# Plastic Analysis of a beam and frame

In plastic analysis and design of a structure, the ultimate load of the structure as a whole is regarded as the design criterion. The term plastic has occurred due to the fact that the ultimate load is found from the strength of steel in the plastic range. This method is rapid and provides a rational approach for the analysis of the structure. It also provides striking economy as regards the weight of steel since the sections required by this method are smaller in size than those required by the method of elastic analysis. Plastic analysis and design has its main application in the analysis and design of statically indeterminate framed structures.

In elastic-plastic materials, stress is proportional to strain up to the yield stress, σy. The yield moment, My is the bending moment corresponding to a bending stress distribution in which the stress equals the yield stress only at the outer-most fibers. In a symmetric cross section of depth d, $M_y = σ_y I/(d/2)$. If the cross section has a solid rectangular shape (dimensions d × b) then $I = bd^3 /12$and $M_y = σ_y (bd^2 /6)$. The section can carry additional moment beyond the yield stress. The plastic moment, Mp is the bending moment corresponding to a bending stress distribution in which the stress equals the yield stress almost everywhere within the cross section. For a solid rectangular cross section, $M_p = σ_y (b d^2/4)$.