Chapter 3: Torsion. Chapter 4: Shear and Moment Diagram. Chapter 5: Stresses In beams

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2 Chapter 3: Torsion Chapter 4: Shear and Moment Diagram Chapter 5: Stresses In beams

3 Torsion Torsion or Torque, T, put simply, is referred to as a twisting moment. θ

4 The derived formulas are: Where: Torsional Shearing Stress T - Torque r Radial Distance J - Polar Moment of Inertia Shafts

5 TORSION ON CIRCULAR SHAFTS 1. For Solid Shafts: Iy r Ix

6 TORSION ON CIRCULAR SHAFTS 1. For Hollow Shafts: Iy R r Ix

7 The derived formulas are: Where: θ Angle of Twist T Torque L Length of Shaft over which the torque is acting J - Polar Moment of Inertia G Modulus of Rigidity

8

9 In solving torsion problems particularly with circular shafts with the derived formulas, the following assumptions must be observed: 1. Circular sections, remain circular. 2. Stresses do not exceed the proportional limit. 3. Plane sections remain plane and do not warp. 4. The projection upon a transverse section of straight radial lines in the sections remain straight. 5. Shaft is loaded by twisting couples in planes that are perpendicular to the axis of the shaft.

10 Torque Diagram 100 Nm 1000 Nm 900 Nm 500 Nm 700 Nm 800 Nm

11 Example : What is the minimum diameter of a solid steel shaft that will not twist through more than 3 o, in a 6 meter length when subjected to a torque of 14 knm? What maximum shearing stress is developed? If the modulus of rigidity of the shaft is 83 GPa.

12 Example : A stepped steel shaft consists of a hollow shaft 2m long, with an outside diameter of 100 mm and an inside diameter of 70 mm, rigidly attached to a solid shaft 1.5 m. long, and 80 mm in diameter. Determine the maximum torque which can be applied without exceeding a shearing stress of 70 MPa or a twist of 2.5 o in the 3.5 m length. Use G = 83 GPa. T T 2 m 1.5 m

13 Example : A steel shaft with a constant diameter of 60 mm is loaded by torques applied to gears attached to it as shown in the figure. Using G = 83 GPa, determine the relative angle of twist of gear D relative to gear A N.m 800 N.m 1200 N.m A 1000 N.m D 3m C 3m B 2m

14 Example : Determine the maximum shearing stress developed in each material and the angle of rotation of the free end of the shaft. Use G al = 28 GPa, G St = 83 GPa and G Br = 35 GPa. d = 100 mm 4 KN-m 1. 5 KN-m Aluminum Steel Bronze 3m 2m d = 75 mm 1.5 m

15 Example : The compound shaft shown is attached to rigid supports. For the bronze segment AB, the diameter is 75 mm, < or = 60MPa, and G br = 35 GPa. For the steel segment BC, the diameter is 50 mm, < or = 80 MPa, and G st = 83 GPa. Compute the maximum torque T that can be applied. A B C Bronze T Steel 2 m 1.5 m

16 Example : A shaft composed of segments AC, CD and DB is fastened to rigid supports and loaded as shown. For steel, G st = 83 GPa, for aluminum, G al = 28GPa, and for bronze, G br = 35 GPa. Determine the maximum shearing stress developed in each segment. 50 mm ф 25 mm ф 25 mm ф 300 N-m 700 N-m A C D Steel Aluminum Bronze B 2 m 1.5 m 1 m

17 Example : A round steel shaft 3 meters long tapers uniformly from a 60 mm diameter at one end to a 30 mm diameter at the other end. Assuming that no significant discontinuity results from applying the angular deformation equation over each infinitesimal length, compute the angular twist for the entire length when the shaft is transmitting a torque of 170 N.m. Use G = 83 GPa. 30 mm ф 60 mm ф 3 meters

18 Example : The steel shaft and an aluminum tube are connected to a fixed support and to a rigid disk as shown in the cross section. Knowing that the initial stresses are zero, determine the maximum torque T that can be applied to the disk if the allowable stresses are 120 MPa in the steel shaft and 70 MPa in aluminum tube. Use G st = 77 GPa and G al = 27 GPa. 8 mm 76 mm 50 mm 500 mm

19 Example : The two steel shafts shown in the figure, each with one end built into a rigid support, have flanges rigidly attached to their free ends. The shafts are to be bolted together at their flanges. However, initially there is a 6o mismatch in the location of the bolt holes, as shown in the figure. Determine the maximum shearing stress in each shaft after the shafts are bolted together. Use G = 83 GPa and neglect deformations of the bolts and flanges. 6 o 50 mm ф 40 mm ф 2 m 1 m

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