Abstract: The present invention relates to a spindle capable of being inserted into a core (1) on which a material in strip form, in particular paper, is wound into a roll. This spindle comprises a shaft (8) connected to a braking device (7), a tubular jaw holder (9) that is rotatable with the shaft, jaws (12) housed in longitudinal slots (13) provided in the jaw holder, elastic members (14) bringing the jaws into engagement with the inner surface of the core (1) when the spindle is inserted into the latter, and complementary clamping means (21) for reinforcing the action of the elastic members (14) on the jaws (12) at the time of braking, acceleration or deceleration of the roll (2), the core (1) of which is borne on the spindle.The present invention also relates to a machine for winding and/or unwinding a material in strip form, the spindles of which are of the structure defined hereabove.