Optiplex 960 Powering Off. Suspect Bad Power Supply

Hi. I have a Dell Optiplex 960 mini tower computer that powers off when under too much load and I think the problem is due to overheating and a faulty power supply. Would appreciate feedback/advice.

In troubleshooting I realized that the power supply fan never comes on - not when under a heavy performance load or even when at startup when I first press the power button (and I remember that it i used to hear the fan at power-on). I have two operating systems (Windows 8.1 and Win 10) each on it's own physical drive and the problem occurs on both.

Specs:
Dell Optiplex 960 Mini-Tower
3.15 gigahertz Intel Core Duo Processor
64 kilobyte primary memory cache
6144 kilobyte secondary memory cache
Bus Clock: 1333 megahertz
Video Card: AMD Radeon R5 200 Series
Drive 0: Win 8.1
Drive 1: Win 10
Drive 2: cd-rom
8gb ram
255 watt dell power supply


In the Windows Event Logs I can only find one event related to the power-offs. This event is almost always present after it happens:
Event 41: The system has rebooted without cleanly shutting down first. This error could be caused if the system stopped responding, crashed, or lost power unexpectedly.

I tried two benchmark apps in the troubleshooting process: SpeedFan and OCCT

SpeedFan: (In Windows 8.1) Starting temps before putting system under any significant load

Before Significant Load Temps just before system powered off

Drive 1 44C 43C
Drive 2 35C 35C
Drive 3 35C 35C
GPU 91C 117C ***
Core 29C 52C ***
Core 1 29C 46C ***

Again, the fan never turned on! Even when the CPU usage was 100% and the temps reached the elevated levels listed above. I'm not a hardware guy. Should the GPU and Core temps reach those levels (117,52,46) without the power supply fan even spinning? And I remember that the power supply fan used to come on regularly.

Can I just order a replacement supply on ebay and install myself?

I tried running the OCCT app in Windows 10 and ran the basic power supply test twice. The system powered off about a minute into the test each time.

Thanks for the feedback!