, Volume 54, Issue 9, pp 1451–1469 | Cite as

Probabilistic assessment of axial force–biaxial bending capacity domains of reinforced concrete sections

  • Salvatore SessaEmail author
  • Francesco Marmo
  • Nicoló Vaiana
  • Luciano Rosati
Stochastics and Probability in Engineering Mechanics


Capacity domains of reinforced concrete elements, computed according to Eurocode 2 provisions, have been investigated from a probabilistic perspective in order to examine the variability of the failure probability over different regions of the domain boundary. To this end, constitutive parameters, such as limit stresses and strains, have been defined as random variables. Discretized ultimate limit state domains have been computed by a fiber-free approach for a set of cross sections. In addition, the failure probability relevant to each point of the domain’s boundary has been evaluated by means of Monte Carlo simulations. The numerical results prove that the failure probability presents a significant variability over the domain boundary; it attains its maximum nearby pure axial force while it drastically decreases in presence of significant bending contributions. Finally, the iso-probability surface, i.e. the locus of the internal forces corresponding to a fixed value of the failure probability, is presented. It permits to establish a probabilistic interpretation of the axial force–biaxial bending capacity check consistently with the underlying philosophy of recent structural codes.


Ultimate limit state Reinforced concrete Limit analysis Capacity surface 



This work was supported by the Italian Ministry of Education, Universities and Research—FFABR Grants—and by the Italian Government—ReLuis 2018 project [AQ DPC/ReLUIS 2014–2018, PR2, Task 2.3]—which are gratefully acknowledged by the authors.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest to disclosure.


  1. 1.
    Alfano G, Marmo F, Rosati L (2007) An unconditionally convergent algorithm for the evaluation of the ultimate limit state of RC sections subject to axial force and biaxial bending. Int J Num Methods Eng 72:924–963MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bing L, Park R, Tanaka H (2000) Constitutive behavior of high-strength concrete under dynamic loads. ACI Struct J 97(4):619–629Google Scholar
  3. 3.
    Casciaro R, Garcea G (2002) An iterative method for shakedown analysis. Comput Methods Appl Mech Eng 191((49–50)):5761–5792ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Chiorean C (2010) Computerised interaction diagrams and moment capacity contours for composite steel-concrete cross-sections. Eng Struct 32(11):3734–3757CrossRefGoogle Scholar
  5. 5.
    Chiorean C (2013) A computer method for nonlinear inelastic analysis of 3d composite steel-concrete frame structures. Eng Struct 57(Suppl C):125–152CrossRefGoogle Scholar
  6. 6.
    Ditlevsen OD, Madsen HO (1996) Structural reliability methods. Wiley, ChichesterGoogle Scholar
  7. 7.
    DM 17-01-2018 (1999) Aggiornamento delle “Norme tecniche per le costruzioni”. Min. Inf. Trasp., ItaliaGoogle Scholar
  8. 8.
    European Union: EN 1992-Eurocode 2 (1992) Design of concrete structuresGoogle Scholar
  9. 9.
    European Union: EN 1998-1-3-Eurocode 8 (1998) Design of structures for earthquake resistanceGoogle Scholar
  10. 10.
    Garcea G, Leonetti L (2011) A unified mathematical programming formulation of strain driven and interior point algorithms for shakedown and limit analysis. Int J Numer Methods Eng 88:1085–1111MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    JCSS (1999) Probabilistic Model Code [ISBN 978-3-909386-79-6]. Joint Committee on Structural SafetyGoogle Scholar
  12. 12.
    Karsan ID, Jirsa JO (1969) Behavior of concrete under compressive loading. J Struct Div 95(12):2543–2563Google Scholar
  13. 13.
    Kent DC, Park R (1971) Flexural members with confined concrete. J Struct Div 97(7):1969–1990Google Scholar
  14. 14.
    Khan AH, Balaji KVGD (2017) A comparative study of probability of failure of concrete in bending and direct compression AS PER IS 456:2000 and AS PER IS 456:1978. ARPN J Eng Appl Sci 12(19):5421–5429Google Scholar
  15. 15.
    Leonetti L, Casciaro R, Garcea G (2015) Effective treatment of complex statical and dynamical load combinations within shakedown analysis of 3D frames. Comp Struct 158:124–139CrossRefGoogle Scholar
  16. 16.
    Mander J, Priestley M, Park R (1988) Observed stress–strain behavior of confined concrete. J Struct Eng US 114(8):1827–1849CrossRefGoogle Scholar
  17. 17.
    Mander J, Priestley M, Park R (1988) Theoretical stress–strain model for confined concrete. J Struct Eng US 114(8):1804–1826CrossRefGoogle Scholar
  18. 18.
    Marmo F, Rosati L (2012) Analytical integration of elasto-plastic uniaxial constitutive laws over arbitrary sections. Int J Numer Methods Eng 91:990–1022MathSciNetCrossRefGoogle Scholar
  19. 19.
    Marmo F, Rosati L (2013) The fiber-free approach in the evaluation of the tangent stiffness matrix for elastoplastic uniaxial constitutive laws. Int J Numer Methods Eng 94:868–894MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Marmo F, Rosati L (2015) Automatic cross-section classification and collapse load evaluation for steel/aluminum thin-walled sections of arbitrary shape. Eng Struct 100:57–65CrossRefGoogle Scholar
  21. 21.
    Marmo F, Rosati L, Sessa S (2008) Exact integration of uniaxial elasto-plastic laws for nonlinear structural analysis. AIP Conf Proc 1020(1):1219–1226ADSCrossRefGoogle Scholar
  22. 22.
    Marmo F, Serpieri R, Rosati L (2011) Ultimate strength analysis of prestressed reinforced concrete sections under axial force and biaxial bending. Comput Struct 89:91–108CrossRefGoogle Scholar
  23. 23.
    McCormac JC (2008) Structural steel design, 4th edn. Pearson Prentice Hall, Upper Saddle RiverGoogle Scholar
  24. 24.
    Melchers RE (2002) Structural reliability, analysis and prediction, 2nd edn. Wiley, ChichesterGoogle Scholar
  25. 25.
    Menun C, Der Kiureghian A (2000) Envelopes for seismic response vectors. I: Theory. J Struct Eng 126(4):467–473CrossRefGoogle Scholar
  26. 26.
    Menun C, Der Kiureghian A (2000) Envelopes for seismic response vectors. II: Application. J Struct Eng 126(4):474–481CrossRefGoogle Scholar
  27. 27.
    Rosati L, Marmo F, Serpieri R (2008) Enhanced solution strategies for the ultimate strength analysis of composite steel-concrete sections subject to axial force and biaxial bending. Comput Methods Appl Mech Eng 197:1033–1055ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Salma Taj, Karthik B, Mamatha N, Rakesh J, Goutham D (2017) Prediction of reliability index and probability of failure for reinforced concrete beam subjected to flexure. Int J Innov Res Sci Eng Technol 6(5):8616–8622Google Scholar
  29. 29.
    Scott BD, Park R, Priestley MJN (1982) Stress-strain behavior of concrete confined by overlapping hoops at low and high strain rates. J Am Concr Inst 79(1):13–27Google Scholar
  30. 30.
    Sessa S, Marmo F, Rosati L (2015) Effective use of seismic response envelopes for reinforced concrete structures. Earthq Eng Struct Dyn 44(14):2401–2423CrossRefGoogle Scholar
  31. 31.
    Sessa S, Marmo F, Rosati L, Leonetti L, Garcea G, Casciaro R (2018) Evaluation of the capacity surfaces of reinforced concrete sections: Eurocode versus a plasticity-based approach. Meccanica 53(6):1493–1512CrossRefGoogle Scholar
  32. 32.
    Sessa S, Marmo F, Vaiana N, Rosati L (2018) A computational strategy for Eurocode 8-compliant analyses of reinforced concrete structures by seismic envelopes. J Earthq Eng.

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Structures for Engineering and ArchitectureUniversity of Naples Federico IINaplesItaly

Personalised recommendations