Hi Jon, thanks for the text and for this little poser
My knee jerk response would be that as long as the bridge infrastructure remained rigid (apart from the loss the stilts) then the gravity would tend to equal itself out around the circumfrence in terms of attracting the bridge toward the centre of the earth.
However, I'd like some more time to think about this but my provisional answer is 'yes'.
Addendum: The only things that could influence the bridge in terms of moving away from its initial condition/state [after all the stilts have been removed] is its lack of rigidity - I'm pretty sure that an analogue to this is where you can take a circular ring magnet and place it around a circular solid iron puck-like object, and as long as the initial condition is attained [Equal separation around puck] then if you let go of the magnet, it will keep it's distance due to the cancelation of net magnetic forces surrounding the puck.
And if it happens with magnets then I'm pretty sure the same will happen with gravitational forces.
Addendum: Mind you, if you're gonna get all pernickety on me and get Quantum Mechanics involved then at certain times there will be an imbalance of forces surrounding the puck/earth but as to whether these anomalies will be sufficient to cause one side to attract the ring/bridge into where it's touching one side of the puck/earth.
I'm kinda thinking that the time scale of these QM anomalies will be so infinitesimally small that there'll be no net movement toward the centre.
OK Mr Sosta, and of course anyone else who fancies trying it - I remember having this one thrown at me many years ago - the only clue I will give is this:- Don't knee-jerk your answer, things are not as they seem, well, mathematically they're not as easy as you first think ....
I shall paraphrase it so it's more in context with this forum:-
In our classifieds section, one of our members has posted up a picture of a marker, and a loader that he wants to sell - he declares that he wants to sell them both for £400 - the marker is valued at £390 more than the loader - how much is the loader worth ?
This is not too difficult really but I'll dreg some more up if this gets answered too quickly .....