# Reliability by Using Weibull Distribution Based on Vibration Fatigue Damage
## 3 Numerical Application
### Fatigue Damage Accumulation
This Weibull family completely represents the observed principal vibration stresses \(\sigma_{1}\) and \(\sigma_{2}\) values. Now, by using the Weibull family results, the random behavior strength can be determined. Since the Weibull parameters only depend on the principal stress values caused by the random vibration \(\sigma_{1}\) and \(\sigma_{2}\) values, the random behavior can be obtained by performing the Weibull analysis using the following steps. Since the determination of the fatigue damage is based on the random behavior of the \(\sigma_{2}\) stress value, here, the random behavior of \(\sigma_{1}\) and \(\sigma_{2}\) is determined by Equations (16) and (17). Then, by using the \(\beta_{D}\) and the \(Y_{i}\) values in Equation (15), the basic Weibull elements [28]\(t_{0i}\) for each \(Y_{i}\) are obtained. Whereas by using the \(\eta_{D}\) and \(\sigma_{1}\) values in Equation (18), the Weibull \(t_{01}\) value from the \(\sigma_{1}\) and \(\sigma_{2}\) stress values are reproduced. This value is calculated as  
\[t_{01}=\frac{69.8077}{304.76}=0.2291\]  
and, by using the \(\beta_{D}\) value in Equation (19), the \(Y_{1}\) value that belongs to the \(t_{01}\) value is determined as  
\[Y_{1}=\ln(0.2291)\times 0.9102=-1.3414\]  
Next, by substituting the \(Y_{1}\) value in Equation (20), the reliability \(R(t)\) that belongs to the \(t_{01}\) element is  
\[R(t)=\exp\{-\exp\{-1.3414\}\}=0.7699\]  
The previous results shown in this section are included in Table 3.  
Here, is important to mention that the reliability obtained \(R(t)=0.7699\) corresponds to a designed component or structure with minimal strength of \(S_{y}=\sigma_{1}=304.76\) MPa. In relation to the mechanical support, it has a \(S_{y}=430\) MPa, then, its reliability is \(R(t)=0.8260\), which is determined as follows. The minimal reliability of the component is obtained when the \(Y_{1}\) value that belongs to the \(S_{y}\) value is used in Equation (20). The steps to determine the reliability of the design component when the \(S_{y}\) value is used as \(\sigma_{1}\) in Equation (18) are, the \(t_{Sy}\) element that belongs to the \(S_{y}\) value is \(t_{sy}=\frac{69.8077}{430}=0.1623\). From Equation (19), the corresponding \(Y_{sy}\) value is \(Y_{sy}=\ln(0.1623)\times 0.9102=-1.6548\). The reliability index for the \(Y_{sy}\) value is calculated by using Equation (20), \(R(t)=\exp\left\{-\exp\{-1.6548\}\right\}=0.8260\). Thus, we conclude that the reliability of the design component is \(R(t)=0.8260\). Additionally, regarding the material's component or structure, it is noticed about that the higher the strength \(S_{y}\) value, the higher the reliability \(R(t)\) will be [28]. Now, as a comparison for the probabilistic cumulative density, the Weibull distribution is used, but in this case, by using the median rank method to estimate the vector \(Y\), which is the base to determine the reliability and the probability of failure. By selecting the fatigue damage accumulated \(D_{i}\) value of each one block of Table 2 and using it in Equation (22), the \(n=29\)\(Y_{i}\) elements are determined. Thus, from Equation (14), the mean is, \(\mu_{y}=-0.5525\). Next, from the results of Table 1, the principal vibration stress values \(\sigma_{1}=304.76\) MPa and \(\sigma_{2}=15.99\) MPa are obtained. With these data, we can now proceed to use the Weibull distribution in order to obtain its failure probability, reliability, and random behavior. From Equation (9), the Weibull shape parameter \(\beta_{D}\) value is  
\[\beta_{D}=\frac{(-4)(-0.5525)}{0.995\times\ln\left(\frac{304.76}{15.99}\right)} =0.7538\]  
Since the principal vibration stress values \(\sigma_{1}=304.76\) MPa and \(\sigma_{2}=15.99\) MPa are maintained, the Weibull parameter value \(\eta_{D}=69.8077\) remains. Next, the remaining steps followed in Section 3 are applied and the results are included in Table 4.