That’s the underside of a church tower roof. It’s a shallow hip roof with (by our analytic-design standards) too-small hip rafters, but it’s also obviously performed just fine. All the 120-year-old rafters are straight, so no past overload is likely. But that’s not what I wanted to talk about.
If you asked me to put a wood hip roof on a square masonry tower, I’d run the rafters over the top of the wall to create a set of eaves that shed water. (And I’d have to provide some kind of tie-down for the rafters to the masonry.) If you told me that the tower parapet had to hide the roof, so that there would be no visible eave, I’d pocket the joists into the masonry. These are both very old details, as in millennia old. So I don’t claim to have invented them. The roof here has a different detail, which is clever but contains a flaw.
The rafters come down to a rim beam, with four of those rim beams forming a square around the edge of the roof. (If the rafters over-ran the wall, there would be four similar beams connecting the ends of the rafters at the edges of the eaves.) That makes the roof into a sort of square pot lid, but doesn’t connect it to the tower walls. That’s where those four short diagonal beams in the tower corners come into play. I’m going to call them squinch beams since the play the role that squinches have traditionally played, although not usually for a roof like this. The rim beams span from squinch beam to squinch beam, carrying the end reaction of the rafters. It’s a clever detail that allows the walls to be almost entirely solid at the top without having the roof project over. The flaw is that there’s a structural gap between the rib beams and the inside face of the walls, which is a natural location for leaks, especially since there’s a gutter formed by the inside of the paraoet and the slope of the roof.