MEM427_HW_5_Solution

# MEM427_HW_5_Solution - MEM427 HW Assignment#5 Solutions...

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Unformatted text preview: MEM427 HW Assignment #5 Solutions Fall 2010 1 / 6 Homework Assignment #5(a) Determine the Jacobian matrix and its determinant for each of the following 4 ‐ node quadrilateral elements. (1,1) (5,1) (5,5) (1,5) (1,1) (5,1) (7,5) (3,5) (1,1) (5,1) (4,5) (2,5) (1,1) (5,1) (5,5) (1,5) (1,1) (5,1) (7,5) (3,5) (1,1) (5,1) (4,5) (2,5) Solution: From lecture notes, we have, for a four ‐ node quadrilateral element 4 4 3 3 2 2 1 1 4 3 2 1 4 3 2 1 Y X Y X Y X Y X S S S S S S S S Y X Y X J in which the shape functions and their derivatives are 1 1 4 1 1 S , 1 1 4 1 2 S , 1 1 4 1 3 S , 1 1 4 1 4 S 1 4 1 1 S , 1 4 1 2 S , 1 4 1 3 S , 1 4 1 4 S 1 4 1 1 S , 1 4 1 2 S , 1 4 1 3 S , 1 4 1 4 S 5 1 5 5 1 5 1 1 4 4 3 3 2 2 1 1 Y X Y X Y X Y X , 2 2 5 1 5 5 1 5 1 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 J 4 2 2 J 5 3 5 7 1 5 1 1 4 4 3 3 2 2 1 1 Y X Y X Y X Y X , 2 1 2 5 3 5 7 1 5 1 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 J 4 2 1 2 J 5 2 5 4 1 5 1 1 4 4 3 3 2 2 1 1 Y X Y X Y X Y X , 2 4 2 4 2 6 5 2 5 4 1 5 1 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 J 3 2 4 2 4 2 6 J (1,1) (5,1) (5,5) (1,5) (1,1) (5,1) (5,5) (1,5) (1,1) (5,1) (7,5) (3,5) (1,1) (5,1) (7,5) (3,5) (1,1) (5,1) (4,5) (2,5) (1,1) (5,1) (4,5) (2,5) MEM427 HW Assignment #5 Solutions Fall 2010 2 / 6 Homework Assignment #5(b) Evaluate the given integral analytically and the using Gauss...
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MEM427_HW_5_Solution - MEM427 HW Assignment#5 Solutions...

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