I've been reading bra theory for a couple of years and the discussions in the "Bra Making Forum" group on Facebook that is hosted by The BraMakery have been as informative as any of the textbooks I've gotten my hands on.

Here's a very detailed description of the purpose of a wire in a bra by Natasha of The BraMakery.

Underwire Theory: Purpose of an Underwire

This is the first post in a new series about the theory behind underwires. This series will cover several topics including wire types, wire spring/splay and choosing an underwire for your shape. This post will cover why underwires are used in bras and give a quick overview of their purpose.