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LECTURE+10+COE-3001-A

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Unformatted text preview: o not change for small rotabons ²་  shear strain (γ) is equal to the decrease in \bad 1/30/13 M. Mello/Georgia Tech Aerospace 5 rd SHEAR STRAIN ON OUTER SURFACE OF THE BAR 0 max max bb = ab d =r dx Maximum shear strain on outer surface rate of twist (φ) per unit length (x) arc length bb’ NOTE: IN CASE OF PURE TORSION: 1/30/13 d = dx L SHEAR STRAINS WITHIN THE BAR =⇢ but, shear strain (γ) at interior radius (ρ) d dx d = dx max r and so, = ⇢ max r r ⇢ = max = L L M. Mello/Georgia Tech Aerospace (from the relabon over on the leP) (shear strains in circular bar vary linearly with radial distance ρ from center) 6 SHEAR STRAIN IN HOLLOW CIRCULAR TUBES IN CASE OF PURE TORSION: Invoke: = ⇢ L min = 1/30/13 max = r1 r1 = L r2 r2 L max d = dx L MAXIMUM STRAIN ON THE EXTERIOR SURFACE (ρ=r2) MINIMUM STRAIN ON THE INTERIOR SURFACE (ρ=r1) M. Mello/Georgia Tech Aerospace 7 CIRCULAR BARS OF LINEAR ELASTIC MATERIALS ²་  Hooke’s law for shear stress shear stress (3- 6) shear strain shear modulus ⇢ ⇢ G max (3- 7) ²་  Subsbtute = max ⌧ = r r ²་  But ⌧max = G max radius (variable) ²་  And so, shear ⇢ ⌧ = ⌧max stress r 1/30/13 ⌧ =G outer radius (ρ=r) max shear stress at outer surface (ρ=r) d ²་  In general case: ⌧ = G⇢ dx ²་  Pure torsion: ⌧ = G⇢ L d (since = constant = ) L dx M. Mello/Georgia Tech Aerospace 8 CIRCULAR BARS OF LINEAR ELASTIC MATERIALS 3D stress element ²་  Shear strain is maximum at the outer surface ²་  Shear stress is therefore maximum at the outer surface ²་  Shear stress linearly decays with radial distance (ρ) unbl it becomes zero along the central axis of the cylinder. r Longitudinal and transverse shear stresses in a circular bar subjected to torsion 1/30/13 M. Mello/Georgia Tech Aerospace 9 COMMON FAILURE MODES REULTING FROM PURE TORSIONAL LOADING ²་  SAY MATERIAL IS WEAKEST IN SHEAR ALONG THE LONGITUDINAL PLANES ²་  e.g.. Wood with wood grain aligned with longitudinal axis ²་  FIRST CRACKS DUE TO TORSION APPEAR ON THE SURFACE IN THE LONGITUDINAL DIRECTION 1/30/13 L = LONGITUDINAL R = RADIAL T = TANGENTIAL DUCTILE TORSION FAILURE (WEAKEST IN SHEAR ALONG TRANSVERSE PLANES) M. Mello/Georgia Tech Aerospace 10...
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