chapPlaneProblems–Applications.docVIP

6． Plane Problems – Applications 6.1 Stress Concentration The stress concentration near a hole is a critical issue concerning the strength of a solid structure. The stress concentration can be measured by the stress concentration coefficients that are the ratios between the most severe stress at the critical point (or termed hot spot) and the remote stress. We use the example of a circular hole under uniaxial remote tension to illustrate this point, as shown in Fig. 6.1(a). The traction free boundary condition leaves only the hoop stress for a point along the rim of the hole. Our analysis in the previous chapter indicated that the hoop stress at the upper and the lower edges of the hole is three times as high as the remote tensile stress. That gives a stress concentration coefficient of 3. One also notices that the hoop stress at the left and right ends of the hole is compressive and equal to . (a) (b) (c) Figure 6.1 Stress concentration near a circular hole. We proceed by examining the case depicted in Fig. 6.1(b), where the remote stress acting in a vertical orientation. The hoop stresses at the left and the right edges of the hole are tensile and equal to , and that at the top and bottom edges of the hole are compressive and equal to . The coefficient of stress concentration is again equal to 3. Adding the two stress states together, one confronts the situation shown in Fig. 6.1(c). This is the axi-symmetric case, the Lamé problem as given in (5.75) in Section 5.5. The hoop stress is equal to everywhere, and the coefficient of stress concentration is 2. It is worthwhile to pointing out that the stress state becomes less critical in Fig. 6.1(c) than either Fig. 6.1(a) or Fig. 6.1(b). Figure 6.2 Stress concentration near a circular hole when remote stresses are opposite. If the horizontal and vertical remote stresses are opposite to each other, as shown in the left graph of Fig. 6.2, the combined stress state becom