The supporting structure of a piece of furniture(noun)
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Use "underframe" in a sentence
"The seat can also be removed from the underframe for use as a pouffe."
"The seat is not permanently fixed to the underframe, but the inner ring in the seat fits over the outer ring of the underframe so that the chair can be tilted into the desired position by the user."
"Then, returning to the bed, I clasped it by its underframe and, with a grunt of effort, dragged it towards the center of the room until the loosened board was fully exposed."