The provisions of this division shall not apply to any work done or to be done, in, upon, along, across, under or over any street pursuant to any street improvement law or street safety law of the state, law of the United States, or ordinance of this city, providing for the improvement thereof or pursuant to any contract in which the city is a party and which has been authorized by the city council; provided that the provisions of sections 12.20.360 through 12.20.550 of this division shall apply to all such work unless the provisions of the specifications therefor are contrary to said sections 12.20.360 through 12.20.550 of this division. (Prior code § 74102)