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Chapter 3 Torsion 

Chapter 3 Torsion . Introduction. -- Analyzing the stresses and strains in machine parts which are subjected to torque T Circular -- Cross-section Non-circular Irregular shapes -- Material (1) Elastic (2) Elasto-plastic -- Shaft (1) Solid (2) Hollow.

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Chapter 3 Torsion 

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  1. Chapter 3 Torsion  Introduction -- Analyzing the stresses and strains in machine parts which are subjected to torque T Circular -- Cross-section Non-circular Irregular shapes -- Material (1) Elastic (2) Elasto-plastic -- Shaft (1) Solid (2) Hollow

  2. 3.1 Introduction • T is a vector  Two ways of expression -- Applications: a. Transmission of torque in shafts, e.g. in automobiles

  3. Assumptions in Torque Analysis: a. Every cross section remains plane and undistorted. b. Shearing strain varies linearly along the axis of the shaft.

  4. 3.2 Preliminary Discussion of the Stresses in a Shaft Where  = distance (torque arm) Since dF =  dA The stress distribution is Statically Indeterminate. Free-body Diagram

  5. -- Must rely on “deformation” to solve the problem. Analyzing a small element:

  6. 3.3 Deformations in a Circular Shaft  =  (T, L)-- the angle of twist (deformation) Rectangular cross section warps under torsion

  7. A circular plane remains circular plane

  8. Determination of Shear Strain  (in radians) The shear strain   

  9.  = c = radius of the shaft Since

  10. 3.4 Stresses in the Elastic Range Hooke’s Law  Therefore, (3.6)

  11. (3.1) (3.6) But (3.9) Therefore, Or,

  12. Substituting Eq. (3.9) into Eq. (3.6) (3.10) (3.9) These are elastic torsion formulas. For a solid cylinder: For a hollow cylinder:

  13. (3-13) Since  

  14. Mohr’s Circle (Sec. 7.4) -- Pure Shear Condition

  15. Ductile materials fail in shear (90o fracture) Brittle materials are weaker in tension (45o fracture)

  16. 3.5 Angle of Twist in the Elastic Range (3.3) (3.15) Therefore, Eq. (3.3) = Eq. (3.15)  Hence,

  17. For Multiple-Section Shafts:

  18. Shafts with a Variable Circular Cross Section

  19. 3.6 Statically Indeterminate Shafts • -- Must rely on both • Torque equations and • (2) Deformation equation, i.e. Example 3.05

  20. 3.7 Design of Transmission Shafts -- Two Parameters in Transmission Shafts: a. Power P b. Speed of rotation where  = angular velocity (radians/s) = 2  = frequency (Hz) [N.m/s = watts (W)] (3.21)

  21. (3.21) (3.9) Therefore, For a Solid Circular Shaft: 

  22. 3.8 Stress Concentrations in Circular Shafts

  23. 3.9 Plastic Deformation sin Circular Shafts (3.4) c = radius of the shaft

  24. d  c dA = 2 d  (3.1) c Knowing dF =  dA (3.26) Where  = ()

  25. (3.9) If we can determine experimentally an Ultimate Torque, TU, then by means of Eq. (3.9), we have RT =Modulus of Rupture in Torsion

  26. 3.10 Circular Shafts Made of an Elasto-Plastic Material Case I: < YHooke’s Law applies,  < max Case I Case II: < YHooke’s Law applies,  = max TY = max elastic torque Case II

  27. Since (3-29) Case III: Entering Plastic Region 0    Y: Case III Y   c: Y – region within the plastic range

  28. By evoking Eq. (3.26) (3.26) (3.31) 

  29. Case IV-- Fully Plastic Y  0: Case IV = Plastic Torque (3-33)

  30. 3.11 Residual Stresses in Circular Shafts

  31. 3.12 Torsion of Noncircular Members A rectangular shaft does not axisymmetry.

  32. From Theory of Elasticity:

  33. 3.13 Thin-Walled Hollow Shafts

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