, Volume 53, Issue 6, pp 1469–1490 | Cite as

The multiple slope discontinuity beam element for nonlinear analysis of RC framed structures

  • Mohsen Rezaee Hajidehi
  • Antonino Spada
  • Giuseppe Giambanco
Novel Computational Approaches to Old and New Problems in Mechanics


The seismic nonlinear response of reinforced concrete structures permits to identify critical zones of an existing structure and to better plan its rehabilitation process. It is obtained by performing finite element analysis using numerical models classifiable into two categories: lumped plasticity models and distributed plasticity models. The present work is devoted to the implementation, in a finite element environment, of an elastoplastic Euler–Bernoulli beam element showing possible slope discontinuities at any position along the beam span, in the framework of a modified lumped plasticity. The differential equation of an Euler–Bernoulli beam element under static loads in presence of multiple discontinuities in the slope function was already solved by Biondi and Caddemi (Int J Solids Struct 42(9):3027–3044, 2005, Eur J Mech A Solids 26(5):789–809, 2007), who also found solutions in closed form. These solutions are now implemented in the new beam element respecting a thermodynamical approach, from which the state equations and flow rules are derived. State equations and flow rules are rewritten in a discrete manner to match up with the Newton–Raphson iterative solutions of the discretized loading process. A classic elastic predictor phase is followed by a plastic corrector phase in the case of activation of the inelastic phenomenon. The corrector phase is based on the evaluation of return bending moments by employing the closest point projection method under the hypothesis of associated plasticity in the bending moment planes of a Bresler’s type activation domain. Shape functions and stiffness matrix for the new element are derived. Numerical examples are furnished to validate the proposed beam element.


Slope discontinuity Nonlinear pushover analysis Lumped plasticity Plastic hinge 



Dr. A. Spada and Prof. G. Giambanco gratefully acknowledge the grant from the Italian Ministry for University and Research (MIUR) for PRIN-15, Project No. 2015LYYXA8, Multiscale mechanical models for the design and optimization of microstructured smart materials and metamaterials.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of PalermoPalermoItaly

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