Torsion Part 1 J TL GJ. Machines that rely on torsion to function. Wind Power. Shear Stress, Angle of Twist,
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1 Machines that rely on torsion to function Torque Wrench Motor Smart Car Maserati MC12 Steam Turbine Wind Power Gearbox Courtesy of Shawn Sheng NRE GRC Shear Stress, ngle of Twist, Tρ τ = J T = OBJECTIVES: You will be able to effectively apply the torsion formulas to calculate stresses and deformation. 1. You will know what each variable represents 2. You will understand and apply the sign convention for torque, shear stress, and shear strain 3. You will know the limitations of the formulas so that you can properly analyze shafts subjected to torque 1
2 EQUTION DEVEOPMENT PROCEDURE: 2. Internal oads 3. Material Relations circular shaft rectangular shaft rigid disks For a circular shaft, plane sections remain plane For a rectangular shaft, these cross sectional planes warp IMITTION Imagine a shaft composed of a series of rigid disks, each disk slips relative to its neighbor to produce deformation rc length, BB = ρ B ρ B 2
3 rc length, BB = ρ Chord length, BB = sin(γ) B B γ rc length, BB = ρ Chord length, BB = sin(γ) IMITTION If γis small, then sin(γ) γ If γis small, then arc length chord length Equating BB ρ= γ ρ γ = K EQ _1 ρ γ = K EQ _1 Conclusion: shear strain varies linearly with the radial position ρ Q: How does angle of twist change with radial position ρ? 3
4 Even though the shear strain is a function of radial position, the angle of twist is not. Sign Convention Positive internal torque C C Shear stresses at points, B, & C for positive internal torque Shear strain for positive internal torque Q: Draw an FBD with a negative internal torque. 2. Internal oads (pplied) ΣM = 0; T o T = 0 (Internal) 4
5 2. Internal oads (pplied) 2. Internal oads Internal torque must be equivalent to the resultant moment from all internal shear forces. T = ρdf 2. Internal oads Internal torque must be equivalent to the resultant moment from all internal shear stresses. T = ρdf df = τ d T = ρτ dk ( EQ _ 2) 5
6 3. Material Relations τ G γ Hooke s aw τ = GγK EQ _ 3 inear elastic material behavior 3. Material Relations τ τ Y G γ Hooke s aw IMITTION Does not apply beyond the yield strength τ Y T = ρτ dk Combine EQ 1-3 ( EQ _ 2) T = ρgγ dkusing EQ _ 3 T = ρgρ dkusing EQ _1 2 G T = G ρ d = J T = Polar moment of inertia, J ngle of twist in radians ( x) dx ( x) J = T Generalize to 0 G x 6
7 Combine EQ 1-3 T = ρτ dk ( EQ _ 2) T = ρgγ dkusing EQ _ 3 T = ρgρ dkusing EQ _1 2 G T = G ρ d = J τ = GγK EQ _ 3 τ = Gρ T τ = Gρ Tρ τ = J Elastic torsion formula T = ngle of twist in radians r i Polar Moment of Inertia for Circular Sections r o J = ρ d 2 d = ρdρdθ = 2π ρdρ J = 2π J = ro ri 3 ρ dρ 4 ro 2 ρ π π = o i 4 OR, in terms of diameters Q: What is Jfor a solid circular section? ri ( r r ) 4 4 ( ) J = π D o D i 32 Q: Would it be valuable in this analysis to determine Jfor noncircular sections? Structures must resist torsion to carry loads Spandrel beam Spandrel beam cross-section Since the floor beam reaction does not occur at the shear center of the spandrel beam there is torsion 7
8 In summary: Internal torque Shear Stress, ngle of Twist, Shear modulus Tρ τ = J T = Radial distance Polar moment of interia Shaft length imitations: circular sections, elastic behavior, small shear strain, angle of twist formula only valid if T, G, and J are constant over Shaft with a slit animation from E MCH 213D T = ρτ dk Combine EQ 1-3 ( EQ _ 2) T = ρgγ dkusing EQ _ 3 T = ρgρ dkusing EQ _1 2 G T = G ρ d = J T = Polar moment of inertia, J ngle of twist in radians τ = GγK EQ _ 3 τ = Gρ T τ = Gρ Tρ τ = J Elastic torsion formula Generalize to = T x dx 0 G x J x 8
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