I know this question is old, but I've often considered STR-based monks, so here's my take on it (note that I'm deliberately focusing on the "STR based monk" aspect of this character concept rather than challenging the notion that this character could be something other than a monk, since the other answers already do that quite well and I wanted to address the STR based monk aspect):

# AC is the problem

Because monks can't use armour without sacrificing various class features like Martial Arts and Unarmored Movement (as pointed out in other answers), you _need_ decent DEX in order to have decent AC (along with WIS, although that's not directly in competition with STR in this case).

This is what makes DEX their primary stat (which is also why the multiclass requirement includes 13 DEX), and even though technically you can use STR for attacks, that's clearly not how they were designed given the multiclass requirement and how their Unarmored Defense feature works.

So your options are go for a STR based monk anyway and just suffer low AC (unless you can find some Bracers of Defense or something), or find a way around the AC problem.

# Solutions to the AC problem

One way is to pick the Tortle race. Since they have an AC of 17 due to their shell, they can be a STR based monk that can have a decent AC without needing a good DEX (and since Tortle's get a +2 bonus to STR and a +1 bonus to WIS, for Ki save DC, they make ideal STR based monks). Of course, last I checked, Friar Tuck wasn't a Tortle, so that won't help in this specific case...

The only other way is homebrew. Going with a homebrew monk archetype that changes the way AC from Unarmored Defense is calculated would solve this problem, but has the added problem of being homebrew. As an example, Way of the Closed Fist on D&DBeyond (there are two versions, I'd recommend the "Direct" version as it is less complicated) specifically says you can use STR instead of WIS to calculated your Unarmored Defense. 

Alternatively, if you're into making homebrew, make your own archetype that does this (I made a Way of the Sumo, not published on the internet anywhere, that has STR based monks in mind, but in my case I allowed Unarmored Defense to use CON instead of DEX). But again, this is all homebrew and thus requires your DM's approval and is definitely not legal for Adventure's League (in case that's relevant).