Principal stress components

Determine the principal stress components. How does the pressure affect the principal stress components Why is it possible, and advisable, to develop an approach to computing the principal stresses by first subtracting the pressure from the stress tensor, that is, forming the deviatoric stress tensor How is the pressure reintroduced, after having determined the principal stress of the deviatoric stress tensor ... 

To explain the influence of stress components on the fonnation of microcracking and slip line patterns, observed in the ZrBa-SiC composite, the maximum principal stress component (<7,) and the maximum shear stress component ( ) have been computed. Figure 10 shows the distribution of the... 

In Equation (16) a, is the hydrostatic component of the stress state, which is equal to the average of the principal stress components ... 

Yield Criteria. A practical 3ueld criterion needs to describe the conditions under which jdeld will occur for a general stress state (eg tension, compression, shear, or some combination). Before we consider the various 3deld criteria which have been proposed, it is useful to very briefly recap the definition of the principal stress components as these are generally the terms which are used to define the yield criteria (see Viscoelasticity for a more thorough introduction to the generalized stress and strain definitions). 

The model was extended beyond the specific case of plane (or biaxial) stress conditions to a general three-dimensional case by Sternstein and Meyers (123). Because they are essentially empirical, the Sternstein et al models have several shortcomings (1) the stress bias, shear stress and it is difficult to reconcile a shear stress component controlling initiation of a craze in a direction perpendicular to the principal stress component physical interpretation, (3) no time dependency for the initiation of crazes. 

The term Uo in Eq. (8.21) has been taken to represent the macroscopic uniaxial stress Uo in the event of uniaxial stretching — or the largest principal stress component Oy in the event of multiaxial states of stress. This does not mean that the times to fracture of tubular and uniaxial specimens are always identical if only Oy is equal Uo. As discussed above the creep functions in these two cases are described by different potential laws, namely by Eqs. (8.16) and (8.17) respectively. 

SCC has been defined as failure by cracking under the combined action of corrosion and stress (Fig. 9.1). The stress and corrosion components interact S3mergistically to produce cracks, which initiate on the surface exposed to the corrodent and propagate in response to the stress state. They may run in any direction but are always perpendicular to the principal stress. Longitudinal or transverse crack orientations in tubes are common (Figs. 9.2 and 9.3). Occasionally, both longitudinal and transverse cracks are present on the same tube (Fig. 9.4). Less frequently, SCC is a secondary result of another primary corrosion mode. In such cases, the cracking, rather than the primary corrosion, may be the actual cause of failure (Fig. 9.5). 

Vertical in-line pumps that are supported only by the attached piping may be subjected to component piping loads that are more than double the values shown in Table 2. lA (2. IB) if these loads do not cause a principal stress greater than 41 MPa (5950 psi) in either nozzle. For calculation purposes, the section properties of the pump nozzles shall be based on Schedule 40 pipe whose nominal size is equal to that of the appropriate pump nozzle. Equations F-6A (F-6B), F-7A (F-7B), and F-8A (F-8B) can be used to evaluate principal stress, longitudinal stress, and shear stress, respectively, in the nozzles. 

For a 3 X 4 X 7 vertical in-line pump, the proposed applied nozzle loadings are as given in Table F-3B. By inspection, FZ5a, AZZ5a, and MXDp are greater than two times the values shown in Table 2. IB. As stated in F.2, these component loads are acceptable provided that the calculated principal stress is less than 5950 psi. The problem is to determine the principal stress for the suction nozzle and the discharge nozzle. 

To visualize the stress states at which failure occurs, a failure surface may be constructed. The six stress components may be resolved into three orthogonal principal stresses. Plotting failure in principal stress space, any stress state which exists in the bounded space containing the... 

Fig. 2.16 In general, the stress state represented on a differential element in a cylindrical coordinate system has nine stress components. The same stress state can be represented as its principal components via a coordinate rotation.

As an illustration, we determine the stress components on the z face illustrated in Fig. 2.18. The cosines of the three angles between each of the principal coordinates... 

The stress vector on the z plane has three components that can be determined from the projections of the principal stresses. These components, written to align with the principal axes, are... 

If the principal stresses had had shear components, which by definition they don t, then, in general, those shear components would have contributed to the stress vector on the rotated z plane. The a vector completely defines the stress state on the rotated z face. However, our objective is to determine the stress-state vector on the z face that aligns with the rotated coordinate system (z,r,G) x--, x-r, and x-e. The a vector itself has no particular value in its own right. Therefore one more transformation from cs to r is required ... 

---