# Combination of Permanent and Variable Load Is Dependent
## 2 Materials and Methods
### Load Groups  
Assume that a roof has n similar girders spaced equally apart from each other, designed for 50 years in one process, i.e., one design and one design load combination apply to all girders. Each girder i has different permanent load \(\mathrm{g_{i}}\) which is the self-weight of the girder plus the weight of the roof structure in the area of the girder length by the spacing. One of the n girders has the maximum permanent load \(\mathrm{g_{max}}\).  
In this case, the variable load is assumed to be a layer of snow uniformly distributed over the roof, as assumed in the codes and as is typically the case in the reality.  
Each year j of the service time has different variable load \(\mathrm{q_{j,i}}\) affecting girder i which is the actual snow load multiplied by the impact area of the girder i. The maximum snow load \(\mathrm{q_{max}}\) occurs one year j into the service life. _This load \(\mathrm{q_{max}}\) occurs simultaneously with all the girders of the roof, including the one with the maximum permanent load \(\mathrm{g_{max}}\)._ Consequently, the maximum loads \(\mathrm{g_{max}}\) and \(\mathrm{q_{max}}\) are simultaneous and the maximum combination load is \(\mathrm{g_{max}}\)\(=\)\(\mathrm{g_{max}}\)\(+\)\(\mathrm{q_{max}}\). An analogous reasoning shows that the next greatest loads \(\mathrm{g_{max-1}}\) and \(\mathrm{q_{max-1}}\) are also simultaneous and the second largest combination load is \(\mathrm{g_{max-1}}\)\(=\)\(\mathrm{g_{max-1}}\)\(+\)\(\mathrm{q_{max-1}}\) Thus, full correlation exists between the loads in the combination although any single pair of G and Q is independent.  
The assumption of the independent load combination wrongly results in that the probability for the maximum loads \(\mathrm{g_{max}}\) and \(\mathrm{q_{max}}\) to occur simultaneously is only 1/50 but is 1.  
The example above shows that maximum variable load \(\mathrm{q_{max}}\) and the maximum permanent load \(\mathrm{g_{max}}\) are simultaneous which suggest the DLC.  
This example shows that the 50-year loads strike every structure. That is, the load combination is not made between random load pairs but rather between a random permanent load and the fixed 50-year variable load.  
The ILC is based on combination of single load pairs of one permanent load and one variable load, i.e., one permanent load is randomly picked and combined with a random variable load, and one may argue that the reasoning above concerning the group effect is irrelevant. However, although the loads are combined by single load pairs the DLC should  
Figure 2: Material safety factors of timber; dash-dotted line, current calculation, the variable load is in the 5-year mode; dotted line, the ILC with the variable load in the 50-year mode; solid line, the DLC; dashed line, the variable load is assumed constant.  
be used as the permanent load distribution and the variable load distributions are fully correlated in the combination in this case. Further, the basic assumption of the ILC is that adjacent load pairs are independent. Consequently, this example shows that the adjacent load pairs are not independent, even though, the permanent loads in the adjacent load pairs are independent but the variable loads are not.