Harper-Nelson, the American champion and favourite to win gold, crashed out after she hit a hurdle and fell to the ground.
In the next heat, Harrison did not even get that far -- the National Collegiate champion, fourth fastest in the world this year, was disqualified for a false start.
Harper-Nelson, Olympic champion on this track seven years ago, clipped the second barrier and was sent tumbling to the ground.
She got up without any serious injury and walked away from the track.
"I got out well and I hit my trail leg," the 31-year-old told reporters.
"You just can't do that in the hurdles. It caught me and took me down. Before I realised it I was going to the ground. I'm sorry I let people down."
Danielle Williams of Jamaica won the first semi-final in a personal best 12.58 seconds with American Sharika Nelvis second in 12.59.
Britain's European champion Tiffany Porter took the second semi-final in 12.62 from Cindy Roleder of Germany.
The third semi-final was less eventful as the fourth American in the event, defending world champion Brianna Rollins, won in 12.71 seconds.
The final is later on Friday at 935 PM local time.
(Reporting by Steven Downes, Editing by Ed Osmond)
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