Laminate bend-twist coupling

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. 

Often, because specially orthotropic laminates are virtually as easy to analyze as isotropic plates, other laminates are regarded as, or approximated with, specially orthotropic laminates. This approximation will be studied by comparison of results for each type of laminate with and without the various stiffnesses that distinguish it from a specially orthotropic laminate. Specifically, the importance of the bend-twist coupling terms D,g and D26 will be examined for symmetric angle-ply laminates. Then, bending-extension coupling will be analj ed for antisym-... 

A specially orthotropic laminate has either a single layer of a specially orthotropic material or multiple specially orthotropic layers that are symmetrically arranged about the laminate middle surface. In both cases, the laminate stiffnesses consist solely of A, A 2> 22> 66> 11> D 2, D22, and Dgg. That is, neither shear-extension or bend-twist coupling nor bending-extension coupling exists. Thus, for plate problems, the transverse deflections are described by only one differential equation of equilibrium ... 

Symmetric angle-ply laminates were described in Section 4.3.2 and found to be characterized by a full matrix of extensional stiffnesses as well as bending stiffnesses (but of course no bending-extension coupling stiffnesses because of middle-surface symmetry). The new facet of this type of laminate as opposed to specially orthotropic laminates is the appearance of the bend-twist coupling stiffnesses D. g and D2g (the shear-extension coupling stiffnesses A. g and A2g do not affect the transverse deflection w when the laminate is symmetric). The governing differential equation of equilibrium is... 

Note the presence of the bend-twist coupling stiffnesses in the boundary conditions as well as in the differential et uation. As with the specially orthotropic laminated plate, the simply supported edge boundary condition cannot be further distinguished by the character of the in-plane boundary conditions on u and v because the latter do not appear in any plate problem for a symmetric laminate. 

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. 

The first problem area of the so-called anisotropic analysis will be broken down into two subareas shear-extension coupling and bend-twist coupling. We have already observed for the most complicated laminate in the design philosophy proposed earlier that the A,q and A26 stiffnesses are both zero. There is no shear-extension coupling in the context of that philosophy. However, in contemporary composite structures analyses, it is relatively easy to include the treatment of shear-extension coupling, so you should not be overwhelmed by that behavioral aspect or by the calculation of its influence. 

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-... 

Figure 4.3 Influence of bend-twist coupling on buckling load of a laminate, (a) Variation of buckling load factor with fibre angle 6 for a laminates with varying bend-twist coupling, (b) Selected D-matrix terms for [—61—61+01+6]s laminate.

Figure 4.5 Mode produced by Eqn (4.6) for bend-twist coupled laminate.

Symmetric laminate with no stretching/shearing and no bending/ twisting coupling... 

Plates Type ill are defined as symmetrically laminated plates whose material properties are different in all directions with respect to the axis of the plate. This class of laminates includes plates in which bending-twisting coupling (non-zero D10 and D26 terms) exists. The general equation for plates Type ill under transverse loading is,... 

When there is no coupling between the bending and twisting coefficients of a symmetrically stacked laminate, i.e. Die = D26 = 0, the laminate is referred to as orthotropic. 

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