Extending a concrete foundation requires a blend of strength and precision. It's a lot more complicated than fastening a couple of Lego blocks. If the extension is for a building addition, it will require an architect. An extension for a porch or deck can be handled by a homeowner, with some help. It will require a lot of digging and a rented concrete mixer or delivery by a concrete truck, depending on the size of the extension.
Dig down at the end of the existing foundation to determine the depth and width of the footings under it; they will be wider than the foundation wall. Dig deep enough to see the depth of a gravel base under the footing. Measure and mark the area for the new foundation with stakes and builder's twine, to the width of the existing footings. Dig a trench of that width and depth, including the gravel base, for the new foundation wall.
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Fill the footing trench with gravel up to the depth of the existing foundation footing. Compact it firmly with a hand tamper. Fill that trench with concrete, up to the top of the existing footing and level it with a trowel. Set upright reinforcing bars in that concrete 18 to 24 inches apart up to the height of the foundation wall. Let that footing set for at least a day.
Build foundation forms with 2-by-6-inch (or wider) boards to the height and width of the existing foundation. The width of the form boards will depend on the height of the foundation; you can use 2-by-8-inch or multiple 2-by-6s. Secure those forms in place with stakes in the ground outside. If the gap between ground and form is wide, nail 1-by-2-inch spacer boards from stakes to the tops of the forms. If using multiple boards, nail 1-by-2 supports on the outside of the form. Forms must be square and solid enough to hold the concrete.
Drill holes every 6 inches or so into the end of the existing foundation with a masonry bit the size of the reinforcing bar. Drill as deep into the foundation as you can. Secure reinforcing bars in the holes with mortar, extending about a foot into the new foundation area.
Pour concrete into the forms, covering both upright and connecting reinforcing bars thoroughly. Use a board to push the concrete firmly into the forms and up against the old foundation. Level the top even with the old foundation and smooth it with a trowel. Use a long board and level to make sure the foundation top is level all the way. Let the concrete cure for a day or two, then remove the forms and backfill with dirt around the new foundation walls.