Over the last months I've done quarter-squats and floor-presses. Both of these were part of designed routines, and each has its own value:
The partial squats, according to Ian King, let you do more weight, and are used to train your neural strength. By hoisting that much more off of the rack and going part way down, the neurological firing going on prepares you for a heavier full-range lift for the next time around.
The floor presses are actually working well for me, in training the weaker part of my lift, which is the upper part. By forcing my chest and tris to relax on the floor, I must come up with quite a surge in power to get the bar up. No slingshot from the bottom of the lift. So, while these aren't really partial-bench presses, but rather a different form, I think they fall in the same category and do indeed have a benefit.
At the beginning of the year I was doing another IK routine (Book of Muscle) and it called for 21's, which incorporate 7 parital lifts at one end of the range and 7 at the end (and 7 full-range in between). This of course was to even out any weak points. It worked well for me.
I've also seen some routines that call for partial reps at the end of the set, to get just a little more stress on the muscle. I've never tried those.
Overall, I'd say they are a good thing if you have a specific plan for their use.
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