what would happen if you scaled up APSFSDS to a 16inch naval gun?

also why do modern ships only use a 5-6inch gun, 

do you know much about squeezebore? could it be scaled down to small arms, giving a ballistic benefit?

Honestly no. In fact I don't even have a good mental model why squeezebore helps boosting muzzle velocity. Intuitively it feels logical (to me), but when I try to bring my physics education in, it does not really becomes obvious to me.

As far as I understand, it seems to me more of a "second order" effect to better utilize propellant energy given the burn characteristics of the powder charge

Maybe someone here as an accurate description at hand.

You are correct, the "squeeze" itself does not really help regarding muzzle velocity.

The basic idea is to have a large cross area inside the barrel for the propellant gases to work on and accelerate the bullet to relative  high velocity.

Then, outside the barrel, a small cross area is in effect, resulting in less air drag. The bullet can keep its velocity better. 

The purest form of this idea is the British "Littlejohn" (after inventor Janecek) system, which is a muzzle adapter. No squeezing inside the barrel.    

More or less you would end where the whole APDSFS story started: the 31cm (~12inch) smothbore variant of the Krupp K5 railway gun (https://en.wikipedia.org/wiki/Krupp_K5) firing  Peenemünder Pfeilgeschosse. Of course not AP, but DSFS.

All those fancy very long range projectile currently advertised trade range for payload. The ads of course emphasize the range aspect and give accuracy as solution to the reduced payload. With the 16inch you have here a wider range for trade-offs. I guesstimate a 200-500kg shell with 100-150km range would be feasible. If you ad some glide feature maybe even >200km.

I think, if the Iowa-class battleships and their 16inch gun would be still around, we would see dozens of such proposals to keep the relevant and a similar number of proposals to replace their 16in turrets by VLS cells.

And we would discuss the pros and cons of these vigorously here in this forum 

Looking at the current discussions on the importance to increase artillery range even if you have to payload 

Thank you. So a "retracting" sabot.

When the German WW2 squezzebore anti-tank guns were developed, was it an engineering decision squezzebore v discarding sabot? Or was the concept of a discarding sabot not yet on the table and only became a natural evolutionary next step after people looked at the squezzebore and understood the pros and cons?

The reason for smaller naval cannon are: if it's long range, use a missile; gunnery systems are way more effectively accurate than their predecessors, so more hits can be made on target, thus keeping the amount of damage caused by hits the same; nobody uses heavily armored ships anymore, so a smaller projectile can do adequate damage; =<155mm systems are relatively easy to make reliable autoloader systems for.

Already the Dreyse needle-gun of 1841 had a discarding sabot. But it also showed the dominating problem: significantly larger shot to shot dispersion.

JPeelen said:

The basic idea is to have a large cross area inside the barrel for the propellant gases to work on and accelerate the bullet to relative  high velocity.

Basically the benefit of a sabot (wide swept bore volume operating on a lightweight projectile for maximum velocity) without the parasitic weight of a sabot. 

smg762 said:

what would happen if you scaled up APSFSDS to a 16inch naval gun?

Thats basically what Project HARP did; experiment with saboted finned projectiles from modified 5", 7", and 16" naval cannons. 

http://www.astronautix.com/a/abriefhistoheharpproject.html

http://www.astronautix.com/m/martlet.html

Ranges were incredible. The 7"/175mm HARP launched a 39lb projectile 100+km into the atmosphere. I don't have the math to estimate what ranges it could achieve if it were fired at a ~45 degree angle like artillery, but 'very far' seems the answer if it could do 100km fired straight up.