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Torsion-7 - KML Torsion(Chapter 3 in Gere KML Torsion is...

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KML 1/18/06 Torsion (Chapter 3 in Gere) KML 9/4/09 Torsion – is the twisting of a straight bar when it is loaded by torques that are applied about the longitudinal axis. Axial (longitudinal) moments are commonly called torques and denoted by T. Shafts – are circular members that are subjected to torques that may transmit power. Pure torsion – when a circular shaft is loaded by torques only, such that every cross section of the shaft has the same internal torque. For small deformations, cross sections do not change shape, and they remain plane and circular since there is pure rotation about the longitudinal axis. Angle of twist, φ – the angle a shaft twists (rotational deformation) under the load of torques. T (torque) L γ L r S γ φ = φ S ρ S - sector length Strain in pure torsion : φ = γ L r max r 0 ; r max < ρ < γ ρ = γ Note: ρ is a variable radius; r is the outer radius Shear in pure torsion : max max max r ; G τ ρ = τ γ = τ Note: shear strain and stress are zero at the center, 0 = ρ Derivation of the relationship between torque and shear stress for elastic circular shafts : Cross-section of shaft in torsion: dA ρ r τ dA dV τ = dA dM τ ρ = max r τ ρ = τ dA r dM 2 max ρ τ = p max A 2 max A I r dA r dM T τ = ρ τ = = shear force acting on dA
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KML 1/18/06 Polar second moment of area – is the measure of the distribution of area with respect to the longitudinal axis. (For reference, sometimes J is used to denote the polar second moment of area). For a solid circular shaft: For an annular shaft: 2 r 32 d I 4 4 p π = π = ( 29 4 i 4 o p d d 32 I - π = Maximum shear stress in the elastic range in terms of torque, T, and second moment of area, I p : p max I r T = τ For solid circular shafts: 3 max d T 16 π = τ Shear stress at any radius, ρ ,: ρ = τ p I T Angle of twist (radians) : ( 29 ... L / r G G I / r T max p max φ φ = γ = = τ p I G
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