The London 2012 bronze medallist, 25, was part of the team that captured world silver last year before picking up a matching medal at this year’s European Championships in Bern.
But after coming fifth in the all-round competition at the final trials event in London earlier this month, he has been dropped in favour of 19-year-old Brinn Bevan in a five-strong men’s line-up that also includes world champion Max Whitlock and three-time Olympic medallist Louis Smith, plus Nile Wilson and Kristian Thomas.
“Gutted I haven’t made the Olympic team,” Purvis wrote on Twitter. “Really thought I’d done enough, gave it my all. Congrats to my team mates and all the best in Rio.”
Whitlock will start favourite on the pommel having clinched the world crown in Glasgow last year – with team-mate Smith set to be one of his biggest rivals. Great Britain have never won Olympic gold in gymnastics, but the four medals claimed by the team at London 2012 matched their haul from all previous Games combined.
Whitlock, who won two of those medals in London with bronze in both the pommel and team events, said: “For me I try to look at it like any other competition. I want to go there, enjoy the experience and do my best. That’s the main thing.
“The results that we have got behind us as a team, gives us a lot of confidence and hopefully we can all have successful competitions out in Rio.”
Commonwealth Games gold medallist Claudia Fragapane heads the women’s squad with Ruby Harrold, Amy Tinkler and sisters Becky and Ellie Downie.
“We are delighted to have selected a group of world-class gymnasts who go to the Games in great form and confident of achieving successful results across the board,” said performance chief Alan Edge.
“This Olympic cycle has seen historic performances at World and European Championships that have firmly established Great Britain amongst the top gymnastics nations.”