Cross-ply laminate stiffnesses

The measured stiffnesses for two- and three-layered special cross-ply laminates are shown with symbols in Figure 4-28, and the theoretical results are shown with solid lines. In all cases, the load was kept so low that no strain exceeded SOOp. Thus, the behavior was linear and elastic. The agreement between theory and experiment is quite good. Both the qualitative and the quantitative aspects of the theory are verified. Thus, the capability to predict cross-ply laminate stiffnesses exists and is quite accurate. 

Figure 4-28a Theoretical and Measured Special Cross-Ply Laminate Stiffnesses (U. S. Standard Units) (After Tsai [4-6])...

A regular antisymmetric cross-ply laminate is delined to have laminae all of equal thickness and is common because of simplicity of fabrication. As the number of layers increases, the bending-extension coupling stiffness B.,., can be shown to approach zero. 

For both odd- and even-layered special cross-ply laminates, the extensional stiffnesses, Aj., are independent of N, the number of layers (although the N individual lamina thickneWs can be summed to get the total laminate thickness t, so N is implicit im quations (4.78) and (4.82)). However, A., and A22 depend on M, the cross-ply ratio, and on F, the lamina stiffness ratio, as shown in Figures 4-22 and 4-23. For a typical glass-fiber-reinforced lamina, F =. 3, so A, varies from. 65Q.nt to... 

Q t as M changes from 1 to 10. Similarly, varies from to 38Aii over the same range of M. The stiffnesses A g Agg are independent of M and F. The remaining stiffnesses A g and Agg are zero for all cross-ply laminates. 

Two- and three-layered special cross-ply laminates were shown to have extrema of behavior in the preceding section. Thus, comparisons between theoretical and measured stiffnesses for such laminates should... 

The individual laminae used by Tsai [4-6] consist of unidirectional glass fibers in a resin matrix (U.S. Polymeric Co. E-787-NUF) with moduli given in Table 2-3. A series of special cross-ply laminates was constructed with M = 1,2,3,10 for two-layered laminates and M = 1,2,5,10 for three-layered laminates. The laminates were subjected to axial loads and bending moments whereupon surface strains were measured. Accordingly, the stiffness relations as strains and curvatures in terms of forces and moments, that is. 

The theoretical and measured stiffnesses are shown in Figure 4-32. As with cross-ply laminates, very good agreement was obtained. Thus, the predictions of laminate stiffnesses are quite accurate. 

Derive the extensional stiffnesses for reguiar symmetric speciai cross-ply laminates, that is, derive Equation (4.78) for the special case in which t = t gn = VN. 

Figure 4-43 Strength of a Cross-Ply Laminate with M =. 2 (After Tsai [4-10]) Strength and Stiffness for Other Cross-Ply Ratios...

Theoretical and measured strengths and stiffnesses of three-layer cross-ply laminates with cross-ply ratios ranging from. 2 to 4 are shown in Figure 4-44. The scatter in the data is partially due to the difficulty of making good tensile specimens the characteristic dog-bone shape is formed by routing that often damages the 90° layer. 

Angle-ply laminates have more complicated stiffness matrices than cross-ply laminates because nontrivial coordinate transformations are involved. However, the behavior of simple angle-ply laminates (only one angle, i.e., a) will be shown to be simpler than that of cross-ply laminates because no knee results in the load-deformation diagram under uniaxial loading. Other than the preceding two differences, analysis of angle-ply laminates is conceptually the same as that of cross-ply laminates. 

The example considered to illustrate the strength-analysis procedure is a three-layered laminate with a [4-15°/-15°/+15°] stacking sequence [4-10]. The laminae are the same E-glass-epoxy as in the cross-ply laminate example with thickness. 005 in (.1270 mm), so that the total laminate thickness is. 015 in (.381 mm). In laminate coordinates, the transformed reduced stiffnesses are... 

Antisymmetric cross-ply laminates were described in Section 4.3.3 and found to have extensional stiffnesses A. , A. 2, A22 = A.. , and Agg bending-extension coupling stiffnesses B., and 822 =-Bn and bending stiffnesses D., D.,2, 822 = and Dgg. The new terms here in comparison to a speciaily orthotro Dic iaminate are B.,and 822- Because of this coupiing, the three equiiibrium differentiai equations are coupied ... 

Bend-twist coupling is a totally different animal. The governing stiffnesses, D g and D2g, simply are never zero for any laminate that is more complicated than a cross-ply laminate. You cannot force those stiffnesses to go to zero unless you do something else to the laminate. You can make them go to zero if you let the laminate be unsymmetric, but that is robbing Peter to pay Paul. In fact, it is not very difficult in most contemporary analyses to include the influence of those bend-twist cou-... 

Deviations of the delamination from the mid-plane imply mixed-mode conditions, i.e., a deviation from a pure Mode I opening load. The degree of mode mixity will depend on the relative stiffness of the two unequal beams. In the cross-ply laminates, this stiffness difference will be small , as long as the unidirectional plies that essentially provide the axial stiffness are equal in both beams. This is a rational justification for a validity criterion for cross-ply laminates that allows deviations from the mid-plane, as long as the delamination does not show a transition into the unidirectional plies. 

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