volume, and Vo = original volume. 39.4.4 Thin Spherical Shell under Internal Pressure A sphere is a symmetrical body.The internal pressure in a thin spherical shell will set up two mutually perpendicular hoop stresses of equal magnitude and a radial stress. When the thickness-to-diameter ratio is less than 1:20, membrane theory permits us to ignore the radial stress component. The stress system then reduces to one of equal biaxial hoop or circumferential stresses. Considering the equilibrium of the half sphere, it can be seen that the force on the half sphere (Fig. 39.3) due to internal pressure pi is F= 2 di pi 4 (39.21)
The resisting force due to hoop stress is given by Fh = dit h (39.22)
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