CIVL222 STRENGTH OF MATERIALS. Chapter 6. Torsion

Transcription

1 CIVL222 STRENGTH OF MATERIALS Chapter 6 Torsion

2 Definition Torque is a moment that tends to twist a member about its longitudinal axis. Slender members subjected to a twisting load are said to be in torsion.

3 Example of Torsion When opening the lid of a common plastic drinks bottle, a torque T applied to the cap is gradually increased until the plastic connectors between the cap and the bottle experience shear failure.

4 Example of Torsion Shafts are structural members with length significantly greater than the largest cross-sectional dimension used in transmitting torque from one plane to another.

5 Example of Torsion

6 Example of Torsion For a non-circular section member or an open section member subjected to torsion: Plane cross sections of the member do not remain plane and the cross sections distort in a manner which is called warping. In other words, the fibers in the longitudinal direction deform unequally.

7 Example of Torsion

8 Example of Torsion For a circular shaft or a closed circular section member subjected to torsion: Plane circular cross sections remain plane and the cross sections at the ends of the member remain flat. The length and the radius of the member remain unchanged. Plane circular cross sections remain perpendicular to the longitudinal axis.

9 Analogy Between Axial Deformation and Torsion Axial Force (P) Elongation (d) Normal Stress (s) Extensional Strain (e) Modulus of Elasticity (E) Torque (T) Twist Angle (f) Shear Stress (t) Shear Strain (g) Shear Modulus (G)

10 Torsion Theory for circular sections

11 Absence of Warping

12 Investigate Deformation

13 Investigate Deformation

14 State of Pure Shear

15 Shear Strain Relate to Angel of Twist

16 Linear Variation of Shear Stress

17 Linear Variation of Shear Stress

18 Shear Stress Surfaces

19 Shear Stress Surfaces

20 Moment dm developed on da

21 Setup the Integration dm over the Area A

22 Relating Torque and Stress

23 Torsion Formula

24 J for Solid Circular Shape

25 J for Hollow Circular Shape

26 Angle of Twist Formula

27 Summery of Key Equations

28 Sign Conventions

29 Sign Conventions

30 Sign Conventions

31 Sign Conventions

32 Example #1

33 Example #2

34 Example #3 The steel shaft of a socket wrench has a diameter of 8.0 mm. and a length of 200 mm (see figure). If the allowable stress in shear is 60 MPa, what is the maximum permissible torque T max that may be exerted with the wrench? Through what angle f (in degrees) will the shaft twist under the action of the maximum torque? (Assume G = 78 GPa and disregard any bending of the shaft.)

35 Example #4 A hollow steel shaft used in a construction auger has outer diameter d 2 =150 mm. and inner diameter d 1 = 115 mm. (see figure). The steel has shear modulus of elasticity G = 80 GPa For an applied torque of 17 kn.m, determine the following quantities: (a) shear stress t 2 at the outer surface of the shaft, (b) shear stress t 1 at the inner surface, and (c) rate of twist f (degrees per unit of length). Also, draw a diagram showing how the shear stresses vary in magnitude along a radial line in the cross section.

36 Example #5 A hollow aluminum tube used in a roof structure has an outside diameter d 2 =100 mm and an inside diameter d 1 =80 mm (see figure). The tube is 2.5 m long, and the aluminum has shear modulus G= 28 GPa. (a) If the tube is twisted in pure torsion by torques acting at the ends, what is the angle of twist f (in degrees) when the maximum shear stress is 50 MPa? (b) What diameter d is required for a solid shaft (see figure) to resist the same torque with the same maximum stress? (c) What is the ratio of the weight of the hollow tube to the weight of the solid shaft?

37 Example #6 Four gears are attached to a circular shaft and transmit the torques shown in the figure. The allowable shear stress in the shaft is 68 MPa. (a) What is the required diameter d of the shaft if it has a solid cross section? (a) What is the required outside diameter d if the shaft is hollow with an inside diameter of 25 mm?

38 Work Done and Power Transmitted When a force moves in a straight line with constant velocity the work done is given by the product the magnitude of the force and the distance through which it has moved. Work done = force distance The power transmitted by this action is defined as the rate at which this work is done, i.e. the work done in unit time. Power = work done time

39 Work Done and Power Transmitted The distance travelled by a rotating body is measured by the number of radians through which it rotates. The work done by a torque acting on a shaft is therefore given by the product of the magnitude of the torque and the amount of rotation in radians. For one revolutions of the shaft: The work done = T 2p (since the shaft turns through 2p radians in one revolution) If the shaft is rotating at N revolutions per minute, then work done = T 2p N (units of work per minute)

40 Work Done and Power Transmitted Usually the torque will measured in Newton meters (N.m) and therefore the units of work will also be N.m. However, it is more usual to give work in joules (J) which are equal numerically to Newton meters. 1 joule = 1 Newton meter Power is measured in watts (W), 1 W = 1 J/s = 1 N.m/s =(1/60) N.m/min

41 Work Done and Power Transmitted Hence, the power transmitted by a shaft rotating at N revolutions per minute and subject to torque of T (N.m) will be given by: Power T 2pN 60 Watts P T Tw P w 2pfT P 2pf w: the shaft s angular velocity (rad/s) : the frequency of shaft s rotation (Hz = 1 revolution/s) w 2pf rpm : revolutions per minute hp: horsepower, 1 hp = 746 W

42 Example #7 A motor drives a shaft at 12 Hz and delivers 20 kw of power (see figure). (a) If the shaft has a diameter of 30 mm, what is the maximum shear stress t max in the shaft? (b) If the maximum allowable shear stress is 40 MPa, what is the minimum permissible diameter d min of the shaft?

43 Example #8 The drive shaft for a truck (outer diameter 60 mm and inner diameter 40 mm) is running at 2500 rpm (see figure). (a) If the shaft transmits 150 kw, what is the maximum shear stress in the shaft? (b) If the allowable shear stress is 30 MPa, what is the maximum power that can be transmitted?