Antisymmetric Laminates

Twisting of a Two-Layered Antisymmetric Laminate under Tension (After Ashton, Hatpin, and Petit [4-3])... 

Antisymmetric laminates must have an even number of layers if adjacent laminae also have alternating signs of the principal material property directions with respect to the laminate axes. If adjacent laminae do not have alternating signs, then the number of layers need not be even. 

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,... 

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 ... 

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. 

Table 4-4 Antisymmetric Laminate with Six Specially Orthotropic Layers...

Show that A g = Asg= D g = D2g = 0 for regular antisymmetric laminates wherein each equal-thickness layer Is made of the same material. 

The real point of this example is that the deflection effects just discussed are very significant for numbers of layers that, in our studies of antisymmetric laminates earlier in this chapter, we concluded that bending-extension coupling had disappeared Here, the exact solution exceeds the orthotropic solution by 165% at 6 layers, 165% at 10 layers, 90% at 20 layers, and 30% at 50 layers. Even at 100 layers, the discrepancy is still 11%. These differences are well within the consideration... 

Robert M. Jones, Harold S. Morgan, and James M. Whitney, Buckling and Vibration of Antisymmetrically Laminated Angle-Ply Rectangular Plates, Journal of Applied Mechanics, December 1973, pp. 1143-1144. 

Classical solutions to laminated shell buckling and vibration problems in the manner of Chapter 5 were obtained by Jones and Morgan [6-47]. Their results are presented as normalized buckling loads or fundamental natural frequency versus the Batdorf shell curvature parameter. They showed that, for antisymmetrically laminated cross-ply shells as for plates, the effect of coupling between bending and extension on buckling loads and vibration frequencies dies out rapidly as the number of layers... 

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. 

An antisymmetric cross-ply laminate consists of an even number of orthotropic laminae laid on each other with principal material directions alternating at 0° and 90° to the laminate axes as in the simple example of Figure 4-19. A more complicated example is given in Table 4-4 (where the adjacent layers do not always have the sequence 0°, then 90°, then 0°, etc.). Such laminates do not have A g, Agg, D g, and Dgg, but do have bending-extension coupling. We will show later that the coupling is such that the force and moment resultants are... 

A regular antisymmetric cross-ply laminate is delined to have laminae all of equal thickness and is common because of simplicity of fabrication. As the number of layers increases, the bending-extension coupling stiffness B.,., can be shown to approach zero. 

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. 

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

Derive the extensional stiffnesses for regular antisymmetric special cross-ply laminates, that is, derive Equation (4.82) for the special case in which t jd = Wn =... 

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

Antisymmetric cross-ply laminates were described in Section 4.3.3 and found to have extensional stiffnesses A. , A. 2, A22 = A.. , and Agg bending-extension coupling stiffnesses B., and 822 =-Bn and bending stiffnesses D., D.,2, 822 = and Dgg. The new terms here in comparison to a speciaily orthotro Dic iaminate are B.,and 822- Because of this coupiing, the three equiiibrium differentiai equations are coupied ... 

We want to study the effect of the number of layers on laminate performance. The fair comparison is to keep the total laminate thickness constant to consider only equal-weight laminates. Then, we vary the number of layers by dividing the laminate into more and more iayers. That is, we construct the sequence of antisymmetric cross-ply laminates with an increasing number of layers but of constant thickness as in Figure 5-12. 

Figure 5-13 Deflection of an Antisymmetric Cross-Ply Laminated Plate under Sinusoidal Transverse Load...

Whitney solved the problem for simply supported edge boundary condition S3 [5-13 and 5-14] (recall that S2 was used for antisymmetric cross-ply laminated plates in Section 5.3.3) ... 

---