Accounting of plastic deformations in the calculation of frames using the displacement method

Method for calculating statically indeterminate frames taking into account plastic deformations, which is based on the use of a schematized diagram of material with hardening is proposed. Two types of standard beams with supports are used during the implementation of the displacement method (DM) like the elastic solution of the problem: "fixed" -"pinned" and "fixed" - "fixed", but unlike the elastic solution, standard beams contain special zones that besides elastic part include elasto-plastic zone (EPZ), plastic zone (PZ) and reinforcement zone (RZ). Therefore, as the stresses in these zones did not exceed the yield stress in the nonlinear frame calculation, we took measures to transform the PZs into equal strength plastic zones (ESPZ). The calculations were made for both types of beams for all unit and load impacts. The frame calculation consists of three stages (elastic, elasto-plastic and plastic). At the elastic and elasto-plastic stages, yield moment and plastic moment diagrams and the corresponding loads are determined. For a practical use of the DM in a nonlinear frame calculation, two simplifying prerequisites are introduced, with the help of which a stress-strain state is modeled in two zones: EPZ and PZ. According to the prerequisites, deformation of fibers occurs without hardening in EPZ and with hardening in PZ. The plastic stage of the calculation is performed at a given length of the PZ using the method of sequential loadings. At each iteration with small loading steps, incremental equations for DM are written, which establish relations between incremental moments and the incremental load, which allows us to build a resulting moment diagram. This diagram represents a sum of the moment diagram obtained at the elastic and elasto-plastic stages and the diagrams of incremental moments at all previous loading steps of plastic stage. According to the resulting diagram, the length of the PZ can be calculated, together with the limiting load. The calculation is considered complete if the length of the PZ does not exceed the specified value within the margin of error. Proposed algorithm is illustrated with an example of static calculation of 2-storey steel frame which perceives horizontal load actions that model a seismic impact.