70 LAMINATED COMPOSITES Table 3.1.The elements of the [Q]matrix for an othotropic or transversely isotropic ply oriented in the direction(Fig 3.2) 01=cQ1+s4Q2+2c2s2(02+2Q6) 02=s01+cQ2+2c2s2(Q2+2Q6) 12=2s2(01+Q2-4Q6)+(c4+s4)02 66=c2s2(Q11+Q22-2Q12)+(c2-s2)2Q6 16=cs(c2Q1-s2Q2-(c2-s2)(Q12+2Q66) 26=cs(s2Q1-c2Q22+(c2-s2)(Q12+2Q66) c=cos s=sin By substituting Eqs.(3.7)and (3.13)into Eq.(3.9),we obtain Kx +@z dz Kxy Kx [dz Qldz Ky (3.16) -hp Kxy + d a (3.17) The stiffness matrices of the laminate are defined as [4 dz [B= z⑨dz (3.18) h [D]70 LAMINATED COMPOSITES Table 3.1. The elements of the [Q ] matrix for an othotropic or transversely isotropic ply oriented in the + direction (Fig 3.2) Q11 = c4Q11 + s4Q22 + 2c2s2 (Q12 + 2Q66) Q22 = s4Q11 + c4Q22 + 2c2s2 (Q12 + 2Q66) Q12 = c2s2 (Q11 + Q22 − 4Q66) + (c4 + s4)Q12 Q66 = c2s2 (Q11 + Q22 − 2Q12) + (c2 − s2)2Q66 Q16 = cs(c2Q11 − s2Q22 − (c2 − s2)(Q12 + 2Q66)) Q26 = cs(s2Q11 − c2Q22 + (c2 − s2)(Q12 + 2Q66)) c = cos  s = sin  By substituting Eqs. (3.7) and (3.13) into Eq. (3.9), we obtain    Nx Ny Nxy    = ) ht −hb    [Q]    o x o y γ o xy    + [Q]z    κx κy κxy       dz = ) ht −hb [Q]dz    o x o y γ o xy    + ) ht −hb z[Q]dz    κx κy κxy    (3.16)    Mx My Mxy    = ) ht −hb z    [Q]    o x o y γ o xy    + [Q]z    κx κy κxy       dz = ) ht −hb z[Q]dz    o x o y γ o xy    + ) ht −hb z2 [Q]dz    κx κy κxy    . (3.17) The stiffness matrices of the laminate are defined as [A] = ) ht −hb [Q]dz [B] = ) ht −hb z[Q]dz (3.18) [D] = ) ht −hb z2 [Q]dz