Unexpected shutdown and restart on 3 different iPhone 5S's

[retz]

Several weeks ago, my iPhone 5s (we'll call this iPhone one), 16GB, running iOS 8.1.1, just over a year old, began randomly shutting off. The first day it only occurred once, then the next day not at all, on the third several times, and on the fourth day iPhone one stayed perpetually on the Apple logo restart screen. I was able to finally get the the phone to respond after attempting to reset it several times at which point I restored the phone from iTunes, restoring to the previous backup. iPhone one stayed operational, installing my apps from iTunes, for around 10 minutes. Then, it shut down again and returned to the Apple logo screen. At this point I took it into the Apple store. The Apple tech could not get the phone to respond, so he switched it out with a new iPhone 5s (iPhone two). I set up iPhone two as a new iPhone (not from a backup, following the advice of the Apple tech) and installed the apps I previously had on the phone one by one. I also updated the phone from iOS 7 up to 8.1.1. iPhone two worked flawlessly for about 30 hours. At this point it began randomly shutting off just like iPhone one. I attempted a factory restore of the phone and did not restore from a backup. Several minutes later, it began restarting again. Next, I spoke with someone at T-mobile who thought there was a small chance that this problem could be caused by a faulty SIM card. To test this idea, I restored iPhone two again and went to the T-mobile store to get a new SIM card. I put the new SIM card in the phone and began setting it up again as a new iPhone, installing apps one by one. iPhone two worked fine for around 5 hours and then started to restart and quickly became unusable. The next day I went to the Apple store again. The Apple tech could not get iPhone two to respond, said she didn't know what the problem was, and so she swapped out the phone for a new one. I set up iPhone three as a new iPhone and installed apps slowly one by one to see if a particular app was causing the problems, but everything installed and worked without a problem. The phone worked like a charm for 7 days. Then, suddenly, iPhone three went to a blue screen (first time this has happened, the other iPhones always shut off immediately to a black screen). Later that day, it shut down again to a black screen. The next morning it shut down a third time to a black screen. Then iPhone three worked fine for around 36 hours. Now, today, it shut down again to a blue screen. I'm guessing iPhone three is doomed too. I cannot figure out any sort of pattern to the shutdowns. It can occur when I'm using any app, at any battery level, at any time of day, and at any location. Does anyone have any ideas as to why this might be happening and what I can do about it?