Laminates coordinate system

Equations (6.1) and (6.2) give the stress—strain response of a single ply in its own coordinate system, where one axis is parallel and the other perpendicular to the hbers. In practice, the plies making up a laminate are oriented in different directions (see Eigure 6.2) following the requirements of the design. It is, therefore, necessary to transform stiffness properties, stresses, and strains from the basic ply coordinate system to other coordinate systems such as the laminate coordinate system. 

In the laminate coordinate system, the stress—strain equations for a single ply can be written in the form ... 

Note that in the laminate coordinate system a single ply is not necessarily orthotropic. As a result, additional stiffness terms gjcs and gys appear. The stiffness properties gxo... 

Equations (6.9) or (6.4) and (6.10) provide the stiffnesses of a ply with hbers oriented at an angle 6 in the laminate coordinate system.The above discussion concentrated on plane stress conditions. In a three-dimensional situation, a 0° ply is dehned with nine stiffness parameters En, E22, E33, G12, G13, G23, J i2, 13, and i 23. Here, as before, 1 denotes the direction parallel to the hbers, 2 is perpendicular to the hbers, and 3 is the out-of-plane direction. Eor orthotropic plies, E22 = E33,... 

Figure 1 Cylindrical section of the vessel, coordinate systems on the laminate plane.

The basic approach [1—4] starts with a single orthotropic ply. In the coordinate system of the ply, with one axis parallel to the fibers and one perpendicular to the fibers, in the plane of the ply, the stiffness properties are assumed known. These stiffness values may be obtained from analytical modelling at lower scales using micromechanics or may be obtained experimentally with 1 and 2 ply coordinates as opposed to the laminate coordinates x and y (see Figure 6.2). 

We have determined the relative locations of nuclear lamins and chromatin in the nuclear periphery in a more quantitative manner (Fig. 11) (Paddy et al, 1990). After converting the coordinate system of chromatin and antilamin data sets into a form more appropriate for describing features relative to a surface (the image intensity in the nuclear periphery is unfolded and flattened in a manner analogous to a Mercator map projection) [see Fig. 7 in Paddy et al (1990) and Fig. 1 in Paddy et al (1992)], we then determined the position of closest approach of individual chromatin fibers to the nuclear periphery. The distance between 93 such chromatin fibers in three nuclei and the nearest feature... 

Other less well-known types of nonlinearities include interaction and intermode . In the former, stress-strain response for a fundamental load component (e.g. shear) in a multi-axial stress state is not equivalent to the stress-strain response in simple one component load test (e.g. simple shear). For example. Fig. 10.3 shows that the stress-strain curve under pure shear loading of a composite specimen varies considerably from the shear stress-strain curve obtained from an off-axis specimen. In this type of test, a unidirectional laminate is tested in uniaxial tension where the fiber axis runs 15° to the tensile loading axis. A 90° strain gage rosette is applied to the specimen oriented to the fiber direction and normal to the fiber direction and thus obtain the strain components in the fiber coordinate system. Using simple coordinate transformations, the shear response of the unidirectional composite can be found (Daniel, 1993, Hyer, 1998). At small strains in the linear range, the shear response from the two tests coincide. 

The orthotropic stress and strain relationships of Equations 8.42 and 8.43 were defined in principal material directions, for which there is no coupling between extension and shear behavior. However, the coordinates natural to the solution of the problem generally will not coincide with the principal directions of orthotropy. For example, consider a simply supported beam manufactured from an angle-ply laminate. The principal material coordinates of each ply of the laminate make angles 0 relative to the axis of the beam. In the beam problem stresses and strains are usually defined in the beam coordinate system (jc,y), which is off-axis relative to the lamina principal axes (L, T). 

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