For 1 d elements continuity is assured since we have

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Unformatted text preview: each defined over a single element. The need to integrate this piecewise smooth function places a requirement on the order of continuity between elements. If the integral contains 1st derivative terms, φ must be continuous between elements but ∂φ / ∂x or ∂ 2φ / ∂x 2 need not to be. For 1-D elements continuity is assured since we have a common node. Chapter 6 Page 15 K. Haghighi Consider 2 adjacent elements: ( φ (1) = N11)φ1 + N(1)φ 3 + N(1)φ 4 3 4 ( φ (2 ) = N12 )φ1 + N(2 )φ 2 + N(2 )φ 3 2 3 Along 1 → 3 : N(2 ) = N(1) = 0 2 4 1 N(1) = L(2 ) , 3 (2 ) = L(2 ) , N 1 1 ( 1 N11) = L(1 ) (2 ) = L(2 ) N 3 3 1 φ (1) = L(1)φ 1 + L(2 )φ 3 1 2 2 φ (2 ) = L(1 )φ 1 + L(3 )φ 3 Chapter 6 Page 16 K. Haghighi L1 + L 2 + L 3 = 1 1 1 1 L(3 ) = 0 => L(2 ) = 1 − L(1 ) (2 ) = 0 => L(2 ) = 1 − L(2 ) L2 3 1 Substituting () 2 2 φ (2 ) = L(1 )φ1 + L(3 )φ 3 = L(2 )φ1 + (1 − L(2 ) ) φ 3 1 1 1 1 1 φ (1) = L(1 )φ1 + L(2 )φ 3 = L(1)φ1 + 1 − L(1 ) φ 3 1 Chapter 6 Page 17 K. Haghighi 2ch(1) ( 2 A11) (1) = 2 =c L1 (1) = 2bh(1) b 2A 2 2ch(2 ) ( 2A12 ) (2 ) = 2 =c L1 (2 ) = 2bh(2 ) b 2A 2 ( ) ( ) 1 1 2 L(1 )φ1 + 1 − L(1 ) φ 3 = L(1 ) φ1 + 1 − L(2 ) φ 3 1 φ (1) = φ (2 ) Chapter 6 Page 18 K. Haghighi...
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This document was uploaded on 01/29/2014.

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