# 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 result of the ASCE method is obviously greater than the other three methods and much greater than the target reliability, while the results of the existing method and the proposed method are close to the target reliability. The reliability calculated with the proposed method is closest to the \(\beta_{T}\), although it is a little smaller than \(\beta_{T}\) when \(\beta_{T}\) is great. In order to investigate the sensitivity of load in the four methods, the influence of the snow load (\(S_{n}/D_{n}\) = 0.5-2.5) and dead load (\(L_{n}/D_{n}\) = 0.5-2.5) is considered. As shown in  
Figure 4: Reliability calculated with different methods for \(\beta_{T}\) = 1.0–4.5. Figure 3: Load and resistance factors calculated using different methods for \(\beta_{T}\) = 1.0–4.0. Figure 5, the results of the Mori method are greater than \(\beta_{T}\) when \(S_{n}/D_{n}\) = 0.5-2.2 and the error is stable. When \(S_{n}/D_{n}\) = 2.3-2.5, the results become much smaller than \(\beta_{T}\). The results of the other three methods are much greater than the target reliability 3.0, in which the result of the proposed method is relatively close to 3.0. In Figure 6, the results of the existing method and the proposed method are much closer to \(\beta_{T}\), compared to the other two methods. In the case of \(L_{n}/D_{n}\) = 0.5-1.4, the reliability calculated using both methods are greater than \(\beta_{T}\), and the error of the proposed method is relatively smaller than that of the existing method. In the case of \(L_{n}/D_{n}\) = 1.5-2.5, the results of the proposed method are slightly smaller than \(\beta_{T}\), while the reliability calculated using the existing method is slightly greater than \(\beta_{T}\). For the ASCE method, the reliability increases to a peak when \(L_{n}/D_{n}\) reaches 0.6, then falls as \(L_{n}/D_{n}\) increases, especially from 0.6-0.7. The results of the Mori method increase initially in the case of \(L_{n}/D_{n}\) = 0.5-1.2, and then drop gently until \(L_{n}/D_{n}\) = 2.2. Afterwards, the reliability inclines sharply from 2.2-2.3, but keeps close to \(\beta_{T}\) from 2.4-fig2.5. In light of Figures 4-6, 3M methods achieve higher precision compared to the ASCE method and the Mori method. In Figure 4, it can be seen that the reliabilities calculated by the proposed method and the existing 3M method are close to \(\beta_{T}\) when \(\beta_{T}\) = 1.0-4.5. In Figures 5 and 6, considering the effect of \(L_{n}/D_{n}\) and \(S_{n}/D_{n}\), the reliability obtained using the existing method is moderately greater than \(\beta_{T}\), but with the inevitable iterative computation for the target mean resistance. Moreover, the existing method cannot be universally deployed due to the limitation of Equation (11). Although reliability computation based on the proposed method might be slightly smaller than \(\beta_{T}\) in some cases, it has two intrinsic advantages: (1) No mathematical limitation; (2) No iteration calculation. Particularly, higher efficiency can be achieved when employing the proposed method for system reliability, and the error is also acceptable. Therefore, the proposed method is considered the most applicable for practical structural reliability design. Figure 5: Reliability calculated by different methods for \(S_{n}/D_{n}\) = 0.5-2.5 (\(\beta_{T}\) = 3).