Laminate force

First, the stress-strain behavior of an individual lamina is reviewed in Section 4.2.1, and expressed in equation form for the k " lamina of a laminate. Then, the variations of stress and strain through the thicyiess of the laminate are determined in Section 4.2.2. Finally, the relation of the laminate forces and moments to the strains and curvatures is found in Section 4.2.3 where the laminate stiffnesses are the link from the... 

The laminate forces and moments are obtained by integrating the stresses of each lamina through the total laminate thickness to obtain a force vector, N, and a moment vector, M, the components of which can be related to the strains, and and curvatures, x, of the laminae, where the superscript indicates that tlie strains are measured... 

Vacuum bag molding is a modification of hand lay-up, in which the layup is completed and placed inside a bag made of flexible film, and all edges are sealed. The bag is then evacuated, so that the pressure eliminates voids in the laminate, forcing excess air and resin from the mold. By increasing external pressure, a higher glass concentration can be obtained, as well as better adhesion between the layers/plies of laminate. Some items for the process can be disposable. 

PBI is being marketed as a replacement for asbestos and as a high temperature filtration fabric with exceUent textile apparel properties. The synthesis of whoUy aromatic polybenzimidazoles with improved thermal stabUities was reported in 1961 (12). The Non-MetaUic Materials and Manufacturing Technology Division of the U.S. Air Force Materials Laboratory, Wright-Patterson Air Force Base, awarded a contract to the Narmco Research and Development Division of the Whittaker Corp. for development of these materials into high temperature adhesives and laminates. 

The in-plane stiffness behaviour of symmetric laminates may be analysed as follows. The plies in a laminate are all securely bonded together so that when the laminate is subjected to a force in the plane of the laminate, all the plies deform by the same amount. Hence, the strain is the same in every ply but because the modulus of each ply is different, the stresses are not the same. This is illustrated in Fig. 3.19. 

The preceding stress-strain and strain-stress relations are the basis for stiffness and stress analysis of an individual lamina subjected to forces in its own plane. Thus, the relations are indispensable in laminate analysis. 

The fundamental analysis of a laminate can be explained, in principle, by use of a simple two-layered cross-ply laminate (a layer with fibers at 0° to the x-direction on top of an equal-thickness layer with fibers at 90° to the x-direction). We will analyze this laminate approximately by considering what conditions the two unbonded layers in Figure 4-3 must satisfy in order for the two layers to be bonded to form a laminate. Imagine that the layers are separate but are subjected to a load in the x-direction. The force is divided between the two layers such that the x-direction deformation of each layer is identical. That is, the laminae in a laminate must deform alike along the interface between the layers or else fracture must existl Accordingly, deformation compatibility of layers is a requirement for a laminate. Because of the equal x-direction deformation of each layer, the top (0°) layer has the most x-direction ress because it is stiffer than the bottom (90°) layer in the x-direction./ Trie x-direction stresses in the top and bottom layers can be shown to have the relation... 

We will proceed in the next section to define the strain and stress variations through the thickness of a laminate. The resultant forces and moments on a laminate will then be obtained in Section 4.2.3 by integrating the stress-strain relations for each layer. Equation (4.6), through the laminate thickness subject to the stress and strain variations determined in Section 4.2.2. 

The resultant forces and moments acting on a laminate are obtained by integration of the stresses in each layer or lamina through the laminate thickness, for example,... 

Note in Figure 4-5 that the stresses vary within each lamina as well as from lamina to lamina, so the integration is not trivial. Actually, N is a force per unit width of the cross section of the laminate as shown in Figure 4-6. 

Similarly, is a moment gemnit width as shown In Figure 4-7. However, Nx, etc., and etc T 6ereferred to as forces and moments with the stipulation of per unit width being dropped for convenience. The entire collection of force and moment resultants for an N-layered laminate is depicted in Figures 4-6 and 4-7 and is defined as... 

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. 

Demonstrate that the force per unit width on a two-layered laminate with orthotropic laminae of equal thickness oriented at -h a and - a to the applied force is... 

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. 

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