Floor joist reinforcement.... ?

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'm working on new bathroom on the first floor - we have new soil pipe which comes in through the ceiling in the corner. The way I would want to position the toilet will mean that the joist will need to be cut approx. 6cm in the deepest place. Can this be done ? Is there a way that the joist can bere-enforced by connecting to other joists or any other way? I have some photos which will make this much clearer to understand.
 

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Why is it half above and half below the ffl? How will that be finished?
 
There will still be floorboards and aquapanel to go on top of the joist - so once that is added it is 2cm above all that. I'm thinking of slightly raising the floor under the toilet to accommodate for that, But main problem is the joist. We need to go either completely below the joist and box it underneath which will not look very good or cut some of the joist if that is possible - but will look much better. I found in US there is a company that sells something that should be ok - http://www.metwoodstore.com/ - maybe I could get something like the first one on the left done - or is there another way to do it?
 
TBH, if you cut it no more than what you are intending i.e. 6 cm, it will not be making an ounce of difference to the strength of the joist mid way, around the edge the joist is already supported from the bottom by joist hanger, so you need not worry about reinforcing it , just go ahead and do it, as planned. The only issue is will your floor boards fit around the pipe , how will they be supported over the joist, what type of floor boards are you using? 26mm thick chipboards or normal timber floor boards?
 
How deep is the joist? If it's 100mm you could trim it to the adjacent ones, if it's 350mm then that's basically an allowable notch.
 
If I can rephrase what I said , it is the position of the notch or cut out along a length of a joist that dictates the strength of any beam ( or joist) that same cut out if it was being made mid span, it would weaken the joist strength significantly, but since in your case the cut is towards one edge and 6cm from top, this would not cause even a minor cause for concern. If you observe how bridges are build where mid span they have taller walls than at ends, it is the same principle that applies here.
 
The shear force is greatest at the ends. The bending force is greatest in the middle. The notch still affect resistance to bending most, so it's better nearer the ends, but you still need to consider both.
 
The shear force is greatest at the ends. The bending force is greatest in the middle. The notch still affect resistance to bending most, so it's better nearer the ends, but you still need to consider both.
yes I agree with what you said, having considered individual application, this is one of many joist that would support the whole floor, so the weight distribution is spread over a wider area, hence the shear force is very limited, hence why I said it should not be too much of a concern in this particular scenario. Still it is always better to consider worst case scenarios.
 
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