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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 Spring '09
 ZHU
 Force, Torsion

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