Coupling stiffnesses

Bearing Load and Cross-Coupling Stiffness Measurements... 

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

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

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 ... 

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. 

Prove that the bending-extenslon coupling stiffnesses. By, are zero for laminates that are symmetric in both material properties and geometry about the middle surface. 

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). 

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... 

Note the presence of the bend-twist coupling stiffnesses in the boundary conditions as well as in the differential et uation. As with the specially orthotropic laminated plate, the simply supported edge boundary condition cannot be further distinguished by the character of the in-plane boundary conditions on u and v because the latter do not appear in any plate problem for a symmetric laminate. 

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 ... 

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. 

Whatjs the value of the bending-extension coupling stiffness invariants... 

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