6-6 - CHAP 6 Finite Elements for Plane Solids 311 6....

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CHAP 6 Finite Elements for Plane Solids 311 6. Calculate the shape function matrix [ N ] and strain-displacement matrix [ B ] of the triangular element shown in the figure Solution: Since the element has three nodes, start with 1 2 3 ( , ) u x y x y . By substituting nodal values, we have 1 1 1 1 2 1 2 2 2 1 3 1 3 3 3 1 uu u u u u u u       Thus, the approximate solution becomes 1 2 1 3 1 1 2 3 ( , ) ( ) ( ) (1 ) u x y u u u x u u y x y u xu yu From the above approximation scheme, we can obtain three shape functions, as 1 2 3 ( , ) 1 ( , ) ( , ) N x y x y N x y x N x y y Then, the matrix of shape functions can be written as 1 0 0 0 [] 0 1 0 0 x y x y x y x y     N The strain displacement matrix can be obtained from the definition of strain as
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