Computer shuts down after some minutes

[Googolplex]

It could be that then. In general immediate shut downs are caused by:

1) CPU reaching TJMax
2) PSU Malfunctioning

Do you have another psu to test? Also, if your i5 has Intel HD graphics you could remove your dedicated graphics card, connect the monitor to the intel hd graphics and check it it shuts down under gaming. The graphics card is the most power hungry thing in there so if you remove it and it does not shut down, we know that the power supply is not good enough.

I have no i5, I have a Q9550 ;-)
And no other PSU, except an older 420W, but that should be even weaker.