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Torsion

  • Anthony Bedford
  • Kenneth M. Liechti
Chapter

Abstract

Bars are subjected to axial torques in many engineering applications. The resulting stresses and deformations are analyzed in this chapter. A state of pure shear stress is first defined. In terms of a cartesian coordinate system with its axes perpendicular to the faces of a cubical element, a state of pure shear stress acting on the element is described by the components of stress σx = 0, σy = 0, σz = 0, τxy = τ, τyz = 0, and τxz = 0. For a linear elastic material, the only resulting nonzero strain component is γxy = τ/G, where G is the shear modulus. By passing an oblique plane through the cubical element and drawing the free-body diagram of the part of the element to one side of the plane, expressions for the normal and shear stresses on oblique planes are obtained. The results show that there is no plane on which the shear stress is larger in magnitude than τ, and the magnitudes of the maximum tensile and compressive stresses are equal to τ. Prismatic cylindrical bars subjected to torques T at the ends are then analyzed. The angle of twist of a bar of length L (the angle through which an end of the bar rotates relative to the other end, expressed in radians) is ϕ = TL/GJ, where J is the polar moment of inertia of the bar’s circular cross section. The shear stress in the bar at a radial distance r from the bar’s axis is τ = Tr/J. Examples of statically indeterminate problems are then presented, showing that the same procedure applied to axially loaded bars in Chapter 3 applies to problems involving torsional loading. Composite bars, consisting of bonded concentric cylindrical bars consisting of different materials, are analyzed, followed by bars with gradually varying cross sections and bars subjected to distributed torsional loads. Torsion of circular bars consisting of material exhibiting elastic-perfectly plastic behavior is covered. A discussion is given of an approximate solution for bars having cross sections consisting of hollow tubes with thin walls that need not be circular. The chapter closes with a discussion of design considerations applying to bars subjected to torsion.

Keywords

Angle of twist Composite bars Design Elastic-perfectly plastic materials Polar moment of inertia Pure shear stress Thin-walled cross section Torque Torsion 

Supplementary material

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Anthony Bedford
    • 1
  • Kenneth M. Liechti
    • 1
  1. 1.University of TexasAustinUSA

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