A pipe-bending rig has an anchorage for holding a proximal end of a pipe and a bending former that presents a convex forming surface toward a side of a pipe when the pipe is held by the anchorage to extend distally from the anchorage. A pipe-bending drive exerts lateral force on the pipe to deflect a distal end of the pipe, which effects plastic bending of the pipe along its length against the forming surface. A carriage is movable along a path with respect to the anchorage and the bending former during deflection of the distal end of the pipe. A link carried by the carriage applies distal force to the pipe to impart longitudinal tension to the pipe as the pipe is being bent against the forming surface.