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MIT2_094S11_hw_4_sol

# MIT2_094S11_hw_4_sol - 2.094 FINITE ELEMENT ANALYSIS OF...

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2.094 F INITE E LEMENT A NALYSIS OF S OLIDS AND F LUIDS S PRING 2008 Homework 4 - Solution Instructor: Prof. K. J. Bathe Assigned: 02/28/2008 Due: 03/06/2008 Problem 1 (20 points): (a) 2 5 1 (, ) ( 3 ) (4 ) 24 hxy x y =− ; 2 6 1 ( 3 ) ) 24 x y =+ 6 1 11 ( 3 ) (2 ) 24 2 x y + ; 5 2 ( 3 ) ) 24 2 x y + 5 3 ( 3 ) ) 24 2 x y ; 6 4 ( 3 ) ) 24 2 x y (b) Note that above interpolation functions satisfy 6 1 (, ) 1 i i = = * Rigid body translation in x-direction 12 6 2.0 uu u ==== " Æ 66 (,) 2 . 0 (,) 2 . 0 i i i ii uxy h xyu h xy == ∑∑ = Æ 0 xx xy T yy uvuv xyyx εε ε γ ⎡⎤ ∂∂∂∂ = ⎢⎥ ⎣⎦ = = * Rigid body translation in y-direction 6 2.0 vv v " Æ 2 . 0 . 0 i i i vxy h xyv Æ 0 xx xy T yy = = Page 1 of 4

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* Rigid body rotation by 60 ° Here we use: ii uy θ =− ⋅ and vx = ( 60 = ° ) Hence {} 6 1 (, ) i uxy h y y xy = =− = , 6 1 i x vxy x hxy = == which corresponds to the rigid body rotation. Also 0 xx xy T yy uvuv xyyx εε ε γ ⎡⎤ ∂∂∂∂ + ⎢⎥ ⎣⎦ = Note: The strains used in the class up to now correspond to infinitesimally small displacements and strains.
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MIT2_094S11_hw_4_sol - 2.094 FINITE ELEMENT ANALYSIS OF...

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