The problem you will have with the 2x10 sistering will occur at the ends. The new larger wood will not rest on the top plates. If you notch them, you lose the 2x10 structure value. A cross beam(lvl's or glulams) under the 2x6 whose ends are supported down to the foundation would be the solution. The part of the beam that hits the header will be supported by a joist hanger engineered for the task. You will need to deal with the garage door header, but it's not that big of an issue.
Ron

The 1/8" steel plate idea sounds theoretically correct, since steel has such great tensile strength, but I have difficulty picturing how it would work. I guess if you had some 16' x 1 1/2" lengths of 1/8" plate you could just drill holes every foot or two and attach with screws to the bottom of the 2/6's. I know I have used similar steel reinforcement in (hidden) furniture applications with good results.

To me, the sistering idea is the more tried and true method, and I would go with at least 2x8. I don't share ron6519's concern about losing structural value by notching because the weak point now is the center of the span and even a notched 2x8 or 10 will help stiffen the center.

Using glue or other adhesive to tightly connect the sistered joist to the existing 2x6's should also add some stiffness. I recently did some sistering to stiffen up a floor for ceramic tile and I was very pleased - felt almost like a slab!

tryinbrian,

agreed on the ends of the joists. The ends of the joists are more critical for shear resistance, which isn't an issue in my case. The moment and deflection are greatest in the center of the span, diminishing between the center and the supports. At a certain point between the center and the supports, the 2x6 will not need additional reinforcement for deflection. (I don't know where this lies exactly, but I think if the center 3/4 of the span were sistered, in this case, it would be OK).

However, if the joist reinforcement doesn't extend to the support, the load must be transfered through the fasteners, to make the members function as one member. The glue and screws would be critical.

Ron6519,

Thanks for the followup. I think the beam would be a good solution. However, I now see I only have about 2-1/2" clearance to play with, due to the garage door rails. Hmmm. Maybe I will sister with a 2x6 and add steel plates to a select few joists as necessary.

I've been thinking about this, and the problems we encountered trying to deal with a similar problem (saggy ceiling in kitchen). One big issue is getting the new boards in place. With 16" on center, the longest measure you get is the diagonal between them. You will find that you can't cut the board very long or you simply will not get it up there.

I'm not sure about the logic of weight on the middle not transferring to, and putting stress on the ends of the board. I would expect without additional support, eventually the 2x10's will develop cracks. May not be in the time you own the house though. Someone had used 2x6's for the roof on our den, and cut the bird mouth wrong - actually it's more likely they needed boards 14'5" and didn't want to spend the money on longer ones. Boards would have been fine except for the notch that effectively changed them to 2x2's. The ones on the ends of the roof were fine, all those in the middle had serious cracks. Roof had a nice bounce in the middle. It probably had been that way for years and years, and probably would have not been a problem for years and years to come. But we are fixing potential problems as we find them. At least the roof does not flex when you walk on it anymore.

If it were my place, I would consider finding joist hangers (or modifying some) that could be used to lap over the sill plate, and support the bottom of the 2x10. I'd also put them in the middle between the 2x6's. A palm pounder is a great tool for nailing joists supports in place.

You may also find the there is a bow in the floor/ceiling already, and you may have to jack up the middle to get the other boards in place. If you have a straight 2x10 and a bowed floor, you will have a heck of a time trying to get it in place. We have two support poles with adjustable ends (use a big wrench) and we've used them a few times to get roof lines back in place.

We are rehabbing the entire house, and have run into a lot of "interesting" situations to deal with.

Another option might be engineered lumber. I know it can be a lot stronger for the width than regular boards. Don't know what is available, just a thought.

Cathy

Here is a link that might be useful:

Reinforcing just the middle of the span will not work.

Don't use any design that relies on glue and/or fasteners to stiffen floor.

If the span is 16 ft. sister 2x10's the full length and support the ends with joist hangers or a ledger.

If the span is 20 ft. you will need 2x12's or double 2x10's.

Lvl's are about 50% stronger and more expensive but would reduce the amount of framing required.