The multiple slope discontinuity beam element for nonlinear analysis of RC framed structures
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.
KeywordsSlope 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.
- 7.Bresler B (1960) Design criteria for reinforced columns under axial load and biaxial bending. ACI J 57:481–490Google Scholar
- 11.Davidster MD (1986) Analysis of reinforced concrete columns of arbitrary geometry subjected to axial load and biaxial bending: a computer program for exact analysis. Concr Int Des Constr 8:56–61Google Scholar
- 12.Di Ludovico M, Lignola GP, Prota A, Cosenza E (2010) Nonlinear analysis of cross sections under axial load and biaxial bending. ACI Struct J 107(4):390Google Scholar
- 18.Italian Building Code (2008) Norme Tecniche per le Costruzioni. Gazzetta Ufficiale della Republica Italiana, RomeGoogle Scholar
- 21.Indian Standard Code of Practice for Plain, Reinforced Concrete (IS:456-2000) Bureau of Indian Standards, New DelhiGoogle Scholar
- 24.Lau CY, Chan SL, So AKW (1993) Biaxial bending design of arbitrarily shaped reinforced concrete column. Struct J 90(3):269–278Google Scholar
- 27.Liu YS, Li GQ (2008) A nonlinear analysis method of steel frames using element with internal plastic hinge. Adv Steel Constr 4(4):341–352Google Scholar
- 35.Paul G, Agarwal P (2012) Experimental verification of seismic evaluation of RC frame building designed as per previous IS codes before and after retrofitting by using steel bracing. Asian J Civ Eng 13(2):165–179Google Scholar
- 40.Soleimani D, Popov EP, Bertero VV (1979) Nonlinear beam model for R/C frame analysis. In: 7th conference on electronic computation, ASCE, St. Louis, Missouri, pp 483–509Google Scholar
- 45.Takeda T, Sozen MA, Nielsen NN (1970) Reinforced concrete response to simulated earthquakes. J Struct Div 96(12):2557–2573Google Scholar