# Reliability by Using Weibull Distribution Based on Vibration Fatigue Damage
## 3 Numerical Application  
### Fatigue Damage Accumulation  
The accumulation fatigue damage study case is performed by analyzing a mechanical support component that is used to install a fiber optic panel into a frame. It is shown in Figure 2.  
The support is made of cold drawn steel AISI 1025, with a modulus of elasticity \(E\) = 200 GPa, Poisson's ratio \(r\) = 0.29, yield strength \(S_{y}\) = 430 MPa, ultimate tensile strength \(S_{ut}\) = 510 MPa, endurance limit \(S_{e}\) = 255 MPa, density \(\rho\) = 7.9 g/cm\({}^{3}\), length \(L\) = 51 mm, width \(W\) = 200 mm, and a wall thickness \(t\) = 3 mm. During its function, the component supports a static load of 80 N. It is submitted to an operating random vibration with an input PSD consisting of frequencies ranging from 10 to 55 Hz at an amplitude of 1.5 mm peak to peak for a period of 2 h. The testing is carried out physically by using a vibration system, and the results are as follows. By using Equations (1) and (2), the dynamic factor \(\sigma_{dynamic}\) and the vibration stresses \(\sigma(vib)_{i}\), are calculated, respectively. The acceleration responses \(A_{resi}\) are obtained from the vibration system but can also be determined by using Equation (3). Then, the vibration cycles applied \(n_{i}\) are determined by the rainflow method and the total cycles \(N_{i}\) are determined by using Equation (4). Finally, the fatigue damage  
Figure 1: Diagram of the Weibull Fatigue Damage Analysis. accumulation is obtained by Equation (5). Table 1 shows the vibration stress and cycle results and Table 2 shows the vibration fatigue damage accumulation results, where the failure is presented when \(D\) = 1 [29]. Figure 3 shows the mechanical support areas where the fatigue damage accumulated was presented. The analysis and estimation of the damage were determined from the acceleration responses that were the base to calculate the principal vibration stresses, \(\sigma_{1}\) and \(\sigma_{2}\), that are employed in the Weibull probabilistic analysis. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \begin{tabular}{c} **Frequency** \\ **(Hz)** \\ \end{tabular} & \begin{tabular}{c} **Accel.** \\ **Response (G)** \\ \end{tabular} & \begin{tabular}{c} **Dynamic Factor** \\ \(\sigma_{dynamic}\) \\ **Equation (2)** \\ \end{tabular} & \begin{tabular}{c} **Vibration Stress** \\ \(\sigma(\textit{vib})_{i}\) \\ **Equation (1)** \\ \end{tabular} & \
\begin{tabular}{c} **Applied** \\ **Vibration Cycles (\(\textit{v}_{i}\))** \\ **Equation (4)** \\ \end{tabular} &
\begin{tabular}{c} **Total Cycles** \\ **(\(\textit{N}_{i}\))** \\ **Equation (4)** \\ \end{tabular} \\ \hline
10 & 0.72 & & 15.99 & 70,384 & 1.36 \(\times\) 10\({}^{20}\) \\ \hline
20 & 2.65 & & 58.86 & 140,195 & 3.04 \(\times\) 10\({}^{13}\) \\ \hline
30 & 5.62 & & 124.84 & 92,619 & 4.42 \(\times\) 10\({}^{9}\) \\ \hline
40 & 9.17 & & 203.69 & 10,807 & 1.40 \(\times\) 10\({}^{7}\) \\ \hline
50 & 13.72 & & 304.76 & 2921 & 1.23 \(\times\) 10\({}^{5}\) \\ \hline
55 & 12.36 & & 274.55 & 762 & 4.20 \(\times\) 10\({}^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Vibration Stress and Cycles Results. Figure 3: Fatigue cracks on the support after vibration testing. Figure 2: Mechanical support. Equation (14), the mean is, \(\mu_{y}=-0.6672\). 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 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  
\[\beta_{D}=\frac{(-4)(-0.6672)}{0.995\times\ln\left(\frac{304.76}{15.99}\right)} =0.9102\]  
and from Equation (11), the logarithm average value is \(\mu_{x}=\ln\sqrt{304.76\times 15.99}=4.2457\). Thus, from Equation (10), the Weibull parameter \(\eta_{D}\) is \(\eta_{D}=\exp\{4.2457\}=69.8077\). Consequently, the Weibull damage family is \(W(0.9102,\,69.8077\) MPa).