Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd
Chapter Objectives To determine the torsional deformation of a perfectly elastic circular shaft. To determine the support reactions when these reactions cannot be determined solely from the moment equilibrium equation. To determine the maximum power that can be transmitted by a shaft. Copyright © 2011 Pearson Education South Asia Pte Ltd

In-class Activities Reading Quiz Applications Torsion Formula Angle of Twist Statically indeterminate torque-loaded members Solid non-circular shafts Stress concentration Concept Quiz Copyright © 2011 Pearson Education South Asia Pte Ltd

READING QUIZ Given the angle of rotation is small and the materials remain linear elastic, which statement below is incorrect for the torsional behavior of a long straight circular shaft. Section-shape remains unchanged Straight member remains straight Plane section remains plane End of member may wrap Copyright © 2011 Pearson Education South Asia Pte Ltd

READING QUIZ (cont) Which one of the statements below is incorrect? Inelastic torsion of a circular shaft leads to non-linear distribution of shear stress plane section remaining plane change of sectional shape wrapping at the end of member Copyright © 2011 Pearson Education South Asia Pte Ltd

READING QUIZ (cont) Which statement below is incorrect? The angle of twist for a thin-walled tube having closed section is proportional to the length of circumference inversely proportional to the average thickness inversely proportional to the square of the enclosed area proportional to the shear modulus G Copyright © 2011 Pearson Education South Asia Pte Ltd

TORSION FORMULA (cont)
Linear distribution of stress: Torsion – shear relationship: Copyright © 2011 Pearson Education South Asia Pte Ltd

TORSION FORMULA (cont)
Polar moment of inertia For solid shaft: For tubular shaft: Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 1 The pipe shown in Fig. 5–12a has an inner diameter of 80 mm and an outer diameter of 100 mm. If its end is tightened against the support at A using a torque wrench at B, determine the shear stress developed in the material at the inner and outer walls along the central portion of the pipe when the 80-N forces are applied to the wrench. Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 1 (cont) Solutions The only unknown at the section is the internal torque T The polar moment of inertia for the pipe’s cross-sectional area is For any point lying on the outside surface of the pipe, Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 2 The two solid steel shafts are coupled together using the meshed gears. Determine the angle of twist of end A of shaft AB when the torque 45 Nm is applied. Take G to be 80 GPa. Shaft AB is free to rotate within bearings E and F, whereas shaft DC is fixed at D. Each shaft has a diameter of 20 mm. Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 2 (cont) Solutions Since the angle of twist of end A with respect to end B of shaft AB caused by the torque 45 Nm, The rotation of end A is therefore Copyright © 2011 Pearson Education South Asia Pte Ltd

STATICALLY INDETERMINATE TORQUE-LOADED MEMBERS
Please refer to the website for the animation: Statically Indeterminate Torque-Loaded Members STATICALLY INDETERMINATE TORQUE-LOADED MEMBERS Procedure for analysis: use both equilibrium and compatibility equations Equilibrium Draw a free-body diagram of the shaft in order to identify all the torques that act on it. Then write the equations of moment equilibrium about the axis of the shaft. Compatibility To write the compatibility equation, investigate the way the shaft will twist when subjected to the external loads, and give consideration as to how the supports constrain the shaft when it is twisted. Copyright © 2011 Pearson Education South Asia Pte Ltd

STATICALLY INDETERMINATE TORQUE-LOADED MEMBERS (cont)
Express the compatibility condition in terms of the rotational displacements caused by the reactive torques, and then use a torque-displacement relation, such as Φ = TL/JG, to relate the unknown torques to the unknown displacements. Solve the equilibrium and compatibility equations for the unknown reactive torques. If any of the magnitudes have a negative numerical value, it indicates that this torque acts in the opposite sense of direction to that indicated on the free-body diagram. Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 3 The solid steel shaft shown in Fig. 5–23a has a diameter of 20 mm. If it is subjected to the two torques, determine the reactions at the fixed supports A and B. Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 3 (cont) Solution It is seen that the problem is statically indeterminate since there is only one available equation of equilibrium and there are 2 unknowns Since the ends of the shaft are fixed, the angle of twist of one end of the shaft with respect to the other must be zero. Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 3 (cont) Solution Using the sign convention established, Using Eq. 1, The negative sign indicates that acts in the opposite direction of that shown in Fig. 5–23b. Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 4 The 6061-T6 aluminum shaft shown in Fig. 5–27 has a cross-sectional area in the shape of an equilateral triangle. Determine the largest torque T that can be applied to the end of the shaft if the allowable shear stress is tallow = 56 MPa. and the angle of twist at its end is restricted to Φallow = 0.02 rad. How much torque can be applied to a shaft of circular cross section made from the same amount of material? Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 4 (cont) Solution By inspection, the resultant internal torque at any cross section along the shaft’s axis is also T. By comparison, the torque is limited due to the angle of twist. Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 4 (cont) Solution For circular cross section, we have The limitations of stress and angle of twist then require Again, the angle of twist limits the applied torque. Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 5 The stepped shaft shown in Fig. 5–33a is supported by bearings at A and B. Determine the maximum stress in the shaft due to the applied torques. The shoulder fillet at the junction of each shaft has a radius of r = 6 mm. Copyright © 2011 Pearson Education South Asia Pte Ltd

EXAMPLE 5 (cont) Solution By inspection, moment equilibrium about the axis of the shaft is satisfied. The stress-concentration factor can be determined by the graph using the geometry, Thus, K = 1.3 and maximum shear stress is Copyright © 2011 Pearson Education South Asia Pte Ltd

CONCEPT QUIZ 2) The keyway shown below is used to connect gears or pulleys to a shaft. The maximum shear stress will occur at the center the circumference the point (dot) indicated None of the above Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd

Similar presentations

Presentation on theme: "Copyright © 2011 Pearson Education South Asia Pte Ltd"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Copyright © 2011 Pearson Education South Asia Pte Ltd

Similar presentations

Presentation on theme: "Copyright © 2011 Pearson Education South Asia Pte Ltd"— Presentation transcript:

Similar presentations

About project

Feedback