# Improved Calculation of Load and Resistance Factors Based on Third-Moment Method
## 2 Computation Procedure of Load and Resistance Factors
### Existing 3M Method for the Computation of Load and Resistance Factors  
\(R^{*}\) and \(Q_{i}^{*}\) are obtained using derivative-based iterations, but explicit expressions of \(R^{*}\) and \(Q_{i}^{*}\) are not available. Some simplifications have been provided to simplify iterative computations. In the existing 3M method based on 3P-lognormal distribution [19], the two-step recursive optimization is used to avoid iterative computation. The process of calculation is presented as follows. In the existing paper, the third-moment (3M) reliability index is employed as the reliability index for the performance function of Equation (3). Substituting the 3M reliability index in the design format obtains:  
\[\beta_{3M}\geq\beta_{T} \tag{10}\]  
where the 3M reliability index, \(\beta_{3M}\), is expressed as [19]:  
\[\beta_{3M}=-\frac{\alpha_{3Z}}{6}-\frac{3}{\alpha_{3Z}}\ln(1-\frac{1}{3} \alpha_{3Z}\beta_{2M}) \tag{11}\]  
\[\alpha_{3Z}=\frac{1}{\sigma_{Z}^{2}}(\alpha_{3R}\sigma_{R}^{3}-\sum\alpha_{3i} \sigma_{Qi}^{3}) \tag{12}\]  
where \(\alpha_{3Z}\), \(\alpha_{3R}\), and \(\alpha_{3i}\) are the skewness of \(G\)(X), \(R\), and \(S_{i}\), respectively. The skewness of random variable can be calculated by:  
\[\alpha_{3}=\frac{E(X-\mu_{X})^{3}}{\sigma_{X}^{3}} \tag{13}\]  
where \(\mu_{X}\) and \(\sigma_{X}\) are the mean value and standard deviation of random variable \(X\), respectively. Substituting Equation (11) into Equation (10), one obtains:  
\[\beta_{2M}\geq\frac{3}{\alpha_{3Z}}\Big{\{}1-\exp\Big{[}\frac{\alpha_{3Z}}{3} \Big{(}-\beta_{T}-\frac{\alpha_{3Z}}{6}\Big{)}\Big{]}\Big{\}} \tag{14}\]  
Denoting the right-hand side of Equation (14) as \(\beta_{2T}\), one obtains:  
\[\beta_{2M}\geq\beta_{2T} \tag{15}\]  
\[\beta_{2T}=\frac{3}{\alpha_{3Z}}\Big{\{}1-\exp\Big{[}\frac{\alpha_{3Z}}{3} \Big{(}-\beta_{T}-\frac{\alpha_{3Z}}{6}\Big{)}\Big{]}\Big{\}} \tag{16}\]  
Equation (15) is equivalent to Equation (10). It indicates that if \(\beta_{2M}\) is at least equal to \(\beta_{2T}\), \(\beta_{3M}\) will be at least equal to \(\beta_{T}\), and the reliability-based design conditions will be satisfied. Hence, \(\beta_{2T}\) and \(\beta_{T}\) can be regarded as the target values of \(\beta_{2M}\) and \(\beta_{3M}\), respectively. Since Equation (15) is equivalent to Equation (10), except that the right-hand side is \(\beta_{2T}\), the load and resistance factors can be easily obtained by substituting \(\beta_{T}\) in Equation (6) with \(\beta_{2T}\). The design formula becomes:  
\[\mu_{R}(1-\alpha_{R}V_{R}\beta_{2T})\geq\sum\mu_{Q_{i}}(1+\alpha_{Q_{i}}V_{Q_ {i}}\beta_{2T}) \tag{17}\]  
and the load and resistance factors are obtained as:  
\[\phi=(1-\alpha_{R}V_{R}\beta_{2T})\frac{\mu_{R}}{R_{n}} \tag{18a}\]  
\[\gamma_{i}=(1+\alpha_{Q_{i}}V_{Q_{i}}\beta_{2T})\frac{\mu_{Q_{i}}}{Q_{ni}} \tag{18b}\]  
At the limit state, according to Equation (5a,b), one obtains:  
\[\beta_{3M}=-\frac{\alpha_{3Z}}{6}-\frac{3}{\alpha_{3Z}}\ln(1-\frac{1}{3} \alpha_{3Z}\beta_{2M}) \tag{11}\]0  
The initial value, \(\mu_{R_{0}}\), is expressed as [19]:  
\[\beta_{3M}=-\frac{\alpha_{3Z}}{6}-\frac{3}{\alpha_{3Z}}\ln(1-\frac{1}{3} \alpha_{3Z}\beta_{2M}) \tag{11}\]1  
Based on the discussion above, the target mean resistance calculated using two-step iteration is expressed as:  
\[\beta_{3M}=-\frac{\alpha_{3Z}}{6}-\frac{3}{\alpha_{3Z}}\ln(1-\frac{1}{3} \alpha_{3Z}\beta_{2M}) \tag{11}\]2  
where \(\mu_{RT}\) is the target mean resistance; \(\beta_{2T_{0}}\) is the original target 2M reliability; \(\sigma_{Z_{0}}\) is the original standard deviation of G(\(X\)); \(\sigma_{Z_{0}}\) and \(\beta_{2T_{0}}\) are obtained using Equations (5a,b) and (16). The steps for determining the load and resistance factors using this method are shown below:  
Calculate \(\mu_{R0}\) using Equation (20). Calculate \(\sigma_{Z0}\), \(\alpha_{3Z0}\), and \(\beta_{2T0}\) using Equations (5a,b), (12), and (16), respectively. Calculate \(\mu_{RT}\) with Equation (21).