Abstract: An apparatus for rounding off edges of openings in a work piece, such as internal openings or cut outs in steel plates. The apparatus includes a belt grinding machine having a continuous grinding belt running over several belt guiding rollers which are provided on a support frame. Two of the rollers are provided on a cradle, that is pivotally supported in the support frame. The grinding belt is, between the rollers on the cradle, disposed for rectilinear working of the work piece, and a device is arranged for operating the cradle in a pendulum like movement about the pivotal support during a grinding operation.