R Jack Abramowitz clarifies (here) the distinction between the permitted and forbiddden part of the melacha of boneh (assembling or building)
While the av melacha (primary labor) involves both assembling separate
pieces into a unified whole and forming a shelter, either activity is
prohibited on Shabbos, even if the other is not accomplished.
This is actually forbidden.
As background, building new items is forbidden because of the melacha of boneh. Rebuilding is somewhat easier but not that much. As R Daniel Braude (Learn Shabbos, pp. 292ff) explains
fastening tightly (e.g., with screws, nail or glue) is forbidden min haTorah
fastening loosely (e.g., attaching parts of an item in a way ...
Those Halachot are quite famous, this is where you should start looking:
מטה של פרקים, אסור להחזירה ולהדקה, ואם תקע חייב חטאת. ואם היה רפויה
A bed made of "joints" (parts that are made to disassemble and reassemble): It is forbidden to assemble it (on Shabbat), and if he did he needs [to offer] a sacrifice. And if it's ...