Simplified analytical Moment-Curvature relationship for hollow circular RC cross-sections

  • Gentile, Roberto (Department of Civil, Environmental, Land, Building Engineering and Chemistry, Polytechnic University of Bari) ;
  • Raffaele, Domenico (Department of Civil, Environmental, Land, Building Engineering and Chemistry, Polytechnic University of Bari)
  • Received : 2018.04.15
  • Accepted : 2018.07.15
  • Published : 2018.10.25


The seismic vulnerability analysis of multi-span bridges can be based on the response of the piers, provided that deck, bearings and foundations remain elastic. The lateral response of an RC bridge pier can be affected by different mechanisms (i.e., flexure, shear, lap-splice or buckling of the longitudinal reinforcement bars, second order effects). In the literature, simplified formulations are available for mechanisms different from the flexure. On the other hand, the flexural response is usually calculated with a numerically-based Moment-Curvature diagram of the base section and equivalent plastic hinge length. The goal of this paper is to propose a simplified analytical solution to obtain the Moment-Curvature relationship for hollow circular RC sections. This based on calibrated polynomials, fitted against a database comprising 720 numerical Moment-Curvature analyses. The section capacity curve is defined through the position of 6 characteristic points and they are based on four input parameters: void ratio of the hollow section, axial force ratio, longitudinal reinforcement ratio, transversal reinforcement ratio. A case study RC bridge pier is assessed with the proposed solution and the results are compared to a refined numerical FEM analysis, showing good match.


  1. Berry, M.P. and Eberhard, M.O. (2005), "Practical performance model for bar buckling", J. Struct. Eng., 131(7), 1060-1070.
  2. Broglio, S., Crowley, H. and Pinho, R. (2010), "Simplified capacity curves for RC bridges", Proceedings of 14th European Conference on Earthquake Engineering, Ohrid, Republic of Macedonia.
  3. Computer and Structures (2016), SAP2000 v18, Structural Analysis Program, Manual, Berkeley, California, USA
  4. Esmaeily, A. and Peterman, R.J. (2007), "Performance analysis tool for reinforced concrete members", Comput. Concrete, 4(5), 331-346.
  5. Fardis, M.N. (2007), "Risk mitigation for earthquakes and landslides. Guidelines for displacement-based design of buildings and bridges", Report No. 5/2007, IUSS Press, Pavia, Italy.
  6. Gentile, R., Porco, F., Raffaele, D. and Uva, G. (2018), "Simplified Moment-Curvature relationships in analytical form for circular RC sections", Bull. NZ Soc. Earthq. Eng., 51(3).
  7. Gentile, R., Raffaele, D. and Uva, G. (2017), "Simplified polynomial formulation for the calculation of the Moment- Curvature diagram of RC rectangular sections", Proceedings of the XVII Italian Conference On Seismic Engineering, Pistoia, Italy, September.
  8. King, D.J., Priestley, M.J.N. and Park, R., (1986), "Computer programs for concrete column design", Research Report 86/12, Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand.
  9. Kircher, C.A., Whitman, R.V. and Holmes, W.T. (2006), "HAZUS Earthquake Loss Estimation Methods", Nat. Hazard Rev., 7, 45-59.
  10. Mander, J.B., Priestley, M.J.N. and Park, R. (1988), "Theoretical stress strain model for confined concrete", J. Struct. Eng., 114(8), 1804-1826.
  11. Miano, A., Fatemeh, J., De Risi, R., Prota, A. and Manfredi, G. (2015), "A case-study on scenario-based probabilistic seismic loss assessment for a portfolio of bridges", Proceedings of the 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12, Vancouver, Canada.
  12. Montejo, L.A. and Kowalsky, M.J. (2007), "Set of codes for the analysis of reinforced concrete members", Technical Report No. IS-07-01, North Carolina State University, USA.
  13. NTC (2008), Ministero delle Infrastrutture e dei Trasporti "DM 14 gennaio 2008 in materia di "norme tecniche per le costruzioni", Gazzetta ufficiale n.29 del 4 febbraio 2008, Supplemento ordinario n.30", Istituto Poligrafico e Zecca dello stato.
  14. NZSEE/MBIE (2017), "The seismic assessment of existing buildings technical guidelines for engineering assessments", Final draft 1, July, New Zealand.
  15. Padgett, J.E., Nielson, B.G. and Desroches, R. (2008), "Selection of optimal intensity measures in probabilistic seismic demand models of highway bridge portfolios", Earthq. Eng. Struct. Dyn., 37, 711-725.
  16. Palermo, A., Liu, R., Rais, A., McHaffie, B., Andisheh, K., Pampanin, S., Gentile, R., Nuzzo, I., Graniero, M., Loporcaro, G., McGann, C. and Wotherspoon, L. (2017), "Performance of road bridges during the 14th November 2016 Kaikoura earthquake", Bull. NZ Soc. Earthq. Eng., 50(2), 253-270.
  17. Perus, I. and Fajfar, P. (2007), "Prediction of the force-drift envelope for RC columns in flexure by the CAE method", Earthq. Eng. Struct. Dyn., 36(15), 2345-2363.
  18. Perus, I., Poljansek, K. and Fajfar, P. (2006), "Flexural deformation capacity of rectangular RC columns determined by the CAE method", Earthq. Eng. Struct. Dyn., 35(12), 1453-1470.
  19. Pinho, R., Monteiro, R., Casarotti, C. and Delgado, R. (2009), "Assessment of continuous span bridges through nonlinear static procedures", Earthq. Spectra, 25(1), 143-159.
  20. Priestley, M.J.N. and Park, R. (1987), "Strength and ductility of concrete bridge columns under seismic loading", ACI Struct. J., 84(1), 61-76.
  21. Priestley, M.J.N., Calvi, G.M. and Kowalsky, M.J. (2007), Displacement based Seismic Design of Structures, IUSS Press, Pavia, Italy.
  22. Priestley, M.J.N., Seible, F. and Calvi, G. (1996), Seismic Design and Retrofit of Bridges, John Wiley and Sons, New York, USA.
  23. Raffaele, D., Porco, F., Fiore, A. and Uva, G. (2014a), "Simplified vulnerability assessment of reinforced concrete circular piers in multi-span simply supported bridges", Struct. Infrastr. Eng., 10(8), 950-962.
  24. Raffaele, D., Porco, F., Uva, G. and Fiore, A. (2014b), "Simplified assessment of seismic retrofitting interventions on RC circular piers in multi-span simply supported bridges", Proceedings of the 7th International Conference on Bridge Maintenance, Safety and Management, Shanghai, China.
  25. Raffaele, D., Uva, G., Porco, F. and Fiore, A. (2014c), "About of seismic capacity of bridge piers: A simplified approach", Proceedings of the 7th International Conference on Bridge Maintenance, Safety and Management, Shanghai, China.
  26. Ranzo, G. and Priestley, M.J.N. (2000), "Seismic performance of large RC circular hollow columns", Proceedings of the 12th World Conference on Earthquake Engineering, Auckland, New Zealand, February.
  27. Sadowski, A.J., Rotter, J.M., Stafford, P.J., Reinke, T. and Ummenhofer, T. (2017), "On the gradient of the yield plateau in structural carbon steels", J. Constr. Steel Res., 130, 120-130.
  28. Zelaschi, C., Monteiro, R. and Pinho, R. (2015), "Improved fragility functions for RC bridge populations", Proceedings of the 5th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering. Crete Island, Greece, May.