8-18. Determine the stress field solution (8.4.47) for the problem of a half space under a concentrated surface moment as shown in Figure 8-23. It is recommended to use the superposition and limiting process as illustrated in the figure. This solution can be formally developed using either Cartesian or polar coordinate stress components. However, a simple and elegant solution can be found by noting that the superposition and limiting process yields the stress function solution fM ¼ d@f=@x, where f is the solution to the Flamant problem shown in Figure 8-21. 8-19*. For the problem of a half space under uniform normal loading as shown in Figure 8-24, show that the maximum shear stress can be expressed by p tmax ¼ sin (y1 y2 ) p Plot the distribution of lines of constant maximum shear stress, and compare the results with the photoelastic fringes shown in Figure 8-28. These results along with several other loading cases have been given by Poulos and Davis (1974). 8-20. Using the formulation and boundary condition results of the thin notch crack problem shown in Figure 8-29, explicitly develop the stress components given by relations (8.4.56) and (8.4.57). 8-21*. Photoelastic studies of the stress distribution around the tip of a crack have produced the isochromatic fringe pattern (opening mode I case) as shown in the figure. Using the solution given in (8.4.57), plot the contours of constant maximum shear stress for both modes I and II. In plotting each case, normalize tmax by the coefficient A or B. For the mode I case, theoretical contours should compare with the photoelastic picture.
(Courtesy of Dynamic Photomechanics Laboratory, University of Rhode Island)
8-22*. Using strength of materials theory (see Boresi, Schmidt, and Sidebottom 1993), the bending stress sy for curved beams is given by sy ¼ M(r B)=[rA(R B)], where A ¼ b a, B ¼ (b a)= log (b=a), R ¼ (a þ b)=2. For the problem shown in Figure 8-30, compare and plot the strength of materials and elasticity predictions for the cases of b/a¼ 2 and 4. Follow the nondimensional plotting scheme used in Figure 8-31.
Two-Dimensional Problem Solution
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