Unsymmetric Laminates

It is interesting to observe that as well as the expected axial and transverse strains arising from the applied axial stress, we have also a shear strain. This is because in composites we can often get coupling between the different modes of deformation. This will also be seen later where coupling between axial and flexural deformations can occur in unsymmetric laminates. Fig. 3.17 illustrates why the shear strains arise in uniaxially stressed single ply in this Example. 

Stiffnesses for single-layered configurations are treated first to provide a baseline for subsequent discussion. Such stiffnesses should be recognizable in terms of concepts previously encountered by the reader in his study of plates and shells. Next, laminates that are symmetric about their middle surface are discussed and classified. Then, laminates with laminae that are antisymmetrically arranged about their middle surface are described. Finally, laminates with complete lack of middle-surface symmetry, i.e., unsymmetric laminates, are discussed. For all laminates, the question of laminae thicknesses arises. Regular laminates have equal-thickness laminae, and irregular laminates have non-equal-thickness laminae. 

Unsymmetric laminates with multiple specially orthotropic layers can be shown to have the force and moment resultants in Equations (4.59) and (4.60) but with different A22, B22, and D22 korn B.,., and Dll, respectively. That is, there are no shear-extensiorTcoupling terms nor any bend-twist coupling terms, so the solution of problems with this kind of lamination is about as easy as with isotropic layers. 

Unsymmetric laminates with multiple generally orthotropic layers or with multiple anisotropic layers have force and moment resultants no... 

Figure 5-36 Unsymmetric Cross-Ply Laminate Example (After Jones [5-19])...

We will see that the unsymmetric laminate has more bending stiffness in the y-direction than the all-0°-layer laminate and almost as much bending stiffness in the x-direction. Thus, the center deflection of the unsymmetrically laminated plate should exceed that of the all-0°-layer laminated plate. However, we are already aware that bending-extension coupling increases deflections, so the center deflection of the unsymmetric laminate should exceed that of the orthotropic laminated plate. 

Unsymmetric Cross-Ply Laminated Graphite-Epoxy Plates (After Jones and Morgan [5-23])... 

In their pioneering paper on laminated plates, Reissner and Stavsky investigated an approximate approach (in addition to their exact approach) to calculate deflections and stresses for antisymmetric angie-ply laminated plates [5-27]. Much later, Ashton extended their approach to structural response of more general unsymmetrically laminated plates and called it the reduced stiffness matrix method [5-28]. The attraction of what is now called the Reduced Bending Stiffness (RBS) method is that an unsymmetrically laminated plate can be treated as an orthotropic plate using only a modified D matrix in the solution, i.e.,... 

Robert M. Jones, Buckling and Vibration of Rectangular Unsymmetrically Laminated Oross-Ply Plates, AIM Journal, December 1973, pp. 1626-1632. 

Robert M. Jones and Harold S. Morgan, Deflection of Unsymmetrically Laminated Cross-Ply Rectangular Plates, Proceedings ol the 12th Annual Meeting of the Society of Engineering Science, 20-22 October 1975, Austin, Texas, pp. 155-167. 

J. E. Ashton, Approximate Solutions for Unsymmetrically Laminated Plates, Journal of Composite Materials, January 1969, pp. 189-191. 

Figure 6-24 Postcuring Shapes of Various Unsymmetric Laminates (After Hyer [6-37])...

The first observation is that the cured shape of an unsymmetric cross-ply laminate is often cylindrical, whereas we would predict it to be a saddle shape (hyperbolic paraboloid) from classical lamination theory (the curvatures can be shown to be = - Ky or - = Ky). A thick lami-... 

Figure 6-25 Unsymmetric Cross-Ply Laminate Shapes (After Hyer [6-38])...

Hyer shovys the conditions under vyhich the cylindrical shape must exist and when the saddle shape must exist [6-38], He approximates the transverse deflection of an unsymmetric cross-ply laminate as... 

A workable theory behind why unsymmetric cross-ply laminates deform as they do has been developed by Hyer (who has extended these papers). Thus, a reasonable understanding of the deformation mechanics exists and can be used to design laminates with specified curvatures. 

Michael W. Hyer, Some Observations on the Cured Shape of Thin Unsymmetric Laminates, Journal of Composite Materials, March 1981, pp. 175-194. 

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-... 

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