g/ Computers and Structures 80(2002)1177-1199 79 global stresses can be transformed into the ply coordi- dg =2()(Zk-Zk-1)dT, nate system through d}=(){d}, where(do= doar, do,x, do The coordinate trans- d=∑(B}(4-2=)dT formation matrixes in(1)and (3), [Te] and [T), are the same as equations(44)and(45)of Ref [l] but with volved s being replaced by the ply angle 0. Substituting N is the total number of lamina plies in the lami Eq.(3) into the right hand sides of equations(14. 1) and nate. (Cy) are the stiffness elements of the kth lam 14.2)of Ref. [u if this ply is a UD lamina, equations ina in the global coordinate system, see Eq.(D). In (42. 1)and (42. 2)of Ref [I] if the ply is a braided fabric Eq.(5), dNxx, dRy, and dNr and dMxx,dMr,and lamina(see Section 6 for additional discussion), or drr are, respectively, the externally applied in-plane equations(43. 2)and(43.) of Ref [I] if the ply is a force and moment increments per unit length on the knitted fabric lamina, the averaged stress increments in laminate the fiber and matrix phases of this lamina can be eval- Apparently, different lamina ply in the laminate uated. It is thus only necessary to determine the in-plane carries different load share. With the increase of the strains and curvature increments. Note that the internal external load, some ply must fail first before the other stress resultants of Eq. (1)must be balanced with the Once some koth lamina fails, the corresponding overall externally applied forces and moments. For instance, we applied load on the laminate is defined as a progressive must have failure strength(e.g. the first-ply or the second- ply fail ure strength, etc. ) As the ply failure has already been dNax= do dz-2 (do d) z, (4.1) d he ne respon dihe failure or de s automaic ml tea. tified. It is either the matrix fail the fiber frac- ture. or the failure of the both constituents that cause doxxzdz (doxx)Zdz,(4.2) the ply failure. Meanwhile, the failed lamina cannot sustain any additional load share, see the experimen- tal evidence shown in Section 6. The additional exter where h is the thickness of the laminate. Substituting(1) nal load must be shared by the remaining un-failed into ( 4. I)and (4.2), and the like and performing the laminae. Namely, Egs. (4.1)and(4.2)should be modi ing integra fied dNxx +dQ Nry +dQ dOxy+dQ dNa s dora dz dMy+dg dMr +d2 Zdz= (doxx),ZdZ. (8.2) el ol2 @is gll oll o dex Oir o2 gb3 g1 o2) oli de Thus, the post-failure analysis is still based on eq oll ol? @l3 @ll 012 gl3 dxe but with reduced overall stiffness elements and equiva- 望望盟②盟|dkn lent thermal loads which are given by el=>(C c=∑(CG)(-4- G (CG)(42-2=1) g=3∑(cA(n- (CG)(2-2-1)global stresses can be transformed into the ply coordi￾nate system through fdrgP k ¼ ð½T  T s Þk fdrgG k ; ð3Þ where fdrgP ¼ fdrxx; dryy ; drxyg T . The coordinate trans￾formation matrixes in (1) and (3), ½Tc and ½Ts, are the same as equations (44) and (45) of Ref. [1] but with in￾volved n being replaced by the ply angle h. Substituting Eq. (3) into the right hand sides of equations (14.1) and (14.2) of Ref. [1] if this ply is a UD lamina, equations (42.1) and (42.2) of Ref. [1] if the ply is a braided fabric lamina (see Section 6 for additional discussion), or equations (43.2) and (43.3) of Ref. [1] if the ply is a knitted fabric lamina, the averaged stress increments in the fiber and matrix phases of this lamina can be eval￾uated. It is thus only necessary to determine the in-plane strains and curvature increments. Note that the internal stress resultants of Eq. (1) must be balanced with the externally applied forces and moments. For instance, we must have dNXX ¼ Z h=2 h=2 drXX dZ ¼ XN k¼1 Z Zk Zk1 ðdrXX Þk dZ; ð4:1Þ dMXX ¼ Z h=2 h=2 drXX Z dZ ¼ XN k¼1 Z Zk Zk1 ðdrXX ÞkZ dZ; ð4:2Þ where h is the thickness of the laminate. Substituting (1) into (4.1) and (4.2), and the like and performing the resulting integrations, we obtain the following equation dNXX þ dXI 1 dNYY þ dXI 2 dNXY þ dXI 3 dMXX þ dXII 1 dMYY þ dXII 2 dMXY þ dXII 3 8 >>>>>>>>>< >>>>>>>>>: 9 >>>>>>>>>= >>>>>>>>>; ¼ QI 11 QI 12 QI 13 QII 11 QII 12 QII 13 QI 12 QI 22 QI 23 QII 12 QII 22 QII 23 QI 13 QI 23 QI 33 QII 13 QII 23 QII 33 QII 11 QII 12 QII 13 QIII 11 QIII 12 QIII 13 QII 12 QII 22 QII 23 QIII 12 QIII 22 QIII 23 QII 13 QII 23 QII 33 QIII 13 QIII 23 QIII 33 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 de0 XX de0 YY 2de0 XY dj0 XX dj0 YY 2dj0 XY 8 >>>>>>>>>>>< >>>>>>>>>>>: 9 >>>>>>>>>>>= >>>>>>>>>>>; ; ð5Þ QI ij ¼ XN k¼1 ðCG ij Þk ðZk  Zk1Þ; QII ij ¼ 1 2 XN k¼1 ðCG ij Þk ðZ2 k  Z2 k1Þ; QIII ij ¼ 1 3 XN k¼1 ðCG ij Þk ðZ3 k  Z3 k1Þ; ð6Þ dXI i ¼ XN k¼1 ðbiÞ G k ðZk  Zk1ÞdT ; dXII i ¼ 1 2 XN k¼1 ðbi Þ G k ðZ2 k  Z2 k1ÞdT : ð7Þ N is the total number of lamina plies in the lami￾nate. ðCG ij Þk are the stiffness elements of the kth lam￾ina in the global coordinate system, see Eq. (1). In Eq. (5), dNXX ; dNYY , and dNXY and dMXX , dMYY , and dMXY are, respectively, the externally applied in-plane force and moment increments per unit length on the laminate. Apparently, different lamina ply in the laminate carries different load share. With the increase of the external load, some ply must fail first before the others. Once some k0th lamina fails, the corresponding overall applied load on the laminate is defined as a progressive failure strength (e.g. the first-ply or the second-ply fail￾ure strength, etc.). As the ply failure has already been defined upon the failure of one constituent material, the corresponding failure mode is automatically iden￾tified. It is either the matrix failure, or the fiber frac￾ture, or the failure of the both constituents that causes the ply failure. Meanwhile, the failed lamina cannot sustain any additional load share, see the experimen￾tal evidence shown in Section 6. The additional exter￾nal load must be shared by the remaining un-failed laminae. Namely, Eqs. (4.1) and (4.2) should be modi- fied to dNXX ¼ Z h=2 h=2 drXX dZ ¼ XN k¼1 k6¼k0 Z Zk Zk1 ðdrXX Þk dZ; ð8:1Þ dMXX ¼ Z h=2 h=2 drXX Z dZ ¼ XN k¼1 k6¼k0 Z Zk Zk1 ðdrXX ÞkZ dZ: ð8:2Þ Thus, the post-failure analysis is still based on Eq. (5), but with reduced overall stiffness elements and equiva￾lent thermal loads which are given by QI ij ¼ XN k¼1 k62fk0g ðCG ij Þk ðZk  Zk1Þ; QII ij ¼ 1 2 XN k¼1 k62fk0g ðCG ij Þk ðZ2 k  Z2 k1Þ; QIII ij ¼ 1 3 XN k¼1 k62fk0g ðCG ij Þk ðZ3 k  Z3 k1Þ; ð9Þ Z.-M. Huang / Computers andStructures 80 (2002) 1177–1199 1179