When a wood beam is notched to receive floor joists, how is the beam analyzed? Are revised section properties used based on the section at the notch (I or inverted T depending on the notch)?
The load-carrying capacities of the beams can be found using notch-factor equation to account for the observed effects of notch corner location, load distribution, beam height, and fractional notch depth.
Even moderately low loads can cause a crack to initiate at the sharp corners and propagate along the grain. Consider estimating the crack-initiation load, which can be obtained by the fracture mechanics analysis.
Don't notch the beam, instead size a beam that can carry the load with the joists sitting on top of it. For apperance fill in between the joists with wood blocking.
If you design carefully the notches should be in the compression (top) side of the beam. The notch factors are generally tension or torsional generated tension.
You can't consider the joists to carry compression loading as a tight enough joint to do this can cause problems with splitting during moisture content changes.
You will need to be very careful designing this joint to account for the tension perpendicular to the joint which will want to cause tearout on the bottom of the beam.
Would the client consider some of the steel joist hangers which hide the fasteners behind the joist. These are more common in heavy timber / post and beam type construction. They allow a split-ring or similar shear carrying device to be installed behind the hanger. Split rings/shear plates have standardized calcs for their effects on the beam design.
There are not many "good" ways to handle this with traditional joinery. The beams just end up oversized. I often wonder if Our forefathers would gag on the thought of Big exposed wood in the interest of style, when post and beam was originally the economical use of wood, with the craftsmanship being lavished on hiding the frame with interior finish by plaster and paneling.
This detail (deep joist cut for taper) invites horizontal shear failure and longitudinal cracking of the joists.
Through-bolting and keys can help a lot resisting (or repairing!) Hor. shear failures. If the design is driven by "looks" maybe the thrubolts and keys will be a plus, for "visual interest". It is for good reason that Codes limit the depth to a percentage of joist depth.
This is more typical proportion of the joist to beam. Note offset joists.