RB211 Series
Hi Andrew,
The primary reason for the three shaft architecture is to have the compressor / turbine stages running nearer to their ideal speeds. Aside from reduced fuel burn, the three shaft design means there are less variable stages in the compressor which reduces complexity and weight. The RB211 is significantly lighter than its competitors which translates into additional aircraft capability. There is also a school of thought which says that the RB211 has better surge margin than the competition due to its three shaft architecture as the IP and HP compressors are not as highly loaded as a HPC in a PW4000 for example.
Another claim that Rolls-Royce make is that the three shaft design results in a stiffer core (less bending) meaning the performance retention is superior to a two shaft design. From a line maintenance perspective, the RB211 is alot easier to work on than its competitors. By mounting the gearbox on the fan case, it frees up alot of room around the core engine meaning access is easier. The fan case mounted gearbox also means it is possible to change the gearbox on-wing rather than having to remove the engine.
If you want additional information, RR sell a book called "The Jet Engine" which is very well written and explains most of this stuff in considerable detail.
All the best,
P2C