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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

If it is true, then everything continues. If it s false, then the program will halt in the debugger (usually by forcing a crash). NOTE: Objective-C provides two assert methods: NSAssert and NSCAssert. NSAssert should be used only in Objective-C methods, while NSCAssert can be used only in C methods. Both can be removed from code by defining NS_BLOCK_ASSERTIONS as a preprocessor macro. The problem with using the built-in C++ assert call is that ideally you want your asserts to be active only while you re debugging. You want a way to disable all of your asserts when you go to ship your app. Asserts are invaluable in helping you track down variables that are out of bounds and other common problems, but in your shipped product, they re just taking up valuable CPU cycles, because your app should be bugfree by that point. Right

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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)

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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The best way to deal with this is to create custom assert code. In this section, you ll create a custom assert macro that can be disabled via a build-time preprocessor define. First, create a header file that s going to hold the macros you re about to write. I m going to call mine MyDebug.h so that I know these are my debugging functions, but you can call it whatever you d like. Then you can include this header wherever you need it. Add the following code to the header file, and then I ll talk about what it s doing:

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.

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