# Combination of Variable Loads in Structural Design
## 2 Materials and Methods
### The Basic Equations  
The author revealed the safety factor calculation procedure in detail earlier [25; 29; 30; 31], and thus, it is explained only concisely here. Five equations are needed in the combination factor calculation. Safety factors, combination factors, and reliabilities can be obtained from these equations and assumed parameters without approximations, i.e., the used model is a fully probabilistic model without approximations.  
When a load \(L\) with the cumulative distribution \(F_{\text{L}}(x;\mu_{\text{L}},\sigma_{\text{L}})\) and safety factor \(\gamma_{\text{L}}\) affects a material with the resistance distribution \(f_{\text{M}}(x;\mu_{\text{M}},\sigma_{\text{M}})\), and safety factor \(\gamma_{\text{M}}\), the equation to calculate the safety factors \(\gamma_{\text{L}}\) and \(\gamma_{\text{M}}\) or the failure probability \(P_{\text{f}}\) is  
\[\int_{0}^{\infty}F_{\text{L}}\bigg{(}x;\frac{\mu_{\text{L}}}{\gamma_{\text{L} }},\frac{\sigma_{\text{L}}}{\gamma_{\text{L}}}\bigg{)}f_{\text{M}}(x;\mu_{ \text{M}}\gamma_{\text{M}},\sigma_{\text{M}}\gamma_{\text{M}})\text{d}x=1-P_{ \text{f}}. \tag{7}\]  
When one load \(L_{1}\) with the cumulative distribution \(F_{\text{1}}(x;\mu_{1},\sigma_{1})\) and safety factor \(\gamma_{1}\) and another load \(L_{2}\) with the density distribution \(f_{2}(x;\mu_{2},\sigma_{2})\) and safety factor \(\gamma_{2}\) is combined in load proportions \(\alpha\) and \(1-\alpha\), the equation to calculate the independent combination load distribution \(F_{\text{i}}\) is  
\[F_{\text{i}}(x;\alpha)=\int_{-\infty}^{\infty}F_{\text{L}}\bigg{(}x-r;\frac{ \mu_{1}\alpha}{\gamma_{1}},\frac{\sigma_{1}\alpha}{\gamma_{1}}\bigg{)}f_{2} \bigg{(}r;\frac{\mu_{2}(1-\alpha)}{\gamma_{2}},\frac{\sigma_{2}(1-\alpha)}{ \gamma_{2}}\bigg{)}\text{d}r. \tag{8}\]  
The dependent distribution \(F_{\text{d}}\) is calculated as follows:  
\[F_{\text{d}}(x;\alpha)=\left|\begin{array}{c}y\leftarrow\text{root}\Big{[} F_{\text{I}}\Big{[}x-r;\frac{\mu_{1}\alpha}{\gamma_{1}},\frac{\sigma_{1} \alpha}{\gamma_{1}}\Big{]}-F_{2}\Big{[}r;\frac{\mu_{2}(1-\alpha)}{\gamma_{2}}, \frac{\sigma_{2}(1-\alpha)}{\gamma_{2}}\Big{]},r\Big{]}\\ F_{\text{d}}\gets F_{2}\Big{[}y;\frac{\mu_{2}(1-\alpha)}{\gamma_{2}}, \frac{\sigma_{2}(1-\alpha)}{\gamma_{2}}\Big{]}\end{array}\right.. \tag{9}\]  
When two variable loads \(1\) and \(2\) with the parameters set at unity are combined, the equation to calculate the independent combination factor \(\psi\) using the ideal material as a reference is  
\[\int_{-\infty}^{\infty}F_{\text{I}}\bigg{(}1-r;\frac{\mu_{1}\alpha}{\gamma_{1 }\psi},\frac{\sigma_{1}\alpha}{\gamma_{1}\psi}\bigg{)}f_{2}\bigg{(}r;\frac{\mu _{2}(1-\alpha)}{\gamma_{2}},\frac{\sigma_{2}(1-\alpha)}{\gamma_{2}}\bigg{)} \text{d}r=1-P_{f}. \tag{10}\]The equation to calculate the dependent combination factor is correspondingly  
\[\begin{split}\left|y\leftarrow\text{root}\Big{[}F_{1}\Big{[}1-r;\frac{ \mu_{1}a}{\gamma_{1}\Psi},\frac{\sigma_{1}a}{\gamma_{1}\Psi}\Big{]}-F_{2}\Big{[} r;\frac{\mu_{2}(1-a)}{\gamma_{2}},\frac{\sigma_{2}(1-a)}{\gamma_{2}}\Big{]},r \Big{]}\right.=1-P_{f}.\end{split} \tag{11}\]