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CURVED BEAMS AND RINGS 38.21
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FIGURE 38.9 A portion of the ring has been isolated here to determine the moment and torque at any section D at angle from the fixed end at A.
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These equations are now employed in the same manner as in Sec. 38.5.1 to obtain a11 a12 a21 a22 T1/wr2 b = 1 M1/wr2 b2 (38.50)
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It turns out that the aij terms in the array are identical with the same coefficients in Eq. (38.28); they are given by Eqs. (38.29), (38.30), (38.32), and (38.33), respectively. The coefficients bk are bk = Xk + where X1 = EI Yk GK (38.51) (38.52) (38.53)
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sin2 + sin cos + 2 sin 2 sin2 sin cos + (1 + cos ) 2
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Y1 = 2 sin
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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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CURVED BEAMS AND RINGS 38.22
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TABLE 38.6 Coefficients bk for Various Span Angles and Uniform Loading
X2 = Y2 =
2 2(1 cos ) sin cos + sin2 2 2 2 2(1 cos ) + sin cos sin2 + sin 2 2
(38.54) (38.55)
Solutions to these equations for a variety of span angles are given in Table 38.6. A solution for the deflection at any point can be obtained using a fictitious load Q at any point and proceeding in a manner similar to other developments in this chapter. It is, however, a very lengthy analysis.
REFERENCES
38.1 Raymond J. Roark and Warren C. Young, Formulas for Stress and Strain, 6th ed., McGraw-Hill, New York, 1984. 38.2 Joseph E. Shigley and Charles R. Mischke, Mechanical Engineering Design, 5th ed., McGraw-Hill, New York, 1989. 38.3 J. P. Den Hartog, Advanced Strength of Materials, McGraw-Hill, New York, 1952.
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.
Source: STANDARD HANDBOOK OF MACHINE DESIGN
PRESSURE CYLINDERS
Sachindranarayan Bhaduri, Ph.D.
Associate Professor Mechanical and Industrial Engineering Department The University of Texas at El Paso El Paso, Texas
39.1 INTRODUCTION / 39.1 39.2 DESIGN PRINCIPLES OF PRESSURE CYLINDERS / 39.2 39.3 DESIGN LOADS / 39.3 39.4 CYLINDRICAL SHELLS STRESS ANALYSIS / 39.4 39.5 THICK CYLINDRICAL SHELLS / 39.12 39.6 THERMAL STRESSES IN CYLINDRICAL SHELLS / 39.14 39.7 FABRICATION METHODS AND MATERIALS / 39.17 39.8 DESIGN OF PRESSURE CYLINDERS / 39.18 REFERENCES / 39.21
39.1 INTRODUCTION
The pressure vessels commonly used in industrial applications consist basically of a few closed shells of simple shape: spherical or cylindrical with hemispherical, conical, ellipsoidal, or flat ends. The shell components are joined together mostly by welding and riveting; sometimes they are bolted together using flanges. Generally, the shell elements are axisymmetrical surfaces of revolution formed by rotation of a straight line or a plane curve known as a meridian or a generator about an axis of rotation. The plane containing the axis of rotation is called the meridional plane. The geometry of such simple shells is specified by the form of the midwall surface, usually two radii of curvature and the wall thickness at every point. The majority of pressure vessels are cylindrical. In practice, the shell is considered thin if the wall thickness t is small in comparison with the circumferential radius of curvature R and the longitudinal radius of curvature R . If the ratio R /t > 10, the shell is considered to be thin shell.This implies that the stresses developed in the shell wall by external loads can be considered to be uniformly distributed over the wall thickness. Many shells used in pressure-vessel construction are relatively thin (10 < R /t < 500), with the associated uniform distribution of stresses throughout the cylinder wall. Bending stresses in the walls of such membrane shells due to concentrated external loads are of higher intensity near the area of application of the load. The attenuation distance from the load where the stresses die out is short. The radial deformation of a shell subjected to internal pressure is assumed smaller than one-half the shell thickness. The shell thickness is designed to keep the maximum stresses below the yield strength of the material.
39.1 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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