Principal material directions

A laminate is a bonded stack of laminae with various orientations of principal material directions in the laminae as in Figure 1-9. Note that the fiber orientation of the layers in Figure 1-9 is not symmetric about the middle surface of the laminate. The layers of a laminate are usually bonded together by the same matrix material that is used in the individual laminae. That is, some of the matrix material in a lamina coats the surfaces of a lamina and is used to bond the lamina to its adjacent laminae without the addition of more matrix material. Laminates can be composed of plates of different materials or, in the present context, layers of fiber-reinforced laminae. A laminated circular cylindrical shell can be constructed by winding resin-coated fibers on a removable core structure called a mandrel first with one orientation to the shell axis, then another, and so on until the desired thickness is achieved. 

If there are two orthogonal planes of material property symmetry for a material, symmetry will exist relative to a third mutually orthogonal plane. The stress-strain relations in coordinates aligned with principal material directions are... 

Principal material directions (PMO) are directions that are parallel to the Intersections of the three orthogonal planes of material property symmetry. Principal material coordinates (PMC) are the set of axes In principal material directions. 

Compare the transformed orthotropic compliances in Equation (2.88) with the anisotropic compliances in terms of engineering constants in Equation (2.91). Obviously an apparenf shear-extension coupling coefficient results when an orthotropic lamina is stressed in non-principal material coordinates. Redesignate the coordinates 1 and 2 in Equation (2.90) as X and y because, by definition, an anisotropic material has no principal material directions. Then, substitute the redesignated Sy from Equation (2.91) in Equation (2.88) along with the orthotropic compliances in Equation (2.62). Finally, the apparent engineering constants for an orthotropic iamina that is stressed in non-principal x-y coordinates are... 

An important implication of the presence of the shear-extension coupling coefficient is that off-axis (non-principal material direction) tensile loadings for composite materials result in shear deformation in addition to the usual axial extension. This subject is investigated further in Section 2.8. At this point, recognize that Equation (2.97) is a quantification of the foregoing implication for tensile tests and of the qualitative observations made in Section 1.2. 

The apparent anisotropic moduli for an orthotropic lamina stressed at an angle 6 to the principal material directions vary with 6 as in Equation... 

That Is, show that an orthotropic material can have an apparent Young s modulus that either exceeds or is less than the Young s moduli in both principal material directions. In doing so, derive the conditions for which each type of behavior exists, i.e., derive the inequalities. Plot E E, for some contrived materials that exemplify these relations. 

What has been accomplished in preceding sections on stiffness relationships serves as the basis for determination of the actual stress field what remains is the definition of the allowable stress field. The first step in such a definition is the establishment of allowable stresses or strengths in the principal material directions. Such information is basic to the study of strength of an orthotropic lamina. 

Determination of the fourth-rank tensor term F. 2 remains. Basically, F.,2 cannot be found from any uniaxial test in the principal material directions. Instead, a biaxial test must be used. This fact should not be surprising because F-,2 is the coefficient of the product of a. and 02 in the failure criterion. Equation (2.140). Thus, for example, we can impose a state of biaxial tension described by a, = C2 = c and all other stresses are zero. Accordingly, from Equation (2.140),... 

Find the Tsai-Hill failure criterion for pure shear loading at various angles B to the principal material directions, i.e., the shear analog of Equation (2.134). 

In conclusion, classical lamination theory enables us to calculate forces and moments if we know the strains and curvatures of the middle surface (or vice versa). Then, we can calculate the laminae stresses in laminate coordinates. Next, we can transform the laminae stresses from laminate coordinates to lamina principal material directions. Finally, we would expect to apply a failure criterion to each lamina in its own principal material directions. This process seems straightfonward in principle, but the force-strain-curvature and moment-strain-curvature relations in Equations (4.22) and (4.23) are difficult to completely understand. Thus, we attempt some simplifications in the next section in order to enhance our understanding of classical lamination theory. 

This section is devoted to those special cases of laminates for which the stiffnesses take on certain simplified values as opposed to the general form in Equation (4.24). The general force-moment-strain-curvature relations in Equations (4.22) and (4,23) are far too comprehensive to easily understand. Thus, we build up our understanding of laminate behavior from the simplest cases to more complicated cases. Some of the cases are almost trivial, others are more specialized, some do not occur often in practice, but the point is that all are contributions to the understanding of the concept of laminate stiffnesses. Many of the cases result from the common practice of constructing laminates from laminae that have the same material properties and thickness, but have different orientations of the principal material directions relative to one another and relative to the laminate axes. Other more general cases are examined as well. 

For laminates that are symmetric in both geometry and material properties about the middle surface, the general stiffness equations. Equation (4.24), simplify considerably. That symmetry has the form such that for each pair of equal-thickness laminae (1) both laminae are of the same material properties and principal material direction orientations, i.e., both laminae have the same (Qjjlk and (2) if one lamina is a certain distance above the middle surface, then the other lamina is the same distance below the middle surface. A single layer that straddles the middle surface can be considered a pair of half-thickness laminae that satisfies the symmetry requirement (note that such a lamina is inherently symmetric about the middle surface). ... 

Because of the analytical complications involving the stiffnesses Ai6, A26, D g, and D26, a laminate is sometimes desired that does not have these stiffnesses. Laminates can be made with orthotropic layers that have principal material directions aligned with the laminate axes. If the thicknesses, locations, and material properties of the laminae are symmetric about the middle surface of the laminate, there is no coupling between bending and extension. A general example is shown in Table 4-2. Note that the material property symmetry requires equal [Q j], of the two layers that are placed at the same distance above and below the middle surface. Thus, both the orthotropic material properties, [Qjjlk. of the layers and the angle of the principal material directions to the laminate axes (i.e., the orientation of each layer) must be identical. 

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. 

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... 

Consider two laminae with principal material directions at -t- a and - a with respect to a reference axis. Prove that fr orthotropic materials... 

A cross-ply laminate in this section has N unidirectionally reinforced thotropic) layers of the same material with principal material directions srnatingly oriented at 0° and 90° to the laminate coordinate axes. The sr direction of odd-numbered layers is the x-direction of the laminate, e fiber direction of even-numbered layers is then the y-direction of the linate. Consider the special case of odd-numbered layers with equal kness and even-numbered layers also with equal thickness, but not essarily the same thickness as that of the odd-numbered layers, te that we have imposed very special requirements on how the fiber sntations change from layer to layer and on the thicknesses of the ers to define a special subclass of cross-ply laminates. Thus, these linates are termed special cross-ply laminates and will be explored his subsection. More general cross-ply laminates have no such con-ons on fiber orientation and laminae thicknesses. For example, a neral) cross-ply laminate could be described with the specification t/90° 2t/90° 2t/0° t] wherein the fiber orientations do not alter-e and the thicknesses of the odd- or even-numbered layers are not same however, this laminate is clearly a symmetric cross-ply lami-e. 

If no laminae have failed, the load must be determined at which the first lamina fails (so-called first-ply failure), that is, violates the lamina failure criterion. In the process of this determination, the laminae stresses must be found as a function of the unknown magnitude of loads first in the laminate coordinates and then in the principal material directions. The proportions of load (i.e., the ratios of to Ny, to My,/ etc.) are, of course, specified at the beginning of the analysik The loaa parameter is increased until some individual lamina fails. The properties, of the failed lamina are then degraded in one of two ways (1) totally to zero if the fibers in the lamina fail or (2) to fiber-direction properties if the failure is by cracking parallel to the fibers (matrix failure). Actually, because of the matrix manipulations involved in the analysis, the failed lamina properties must not be zero, but rather effectively zero values in order to avoid a singular matrix that could not be inverted in the structural analysis problem. The laminate strains are calculated from the known load and the stiffnesses prior to failure of a lamina. The laminate deformations just after failure of a lamina are discussed later. 

The analysis of such a laminate by use of classical lamination theory revolves about the stress-strain relations of an individual orthotropic lamina under a state of plane stress in principal material directions... 

Rather than a plane-stress state, a three-dimensional stress state is considered in the elasticity approach of Pipes and Pagano [4-12] to the problem of Section 4.6.1. The stress-strain relations for each orthotropic layer in principal material directions are... 

Each layer is orthotropic (but the principal material directions of each layer need not be aligned with the plate axes), linear elastic, and of constant thickness (so the entire plate is of constant thickness). 

Pagano studied cylindrical bending of symmetric cross-ply laminated composite plates [6-21]. Each layer is orthotropic and has principal material directions aligned with the plate axes. The plate is infinitely long in the y-direction (see Figure 6-16). When subjected to a transverse load, p(x), that is, p is independent of y, the plate deforms into a cylinder ... 

The treatment of transverse shear stress effects in plates made of isotropic materials stems from the classical papers by Reissner [6-26] and Mindlin [6-27. Extension of Reissner s theory to plates made of orthotropic materials is due to Girkmann and Beer [6-28], Ambartsumyan [6-29] treated symmetrically laminated plates with orthotropic laminae having their principal material directions aligned with the plate axes. Whitney [6-30] extended Ambartsumyan s analysis to symmetrically laminated plates with orthotropic laminae of arbitrary orientation. 

The basic approaches as summarized by Ashton and Whitney [6-31] will now be discussed. First, a symmetric laminate with orthotropic laminae having principal material directions aligned with the plate axes will be treated. The transverse normal strain can be found from the orthotropic stress-strain relations, Equation (2.15), as... 

The topic of invariant transformed reduced stiffnesses of orthotropic and anisotropic laminae was introduced in Section 2.7. There, the rearrangement of stiffness transformation equations by Tsai and Pagano [7-16 and 7-17] was shown to be quite advantageous. In particular, certain invariant components of the lamina stiffnesses become apparent and are heipful in determining how the iamina stiffnesses change with transformation to non-principal material directions that are essential for a laminate. 

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