To avoid undue hardship, nothing in this chapter shall be deemed to require a change in the plans, construction, or designated use of any building on which actual construction was lawfully begun prior to the effective date of adoption or amendment of this Code and upon which actual building construction has been carried on diligently. Actual construction is hereby defined to include the placing of construction materials in permanent position and fastened in a permanent manner. Where demolition or removal of an existing building has been substantially begun preparatory to rebuilding, such demolition or removal shall be deemed to be actual construction, provided that the work shall be carried out diligently.