Specially Orthotropic Laminated Plates

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  

Note that because the in-plane deformations, u and v, are not present in the differential equation, the simply supported edge boundary condition takes on an especially simple form as compared to Equation (5.11). 

If the transverse loading is represented by the Fourier sine series in Equation (5.25), the solution to this fourth-order partial differential equation and subject to its associated boundary conditions is remarkably simple. As with isotropic plates, the solution can easily be verified to be 

That is. Equation (5.29) satisfies the differential equation. Equation (5.27), and the boundary conditions. Equation (5.28), so is the exact solution if 

For a uniform transverse load, the solution is easily shown to be 

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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 solution to the governing differential equation, Equation (5.32), is not as simple as for specially orthotropic laminated plates because of the presence of D. g and D2g. The Fourier expansion of the deflection w. Equation (5.29), is an example of separation of variables. However, because of the terms involving D.,g and D2g, the expansion does not satisfy the governing differential equation because the variables are not separable. Moreover, the deflection expansion also does not satisfy the boundary conditions. Equation (5.33), again because of the terms involving D. g and D2g. 

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. 

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. 

Engineering Sciences Data Unit (ESDU), Natural frequencies of rectangular specially orthotropic laminated plates. Item No. 83036, V4, 1989. 

For plate problems, whether the specially orthotropic laminate has a single layer or multiple layers is essentially immaterial the laminate need only be characterized by 0 2, D22. and Dgg in Equation (5.2). That is, because there is no bending-extension coupling, the force-strain relations, Equation (5.1), are not used in plate analysis for transverse loading causing only bending. However, note that force-strain relations are needed in shell analysis because of the differences between deformation characteristics of plates as opposed to shells. 

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-... 

Obvious and sometimes drastic simplifications occur when the laminate is symmetric about the middle surface (By = 0), specially orthotropic (all the terms with 16 and 26 subscripts vanisn in addition to the By), homogeneous ( = 0 and Djj = Ayr/12), or isotropic. In all those cases. Equations (5.6) and (5.7) are coupled to each other, but uncoupled from Equation (5.8). That is. Equation (5.8) contains derivatives of the transverse displacement w only, and Equations (5.6) and (5.7) contain both u and V but not w. Accordingly, only Equation (5.8) must be solved to determine the transverse deflections of a plate with the aforementioned... 

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. 

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. 

There are several resources available for designing filament-wound cylinders. In general, filament-wound cylinders are classified as cylindrically orthotropic. Adjacent helical plies, loading conditions can be determined by following the principles of laminated plate theory [7]. When applying laminated plate theory, the plate consists of the cylinder wall. In this case, the effect of cylinder curvature is neglected, and the Q and z axes are considered the planar axes of the plate. Failure criteria applied in laminated plate theory, such as maximum stress or strain, or the quadratic Tsai-Wu failure criteria [7] may also be applied. Several specialized loading cases have been studied. 

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