Do you mean an Ab4? As mentioned above Ab5 is really freaking high. 4 notes higher than your current max F4 would be an Ab4
I'm really new to this forum and singing in general (I only started taking weekly voice lessons 2 months ago), but i felt like replying because your frustration sounds a lot like how I used to feel!
Prior to taking lessons, my highest consistent note in a full 'chest' voice was an F4, with a lot of straining. I could strain up to an F#4 but half the time my voice would break. Today I hit a C5 in full voice during my vocal exercises - check it out ->
^that was a bit of an extreme case, though. All things considered I'm still reeeaaaallllly new to this and I don't quite have the muscle coordination yet to articulate words on a C5. But I can definitely sing on an Ab4, which already is higher than I ever dreamed I'd be able to sing!
My top "eureka" tips that came to me over the past 9 or so weeks since I started training would be (in order):
1. Support - draw the breath in real low and push down when singing high. Make sure the diaphragmatic squeezing is actually getting airflow out through the vocal cords though, and not just constricting your abs like you're constipated. That was the biggest breakthough for me. For the first few weeks my voice teacher would say, "Push with your stomach!" and I'd clamp my abs so hard my whole body went sore, but I still couldn't stop constricting. Until I realised I needed to actually to have a healthy airflow through my vocal cords.
2. Frontal resonance - Or "mask", which is a term my teacher doesn't really like, lol. But bringing the resonance from the back of the throat to the front part really helps to slip naturally into your higher register. I was practicing this technique with some nasal twang on scales up to F4 and suddenly discovered if I just kept the same feeling of forward resonance and tried singing higher I could do an F#4, and then a G4, etc. I guess this is how I found my "mixed voice", and I can tell you it feels totally different from what I imagined a mixed voice would feel like. It doesn't quite feel like chest voice but it's a completely different beast from falsetto. The sound feels like... it's built in separate pieces. I can't describe better than that. For me, my vowels hit the hard palate, the brightness rings in the nasal twangy area, and the chesty resonance is in the back of the mouth pushing against the soft palate.
3. Vowel modification - This one saved my life. Trying to do a scale on the word 'give' was a real killer around E4-F4 without widening to prevent constriction. Now I'm doing scales up to Ab4.
Hopefully these help! I'd say the best thing you could do would be to find yourself a good vocal teacher, I seem to have gotten really lucky with mine! Although, even he was shocked at how fast I've progressed in such a short space of time
Bear in mind I'm by no means a good singer yet, I've been focussing on expanding my range but my tone quality is still pretty amateurish. If you want to hear me singing something around an Ab4 though, let me know and I could probably do a short recording to show how I do it.
Best of luck!!!