# Combination of Variable Loads in Structural Design
## 2 Materials and Methods
### The Reference Calculation  
The Eurocodes [1; 2], especially the section on 'combination of actions' regarding variable loads, are used as a reference. However, the Eurocodes' safety factors and parameters do not correspond with the target reliability, with the reliability of variable loads being particularly below the target [25]. Therefore, the safety factors are calculated here based on a reliability model which matches the target reliability in the reference calculation, i.e., the calculation does not include errors in the safety factor and parameter settings regarding the target reliability.  
In codes, the combination factors, though different, are the same for all materials [26; 27; 28]. Here, timber is selected as the reference material as it has a mean variability of \(V_{\text{M}}=0.2\), compared to \(V_{\text{M}}=0.1\) for steel and \(V_{\text{M}}=0.3\) for concrete. A check is performed to confirm that the combination factors based on timber are comparable to those obtained based on ideal material, \(V_{\text{M}}=0\), steel, and concrete, too.  
In the SLS, the calculations are based on the ideal material, \(V_{\text{M}}=0\).  
In codes, the combination factors are the same regardless of the amount of permanent load in the combination. In this study, the combination factors are first calculated without the permanent load. A check is then made to confirm the calculation outcome remains as long as the permanent load is added. To simplify calculations, the reliability model is designed such that the material factor remains constant for both permanent and variable loads. This means that the material factor for timber is not affected by the amount of permanent load in the load combination.  
The load safety factors can be fixed arbitrarily and setting the permanent load factor the same as the variable load factor, i.e., \(\gamma_{\text{G}}=\gamma_{\text{Q}}\) is feasible as it simplifies calculation. In the author's opinion, \(\gamma_{\text{G}}=\gamma_{\text{Q}}=1\) would be the best option. However, \(\gamma_{\text{G}}=\gamma_{\text{Q}}=1.2\) produces a similar outcome and is selected here as it corresponds to the common partial safety factor design model.  
The basic calculation is that variable loads \(1\) and \(2\) are combined in proportions \(\alpha\) and \(1-\alpha\), \(\alpha=\mu_{1}/(\mu_{1}+\mu_{2})\) where \(\mu_{1}\)and \(\mu_{2}\) are the means of the loads. A combination factor is fixed in one load and an equation for the material factor of timber \(\gamma_{\text{M}}\) is set for one-year loads. The condition for this design is that the safety factor for the service time combination load must be equal to the safety factor for a single load where the combination factor \(\psi\) is solved. A check is then made to confirm the obtained combination factor is the same or approximately the same as a permanent load is added. The same check is carried out when the calculation is made for ideal material, steel, and concrete. With the loads being often similar it is irrelevant to ascertain load the combination factor is fixed. If the loads are not similar, the combination factor is fixed to both loads and the bigger factor is selected.