Nonlinear pile-head macro-element for the seismic analysis of structures on flexible piles


Performance-based design (PBD) procedures require accurate estimates of both maximum and residual displacements in structural systems. Macro-element models are already proven tools for designing structures on shallow foundations according to PBD, since they represent a very cost-effective solution in terms of balance between physical behaviour, simulation accuracy and computational cost. This work extends the macro-element approach to the analysis of laterally loaded pile-shafts and soil-pile-structure interaction. The lateral response of the entire soil-pile system to seismic actions is thus condensed at the pile-head, being represented by a zero-length element located at the base of the columns and subjected to the foundation input motion. The macro-element model is presented, based on the three fundamental features of the response of laterally loaded piles: initial elastic behaviour, gap opening/closure effects and failure conditions. These three characteristic behaviours are all made compatible by using an inelastic model which accounts for the evolution from initial nonlinear elastic behaviour to full plastic flow at failure. Such inelastic model is based on a bounding surface plasticity theory formulation that ensures a smooth transition from the initial elastic pile-head response up to nonlinear behaviour and plastic mechanism formation. In order to validate the macro-element, its response is favourably compared with numerical results from advanced simulations of pile lateral behaviour and with load tests on real piles.

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Code availability

Macro-element implemented in structural analysis software SeismoStruct.


  1. Borja RI, Lin CH, Montáns FJ (2001) Cam-Clay plasticity, Part IV: implicit integration of anisotropic bounding surface model with nonlinear hyperelasticity and ellipsoidal loading function. Comput Methods Appl Mech Eng 190:3293–3323.

    Article  Google Scholar 

  2. Budhu M, Davies TG (1987) Nonlinear analysis of laterally loaded piles in cohesionless soils. Can Geotech J 24:289–296.

    Article  Google Scholar 

  3. Budhu M, Davies TG (1988) Analysis of laterally loaded piles in soft clays. J Geotech Eng 114(1):21–39.

    Article  Google Scholar 

  4. Cavalieri F, Correia AA, Crowley H, Pinho R (2020) Seismic fragility analysis of URM buildings founded on piles: influence of dynamic soil–structure interaction models. Bull Earthq Eng 18(9):4127–4156.

    Article  Google Scholar 

  5. Chatzigogos CT (2007) Comportement sismique des fondations superficielles: vers la prise en compte d’un critère de performance dans la conception, PhD Thesis, École Polytechnique, Palaiseau, France

  6. Chatzigogos CT, Pecker A, Salençon J (2009) Macroelement modeling of shallow foundations. Soil Dyn Earthq Eng 29(5):765–781.

    Article  Google Scholar 

  7. Chatzigogos CT, Figini R, Pecker A, Salençon J (2011) A macroelement formulation for shallow foundations on cohesive and frictional soils. Int J Numer Anal Methods Geomech 35(8):902–931.

    Article  Google Scholar 

  8. Correia AA (2011) A pile-head macro-element approach to seismic design of monoshaft-supported bridges, PhD Thesis, ROSE School, IUSS Pavia, Italy

  9. Correia AA, Pecker A, Kramer SL (2021) Failure mechanism for laterally loaded flexible piles based on yield design theory, Géotechnique (submitted for publication)

  10. Cremer C, Pecker A, Davenne L (2001) Cyclic macro-element for soil-structure interaction: material and geometrical non-linearities. Int J Numer Anal Methods Geomech 25(13):1257–1284.

    Article  Google Scholar 

  11. Cremer C, Pecker A, Davenne L (2002) Modelling of nonlinear dynamic behaviour of a shallow strip foundation with macro-element. J Earthq Eng 6(2):175–211.

    Google Scholar 

  12. Dafalias YF (1986) Bounding surface plasticity. I: mathematical foundation and hypoplasticity. J Eng Mech 112(9):966–987.

    Article  Google Scholar 

  13. Davies TG, Budhu M (1986) Non-linear analysis of laterally loaded piles in heavily overconsolidated clays. Géotechnique 36(4):527–538.

    Article  Google Scholar 

  14. Elgamal A, Yang Z, Parra E, Ragheb A (2003) Modeling of cyclic mobility in saturated cohesionless soils. Int J Plast 19:883–905.

    Article  Google Scholar 

  15. EN 1998–5 (2004) Eurocode 8: design of structures for earthquake resistance—Part 5: foundations, retaining structures and geotechnical aspects, EN 1998–5:2004, European Committee for Standardization (CEN), Belgium

  16. Figini R (2010) Non-linear dynamic soil-structure interaction: application to seismic analysis and design of structures on shallow foundations, PhD Thesis, Politecnico di Milano, Milan, Italy

  17. Figini R, Paolucci R, Chatzigogos CT (2012) A macro-element model for non-linear soil-shallow foundation-structure interaction under seismic loads: theoretical development and experimental validation on large scale tests. Earthq Eng Struct Dyn 41(3):475–493.

    Article  Google Scholar 

  18. Gazetas G (1991) Foundation vibrations. In: Fang HY (ed) Foundation engineering handbook, 2nd edn. Van Nostrand Reinhold, New York, pp 553–593

    Google Scholar 

  19. Hutchinson TC, Boulanger RW, Chai YH, Idriss IM (2002) Inelastic seismic response of extended pile shaft supported bridge structures, Report PEER 2002/14, Pacific Earthquake Engineering Research Center, University of California, Berkeley, USA

  20. Krieg RD, Krieg DB (1977) Accuracies of numerical solution methods for the elastic-perfectly plastic model. J Press Vessel Technol 99(4):510–515.

    Article  Google Scholar 

  21. Lemnitzer A, Khalili-Tehrani P, Ahlberg ER, Rha C, Taciroglu E, Wallace JW, Stewart JP (2010) Nonlinear efficiency of bored pile group under lateral loading. J Geotech Geoenviron Eng 136(12):1673–1685.

    Article  Google Scholar 

  22. Lu J, Elgamal A, Yang Z (2011) OpenSeesPL: 3D lateral pile-ground interaction, available at, University of California, San Diego, USA

  23. McKenna F, Fenves GL, Scott MH, Jeremic B (2000) OpenSees: Open system for earthquake engineering simulation, available at, Pacific Earthquake Engineering Research Center, University of California, Berkeley, USA

  24. Mróz Z, Zienkiewicz OC (1984) Uniform formulation of constitutive equations for clays and sands. In: Desai CS, Gallagher RH (eds) Mechanics of engineering materials. Wiley, Chichester, pp 415–449

    Google Scholar 

  25. Ortiz M, Martin JB (1989) Symmetry-preserving return mapping algorithms and incrementally extremal paths: a unification of concepts. Int J Numer Methods Eng 28(8):1839–1853.

    Article  Google Scholar 

  26. Ortiz M, Simo JC (1986) An analysis of a new class of integration algorithms for elastoplastic constitutive relations. Int J Numer Methods Eng 23(3):353–366.

    Article  Google Scholar 

  27. Phillips A (1972) On rate-independent continuum theories of graphite and their experimental verification. Nucl Eng Des 18(2):203–211.

    Article  Google Scholar 

  28. Prévost JH (1987) “Modeling the behavior of geomaterials”, course notes, short-course on nonlinear soil mechanics and dynamic soil-structure interaction. Zace Services Ltd., Lausanne

    Google Scholar 

  29. Priestley MJN, Calvi GM, Kowalsky MJ (2007) Displacement-based seismic design of structures. IUSS Press, Pavia

    Google Scholar 

  30. Randolph MF (1981) The response of flexible piles to lateral loading. Géotechnique 31(2):247–259.

    Article  Google Scholar 

  31. Seismosoft (2020) SeismoStruct 2020—a computer program for static and dynamic nonlinear analysis of framed structures, available at:

  32. Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New York

    Google Scholar 

  33. Simo JC, Ortiz M (1985) A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations. Comput Methods Appl Mech Eng 49(2):221–245.

    Article  Google Scholar 

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

    Article  Google Scholar 

  35. Stewart JP, Taciroglu E, Wallace JW, Ahlberg ER, Lemnitzer A, Rha C, Khalili-Tehrani P, Keowen S, Nigbor RL, Salamanca A (2007) Full scale cyclic large deflection testing of foundation support systems for highway bridges. Part I: drilled shaft foundation Report UCLA SGEL-01, structural and geotechnical engineering laboratory, University of California, Los Angeles, USA

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This research was partially funded by the Fundação para a Ciência e a Tecnologia (FCT), Portugal, through grant SFRH/BD/29311/2006.

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Correspondence to António A. Correia.

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Correia, A.A., Pecker, A. Nonlinear pile-head macro-element for the seismic analysis of structures on flexible piles. Bull Earthquake Eng (2021).

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  • Macro-element
  • Soil-pile-structure interaction
  • Pile-head
  • Lateral response
  • Gap
  • Bounding surface plasticity