notes_13_pure_twist

notes_13_pure_twist - Lecture 4 - 2003 Pure Twist pure...

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Lecture 4 - 2003 Pure Twist pure twist around center of rotation D => neither axial ( σ ) nor bending forces (Mx, My) act on section; as previously, D is fixed, but (for now) arbitrary point. as before: a) equilibrium of wall element: d q + d σ t = 0 ds dx b) compatibility (shear strain) d u + d v = γ = 0 small deflections ds dx c) tangential displacement ( δ v) in terms of η , ζ and φ ( geometry) δ v = δη cos α () + h p δφ + δζ sin α δ x δ x δ x δ x N.B. h p =>h D from definition of problem further assumptions: 1) preservation of cross section shape => ζ = ζ (x); η = η (x) φ = φ (x) 2) shear though finite is small ~ 0 => d u = d v ds dx 3) Hooke's law holds => σ = E δ u axial stress δ x ----------------------------- from equilibrium -------------------------------- pure twist σ dA = N x τ⋅ h p dA = qh p ds = T p σ dA = N x = 0 σ⋅ y dA = -M.z dA = q cos α cos α ds = V y y dA = M z = 0 sin α ds = V z z dA = M y = 0 z dA = M y dA = q sin α pure twist also => only φ is finite i.e. other displacements (and derivatives) ζ = η = 0 => δ v δη + δζ sin α = δ x cos α δ x + h p δφ becomes δ v = h D δφ δ x δ x δ x δ x using negligible shear assumption d u = d v => d u = h D δφ and integration along s => ds dx ds δ x 1 notes_13_pure_twist.mcd
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u = δφ h D ds + u 0 x δ x () which showed u linear with y and z => plane sections plane. previously u = −η ' Y − ζ 'Z + u 0 x here - only if h D is constant so it can come outside h D 1 ds - is u (longitudinal displacement) linear. u is defined as warping displacement (function). stress analysis can be made analogous for torsion and bending IF the integrand h D ds thought to be a coordinate. calculation of stresses will involve statical moments, moments of inertia and products of inertia which will be designated "sectorial" new coordinate = wrt arbitrary origin and ω wrt normalized sectorial coordinate (as before like wrt center of area) d = h D ds = d ω the warping function then becomes:
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notes_13_pure_twist - Lecture 4 - 2003 Pure Twist pure...

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