## 2 Complex Load and Reliability Modeling
### Reliability Modeling under Complex Load  
In the reliability analysis, the Monte Carlo simulation method was used to conduct a large number of random samplings of the variables, and the numerical simulation was analyzed based on the sampling results. The reliability analysis is related to the random variables, including the material elastic modulus (\(EX\)), the material yield strength (\(R\)), the hook load (\(GZ\)), the wind pressure of the stand (\(W_{z}\_L\)), and the back wind pressure of each derrick section (\(W_{z}\_(i\)) (\(i\) = 0, 1, 2, 3, 4)). \(W_{z}\_L\) is the wind pressure of the stand and \(W_{z}\) (\(i\)) is the wind pressure of each section. \(W_{z}\_L=W_{0}k_{z}\), \(W_{z}(i)=W_{0}k_{z}\). The reliability analysis flow chart is shown in Figure 2.  
**Step 1**: Establish the finite element model of the offshore derrick by APDL and select one of the composite loads (See Appendix A). **Step 2**: Determine the number of cycles \(N\). To ensure that the mean and variance of the numerical simulation result samples are stable and accurate, a sufficient number of cycles should be selected. According to the relevant literature [15; 16], the number of cycles \(N\) selected in this paper was 300. **Step 3:** Input random variables and sample randomly. Randomly sample \(x\) from the inputted random variables \(X\) = (\(EX\), \(R\), \(GZ\), \(W_{z}\_L\), \(W_{z}\) (0), \(W_{z}\) (1), \(W_{z}\) (2), \(W_{z}\) (3), \(W_{z}\) (4)) for subsequent finite element analysis. **Step 4:** Strength analysis by APDL. The finite element simulation analysis is performed by inputting the sample \(x\). Obtain a 1\(\times\)\(M\) matrix A\({}_{1}\) = [\(\sigma_{j1}\), \(\sigma_{j2}\),..., \(\sigma_{jM}\)], where \(\sigma_{ji}\) is the stress of the \(i\)-th element in the \(j\)-th strength analysis (0 \(<i\leq M\)), and M is the total number of elements. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Location**} & **Area of Bearing** & **Wind Pressure** & **Shape Coefficient of Wind** \\  & **Wind Load F/m\({}^{2}\)** & **Coefficients \(k_{z}\)** & **Pressure \(k_{p}\)** \\ \hline Section IV & 33.83 & 1.64 & 0.338 \\ Section III & 50.45 & 1.50 & 0.246 \\ Section II & 67.08 & 1.33 & 0.203 \\ Section I & 83.71 & 1.07 & 0.183 \\ Bottom & 65.32 & 0.42 & 0.395 \\ Stand & 109.46 & 1.26 & 1.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Value of coefficients and area of bearing wind load. Repeat steps (3) and (4) \(N\) times to obtain the stress matrix A of \(N\) strength analysis under complex loads of all \(M\) elements; A is an \(N\times M\) matrix. \[\text{A}= \begin{pmatrix}\sigma_{11}&\sigma_{12}&\dots&\sigma_{1M}\\ \sigma_{21}&\sigma_{22}&\dots&\sigma_{2M}\\ \vdots&\vdots&\vdots&\vdots\\ \sigma_{N1}&\sigma_{N2}&\dots&\sigma_{NM}\end{pmatrix}  \tag{6}\]  
**Step 5**: Process the stress sample matrix A by Python. Substitute the matrix A into Equations (7)-(9) to obtain the matrices \(U\) and \(S\), which are composed of each element's stress mean and variance (See Appendix B). \[\frac{1}{N}\big{(}1\quad 1\quad\dots\quad 1\quad 1\big{)}\times\text{A}=\big{(} \mu_{1}\quad\dots\quad\mu_{M}\big{)}=U \tag{7}\]  
\[\Gamma=\text{A}- \begin{pmatrix}\mu_{1}&\mu_{2}&\dots&\mu_{M}\\ \mu_{1}&\mu_{2}&\dots&\mu_{M}\\ \vdots&\vdots&\vdots&\vdots\\ \mu_{1}&\mu_{2}&\dots&\mu_{M}\end{pmatrix}  \tag{8}\]  
\[\frac{1}{N}(\Gamma^{\top}\times\Gamma)= \begin{pmatrix}s_{1}^{2}&&\\ &\ddots&\\ &&s_{M}^{2}\end{pmatrix} =S \tag{9}\]  
**Step 6**: Calculate the reliability index of each element by Python. In the reliability model of the finite element, the stress-strength limit state function \(Z_{i}\) of the beam element in the derrick (element number is \(i\), \(0<i\leq\text{M}\)) is shown in Equation (10); see [17; 18; 19; 20]. The safety margin \(Z_{i}\) is a random variable, and it depends on the random variable of \(R\) (yield strength) and \(S_{i}\) (stress of \(i\)-th element). The parameters of this system are the mean of the safety margin \(\mu_{Zi}\) and standard deviation of the safety margin \(s_{zi}\).