This is a traditional trapper cabin style with the ridge and purlins notched into the gables. Some small poles were spiked in place for rafters.
Again, small poles nailed to the plate and the ridge.
This is more of a traditional log cabin roof, here the ridge and purlins were installed with a post. Some heavy 2x planking will be spiked to the ridge, plate, and purlin or you might install dimensional rafters.
Here the crew is laying out the rafters. You notice there is a flat side on top. They are working to shim everything level. Then, they will use a standard scribe setting on each log and the roof will be level. To cut the notches after completion of the scribe, the logs have to come back off the building. The location must be marked so they get put back in the same location.
Cutting a birds-mouth notch in a rafter. Even with the flat top of the plate log, this rafter was likely scribed to fit.
This video shows rafter logs that were spiked or lagged to the ridge and plate. This option might work for your project.
Potential Solutions for your project....
Based on your photo and description of the construction, I imagine that when you pull the roof off the log building, you will find that the roof provided a great deal of support; the walls might wobble a bit. One way or another, you will need to stabilize the structure. Think installing a top plate on a traditional structure. You could use logs or timbers.

Based on your photo and description of the construction, I imagine that when you pull the roof off the log building, you will find that the roof provided a great deal of support; the walls might wobble a bit. One way or another, you will need to stabilize the structure. Think installing a top plate on a traditional structure. You could use logs or timbers.

To stabilize, you could install the bottom four logs in this structure using a round locking notch at the corners. Then, lag, spike the structure to your your current cabin. In this case I would use something at least 8". If you didn't want to lock notch you could spike or lag the logs together.
The king post needs to be secured to the plate and ridge, mortise and tenon would work well. You would need to check snow loads on the roof based on the size of your ridge. I don't think 8" ridge will get you 20 feet.

The easiest rafter design is to spike or lag your rafters to the plate and ridge.

You can scribe your rafters to the ridge and plate and for even more precision you can flatten the tops. Dealing with small lags might be problematic to scribe.
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Looking at your project, you might consider a timber framed structure. You could use 6" x 6" Timbers and construct a rectangular base that can be lagged to the top of your structure. When this is in place, it would tighten things up. Then, you add king posts, ridge, and rafters. The upside of working with timbers is you can do this work in your driveway, test fit everything up, and, then, transport them when you are ready to do the roof. There is a sawmill in Spring Valley, WI that cuts well and has good pricing on timbers.

Timber Frame base is spiked to the top of the structure. These timbers are 6" x 6" I added in a center timber for support. Based on snow load you might have to drop a support under the center. I have a shed at my home in Red Wing with a similar design. I installed a 1/2 loft for storage.

Here we show rafters that are 6" x 6". Many structures use rafters that are 3" x 5", It depends on what you want to see on the underside. This design works well for nailing tongue and groove on the top, then building up the insulation and roof from there. It also eliminates having to fill a gap at the top of your wall (which is a pain in the butt),.

You can scribe your rafters to the ridge and plate and for even more precision you can flatten the tops. Dealing with small lags might be problematic to scribe.
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Looking at your project, you might consider a timber framed structure. You could use 6" x 6" Timbers and construct a rectangular base that can be lagged to the top of your structure. When this is in place, it would tighten things up. Then, you add king posts, ridge, and rafters. The upside of working with timbers is you can do this work in your driveway, test fit everything up, and, then, transport them when you are ready to do the roof. There is a sawmill in Spring Valley, WI that cuts well and has good pricing on timbers.

Timber Frame base is spiked to the top of the structure. These timbers are 6" x 6" I added in a center timber for support. Based on snow load you might have to drop a support under the center. I have a shed at my home in Red Wing with a similar design. I installed a 1/2 loft for storage.

Here we show rafters that are 6" x 6". Many structures use rafters that are 3" x 5", It depends on what you want to see on the underside. This design works well for nailing tongue and groove on the top, then building up the insulation and roof from there. It also eliminates having to fill a gap at the top of your wall (which is a pain in the butt),.



