Max span of 4x2

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My small 2 bed flat is severely lacking in storage space. The guest/office room is currently used as a store / laundry drying room and I'm wanting to get that sorted.

My flat is an old tenement building so it has high ceilings (approx 3.3m) and I'm considering getting a high level shelf put in above the door to the guest/office room that spans wall to wall, approx 3m long. The shelf would run parallel with the stud wall that has the door, the other walls are brick/stone, and I would like to avoid any vertical supports dropping into the room.

Weight of items would just be random stuff like small power tools or empty suitcases, etc and occasionally me (65kg) as I look for stuff up there. Would 4x2 spanning 3 meters be strong enough or would sag likely be an issue for that span?
 
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c16 timber spaced at 400 centres would take a loading of 70kg/m2 which in theory would deflect 9mm but structurally be well within allowed limits
 
But laid flat, 4x2 will only have enough room for a thin vase, a 5x4 photo frame of grandma, some old trainers placed toe to heel and an anorexic cat to shimmy along.

Or are we talking mezzanine? If so, I only know a few common sentences.
 
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scaffold boards are brilliant for shelves at 10ft/3m and will take far more load without bounce
they are 227 wide and 38mm thick or 9x1.5"
 
Not if it’s laid flat like a shelf though
I was thinking 3 2x4's at 400 centre spanning the 3m, laid edge-ways, then boarded over the top.
But laid flat, 4x2 will only have enough room for a thin vase, a 5x4 photo frame of grandma, some old trainers placed toe to heel and an anorexic cat to shimmy along.

Or are we talking mezzanine? If so, I only know a few common sentences.
Not sure what the exact definition of a mezzanine is but I guess you could call it that, it would be similar apart from primarily being used for storage, not people.
 

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