# Improved Calculation of Load and Resistance Factors Based on Third-Moment Method
## 4 Comparison of the ASCE Method, the Existing 3M Method, the Proposed Method, and the Mori Method
#### Results Comparison of Four Methods  
The results of load and resistance factors obtained using different methods are listed in Table 4. The results show that the resistance factors, \(\phi\), are in great agreement. The results of the dead load factor, \(\gamma_{D}\), are similar, while the live load factor, \(\gamma_{L}\), determined using the Mori method is obviously greater than those determined using the other methods, and the snow load factor, \(\gamma_{S}\), obtained using the Mori method is the smallest one compared to the results obtained using other methods. In order to compare the accuracy of the four methods, the MC simulation (100,000 times when \(\beta_{T}\leq 4\) and 1,000,000 times when \(\beta_{T}>4\)) is used to calculate the reliability index. For the MC simulation, the calculation of \(\mu_{R}/D_{n}\) is necessary (e.g., the proposed method):  
\[\frac{\mu_{R}}{D_{n}}=\frac{\mu_{R}}{R_{n}}.\frac{R_{n}}{D_{n}}=\frac{\mu_{R}} {R_{n}}.\frac{1}{\phi}\bigg{(}\gamma_{D}.\frac{D_{n}}{D_{n}}+\gamma_{L}.\frac {L_{n}}{D_{n}}+\gamma_{S}.\frac{S_{n}}{D_{n}}\bigg{)}=3.05\]  
With the values of \(\mu_{R}/D_{n}\mu_{D}/D_{n}\mu_{L}/D_{n}\), and \(\mu_{S}/D_{n}\), the results of the MC simulation are shown in Table 5; reliability of 3.01 of the proposed method is closest to the target reliability of 3.0. Therefore, the proposed method is considered accurate enough. The existing method is also accurate and safe, but with some limitations, and iterative computation is inevitable. The ASCE method is simpler, but the reliability computed by this method is much greater than the target reliability, which is safe but wastes a lot of structural materials. Conversely, the reliability calculated by the Mori method is much smaller than the target reliability, indicating that the design is not safe. In ASCE 7-10 (1.5.1), structures are classified into four different risk categories. For different risk categories and damage types, the acceptable reliability indexes (\(\beta_{T}\) =2.5-4.5) are provided in ASCE 7-10 C1.3.1 for a 50-year service period. In this paper, \(\beta_{T}\) =1.0-4.5 is chosen to analyze the application of four different methods. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\phi\) & \(\gamma_{D}\) & \(\gamma_{L}\) & \(\gamma_{S}\) \\ \hline ASCE method & 0.877 & 1.600 & 0.598 & 1.229 \\ Existing 3M method & 0.865 & 1.439 & 0.500 & 1.191 \\ Proposed method & 0.848 & 1.383 & 0.481 & 1.164 \\ Mori method & 0.924 & 1.328 & 1.352 & 0.444 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of load and resistance factors calculated using different methods. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**RVs** & \(\mu_{Q}/D_{n}\) & \(V_{i}\) & \(\sigma_{Q}=\mu_{Q}/D_{n}\)-\(V_{i}\) & \(\alpha\beta i\) & \(\mu_{R}/R_{n}\) or \(\mu_{Qi}/Q_{in}\) & \(Q_{in}/D_{n}\) & **Distribution** \\ \hline \(R\) & – & 0.09 & – & 0.27 & 1.06 & – & Lognormal \\ \(D\) & 1 & 0.25 & 0.25 & 0 & 1.0 & 1 & Normal \\ \(L\) & 0.175 & 0.59 & 0.103 & 1.18 & 0.35 & 0.5 & Gamma \\ \(S\) & 0.6874 & 0.21 & 0.144 & 1.14 & 0.982 & 0.7 & Gumbel \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Basic random variables for Equation (44). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **ASCE Method** & **Existing 3M Method** & **Proposed Method** & **Mori Method** \\ \hline \(\beta\) & 3.53 & 3.08 & 3.01 & 2.656 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Reliability of MC simulation with different methods. As shown in Figure 3, load and resistance factors calculated using the existing method, the ASCE method, the proposed method, and the Mori method are inconsistent. In addition, the difference increases with the increase in the target reliability, \(\beta_{T}\). For resistance factor, the difference among the four methods is slight. Figure 4 displays the comparison of reliability calculated by different methods for \(\beta_{T}\) = 1.0-4.5.