# Haghighi using matrix notation ux y ni 0 n j 0 nk

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Unformatted text preview: . Haghighi using a linear variation for each of u & v w/in the elem., we may write u(x, y ) = NiU2i−1 + N jU2 j−1 + NkU2k −1 v(x, y ) = NiU2i + N jU2 j + NkU2k or in general u = NiU2i−1 + 0 U2i + N jU2 j−1 + 0 U2 j + NkU2k −1 + 0 U2k v = 0 U2i−1 + NiU2i + 0 U2 j−1 + N jU2 j + 0 U2k −1 + NkU2k Chapter 23 Page 11 K. Haghighi using matrix notation: ⎧u(x, y )⎫ ⎡Ni 0 N j 0 Nk ⎨ ⎬= ⎢0 N 0 N i j0 ⎩ v(x, y )⎭ ⎣ or in a compact form {} ⎧ U2i−1 ⎫ ⎪U ⎪ ⎪ 2i ⎪ 0 ⎤ ⎪ U2 j−1 ⎪ Nk ⎥ ⎨ U2 j ⎬ ⎦⎪ ⎪ ⎪ U2k −1 ⎪ ⎪ ⎪ ⎩U2k ⎭ ⎧u(x, y )⎫ (e ) ⎨ ⎬ = [N]2×6 U 6×1 ⎩ v(x, y )⎭ elem. nodal disp’s. Chapter 23 Page 12 K. Haghighi Strain-disp. relations now reduce to ∂u ∂v ∂ u ∂v & e xy = e xx = e yy = + ∂x ∂y ∂y ∂x or (w = 0 , ( ( ( u, v ≠ f (z )) ) 1 e xx = biU2i−1 + b jU2 j−1 + bkU2k −1 2A 1 e yy = ciU2i + c jU2 j + ckU2k 2A 1 e xy = ciU2i−1 + biU2i + c jU2 j−1 + b jU2 j 2A + ckU2k −1 + bkU2k ) ) Chapter 23 Page 13 K. Haghighi in the matrix form ⎡bi 0 b j 0 bk ⎧e xx ⎫ ⎪ ⎪ 1⎢ ⎨e yy ⎬ = ⎢ 0 ci 0 c j 0 ⎪e ⎪ 2 A ⎢ c b c b c j j k ⎣i i ⎩ xy ⎭ or {e}3×1 = [B]3×6 3 unknown strain comp. (2D prob.) {U(e)}6×1 ⎧ U2i−1 ⎫ ⎪ ⎪U ⎪ 2i ⎪ 0⎤ ⎥ ⎪ U2 j−1 ⎪ ⎪ ⎪ ck ⎥ ⎨ ⎬ ⎪U2 j ⎪ bk ⎥ ⎦ ⎪ U2k −1 ⎪ ⎪ ⎪ ⎪U2k ⎪ ⎭ ⎩ only for ∆ elem. Chapter 23 Page 14 K. Haghighi The Element Matrices: Recall that [k(e)]= ∫ [B]T [D][B]dV V from {e} = [B]{U} but [B] & [D] consist of all constant terms So, (e ) = [B]T [D] [B] tA k 6×3 3× 3 3×6 elem. area elem. thickness Chapter 23 Page 15 K. Haghighi Note: t actual t of body for plane 1 for plane ε Also, the elem. force vector a {} σ c b (e ) = [B]T [D]{ε }dV + [N]T ⎧X⎫dV + [N]T ⎧p x ⎫dΓ f T ∫ ∫ ⎨Y⎬ ∫ ⎨p ⎬ V V ⎩⎭ V ⎩ y⎭ {} ⎧u( x, y) ⎫ defined by⎨ = [N] 2×6 U(e ) 6×1 ⎬ ⎩v( x, y)⎭2×1 Chapter 23 Page 16 K. Haghighi but a = ∫ [B]T [D]{εT} = [B]T [D]{εT}tA V ⌠ ⎡ NiX ⎤ ⌠ ⎡ Ni 0 ⎤ ⎮ ⎢N Y ⎥ ⎮⎢ 0 N ⎥ i⎥ ⎮⎢ i ⎥ ⎮⎢ ⎮ ⎢ Nj X ⎥ ⎮ ⎢ Nj 0 ⎥ ⎧X ⎫ T ⎧X...
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## This document was uploaded on 01/29/2014.

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