After approval of such general plans and specifications by the Secretary, and after notification of such approval, the bridge owner shall, in such manner and within such times as the Secretary may prescribe, take bids for the alteration of such bridge in accordance with such general plans and specifications. All bids, including any bid for all or part of the project submitted by the bridge owner, shall be submitted to the Secretary, together with a recommendation by the bridge owner as to the most competent bid or bids, and at the same time the bridge owner shall submit to the Secretary a written guaranty that the total cost of the project, including the cost of such work as is to be performed by the bridge owner and not included in the work to be performed by contract, shall not exceed the sum stated in said guaranty. The Secretary may direct the bridge owner to reject all bids and to take new bids, or may authorize the bridge owner to proceed with the project, by contract, or partly by contract and partly by the bridge owner, or wholly by the bridge owner. Upon such authorization and fixing of the proportionate shares of the cost as provided in section 516 of this title, the bridge owner shall, within a reasonable time to be prescribed by the Secretary, proceed with the work of alteration; and the cost thereof shall be borne by the United States and by the bridge owner, as provided in sections 516 and 517 of this title: Provided, That where funds have been appropriated for part only of a project, the bridge owner may take bids for part only of the work. In the event the bridge owner proceeds with the alteration through the taking of successive partial bids, the bridge owner shall, if required by the Secretary, submit a revised guaranty of cost after bids are accepted for successive parts of the work.