## 3 Proposed Stress-Strength Method
### Numerical Application of the Proposed Method
#### 3.3.1 Stress Distribution Based on the Time-Dependent Stress
Because the changes of the deflections \(\delta_{i}\) generate the internal bending moment \(M_{\max-t=i}\) to change, then for any \(\delta_{i}\), by performing the static analysis the corresponding \(M_{\max-t=i}\) must be estimated and utilized to determine its equivalent \(\sigma_{eq-i}\) value (See Equation (32)). In Table 2, the \(\sigma_{eq-i}\) value that corresponds to \(\delta_{i=10.5}\) is also given. Moreover, because the strain \(\varepsilon_{i}\) determines the equivalent stress \(\sigma_{eq-i}\), then the maximum strain \(\varepsilon_{max}\) that corresponds to \(N_{f}=10^{6}\) is needed. Based on this, let us develop the numerical application. _Step 6_. Based on [PERSON] et al. [20], the data for the stress-strain analysis is shown in Table 3.  
In Table 3, the strain values are dimensionless and the stress data is given MPa. After this analysis, the stress-strain behavior of the material was determined. But, due to the analysis of the main cycles is performed at \(N_{f}=10^{6}\), then the strain-life curve is given next. _Step 7_. Based on the true stress-true strain values given in Table 3, the strain-life curve \(\varepsilon N\) analysis is as follows. _Step 7.1_. Based on the true stress-true strain values, the stress amplitude and the constant \(b\) value are determined as follows:  
\[S_{a}=S_{f}^{\prime}\left(2N_{f}\right)^{b} \tag{28}\]  
By taking \(S_{a}=S_{f}^{\prime}\) at \(N_{f}=1\), and by considering that \(N(S_{a})_{10^{6}}=S_{e}\)[2,25], from Equation (28), the constant \(b\) value is given as:  
\[b=\frac{\log\left(\frac{\sigma_{a}}{\sigma_{f}^{\prime}}\right)}{\log\left(2N _{f}\right)} \tag{29}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Time \(f(t)\)** & **Time \(t=i\) (s)** & \(E\) (\(t=i\)) & \(\delta_{max}\) & \(\varepsilon_{(t=i)}\) & \(\sigma_{1(b)}\) & \(\sigma_{eq-i}\) \\
**Equation (8)** & **Equation (14)** & **Equation (7)** & **Equation (13)** & **Step 8** \\ \hline
0.00 & 0 & 213.00 & 0.0227465 & 0.0004969 & 105.84 & \\
30 days & 2,592,000 & 211.77 & 0.0228782 & 0.0004998 & 105.24 & \\
0.5 year & 15,768,000 & 205.75 & 0.0235472 & 0.0005144 & 102.36 & \\
1 year & 31,536,000 & 198.99 & 0.0243478 & 0.0005319 & 99.31 & \\
5 years & 157,680,000 & 157.54 & 0.0307527 & 0.0006718 & 83.98 & \\
10 years & 315,360,000 & 125.00 & 0.0387589 & 0.0008467 & 74.90 & \\
**10.50 years** & 330,976,627 & 122.49 & **0.0395518** & **0.0008640** & **74.28** & **184.03** \\
25 years & 788,400,000 & 77.17 & 0.0627775 & 0.0013714 & 64.63 & \\
50 years & 1,576,800,000 & 47.12 & 0.1028084 & 0.0022459 & 59.50 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Time-dependent stress data. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Tested Stress** & **Tested Strain** & **True Strain** & **True Stress** \\ \hline \(S_{pl}\) & 398.0000 & 0.002002 & \(S_{pl}^{\prime}\) & 0.002000 & 398.0000 \\ \(S_{y}\) & 427.0000 & 0.004008 & \(S_{y}^{\prime}\) & 0.004000 & 427.0000 \\ \(S_{ut}\) & 578.0000 & 0.163113 & \(S_{ut}^{\prime}\) & 0.151100 & 681.0000 \\ \(S_{f}\) & 488.0000 & 0.220200 & \(S_{f}^{\prime}\) & 0.199015 & **709.5745** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Data of the selected structural element. Consequently, \(b=-0.093\), and by substituting it into Equation (28), the stress amplitudes \(S_{a}\) values are shown in Table 4.  
In Table 4, the strain values are dimensionless and the stress data is given MPa.