Symmetric laminate bending

The previous section has illustrated a simple convenient means of analysing in-plane loading of symmetric laminates. Many laminates are of this type and so this approach is justified. However, there are also many situations where other types of loading (including bending) are applied to laminates which may be symmetric or non-symmetric. In order to deal with these situations it is necessary to adopt a more general type of analysis. 

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. 

Whitney and Pagano [6-32] extended Yang, Norris, and Stavsky s work [6-33] to the treatment of coupling between bending and extension. Whitney uses a higher order stress theory to obtain improved predictions of a, and and displacements at low width-to-thickness ratios [6-34], Meissner used his variational theorem to derive a consistent set of equations for inclusion of transverse shearing deformation effects in symmetrically laminated plates [6-35]. Finally, Ambartsumyan extended his treatment of transverse shearing deformation effects from plates to shells [6-36]. 

Equations (6.17) and (6.21) describe the response of a laminate when the stretching and bending behaviors are decoupled. This would be flie case of symmetric laminates where plies symmetrically located with respect to the mid-plane of the laminate have the same material, thickness, and orientation. For general laminates, however, membrane and bending behaviors are coupled. In such a case, the inplane forces IV,... 

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

Ply orientations in a laminate are taken with reference to a particular loading direction, usually taken to be the direction of the maximum applied load, which, more often than not, coincides with the fibre direction to sustain the maximum load, and this is defined as the 0° direction. It is usual to choose balanced, symmetric laminates in design. A balanced laminate is one in which there are equal numbers of-1-0 and - 0 plies a symmetric laminate is one in which the plies are symmetric in terms of geometry and properties with respect to the laminate mid-plane. Hence, a laminate with a stacking sequence 0/90/-I-45/-45/-45/-I-45/90/0, which is written (0/90/ 45), is both balanced and symmetric. Balanced, symmetric laminates have a simplicity of response. In contrast, an unbalanced, asymmetric laminate will, in general, shear, bend, and twist under a simple axial loading. 

The temperature-dependent Euler buckling load Pg(T) depended on the temperature-dependent bending stiffness EI(T), as demonstrated by Eq. (7.12). Owing to a symmetric laminate architecture, the neutral axis was always located at mid-depth, and therefore, the moment of inertia, I, was temperature-independent and the specimen stiffness could be determined as ... 

The laminated plates discussed in this code are symmetric laminates, there is no extension/bending coupling, hence By=0, and classical thin plate theory can be used. Membrane action is not considered. 

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,... 

In the case of symmetric laminate, the in-plane deformations become uncoupled from bending, i.e. [S] = 0. The fundamental equation of classical lamination theory reduces to... 

For symmetric laminates it is possible to define effective in-plane moduli in terms of the in-plane stiffness or extensional compliance matrix, since there is no coupling between in-plane and bending response. The effective... 

Similarly, the flexural elastic moduli of symmetric laminates are readily obtained from the bending compliance matrix. 

For a symmetric laminate, symmetric about the z = 0 plane, the By = 0, and there is no coupling between extension and bending. 

Because of the analytical complications involving the stiffnesses Ai6, A26, D g, and D26, a laminate is sometimes desired that does not have these stiffnesses. Laminates can be made with orthotropic layers that have principal material directions aligned with the laminate axes. If the thicknesses, locations, and material properties of the laminae are symmetric about the middle surface of the laminate, there is no coupling between bending and extension. A general example is shown in Table 4-2. Note that the material property symmetry requires equal [Q j], of the two layers that are placed at the same distance above and below the middle surface. Thus, both the orthotropic material properties, [Qjjlk. of the layers and the angle of the principal material directions to the laminate axes (i.e., the orientation of each layer) must be identical. 

The aforementioned coupling that involves Aig, Ags, Dig, and D2g takes on a special form for symmetric angle-ply laminates. Those stiffnesses can be shown to be largest when N = 3 (the lowest N for which this class of laminates exists) and decrease in proportion to 1/N as N increases (see Section 4.4.4). Actually, in the expressions for the extensional and bending stiffnesses Aig and Dig,... 

Antisymmetry of a laminate requires (1) symmetry about the middle surface of geometry (i.e., consider a pair of equal-thickness laminae, one some distance above the middle surface and the other the same distance below the middle surface), but (2i some kind of a reversal or mirror image of the material properties [Qjjlk- In fact, the orthotropic material properties [Qjj], are symmetric, but the orientations of the laminae principal material directions are not symmetric about the middle surface. Those orientations are reversed from 0° to 90° (or vice versa) or from + a to - a (a mirror image about the laminate x-axis). Because the [Qjj]k are not symmetric, bending-extension coupling exists. 

Prove that the bending-extenslon coupling stiffnesses. By, are zero for laminates that are symmetric in both material properties and geometry about the middle surface. 

Derive the bending stiffnesses for regular symmetric special cross-ply laminates, that is, derive Equation (4.80) for the special case in which t = t = VN. 

The particular cross-ply laminate to be examined [4-10] has three layers, so is symmetric about its middle surface. Thus, no coupling exists between bending and extension. Under the condition N = N and all other loads and moments are zero, the stresses in the (symmetric) outer layers are identical. One outer layer is called the 1-layer and has fibers in the x-direction (see Figure 4-39). The inner layer is called the 2-layer and has fibers in the y-direction. The other outer layer is the 3-layer, but because of symmetry there is no need to refer to it. The cross-ply ratio, M, is, 2, so the thickness of the inner layer is ten times that of each of the outer layers (actually, the inner layer" is ten like-oriented lamina Each lamina is. 005 in (.1270 mm) thick, so the total laminate thickness is. 060 in (1.524 mm). 

Consider an angle-ply laminate composed of orthotropic laminae that are symmetrically arranged about the middle surface as shown in Figure 4-48. Because of the symmetry of both material properties and geometry, there is no coupling between bending and extension. That is, the laminate in Figure 4-48 can be subjected to and will only extend in the x-direction and contract in the y- and z-directions, but will not bend. 

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... 

Thus, the error from ignoring the bend-twist coupiing terms is about 24%, certainly not a negligible error. Hence, the specially orthotropic laminated plate is an unacceptable approximation to a symmetric angle-ply laminated plate. Recognize, however, that Ashton s Rayleigh-Ritz results are also approximate because only a finite number of terms were used in the deflection approximation. Thus, a comparison of his results with an exact solution would lend more confidence to the rejection of the specially orthotropic laminated plate approximation. 

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