Extensional stiffnesses

Thus the addition of the stiff carbon fibers has a very great effect in stiffening the epoxy matrix. Eor the commonly used fiber volume fraction of 0.6 the unidirectional carbon—epoxy lamina has a predicted extensional stiffness E = 145 GPa (2.1 x 10 psi)-... 

A] is the Extensional Stiffness Matrix although it should be noted that it also contains shear terms. 

Having obtained all the terms for the extensional stiffness matrix [A], this may then be inverted to give the compliance matrix [a]. 

This is called the Extensional Stiffness Matrix and the similarity with that derived earlier for the single ply should be noted. 

Similarly, under suitable constraints, deformation is possible in which only one extensional strain arises or is applied. Again, work is produced by the corresponding stress alone. Thus, because the work done is determined by the diagonal elements of the stiffness matrix, those elements must be positive, that is,... 

Knowledge of the variation of stress and strain through the laminate thickness is essential to the definition of the extensional and bending stiffnesses of a laminate. The laminate is presumed to consist of per-... 

Verify for a single layer of isotropic material with material properties E and v and thickness t that the extensional and bending stiffnesses are... 

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. 

The extensional and bending stiffnesses for the general case are calculated from Equation (4.24) wherein for the k " layer... 

The logic to establish the various stiffnesses will be traced to illustrate the general procedures. First, consider the extensional stiffnesses N... 

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,... 

Quasi-isotroplc laminates do not behave like Isotropic homogeneous materials. Discuss why not, and describe how they do behave. Why is a two-ply laminate with a [0°/90°] sacking sequerx and equat-thickness layers not a quasi-isotropic laminate Determine whether the extensional stiffnesses are the same irrespective of the laminate axes for the two-ply and three-ply cases. Hint use the invariant properties In Equation (2.93). 

For both odd- and even-layered special cross-ply laminates, the extensional stiffnesses, Aj., are independent of N, the number of layers (although the N individual lamina thickneWs can be summed to get the total laminate thickness t, so N is implicit im quations (4.78) and (4.82)). However, A., and A22 depend on M, the cross-ply ratio, and on F, the lamina stiffness ratio, as shown in Figures 4-22 and 4-23. For a typical glass-fiber-reinforced lamina, F =. 3, so A, varies from. 65Q.nt to... 

Figure 4-22 Extensional Stiffness, A, versus Cross-Ply Ratio, M (After Tsai [4-6])...

The extensional stiffnesses, Ajj, are shown in Figure 4-29 as a function of the lamination angle. The terms A12, A22, and Agg are independent of the number of layers, N. However, A g and A26 depend on N. When N is odd, they are inversely proportional to N. When N is even, they are zero. Thus, the biggest values of A.,e and A26 occur when N = 3. 

Derive the extensional stiffnesses for reguiar symmetric speciai cross-ply laminates, that is, derive Equation (4.78) for the special case in which t = t gn = VN. 

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 ... 

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

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