As the subject suggests, I am wondering if my first floor bath remodel
requires reinforcing my floor joists. I'll probably end up asking an
engineer I'll be consulting anyway, but I would like to have some idea
how these calculations work. The details:

The bathtub weighs 376 lbs and can hold 47 gallons. If a pint's a
pound the world around, that would be another 376 lbs when full. If
two 300 pound people are in the bathtub (worst case), that would make
a total load of 1350 lbs. The bathtub's dimensions are 32" by 66".

The floor system consists of 2" x 9.5" (old nominal 2x10) joists 16"
o.c. spanning 12 feet from the exterior wall to a girder. The joists
run parallel to the bathtub, so there are only two of them directly
underneath. The bathtub is basically centered over the span, as it is
3 feet from the exterior wall. The subfloor is 7/8" thick T+G 1x4s,
on top of which will be 3/8" ACX plywood and linoleum, then the tub.

Does this floor system need reinforcing, and if so, how? Note that I
will be careful not to cut away any of the joists when installing the
plumbing. The sidewall of the bathtub will eventually be an interior
shear wall, as it is over one of the concrete retaining walls of my
half basement, in case that would be of value.

Thanks, Wayne