# Improved Calculation of Load and Resistance Factors Based on Third-Moment Method
## 2 Computation Procedure of Load and Resistance Factors  
### Determination of Load and Resistance Factors  
Based on the LRFD format, the performance function in structural design can be expressed as:  
\[\phi R_{n}\geq\sum\gamma_{i}Q_{ni} \tag{1}\]  
where \(\phi\) is the resistance factor, \(\gamma_{i}\) is the partial load factor applied to load, \(R_{n}\) is the nominal value of the resistance, and \(Q_{ni}\) is the nominal value of the load.  
In reliability-based design, the load and resistance factors \(\phi\) and \(\gamma_{i}\) should be determined with a specified reliability, namely target reliability. Therefore, the computation of Equation (1) should be achieved by a performance function, as presented below:  
\[G(X)=R-\sum Q_{i} \tag{2}\]  
where \(R\) and \(Q_{i}\) are random variables representing uncertainty in the resistance and load effects, respectively.  
For a given target reliability, \(\beta_{T}\), or target failure probability, \(P_{fT}\), Equation (2) can be expressed as:  
\[\beta\geq\beta_{T}\text{ or }P_{f}\leq P_{fT} \tag{3}\]  
where \(\beta\) and \(P_{f}\) are the reliability and the failure probability, respectively.  
If \(R\) and \(Q_{i}\) are mutually independent normal random variables, the second-moment (2M) method is correct and the design formula is expressed as:  
\[\beta_{2M}\geq\beta_{T} \tag{4}\]  
\[\beta_{2M}=\frac{\mu_{Z}}{\sigma_{Z}} \tag{5a}\] \[\mu_{Z}=\mu_{R}-\sum\mu_{Q_{i}},\sigma_{Z}=\sqrt{\sigma_{R}^{2}+\sum\sigma_{Q_{ i}}^{2}} \tag{5b}\]where \(\beta_{2M}\) is the 2M reliability index; \(\mu_{Z}\) and \(\sigma_{Z}\) are the mean value and standard deviation of the performance function \(G\)(X), respectively; \(\mu_{R}\) and \(\sigma_{R}\) are the mean value and standard deviation of deviation of \(R\), respectively; \(\mu_{Qi}\) and \(\sigma_{Qi}\) are the mean value and standard deviation of \(Q_{i}\), respectively.  
Substituting Equation (5a,b) in Equation (4), the load and resistance factors can be expressed as:  
\[\mu_{R}(1-\alpha_{R}V_{R}\beta_{T})\geq\sum\mu_{Q_{i}}(1+\alpha_{Q_{i}}V_{Q_{ i}}\beta_{T}) \tag{6}\]  
Comparing Equation (6) with Equation (1), the load and resistance factors can be expressed as:  
\[\phi=(1-\alpha_{R}V_{R}\beta_{T})\frac{\mu_{R}}{R_{n}} \tag{7a}\] \[\gamma_{i}=\big{(}1+\alpha_{Q_{i}}V_{Q_{i}}\beta_{T}\big{)}\frac{\mu_{Q_{i}}} {Q_{ni}} \tag{7b}\]  
where \(V_{R}\) and \(V_{Qi}\) are the coefficient of variation for \(R\) and \(Q_{i}\), respectively; \(\alpha_{R}\) and \(\alpha_{Qi}\) are the sensitivity coefficients of \(R\) and \(Q_{i}\), respectively, where:  
\[\alpha_{Q_{i}}=\frac{\sigma_{Q_{i}}}{\sigma_{Z}},\ \alpha_{R}=\frac{\sigma_{R}}{ \sigma_{Z}} \tag{8}\]  
As introduced above, the 2M method is based on the assumption that all of the variables obey normal distribution and are independent of each other. When \(R\) and \(Q_{i}\) are other random variables, the 2M reliability computed by Equation (5a) is inaccurate. Therefore, other methods were proposed. Typically, the FORM is widely implemented, in which the load and resistance factors can be obtained as:  
\[\phi=\frac{R^{*}}{R_{n}},\ \gamma_{i}=\frac{Q_{i}^{*}}{Q_{ni}} \tag{9}\]  
where \(R^{*}\) and \(Q_{i}^{*}\) are the values of the variables \(R\) and \(Q_{i}\), respectively, at the design point of the FORM.