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The two normal stresses are equal when the element is aligned in the directions given by Eq. (36.6). The act of referring stress components to another reference system is called transformation of stress. Such transformations are easier to visualize, and to solve, using a Mohr s circle diagram. In Fig. 36.2 we create a coordinate system with normal stresses plotted as the ordinates. On the abscissa, tensile (positive) normal stresses are plotted to the right of the origin O, and compression (negative) normal stresses are plotted to the left. The sign convention for shear stresses is that clockwise (cw) shear stresses are plotted above the abscissa and counterclockwise (ccw) shear stresses are plotted below. The stress state of Fig. 36.1a is shown on the diagram in Fig. 36.2. Points A and C represent x and y, respectively, and point E is midway between them. Distance AB is xy and distance CD is yx. The circle of radius ED is Mohr s circle. This circle passes through the principal stresses at F and G and through the extremes of the shear stresses at H and I. It is important to observe that an extreme of the shear stress may not be the same as the maximum.

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Listing 8-1 Twelve-channel radio remote control system program (Continued).

FIGURE 36.2 Mohr s circle diagram for plane stress. (From Applied Mechanics of Materials, by Joseph E. Shigley. Copyright 1976 by McGraw-Hill, Inc. Used with permission of the McGrawHill Book Company.)

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36.1.1 Programming To program a Mohr s circle solution, plan on using a rectangular-to-polar conversion subroutine. Now notice, in Fig. 36.2, that ( x y)/2 is the base of a right triangle, xy is the ordinate, and the hypotenuse is an extreme of the shear stress. Thus the conversion routine can be used to output both the angle 2 and the extreme value of the shear stress. As shown in Fig. 36.2, the principal stresses are found by adding and subtracting the extreme value of the shear stress to and from the term ( x + y)/2. It is wise to ensure, in your programming, that the angle indicates the angle from the x axis to the direction of the stress component of interest; generally, the angle is considered positive when measured in the ccw direction.

Reading though the documentation and the sample code, I gradually realized that the Audio Queue Services approach was designed around reading audio from a file, as with previous methods I d tried. I pressed on anyway. The sample code didn t seem to grab the entire audio file at once instead, it used a collection of buffers that would be successively filled with part of an audio file and then played. By cycling through the buffers, the entire file would gradually be played. That sounded like a promising approach, since I expected that once I knew what I was doing, I could modify the code to take data from the network instead of from a file. The general approach is a repeating cycle in which you read a chunk of audio data from a file into a buffer and add it to an audio queue. The audio queue plays the buffer and then calls a callback function, which repeats the process until no more file data remains. Figure 3-2 illustrates this cycle.

The general three-dimensional stress element in Fig. 36.3a has three normal stresses x, y, and z, all shown as positive, and six shear-stress components, also shown as positive. The element is in static equilibrium, and hence

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Note that the first subscript is the coordinate normal to the element face, and the second subscript designates the axis parallel to the shear-stress component. The negative faces of the element will have shear stresses acting in the opposite direction; these are also considered as positive. As shown in Fig. 36.3b, there are three principal stresses for triaxial stress states. These three are obtained from a solution of the equation

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