you wouldn do that since you need the ability to take the rod out fast if the reactor is .... well pushing a bit hard.
And if it was possible to split an rod they would have don that ages ago... but they don't.
You don't take fuel rods out normally, certainly not when the reactor is "pushing a bit hard", it's not an everyday procedure, US carriers are only re-fueled once in their lifetime.
They do split fuel assemblies, there's nothing particularly novel about the idea, it's not common since in most reactors there is enough height above the reactor to remove a full size rod without difficulty, thus it's quicker and more convenient to use a single rod.
To scram ( shut don an out of controller reactor) you don't remove fuel rods you insert all control rods or inject neutron absorbers (reactor poisons ie Boron etc) and increase coolant volumes to quickly remove the decay heat, and cross your fingers......
I'm interested in bullfrogs idea but in the event of refuelling how is the first half of the fuel rod removed, it wouldn't be possible to grab the top of the fuel element as the fuel channels wouldn't be much bigger than the fuel rod. I would also be concerned about a Neutron imbalance if half a rod is still in the reactor......
Fuel elements in sections have been used in the past but mainly in early pile stile reactors, horizontally loaded and discharged through a rear charge face, these reactors were primarily designed for weapons grade material production and were not very fuel and energy inefficient and fuel was more cartridges than rods.... Examples are Windscale UK, Chicago Pile 1 and the Handford Early piles.