Plates, laminated

FIGURE 9.9 Laminated plate made up of several layers. 

33 are still assumed valid for the laminated plate. However, the stress resultants for arbitrary lamina notation are now defined as 

Similar resnltants for Ny, N, My, and are obtained by snmming integrals of Oy, x y, zOy, aod zx, respectively. Then it can be shown that 

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

the applied external loads to the laminate are known it is more useful to know the average extensional strain e)ni and curvature k, in order to calculate individual ply stresses  

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Classical laminated plate tlieoiy is used to determine the stiffness of laminated composites. Details of the Kitchoff-Love hypothesis on which the theory is based can be found in standard texts (1,7,51). Essentially, the strains in each ply of the laminate ate represented as middle surface strains plus... 

Fig. 13. Schematic of a laminate with the coordinates and ply notation used ia laminated plate theory.

The strength of laminates is usually predicted from a combination of laminated plate theory and a failure criterion for the individual larnina. A general treatment of composite failure criteria is beyond the scope of the present discussion. Broadly, however, composite failure criteria are of two types noninteractive, such as maximum stress or maximum strain, in which the lamina is taken to fail when a critical value of stress or strain is reached parallel or transverse to the fibers in tension, compression, or shear or interactive, such as the Tsai-Hill or Tsai-Wu (1,7) type, in which failure is taken to be when some combination of stresses occurs. Generally, the ply materials do not have the same strengths in tension and compression, so that five-ply strengths must be deterrnined ... 

Actually, because of the stress and deformation hypotheses that are an inseparable part of classical lamination theory, a more correct name would be classical thin lamination theory, or even classical laminated plate theory. We wiiruS ffi bmmon term classical lamination theory, but recognize that it is a convenient oversimplification of the rigorous nomenclature. In the composite materials literature, classical laminationtheoryls en abbreviated as CLT. 

Laminated plates are one of the simplest and most widespread practical applications of composite laminates. Laminated beams are, of course, simpler. However, such essentially one-dimensional structural elements do not display well the unique two-dimensional capabilities and characteristics of composite laminates. 

Figure 5-1 Basic Questions of Laminated Plate Analysis...

A simply supported rectangular plate is used consistently in all sections to illustrate the kinds of results that can be obtained, i.e., the influence of the various stiffnesses on laminated plate behavior. In addition, only the simplest types of loading will be studied in order to avoid the solution difficulties inherent to complex loadings. Accordingly, in the interest of simplicity, just the bare thread of laminated plate results will be displayed. 

Note that no assumptions involve fiber-reinforced composite materials explicitly. Instead, only the restriction to orthotropic materials at various orientations is significant because we treat the macroscopic behavior of an individual orthotropic (easily extended to anisotropic) lamina. Therefore, what follows is essentially a classical plate theory for laminated materials. Actually, interlaminar stresses cannot be entirely disregarded in laminated plates, but this refinement will not be treated in this book other than what was studied in Section 4.6. Transverse shear effects away from the edges will be addressed briefly in Section 6.6. 

Boundary conditions used to be thought of as a choice between simply supported, clamped, or free edges if all classes of elastically restrained edges are neglected. The real situation for laminated plates is more complex than for isotropic plates because now there are actually four types of boundary conditions that can be called simply supported edges. These more complicated boundary conditions arise because now we must consider u, v, and w instead of just w alone. Similarly, there are four kinds of clamped edges. These boundary conditions can be concisely described as a displacement or derivative of a displacement or, alternatively, a force or moment is equal to some prescribed value (often zero) denoted by an overbar at the edge ... 

DEFLECTION OF SIMPLY SUPPORTED LAMINATED PLATES UNDER DISTRIBUTED TRANSVERSE LOAD... 

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. 

Figure 5-13 Deflection of an Antisymmetric Cross-Ply Laminated Plate under Sinusoidal Transverse Load...

Whitney solved the problem for simply supported edge boundary condition S3 [5-13 and 5-14] (recall that S2 was used for antisymmetric cross-ply laminated plates in Section 5.3.3) ... 

Figure 5-22 Buckling of Rectangular Specially Onholropic Laminated Plates under Uniform Compression, N,...

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