Vol.:(0123456789) 1 3 Innovative Infrastructure Solutions (2018) 3:42 https://doi.org/10.1007/s41062-018-0143-6 TECHNICAL PAPER Computing redistribution moments in the plastic stage by using linear analysis M. A. Farouk 1  · Majed Alzara 1  · Mahmoud Samir El‑kady 1,2 Received: 31 January 2018 / Accepted: 29 March 2018 / Published online: 14 May 2018 © Springer International Publishing AG, part of Springer Nature 2018 Abstract In this research, a suggested linear model was investigated to analyze the plastic stage for indeterminate skeletal steel struc- tures. The aim of this model is to facilitate the analysis of the structure element in the plastic stage without resorting to the complicated calculations of the material nonlinearity. The suggested model was represented by considering the full plastic sections in the element as a concentrated plastic hinge. The plastic hinge was modeled instead of the plastic zones as a pin support or an intermediate hinge with a rotational spring. Computing the stifness of rotational spring was based on the acceptance criteria in the nonlinear static analysis according to FEMA 356 (2000). The linear structural methods can be used after that to calculate the deformations and moments in plastic stage. In this paper and due to the simple cases which are ana- lyzed, the forced method of structural analysis can be used. But for structural elements which are more complicated than the present cases where the plastic hinges are separated on more positions, the fnite element analysis is the best. The suggested model can be used to predict the mechanism of failure, to evaluate the deformations after occurring the plastic moment as well as to compute the elastic redistribution moments. The suggested model was verifed by comparing the experimentally and analytically results of steel beam deformations which made by El Damatty (J Steel Compos Struct 3:421–438, 2003) with the obtained results of the suggested model, and the suggested model gave good results. Moreover, a W-shaped fxed steel beam was analyzed by fnite element method by using ANSYS program, the suggested model and elastic analysis to compute the induced moments in plastic stage and evaluated the elastic redistribution moments. The suggested model gave matching values of the induced moments of the fxed compared with the fnite element results. Keywords Plastic hinge · Finite element analysis · Suggested model Introduction Predicting the failure load and computing bending moments in plastic stage for statically indeterminate structures such as steel beams can be performed experimentally or analytically. Analytical procedures require high efort and complicated calculations needing special software programs. These pro- grams take into account the materials’ nonlinearity through the loading stages, such as ANSYS [3], ABAQUS [4] or ADINA [5] programs. When the elastic analysis methods are used to compute the bending moments in the plastic stage, the moment redistribution in plastic zones must be taken into consideration. For approximate analysis, the plastic theory is a simple method which can be used to analyze the element in plastic stage. This method is used to predict both the failure load and the mechanism of failure as well as it can be used to evaluate the elastic redistribution moments in indeterminate structures. In the plastic theory, plastic analysis is based on the idealization of the stress–strain curve as elastic-perfectly- plastic. To illustrate the beam behavior in plastic theory, consider fxed beam subjected to distributed load as shown in Fig. 1. The stress distribution across any cross section is linear as in Fig. 2. As w is increased gradually, the bend- ing moment at every section increases and the stresses also increase. When the bending moment at beam ends reaches to yield moment, the stresses in the extreme fbers reach the yield stress. As the load increases, more and more fbers reach the yield stress and the stress distribution is as shown in Fig. 2. Eventually the whole of the cross section reaches the yield * M. A. Farouk moh.anour@yahoo.com 1 Civil Engineering Department, Jouf University, Al-Jouf, Saudi Arabia 2 Structural Engineering Department, Faculty of Engineering, Zagazig University, Zagazig, Egypt