Be good to step back a little and make sure we are clear what is going on, what does "shutting down" exactly mean? What exactly happens on the screen? Is it panicing? Shutting down normally? The screen just goes totally black?
If it's panicing you'll see a screen like
https://www.lifewire.com/troubleshoo...anics-2260849?
Is the Mac plugged in to it's AC adapter? and Mac battery meter showing it is connected to the AC adapter and and showing a good charge on the battery
I would be testing using the Thunderbolt to Firewire adapter (better to fry that than the port in your MacBook) and as BsScout says try a different cable and several of your adapters.
Where did this MacBook Pro come from? If its faulty do you have an option to return it?