EXAMPLE 8-10: Disk Under Diametrical Compression–Cont’d stress contours are plotted using MATLAB. The corresponding photoelastic results are also shown in the figure. In general, the theoretical contours match quite well with the experimental results except for the regions near the loading points at the top and bottom of the disk. This lack of correspondence is caused by the fact that the photoelastic isochromatics were generated with a loading distributed over a small but finite contact area, and thus the maximum shear stress occurs slightly below the contact surface, as per earlier discussions of Figure 8-28. A numerical analysis of this problem using the finite element method is developed in Chapter 15; see Example 15-3 and Figure 15-6.
EXAMPLE 8-11: Rotating Disk Problem As a final example in this section, consider the problem of a thin uniform circular disk subject to constant rotation ! as shown in Figure 8-37. The rotational motion generates centrifugal acceleration on each particle of the disk, and this then becomes the source of external loading for the problem. No other additional external loadings are considered. It is convenient to handle the centrifugal force loading by relating it to a body force density through the disk. For the case of constant angular velocity, the body force is only in the radial direction given by Fr ¼ r!2 r
(8:4:73)
where r is the material mass density. This problem is axisymmetric, and thus the equilibrium equations reduce to dsr sr sy þ þ r!2 r ¼ 0 dr r
(8:4:74)
The solution can be efficiently handled by using a special stress function that automatically satisfies the equilibrium equation. The particular stress-stress function relation with this property is given by sr ¼ j=r dj sy ¼ þ r!2 r 2 dr
(8:4:75)
where j ¼ j(r) is the stress function. y
w
a
FIGURE 8-37
x
Rotating circular disk. Continued
Two-Dimensional Problem Solution
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