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torsion1 - mer ATA Ouzhan DEDE TORSION 1 Mechanics of...

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Ömer ATAŞ 23/12/2009 Oğuzhan DEDE TORSION 1- Mechanics of Materials Approach 2- Prandtl Stress Function Approach -more sufficient for axial cylinder problem Sufficient insufficient sufficient 1-) Mechanics of Material Approach to Torsion of Circular Bars
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Assumptions: 1-) All plane sections perpendicular to torsion axis z remain perpendicular. 2-) Cross sections are undistorted in their individuals planes -Shearing strain Ɣ varies linearly from at the center to a maximum at the outer surface 3-) Material is homogenous T= 𝜏 . ?𝑑𝐴 T= ? ? ? 0 . 𝜏 𝑚𝑎? . ?𝑑𝐴 = 𝜏 𝑚𝑎? ? . ? 2 𝑑𝐴 Where ? 2 𝑑𝐴 is polar moment of inertia J= ? 2 𝑑𝐴 and τ max = 𝑇 . ? 𝐽 τ(ρ)= 𝑇 . ? 𝐽 Note: Polar moment of inertia for circle of radius r ; J= ? . ? 4 2 Angle of Twist Φ : Φ .r = Ɣ max .L Ɣ max = 𝜏 𝑚𝑎? ? ( Hooke’s law) so, Ɣ max = 𝑇 . ? 𝐽 . ? Φ .r = 𝑇 . ? 𝐽 . ? . 𝐿 Φ = 𝑇 . 𝐿 𝐽 . ? J.G : Torsional rigidity
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2) theory of elasticity approach to torsion of bar y P α v r -u p θz A α x u=- (rθz)sinα= - yθz v=(rθz)cosα =x θz
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