L13_ Thin-wall shafts-fall 10

L13_ Thin-wall shafts-fall 10 - EAS 209-Fall 2010...

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Unformatted text preview: EAS 209-Fall 2010 Instructors: Christine Human 9/17/2010 1 Lecture 13- Thin-walled Shafts Lecture 13 Thin-walled Hollow Shafts In previous lectures we have derived equations for the shear stress τ and angle of twist φ for both hollow and solid circular shafts. Today’s Objective: Develop equations for shear stress τ and angle of twist φ in thin-walled shafts. We will start with circular shafts and then consider non-circular thin-walled shafts. Today’s Homework: EAS 209-Fall 2010 Instructors: Christine Human 9/17/2010 2 Lecture 13- Thin-walled Shafts Circular thin-walled shafts Consider a thin-walled circular hollow shaft with inner radius r 1 , outer radius r 2 , thickness t=r 2-r 1 , and median radius r m =½( r 1 +r 2 ). The strain distribution for a circular shaft increases linearly as we move from the center axis to the surface of the member. τ ρ τ max γ ρ t r m L ρφ γ = In the linear elastic range, τ =G γ, hence the stress is distributed in the same fashion. The maximum and minimum stresses are J Tr 1 min = τ , J Tr 2 max = τ where ) ( 2 4 1 4 2 r r J − = π EAS 209-Fall 2010 Instructors: Christine Human 9/17/2010 3 Lecture 13- Thin-walled Shafts We can write the inner and outer radius in terms of the median radius 2 / 2 / 1 2 t r r t r r m m − = + = Hence the polar moment of inertia becomes t r r t r t r t r J m m m m m 3 2 2 2 2 ) 4 ( 2 ) 4 ( 2 π π π = ≈ + = (we neglect the t 2 term because it is very small compared to r m 2 for a thin-walled shaft)....
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L13_ Thin-wall shafts-fall 10 - EAS 209-Fall 2010...

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