The 28-year-old lock has represented Wales at under-21 and A levels and has previously played his club rugby for Northampton Saints and Neath.
The 6ft 7ins powerhouse has made 75 appearances for the Ospreys since 2003, scoring six tries in the process.
Andy said: "When I spoke with Sean Lineen (head coach), I was really pleased with the club's ambition and what he had to say.
"They've had a tough season but given the changes that have happened so far and the new signings they've made, the future sounds very positive and I'm looking forward to being a part of what's ahead."
Sean said: "Andy is just the type of player Glasgow need, experienced and hard nosed and I know he will make a huge difference to our forward play."