Symmetric laminate layers

The important point to note from this Example is that in a non-symmetrical laminate the behaviour is very complex. It can be seen that the effect of a simple uniaxial stress, or, is to produce strains and curvatures in all directions. This has relevance in a number of polymer processing situations because unbalanced cooling (for example) can result in layers which have different properties, across a moulding wall thickness. This is effectively a composite laminate structure which is likely to be non-symmetrical and complex behaviour can be expected when loading is applied. 

Symmetric Laminates with Muitipie isotropic Layers... 

Figure 4-15 Unbonded View of a Three-Layered Symmetric Laminate with Isotropic Layers...

Table 4-1 Symmetric Laminate with Six Multiple Isotropic Layers...

Table 4-2 Symmetric Laminate with Five Specially Orthotropic Layers...

A very common special case of symmetric laminates with multiple specially orthotropic layers occurs when the laminae are all of the same thickness and material properties, but have their major principal material... 

The particular cross-ply laminate to be examined [4-10] has three layers, so is symmetric about its middle surface. Thus, no coupling exists between bending and extension. Under the condition N = N and all other loads and moments are zero, the stresses in the (symmetric) outer layers are identical. One outer layer is called the 1-layer and has fibers in the x-direction (see Figure 4-39). The inner layer is called the 2-layer and has fibers in the y-direction. The other outer layer is the 3-layer, but because of symmetry there is no need to refer to it. The cross-ply ratio, M, is, 2, so the thickness of the inner layer is ten times that of each of the outer layers (actually, the inner layer" is ten like-oriented lamina Each lamina is. 005 in (.1270 mm) thick, so the total laminate thickness is. 060 in (1.524 mm). 

Schematic presentation of symmetric three-layered design of B4C/B4C30wt%SiC laminate. 

Morye and Wool [82] used symmetric and non-symmetric stacking sequences with glass/flax hybrid composites and also varied the glass/flax fiber ratio, namely, 100/0, 80/20, 60/40, 40/60, and 0/10. For non-symmetric composites, flexural and impact tests were performed with the top face of the composite being either glass fibers or flax fibers. The mechanical properties of the composites were found to depend on the fiber layer arrangement in the composite, and the non-symmetrical laminates with flax at the loaded top face presented superior performance under flexure or impact. 

Laminates that are symmetric about their geometric midplane require a listing of only the plies located on one side of the plane of symmetry, with a subscript s placed after the closing bracket to denote symmetry. If the symmetric laminate has an odd number of layers then a bar is placed over the last number in the code that represents the layer that is shared by both halves of the laminate. 

A laminate is a bonded stack of laminae with various orientations of principal material directions in the laminae as in Figure 1-9. Note that the fiber orientation of the layers in Figure 1-9 is not symmetric about the middle surface of the laminate. The layers of a laminate are usually bonded together by the same matrix material that is used in the individual laminae. That is, some of the matrix material in a lamina coats the surfaces of a lamina and is used to bond the lamina to its adjacent laminae without the addition of more matrix material. Laminates can be composed of plates of different materials or, in the present context, layers of fiber-reinforced laminae. A laminated circular cylindrical shell can be constructed by winding resin-coated fibers on a removable core structure called a mandrel first with one orientation to the shell axis, then another, and so on until the desired thickness is achieved. 

Stiffnesses for single-layered configurations are treated first to provide a baseline for subsequent discussion. Such stiffnesses should be recognizable in terms of concepts previously encountered by the reader in his study of plates and shells. Next, laminates that are symmetric about their middle surface are discussed and classified. Then, laminates with laminae that are antisymmetrically arranged about their middle surface are described. Finally, laminates with complete lack of middle-surface symmetry, i.e., unsymmetric laminates, are discussed. For all laminates, the question of laminae thicknesses arises. Regular laminates have equal-thickness laminae, and irregular laminates have non-equal-thickness laminae. 

For laminates that are symmetric in both geometry and material properties about the middle surface, the general stiffness equations. Equation (4.24), simplify considerably. That symmetry has the form such that for each pair of equal-thickness laminae (1) both laminae are of the same material properties and principal material direction orientations, i.e., both laminae have the same (Qjjlk and (2) if one lamina is a certain distance above the middle surface, then the other lamina is the same distance below the middle surface. A single layer that straddles the middle surface can be considered a pair of half-thickness laminae that satisfies the symmetry requirement (note that such a lamina is inherently symmetric about the middle surface). ... 

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. 

Thus, for many-layered symmetric angle-ply laminates, the values of Afg, A2g, Dig, and D2g can be quite small when compared to the other Ajj and D j, respectively. 

The general case of a laminate with multiple anisotropic layers symmetrically disposed about the middle surface does not have any stiffness simplifications other than the elimination of the Bjj by virtue of symmetry. The Aig, A2g, Dig, and D2g stiffnesses all exist and do not necessarily go to zero as the number of layers is increased. That is, the Aig stiffness, for example, is derived from the Q matrix in Equation (2.84) for an anisotropic lamina which, of course, has more independent... 

Although the word balanced is ambiguous and not definitive, the common meaning for a balanced laminate is a laminate in which all equal-thickness laminae at angles 0 other than 0° and 90° to the reference axis occur only in 0 pairs. The individual -n O and - 0 layers are not necessarily adjacent to each other. Note also that balanced laminates are required to be symmetric about the laminate middle surface, so there must be two + Q laminae and two - 0 laminae for each 0 pair. The behavioral characteristics of a balanced laminate are that shear-... 

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