Why do my halogen lamps keep burning out?
@Andyaka, I know you don't need the extra reputation, but it would be
  cleaner for future viewers. Please copy your comments into an answer,
  so I can accept it.


Getting to the bottom of why the halogen bulbs have failed may be difficult. There is no technical reason for them to unnaturally fail but it could be a bad batch but my advice is not to replace the halogen bulb with another halogen type.

I'm swapping out all my old bulbs for LEDs everywhere in my house for reasons of long-term economy. It looks like your bulbs are G4: -



And there are plenty of LED replacements coming into the market place now. This wasn't true 3 years ago. Anyway here are a couple: -