Ch_23_annotated

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 01/29/2014.

Ask a homework question - tutors are online