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-strain
400 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
9.4 SUBLAMINATE 401 0. L(1+E) h(1+e) Figure 9.5:Illustration of Step 1.The ply stress and the corresponding average stress on the sublaminate. relationship (Eq.9.15)may be written as J12 J16 J21 J J26 (9.19) 16 J62 e}=[J31 J32 J36] (9.20) 「J4 J46 6 (9.21) J56 69 The strains are uniform across the thickness(Eq.9.18).Under these conditions Kx,Ky,Kxy are zero,and we have (see Egs.3.21 and 3.7) (9.22) where [A]is the tensile stiffness matrix of the sublaminate.(The summation in Eq.3.20 is performed from 1 to Ks,where Ks is the number of layers or ply groups in the sublaminate;see Fig.9.4).For the sublaminate Egs.(9.17),(9.18),and(9.22) yield [A (9.23) J11 J12 J16 J21 J26 J61 J62 J6小
9.4 SUBLAMINATE 401 σx σx L(1+ )x hs (1+ )z σx σx hs L L σx σy τxy Figure 9.5: Illustration of Step 1. The ply stress and the corresponding average stress on the sublaminate. relationship (Eq. 9.15) may be written as x y γ xy = J11 J12 J16 J21 J22 J26 J61 J62 J66 σ x σ y τ xy (9.19) {z} = [J31 J32 J36] σ x σ y τ xy (9.20) 1 γ yz γ xz6 = J41 J42 J46 J51 J52 J56! σ x σ y τ xy . (9.21) The strains are uniform across the thickness (Eq. 9.18). Under these conditions κx, κy, κxy are zero, and we have (see Eqs. 3.21 and 3.7) Nx Ny Nxy = [A] o x o y γ o xy = [A] x y γxy , (9.22) where [A] is the tensile stiffness matrix of the sublaminate. (The summation in Eq. 3.20 is performed from 1 to Ks, where Ks is the number of layers or ply groups in the sublaminate; see Fig. 9.4). For the sublaminate Eqs. (9.17), (9.18), and (9.22) yield x y γ xy = hs [A] −1 % &' ( J11 J12 J16 J21 J22 J26 J61 J62 J66 σ x σ y τ xy . (9.23)