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8b-Quad-shp-fns-2010s - MAE M168/CEE M135C Introduction to...

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© William Klug MAE M168/CEE M135C Introduction to Finite Element Methods Lecture 8b 1-D FEM: Quadratic Shape Functions

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© William Klug Summary & Review Local (linear) shape functions Global shape functions assembled from local ( ) 1 ( ) ; ( ) i j j e e x N x x x N x l l = = i j l e N i (x) N j (x) x N(x) 0 X i 1.0 l N i (X) N j (X) N(X) X j L X x i = 0 x j = l O ( ) ( ) ( ) i i j j u x N x u N x u = + Local coords Global coords N 3 ( X ) = N 3 (2) X ( ) = X X 2 ( ) l , N 3 (3) X ( ) = X 4 X ( ) l , 0, X inelement 2 X inelement3 otherwise 1 2 3 4 1 2 3 4 5 (2) 2 N (2) 3 N 1 2 3 4 1 2 3 4 5 (3) 3 N (3) 4 N + 1 2 3 4 1 2 3 4 5 (2) (3) 3 3 3 N N N = + = ( x 1 , D 1 ) ( x 2 , D 2 ) ( x 3 , D 3 ) ( x 4 , D 4 ) ( x 5 , D 5 ) x x 1 x 2 x 3 x 4 x 5 ( ) I u x
© William Klug Quadratic Interpolation ( x 1 , D 1 ) ( x 2 , D 2 ) ( x 3 , D 3 ) ( x 4 , D 4 ) ( x 5 , D 5 ) x x 1 x 2 x 3 x 4 x 5 ( ) I u x linear interpolation quadratic interpolation ( x 1 , D 1 ) ( x 2 , D 2 ) ( x 3 , D 3 ) ( x 4 , D 4 ) ( x 5 , D 5 ) x x 1 x 2 x 3 x 4 x 5 ( ) I u x piecewise quadratic segments ( x 1 , D 1 ) ( x 2 , D 2 ) ( x 3 , D 3 ) ( x 4 , D 4 ) ( x 5 , D 5 ) x x 1 x 2 x 3 x 4 x 5 u ( x ) ( ) 1,2, ..... i i D u x i = = Original curve

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© William Klug Quadratic Shape Functions Consider four-element model of the bar. Three data points define a unique quadratic polynomial. Elements are required to have three nodes i , j and k . (1) (2) (3) (4) L i k j l
© William Klug Quadratic Function Piecewise approximation of u in each element is to be a quadratic function. In any element u is a linear combination of three displacement modes ( ) 2 1 2 3 ( ) 0 u u x x x x l α α α = + + l û ( x ) û L X i j 0 k

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© William Klug Deriving Shape Functions
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8b-Quad-shp-fns-2010s - MAE M168/CEE M135C Introduction to...

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