OK dude here it is. If you only got enough food, water, beds, defense for the people you'll just get around 80 happiness. Any extra food or water doesn't add to happiness just adds extra into the workshop. The less people you have the easier it is to get 100 happiness. Also defense should be higher than your number of people, that's just minimum. The higher it is the less you'll be attacked or lose attacks and less you'll have to go back and repair stuff. I make my defense over 100 (if I can) or the the number of food and water added together.
The stores are the only thing that directly raises happiness besides junkyard dog (which is a random encounter a guy [Gene] will try and sell you a dog) after happiness reaches 80. Here's the formula (assuming you got all enough food, water, beds [under a roof], defense for people, 1 of each per person).
So basically use the formula: people x 20 and then build (& assign) enough stores to be equal to or greater than the formula amount. So from earlier you said you have 16 beds I'll just assume you also have 16 people. So you need to add enough stores to be equal to or greater than 320 (16 people x 20). The stores must be assigned to a person as well. So 8 level 3 restaurants would do the trick for instance or any combination of any of this stuff (remember the less people the easier it is so you can dismiss people or try at another settlement with less people). Also keep in mind that after you add stores don't expect your happiness to raise immediately, it takes time. So perhaps rest for 24 hours until you see an up arrow meaning everything is working as expected.
(Also since there's a bug with settlements it's advised not to leave until you reach 100 happiness.Bethesda claims to have fixed this bug with 1.03 patch, i for one know in the pip boy under workshops it'll report the wrong data but idk if that affects happiness anymore since they have claimed to fix that part of it. I did leave the settlement with an up arrow but not at 100. I however walked out of the settlement manually before fast traveling. Do this at your own risk, if you do this and see that yr arrow stops showing up or goes down go back to the settlement in question right away to fix the bug!)
Stores
Food and Drink
- Drink Stand 15 (Rank 1)
- Bar 20 (Rank 2)
- Restaurant 40 (Rank 3)
Clinic
- First Aid Station 10 (Rank 1)
- Clinic 20 (Rank 2)
- Surgery Center 30 (Rank 3)
Clothing
- Clothing Stand 8 (Rank 1)
- Clothing Shop 10 (Rank 2)
- Clothing Emporium 10 (Rank 3)
Trader
- Trading Stand 10 (Rank 1)
- Trading Shop 10 (Rank 2)
- Trading Emporium 15 (Rank 3)
Junkyard Dog 10
Individuals online have said use decorations or keep generators away from beds etc. Noise and synths in settlements is something else mentioned a lot that'll reduce happiness, not true. All that doesnt seem to matter. The settlement being attacked will reduce happiness but that more than likely due to resources getting damaged. Also another thing the trophy says you need a large size settlement that doesn't seem to matter either. But just in case, to the right of happiness (in workshop build mode) is the size meter it should be yellow (or orange depending on tv settings) if it's large, if it's not large it'll be green (the meter will be at least 80% full if not more).