Linear_Elasticity_04_Torsion - Section 4.4 Solid Mechanics...

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Unformatted text preview: Section 4.4 Solid Mechanics Part I Kelly 121 4.4 Torsion In this section, the geometry to be considered is that of a long slender circular bar and the load is one which twists the bar. Such problems are important in the analysis of twisting components, for example lug wrenches and transmission shafts. 4.4.1 Basic relations for Torsion of Circular Members The theory of torsion presented here concerns torques 1 which twist the members but which do not induce any warping , that is, cross sections which are perpendicular to the axis of the member remain so after twisting. Further, radial lines remain straight and radial as the cross-section rotates – they merely rotate with the section. For example, consider the member shown in Fig. 4.4.1, built-in at one end and subject to a torque T at the other. The x axis is drawn along its axis. The torque shown is positive, following the right-hand rule (see §4.3.4). The member twists under the action of the torque and the radial plane ABCD moves to D C AB ′ . Figure 4.4.1: A cylindrical member under the action of a torque Whereas in the last section the measure of deformation was elongation of the axial members, here an appropriate measure is the amount by which the member twists, the rotation angle φ . The rotation angle will vary along the member – the sign convention is that φ is positive in the same direction as positive T as indicated by the arrow in Fig. 4.4.1. Further, whereas the measure of strain used in the previous section was the normal strain xx ε , here it will be the engineering shear strain xy γ (twice the tensorial shear strain xy ε ). A relationship between γ (dropping the subscript) and φ will next be established. As the line BC deforms into C B ′ , Fig. 4.4.1, it undergoes an angle change α . As defined in §3.6.2, the shear strain γ is the change in the original right angle formed by BC and a tangent at B (indicated by the dotted line – this is the y axis to be used in...
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 1 taught by Professor Staff during the Fall '11 term at Auckland.

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Linear_Elasticity_04_Torsion - Section 4.4 Solid Mechanics...

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