Antisymmetric Cross-Ply Laminated Plates

Antisymmetric cross-ply laminates were found in Section 4.3.3 to have extensional stiffnesses A i, A., 2, A22 = Aj, and Aeei bending-extension coupling stiffnesses and 822 =-Bn and bending stiffnesses Di2. D22 = Dn. and Dge. The new terms here in comparison to a specially orthotropic laminate are and 822- Because of this bending-extension coupling, the three buckling differential equations are coupled  

Jones solved the problem for simply supported edge boundary condition S2 [5-19]  

Note that if Bn is zero, then T13 and T23 are also zero, so Equation (5.81) reduces to the specially orthotropic plate solution. Equation (5.65), if D11 =D22- Because Tn, T12, and T22 are functions of both m and n, no simple conclusion can be drawn about the value of n at buckling as could be done for specially orthotropic laminated plates where n was determined to be one. Instead, Equation (5.81) is a complicated function of both m and n. At this point, recall the discussion in Section 3.5.3 about the difference between finding a minimum of a function of discrete variables versus a function of continuous variables. We have already seen that plates buckle with a small number of buckles. Consequently, the lowest buckling load must be found in Equation (5.81) by a searching procedure due to Jones involving integer values of m and n [5-20] and not by equating to zero the first partial derivatives of N with respect to m and n. 

One of the major complications in the plate buckling solution is the need to investigate the influence of buckle mode shape on the buckling load itself. That is, the plate buckling load in Equation (5.81) is a function 

However, because we are usualiy interested oniy in the lowest buckling load for a column, m is always one. For plates, both m and n enter the buckling equation as well as the plate aspect ratio, a/b, so the lowest buckling load does not typically occur for m = 1 and n = 1. Thi, we must find the absolute minimum of the values of the buckling load, N, or more generally, X, for a wide range of m and n values. 

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Figure 5-13 Deflection of an Antisymmetric Cross-Ply Laminated Plate under Sinusoidal Transverse Load...

Whitney solved the problem for simply supported edge boundary condition S3 [5-13 and 5-14] (recall that S2 was used for antisymmetric cross-ply laminated plates in Section 5.3.3) ... 

As for the deflection problem in Section 5.3.3, the effect of the number of layers on the buckling load is found by dividing a constantthickness, equal-weight cross-ply laminate into more and more laminae as in Figure 5-12. Results for graphite-epoxy antisymmetric cross-ply laminated plates for which Ei/E2 = 40, Gi2/E2 = - - v,2 = -25 are... 

Figure 5-27 Relative Uniaxial Buckling Loads of Square Antisymmetric Cross-Ply Laminated Plates (After Jones [5-19])...

Note that if B g and 825 are zero, then and T23 are also zero, so Equation (5.92) reduces to the specially orthotropic plate solution. Equation (5.65). The character of Equation (5.92) is the same as that of Equation (5.81) for antisymmetric cross-ply laminated plates, so the remarks on finding the buckling load in Section 5.4.3 are equally applicable here. 

The buckling load will be determined for plates with various laminations specially orthotropic, symmetric angle-ply, antisymmetric cross-ply, and antisymmetric angle-ply. The results for the different lamination types will be compared to find the influence of bend-twist coupling and bending-extension coupling. As with the deflection problems in Section 5.3, different simply supported edge boundary conditions will be used in the several problems addressed for convenience of illustration. 

Classical solutions to laminated shell buckling and vibration problems in the manner of Chapter 5 were obtained by Jones and Morgan [6-47]. Their results are presented as normalized buckling loads or fundamental natural frequency versus the Batdorf shell curvature parameter. They showed that, for antisymmetrically laminated cross-ply shells as for plates, the effect of coupling between bending and extension on buckling loads and vibration frequencies dies out rapidly as the number of layers... 

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