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The phrase "a second building of" is correct and usable in written English.
It can be used when referring to the construction or establishment of an additional structure or facility, often in the context of expansion or replication.
Example: "The city council approved plans for a second building of the community center to accommodate the growing population."
Alternatives: "an additional structure of" or "a duplicate facility of".
Exact(1)
EBI already has planning permission for a second building of the same size next door to the existing premises.
Similar(59)
A sixth building of 255,000 square feet at the five-building complex called Merritt Seven on Route 7 will be ready for occupancy next summer.
Construct a third building of 10,300 square feet (956.901 m2) for their third year of growth.
That was a second building block of the governor's victory.
Half of a second building houses a cold storage morgue for body remains.
That structure is the second building of an envisioned three-building complex.
A third building houses Jenners of Edinburgh, a department store, as well as more cafes and restaurants.
A third building, in the Huriya neighborhood of Baghdad, was badly damaged by an explosion whose cause was uncertain.
James Lee, a senior vice president at Opus, said his company was moving ahead on a second speculative building, of 350,000 square feet, at the park.
Now the market is entering a second round of building.
The seventeenth century saw a second wave of building at Brasenose, started under the Principalship of Samuel Radcliffe.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com