The department of transportation, pursuant to its rules with respect to state highways, and local authorities, with respect to public highways under their jurisdiction, may, upon application in writing and good cause being shown therefor, issue a special permit in writing, or electronically, authorizing the applicant to operate or move a vehicle or combination of vehicles of a size, weight of vehicle, or load exceeding the maximum set forth in RCW
46.44.010,
46.44.020,
46.44.030,
46.44.034, and
46.44.041 upon any public highway under the jurisdiction of the authority granting such permit and for the maintenance of which such authority is responsible.