IPhone 5 will not ring, it only vibrates

First make sure that the Silent switch is turned on. Check on the left side of the iphone and make sure that the silent switch does not show red.

If it is not in silent mode, change try changing the ringtone and see if it works. If not reset the iphone from settings->general->reset->reset all settings.

If both methods does not work contact an Apple Genius.

It's my understanding that this is by design.

Do Not Disturb mode is meant to be active when the phone is not actively in use. The idea behind it is that, if the phone is unlocked and you are working with it, you do in fact WANT to be notified of a new item. However, when the phone is locked and dormant, the logic is that you no longer wish to be disturbed, and it will suppress vibration / notification indicators.

So for example, if you are sleeping, and DND is on, then it will remain silent, and not vibrate. If you roll over, pick up your phone to check messages, and it's active, the DND will still allow vibrate to indicate new messages while you are using it.