Kelly moves to the Tykes in a free transfer and has penned a three-year contract.
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Reds boss Conor Hourihane told the club's website of the Northern Ireland U21 international: “We’re really pleased to add Patrick to the group.
“He’s a dynamic, intelligent midfielder who fits the way we want to play. He’s had a brilliant grounding at youth international level and has already tasted success in a promotion-winning team, which will stand him in good stead.
“We’re looking forward to working with him and helping him settle into the squad.”
Kelly added, “It’s a really good feeling.
“I’m really happy to get it over the line and I just can’t wait to get going.”