Z-M. Huang / Comput and Structures 80(2002)1177-1199 the failure behavior and ultimate strength of laminated dg2=∑(MB)(4-2-)dT, composites reinforced with various fiber preforms and (10) having dierent constituent materials. In the present dg=∑(B)(z-2-)dT are analyzed. a number of multidirectional laminates were subjected to in-plane biaxial loads. Those problems were designed to verify and compare the efficiency of the In the above, kol represents all those laminae which outmost composite strength theories in the current lit have already failed and the symbol"stands for"not erature. No experimental data for any of the exercised belonging to. Continued in this way, the ultimate fail laminates were given in Ref [6]. Predictions from the ure strength of the laminate is determined using the most famous theories in current use were presented in overall applied load at which all the plies have failed. Ref [4]. Although no comparison with experiments was It must be pointed out that the stiffness reduction, as able to make, one can still recognize that quite large per Eqs.(9)and (10), up to the last ply failure is appli- discrepancy existed among the predictions of the differ- load. If, however, the laminate is involved with a lateral noted that almost all of the theories taken part in the (out-of-plane) load, the stiffness reduction must be per exercises were phenom gical [3, 4]. Namely, a com- formed more carefully. For instance, let the laminate be plete set of critical strength parameters as well as elastic subjected to only a bending load. Using the stiffness properties measured from the composite (lamina or lam- reduction procedure given in Eqs. (9)and (10), the inate)had been employed in the predictions. In the con- predicted failure process (as well as load-deflection trast, the present constitutive relationship makes use of curve)may no longer be correct after the failure of an the constituent materials intermediate ply, which corresponds to the ultimate Four material systems, i.e. the E-Glass 21 x K43 failure. The reason is that in reality the ultimate bending Gevetex and LY556/HT907/DY063 epoxy, the Sile- load (which is defined as the maximum load that the nka E-glass 1200tex and Mr750/HY917/DY063 epoxy laminate can sustain) is generally attained by the lami- the graphite AS4 and 3501-6 epoxy, and the graph nate before its last ply failure. After the ultimate bending ite T300 and BSL914C epoxy, were used in the exercise ad, the load-deflection curve of the laminated beam is [6]. The objective is to predict the failure envelop loa downward, a phen named material softening. or stress-strain response of the laminates up to failure However, using the incremental solution strategy de- scribed above the predicted load-deflection curve is always upward till the last-ply failure. Under the pure UD laminae together with their fiber volume fractions bending condition, the middle plane strain increm used for the exercise problems have been specified in deox etc, are negligibly small. The remaining bending Ref. [6]. They are summarized in Tables 1-4. In the curvature will have very little, if any, stress contribution present calculations, all the fibers are assumed to be to the last ply failure or the last two-ply failures, ac linearly elastic until rupture. The matrix of each UD cording to Eqs. (1),(2.1), and(2.2). Thus, the last ply lamina, however, must be considered as elastic-plastic will not fail at all, but the deflection(curvature)can be This is because the in-plane shear stress-shear strain increased unlimitedly. In a subsequent example, we will curve of every lamina provided in Ref. [6]is nonlinear see experimental evidence that some plies of the beam From Eq.(4)of Ref. [I], we can clearly see that the could not be forced to failure due to the limitation of the composite can display a nonlinear deformation if and testing apparatus to excessive beam deflection. Namely, only if at least one of its constituent materials undergoes only partial layers of the laminated beam failed after the such a deformatio bending test. This is consistent with our theory. This All the thermo-elastic properties of the constituent also indicates that if a fiexural load is involved, the only materials of the four laminae as well as their fiber vol- use of a stress failure criterion will not be enough to ume fractions are remained the same as those provided determine the ultimate failure; an additional critical in Ref. [6]. However, the bridging parameters a and B order that the ultimate strength of the laminate can be used in defining the independent bridging elements a33 deflection/curvature condition has to be adopted and a2?(see Eq (I1) of Ref [I] have to be calibrated against the transverse and in-plane shear moduli of the laminae in such a way that the predicted moduli agree airly well with the measured values. The parameter B has 3. Biaxial failure of multidirectional laminates been chosen as 0.45 for all the four laminae. whereas the parameters a has been taken to be 0.3 for one lamina and In this and the subsequent sections, we will apply the 0.35 for the others, as indicated in Tables 1-4. Both the general constitutive relationship developed to identify ridging parameters of each lamina remain unchangeddXI i ¼ XN k¼1 k62fk0g ðbiÞ G k ðZk  Zk1ÞdT ; dXII i ¼ 1 2 XN k¼1 k62fk0g ðbiÞ G k ðZ2 k  Z2 k1ÞdT : ð10Þ In the above, fk0g represents all those laminae which have already failed and the symbol ‘‘62’’ stands for ‘‘not belonging to’’. Continued in this way, the ultimate fail￾ure strength of the laminate is determined using the overall applied load at which all the plies have failed. It must be pointed out that the stiffness reduction, as per Eqs. (9) and (10), up to the last ply failure is appli￾cable only when the laminate is subjected to an in-plane load. If, however, the laminate is involved with a lateral (out-of-plane) load, the stiffness reduction must be per￾formed more carefully. For instance, let the laminate be subjected to only a bending load. Using the stiffness reduction procedure given in Eqs. (9) and (10), the predicted failure process (as well as load–deflection curve) may no longer be correct after the failure of an intermediate ply, which corresponds to the ultimate failure. The reason is that in reality the ultimate bending load (which is defined as the maximum load that the laminate can sustain) is generally attained by the lami￾nate before its last ply failure. After the ultimate bending load, the load–deflection curve of the laminated beam is downward, a phenomenon named material softening. However, using the incremental solution strategy de￾scribed above, the predicted load–deflection curve is always upward till the last-ply failure. Under the pure bending condition, the middle plane strain increments, de0 XX etc., are negligibly small. The remaining bending curvature will have very little, if any, stress contribution to the last ply failure or the last two-ply failures, ac￾cording to Eqs. (1), (2.1), and (2.2). Thus, the last ply will not fail at all, but the deflection (curvature) can be increased unlimitedly. In a subsequent example, we will see experimental evidence that some plies of the beam could not be forced to failure due to the limitation of the testing apparatus to excessive beam deflection. Namely, only partial layers of the laminated beam failed after the bending test. This is consistent with our theory. This also indicates that if a flexural load is involved, the only use of a stress failure criterion will not be enough to determine the ultimate failure; an additional critical deflection/curvature condition has to be adopted in order that the ultimate strength of the laminate can be assessed. 3. Biaxial failure of multidirectional laminates In this and the subsequent sections, we will apply the general constitutive relationship developed to identify the failure behavior and ultimate strength of laminated composites reinforced with various fiber preforms and having different constituent materials. In the present section, the failure exercise problems as posed in Ref. [6] are analyzed. A number of multidirectional laminates were subjected to in-plane biaxial loads. Those problems were designed to verify and compare the efficiency of the outmost composite strength theories in the current lit￾erature. No experimental data for any of the exercised laminates were given in Ref. [6]. Predictions from the most famous theories in current use were presented in Ref. [4]. Although no comparison with experiments was able to make, one can still recognize that quite large discrepancy existed among the predictions of the differ￾ent theories for any of the exercise problems [4]. It is noted that almost all of the theories taken part in the exercises were phenomenological [3,4]. Namely, a com￾plete set of critical strength parameters as well as elastic properties measured from the composite (lamina or lam￾inate) had been employed in the predictions. In the con￾trast, the present constitutive relationship makes use of the constituent materials. Four material systems, i.e. the E-Glass 21  K43 Gevetex and LY556/HT907/DY063 epoxy, the Sile￾nka E-glass 1200tex and MY750/HY917/DY063 epoxy, the graphite AS4 and 3501-6 epoxy, and the graph￾ite T300 and BSL914C epoxy, were used in the exercise [6]. The objective is to predict the failure envelope or stress–strain response of the laminates up to failure [6]. Measured properties of the fibers, matrixes, and four UD laminae together with their fiber volume fractions used for the exercise problems have been specified in Ref. [6]. They are summarized in Tables 1–4. In the present calculations, all the fibers are assumed to be linearly elastic until rupture. The matrix of each UD lamina, however, must be considered as elastic–plastic. This is because the in-plane shear stress–shear strain curve of every lamina provided in Ref. [6] is nonlinear. From Eq. (4) of Ref. [1], we can clearly see that the composite can display a nonlinear deformation if and only if at least one of its constituent materials undergoes such a deformation. All the thermo-elastic properties of the constituent materials of the four laminae as well as their fiber vol￾ume fractions are remained the same as those provided in Ref. [6]. However, the bridging parameters a and b used in defining the independent bridging elements a33 and a22 (see Eq. (11) of Ref. [1]) have to be calibrated against the transverse and in-plane shear moduli of the laminae in such a way that the predicted moduli agree fairly well with the measured values. The parameter b has been chosen as 0.45 for all the four laminae, whereas the parameters a has been taken to be 0.3 for one lamina and 0.35 for the others, as indicated in Tables 1–4. Both the bridging parameters of each lamina remain unchanged 1180 Z.-M. Huang / Computers andStructures 80 (2002) 1177–1199