Generally orthotropic laminate

P(3) Laminates that are subject to bending-stretching-torsion coupling are defined as generally orthotropic. Typically, generally orthotropic laminates have a symmetrical and balanced stacking sequence of plies, with an appreciable amount (more than about 20%) of reinforcement orientated at other than 0° and 90° (e.g. +45° or -45°). 

For a ngle generally orthotropic layer of thickness, t, and lamina stiffnesses, Q, given in Equation (2.80), the laminate stiffnesses are... 

Table 4-3 Symmetric Laminate with Five Generally Orthotropic Layers...

The stiffnesses of an antisymmetric laminate of anisotropic laminae do not simplify from those presented in Equations (4.22) and (4.23). However, as a consequence of antisymmetry of material properties of generally orthotropic laminae, but symmetry of their thicknesses, the shear-extension coupling stiffness A.,6,... 

Because o< the existance of bending-extension coupling, the terminology generally orthotropic and specially orthotropic have meaning only with reference to an individual layer and not to a laminate. 

The bending-extension coupling stiffnesses, Bjj, vary for different classes of antisymmetric laminates of generally orthotropic laminae, and, in fact, no general representation exists other than in the following force and moment resultants ... 

Unsymmetric laminates with multiple generally orthotropic layers or with multiple anisotropic layers have force and moment resultants no... 

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.,... 

For many composite applications the laminate is not orthotropic, but anisotropic, though this can exist in many particular forms, one of which occurs when an orthotropic laminate is loaded in a direction which does not coincide with one of the principal axes, or when the lay-up is symmetric but not balanced about a principal reference axis. A [0°/+45°/90°]s laminate is balanced and quasi-isotropic. As a general rule, all laminates should be symmetrically laid up about their mid-plane to avoid stretching bending coupling. 

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. 

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. 

For a unidirectional laminate the elastic stress-strain relations define an orthotropic material for which the generalized form of Hooke s Law, relating the stress o to the strain e,... 

The orthotropic stress and strain relationships of Equations 8.42 and 8.43 were defined in principal material directions, for which there is no coupling between extension and shear behavior. However, the coordinates natural to the solution of the problem generally will not coincide with the principal directions of orthotropy. For example, consider a simply supported beam manufactured from an angle-ply laminate. The principal material coordinates of each ply of the laminate make angles 0 relative to the axis of the beam. In the beam problem stresses and strains are usually defined in the beam coordinate system (jc,y), which is off-axis relative to the lamina principal axes (L, T). 

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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