prob_06_04_04 - Problem 6.4-4 Sketch an eight-node plane...

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Problem 6.4-4 Sketch an eight-node plane element for which J is a function of η but not ξ , if (a) the element is rectangular, and (b) the element has two curved sides N 1 1 4 1 ξ () 1 η 1 4 1 ξ 2 1 η 1 4 1 ξ 1 η 2 = N 1 ξ 1 4 1 η ξ 2 1 η + 1 4 1 η 2 + = N 1 η 1 4 1 ξ 1 4 1 ξ 2 + η 2 1 ξ + = N 2 1 4 1 ξ + 1 η 1 4 1 ξ 2 1 η 1 4 1 ξ + 1 η 2 = N 2 ξ 1 4 1 η ξ 2 1 η + 1 4 1 η 2 = N 2 η 1 4 1 ξ + 1 4 1 ξ 2 + η 2 1 ξ + + = N 3 1 4 1 ξ + 1 η + 1 4 1 ξ + 1 η 2 1 4 1 ξ 2 1 η + = N 3 ξ 1 4 1 η + 1 4 1 η 2 ξ 2 1 η + + = N 3 η 1 4 1 ξ + η 2 1 ξ + + 1 4 1 ξ 2 = N 4 1 4 1 ξ 1 η + 1 4 1 ξ 2 1 η + 1 4 1 ξ 1 η 2 = N 4 ξ 1 4 1 η + ξ 2 1 η + + 1 4 1 η 2 + = N 4 η 1 4 1 ξ 1 4 1 ξ 2 η 2 1 ξ + = N 5 1 2 1 ξ 2 1 η = N 5 ξ ξ 1 η = N 5 η 1 2 1 ξ 2 = N 6 1 2 1 ξ + 1 η 2 = N 6 ξ 1 2 1 η 2 = N 6 η η 1 ξ + = N 7 1 2 1 ξ 2 1 η + = N 7 ξ ξ 1 η + = N 7 η 1 2 1 ξ 2 = N 8 1 2 1 ξ 1 η 2 = N 8 ξ 1 2 1 η 2 = N 8 η η 1 ξ =
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Consider a square element, side nodes at the middle x 1 2 := x 2 4 := x 3 4 := x 4 2 := x 5 3 := x 6 4 := x 7 3 := x 8 2 := y 1 2 := y 2 2 := y 3 4 := y 4 4 := y 5 2 := y 6 3 := y 7 4 := y 8 3 := J11 i N i ξ x i () = J12 i N i ξ y i = J21 i N i η x i = J22 i N i η y i = J11 1 4 1 η ξ 2 1 η + 1 4 1 η 2 + 2 1 4 1 η ξ 2 1 η + 1 4 1 η 2 4 + 1 4 1 η + 1 4 1 η 2 ξ 2 1 η + + 4
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This note was uploaded on 01/11/2011 for the course MAE 5020 taught by Professor Folkman during the Fall '10 term at Utah State University.

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prob_06_04_04 - Problem 6.4-4 Sketch an eight-node plane...

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