How to fix an Eames Sidechair (fiberglass)  

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What I have done in the past is use appropriate length steel nails and channeled out enough 'within' the breaks and then solidly glued in the nails. Sort of like internal reinforcement and the nails acting as an internal splint.
Given that it has a textured finish anyway, just fibreglass over the top of the repair when you've completed the above. Then paint it over, again because it's textured it would matter if you hand painted it so long as you colour match it properly.
And finally lightly coat it with a pressure pack can of varnish.
It's worked for me.

Thank you very much!
What kind of glue should I use? Something like Epoxy for boat repair? Most of those repair kits also come with a stripe of fibreglass which I could use for the second part of your advice.
What do you mean by "channel out"? To remove some fibre out of the breaks so the nails fit in?

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you're welcome!
The strongest possible glue you can buy, usually a 2 pac adhesive works the best.
And yes remove some of the fibre out and make a nice channel for the nail to fit in.
What you are creating is not unlike when you have a serious broken bone and the surgeon implants a metal pin to support the bone

Some days old this thread, but...
...this may be useful for some of you:
We made a short video again, this time Lounge Chair shock mount replacement.
Please follow the link below and enjoy
http://youtu.be/V4D1-SVQWyk?hd=1

Use "cat Hair" bondo product ...
Use "cat Hair" bondo product and paint the sitting surface then clear it.
cat hair is a bondo binding agent where you mix a hardener into
it has fiber pieces floating in it
if you mix enough floetrol or reducer into a red paint it will still show the fibers
the clear coat will shine it a bit