Antisymmetric laminate buckling

Robert M. Jones, Harold S. Morgan, and James M. Whitney, Buckling and Vibration of Antisymmetrically Laminated Angle-Ply Rectangular Plates, Journal of Applied Mechanics, December 1973, pp. 1143-1144. 

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... 

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. 

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 ... 

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])...

Whitney solved the problem for simply supported edge boundary condition S3 [5-13 and 5-14] (note that this boundary condition differs significantly from the S2 condition used for buckling of antisymmetric cross-ply laminated plates in Section 5.4.3)) ... 

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. 

Figure 5-28 Buckling of Square Antisymmetric Angle-Ply Laminated Plates under Uniform Uniaxial Compression,...

Normalized fundamental natural frequencies for the exampie un-symmetrical cross-piy laminated graphite-epoxy piates are shown in Figure 5-40. The vibration resuits are analogous to the buckling resuits in the same manner as explained for antisymmetric iaminates, nameiy, iesser differences for vibration frequencies because of the square-root factor. 

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