EAS209 L14_ Thin-wall shafts-spring 11

EAS209 L14_ Thin-wall shafts-spring 11 - EAS 209-Spring...

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EAS 209-Spring 2011 Instructors: Christine Human 1/18/2011 1 Lecture 14- Thin-walled Shafts Lecture 14 Thin-walled Hollow Shafts In previous lectures we have derived equations for the shear stress τ and angle of twist 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-Spring 2011 Instructors: Christine Human 1/18/2011 2 Lecture 14- 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 stress distribution for a circular shaft increases linearly as we move from the center axis to the surface of the member. 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 J 2 ( r m t /2) 4 ( r m t /2) 4 J r m t 2 (4 r m 2 t 2 ) 2 r m 3 t τ ρ τ max J Tr 1 min , J Tr 2 max where ) ( 2 4 1 4 2 r r J
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EAS 209-Spring 2011 Instructors: Christine Human 1/18/2011 3 Lecture 14- Thin-walled Shafts (we neglect the t 2 term because it is very small compared to r m 2 for a thin-walled shaft). Note
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EAS209 L14_ Thin-wall shafts-spring 11 - EAS 209-Spring...

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