1" tube with 1/8" wall would put you at about 100MPa so depending on material and your ability to live life on the edge you're technically OK as even plain carbon steel is around 220-250Mpa. So either make sure it's secured to a wall or something to prevent it from being bumped over or take the advice of others and add some reinforcement. Personally I would get myself some 3/4" ply and screw it to the frame all around and cut some access panels in it for doors, that would make it plenty rigid, plus who likes unfinished open stands?
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