400 FINITE ELEMENT ANALYSIS K 尽： Figure 9.4:Illustration of a sublaminate. The x,y,z coordinate system is shown in Figure 9.4.The average stresses are defined as 1 石xhsJh) xdz= 石：=02 1 1 yhsJh Ny Tyz=Tyz (9.17) 1 元xy=hsJh) xydi=hs Nxy Txz Txz- The second equalities in the left-hand column are written by virtue of Eq.(3.9). The average strains are 1「 e.dz Ex -Ex 1 7= Ey=Ey (9.18) 7=元小 Yxadz Txy=Yxy where hs is the thickness of the sublaminate(Fig.9.4).The terms in the right-hand columns of Eqs.(9.17)and(9.18)show that the stresses o,tyz,txz and the strains er,Ey,Yry do not vary across the thickness. In the following we derive the elements of the compliance matrix. 9.4.1 Step 1.Elements of [due to In-Plane Stresses In this step we determine the elements in the first,second,and sixth columns of the matrix [To this end,we impose the average in-plane stresses y,and Try on the sublaminate(Fig.9.5,top).Since,yr are zero,the stress-strain400 FINITE ELEMENT ANALYSIS hs z y x z y x hs 1 … ks Ks zk −1 zk Figure 9.4: Illustration of a sublaminate. The x,y,z coordinate system is shown in Figure 9.4. The average stresses are defined as σ x = 1 hs ) (hs) σxdz= 1 hs Nx σ z = σz σ y = 1 hs ) (hs) σydz= 1 hs Ny τ yz = τyz τ xy = 1 hs ) (hs) τxydz= 1 hs Nxy τ xz = τxz. (9.17) The second equalities in the left-hand column are written by virtue of Eq. (3.9). The average strains are z = 1 hs ) (hs) zdz x = x γ yz = 1 hs ) (hs) γyzdz  y = y γ xz = 1 hs ) (hs) γxzdz γ xy = γxy, (9.18) where hs is the thickness of the sublaminate (Fig. 9.4). The terms in the right-hand columns of Eqs. (9.17) and (9.18) show that the stresses σz, τyz, τxz and the strains x, y, γxy do not vary across the thickness. In the following we derive the elements of the compliance matrix. 9.4.1 Step 1. Elements of [J ] due to In-Plane Stresses In this step we determine the elements in the first, second, and sixth columns of the matrix [J ]. To this end, we impose the average in-plane stresses σ x, σ y, and τ xy on the sublaminate (Fig. 9.5, top). Since σ z, τ yz, τ xz are zero, the stress–strain