Cleaning a fabric shell chair  

honestly....
and im sure many will turn their nose up to it.....but i used to restore and customize old cars and the one thing that got the dirt out the best,easiest, and cheapest was Dow Scrubbing bubbles foaming carpet cleaner and a very soft nylon bristle hand brush....after all that wipe with wet cloth to remove residue....did it before and ill do it again!!!
p.s.always test in an inconspicuous area......oh and steam may cause a fiberglass shell to soften...yikes

People would turn thier...
People would turn thier noses up about cleaning the fabric?
I gave them a good vacuume and they aren't as dirty as I first thought.
Thanks for the scrubbing bubbles advice, I'll try it.
L

My method
This works well for me: I put a small amount of liquid laundry detergent in a spray bottle, fill it up with hot water, and then shake it up. I then get my wet/dry vacuum and take the filter out. I do a small section at a time - spray on the soapy water, and immediately vacuum it up. It works like a steam cleaner and I get good results with it.

i meant they might turn...
i meant they might turn their noses up to using scrubbing bubbles...its kinda "shade tree"....