Summer house roof - How do I keep it strong yet shallow

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Hi All, once again turning to you for some advice.

I am building a 3 x 3 m corner summer house, which has a diagonal of about 4m

I need to stay below 2.5m to avoid planning restrictions and struggling

The option I'm looking at at the moment is....
Base - 3" (3 x 2 on concrete with a membrane)
Doors 79.5" (french doors, this is the standard size, minimum I seem to be able to get)
Door Lintel (Kingpin?) 6" x 2" to ensure doesn't bend and cause issues with door frame
Top of wall studwork 4" x 4" (or 2 x 2 x 4's sistered)
Roof joists 8" x 2 as per span tables for C24 - 390m span
Roof felt etc

this adds up to 101.5 Inches i.e. 2.57m

I am trying to work out if I can sister up the roof joists to make them 6X2s or if I'm over doing it somewhere else. Also hear of putting steel between two joists and glue/bolt for strength that could reduce their depth, ... you can tell I've been on google.

Its for my wife and I dont have a great success rate at DIY so need to get this right :-)

Thanks any thoughts or suggestions.
 
Fix the 4x4 at the same height as the lintel and use 6x2 rafters, 150mm saved.

Also you won’t have the 3x2 under the door would you? That’s another 50mm off.
 
You haven't said how you will board the roof? OSB will massively increase the strength and provide inherent cross bracing if you screw it down right.
 
Thank you Ian & Motorbiking for taking time to reply, much appreciated.

I’ll switch to 6 x 2 rafters thank you that’s saved a bit, looking at span tables for c24, I cant go any thinner + it allows for 100mm insulation + 50mm air gap as doing a “cold roof”.. everydays a school day


I’ve found somewhere that does 1850 door height so that better too

I’ll have the 3” under the door as that’s a base that allows 60mm of insulation from concrete slab.

I’ll put 12mm or 18mm osb or plywood top of the rafters screwed every 6”, but

I’ll cover the roof with EDPM (maybe GRP as I can probably get a smoother finish).

Hoping this approach isn’t too dodgy, though looking at my shed it’s fine on a 2x2 frame
 
You don't need a rubber roof, if you use underlay felt and normal shed felt, but over lay it, it will last. Its also a lot easier to repair and maintain.
 
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