An MRI exam revealed the extent of the injury, which coach Steve Spagnuolo said Monday would not require surgery. The Rams lost 9-7 at Washington on Sunday.
Adam Goldberg finished the game at Smith's position and would likely replace him as the starter. Smith said he was hurt during the Rams' touchdown drive in the second quarter, although he stayed in for several more plays before leaving the game.
"I don't remember whether it popped or not," Smith said. "In the heat of the battle, you don't really feel it, but I knew something wasn't right."
Goldberg started six games at four offensive line positions last year and started at right tackle much of the preseason before Smith was elevated to first string. He has experience against the Packers after playing with the Vikings for three years.
"The goal is just to play well when called upon," Goldberg said. "It's nothing I'm not used to."