## 3 TR Rule and Its Approximate Combination Rules
### TR1 Combination Rule and Simplified Combination Coefficients  
In the actual structure design, the probability distribution function of the combined effect is difficult to obtain, but the lower bound of the combination extreme can be obtained from the upper bound of the probability distribution of the combined effect, and the mean value of the lower bound of the combination extreme can be calculated according to the conditional probability. Then, the simplified combination Equation of the TR1 rule of the combination effect is derived. Take the extreme value combination of two effect components as an example. From the research of [PERSON] et al. (1986), the probability distribution function of the extreme value of the effect component scalar sum has the following relationship:  
\[F_{\hat{X}_{1}\hat{X}_{2}}(x,x)\leq\text{min}\big{\{}F_{\hat{X}_{1}},F_{\hat{X }_{2}}\big{\}}, \tag{22}\]  
In the Equation, \(F_{\hat{X}_{1}}\) and \(F_{\hat{X}_{2}}\) are the probability distribution functions of \(\hat{X}_{1}\) and \(\hat{X}_{2}\), respectively. When \(R_{1}(t)\) reaches the maximum, the derivative of the conditional probability distribution of \(R_{2}(t)\) is the conditional probability density \(f_{R_{2}|R_{1}}(R_{2})\). The conditional probability density is equal to the ratio of the joint probability density to the conditional edge probability density. In the case of a one-sided safety limit, \(f_{R_{2}|R_{1}}(R_{2})\) is:  
\[f_{R_{2}|R_{1}}(R_{2})=\frac{1}{\sqrt{2\pi}\cdot\Big{(}\sigma_{2}\cdot\sqrt{1- \rho_{12}^{2}}\Big{)}}\cdot\exp\Bigg{[}\frac{-1}{\Big{(}2\sigma_{2}\cdot\sqrt {1-\rho_{12}^{2}}\Big{)}^{2}}\cdot\bigg{(}R_{2}-\bigg{(}\mu_{2}+\rho_{12}^{2} \cdot\frac{\sigma_{2}}{\sigma_{1}}(R_{1\text{max}}-\mu_{1})\bigg{)}\bigg{)}^{2 }\Bigg{]}, \tag{23}\]Calculate the expectation of Equation (23), and obtain the conditional probability expectation value of \(R_{2}(t)\):  
\[E\Big{[}R_{2}^{(1)}\Big{]} =\mu_{2}+\rho_{12}\cdot\frac{\sigma_{2}}{\sigma_{1}}(R_{1\max}-\mu_ {1}) \tag{24}\] \[=\mu_{2}+\rho_{12}g\sigma_{2}\]  
In Equations (20) and (21), when \(n\) is equal to 2, the lower bound \(R_{\max}\) of the combined effect extreme value of the two effect components combined is derived according to the TR1 rule:  
\[R_{\max}=\max\left\{\hat{X}_{1},\hat{X}_{2}\right\}, \tag{25}\]  
\[\hat{X}_{1}=R_{1\max}+E\Big{[}R_{2}^{(1)}\Big{]}, \tag{26}\]  
\[\hat{X}_{2}=R_{2\max}+E\Big{[}R_{1}^{(2)}\Big{]}, \tag{27}\]  
Substituting Equation (24) into Equation (26), we get:  
\[ \begin{array}{ll}\hat{X}_{1}&=R_{1\max}+\mu_{2}+\rho_{12}g\sigma_{2}\\ &=R_{1\max}+\lambda_{2}\cdot R_{2\max}\end{array}  \tag{28}\]  
In the Equation, \(\lambda_{2}\) is called the combination coefficient of extreme value \(R_{2\max}\), and its expression is as follows:  
\[\lambda_{2}=1-\frac{\left(1-\rho_{12}\right)}{\frac{\mu_{2}}{\hat{g}\sigma_{2 }}+1}, \tag{29}\]  
Similarly:  
\[\hat{X}_{2}=\lambda_{1}\cdot R_{1\max}+R_{2\max}, \tag{30}\]  
In the Equation, \(\lambda_{1}\) is called the combination coefficient of extreme value \(R_{1\max}\), and its expression is as follows:  
\[\lambda_{1}=1-\frac{\left(1-\rho_{21}\right)}{\frac{\mu_{1}}{\hat{g}\sigma_{1 }}+1}, \tag{31}\]  
Substitute Equations (28) and (30) into Equation (25) to obtain a simplified combination Equation of the TR1 rule for the extreme value combination of the two effect components  
\[R_{\max}=\max\left\{R_{1\max}+\lambda_{2}\cdot R_{2\max},\lambda_{1}\cdot R_{ 1\max}+R_{2\max}\right\}, \tag{32}\]  
If the mean values of \(R_{1}(t)\) and \(R_{2}(t)\) are both zero, the combination coefficient is equal to \(\rho_{12}\), and the combination coefficient under this specific condition is identical to the combination coefficient proposed by [PERSON] [4]. The combination coefficients of the extreme values of the two effect components combined according to the TR1 rule are proposed above, and the simplified Equation for combination of the two effect components is derived according to the TR1 rule.