Timber Framers Guild Forums Tools Timber Engineering Engineering Check

As mentioned the curved cut is done to remove the "point of fracture" from the floor joists.
In order to support a floor load the beam has to be sized correctly for the intended load. That is at the mid point of the beam where the bending will be the most. The ends of the beam have to be sized to prevent the beam from failing in shear. That is as if the beam was sheared vertically.... Solving for shear is usually quite easy as wood is strong perpendicular to the grain...
Modifying a straight beam with curve cut ends doesn't weaken the beam as the middle is thick where it needs to be for strength, and thick enough at the ends.
And as mentioned not reducing the tie or girder it's set into, to full depth of the joist makes it stronger as less of the tie or girder has to be removed...