# Comparison of Stress Concentration Factors Obtained by Different Methods
## 5 The Problem of Determining Stress Concentration Factors  
The fundamental problem of determining stress concentration factors is the creation or existence of a certain system solution for their fast, simple and efficient determination. This is particularly relevant not only for existing, but also for new mutual configurations or combinations of shape, dimension and loading. The results are most often represented graphically, i.e., by tables, diagrams or nomograms. It is also possible to express them analytically by means of appropriate dependencies or equations. Among the more widespread methods, methods based on experimental and numerical analyses, or combinations thereof, may be mentioned. Whatever the approach used, however, the important criterion must be to achieve acceptable precision and, at the same time, efficiency and statistical significance. Experimental methods are mainly dominated by methods based on transmission photoelasticity. However, a particular problem with this method is that the stress analysis of a certain group of objects, due to the evaluation methodology, is made difficult or complicated by complex stress states. These are situations where, depending on the shape-discontinuity-dimensions-load configuration, the stresses may be not only uniaxial but also biaxial and in some cases even triaxial. In the case of triaxial strain, the application of spatial photoelasticity would be required, but this is a rather labor-intensive and challenging procedure. The reflection method (Photostress(r)) is much less suitable for these situations. A relatively unsuitable method for such cases is the strain-gauge method, which is a fairly widespread and well-established method in other stress analysis cases. The inappropriateness of this method lies in the applied principle of measuring small deformations and the consequent inability to capture large stress and strain gradients in the conditions of small stress concentrator regions. Moreover, the situation is more complex, e.g., in the case of 3D axially symmetric and other objects. In this case, it is particularly challenging to obtain data on the gradients of the quantities of interest, and the problem is not only related to the strain-gauge method, but also to reflection photoelasticity. However, any applied experimental method requires, in addition to the respective hardware and software means of the measurement chains, mainly a corresponding physical model as a material object for each shape-discontinuity-dimension-stress configuration separately. But then such procedures are unacceptably time-consuming and economically demanding. Some advances in the creation of physical models for these purposes can be brought about by modern additive 3D printing technologies. Rotationally symmetric and spatially symmetric objects, e.g., according to Figure 1c are typical examples of objects with a complex stress state. According to our classification, they are axially symmetric 3-dimensional tension bar of circular cross-section with a U-shaped groove. In this case, the largest increase in stress from the nominal value \((\sigma_{\text{nom}})_{1}\) occurs at the root of the notch, assuming perfectly elastic tension at all points along the perimeter of the cross-section. This maximum stress has a perpendicular direction to the smallest cross-section of the bar. This is the axial stress \((\sigma_{\text{max}})_{1}\) denoted by the index 1. At the root of the notch for \(x=0\), in addition to this stress, there is a peak circumferential stress of magnitude \((\sigma_{\text{max}})_{2}\). If the free unloaded surface of the root of the notch is assumed, then the third principal stress in the direction of the radius of the bar, i.e., in the radial direction, is zero \((\sigma_{3}=0)\). This location is then corresponded by the corresponding equivalent stress concentration factor \(\alpha_{\text{eq}}\) and stress factor \(\
u_{(\text{x}=0)}\). However, the stress factor varies from location to location.