In the event the whole or any part of the cost of such bridge or viaduct or the approaches thereto, or the right-of-way therefor, or any interest therein, or any other property and rights upon which any such bridge, viaduct or its approaches are now or may hereafter be constructed, operated and maintained, shall be sought to be condemned by such city and paid for by special assessments against any private property, subject to special assessment as may now or hereafter be prescribed by the charter of such city, the common council, or two houses of legislation, of such city shall so declare by ordinance, and the city, if it so desires, may proceed with such condemnation in the manner now prescribed or which hereafter may be prescribed by the charter of such city for the taking or damaging of private property for public use; but no part of such structure sought to be condemned shall be removed by the owner thereof, and it shall not be necessary to file detailed plans of such structure with such proceedings, but a general description of such structure may be made in the ordinance. All special assessments shall be made and collected in accordance with the charter provisions above referred to.