# Combination of Variable Loads in Structural Design
## 3 Results
### Combination Factors in the SLS  
The target reliability index of the Eurocodes in the SLS is \(\beta_{1}=2.9\), i.e., \(P_{\text{f,50}}=1/11\). The characteristic load for the variable load is the same in the SLS and in the ULS, i.e., it is the 50-year return load. As explained above, the imposed loads are combined dependently without a combination factor, \(\psi=1\) as the loads are correlated. Two independent loads like imposed load and wind combined in the SLS with each other have an unrealistically high combination factor \(\psi=1.1\). This high value means that the combination load is higher than the arithmetic sum of the partial loads. The high value is attributed to a mismatch between the characteristic variable load and the reliability index \(\beta_{1}=2.9\). This contradiction diminishes if the characteristic load is changed for a longer return time [25] and the robust upper tail of the Gumbel distribution is cut off. If the calculation is made for ideal parameter settings for \(\beta_{1}=2.9\) and for the Gumbel distribution, the parameters are \(\mu_{Q}=0.360\) and \(\sigma_{Q}=0.144\), i.e., the return time of the characteristic load is 536 years and the combination factor is \(\psi=0.65\), meaning that the code value is about \(\psi=0.80\). The combination factors for snow in the SLS are unrealistically high if the current load parameters are used and about the same as in the ULS if ideal parameters are used. The calculations above indicate that the current approach to load combination in the SLS is questionable, as the fixed \(\beta_{1}\) value and the characteristic load of the variable load arbitrarily affect the outcome. A new perspective suggests that SLS loads should be combined without using a constraint, like \(\beta\), to guide the combination. In the ULS, whether there is one variable load or multiple variable loads, all loads are multiplied by a safety factor defined by \(\beta\), making \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Time/** & **One** & **One** & **One** & **Two** & **Four** & **One** \\ \(\psi\) & **Day** & **Week** & **Month** & **Months** & **Months** & **Year** \\ \hline \(\psi_{\text{d,0.5}}\) & 0.51 & 0.67 & 0.79 & 0.86 & 0.92 & 1.00 \\ \(\psi_{\text{d,code}}\) & 0.70 & 0.80 & 0.85 & 0.90 & 0.95 & 1.00 \\ \(\psi_{\text{b,0.5}}\) & 0.30 & 0.39 & 0.47 & 0.50 & 0.54 & 0.61 \\ \(\psi_{\text{i,code}}\) & 0.55 & 0.60 & 0.65 & 0.65 & 0.70 & 0.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The dependent combination factors for snow for timber \(\psi_{\text{d,0.5}}\), highlighted code factors \(\psi_{\text{d,code}}\), the corresponding factors calculated independently \(\psi_{\text{b,0.5}}\), and code factors \(\psi_{\text{i,code}}\). the combination load always represent the extreme load. Conversely, in the SLS, when one variable load is combined with the permanent load, the mean loads are added up, and no combination factor or constraint is applied. However, when the permanent load is combined with multiple variable loads, combination factors apply, meaning that the combination load is the extreme load. The use of combination factors imposes a constraint on the load combination through the fixed \(\beta_{1}\) value, which is flawed. Instead, SLS loads are mean loads which are combined by adding up the loads as such regardless of whether the loads are independent or dependent or the combination is dependent or independent. No constraint, such as a combination factor, should be applied when combining loads in the SLS that act full-time. If one load acts year-round while the other acts part-time, like snow, a combination factor is applied due to conversion of the distributions to the same active time, with one-month, one-week, and one-day \(\psi_{0.5}\) values being 0.6, 0.4, and 0.1, respectively, obtained from dependent combinations. If both loads act part-time or intermittence is considered, a further reduction may be needed. Setting these factors at unity is also desirable.