Stiffnesses bending-extension

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

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 ... 

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... 

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). 

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 ... 

We will see that the unsymmetric laminate has more bending stiffness in the y-direction than the all-0°-layer laminate and almost as much bending stiffness in the x-direction. Thus, the center deflection of the unsymmetrically laminated plate should exceed that of the all-0°-layer laminated plate. However, we are already aware that bending-extension coupling increases deflections, so the center deflection of the unsymmetric laminate should exceed that of the orthotropic laminated plate. 

As an aid to understanding the deflection behavior in Figure 5-36, the normalized extensional, bending-extension coupling, and bending stiffnesses are plotted versus the number of layers in Figure 5-. The stiffnesses in the x-direction (with which most fibers are aligned), and... 

The extensional and bending stiffnesses are those of an orthotropic material, but the bending-extension coupling stiffnesses are not all zero (B,6 and 026 remain). 

Accordingly, all the bending-extension coupling stiffnesses, Bg, in Table 7-6 vanish. The Ajj and Dg are those of an anisotropic matenal. 

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