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MIT2_094S11_lec19

# MIT2_094S11_lec19 - 2.094 Finite Element Analysis of Solids...

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2.094 Finite Element Analysis of Solids and Fluids Fall ‘08 Lecture 19 - Slender structures Prof. K.J. Bathe MIT OpenCourseWare Beam analysis, t L 1 (e.g. t L = 1 1 100 , 1000 , · · · ) Reading: Sec. 5.4, 6.5 (plane stress) 2 L 0 (19.1) J = 0 1 t 2 (1 r ) (1 + s ) (19.2) h 2 = 4 1 (1 r ) (1 s ) (19.3) h 3 = 4 = v v 3 2 3 2 Beam theory assumptions (Timoshenko beam theory): u u (19.4) (19.5) (19.6) = v 2 t = u 2 + θ 2 2 t = u 2 2 θ 2 u v u v 3 3 2 2 1 2 L 0 1 4 (1 s ) 2 L 0 2 (1 ) r t (1 + s ) 4 0 etc (19.7) B = 1 4 2 (1 ) r t (1 + s ) 1 4 2 (1 ) r t 2 (1 s ) 0 1 2 L 1 4 2 (1 ) r t 81 1 4 4 L 1 4

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MIT 2.094 19. Slender structures u 2 v 2 θ 2 ∂u ∂x 1 t 0 s L 2 L B beam = · · · 0 (19.8) 0 0 0 ∂v ∂y 0 1 1 (1 r ) ∂y ∂x L 2 ∂u + ∂v 1 v ( r ) = 2 (1 r ) v 2 1 st u ( r ) = 2 (1 r ) u 2 4 (1 r ) θ 2 at r = 1, v ( 1) = v 2 st u ( 1) =
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