# Fatigue Failure of a Pressing Machine
## 2 Analysis of the Failure
The nominal stress is based on the measured values of tensile stress (40 MPa) and additional bending stress at the thread point (30 MPa), to which the stress from the bolt pre-tension (74 MPa recalculated according to the manufacturer's data) is added. Point 1 in Figure 10 represents the stress at the root of the thread for a linear material model, point 2 for the actual tensile curve of the guide bar material. Figure 4:2 The resulting stresses obtained by measured values of tensile and bending loading (measured on the outer surface of the guide bar according to Figure 6). Figure 4:3 Exact values of thread geometry obtained using a Keyence VHX 6000 digital microscope. Taking the above-mentioned into consideration, it turns out to be crucial to assess the fatigue strength of the critical point of the guide bar's internal thread. Although the decisive variables are not deterministic values, they could be determined at a certain interval or with a certain probability. Therefore, a more credible approach was taken to assess the risk of premature fatigue fracture in probabilistic form [8]. The methodology of this approach is based on the interaction of two sets of uncertainties:  
1. a set of stress amplitudes that can be characterized by Gaussian (or Student, etc.) distribution and describing the mean value of stress amplitude and its scatter obtained by the longer-term measurement of stresses in operation and
2. a set of fatigue limit values of the guide rod material, which is also characterized by the Gaussian (or Student, etc.) distribution, and will be described by the mean value of fatigue limit and its possible scatter for this type of material. The relative position of these two statistical sets indicates the probability with which the amplitude of potential stresses in the root of the thread exceeds the fatigue strength of the guide bar material (Figure 11). Figure 4:4 Statistical interpretation of the risk of early fatigue fracture for a critical location at the point of the internal thread in the guide bar of the press machine. Figure 4:5 Application of the Neuber method of transforming the value of the nominal stress of 144 MPa into the root of the thread (**1—**633 MPa, **2—**352 MPa). The value of the penetration of such defined functions depends on the mutual positioning of the sets of variables considered and the widths of probability intervals of their occurrence (i.e., their variability in operation or material). The risk of premature fatigue fracture can also be mathematically described and numerically expressed. A distribution function of stress in notch (resulting from the uncertainty of designation of notch coefficient and fluctuation of the stress in the individual working cycles in Figure 12, uncertainty of notch factor \(K_{\text{ef}}\) and mean value of stress cycle taken into account based on [PERSON] (1970) or [PERSON] (1958) approach) is given by the following parameters of normal distribution: \(\sigma_{c}:\overline{\sigma}_{c},s_{\sigma_{c}}\). Mean value: \(\overline{\sigma}_{c}=210\) MPa. Standard deviation: \(s_{\sigma_{c}}=10\) MPa. The corresponding standard deviation is calculated using the following formula  
Figure 4:6  
where \(\Delta\) is the maximal range of the stress amplitude value determined by measurement directly in operation (Figures 6 and 12) and \(u^{95\%}\) is the normal distribution quantile for a 95% probability of occurrence. Distribution of fatigue limit values (resulting from the statistical nature variance of cyclic material properties, as documented in Figure 13): \(\sigma_{a}:\overline{\sigma}_{a},s_{\sigma_{a}}\). Mean value [9; 10]: \(\overline{\sigma}_{a}=220\) MPa. Standard deviation: For most steels, the standard deviations determined by this relationship are in the range of 5% to 7% of the stress amplitude on the regression line [9; 10]. In our considered case \(s_{\sigma_{a}}=11\) MPa.