Laminated plates bending

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-... 

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. 

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. 

We will see that the unsymmetric laminate has more bending stiffness in the y-direction than the all-0°-layer laminate and almost as much bending stiffness in the x-direction. Thus, the center deflection of the unsymmetrically laminated plate should exceed that of the all-0°-layer laminated plate. However, we are already aware that bending-extension coupling increases deflections, so the center deflection of the unsymmetric laminate should exceed that of the orthotropic laminated plate. 

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.,... 

J. M. Whitney, Bending-Extension Coupling in Laminated Plates Under Transverse Loading, Journal of Composite Materials, January 1969, pp. 20-28. 

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]. 

Bartholomew P. Ply stacking sequences for laminated plates having in-plane and bending orthotropy. Fibre Sci Technol 1977 10(4) 239—53. 

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,... 

Engineering Sciences Data Unit (ESDU), Thickness selection for laminated plates subjected to in-plane loads and bending moments. Item No. 92033, S14, 1992. 

Once the orientations of a laminate are specified, the stacking sequence will control the flexural rigidities of a laminated plate. Thus, the stability, vibration, and static bending behavior are controlled by the relative dispersion and thickness distribution of these oriented layers. 

Before we consider laminated plates with several layers, consider a laminated composite beam of the sandwich type shown in Figure 9.6a. Let the heights (thicknesses) of the layers be denoted by and h2 as shown, where h, is the thickness of the faces and is the thickness of the core. Assume pure bending and assume that the bending strain is linear, that is, that planes remain plane as shown in Figure 9.6b. For linear elastic behavior... 

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... 

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