Transport Scotland said the decision was based on inspections by engineers and the "advice and assessment" of independent experts.
A 20mm crack in a truss under the southbound carriageway was detected on Tuesday, close to the bridge's north tower.
Repairs are already under way and authorities said it should be reopened "for the return to work in January".
Engineers said allowing vehicles to still use the bridge would "increase the risk of causing extensive secondary damage".
The closure, which began at midnight, has already caused big delays for some rush-hour drivers.
Traffic Scotland said Friday morning queues on the A985 had reached 11 miles on the approach to the Kincardine Bridge.
There was also a six-mile queue near the Clackmannanshire Bridge.
As well as using those bridges, drivers are being urged to travel on the M9 or A9, and use public transport if possible.
Extra bus and rail services will now be laid on between Fife and Edinburgh.
A full rail plan for the duration of the closure and a special travel webpage are also being prepared.
Transport Minister Derek MacKay said the decision to close the bridge was "essential for the safety of the travelling public" and "not taken lightly".
"Every effort is being made to open the bridge as quickly as possible but safety is the main priority, however these works are weather dependent given the height and location of the bridge," said Mr MacKay.
The problems only occurred in the last few weeks, he added.
Mark Arndt , from Amey, the company which maintains the bridge, said the work was a "complex engineering challenge".
"The component failure is in a difficult to access location and our response is also highly dependent on weather conditions," he said.
More than 70,000 vehicles normally use the bridge every week day.
Police and the ambulance and fire services will still be able to cross in emergency situations.
Motorists are also being advised to follow @forthroadbridge and @trafficscotlandfor the latest developments.
A new bridge - the Queensferry Crossing - is due to open over the Firth of Forth next year, costing the Scottish Government up to £1.4bn.
The existing Forth Road Bridge will eventually become dedicated for public transport, cycling and walking.
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