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Mechanics Of Materials

by Karthic Easwar
Type: NoteInstitute: PSG College of Arts and Science Course: B.Tech Specialization: Mechanical EngineeringOffline Downloads: 32Views: 945Uploaded: 4 months agoAdd to Favourite

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Karthic Easwar
Karthic Easwar
2.001 - MECHANICS AND MATERIALS I Lecture #24 12/4/2006 Prof. Carol Livermore Torsion (Twisting) Mxx = Mt Examples where torsion is important: - Screwdriver - Drills - Propellers Use superposition Example: Unifrom along length, has circular symmetry 1
ϕ(x) = Angle of Twist (Total) at Point x (Built Up) Compatibility: d = ϕ(x + dx)R − ϕ(x)R d = γdx So: γdx = ϕ(x + dx)R − ϕ(x)R γdx = dϕR dγ γ=R dx What is γ? So: γ = γrθ Shear Strain Thus: γrθ = R 2 dϕ dx
And: γ R dϕ = 2 2 dx θx = rr = θθ = xx = rθ = rx = 0 Constitutive Relations: σθx = 2Gθx = Gγxθ = Gr dϕ , where G is the shear modulus. dx E 2(1 + ν) G= σxx = σθθ = σrr = σrθ = σrx = 0 σθx = Gr Recall beam bending σxx = dϕ dx −Ey ρ . Equilibrium:  Mx = 0  rdF = 0 −Mt + A dF = σθx dA  rσθx dA = 0 −Mt +  A Mt = rGr A 3 dϕ dA dx
Mt = dϕ dx  Gr2 dA A Recall beam bending: M= 1 ρ  Ey 2 dA A Note: dϕ happens dx : what 2 Gr dA: effective stiffness A Mt : what we apply If G is constant ”Special Case” Mt =  J≡ dϕ G dx  r2 dA A r2 dA Polar Moment of Inertia A So for ”special case” G is constant. Mt = GJ dϕ dx Recall beam bending: M= EI ρ If G is not constant:  (GJ)ef f = Gr2 dA A When G is constant: Mt = GJ dϕ dϕ and generally σθx = Gr dx dx Mt σθx = GJ Gr So: σθx = 4 Mt r J

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