It is understood that Fitzpatrick, 52, died while competing in a cycle race in San Francisco when he collided with a lorry on a downhill stretch.
In a statement Arup said: "Tony dedicated his working life and his unbounded energy to Arup and engineering, and leaves a large gap in our lives. He will be missed by all who knew him. We extend our deepest sympathy to his widow and to his children."
Fitzpatrick came to the public's attention as the man who took the wobble out of London's Millennium bridge across the Thames in 2001.
His achievements include the structural design of Foster and Partners' Hongkong and Shanghai Bank in Hong Kong, its Canary Wharf sister, Jean Nouvel's Tour Sans Fin in Paris, Renzo Piano's London Bridge Tower and Richard Rogers Partnership's Terminal 5 at Heathrow.
His death is a tragic loss to the industry
Graham Watts, chief executive, the Construction Industry Council
Technically he was top-notch. I owe him a lot
Chris Wise, principal of Expedition Engineering and former Arup colleague
I am still shocked by the news. He was a good friend and a fantastic engineer; he was totally one-off, there was nobody quite like him
Ken Shuttleworth, Foster and Partners
Fitzpatrick had to call on all these qualities when Arup’s reputation was put on the line by the “wobbly bridge” debacle. Fitzpatrick came up with the vibration control mechanism that stopped London’s Millennium Bridge from wobbling and pulled his firms chestnuts out of the fire.
Bob Emerson, the engineer’s group chairman, said: “He had an uncanny ability to make incisive contributions that changed the design direction of many important projects. His strategic vision will be missed.”
Fitzpatrick was also in the front ranks of those advocating the swift adoption of new technologies. Last year he gave an interview to Building using a video link from Arup’s San Francisco office about working without paper.
Chris Wise, a former colleague, said of him: “He was a man who lived for everything that he did. He was fantastically motivated and wanted everybody to live life like he did.”