tailieunhanh - 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 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. 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 . 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 My for bending of a rectangular beam Eq. . 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. . 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. . 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 . Derive the relation for the shear stress distribution on the X axis for the equilateral triangle in Fig. . . Derive Eqs. for the equilateral triangle. . A square shaft may be used to transmit power from a farm tractor to farm implements. A 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 by using the octahedral shear stress criterion of failure. . A square shaft has sides and the same .