Just a WAG (wild *** guess) but I think a restriction would be the way to go.
Think of it this way; if you add a cost all that does is give the solver a reason to minimize the cost. But what if it can't? Lets say a route for a vehicle is restricted due to barriers of some sort. It would then route under/over the bridge in question just because it could. The cost is what it is. But the real problem is you routed a truck over a bridge that can't support it's weight or under one that isn't tall enough for the load to pass. Ouch...
A restriction like XXX,XXX lbs or XX feet would ensure you from what I like to call a 'resume generating event'...