# Combination of Variable Loads in Structural Design
## 4 Discussion  
The calculations here are based on dependent or independent load combinations depending on the nature of loads. However, the conclusions remain valid even when the dominant approach of independent combination is applied:  
* Although the combination of imposed loads is assumed to be independent, removing the combination factor is justified to avoid obvious contradictions. Such removal simplifies codes and does not affect the design outcome, as the load reduction can still be applied to the exposed area. * In this article, wind load is independently combined in the ULS. * Snow load is combined dependently here. If combined independently, slightly lower combination factors are obtained. In any case, the current factors in codes should be increased, as they are \(\psi_{0.5}\) values. However, the overall effect is negligible. These points illustrate that the approach to combining loads should be re-evaluated, particularly in light of the contradictions, potential simplifications, and cultivating international consensus [4] that can be achieved with improved safety. Current combination factors in structural codes are the lowest values obtained from a load proportion of 0.5. However, the code values should be higher since other load proportions must also be considered. If only the \(\psi_{0.5}\) values were used in codes, the outcome would be correct at a load proportion of 0.5 but unsafe by up to about 5% in every other case. This is illustrated in Figure 1, which includes a left load figure and a right error figure. The following can be seen in the left load figure:  
* The solid line represents the combination factor \(\psi\) as a function of load proportion \(\alpha\) = 0-50% in the ultimate case of a wind-imposed load combination, \(V_{\text{Q}}\) = 0.4, where the combination factor is lowest, \(\psi_{0.5}\) = 0.61 and \(\psi_{\text{code}}\) = 0.75. * The dashed line shows the total load assuming no permanent load is included in the combination, and the correct combination factor is applied for each design case, i.e., for each load proportion \(\alpha\). * The dash-dotted line shows the total load when the combination factor used in the codes is \(\psi_{\text{code}}\) = 0.75 (instead of the lowest value \(\psi_{0.5}\) = 0.61). The following can be seen in the right error figure:
* The dashed line indicates the error in the total load if the combination factor obtained from \(\alpha\) = 0.5, \(\psi_{0.5}\) = 0.61, is used in the codes. * The solid line shows the error when \(\psi_{\text{code}}\) = 0.75 is used in the codes. Figure 1 shows the following:  
* The combination factor in codes \(\psi_{\text{code}}\) should be higher than the lowest factor \(\psi_{0.5}\) at a load proportion of 0.5; an approximate relationship is \(\psi_{\text{code}}\approx(2\psi_{0.5}+1)/3\). * The current model for variable load calculation with only one parameter approximately includes up to about 7% safe and 2% unsafe errors. * Combination factors are effective in mitigating excess safety in variable load combinations, when the loads are of similar magnitude, typically within the range of load proportions \(\alpha\approx\) 0.3-0.5. In these scenarios, the calculation results in a reliability gain of 6-12% for the variable loads. When considering that the combination includes  
Figure 1: The solid line in the left load figure shows the combination factor \(\psi\) as a function of load proportion \(\alpha\); the dashed line shows the total load; the dash-dotted line shows the total load when \(\psi\) = 0.75. The solid line in the right error figure shows the error in the total load if factor \(\psi\) = 0.75 is selected for the codes; the dashed line shows the error when \(\psi\) = 0.61 is selected for the codes. 20-50% permanent load, the overall reliability gain ranges from 4 to 10% (the upper value is theoretical, light floor with equal wind and imposed load). However, load cases with proportions \(\alpha\approx\) 0.3-0.5 are rare.