Angle-ply laminates

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,... 

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 purpose of the remainder of this section is to discuss two important classes of antisymmetric laminates, the antisymmetric cross-ply laminate and the antisymmetric angle-ply laminate. Neither laminate is used much in practice, but both add to our understanding of laminates. 

A regular antisymmetric angle-ply laminate has laminae all of the same material and thickness for ease of fabrication. This class of laminates can be further restricted to have a single value of a as opposed to the several orientations, materials, and thicknesses in Table 4-5. 

The force and moment resultants for an antisymmetric angle-ply laminate are... 

Discuss wfhether this relation is valid for anisotropic materials. That is, denwistrate whether a a angle-ply laminate of the same anisotropic laminae that are symmetric geometrically is antisymmetric or not. The transformation equations for anisotropic materials are given in Section 2.7. 

Show that B g and B g for an antisymmetric angle-ply laminate with equal-thickness layers of the same material approach zero as the even number of (equal-thickness) layers increases if the total laminate thickness is held constant. What happens if equal-thickness layers are added so the total laminate thickness increases In both cases, develop equations for B g and B2g that you can study, modify, and use to determine your answers. 

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,... 

Special Angle-Ply Laminates with N Even (Antisymmetric)... 

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

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. 

Figure 4-45 Strength of Angle-Ply Laminates (After Tsai [4-10])...

For angle-ply laminates, no such knee or change in slope occurs in the load-deformation behavior. Simultaneous failure (fracture) of all layers occurs. 

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. 

Figure 4-48 Symmetric Angle-Ply Laminate Geometry and Stresses (After Pipes and Pagano [4-12])...

Consider an angle-ply laminate composed of orthotropic laminae that are symmetrically arranged about the middle surface as shown in Figure 4-48. Because of the symmetry of both material properties and geometry, there is no coupling between bending and extension. That is, the laminate in Figure 4-48 can be subjected to and will only extend in the x-direction and contract in the y- and z-directions, but will not bend. 

Figure 4-50 CLT Approach to Analysis of a Two-Layered Angle-Ply Laminate...

Figure 4-51 Free-Body Diagram for an Angle-Ply Laminate 4.6.2 Elasticity Formulation...

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