20_Torsional_Loading

20_Torsional_Loading - 1 © 2002-2003 L.D Sturges EM 324 Mechanics of Materials Torsional Loading Torsion – the twisting of a straight bar when

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Unformatted text preview: 1 © 2002-2003, L.D. Sturges EM 324 Mechanics of Materials Torsional Loading Torsion – the twisting of a straight bar when it is loaded by moments (torques) that tend to produce rotation about the longitudinal axis . Synonyms : Torques, twisting moments. © 2002-2003, L.D. Sturges EM 324 Mechanics of Materials Torsional Loading (cont) Definition : Positive torque – the torque vector points outward from the “cut” surface. According to the right-hand-rule , this is a counter-clockwise torque when viewed from the “part discarded.” © 2002-2003, L.D. Sturges EM 324 Mechanics of Materials Torsional Loading (cont) Approximations/Assumptions : • Circular shaft (solid or hollow). • Straight; constant diameter. • Shearing strains are small. • Plane section before twisting – remains plane. • A diameter before twisting – remains straight. 2 © 2002-2003, L.D. Sturges EM 324 Mechanics of Materials Torsional Shearing Strain Take a short segment of a shaft rotated by a negative torque T (not shown). The right end of the shaft will rotate through an angle θ as shown. © 2002-2003, L.D. Sturges EM 324 Mechanics of Materials Torsional Shearing Strain (cont) Inscribe a line AB down the side of the shaft. The right end of the line will rotate taking B to B' and making an angle of φ between the lines AB and AB' . © 2002-2003, L.D. Sturges EM 324 Mechanics of Materials Torsional Shearing Strain (cont) The length of the line/arc BB' is A square element of material at B will deform into a parallelogram. ' B B L c φ θ = = 3 © 2002-2003, L.D. Sturges EM 324 Mechanics of Materials Torsional Shearing Strain (cont) Then and the shearing strain of an element on the outside surface of the shaft is ' B B L c φ θ = = x δ = y δ = x ε = y ε = xy γ φ = ' c B B c L L θ γ φ = = = where θ is in radians . © 2002-2003, L.D. Sturges EM 324 Mechanics of Materials Torsional Shearing Strain (cont) Similarly looking at the line ED (at a distance ρ from the center) which rotates to ED' would give Therefore ' c B B c L L θ γ φ = = = ' DD L L ρ ρθ γ φ = = = c L L c ρ γ γ θ ρ = = and © 2002-2003, L.D. Sturges EM 324 Mechanics of Materials Torsional Shearing Strain (cont) That is, the shear strain increases linearly with respect to the distance from the center of the shaft, and is a maximum at the outer surface of the shaft ( ρ = c ). c c ρ γ ρ γ = c L L c ρ γ γ θ ρ = = max c c L θ γ γ = = 4 © 2002-2003, L.D. Sturges EM 324 Mechanics of Materials Torsional Shearing Stresses If the material is linearly elastic and τ < τ p (the stress is lower than the proportional limit), then G τ γ = or G L ρ ρθ τ = c Gc L θ τ = and c c ρ τ ρ τ = © 2002-2003, L.D. Sturges EM 324 Mechanics of Materials...
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This note was uploaded on 03/27/2008 for the course EM 324 taught by Professor Boylan during the Spring '08 term at Iowa State.

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20_Torsional_Loading - 1 © 2002-2003 L.D Sturges EM 324 Mechanics of Materials Torsional Loading Torsion – the twisting of a straight bar when

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