Here is the formula for calculation of a roof purlin.
Construction purlin flat roof.
In lightweight timber roof construction under purlins were used to support rafters over longer spans than the rafters alone could span.
8 x section modulus of a 4 2 of 1 3125 x fb fiberstress in bending x 1 15 duration of load x 1 15 cr repetitive member factor as the purlins are spaced 24 o c.
The rafters are generally spaced every 16 inches on center.
Fasten the purlin with 16d common nails into each vertical rafter.
Snap a chalk line horizontally across the roof two feet down from the top.
Cos of live load dead load x cos of the purlin spacing x purlin span squared 8 x sm section modulus of the member x duration of load 1 15 in areas with snowfall x cr repetitive member factor of 1 15 where members are 24 on center or less fb fiberstress required.
Roof live load roof dead load x spacing x span of purlin 2.
For example an 8 x 4 under purlin would support the center of a row of 6 x 2 rafters that in turn would support 3 x 2 roof purlins to which the roof cladding was fixed.
The roof support system is a ridgepole and either rafters which run perpendicular from the top of the walls to the ridge or purlins which run parallel to the ridge.
Lay the first purlin at the ridge of the roof down to the chalk line beginning at either corner.
Or less x 1 1 cfu the flat use factor for applying the load to the wide face of a member.