Elastoplastic and Large Deflection Analysis of Steel Frames by One Element per Member. I: One Hinge along Member

The ultimate load of a typical steel frame is dependent on the geometrically nonlinear and material yielding effects. The complexity for considering material yielding by the plastic hinge approach is the unknown location of the plastic hinge, which can occur at the ends or any position along the element length. For the latter case, a member is divided into many elements in order to approximate the location of a plastic hinge. This process is tedious, inconvenient to use, and involves extensive computer time. Further, the strength check for sectional capacity using the LRFD code requires an assumption of the K factor or the effective length ratio, which further complicates a computer analysis for the ultimate load of a steel frame. To describe the formation of a plastic hinge along an element in a member at the ultimate limit state, a single element capable of modeling the $P-\delta$ effect as well as the formation of the plastic hinge is needed. This paper adopts a simple concept of superimposition of triangular deflected shapes due to the formation of plastic hinge to the fifth order deflection shape for elastic deflection to yield the final deflection of the element, the plastic pointwise equilibrium polynomial (PPEP) element. Equilibrium of moment and shear at midspan of an element is maintained for accurate modeling of the $P-\delta$ effect in the tangent and secant stiffness. The robustness, accuracy and reliability of the developed element are demonstrated in a number of worked examples.