# Vibration Fatigue Life Reliability Cable Trough Assessment by Using Weibull Distribution
## 3 Proposed Random Vibration--Stress Method
### Weibull Stress Random Analysis  
Due to the statistical nature of random vibration, the addressed \(\sigma_{1}\) and \(\sigma_{2}\) values are also random. Additionally, because their values are the input to determine the parameters of the associated Weibull distribution, then to determine their random behavior is necessary.  
The expected behavior of the \(\sigma_{1}\) and \(\sigma_{2}\) parameters is determined by the following steps:  
Step 1. By using the required reliability \(R(t)\) index in Equation (3), determine the sample size \(n\) value. Because in our case \(R(t)=95\%\)[21], then from Equation (3) \(n\approx 20\).  
Step 2. By using the n value from Equation (3) in the median rank approach function stated by Equation (15) [20], the corresponding cumulated failure percentile \(F(t_{i})\) is determined as  
\[F(t_{i})=\frac{i-0.3}{n+0.4} \tag{15}\]  
Step 3. By using the \(F(t_{i})\) elements in the linearized form of the reliability function given in Equation (11), determine the corresponding \(Y_{i}\) elements as in Equation (16), and then compute its corresponding arithmetic mean value as in Equation (17) [20].  
\[Y_{i}=LN(-LN(1-((i-0.3)/(n+0.4)))) \tag{16}\]  
\[\mu_{y}=\sum_{i=1}n\frac{Y_{i}}{n} \tag{17}\]  
Step 4. By using the \(\mu_{y}\) value and the \(\sigma_{1}\) and \(\sigma_{2}\) values in Equation (12), determine the corresponding Weibull shape \(\beta_{s}\) parameter. Similarly, by using the \(\sigma_{1}\) and \(\sigma_{2}\) values in Equation (14), determine the corresponding \(\mu_{x}\) value, and then by using it in Equation (13), determine the corresponding Weibull scale \(\eta_{s}\) parameter. These \(\beta_{s}\) and \(\eta_{s}\) parameters represent the Weibull stress family that is used to model the random behavior of the estimated principal stresses \(\sigma_{1}\) and \(\sigma_{2}\) 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 the reliability of the component will be at least (as minimum) the desired \(R(t)\) index stated in step 1.  
From the Weibull analysis, by using the \(\beta_{s}\) and \(\eta_{s}\) parameters, the minimum strength \(\sigma_{1i}\) values are determined by using the \(t_{0i}\) value that corresponds to each \(Y_{i}\) element as [20],  
\[t_{0i}=exp\{Y_{i}/\beta_{s}\} \tag{18}\]  
Thus, the \(\sigma_{2i}\) value is determined as  
\[\sigma_{2i}=\eta_{s}*t_{0i} \tag{19}\]  
Additionally, the \(\sigma_{1i}\) value is determined as Finally, the reliability index that corresponds to the \(Y_{1}\)[20] value is determined as  
\[R(t)=\exp\{-\exp\{Y_{1}\}\} \tag{23}\]  
Note 2. Here, observe the \(R(t)\) index determined in Equation (23) 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 (21), and the corresponding \(Y_{Sy}\) value of Equation (22) in Equation (23), the minimum expected reliability of a component that presents a strength of \(S_{y}\), is determined. Please also notice from the proposed Weibull analysis any desired strength value can be used to determine its corresponding reliability. Now, a numerical application is presented.