Prolonged torrential rain caused a long delay before Bradshaw could make her first attempt at the qualifying height.
And she cleared 4.55m at the first time of asking.
But defending champion, the USA’s Sandi Morris, is out.
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Bradshaw said: “It was a bloodbath and it was just about whoever could keep your head in the game and keep your body in the right place.
“There were lots of run throughs, lots of people close to hurting themselves, Sandi actually hurt herself and it could have been a lot worse.
“To lose someone like her from the final is a massive loss and I am gutted for her she can’t defend her medal.”