Coupling bending-extension

Which are commonly called B and D, respectively, in ordinary isotropic plate theory. What are the bending-extension coupling stiffnesses ... 

Derive the summation expressions for extensional, bending-extension coupling, and bending stiffnesses for laminates with constant properties in each orthotropic lamina that is, derive Equation (4.24) from Equations (4.20) and (4.21). 

Determine the extensional, bending-extension coupling, and bending stiffnesses of an equal-thickness bimetallic strip as shown in Figure 1-3 (a beam made of two different isotropic materials with E, v, a, E2, V2, and 02). Use the middle surface of the beam as the reference surface. 

Second, consider the bending-extension coupling stiffnesses... 

Antisymmetry of a laminate requires (1) symmetry about the middle surface of geometry (i.e., consider a pair of equal-thickness laminae, one some distance above the middle surface and the other the same distance below the middle surface), but (2i some kind of a reversal or mirror image of the material properties [Qjjlk- In fact, the orthotropic material properties [Qjj], are symmetric, but the orientations of the laminae principal material directions are not symmetric about the middle surface. Those orientations are reversed from 0° to 90° (or vice versa) or from + a to - a (a mirror image about the laminate x-axis). Because the [Qjj]k are not symmetric, bending-extension coupling exists. 

Because o< the existance of bending-extension coupling, the terminology generally orthotropic and specially orthotropic have meaning only with reference to an individual layer and not to a laminate. 

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 ... 

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. 

The bending-extension coupling stiffnesses B g and Bgg can be shown to go to zero as the number of layers in the laminate increases for a fixed laminate thickness. 

Bending-Extension Coupling Stiffness, versus Cross-Ply Ratio, M (After Tsai [4-6])... 

The bending-extension coupling stiffnesses, By, are zero for an odd number of layers, but can be large for an even number of layers. The values of B e/(tAii) are shown as a function of lamination angle in Figure 4-30. Because B e is inversely proportional to N, the largest value of B e occurs when N = 2. The quantity plotted can be shown to be... 

From Figure 4-30, bending-extension coupling is largest when 9 = 45 and N = 2. 

Derive the bending-extension coupling stiffnesses for regular special antisymmetric cross-ply laminates, that is, derive Equation (4.83) for the special case in which odd = v n = f/N (for which also M = 1). 

Specially orthotropic plates, i.e., plates with multiple specially orthotropic layers that are symmetric about the plate middle surface have, as has already been noted in Section 4.3, force and moment resultants in which there is no bending-extension coupling nor any shear-extension or bend-twist coupling, that is,... 

For plate problems, whether the specially orthotropic laminate has a single layer or multiple layers is essentially immaterial the laminate need only be characterized by 0 2, D22. and Dgg in Equation (5.2). That is, because there is no bending-extension coupling, the force-strain relations, Equation (5.1), are not used in plate analysis for transverse loading causing only bending. However, note that force-strain relations are needed in shell analysis because of the differences between deformation characteristics of plates as opposed to shells. 

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-... 

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 ... 

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 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 ... 

Results for a square plate under sinusoidal transverse load with a variable modulus ratio, E1/E2, and a 45° lamination angle are shown in Figure 5-17. There, the effect of bending-extension coupling on deflections is significant for all modulus ratios except those quite close to Ei/E2=1. 

The buckling load will be determined for plates with various laminations specially orthotropic, symmetric angle-ply, antisymmetric cross-ply, and antisymmetric angle-ply. The results for the different lamination types will be compared to find the influence of bend-twist coupling and bending-extension coupling. As with the deflection problems in Section 5.3, different simply supported edge boundary conditions will be used in the several problems addressed for convenience of illustration. 

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