# Reliability by Using Weibull Distribution Based on Vibration Fatigue Damage
## 2 Fatigue Damage
### Weibull Fatigue Damage Analysis  
The expected behavior of the \(\sigma_{1}\) and \(\sigma_{2}\) parameters is determined by the following steps:  
Step 1. By using the fatigue damage accumulation results from the component's operation, the times that the vibration profile (PSD) loading were applied to the mechanical component are taken as \(n\).  
Step 2. The accumulation fatigue damage \(D_{i}\) determined by Equation (5) in the end of each one of the n vibration loadings is used by Equation (12) as the corresponding cumulated failure percentile \(R(D_{i})\), as  
\[R(D_{i})=1-D_{i} \tag{12}\]  
Step 3. By using the \(R(D_{i})\) elements in the linearized form of the reliability function given in Equation (8), the corresponding \(Y_{i}\) elements are determined as in Equation (13). Then, its corresponding arithmetic mean value is computed as in Equation (14).  
\[Y_{i}=LN(-LN(1-D_{i})) \tag{13}\]  
\[\mu_{y}=\sum_{i=1}^{nt}\frac{Y_{i}}{n} \tag{14}\]  
Step 4. By plugging the \(\mu_{y}\) value and the \(\sigma_{1}\) and \(\sigma_{2}\) values into Equation (9), the corresponding Weibull shape \(\beta_{D}\) parameter is determined. Similarly, by plugging the \(\sigma_{1}\) and \(\sigma_{2}\) values into Equation (11), the corresponding \(\mu_{x}\) value is determined. Then, by using this corresponding \(\mu_{x}\) value in Equation (10), the corresponding Weibull scale \(\eta_{D}\) parameter is determined. These \(\beta_{D}\) and \(\eta_{D}\) parameters represent the Weibull fatigue damage family that is used to model the random behavior of the estimated \(\sigma_{1}\) and \(\sigma_{2}\) principal stress values.  
Note 1. Here, notice the random behavior of the \(\sigma_{1}\) and \(\sigma_{2}\) values. In the proposed Weibull analysis, let us use the \(\sigma_{1i}\) values as the minimum required strength that the component's material must present in order to ensure that the reliability of the component will meet at least (as a minimum) the desired \(R(t)\) index.  
From the Weibull analysis, by using the \(\beta_{D}\) and \(\eta_{D}\) parameters, the minimum strength \(\sigma_{1i}\) values are determined by using the \(t_{0i}\) value that correspond to each \(Y_{i}\) element as  
\[t_{0i}=\exp\{Y_{i}/\beta_{D}\} \tag{15}\]  
Thus, the \(\sigma_{2i}\) value is determined as  
\[\sigma_{2i}=\eta_{D}\times t_{0i} \tag{16}\]  
and the \(\sigma_{1i}\) value is determined as  
\[\sigma_{1i}=\eta_{D}/t_{0i} \tag{17}\]  
Additionally, from Equation (18), by using the known \(\sigma_{1}\) value, the \(t_{01}\) element that belongs to the \(\sigma_{1}\) and \(\sigma_{2}\) values determined in Section 2.1 is determined as  
\[t_{01}=\eta_{D}/\sigma_{1} \tag{18}\]Now, the \(t_{01}\) and the \(\beta_{D}\) values are used to determine the corresponding \(Y_{1}\) value as follows,  
\[Y_{1}=\ln(t_{01})\times\beta_{D} \tag{19}\]  
Finally, the reliability index that corresponds to the \(Y_{1}\) value is determined as  
\[R(t)=\exp\{-\exp\{Y_{1}\}\} \tag{20}\]  
Note 2. Here, observe that the \(R(t)\) index determined in Equation (20) by using the \(\sigma_{1}\) value according to our proposed method corresponds to a component with strength equivalent to the \(\sigma_{1}\) value. Thus, if we define the material \(S_{y}\) parameter as the actual strength of the component, then by using this \(S_{y}\) value in Equation (18) and the corresponding \(Y_{Sy}\) value of Equation (19) in Equation (20), the minimum expected reliability of a component that presents a strength of \(S_{y}\) is determined. Please also notice from the proposed Weibull analysis that any desired strength value can be used to determine its corresponding reliability. The diagram with the steps required to determine the probability of failure \(F(t)\) and the reliability \(R(t)\) based on the vibration fatigue damage \(D\) is shown in Figure 1.  
Next, a numerical application is presented where the proposed method is applied to a mechanical component that is submitted to random vibration stress due to its field application.