The council had previously placed temporary red and white plastic barriers on the bridge to stop large vehicles using it and was hoping to make the obstructions permanent today.
A council spokesman said: "We are aware of the situation at the farm, and we are trying to resolve the matter with the farmer.
"While we understand the farmer's concerns, our staff are just trying to do their job.
"We understand his issue regarding visitors accessing the farm and we are looking at possible alternative routes with him.
"This bridge is unsafe for large vehicles and we have tried alternative methods over the last few years to prevent large vehicles getting through, but at the same time, still maintaining access to cars.
"Unfortunately this hasn't worked, and to ensure public safety we wanted to close the bridge."
It is understood the police have been called, but none have yet arrived at the scene.
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