Problem 3. Solid circular shafts made of titanium 6Al-4V (annealed) are subjected in service to bending,...

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Problem 3. Solid circular shafts made of titanium 6Al-4V (annealed) are subjected in service to bending, with a moment M = 6.0kNm. Half-circular surface cracks, as in Figure below, may exist in the part. From nondestructive inspection, it is expected that no cracks larger than a = 5 mm are present. (a) What shaft diameter is required to resist yielding with a safety factor of 2.0 if no crack is present? (b) For an inspection-size crack, what shaft diameter is required to resist brittle fracture with a safety factor of 3.0? (c) What shaft diameter should actually be used? (a) P 111 2b 2t M P M (b) P M K = FS SS (tension, P) SS (bending, M) (c) (d) Case St Sb F for small a Limits for 10% on F P 2 a a (a) = 0.637 <0.5 4bt t b P 3M a a (b) 0.728 < 0.4, < 0.3 2bt bt2 b P 6M a a (c) 0.722 bt bt2 <0.35, < < 0.2 t 4P 32M a (d) 0.728 < 0.2 or 0.351 d d Note: 'Different limits for tension or bending, respectively. Figure 8.17 Stress intensity factors for (a) an embedded circular crack under uniform tension normal to the crack plane, and related cases: (b) half-circular surface crack, (c) quarter-circular corner crack, and (d) half-circular surface crack in a shaft, where the latter is more precisely a portion of a circular arc with center on the surface. (Based on [Newman 86] and [Raju 86].) Problem 3. Solid circular shafts made of titanium 6Al-4V (annealed) are subjected in service to bending, with a moment M = 6.0kNm. Half-circular surface cracks, as in Figure below, may exist in the part. From nondestructive inspection, it is expected that no cracks larger than a = 5 mm are present. (a) What shaft diameter is required to resist yielding with a safety factor of 2.0 if no crack is present? (b) For an inspection-size crack, what shaft diameter is required to resist brittle fracture with a safety factor of 3.0? (c) What shaft diameter should actually be used? (a) P 111 2b 2t M P M (b) P M K = FS SS (tension, P) SS (bending, M) (c) (d) Case St Sb F for small a Limits for 10% on F P 2 a a (a) = 0.637 <0.5 4bt t b P 3M a a (b) 0.728 < 0.4, < 0.3 2bt bt2 b P 6M a a (c) 0.722 bt bt2 <0.35, < < 0.2 t 4P 32M a (d) 0.728 < 0.2 or 0.351 d d Note: 'Different limits for tension or bending, respectively. Figure 8.17 Stress intensity factors for (a) an embedded circular crack under uniform tension normal to the crack plane, and related cases: (b) half-circular surface crack, (c) quarter-circular corner crack, and (d) half-circular surface crack in a shaft, where the latter is more precisely a portion of a circular arc with center on the surface. (Based on [Newman 86] and [Raju 86].)