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

  • Novel Computational Approaches to Old and New Problems in Mechanics
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A Correction to this article was published on 01 February 2018

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Abstract

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.

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  • 01 February 2018

    This article has been published with an erroneous interpretation of the name of author Mohsen Rezaee Hajidehi. Please find within this document the correct interpretation of the name of author Mohsen Rezaee Hajidehi that should be regarded by the reader as the final version.

References

  1. Bagarello F (1995) Multiplication of distributions in one dimension: possible approaches and applications to $\delta $-function and its derivatives. J Math Anal Appl 196(3):885–901

    Article  MathSciNet  MATH  Google Scholar 

  2. Besson J, Cailletaud G, Chaboche JL, Forest S (2009) Non-linear mechanics of materials, vol 167. Springer, Berlin

    MATH  Google Scholar 

  3. Biondi B, Caddemi S (2005) Closed form solutions of Euler–Bernoulli beams with singularities. Int J Solids Struct 42(9):3027–3044

    Article  MATH  Google Scholar 

  4. Biondi B, Caddemi S (2007) Euler–Bernoulli beams with multiple singularities in the flexural stiffness. Eur J Mech A Solids 26(5):789–809

    Article  MathSciNet  MATH  Google Scholar 

  5. Borzi B, Pinho R, Crowley H (2008) Simplified pushover-based vulnerability analysis for large-scale assessment of RC buildings. Eng Struct 30(3):804–820

    Article  Google Scholar 

  6. Bracci JM, Kunnath SK, Reinhorn AM (1997) Seismic performance and retrofit evaluation of reinforced concrete structures. J Struct Eng 123(1):3–10

    Article  Google Scholar 

  7. Bresler B (1960) Design criteria for reinforced columns under axial load and biaxial bending. ACI J 57:481–490

    Google Scholar 

  8. Buda G, Caddemi S (2007) Identification of concentrated damages in Euler–Bernoulli beams under static loads. J Eng Mech 133(8):942–956

    Article  Google Scholar 

  9. Campione G, Cavaleri L, Di Trapani F, Macaluso G, Scaduto G (2016) Biaxial deformation and ductility domains for engineered rectangular RC cross-sections: a parametric study highlighting the positive roles of axial load, geometry and materials. Eng Struct 107:116–134

    Article  Google Scholar 

  10. Cavaleri L, Di Trapani F (2014) Cyclic response of masonry infilled RC frames: experimental results and simplified modeling. Soil Dyn Earthq Eng 65:224–242

    Article  Google 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 8:56–61

    Google 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):390

    Google Scholar 

  13. Fajfar P, Gašperšič P (1996) The N2 method for the seismic damage analysis of RC buildings. Earthq Eng Struct Dyn 25(1):31–46

    Article  Google Scholar 

  14. Fiore A, Spagnoletti G, Greco R (2016) On the prediction of shear brittle collapse mechanisms due to the infill-frame interaction in RC buildings under pushover analysis. Eng Struct 121:147–159

    Article  Google Scholar 

  15. Fossetti M, Papia M (2012) Dimensionless analysis of RC rectangular sections under axial load and biaxial bending. Eng Struct 44:34–45

    Article  Google Scholar 

  16. Gencturk B, Hossain K, Lahourpour S (2016) Life cycle sustainability assessment of RC buildings in seismic regions. Eng Struct 110:347–362

    Article  Google Scholar 

  17. Ghobarah A (2000) Seismic assessment of existing RC structures. Prog Struct Mater Eng 2(1):60–71

    Article  Google Scholar 

  18. Italian Building Code (2008) Norme Tecniche per le Costruzioni. Gazzetta Ufficiale della Republica Italiana, Rome

  19. Izzuddin BA, Elnashai AS (1993a) Adaptive space frame analysis: part 1: a plastic hinge approach. Proc Inst Civ Eng Struct Build 99(3):303–316

    Article  Google Scholar 

  20. Izzuddin BA, Elnashai AS (1993b) Adaptive space frame analysis: part 2: a distributed plasticity approach. Proc Inst Civ Eng Struct Build 99:317–326

    Article  Google Scholar 

  21. Indian Standard Code of Practice for Plain, Reinforced Concrete (IS:456-2000) Bureau of Indian Standards, New Delhi

  22. Kwak HG, Kwak JH (2010) An improved design formula for a biaxially loaded slender RC column. Eng Struct 32(1):226–237

    Article  Google Scholar 

  23. Lai SS, Will GT, Otani S (1984) Model for inelastic biaxial bending of concrete members. J Struct Eng 110(11):2563–2584

    Article  Google Scholar 

  24. Lau CY, Chan SL, So AKW (1993) Biaxial bending design of arbitrarily shaped reinforced concrete column. Struct J 90(3):269–278

    Google Scholar 

  25. Lemaitre J, Chaboche JL (1994) Mechanics of solid materials. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  26. Liu SW, Liu YP, Chan SL (2014) Direct analysis by an arbitrarily-located-plastic-hinge element—part 1: planar analysis. J Constr Steel Res 103:303–315

    Article  Google 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):341–352

    Google Scholar 

  28. Mazza F (2014) A distributed plasticity model to simulate the biaxial behaviour in the nonlinear analysis of spatial framed structures. Comput Struct 135:141–154

    Article  Google Scholar 

  29. Meyer C (1998) Modelling and analysis of reinforced concrete structures for dynamic loading. Springer, Berlin

    Book  Google Scholar 

  30. Munoz PR, Hsu CTT (1997) Behavior of biaxially loaded concrete-encased composite columns. J Struct Eng 123(9):1163–1171

    Article  Google Scholar 

  31. Mwafy AM, Elnashai AS (2001) Static pushover versus dynamic collapse analysis of RC buildings. Eng Struct 23(5):407–424

    Article  Google Scholar 

  32. Neuenhofer A, Filippou FC (1997) Evaluation of nonlinear frame finite-element models. J Struct Eng 123(7):958–966

    Article  Google Scholar 

  33. Pantò B, Rapicavoli D, Caddemi S, Caliò I (2017) A smart displacement based (SDB) beam element with distributed plasticity. Appl Math Model 44:336–356

    Article  MathSciNet  Google Scholar 

  34. Park R, Paulay T (1975) Reinforced concrete structures. Wiley, Hoboken

    Book  Google 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):165–179

    Google Scholar 

  36. Santoro MG, Kunnath SK (2013) Damage-based RC beam element for nonlinear structural analysis. Eng Struct 49:733–742

    Article  Google Scholar 

  37. Scott MH, Fenves GL (2006) Plastic hinge integration methods for force-based beam-column elements. J Struct Eng 132(2):244–252

    Article  Google Scholar 

  38. Sfakianakis MG (2002) Biaxial bending with axial force of reinforced, composite and repaired concrete sections of arbitrary shape by fiber model and computer graphics. Adv Eng Softw 33(4):227–242

    Article  MATH  Google Scholar 

  39. Simo JC, Taylor RL (1985) Consistent tangent operators for rate-independent elastoplasticity. Comput Methods Appl Mech Eng 48(1):101–118

    Article  ADS  MATH  Google 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 483–509

  41. Spacone E, Filippou FC, Taucer FF (1996) Fibre beam-column model for non-linear analysis of R/C frames: part I. Formulation. Earthq Eng Struct Dyn 25(7):711–726

    Article  Google Scholar 

  42. Spada A, Giambanco G, Rizzo P (2009) Damage and plasticity at the interfaces in composite materials and structures. Comput Methods Appl Mech Eng 198(49):3884–3901

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Spada A, Giambanco G, Rizzo P (2011) Elastoplastic damaging model for adhesive anchor systems. I: theoretical formulation and numerical implementation. J Eng Mech 137(12):854–861

    Article  Google Scholar 

  44. Takayanagi T, Schnobrich WC (1979) Non-linear analysis of coupled wall systems. Earthq Eng Struct Dyn 7(1):1–22

    Article  Google Scholar 

  45. Takeda T, Sozen MA, Nielsen NN (1970) Reinforced concrete response to simulated earthquakes. J Struct Div 96(12):2557–2573

    Google Scholar 

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Acknowledgements

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.

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Authors and Affiliations

  1. University of Palermo, Viale delle Scienze, Ed. 8, Palermo, Italy

    Mohsen Rezaee Hajidehi, Antonino Spada & Giuseppe Giambanco

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  1. Mohsen Rezaee Hajidehi
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  2. Antonino Spada
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  3. Giuseppe Giambanco
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Correspondence to Antonino Spada.

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This article has been corrected.

A correction to this article is available online at https://doi.org/10.1007/s11012-018-0827-1.

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Rezaee Hajidehi, M., Spada, A. & Giambanco, G. The multiple slope discontinuity beam element for nonlinear analysis of RC framed structures. Meccanica 53, 1469–1490 (2018). https://doi.org/10.1007/s11012-018-0817-3

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  • DOI: https://doi.org/10.1007/s11012-018-0817-3

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