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Advanced mechanics of materials (1993-John Willey) Episode 6

Tham khảo tài liệu 'advanced mechanics of materials (1993-john willey) episode 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 282 6 TORSION Figure E6.7 Solid circular cross section a Yield at r b. b Yield at r by. c Residual shear stress. c Likewise the twisting moment TE due to the shear stress up to r by that is in the elastic core by Eq. 6.15 with J ftby 2 is TE - e The torque Tp due to the stress in the yielded plastic region of the bar r by is Cb 2 Tp ttỵ 2tit dr 7ĩTy b3 by f J by 3 The total elastic-plastic moment TEp is by Eqs. e and f TEp Te Tp 7ĨTy b3 - by g d The fully plastic or limiting moment TL is given by Eq. g letting by - 0. Thus _ 2 TL 7TCYb3 h PROBLEMS 283 This result agrees with that of Table 6.3. Dividing Eq. h by Eq. a we obtain TL ịTY i Thus the limiting moment TL is 4 3 times larger than the torque that initiates yield recall the relation Mp 1.5 My for bending of a rectangular beam Eq. 4.42 . e When the limiting torque TL is released the bar springs back unwinds elastically and the shear stress-strain curve will follow a path similar to CA in Fig. 4.4iz. Since the limiting torque is TL 4Ty 3 the elastic stress recovery at r b is 4 y 3. Hence since the shear stress is initially Ty at r b a residual shear stress Ty 3 in the opposite sense of Ty remains at r b. However at the center of the cross section r 0 the residual stress remains Ty in the original sense as shown in Fig. E6.7c. Since the recovery is elastic the shear stress varies linearly from r 0 to r b. Hence the residual shear stress is zero at r 3b 4. PROBLEMS Sections 6.1-6.3 6.1. Derive the relation for the shear stress distribution on the X axis for the equilateral triangle in Fig. 6.9. 6.2. Derive Eqs. 6.49 for the equilateral triangle. 6.3. A square shaft may be used to transmit power from a farm tractor to farm implements. A 25.0-mm square shaft is made of a steel having a yield stress of Y 380 MPa. Determine the torque that can be applied to the shaft based on a factor of safety of SF 2.00 by using the octahedral shear stress criterion of failure. 6.4. A square shaft has 42.0-mm sides and the same .