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The ideal spot for my drain pipe is 3/8 inch into the side of a floor joist. I see all kinds of rules for notching and boring the horizontal of a joist, but don't see anything about notching the vertical side. Can I do this, and reinforce it on the other side?
 

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You can notch under certain limitations.
The top and bottom 2" of a joist are the sacred compression and tension zones. I am sure you could find a way to do it with reinforcing the other side but you will need an engineer to tell you what kind of reinforcement.
you can prob find another solution. You might want to ask this over in plumbing section. i know they make offset toilet flanges for this reason.

IRC 2012
http://publicecodes.cyberregs.com/icod/irc/2012/icod_irc_2012_5_par025.htm
R502.8.1 Sawn lumber.
Notches in solid lumber joists, rafters and beams shall not exceed one-sixth of the depth of the member, shall not be longer than one-third of the depth of the member and shall not be located in the middle one-third of the span. Notches at the ends of the member shall not exceed one-fourth the depth of the member. The tension side of members 4 inches (102 mm) or greater in nominal thickness shall not be notched except at the ends of the members. The diameter of holes bored or cut into members shall not exceed one-third the depth of the member. Holes shall not be closer than 2 inches (51 mm) to the top or bottom of the member, or to any other hole located in the member. Where the member is also notched, the hole shall not be closer than 2 inches (51 mm) to the notch.
R502.8 Cutting, drilling and notching.
Structural floor members shall not be cut, bored or notched in excess of the limitations specified in this section. See Figure R502.8.


For SI: 1 inch = 25.4 mm.

FIGURE R502.8 CUTTING, NOTCHING AND DRILLING
IRC 2012 http://publicecodes.cyberregs.com/icod/irc/2012/icod_irc_2012_5_par025.htm
 

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If you make 2x lumber into 1x lumber, the repair - I am assuming and have no engineer's number for this - must be substantial. If the location is that good for the plumbing, you can add a new joist and have no worries.
 

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I've never seen any type of vertical plumbing that was not easily moved 3/8".Even if you have to cut a joint to reposition it ,it's a simple task.Maybe you have a unique circumstance but I can't imagine what it would be?
 
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