Whenever one or more parcels of land, upon which is located the bed of such a mapped street or public drainage way, flood control basin or public area reserved pursuant to section 17-100 cannot yield a reasonable return to the owner unless a building permit is granted, the board of adjustment may, in a specific case, by an affirmative vote of a majority of the full authorized membership of the board, direct the issuance of a permit for a building or structure in the bed of such mapped street or public drainage way or flood control basin or public area reserved pursuant to section 17-100 which will as little as practicable increase the cost of opening such street, or tend to cause a minimum change of the official map, if any, and the board shall impose reasonable requirements as a condition of granting the permit so as to promote the health, morals, safety and general welfare of the public.