Offering up sufferings is such a beautiful way to get closer to God.
I’ve never actually heard, or used a formula for it-- I usually just say, internally, “Lord, this is for ______” whatever the intention is. As AquinasXVI noted, we aren’t limited to physical suffering in terms of what we offer up-- sometimes the most difficult things to offer up are those little, unplanned contradictions, like hitting every stinkin’ $#(^$^& traffic light when you’re already running late.

Offering up those kinds of things not only help whatever intention we’re praying for, but also help those of us who are prone to getting rather cranky about things like traffic lights to be more cheerful…
Also, sometimes it’s helpful to “plan out” in some way what you’re offering up-- sometimes I’ll have such a full morning of errands and housework that offering it up will slip my absent mind. But at least having the general intention that I always change diapers in reparation for my sins, or always straighten the trash can liner to pray for so-and-so’s vocation, helps here.
This is especially on my mind right now because I’m having a baby next week. The pain, to put it mildly, is excruciating. And, unfortunately, it’s one of those situations where offering it up doesn’t actually minimize the pain. (Trust me, I’ve tried it.) But at the same time, it seems foolish to waste such an opportunity to please Our Lord. So I’ve already sort of got a list of intentions that the labor will be for-- and if in the thick of things in the delivery room, I don’t actually remember to repeat the intentions, I trust that my Guardian Angel will fill in the gap for me.
Margaret