The Ohio Building Code, Plumbing Code, Residential Code, Mechanical Code and more are now available at: Ohio E-Codes. There is also a link from Department of Commerce, Board of Building Standards. The Building Code is dated 2005, and the library has received 2006 updates, so the online one may not be the most current. The online version is not searchable and can only be viewed section by section. The sections do not have history information.
