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After we exposed the rim joist, we found a few other deficiencies.  We jacked and straightened the rim joist (more of a beam in this case).  The rim joist was made up of a double 2×8, this was not strong enough to support the loads between the piers, so we would need to address this as well.  The rim joist was also rolling out and away from the floor joists, so the connection between the rim joist and the floor joists would also need to be beefed up.  So we jacked and straightened some more, and with ratchet straps and some gentle coaxing we were able to get the rim joist straight and level, and tight up to the joist ends.  After this we added joist hangers to both ends of the joists and a row of solid blocking down the center of the joists.  The blocking helps to tie everything together so the joists act as one “system,”  and makes the floor system much stiffer.  After this was all done, we were ready to replace the piers.
The original ‘piers’ were just cinder blocks stacked on the dirt, so these were removed, and new holes dug.  This part of the house is built on tailings from a near by quarry, so the digging was pretty tough, but we were able to get the footings down onto good free draining gravel-type soil.  Once the holes were dug, and piers and concrete placed, new 6×6 columns were installed, and the old addition was once again solidly on the ground.  As a final step, a LVL was bolted to the existing rim joist to re-enforce the existing structure.