On the rivet
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Extreme physical [exertion] to the point of almost giving up. Derived from bike racing in the [olden days] when bike seats had rivets, and in an effort not to [get dropped] one would slide forward on the saddle to exert maximum pressure on the pedals so as to hang on the the amphetamine-loaded Belgian in front of you.
On the rivet meaning & definition 1 of On the rivet.