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)