# Fatigue Failure of a Pressing Machine
## 2 Analysis of the Failure
The measured typical stresses in the critical cross-section on the outer surface of the guide bar are displayed in Figure 6.  
Figure 5: Deployment of strain gauges along the outer cross-sectional area of the guide bar. Figure 6: Diagrams of maximum normal stresses from bending moment and stress from normal force measured for two operational modes (mode 1—top, mode 2—bottom). The fatigue crack initiation site was at the root of the internal thread (the first engaged thread, see Figure 1). The criticality of this cross-section was predetermined by the change in cross-sectional stiffness at the location between the end of the bolt and the bottom of the threaded hole (Figure 2). In this area, the annular cross-section caused additional bending loading under the tensile loading from the prestressed bolt and the pressing force developed by the hydraulic cylinder. The resulting value of the stress (from the bolt pre-tension, additional bending moment, and pressing force) had to be recalculated (transformed) into the notch root of the thread. The effective method for such transformation, applicable in engineering practice, is [PERSON]'s method [12; 13]. The equation below requires the knowledge of nominal stress/strain value \(\sigma_{n}/\varepsilon_{n}\), deformation as well as the effective notch coefficient of the thread root in the hole of the guide bar:  
\[\sigma_{notch}\varepsilon_{notch}=\beta^{2}\sigma_{nom}\varepsilon_{nom} \tag{2}\]  
The value of nominal stress (as well as the deformation) can be determined as the addition of stresses from the working cycle (see Figure 6) and calculated stress from the pre-tension of the bolt. The magnitude of notch coefficient is given by [PERSON]'s formula as  
\[K_{ef}=K_{\sigma}\sqrt{\frac{\rho}{\rho_{F}}} \tag{3}\]  
where \(K_{\sigma}\) is the stress concentration factor, \(\rho\) is the radius of the notch root, and \(\rho_{F}=\rho+m\rho^{*}\) is the fictitious radius defined by [PERSON] [13] (material parameter \(m=2\), structural length \(\rho^{*}=0.095\div 0.115\) mm, \(\rho=0.45\) mm)  
The value of the stress concentration factor \(K_{\sigma}\) can be determined in several ways, e.g., by using the approach of equivalent ellipse and applying the expression for an elliptical groove [6] or using the FEM model. Let us note that the usage of modeling with FEM frequently requires a careful choice of mesh, types of elements, and boundary conditions [14] (see Figure 7). When using a sufficiently fine mesh requiring as many as five elements at the thread root, a linear element (Solid 8 Node 185) could be used. At the interface of threads, an 8-node contact element (Conta 174) was employed in combination with a 3D Target 170. The setting-up of the conditions of contact and boundary conditions of the calculation was based on reaching the same stress values on the surface of the guide bar as were obtained by direct measurement. The results of the precision mathematical model of the critical node under the load of tensile force and bending moment corresponding to the measured values of stress according to Figure 6 are in Figure 4:0 For the value of the nominal stress from the tensile force of 40 MPa and the bending at the thread point of 30 MPa, the value of the stress concentration factor based on FEM simulation is \(K_{\sigma}=5.43\) and using Equation (3) for the notch factor is \(K_{\sigma}=4.4\div 4.53\). Figure 4:1 Finite element mesh at the internal M80 thread location of guide bar. By measuring the exact geometry in the root of the thread using a digital microscope (Figure 9) and using the elliptical groove concept according to [PERSON] [15; 16], it is possible to analytically determine the value of the stress concentration factor, \(K_{\sigma}\) = 5.95. This value is in good agreement with the results of the FEM model, which we will use in the next analysis. The value of the stress in the root of the internal thread is obtained by applying the Neuber method (Equation (2)).