If the mayor refuses or fails to sign any ordinance, resolution, contract, warrant, demand or other document or instrument requiring his signature for five (5) consecutive days, then a majority of the members of the council may, at any regular or special meeting, authorize the vice mayor or, in his absence, an acting mayor to sign such ordinance, resolution, contract, warrant, demand or other document or instrument which, when so signed, shall have the same force and effect as if signed by the mayor. (1988 Code § 2-2-6)