# Vibration Fatigue Life Reliability Cable Trough Assessment by Using Weibull Distribution
## 4 Numerical Application
### Material's Strength for Variant Stress Analysis
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(n\) & \(Y_{i}\) & \(\mu_{y}\) & \(t_{oi}\) & \(R(t_{oi})\) & \(\sigma_{2i}\) & \(\sigma_{1i}\) \\
**Equation (3)** & **Equation (22)** & **Equation (17)** & **Equation (18)** & **Equation (23)** & **Equation (19)** & **Equation (20)** \\ \hline \(1\) & \(-3.355\) & \(-0.168\) & \(0.140\) & \(0.966\) & \(87.902\) & \(4459.989\) \\  & \(\mathbf{-3.312}\) & \(\mathbf{-1.938}\) & \(\mathbf{0.144}\) & \(\mathbf{0.964}\) & \(\mathbf{90.124}\) & \(\mathbf{4350.000}\) \\  & \(\mathbf{-2.970}\) & \(\mathbf{-0.149}\) & \(\mathbf{0.176}\) & \(\mathbf{0.950}\) & \(\mathbf{110.090}\) & \(\mathbf{3561.071}\) \\ \(2\) & \(-2.442\) & \(-0.122\) & \(0.240\) & \(0.917\) & \(149.993\) & \(2613.729\) \\ \(3\) & \(-1.952\) & \(-0.098\) & \(0.319\) & \(0.868\) & \(199.757\) & \(1962.581\) \\ \(4\) & \(-1.609\) & \(-0.080\) & \(0.390\) & \(0.819\) & \(244.211\) & \(1605.334\) \\ \(5\) & \(-1.340\) & \(-0.067\) & \(0.457\) & \(0.770\) & \(285.834\) & \(1371.566\) \\ \(\mathbf{-1.094}\) & \(\mathbf{-0.640}\) & \(\mathbf{0.527}\) & \(\mathbf{0.716}\) & \(\mathbf{330.000}\) & \(\mathbf{1188.000}\) \\ \(6\) & \(-1.116\) & \(-0.056\) & \(0.521\) & \(0.721\) & \(325.909\) & \(1202.913\) \\ \(7\) & \(-0.921\) & \(-0.046\) & \(0.583\) & \(0.672\) & \(365.252\) & \(1073.342\) \\ \(8\) & \(-0.747\) & \(-0.037\) & \(0.646\) & \(0.623\) & \(404.468\) & \(969.274\) \\ \(9\) & \(-0.587\) & \(-0.029\) & \(0.709\) & \(0.574\) & \(444.068\) & \(882.837\) \\ \(10\) & \(-0.438\) & \(-0.022\) & \(0.774\) & \(0.525\) & \(484.538\) & \(809.101\) \\ \(11\) & \(-0.297\) & \(-0.015\) & \(0.841\) & \(0.475\) & \(526.385\) & \(744.778\) \\ \(12\) & \(-0.160\) & \(-0.008\) & \(0.911\) & \(0.426\) & \(570.190\) & \(687.561\) \\ \(13\) & \(-0.026\) & \(-0.001\) & \(0.985\) & \(0.377\) & \(616.668\) & \(635.739\) \\  & \(\mathbf{0.000}\) & \(\mathbf{0.000}\) & \(\mathbf{1.000}\) & \(\mathbf{0.368}\) & \(\mathbf{626.131}\) & \(\mathbf{626.131}\) \\ \(14\) & \(0.107\) & \(0.005\) & \(1.065\) & \(0.328\) & \(666.766\) & \(587.973\) \\ \(15\) & \(0.243\) & \(0.012\) & \(1.153\) & \(0.279\) & \(721.817\) & \(543.130\) \\ \(16\) & \(0.384\) & \(0.019\) & \(1.252\) & \(0.230\) & \(783.852\) & \(500.145\) \\ \(17\) & \(0.535\) & \(0.027\) & \(1.368\) & \(0.181\) & \(856.261\) & \(457.851\) \\ \(18\) & \(0.704\) & \(0.035\) & \(1.510\) & \(0.132\) & \(945.484\) & \(414.645\) \\ \(19\) & \(0.910\) & \(0.046\) & \(1.704\) & \(0.083\) & \(1066.630\) & \(367.550\) \\  & \(\mathbf{1.094}\) & \(\mathbf{0.640}\) & \(\mathbf{1.897}\) & \(\mathbf{0.050}\) & \(\mathbf{1188.000}\) & \(\mathbf{330.000}\) \\ \(20\) & \(1.216\) & \(0.061\) & \(2.037\) & \(0.034\) & \(1275.318\) & \(307.406\) \\ \hline \hline \end{tabular} Bold: The principal stresses above and below the \(\eta\) parameter, the \(R(t)\) index of \(95\%\), and the \(R(t)\) index of the used \(4350\) Psi.  
\end{table} 
Table 4: Weibull statistics for vibration stress. In addition, from Section 4.2, the corresponding \(\beta\) and \(\eta\) values are, \(\beta\) = 1.7087 and \(\eta\) = 626.13 Psi. Step 2. Determine the strength average \(\sigma_{M}\) of the used material. In this work, the ABS polymeric used material has a \(\sigma_{M}\) = 3625 Psi. Step 3. In Equation (26) [20], the \(\sigma_{M}\), \(\mu\) and \(\eta\) values, the strength \(\eta_{s}\) scale parameter is,  
\[\eta_{s}=\eta\left(\frac{\sigma_{M}}{\mu}\right) \tag{26}\]  
\[\eta_{s}=626.31\left(\frac{3625}{759}\right)=2991.269\ \text{Psi}\]  
Then, from Equation (24) with the value \(\beta\) = 1.7087, the reliability \(R(t)\) is,  
\[R(t)=\frac{2991.269^{1.7087}}{2991.269^{1.7087}+626.13^{1.7087}}=0.935\]  
Thus, the component with a strength of 3625 Psi will have a reliability of \(R(t)\) = 0.935, when it is submitted to variant stress (random vibration). Now, let us present the steps to determine the cycle to failure (\(N\)) that corresponds to the random cycles to failure behavior.