You have what we call in the oilfield industry, a fingered collet. You can treat individual fingers as cantilever beams and apply such simple equations plus friction of wedged blocks. It is a problem of statics, found in most textbooks.
There are two types of collets, open and closed. The closed is modelled as a statically indeterminate beam, but otherwise operates the same. I've seen this problem handled as a circular beam in bending, Roarks also has some discussion on this with formulae in the table.
Also try the various oilfield technology companies website or forums. This is a typical application for downhole completions, packers, landing equipment, nipples, etc.
Good luck with your search or research!
Kenneth J Hueston, PEng
Principal
Sturni-Hueston Engineering Inc
Edmonton, Alberta Canada