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On our Sister Stack, Earth Sciences, we find the following question: How long could a steel artifact last?. It reads:
Would man-made steel artifacts be able to be preserved like fossils for millions of years, or would corrosion make them dissipate? What about metals such as aluminum that don't corrode as easily?
The accepted answer states:
The short answer is no.
Most metals do not fossilize well, they are too reactive/water soluble. Metals are usually what is doing the fossilizing, dissolved metal in groundwater is being attracted to the electrical properties of organic material and filling in or swapping places with a porous material. You might have a stain left or some of the shape as a natural casting, but not material preserved. The exception might be in amber, but it could also be really bad: amber preservation is tricky chemistry, so I am not comfortable saying yes or no. Metal objects have good short term preservation but horrid long term preservation. there are some low reactivity metals that might survive like gold, but not as fossils. But even gold is water soluble over millions of years, so even that would require some unusual conditions.
This is explaining the worst case situation. Over the course of thousands of years, some metal may remain on the surface and look vaguely like the superstructure of a city, but some portion of it will have been dissolved back into the ground.
But my point here is this: it's still metal. Even rust is still metal with too much oxygen. The only practical difference between a steel beam hot off the press and ready for construction and one that's been left to rot for thousands of years is that some portion of the latter (due heavily on the circumstances of its situation, even with the precision you gave us) will simply be part of the dirt beneath it's original constructed location.
Ready for your intrepid survivor to dig up and smelt.
It should be said that in an area as rainy as the U.S. Great Lakes region there will be a substantial loss of above-ground metal.
Nothing is built to last forever
Part of your problem will be working with the rubble. Modern highway bridges have an expected lifespan of only 75-100 years. Modern skyscrapers have an expected lifespan of about 140 years. None of this is because the metal is dissolving into the ground, but (a) weakening infrastructure from use raises the likelihood that the construction might partially or wholly collapse and (b) the economic value of doing so makes sense (it might have nothing to do with whether or not the construction could last longer). From a more realistic perspective, such buildings will last 500-1,000 years.
Structures are built to resist environmental loading events – including earthquakes, windstorms and snowstorms – that have a mean recurrence interval of 50 years. This means the design basis uses events that on average will occur once every 50 years – though of course it is possible for a 50-year storm to occur in consecutive years.
The combination of using a 50-year recurrence for design loading events and safety factors in construction typically results in a design exceedance interval of about 500 years, with special buildings (as mentioned above) having intervals of 1,000 years or more. This means we would expect a typical structure to fail once in every 500 to 1,000 years. (Source)
What does all this tell us? While it's unlikely that you'll have standing buildings after 3,000-5,000 years, it's not impossible. This means that there should still be plenty of above-ground metal that can be salvaged — assuming that working through a potentially collapsed pile of rubble is practical.
But what does "practical" mean?
There's a huge difference between what a growing society of 1,000 or 10,000 people need and what a single person needs. A single person would likely find plenty of salvageable metal in collapsed structures, decaying vehicles, etc. The needs of a larger community might find the economic practicality of working through the rubble, no matter how much metal exists, more work than it's worth.
Whether or not any of this matters depends more on where you're taking your story than on worldbuilding. From a worldbuilding perspective, the metal is there, waiting to be smelted and used. Your story gets to deal with the ratio of value-vs.-effort.
But one thing you can't ignore, a steel beam can't simply be tossed on a cart and hauled away by a couple of horses. There may be some short pieces, but even a short piece is honking heavy. Without cutting torches (or some whomping good hacksaws...), a piece sitting right in front of you, waiting to be used, might be entirely inaccessible, anyway.
Some metal will have dissolved, rusted, or simply become inaccessible without some form of mining technology (whether it's picks and buckets or earth moving equipment). The metal's still there, it just needs to be mined like any good ore and smelted again. Metal, being heavy, is unlikely to travel far. Don't let gold in rivers fool you. That's really small bits and flakes for the most part... the mother load being a seam of gold in rock someplace. Same deal with steel. With the exception of small amounts likely not worth panning for, most of it will be directly below where it was on the surface.
A great deal of metal won't be accessible without serious tools. This ranges from pieces of I-beam sitting in the middle of the street that can't be lifted or transported due to the nature of the individual/team that found it to stuff that's so buried in rubble that the effort of freeing it might be more than the value it represents.
What's left is what you can salvage. It's still a considerable amount — unless you're trying to provide the needs for a very large community. There's metal everywhere: power lines, towers and poles, vehicles, rails and rail cars, various factories and plants... Apartment and office buildings are common, but by no means the simplest source. Heck, start by looking for gravel pits or cement plants. Look for railways. Look for the protective sidings on road ways. There's metal everywhere, and while 3,000-5,000 years will have made a lot of it hard to find, if we're only talking about small communities with small needs... I should think you don't have a problem at all.
If there's a problem, it's going to be finding a predictable source of liquid acetylene and oxygen. But, now that I think about it, while I doubt those big, heavy tanks of oxy and acetylene you find at hardware stores won't have much in them after a couple 'o thousand years... what they will be is a movable source of big chunks of metal.
Yeah... it's everywhere.