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11.2 STIFFNESS COEFFICIENTS When one inverts Equation 11.1 written in the coordinate axesl,t of a ply,one obtains VuEt (1-vevu) (1-vuvn) 0 vuE E o (1-vuvie) (1-vuvn) 0 0 0 GelYu where the "stiffness"coefficients appear,as opposed to the Equation 11.1 where the "compliance"coefficients appear.To simplify the writing,one can denote Ee VuEe 0 eEr E 0 (11.6) 0 0 An identical procedure can be followed to arrive at the stress-strain relation: Ox 2cs Oy s -2cs -CS (c2-s) [T] (11.7) -CS 2 Yu 2cs -2cs (c2-s) Yxy [T'] Recall that axes x,y are derived from the axes t,t by a rotation 0 about the third axis z.Substituting Equations 11.7 into 11.6,one obtains ViEe 0 = [T] E 0 Ey Tx 0 0 G Yxy which can be rewritten as: 6￥ E E12 E 6 En to) E 制 Yxy 2003 by CRC Press LLC11.2 STIFFNESS COEFFICIENTS When one inverts Equation 11.1 written in the coordinate axes l,t of a ply, one obtains where the “stiffness” coefficients appear, as opposed to the Equation 11.1 where the “compliance” coefficients appear. To simplify the writing, one can denote (11.6) An identical procedure can be followed to arrive at the stress–strain relation: (11.7) Recall that axes x,y are derived from the axes ,t by a rotation q about the third axis z. Substituting Equations 11.7 into 11.6, one obtains which can be rewritten as: s st Ót t ˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ E 1 ntntl ( ) – -------------------- ntE 1 ntnt ( ) – --------------------- 0 ntEt 1 ntnt ( ) – --------------------- Et 1 ntnt ( ) – --------------------- 0 0 0 Gt e et Óg t˛ Ô Ô Ô Ô Ì ˝ Ô Ô Ô Ô Ï ¸ = sl st Ót t˛ Ô Ô Ì ˝ Ô Ô Ï ¸ E ntE 0 ntEt Et 0 0 0 Gt e et Óg t˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = sx sy Ótxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ c 2 s 2 2cs s 2 c 2 –2cs –cs cs c2 s 2 ( ) – s st Ót t˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = : T1 [ ] Ï Ô Ô Ô Ì Ô Ô Ô Ó e et Óg t˛ Ô Ô Ì ˝ Ô Ô Ï ¸ c 2 s 2 –cs s 2 c 2 cs 2cs –2cs c2 s 2 ( ) – ex e y Óg xy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = T1 [ ]¢ Ï Ô Ô Ô Ô Ì Ô Ô Ô Ô Ó sx sy Ótxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ T1 [ ] E ntE 0 ntEt Et 0 0 0 Gt T1 [ ]¢ ex e y Óg xy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = sx sy Ótxy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ E11 E12 E13 E21 E22 E23 E31 E32 E33 ex e y Óg xy ˛ Ô Ô Ì ˝ Ô Ô Ï ¸ = TX846_Frame_C11 Page 228 Monday, November 18, 2002 12:26 PM © 2003 by CRC Press LLC