Orthotropic Plate

Al = the average (membrane) normal stress force-resultant in the L direction per unit length in the T direction, Aj = the average (membrane) normal stress force resultant in the T direction per unit length in the L direction, and Alt= the average (membrane) shear stress force resultant on the L normal plane in the T direction per unit length in the T direction. 

Similarly, bending moment stress resultants are defined as -  

Ml = the bending moment resultant on the L normal edge in the T direction per unit length in the T direction, 

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Specially orthotropic plates, i.e., plates with multiple specially orthotropic layers that are symmetric about the plate middle surface have, as has already been noted in Section 4.3, force and moment resultants in which there is no bending-extension coupling nor any shear-extension or bend-twist coupling, that is,... 

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. 

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. 

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.,... 

Figure 6-6 Effect of Material Properties on Circumferential Stress Gq at the Edge of a Circular Hole in an Orthotropic Plate under a, (After Greszczuk [6-11])...

K. Girkmann and R. Beer, Application of Eric Reissneris Refined Plate Theory to Orthotropic Plates, Osterr. Ingenieur-Archiv., Vol. 12, 1958, pp. 101-110. Robert M. Jones (Translator), Department of Theoretical and Applied Mechanics, University of liiinofs, Urbana, 1962. 

Berbinau P, Soutis C. A new approach for solving mixed boundary value problems along holes in orthotropic plates. Int J SoUds Struct 2001 38(1) 143—59. 

Hyer MW, Klang EC, Cooper DE. The effects of pin elasticity, clearance, and friction on the stresses in a pin-loaded orthotropic plate. J Compos Mater 1987 21(3) 190—206. Naik RA, Crews Jr JH. Stress analysis method for a clearance-fit bolt under bearing loads. AIAA J 1986 24(8) 1348-53. 

P(l) Plates Type 11, a particular type of specially orthotropic plates, are defined as plates whose material properties are such that Di6=D26=0.0 and Dll + D22. 

J. Majak, M. Pohlak, and M. Eerme, Application of the haar wavelet based discretization technique to orthotropic plate and shell problems, Mechanics of Composite Materials, vol. 45, pp. 631 -642, 2009. 

IRSCHIK, H. A Boundary integral equation method for bending of orthotropic plates. Int. J. Solids, Structures 20 (1984),... 

Before we consider a laminated plate, consider a homogeneous orthotropic plate made up of one layer, as shown in Figure 9.7. Unlike the beam the bending stress can now vary in the two directions L and T, or in general in the x and y directions, just like... 

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-... 

Each layer is orthotropic (but the principal material directions of each layer need not be aligned with the plate axes), linear elastic, and of constant thickness (so the entire plate is of constant thickness). 

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. 

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... 

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 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. 

The stiffnesses in Equation (5.39) are equivalent to the stiffnesses of an equivalent orthotropic material with principal material axes of orthotropy at 45° to the plate sides. The orthotropic bending stiffnesses of the equivalent material can be shown to be... 

Because exact solutions for skew isotropic plates are readily available, Ashton was able to get some exact solutions for anisotropic rectangular plates by the special identification process outlined in the preceding paragraph. Specifically, values for the center deflection of a uniformly loaded square plate are shown in Table 5-2. There, the exact solution is shown along with the Rayleigh-Ritz solution and the specially orthotropic solution. For the case already discussed where D22/D11 -1, (D.,2-i-2De6)/Di. = 1.5, and D.,g/Di., = D2 Di., =-.5, the exact solution is... 

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. 

For a specially orthotropic square boron-epoxy plate with stiffness ratios 0 /022= 10 and (Di2-t-2D66) = 1, the four lowest frequencies are displayed in Table 5-3 along with the four lowest frequencies of an isotropic plate. There, the factor k is defined as... 

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