The 27-year-old, who retired in 2013 but returned to competition in 2014, won pommel horse silver and team bronze at his home London Olympics in 2012 as well as pommel horse bronze in Beijing in 2008.
Smith was warned by British Gymnastics after he made comments on social media questioning the judging of the national championships where he lost out to younger teammate Max Whitlock.
Whitlock, the 23-year-old world champion and double 2012 Olympic bronze medallist, will join Smith in Rio for what will be his second Games along with Kristian Thomas, a team bronze medallist in 2012.
They are joined by Brinn Bevan, whose participation had looked in doubt last November when the teenager suffered a double leg break after landing on a concrete floor in a vault that went wrong.
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On the women's side, sisters Becky and Ellie Downie are included while 16-year-old Amy Tinkler is set to be the youngest Team GB member in Rio.
"Gymnastics in Britain is stronger than it has ever been," team Chef de Mission Mark England said in a statement.
"Our gymnasts have enjoyed a very successful four years since London 2012 and the quality of athletes selected today is right up there with the best in the world."
(Reporting by Alan Baldwin; editing by Mark Heinrich)
