A machine for bending an elongated workpiece without wrinkling has a roller-like bending die (2) with a vice (3) adapted to retain one end of the elongated workpiece (P) to be bent. The roller-like bending die (2), being provided with a central hole (5) and a groove profile (6) and rotatable about a rotation axis (y), bears, centrally and coaxially in its groove profile (6), a plate element (18) oscillating with respect to the bending die (2) around a rotation axis (y) thereof. The plate element (18) having a peripheral edge (19) which continues the groove profile (6) of the bending die (2) consists of at least two pieces (20, 21) which are adapted to be connected to one another to surround the bending die (2).