cutting a joist to fit an attic stairs

Hi,

I am looking to fit a stairs into the attic.

The only thing is I will have to cut a joist

When I cut the middle joist in the attached, the gap is 550mm which is perfect.

I was thinking of cutting a section out of the joist, maybe 3/4 way through. and fitting a cross beam with 1/4 notched out, cut the beam and then fit a second cross beam to strengthen the ope.

I could then fit the stair box and bobs yer uncle.

Is this a reasonable approach do people think ?

Thanks