3.2 STIFFNESS MATRICES OF THIN LAMINATES 75 The relationships between the strains and curvatures and the forces and mo- ments(Eq.3.22)now simplify to 11 112 16 u12 a22 126 N (3.31) 416 126 a66 Nxy Kx d2 d16 d2 d2 d26 (3.32) Kxy d16 d26 d66 Mxy Balanced laminate.In a balanced laminate,for every unidirectional ply in the direction(measured counterclockwise from the x coordinate)there is an identical ply in the direction.The elements of the stiffness matrix [O]are given in Table 3.1(page 70).From this table we deduce that the elements of the stiffness matrices of plies in the and-directions are related as follows: D1(49)=D11(-9) Q22+9)=022(-6) 012(+9)=12(-0) 26(+9)=06(-e (3.33) 016+e)=-②16-e)O26+e)=-026-9) By substituting these elements into the expression of the stiffness matrix in Eq.(3.20),we find that A6=h6=0. (3.34) The structure of the stiffness matrix given in Table 3.4 shows that there is no extension-shear coupling in a balanced laminate (Table 3.5).(Note that A6 and A6 are zero only in the x-y coordinate system.) Elements A6 and A6 are zero for symmetrical and unsymmetrical laminates. Correspondingly,for symmetrical balanced laminates the a16 and a26 elements of the compliance matrix are zero(a16 0and a26 =0).However,for unsymmetrical balanced laminates none of the elements of the compliance matrix is zero (see Eq.3.22). Orthotropic laminate.In orthotropic laminates we are interested in two mu- tually perpendicular directions,called orthotropy directions,in the plane of the laminate.Normal forces and bending moments applied in these directions do not cause shear or twist of the laminate.Hence,there are no extension-shear, bending-twist,and extension-twist couplings. A laminate is orthotropic when every ply is orthotropic and the orthotropy di- rections coincide with the x and y directions.Fiber-reinforced plies are orthotropic under the following conditions: when the ply is made of unidirectional fibers and all the fibers are aligned with one of the laminate's orthotropy directions(Fig.3.13);3.2 STIFFNESS MATRICES OF THIN LAMINATES 75 The relationships between the strains and curvatures and the forces and mo￾ments (Eq. 3.22) now simplify to    o x o y γ o xy    =    a11 a12 a16 a12 a22 a26 a16 a26 a66       Nx Ny Nxy    (3.31)    κx κy κxy    =    d11 d12 d16 d12 d22 d26 d16 d26 d66       Mx My Mxy    . (3.32) Balanced laminate. In a balanced laminate, for every unidirectional ply in the + direction (measured counterclockwise from the x coordinate) there is an identical ply in the − direction. The elements of the stiffness matrix [Q] are given in Table 3.1 (page 70). From this table we deduce that the elements of the stiffness matrices of plies in the + and − directions are related as follows: Q11(+) = Q11(−) Q22(+) = Q22(−) Q12(+) = Q12(−) Q66(+) = Q66(−) (3.33) Q16(+) = −Q16(−) Q26(+) = −Q26(−). By substituting these elements into the expression of the stiffness matrix in Eq. (3.20), we find that A16 = A26 = 0. (3.34) The structure of the stiffness matrix given in Table 3.4 shows that there is no extension–shear coupling in a balanced laminate (Table 3.5). (Note that A16 and A26 are zero only in the x–y coordinate system.) Elements A16 and A26 are zero for symmetrical and unsymmetrical laminates. Correspondingly, for symmetrical balanced laminates the a16 and a26 elements of the compliance matrix are zero (a16 = 0 and a26 = 0). However, for unsymmetrical balanced laminates none of the elements of the compliance matrix is zero (see Eq. 3.22). Orthotropic laminate. In orthotropic laminates we are interested in two mu￾tually perpendicular directions, called orthotropy directions, in the plane of the laminate. Normal forces and bending moments applied in these directions do not cause shear or twist of the laminate. Hence, there are no extension–shear, bending–twist, and extension–twist couplings. A laminate is orthotropic when every ply is orthotropic and the orthotropy di￾rections coincide with the x and y directions. Fiber-reinforced plies are orthotropic under the following conditions:
when the ply is made of unidirectional fibers and all the fibers are aligned with one of the laminate’s orthotropy directions (Fig. 3.13);