Adding Attic Floor

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I need to add more attic storage space, so I want to put a floor down in an unfinished section of the attic. Joists are 2"x6" and 16" on center with 10" of blown insulation. My plan is to strap another 2"x6" perpendicular to the joists giving me 11" of space for the insulation, and then putting down 23/32" plywood for the new floor.

My concern is the weight load on the original joist. Will a 2x6 joist support this (it's a ceiling joist, not a floor joist, so attachment is different), or do I need to sister another 2x6 to the existing joist before I add the perpendicular member?

Or is there a better method?
 
6 inch is pretty good for loft joists. I think you just need to make sure they're bearing on something solid at both ends. How long is the span?
Photos might help if it's possible
 
You'd be better of adding an 8x2 alongside the existing joists, and then using either 100mm of celotex or equivalent, or 170mm of rockwool, as you'll have floorboards over it.
 
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