Eames Aluminum Group Lounge Chair Assembly  

you need the hydralic apparatus from vitra
at 03:25
http://www.youtube.com/watch?v=0focEFg-Z9k
or you need to come up with a manual setup that would yield the same result.
look how they use a rubber mallet to hit the back rest rail in between the side rails while the side rails are still mounted on the hydraulic apparatus.
perhaps some other folks here might know a manual equivalence that could work. regardless, please let us know how it turns out.
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great video, thanks!
If you had a pipe sash clamp you could reverse the head and wind it out to get the same action. Perhaps put some wood blocks with a v shape cut in them to fit against the frame.

Scissor jack
from your car's toolkit.

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Or if that won't work try and get a slightly over long length of 2x2, cut notches in the ends, cut it in half then put the two pieces like an A frame between the frame and push them down together, might need a helper.

Bar clamp
Like someone said a pipe clamp will work. Simply invert the screw end and it will work as a spreader. I believe the back bar on the chair is held to the sides by roll pins.
http://www.amazon.com/gp/product/B0006694EQ?ie=UTF8&ref=aw_bottom_links&...