9.4 SUBLAMINATE 403 where zk,Z-1 are illustrated in Figure 9.4.Since Tyz=Txz=0,we have that yyz and yxz are zero (Eqs.2.26 and 2.21),and Eq.(9.21)now becomes 63 (9.30) To satisfy this equation,the preceding elements of the compliance matrix must be zero: J J42 J46 001 51J52 J56」 0 00 (9.31) 9.4.2 Step 2.Elements of [due to Out-of-Plane Normal Stresses In this step we determine the elements in the third column of the matrix []To accomplish this we consider a sublaminate in which there are only o:stresses.To form such a sublaminate,first we consider a sublaminate restrained along its edges (ex =ey=yxy =0)and apply a uniform stress o:=:(Stage 1,Fig.9.6),which introduces in-plane stresses,,y.Second,we apply in-plane stressesx, -y,-Txy(Stage 2,Fig.9.6).Third,we superimpose Stages 1 and 2 and arrive at the sublaminate inside which the in-plane average stresses oy,Try and the transverse shear stresses y,x are zero and the stress-strain relationships (Eq.9.15)are 「J3 J23 (9.32) Yxy J63 e2=J33可： (9.33) - (9.34) Stage 1.Following the outline above,we apply o:to the sublaminate,which is restrained along the edges(x=y=y=0).For a single layer,Eqs.(2.27)and 个个 个个个个个 6 ↓↓↓↓↓ ↓↓↓↓ Stage 1 Stage 2 Stage 3 Figure 9.6:Illustration of Step 2.Sublaminate restrained along its edges subjected to o:and the resulting stressx(Stage 1);unrestrained sublaminate subjected tox(Stage 2);unrestrained sublaminate subjected to o:(Stage 3);(y and Txy are not shown).9.4 SUBLAMINATE 403 where zk, zk−1 are illustrated in Figure 9.4. Since τ yz = τ xz = 0, we have that γyz and γxz are zero (Eqs. 2.26 and 2.21), and Eq. (9.21) now becomes  0 0  = J41 J42 J46 J51 J52 J56!    σ x σ y τ xy    . (9.30) To satisfy this equation, the preceding elements of the compliance matrix must be zero: J41 J42 J46 J51 J52 J56! = 000 000! . (9.31) 9.4.2 Step 2. Elements of [J ] due to Out-of-Plane Normal Stresses In this step we determine the elements in the third column of the matrix [J ]. To accomplish this we consider a sublaminate in which there are only σz stresses. To form such a sublaminate, first we consider a sublaminate restrained along its edges (x = y = γxy = 0) and apply a uniform stress σz = σ z (Stage 1, Fig. 9.6), which introduces in-plane stresses σ x, σ y, τ xy. Second, we apply in-plane stresses −σ x, −σ y, −τ xy (Stage 2, Fig. 9.6). Third, we superimpose Stages 1 and 2 and arrive at the sublaminate inside which the in-plane average stresses σ x, σ y, τ xy and the transverse shear stresses τ yz, τ xz are zero and the stress–strain relationships (Eq. 9.15) are    x  y γ xy    =    J13 J23 J63    σ z (9.32) z = J33σ z (9.33)  γ yz γ xz = J43 J53! σ z. (9.34) Stage 1. Following the outline above, we apply σz to the sublaminate, which is restrained along the edges (x = y = γxy = 0). For a single layer, Eqs. (2.27) and σx σx + = σx σx σz σz Stage 1 Stage 2 Stage 3 Figure 9.6: Illustration of Step 2. Sublaminate restrained along its edges subjected to σz and the resulting stress σ x (Stage 1); unrestrained sublaminate subjected to σ x (Stage 2); unrestrained sublaminate subjected to σz (Stage 3); (σ y and τ xy are not shown)