How do you analyze beams in elastic-plastic bending beyond yield?

As load increases beyond elastic limit, outer fibers yield while core remains elastic, creating an elastic-plastic boundary. Shape factor k = Mp/My (plastic/yield moment) is 1.5 for rectangular and 1.7 for circular sections. Full plastic moment Mp = σy × Z (plastic section modulus). Plastic hinge forms when entire section yields. Used in collapse analysis, limit state design, and understanding ductile failure in overload conditions.