Apple just released a new update for iOS 7.1.2 yesterday and it seems that for some people who are trying to install the update through the Settings, the process is freezing before they finish. The same happened to me this afternoon, the update bar was freezing for more than an hour until I have started to panic and decided to do something about it.
I have tried the hard reboot trick, and it worked. All you have to do is just hold down the home button and sleep/wake button, just like you would take a screenshot, and don't let go. Keep holding them down until the screen turns off and then comes back again with Apple logo.
After the same is done, you need to make sure your iOS was updated by going to Settings > General > Software Update. If not updated to iOS 7.1.2, repeat the update process again.