The "project and get side rails" way would be like the attached.

You can now select the arcs and run sweep and use the 2 open edges (which are projected trims from the 2 lines) as the rails.

The resulting sweep will hug the surfaces and will have the shape you want at each end of it, but it won't be guaranteed to be smooth on the inside areas. Generally only fillet or blend will yield something that is a smooth connection between 2 existing surfaces.

- Michael