Profile graded

Total US production of paper in 2005 was 37.6 million metric tons, and production of paperboard was 45.7 million tons at operating rates of 92% and 97% respectively. Production data and projections are analyzed by RISI,1 and periodic updates on industry issues, grade profiles and production statistics are published in Pulp and Paper, Paper Age, Pulp and Paper International and other trade journals. The US paper industry reuses more than 51% of the product it produces with an ultimate goal of 55%.2... 

Energy-dispersive x-ray microanalysis was used to qualitatively verify the graded profile. A cross-sectioned sample, polished to a 1pm finish, was prepared. The measurement was conducted from the near-surface region to the center of the sample with step size of 50 pm using a JEOL 35C scanning electron microscope. The x-ray emission intensities for TiKot, AlKa, and ZrLot were collected at each point. 

With the exception of the bridge deck profiles, most pultruded GFRP structural-grade profiles have simple open or closed cross-sectional shapes that are similar to structnral steel and aluminium profiles (see Fig. 13.2). Nonetheless, it should be appreciated that composite material profiles with more complicated cross-sectional shapes may be pultruded. These profiles, known as custom profiles, are nsed less frequently in construction and will not be considered herein. 

Pultruded GFRP structural grade profiles (reproduced with permission from Strongwell, www.strongwell.com). 

Tests to characterise the flexural, torsional, buckling and collapse responses of pultruded GFRP structural grade profiles... 

Fig. 1-1 Nomenclature and coordinates for describing planar waveguides. A representative graded profile varies over the core and is uniform over the cladding, assumed unbounded.

While this is a trivial observation for a step profile, and of no particular advantage, the introduction of the ray invariant simplifies the description of ray paths on graded profiles, as we show in Section 1-7. The relationship between the ray invariant and the direction of propagation enables us to classify rays according to their value of We deduce from Eq. (1-5) that... 

The description of ray propagation on the step profile is readily generalized to allow for a graded profile. For simplicity, we consider profiles of the type in Fig. I-l when only the core is graded. Our results are readily generalized to profiles that are graded in the cladding as well. 

Fig. 1-8 Sinusoidal-like bound-ray path within the core of a graded-profile planar waveguide.

Graded profiles tend to equalize the transit times of different rays. There is a simple explanation for this. As a ray propagates further from the axis, (x) decreases, thus increasing the local speed of light c/n(x). This increase in speed compensates, in part, for the extra distance travelled by the ray off axis. For the hyperbolic secant profile this equalization is exact, as we show in Section 1-12. 

To illustrate the use of the formal results for graded profile waveguides, we consider examples of profiles which have analytical solutions for some or all of the ray-path quantities of interest, including solutions within the weak-guidance approximation. The more important parameters are also included in Table 1-1. 

Fig. 2-1 Nomenclature for describing circular fibers. Cartesian coordinates X, y, z and cylindrical polar coordinates r, , z are oriented so that the z-axis lies along the fiber axis. A representative graded profile varies over the core and is uniform over the cladding, assumed unbounded.

Integration of Eq. (2-13c) leads to the ray invariant for graded-profile fibers... 

This invariant is identical to Eq. (1-25) for graded-profile planar waveguides, and is associated with the translational invariance of the fiber. 

Fig. 2-4 Ray paths within the core of a graded-profile fiber showing (a) a meridional path and (b) a skew path, together with their projections onto the core cross-section. The angle 0 (r) between the projection and the azimuthal direction is shown in (c).

A simple method for classifying rays on graded-profile fibers uses the ray equation to determine the range of values of the radial coordinate r for which rays can propagate. This is accomplished by expressing the radial component of the ray-path equation in Eq. (2-13a) as a relationship between r and z. We use Eq. (2-16) to replace ds by dz, and substitute for d/ds from Eq. (2-17). This leads to... 

Fig. 2-6 Section of a tunneling ray path on a graded-profile fiber. In (a) the core path touches the turning-point caustic at P. Radiation originates at Q in the cladding and propagates along QR tangential to the radiation caustic. The projection onto the fiber cross-section is shown in (b).

Each ray of the graded-profile fiber is characterized by the invariants and /. For most purposes in subsequent chapters the ray trajectory is unimportant, and it is sufficient to know only the values of the ray-path parameters. The... 

In Section 1-8 we introduced the notion of the complement of the local critical angle for graded-profile planar waveguides. By analogy, the complement of the local critical angle, 0c( ), for fibers is defined in terms of the profile by... 

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