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 plane66 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 in￾plane 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 dis￾placement 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