68 LAMINATED COMPOSITES 个2 个2 Deformed Reference Plane 1 R = R,= 1 Undeformed Reference Plane w=五+h-人-上 L Ly K=Kv0 Figure 3.10:The curvatures Kx.Ky,and Ky of the reference plane. where N and M are the in-plane forces and moments(per unit length),and ht and hb are the distances from the reference plane to the plate's surfaces (Fig.3.12). The transverse shear forces(per unit length)are(Fig.3.11,right) (3.10) We now recall that for plane-stress condition the stress-strain relationships for each ply are (Eq.2.126) 11 012 016 (3.11) N四 E M yM王 N Figure 3.11:The in-plane forces acting at the reference plane (left)and the moments and the transverse shear forces(right)
68 LAMINATED COMPOSITES x y x y Undeformed Reference Plane c b a d c d a b Ly Lx z z Deformed Reference Plane fd fc fb fa yx cadb xy LL ffff κ + −− = κκ yx 0== b ′ b ′ a′ a′ c ′ c ′ d ′ d ′ x Rx κ 1 = y Ry κ 1 = Figure 3.10: The curvatures κx, κy, and κxy of the reference plane. where N and M are the in-plane forces and moments (per unit length), and ht and hb are the distances from the reference plane to the plate’s surfaces (Fig. 3.12). The transverse shear forces (per unit length) are (Fig. 3.11, right) Vx = ) ht −hb τxzdz Vy = ) ht −hb τyzdz. (3.10) We now recall that for plane-stress condition the stress–strain relationshipsfor each ply are (Eq. 2.126) σx σy τxy = Q11 Q12 Q16 Q12 Q22 Q26 Q16 Q26 Q66 x y γxy . (3.11) x y x y x y Nx Nxy Nyx Ny Mxy Mx My Myx Vy Vx Figure 3.11: The in-plane forces acting at the reference plane (left) and the moments and the transverse shear forces (right)
3.2 STIFFNESS MATRICES OF THIN LAMINATES 69 Reference Plane Figure 3.12:Distances from the reference plane. By introducing the notation Cu 012 216 [Q]= Q12 022 026 (3.12) ②16 026 266 we write the stress-strain relationships for a ply as (3.13) where [O is the stiffness matrix of the ply in the x-y coordinate system.The ele- ments of this stiffness matrix are obtained from the elements of the stiffness matrix [O]in the xi-x2 coordinate system by the transformation given by Eq.(2.195) By replacing [O]and [O]by [O]and [O],respectively,Eq.(2.195)yields 11 On O Q11 Q12 Q16 12 6 =[T] Q12 Q22 Q26 (3.14) 亘26 O66 Q16 Q26 Q66 where [T and [T]are given by Eq.(2.196)and are reiterated below, c2 2es c2 s2 cs [T]= -2cs [T]= s2 c2 -CS (3.15) 一CS cs c2-s2 -2cs 2cs c2-s2 and c cos s sin with defined in Figure 3.2.For an orthotropic ply the local coordinates x1,x2 are in the orthotropy directions.For transversely isotropic plies these local coordinates are parallel and perpendicular to the fibers(Fig.2.15) For orthotropic and transversely isotropic materials the elements of the stiffness matrix in the global coordinate system are given in Table 3.1 in terms of the elements of the stiffness matrix in the local coordinate system
3.2 STIFFNESS MATRICES OF THIN LAMINATES 69 Reference Plane ht hb K … 1.. zk –1 zk zK–1 z1 zK z0 k Figure 3.12: Distances from the reference plane. By introducing the notation [Q] = Q11 Q12 Q16 Q12 Q22 Q26 Q16 Q26 Q66 , (3.12) we write the stress–strain relationships for a ply as σx σy τxy = [Q] x y γxy , (3.13) where [Q] is the stiffness matrix of the ply in the x–y coordinate system. The elements of this stiffness matrix are obtained from the elements of the stiffness matrix [Q] in the x1 − x2 coordinate system by the transformation given by Eq. (2.195). By replacing [Q] and [Q ] by [Q] and [Q], respectively, Eq. (2.195) yields Q11 Q12 Q16 Q12 Q22 Q26 Q16 Q26 Q66 = [Tσ ] −1 Q11 Q12 Q16 Q12 Q22 Q26 Q16 Q26 Q66 [T ] , (3.14) where [Tσ ] and [T ] are given by Eq. (2.196) and are reiterated below, [Tσ ] = c2 s2 2cs s2 c2 −2cs −cs cs c2 − s2 [T ] = c2 s2 cs s2 c2 −cs −2cs 2cs c2 − s2 , (3.15) and c = cos , s = sin with defined in Figure 3.2. For an orthotropic ply the local coordinates x1, x2 are in the orthotropy directions. For transversely isotropic plies these local coordinates are parallel and perpendicular to the fibers (Fig. 2.15). For orthotropic and transversely isotropic materials the elements of the stiffness matrix in the global coordinate system are given in Table 3.1 in terms of the elements of the stiffness matrix in the local coordinate system.
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
3.2 STIFFNESS MATRICES OF THIN LAMINATES 71 The elements of these matrices are (i,j=1,2,6) (3.19) The [A],[B],and [D]matrices are the stiffness matrices of the laminate,and[] is the stiffness matrix of the ply.Since [O]is constant across each ply,the integrals in the equations above (Eq.3.19)may be replaced by summations(Fig.3.12)as follows(i,j=1,2,6)月 g=20(4-4 B=20- (3.20) 1 K D=2@,k-小 where K is the total number of plies(or ply groups)in the laminate;zk,Zk-1 are the distances from the reference plane to the two surfaces of the kth ply;and are the elements of the stiffness matrix of the kth ply. With the preceding definitions of the stiffness matrices,the expressions for the in-plane forces and moments(Egs.3.16 and 3.17)become N Au 42A16 B11 B12 B16 42 b226 B12 B22 B26 N A6 426 A66 B16 B26 Boo Y8 (3.21) M B11 B12B16 Di1 D12 D16 Kx M B12 B2 B26 D12 D22 D26 Ky Mxy B16 B26 B66 D16 D26 D66」 Kxy The vectors on the left and right hand side represent generalized forces and strains.Hereafter,we simply refer to these as forces and strains. By inverting Eqs.(3.21),we obtain the strains and curvatures in terms of the in-plane forces and moments: C11 012 C16 P11 B12 B16 N c12 c22 026 P21 B22 P26 N c16 a26 066 P61 F62 B66 B B21 B61 S d12 d16 (3.22) 12 B22 Be2 612 i26 M Kxy LB1 p26 B66 816 d26 866」
3.2 STIFFNESS MATRICES OF THIN LAMINATES 71 The elements of these matrices are (i, j = 1, 2, 6) Ai j = ) ht −hb Qi jdz Bi j = ) ht −hb zQi jdz Di j = ) ht −hb z2Qi jdz. (3.19) The [A], [B], and [D] matrices are the stiffness matrices of the laminate, and [Q] is the stiffness matrix of the ply. Since [Q] is constant across each ply, the integrals in the equations above (Eq. 3.19) may be replaced by summations (Fig. 3.12) as follows (i, j = 1, 2, 6): Ai j = * K k=1 (Qi j)k(zk − zk−1) Bi j = 1 2 * K k=1 (Qi j)k z2 k − z2 k−1 (3.20) Di j = 1 3 * K k=1 (Qi j)k z3 k − z3 k−1 , where K is the total number of plies (or ply groups) in the laminate; zk, zk−1 are the distances from the reference plane to the two surfaces of the kth ply; and (Qi j)k are the elements of the stiffness matrix of the kth ply. With the preceding definitions of the stiffness matrices, the expressions for the in-plane forces and moments (Eqs. 3.16 and 3.17) become Nx Ny Nxy Mx My Mxy = A11 A12 A16 B11 B12 B16 A12 A22 A26 B12 B22 B26 A16 A26 A66 B16 B26 B66 B11 B12 B16 D11 D12 D16 B12 B22 B26 D12 D22 D26 B16 B26 B66 D16 D26 D66 o x o y γ o xy κx κy κxy . (3.21) The vectors on the left and right hand side represent generalized forces and strains. Hereafter, we simply refer to these as forces and strains. By inverting Eqs. (3.21), we obtain the strains and curvatures in terms of the in-plane forces and moments: o x o y γ o xy κx κy κxy = α11 α12 α16 β11 β12 β16 α12 α22 α26 β21 β22 β26 α16 α26 α66 β61 β62 β66 β11 β21 β61 δ11 δ12 δ16 β12 β22 β62 δ12 δ22 δ26 β16 β26 β66 δ16 δ26 δ66 Nx Ny Nxy Mx My Mxy . (3.22)