Angle-ply laminate stiffnesses

Figure 4-32a Theoretical and Measured Special Angle-Ply Laminate Stiffnesses (U. S. Standard Units) (After Tsai [4-6])...

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,... 

In preceding sections, laminate stiffnesses were predicted on the basis of combination of lamina stiffnesses in accordance with classical lamination theory. However, the actual, practical realization of those laminate stiffnesses remains to be demonstrated. The purpose of this section is to compare predicted laminate stiffnesses with measured laminate stiffnesses to determine the validity of classical lamination theory. Results for two types of laminates, cross-ply and angle-ply laminates, are presented. 

The laminate behavior of these special angle-ply laminates can be described with the number of layers, N, the laminae orientation, a, and the laminae stiffnesses, Q , in addition to the total laminate thickness, t. The laminate stiffnesses,... 

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

Derive the stiffnesses for antisymmetric special angle-ply laminates in Equations (4.94) - (4.96). 

Angle-ply laminates have more complicated stiffness matrices than cross-ply laminates because nontrivial coordinate transformations are involved. However, the behavior of simple angle-ply laminates (only one angle, i.e., a) will be shown to be simpler than that of cross-ply laminates because no knee results in the load-deformation diagram under uniaxial loading. Other than the preceding two differences, analysis of angle-ply laminates is conceptually the same as that of cross-ply laminates. 

For two- and three-layered cross-ply and angle-ply laminates of E-glass-epoxy, Tsai [4-10] tabulates all the stiffnesses, inverse stiffnesses, thermal forces and moments, etc. Results are obtained for various cross-ply ratios and lamination angles, as appropriate, from a short computer program that could be used for other materials. 

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... 

Antisymmetric angle-ply laminates were described in Section 4.3.3 and found to have extensional stiffnesses A i,A.,2, A22, and Agg bending-extension coupling stiffnesses B. g and 625 and bending stiffnesses D.. , D. 2, D22. and Dgg. Thus, this laminate exhibits a different type of bending-extension coupling than does the antisymmetric cross-ply laminate. The coupled governing differential equations of equilibrium are... 

Antisymmetric angle-ply laminates were found in Section 4.3.3 to have extensional stiffnesses bending-extension... 

Strength calculations on angle-ply laminates are based upon the elastic analysis described in 8.4.2. The strains produced in the laminate by a given set of applied stresses are first calculated, using the computed laminate stiffness constants. Stresses corresponding to these strains are then calculated for each layer of the laminate. These stresses are then expressed in terms of stresses parallel to and normal to the fibres, and the combination of stresses is compared with strength criteria for unidirectional material, which should preferably be obtained by experiment. Results for a three-direction carbon fibre-epoxy composite (0, 0) under tensile stress in the 0 = O direction are shown in Fig. 8.23. The composite contains 60 vol % fibre R is the fraction of plies which are angled at 0 and 1 — R is the fraction at W = 0. 

The expressions for the strain energy release rate associated with local delaminations growing from the tips of angle ply matrix cracks in orthotropic laminates loaded in tension are presented. Strain energy release rate and the laminate residual stiffness properties are predicted as functions of matrix crack density and delamination length. 

Equations (6.9) or (6.4) and (6.10) provide the stiffnesses of a ply with hbers oriented at an angle 6 in the laminate coordinate system.The above discussion concentrated on plane stress conditions. In a three-dimensional situation, a 0° ply is dehned with nine stiffness parameters En, E22, E33, G12, G13, G23, J i2, 13, and i 23. Here, as before, 1 denotes the direction parallel to the hbers, 2 is perpendicular to the hbers, and 3 is the out-of-plane direction. Eor orthotropic plies, E22 = E33,... 

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