If any street, avenue or alley, or the right to use and control the same for purposes of public travel, shall belong to any city and such city shall establish a grade therefor, which grade requires any cut or fill, damaging abutting property, the damages to arise from the making of such grade may be ascertained in the manner provided in this chapter, but such city may provide that the compensation to be made for such damage, together with the accruing costs, shall be added to the cost of the labor and material necessary for the grading thereof, and shall be paid by assessment upon the property within the local assessment district defined by law or the charter or ordinances of such city in the same manner and to the same extent as other expenses of such improvement are assessed and collected. In such cases it shall not be necessary to procure the appointment of commissioners to take the other proceedings herein provided for making such assessments, but all the proceedings for the assessment and collection of such damages and costs, shall, if so ordained by such city, be governed by the charter provisions, law or ordinances in force in such city for the assessment and collection of the costs of such improvements upon property locally benefited thereby: PROVIDED, HOWEVER, That this section shall not apply to the original grading of such street, avenue or alley.