How can you effectively position yourself in a teamfight as an AD carry without getting destroyed? I usually play Ezreal, Corki and Graves.
Make sure that you stay as far away from the heart of the team fight as possible while still being able deal damage. The enemy WILL send people at you to kill you so stick around someone tanky.There are many champions whose role is to protect you.
To summarize, stay on the outside, and get someone on your team to protect you when the other team focuses you.
Positioning is key in every role for team fights, but perhaps the most important are the AD Carries, since your team is counting on you to dish out all that sustained DPS.
Stay in a safe area where the enemy team cannot get to you at all if possible, or have to go through your whole team to get to you if not possible. Your first priority as AD carry is do not die. This means that you will generally attack whoever is closest to you.
If you have escapes like Ezreal, Corki, Trist, Caitlyn use them to give yourself better positioning. If you are Ashe, you might have to continually kite the enemies.
Just keep focus on the closest target in range and move on from there.