A method and apparatus for forming rivet joints with a desired amount of clearance to allow pivotal motion of the parts interconnected by such joints. Parts to be riveted together are aligned with each other, and a rivet is placed into aligned holes. A parts clamp urges the parts and a preformed head of the rivet against a rivet support anvil to establish an initial condition. A parts support anvil is placed in a required position with respect to the rivet support anvil prior to formation of the second head on the opposite end of the rivet, to provide the desired amount of clearance. The parts support anvil is moved by pushing against a set of arched leaf springs to adjust their effective length, using a hydraulic motor controlled in response to an electrical signal from a transducer that measures movement of parts by the parts support anvil.