STRUCTURAL ANALYSIS 179 Substituting from equations(6.10)into(6.11),and remembering that the strains are the same in all plies,the following result is readily obtained: Nx AuEx +AxEy +Axs Yxy Ny =AxyEx +Ayyy +Ays Yo (6.12) N:AxsEx +Aysy +Ass Yxy where: Ag=>Qu(0x)(hk-hg-1) (6.13) k= The quantities Ay are the terms of the laminate "in-plane stiffness matrix." Given the single-ply moduli and the laminate lay-up details,they can be calculated routinely by using equations (6.4),(6.8),and (6.13).Equation (6.12)are generally taken as the starting point for any laminate structural analysis. 6.2.3.2 Laminate Stress-Strain Law.As was just implied,it seems to be the current fashion in laminate mechanics to work in terms of the stress resultants, rather than the stresses.However,for some purposes,the latter are more convenient.From the stress resultants,the average stresses(averaged through the thickness of the laminate)are easily obtained;writing these stresses simply as ox, dy,and Try then: 0x= t (6.14) Hence,in terms of these average stresses,the stress-strain law for the laminate becomes: Ox Atex Aryey Ars Yoy y=Atyex +Ayyey+Ays Yo (6.15) Tg=AtEx十A,Ey+AxY where: 2(0)hk-hk-i） (6.16) In some cases,equation (6.15)is more convenient than is equation (6.12). 6.2.3.3 Orthotropic Laminates.An orthotropic laminate,having the laminate axes as the axes of orthotropy,is one for which Axs=Ays=0;STRUCTURAL ANALYSIS 179 Substituting from equations (6.10) into (6.11), and remembering that the strains are the same in all plies, the following result is readily obtained: Nx = Axxex + Axyey + AxsYxy Ny = Axyex + Ayyey "t- Ays Yxy (6.12) Ns = Axsex -t'-Ayse, y + Ass'Yxy where: Aij = XZ, Qij( Ok)(hk -- hk-1) (6.13) k=l The quantities A o are the terms of the laminate "in-plane stiffness matrix." Given the single-ply moduli and the laminate lay-up details, they can be calculated routinely by using equations (6.4), (6.8), and (6.13). Equation (6.12) are generally taken as the starting point for any laminate structural analysis. 6.2.3.2 Laminate Stress--Strain Law. As was just implied, it seems to be the current fashion in laminate mechanics to work in terms of the stress resultants, rather than the stresses. However, for some purposes, the latter are more convenient. From the stress resultants, the average stresses (averaged through the thickness of the laminate) are easily obtained; writing these stresses simply as o'x, try, and 7xy then: Nx Nx N, O'x=-- , O'y = --, "/'xy = -- (6.14) t t t Hence, in terms of these average stresses, the stress-strain law for the laminate becomes: O-x = Axxex + Axyey "l- Axs ]txy Or y = Axy 8 x -t- Ayy ~ y --I- Ays 'Yxy (6.15) where: "rxy = A*xsex + Aysey + Ass'Yxy AU 1 A~j =--= - ~ Qij(Ok)(hk -- hk-1) (6.16) t tk= 1 In some cases, equation (6.15) is more convenient than is equation (6.12). 6.2.3.30rthotropic Laminates. An orthotropic laminate, having the laminate axes as the axes of orthotropy, is one for which Axs = Ays = O;