Copyrighted Materials 0CrpUPress o CHAPTER THREE Laminated Composites Composites are frequently made of layers(plies)bonded together to form a lam- inate (Fig.3.1).A layer may consist of short fibers,unidirectional continuous fibers,or woven or braided fibers embedded in a matrix(Figs.1.1 and 1.2).A layer containing woven or braided fibers is referred to as fabric. Adjacent plies having the same material and the same orientation are referred to as a ply group.Since the properties and the orientations are the same across the ply group,a ply group may be treated as one layer. 3.1 Laminate Code An x,y,z orthogonal coordinate system is used in analyzing laminates with the z coordinate being perpendicular to the plane of the laminate(Fig.3.2). The orientations of continuous,unidirectional plies are specified by the angle (in degree)with respect to the x-axis (Fig.3.2).The angle is positive in the counterclockwise direction.The number of plies within a ply group is specified by a numerical subscript.For example,the laminate consisting of unidirectional plies and shown in Figure 3.3 is designated as [453/04/902/60] This laminate contains four ply groups,the first containing three plies in the 45-degree direction,the second containing four plies in the 0-degree direction,the third containing two plies in the 90-degree direction,the fourth containing one ply in the 60-degree direction. Symmetrical laminate.When the laminate is symmetrical with respect to the midplane it is referred to as a symmetrical laminate.Examples of symmetrical laminates are shown in Figure 3.4.The laminates represented in Figure 3.4 are 63
CHAPTER THREE Laminated Composites Composites are frequently made of layers (plies) bonded together to form a laminate (Fig. 3.1). A layer may consist of short fibers, unidirectional continuous fibers, or woven or braided fibers embedded in a matrix (Figs. 1.1 and 1.2). A layer containing woven or braided fibers is referred to as fabric. Adjacent plies having the same material and the same orientation are referred to as a ply group. Since the properties and the orientations are the same across the ply group, a ply group may be treated as one layer. 3.1 Laminate Code An x, y, z orthogonal coordinate system is used in analyzing laminates with the z coordinate being perpendicular to the plane of the laminate (Fig. 3.2). The orientations of continuous, unidirectional plies are specified by the angle (in degree) with respect to the x-axis (Fig. 3.2). The angle is positive in the counterclockwise direction. The number of plies within a ply group is specified by a numerical subscript. For example, the laminate consisting of unidirectional plies and shown in Figure 3.3 is designated as [453/04/902/60] . This laminate contains four ply groups, the first containing three plies in the 45-degree direction, the second containing four plies in the 0-degree direction, the third containing two plies in the 90-degree direction, the fourth containing one ply in the 60-degree direction. Symmetrical laminate. When the laminate is symmetrical with respect to the midplane it is referred to as a symmetrical laminate. Examples of symmetrical laminates are shown in Figure 3.4. The laminates represented in Figure 3.4 are 63
64 LAMINATED COMPOSITES Plygroup 2 Fabric 1 Layer 1 Layer 2 Plygroup 1 Figure 3.1:Laminated composite. 个3 S 工1V Figure 3.2:The x,y,z laminate coordinate system,the xi,x2,x3 ply coordinate system,and the ply angle. 60 60 90 90 0 0 0 0 45 45 459 45 Figure 3.3:Description of the layup in a laminate consisting of unidirectional plies [453/04/902/601. -45 45 -45 -45 0 -45 0 45 0 45 0 -45 -45 -45 -45 45 [-452z/02l [45/-45z/45] Figure 3.4:Examples of symmetrical laminates
64 LAMINATED COMPOSITES z y x Plygroup 2 Fabric 1 Layer 1 Layer 2 Plygroup 1 Figure 3.1: Laminated composite. x z y Θ x1 Θ x2 x3 x Figure 3.2: The x, y, z laminate coordinate system, the x1, x2, x3 ply coordinate system, and the ply angle. 60 0 0 0 90 90 0 x y 60o 45o 45 45 45 Figure 3.3: Description of the layup in a laminate consisting of unidirectional plies [453/04/902/60]. 0 0 0 0 45 45 45 45 [45/–452/45]s –45 –45 –45 –45 –45 –45 –45 –45 [–452/02]s Figure 3.4: Examples of symmetrical laminates
3.2 STIFFNESS MATRICES OF THIN LAMINATES 65 -45 45 -45 -45 -30 30 30 -30 90 -30 90 30 45 -45 45 45 [45,/90/301-30/-452] [45/-45/30/-301s Figure 3.5:Examples of balanced laminates. specified as [-452/04/-452]=[-452/02ls [45/-452/452/-452/45]=[45/-452/45]s The subscript s indicates symmetry about the midplane. Balanced laminate.In balanced laminates,for every ply in the+direction there is an identical ply in the-direction.Examples of balanced laminates are shown in Figure 3.5. Cross-ply laminates.In cross-ply laminates fibers are only in the 0-and 90-degree directions(Fig.3.6).Cross-ply laminates may be symmetrical or unsymmetrical. Since there is no distinction between the +0 and-0 and between the +90-and -90-degree directions,cross-ply laminates are balanced. Angle-ply laminate.Angle-ply laminates consist of plies in the and-6 di- rections.Angle-ply laminates may by symmetrical or unsymmetrical,balanced or unbalanced.Examples of angle-ply laminates are shown in Figure 3.7. /4 laminate./4 laminates consist of plies in which the fibers are in the 0-,45-, 90-,and-45-degree directions.The number of plies in each direction is the same (balanced laminate).In addition,the layup is also symmetrical. 3.2 Stiffness Matrices of Thin Laminates Thin laminates are characterized by three stiffness matrices denoted by [A],[B], and [D].In this section we determine these matrices for thin,flat laminates under- going small deformations.The analyses are based on the laminate plate theory and are formulated using the approximations that the strains vary linearly across the 0 0 90 90 90 90 0 [902/0] [0/901g Figure 3.6:Examples of cross-ply laminates
3.2 STIFFNESS MATRICES OF THIN LAMINATES 65 30 90 90 –30 –30 –30 [45 /90 2 2/30/ –30/–452] 45 45 30 45 30 45 [45/ –45/30/–30]s –45 –45 –45 –45 Figure 3.5: Examples of balanced laminates. specified as [−452/04/−452] ≡ [−452/02]s [45/−452/452/−452/45] ≡ [45/−452/45]s . The subscript s indicates symmetry about the midplane. Balanced laminate. In balanced laminates, for every ply in the + direction there is an identical ply in the − direction. Examples of balanced laminates are shown in Figure 3.5. Cross-ply laminates. In cross-ply laminates fibers are only in the 0- and 90-degree directions (Fig. 3.6). Cross-ply laminates may be symmetrical or unsymmetrical. Since there is no distinction between the +0 and −0 and between the +90- and −90-degree directions, cross-ply laminates are balanced. Angle-ply laminate. Angle-ply laminates consist of plies in the + and − directions. Angle-ply laminates may by symmetrical or unsymmetrical, balanced or unbalanced. Examples of angle-ply laminates are shown in Figure 3.7. π/4 laminate. π/4 laminates consist of plies in which the fibers are in the 0-, 45-, 90-, and −45-degree directions. The number of plies in each direction is the same (balanced laminate). In addition, the layup is also symmetrical. 3.2 Stiffness Matrices of Thin Laminates Thin laminates are characterized by three stiffness matrices denoted by [A], [B], and [D]. In this section we determine these matrices for thin, flat laminates undergoing small deformations. The analyses are based on the laminate plate theory and are formulated using the approximations that the strains vary linearly across the 0 [90 /02 2] 90 0 90 0 [0/90]s 90 90 0 Figure 3.6: Examples of cross-ply laminates
66 LAMINATED COMPOSITES -45 30 -45 30 45 30 45 -30 -30 Figure 3.7:Examples of angle-ply laminates. -45 45 -30 [-45z/452/-452] [-303/30] laminate,(out-of-plane)shear deformations are negligible,and the out-of-plane normal stress o:and the shear stresses tx,ty are small compared with the in- plane ox,oy,and txy stresses.These approximations imply that the stress-strain relationships under plane-stress conditions may be applied.The x,y,z refer to a coordinate system with the x and y coordinates in a suitably chosen reference plane,and z is perpendicular to this reference plane(Fig.3.8). Frequently,though not always,for convenience the reference plane is taken to be the midplane of the laminate.Unless the laminate is symmetrical with respect to the reference plane,the reference plane is not a neutral plane,and the strains in the reference plane are not zero under pure bending.The strains in the reference plane are (see Eqs.2.2,2.3,and 2.11) ax eo=duo y8= (3.1) ay 0x where u and v are the x,y components of the displacement and the superscript 0 refers to the reference plane. We adopt the Kirchhoff hypothesis,namely,that normals to the reference surface remain normal and straight(Fig.3.9).Accordingly,for small deflections the angle of rotation of the normal of the reference plane xx is aw° Xx:=3x (3.2) where wo is the out-of-plane displacement of the reference plane.The total dis- placement in the x direction is w° u=°-zXxz=°-z ax (3.3) Similarly,the total displacement in the y direction is w° v=v°-Z (3.4) ay Figure 3.8:The coordinate system. Reference plane
66 LAMINATED COMPOSITES [–452/452/–452] 30 [–303/303] 45 45 30 30 –30 –30 –30 –45 –45 –45 –45 Figure 3.7: Examples of angle-ply laminates. laminate, (out-of-plane) shear deformations are negligible, and the out-of-plane normal stress σz and the shear stresses τxz, τyz are small compared with the inplane σx, σy, and τxy stresses. These approximations imply that the stress–strain relationships under plane-stress conditions may be applied. The x, y, z refer to a coordinate system with the x and y coordinates in a suitably chosen reference plane, and z is perpendicular to this reference plane (Fig. 3.8). Frequently, though not always, for convenience the reference plane is taken to be the midplane of the laminate. Unless the laminate is symmetrical with respect to the reference plane, the reference plane is not a neutral plane, and the strains in the reference plane are not zero under pure bending. The strains in the reference plane are (see Eqs. 2.2, 2.3, and 2.11) o x = ∂uo ∂x o y = ∂vo ∂y γ o xy = ∂uo ∂y + ∂vo ∂x , (3.1) where u and v are the x, y components of the displacement and the superscript 0 refers to the reference plane. We adopt the Kirchhoff hypothesis, namely, that normals to the reference surface remain normal and straight (Fig. 3.9). Accordingly, for small deflections the angle of rotation of the normal of the reference plane χxz is χxz = ∂wo ∂x , (3.2) where wo is the out-of-plane displacement of the reference plane. The total displacement in the x direction is u = uo − zχxz = uo − z ∂wo ∂x . (3.3) Similarly, the total displacement in the y direction is v = vo − z ∂wo ∂y . (3.4) x z y Reference plane Figure 3.8: The coordinate system
3.2 STIFFNESS MATRICES OF THIN LAMINATES 67 Reference plane Reference plane A Figure 3.9:Deformation of a plate in the x-z plane. By definition,the strains are(Egs.2.2,2.3,2.11) au av Ex= Ey= Yxy (3.5) ax ay ay Substituting Egs.(3.3)and (3.4)into these expressions,we obtain 0u002w° Er= -Z- ax ax2 8u0 02w° Ey= -7- (3.6) ay 8y2 au°8v°282w0 Yxy= -Z ay dx axav These equations can be written in the following form: (3.7) where e,e,y are the strains in the reference plane (Eq.3.1),and Kr,Ky,and Kry are the curvatures of the reference plane of the plate (Fig.3.10)defined as 82w0 82w0 282w° Kx=- Ky三一 Kxy=一 (3.8) 8x2 ay2 axay The in-plane forces and moments acting on a small element are (Fig.3.11) h h Nt= Oxdz Ny ovdz Nxy Txydz -hb (3.9) h M M,=
3.2 STIFFNESS MATRICES OF THIN LAMINATES 67 x z B A u w χxz Reference plane Reference plane x w ∂ ∂ o A′ B′ Figure 3.9: Deformation of a plate in the x–z plane. By definition, the strains are (Eqs. 2.2, 2.3, 2.11) x = ∂u ∂x y = ∂v ∂y γxy = ∂u ∂y + ∂v ∂x . (3.5) Substituting Eqs. (3.3) and (3.4) into these expressions, we obtain x = ∂uo ∂x − z ∂2wo ∂x2 y = ∂vo ∂y − z ∂2wo ∂y2 (3.6) γxy = ∂uo ∂y + ∂vo ∂x − z 2∂2wo ∂x∂y . These equations can be written in the following form: x y γxy = o x o y γ o xy + z κx κy κxy , (3.7) where o x , o y , γ o xy are the strains in the reference plane (Eq. 3.1), and κx, κy, and κxy are the curvatures of the reference plane of the plate (Fig. 3.10) defined as κx = −∂2wo ∂x2 κy = −∂2wo ∂y2 κxy = −2∂2wo ∂x∂y . (3.8) The in-plane forces and moments acting on a small element are (Fig. 3.11) Nx = ) ht −hb σxdz Ny = ) ht −hb σydz Nxy = ) ht −hb τxydz Mx = ) ht −hb zσxdz My = ) ht −hb zσydz Mxy = ) ht −hb zτxydz, (3.9)
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)
72 LAMINATED COMPOSITES The [a],[B],and [matrices are related to the [A],[B],and [D]matrices by 11 a12 C16 B11B12B16 A A2 A16 B11 B12B161 C12 a22026 21 22 f26 A2 2k6B2 B22 B26 C16 C26 C66 P61 B62 B66 6 A426A66 B16 B26 B66 11 P21 B61 611 612 616 B11 B12 B16 D11 D12 D16 P12 P22 P62 612 622 d26 B12 B22 B26 D12 D22 D26 16 B26 B66 d16 826 866 B16 B26 B66 D16 D26 D66 (3.23) 3.2.1 The Significance of the [A],[B],and [D]Stiffness Matrices The [A,B],and [D]matrices represent the stiffnesses of a laminate and describe the response of the laminate to in-plane forces and moments. Ai are the in-plane stiffnesses that relate the in-plane forces Nr,Ny,Nry to the in-plane deformations e Dij are the bending stiffnesses that relate the moments M,My,Mry to the curvatures Kx,Ky,Kxy. Table 3.2.llustration of the coupling terms A16,D6,B16,B11,B12,B66 for composite materials.When the element shown in the last column is zero,there is no coupling.(The coupling terms A26,D26,B26,B22 can be illustrated in a similar manner by applying a force Ny and a moment My in the y-z plane.) Coupling No Coupling Element Extension-shear A6 Bending-twist M D16 Extension-twist B16 In-plane-out-of-plane N Bi1 B12 N Bo6
72 LAMINATED COMPOSITES The [α], [β], and [δ] matrices are related to the [A], [B], and [D] matrices by 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 = 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 −1 . (3.23) 3.2.1 The Significance of the [A], [B ], and [D ] Stiffness Matrices The [A], [B], and [D] matrices represent the stiffnesses of a laminate and describe the response of the laminate to in-plane forces and moments. Ai j are the in-plane stiffnesses that relate the in-plane forces Nx, Ny, Nxy to the in-plane deformations o x , o y , γ o xy. Di j are the bending stiffnesses that relate the moments Mx, My, Mxy to the curvatures κx, κy, κxy. Table 3.2. Ilustration of the coupling terms A16, D16, B16, B11, B12, B66 for composite materials. When the element shown in the last column is zero, there is no coupling. (The coupling terms A26, D26, B26, B22 can be illustrated in a similar manner by applying a force N y and a moment M y in the y–z plane.) Coupling No Coupling Element Extension--shear Nx Nx Nx Nx A16 Bending--twist Mx Mx Mx Mx D16 Extension--twist Nx Nx Nx Nx B16 In-plane--out-of-plane Nx Nx Nx Nx Nx Nx Nx Nx Nx Nx Nx Nx B11 B12 B66