Symmetric laminate stiffnesses

The in-plane stiffness behaviour of symmetric laminates may be analysed as follows. The plies in a laminate are all securely bonded together so that when the laminate is subjected to a force in the plane of the laminate, all the plies deform by the same amount. Hence, the strain is the same in every ply but because the modulus of each ply is different, the stresses are not the same. This is illustrated in Fig. 3.19. 

Thus the stiffness matrix for a symmetric laminate may be obtained by adding, in proportion to the ply thickness, the corresponding terms in the stiffness matrix for each of the plies. 

Summary of Steps to Predict Stiffness of Symmetric Laminates... 

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 influence of the orientation of the laminae on the stiffness of the composite is illustrated in Figure 15.15b, where generic stress-strain curves for unidirectional cross-ply random laminates are shown. In the design of laminates it is necessary to define not only the orientation of the plies but also the stacking sequence, i.e., the order in which the plies are placed through the thickness. Figure 15.16 shows examples of symmetrical and non-symmetrical laminates. The most standard ply orientations are 0°,... 

The temperature-dependent Euler buckling load Pg(T) depended on the temperature-dependent bending stiffness EI(T), as demonstrated by Eq. (7.12). Owing to a symmetric laminate architecture, the neutral axis was always located at mid-depth, and therefore, the moment of inertia, I, was temperature-independent and the specimen stiffness could be determined as ... 

A laminate in which the stacking sequence below the midplane is a mirror image of the sequence above the midplane is called a symmetric laminate. For symmetric laminates the coupling terms of its stiffness matrix. By, vanish. 

There is an important group of laminates that exhibit in-plane isotropic elastic response. These laminates are called quasi-isotropic. This group includes all symmetric laminates with IN (N > 2) laminae with the same thickness and N equal angles between fibre orientations (A0 = nIN), i.e. = 60° for N = 3, A6 = 45° for N = 4, Ad = 30° for = 6 and so on. It is possible to prove that the in-plane stiffness or extensional matrix of quasi-isotropic laminates is given in reduced form as [14]... 

For symmetric laminates it is possible to define effective in-plane moduli in terms of the in-plane stiffness or extensional compliance matrix, since there is no coupling between in-plane and bending response. The effective... 

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. 

The aforementioned coupling that involves Aig, Ags, Dig, and D2g takes on a special form for symmetric angle-ply laminates. Those stiffnesses can be shown to be largest when N = 3 (the lowest N for which this class of laminates exists) and decrease in proportion to 1/N as N increases (see Section 4.4.4). Actually, in the expressions for the extensional and bending stiffnesses Aig and Dig,... 

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... 

Prove that the bending-extenslon coupling stiffnesses. By, are zero for laminates that are symmetric in both material properties and geometry about the middle surface. 

Derive the extensional stiffnesses for reguiar symmetric speciai cross-ply laminates, that is, derive Equation (4.78) for the special case in which t = t gn = VN. 

Derive the stiffnesses for symmetric special angle-ply laminates In Equations (4.91)-(4.93). 

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-... 

Symmetric angle-ply laminates were described in Section 4.3.2 and found to be characterized by a full matrix of extensional stiffnesses as well as bending stiffnesses (but of course no bending-extension coupling stiffnesses because of middle-surface symmetry). The new facet of this type of laminate as opposed to specially orthotropic laminates is the appearance of the bend-twist coupling stiffnesses D. g and D2g (the shear-extension coupling stiffnesses A. g and A2g do not affect the transverse deflection w when the laminate is symmetric). The governing differential equation of equilibrium is... 

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