It is recommended that if you are not willing to smile a full smile, then just do a smaller one instead. You do not have to expose your teeth in full, but can hide your braces by smiling from the corners of your mouth. This will look a lot more natural than trying to fake a smile by exposing a small portion of your teeth.
A small smile does not have to be tight because it will look unnatural. Make sure you choose a nice colour for your braces so that you will take pride in having a unique and special smile.
There is no reason to keep your braces hidden away, because a lot of kids at your school will most probably have them as well. A lot of people have to go through this phase during their school years, which makes them even more memorable. It is important to just be yourself and always look at the brighter side of things.
It will certainly not harm you or your friendship with anyone and you never know, you just may be one of those kids which really shine when wearing braces. Try to get an honest opinion from your family and close friends.
If you consider your braces to give you a horrible look, not smiling at all could also be the answer to have a picture taken on picture day. It may not look as bright but will get the job done.
If the reason you are not smiling is because of your friends making fun of you more than other people, it may be time to confront them. If that does not work, getting new friends should be your top most priority because real friends should always have your back when you are in need.