Stress intensification factor

As mentioned before, there is an important factor, p, in the hot pressing models, which is known as stress intensification factor or stress multiplication factor. It is used to relate the mean stress applied to the grain boundary, pe, to the externally appfied stress, p. The significance of (p is such that while pa is the stress that is measured, pe is its counterpart that influences the rate of matter transport. The factor (p is geometrically dependent on the porosity and the shape of the pores of the ceramic compacts. 

When a hydrostatic pressure pa is applied to the external surface of a powder compact, it can be represented by using a model, as shown schematically in Fig. 5.25a [1]. In this case, the applied pressure exerts a load on the surface of the solid, which is Fa = AxPa, where Aj is the total external cross-sectional area of the solid, including areas pores. Due to presence of porosity at the grain boundaries, the actual grain boundary area Ag is smaller than the total external area. If a force balance is maintained across any plane of the solid, the following expression is valid  

If the pores are assumed to be spherical and randomly distributed in a porous solid, the factor p can be obtained through the following steps. It is assumed that there is 

This equation is valid for polycrystalline ceramics with equilibrium shapes of the isolated pores to be nearly spherical, i.e., dihedral angles are larger than 150°. Once the pores become nonspherical, the expression of (j) could have very complicated form. If the shapes of the pores are dramatically changed, as shown in Fig. 5.25b [1], although the volume is the same, the value of (j) could be changed, because it is now dependent on not only the porosity but also the shape of the pores. When the pore shape is not spherical, the dihedral angles are reduced, so that the actual area of the grain boundary is decreased, therefore, (f will be decreased. 

An analysis of simulation results of a continuous network of pores with equilibrium shapes indicates that (f can be approximated using a simple expression, which is [59, 61, 62]  

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Fracture Mechanics. Linear elastic fracture mechanics (qv) (LEFM) can be appHed only to the propagation and fracture stages of fatigue failure. LEFM is based on a definition of the stress close to a crack tip in terms of a stress intensification factor K, for which the simplest general relationship is... 

The concepts behind the analysis are not difficult. The piping system is simply a stmcture composed of numerous straight and curved sections of pipe. Although, for straight pipe, elementary beam theory is sufficient for the solution of the problem, it is not adequate for curved pipe. However, by the iatroduction of a flexibiUty factor, to account for iacreased flexibiUty of curved pipe over straight pipe, and a stress intensification factor, /, to account for... 

Flexibility and Stress-Intensification Factors. The flexibihty factor k (>1.0) is defined as the ratio between the rotation per unit length of the part in question produced by a given moment to the rotation of a straight pipe (of the same size and schedule) produced by the same moment. A close approximation of the flexibiUty factor that agrees quite well with theory and experiment for bends is as follows ... 

In the absence of more direc tly applicable data, the flexibility factor k and stress-intensification factor i shown in Table 10-54 may be used in flexibihty calculations in Eq. (10-101). For piping components or attachments (such as valves, strainers, anchor rings, and bands) not covered in the table, suitable stress-intensification factors may be assumed by comparison of their significant geometry with that of the components shown. 

Comprehensive analysis shall take into account stress-intensification factors for any component other than straight pipe. Credit may be taken for the extra flexibility of such a component. 

TABLE 10-54 Flexibility Factor kand Stress-Intensification Factor i ... 

The flexibility factor k applies to bending in any plane. The flexibility factors k and stress intensification factors shall not be less than unity factors for torsion equal unity. Both factors apply over the effective arc length (shown by heavy centerlines in the sketches) for curved and miter bends and to the intersection point for tees. 

Sweepolet is a contoured, integrally reinforced butt-weld branch connection with a low stress Intensification factor for low stresses and Iona fatigue life. The attachment weld is easily examined by radiography, ultrasound and other standard non-destructive techniques... 

The stress intensification factors in Appendix D of ASME B31.3 have been developed from fatigue tests of representative piping components and assemblies manufactured from ductile ferrous materials. The allowable displacement stress range is based on tests of carbon and austenitic stainless steels. Caution should be exercised when using eqs. (la) and (lb) (para. IP-2.2.10) for allowable displacement stress range for some nonferrous materials (e.g., certain copper and aluminum alloys) for other than low-cycle applications. 

Standard Test Method for Determining Stress Intensification Factors (/-Factors) for Metallic Piping Components.B31J-2008... 

Tees Tees may be cast, forged, or hot- or cold-formed from plate or pipe. Tees are typically stocked with both header (run) ends of the same size. In general, run ends of different sizes are not typically stocked or specified however, occasionally run ends of different sizes are specified in threaded or socket-welded sizes. Branch connections may be full size or reducing sizes. Branch reductions two sizes smaller than the header are routinely stocked, and it is not typically difficult to purchase reducing tees with branches as small as those listed in ASME B16.9 (i.e., approximately one-half the header size). Economics, stress intensification factors, and nondestructive examination requirements typically dictate the branch connection type. 

The out-of-plane stress intensification factor (SIF) for a reducing branch connection with branch-to-run diameter ratio of 0.5 < dlT) < 1.0 may be nonconservative. A smooth concave weld contour has been shown to reduce the SIF. Selection of the appropriate SIF is the designers responsibility. 

Stress intensification factors for branch connections are based on tests with at least two diameters of straight run pipe on each side of the branch centerline. More closely loaded branches may require special consideration. 

Tj, = thickness of pipe matching branch, in (mm) io = out-plane stress-intensification factor (Table 10-48) i, = in-plane stress-intensification factor (Table 10-48)... 

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