Bulletin of Earthquake Engineering

, Volume 17, Issue 1, pp 313–336 | Cite as

An investigation of P-delta effect in conventional seismic design and direct displacement-based design using elasto-plastic SDOF systems

  • Nasir Pourali
  • Horr KhosraviEmail author
  • Mehdi Dehestani
Original Research


The seismic responses of structures are always influenced by P-Δ effects. The significance of this effect may be negligible when the structure responds elastically but is very important when the structure responds into inelastic range. The P-Δ effect usually increases the displacement response of structures. It may even cause dynamic instability when the structure subjected to severe earthquake ground motions. In conventional seismic design codes, the P-Δ effect usually leads to an increase in design base shear. The same approach is used in direct displacement-based design. Various researchers have proposed different relationships for base shear increase which can be used in different seismic design approaches such as force-based design and performance-based design. In this paper, the proposed expressions are reviewed extensively and their adequacy is evaluated by detail. Then two main purposes are pursued: (1) the development of new expressions for strength amplification factor due to P-Δ effect for current force-based design seismic codes; and (2) the modification of equivalent viscous damping or the required additional base shear considering the P-Δ effect in direct displacement-based design procedure. For this purpose, a new algorithm based upon elasto-plastic SDOF system analyses is presented. The algorithm is implemented 102,500 times overall for different periods, ductility levels, stability indices and different earthquake ground motions that each implementation needs a large amount of trial and error process in linear and nonlinear SDOF systems. The results seem to present a good development of the P-Δ effect relations for the seismic design procedures.


P-delta effect Direct displacement-based design Performance based design Force-based design Seismic design codes Nonlinear dynamic analysis 



The authors are grateful to anonymous reviewers whose encouragements and valuable comments lead to an improvement in this paper.


  1. AASHTO (2011) Guide specifications for LRFD seismic bridge design, 2nd edn. Published by the American Association of State Highway and Transportation Officials Washington, DCGoogle Scholar
  2. Adam C, Jäger C (2012) Seismic collapse capacity of basic inelastic structures vulnerable to the P-delta effect. Earthq Eng Struct Dyn 41(4):775–793CrossRefGoogle Scholar
  3. Amara F, Bosco M, Marino EM, Rossi PP (2014) An accurate strength amplification factor for the design of SDOF systems with P-Δ effects. Earthq Eng Struct Dyn 43(4):589–611CrossRefGoogle Scholar
  4. Applied Technology Council (ATC-55) (2005) Improvement of nonlinear static seismic analysis procedures (FEMA 440), Washington, DCGoogle Scholar
  5. ASCE/SEI (2016) Minimum design loads and associated criteria for buildings and other structures. ASCE 7–16, American Society of Civil Engineers/Structural Engineering Institute: Reston, VAGoogle Scholar
  6. Belleri A, Torquati M, Marini A, Riva P (2017) A novel framework to include P-Δ effects in displacement-based seismic assessment. J Earthq Eng 21(3):486–492CrossRefGoogle Scholar
  7. Bernal D (1987) Amplification factors for inelastic dynamic p-Δ effects in earthquake analysis. Earthq Eng Struct Dyn 15(5):635–651CrossRefGoogle Scholar
  8. Browning JP (2001) Proportioning of earthquake-resistant RC building structures. J Struct Eng 127(2):145–151CrossRefGoogle Scholar
  9. Building Seismic Safety Council (BSSC) (2004) The 2003 NEHRP recommended provisions for new buildings and other structures. Part 1: Provisions. FEMA 450Google Scholar
  10. CEN EN 1998-1 (2005) Eurocode 8: design of structures for earthquake resistance, part 1: general rules, seismic actions and rules for buildings. European Committee for Standardization, BrusselsGoogle Scholar
  11. Chopra AK, Goel RK (2001) Direct displacement-based design: use of inelastic vs. elastic design spectra. Earthq Spectra 17(1):47–64CrossRefGoogle Scholar
  12. Dicleli M, Buddaram S (2007) Comprehensive evaluation of equivalent linear analysis method for seismic-isolated structures represented by SDOF systems. Eng Struct 29(8):1653–1663CrossRefGoogle Scholar
  13. Dwairi HM, Kowalsky MJ, Nau JM (2007) Equivalent damping in support of direct displacement-based design. J Earthq Eng 11(4):512–530CrossRefGoogle Scholar
  14. Freeman SA (1998) The capacity spectrum method. In: Proceedings of the 11th European conference on earthquake engineering, ParisGoogle Scholar
  15. Grant DN, Blandon CA, Priestley MJ (2004) Modeling inelastic response in direct displacement-based design. Report no. ROSE 2004/02, European School of Advanced Studies in Reduction of Seismic Risk, Pavia, ItalyGoogle Scholar
  16. Gulkan P, Sozen M (1974) Inelastic responses of reinforced concrete structure to earthquake motions. ACI J Proc 71(12):604–610Google Scholar
  17. Gupta A, Krawinkler H (2000) Dynamic P-delta effects for flexible inelastic steel structures. J Struct Eng 126(1):145–154CrossRefGoogle Scholar
  18. Ibarra LF, Krawinkler H (2005) Global collapse of frame structures under seismic excitations. Report no. TB 152. Pacific Earthquake Engineering Research Center, Berkeley, CAGoogle Scholar
  19. Iwan WD (1980) Estimating inelastic response spectra from elastic spectra. Earthq Eng Struct Dyn 8(4):375–388CrossRefGoogle Scholar
  20. Iwan WD, Gates NC (1979) The effective period and damping of a class of hysteretic structures. Earthq Eng Struct Dyn 7(3):199–211CrossRefGoogle Scholar
  21. Jacobsen LS (1930) Steady forced vibrations as influenced by damping. ASME Trans 52(15):169–181Google Scholar
  22. Jara M, Casas JR (2006) A direct displacement-based method for the seismic design of design of bridges on bilinear isolation devices. Eng Struct 28(6):869–879CrossRefGoogle Scholar
  23. Judi HJ, Fenwick RC, Davidson BJ (2001) Direct displacement based design-a definition of damping. In: The New Zealand society for earthquake engineering technical conferenceGoogle Scholar
  24. Kowalsky MJ (1994) Displacement based design: a methodology for seismic design applied to RC bridge columns. Dissertation. University of California, San Diego (CA)Google Scholar
  25. Kowalsky MJ (2001) RC structural walls designed according to UBC and displacement-based methods. J Struct Eng 127(5):506–516CrossRefGoogle Scholar
  26. Kowalsky MJ (2002) A displacement-based approach for the seismic design of continuous concrete bridges. Earthq Eng Struct Dyn 31(3):719–747CrossRefGoogle Scholar
  27. Kwan WP, Billington SL (2003) Influence of hysteretic behavior on equivalent period and damping of structural systems. J Struct Eng 129(5):576–585CrossRefGoogle Scholar
  28. Lignos D, Krawinkler H (2009) Sidesway collapse of deteriorating structural systems under seismic excitations. Report no. TB 177, the John A. Blume Earthquake Engineering Research Center, Stanford University, Stanford, CAGoogle Scholar
  29. Liu T, Zordan T, Briseghella B, Zhang Q (2014a) An improved equivalent linear model of seismic isolation system with bilinear behavior. Eng Struct 61:113–126CrossRefGoogle Scholar
  30. Liu T, Zordan T, Briseghella B, Zhang Q (2014b) Evaluation of equivalent linearization analysis methods for seismically isolated buildings characterized by SDOF systems. Eng Struct 59:619–634CrossRefGoogle Scholar
  31. López SE, Ayala AG, Adam C (2015) A novel displacement-based seismic design method for framed structures considering P-Delta induced dynamic instability. Bull Earthq Eng 13(4):1227–1247CrossRefGoogle Scholar
  32. MacRae GA (1994) P-Δ effects on single-degree-of-freedom structures in earthquakes. Earthq Spectra 10(3):539–568CrossRefGoogle Scholar
  33. Mahin S, Boroschek R (1991) Influence of geometric nonlinearities on the seismic response and design of bridge structures. Background report to California Department of Transportation, Sacramento, CaliforniaGoogle Scholar
  34. Miranda E, Akkar SD (2003) Dynamic instability of simple structural systems. J Struct Eng 129(12):1722–1726CrossRefGoogle Scholar
  35. Moehle JP (1992) Evaluation and rehabilitation of multi-level, multi-column concrete freeways. In: International symposium on earthquake disaster prevention, pp 54–65Google Scholar
  36. Montgomery CJ (1981) Influence of P-delta effects on seismic design. Can J Civ Eng 8(1):31–43CrossRefGoogle Scholar
  37. Panagiotakos TB, Fardis MN (1999) Deformation-controlled earthquake-resistant design of RC buildings. J Earthq Eng 3(04):495–518Google Scholar
  38. Paulay T (1978) A consideration of P-delta effects in ductile reinforced concrete frames. Bull N Z Natl Soc Earthq Eng 11(3):151–160Google Scholar
  39. Paulay T, Priestley MJN (1992) Seismic design of reinforced concrete and masonry buildings. Wiley, New York, p 240CrossRefGoogle Scholar
  40. Pettinga JD, Priestley MJN (2007) Accounting for P-delta effects in structures when using direct displacement-based design. Research report ROSE (European School for Advanced Studies in Reduction of Seismic Risk), IUSS Press, PaviaGoogle Scholar
  41. Priestley MJN (1993) Myths and fallacies in earthquake engineering-conflicts between design and reality. Bull N Z Natl Soc Earthq Eng 26(3):329–341Google Scholar
  42. Priestley MJN, Kowalsky MJ (2000) Direct displacement-based seismic design of concrete buildings. Bull N Z Natl Soc Earthq Eng 33(4):421–444Google Scholar
  43. Priestley MJN, Calvi GM, Kowalsky MJ (2007) Displacement-based seismic design of structures. IUSS Press, Pavia, p 114Google Scholar
  44. Raghunandan M, Liel AB (2013) Effect of ground motion duration on earthquake-induced structural collapse. Struct Saf 41:119–133CrossRefGoogle Scholar
  45. Rosenblueth E (1965) Slenderness effects in buildings. J Struct Div 91(1):229–252Google Scholar
  46. Rosenblueth E, Herrera I (1964) On a kind of hysteretic damping. J Eng Mech Div 90(4):37–48Google Scholar
  47. Shibata A, Sozen MA (1976) Substitute-structure method for seismic design in R/C. J Struct Div 102(ST1):1–18Google Scholar
  48. Sullivan TJ, Calvi GM, Priestley MJ, Kowalsky MJ (2003) The limitations and performances of different displacement based design methods. J Earthq Eng 7(spec01):201–241Google Scholar
  49. Tremblay R, Côté B, Léger P (1999) An evaluation of P-Δ amplification factors in multistorey steel moment resisting frames. Can J Civ Eng 26(5):535–548CrossRefGoogle Scholar
  50. Wei B, Xu Y, Li J (2011) Treatment of P-Δ effects in displacement-based seismic design for SDOF systems. J Bridge Eng 17(3):509–518CrossRefGoogle Scholar
  51. Wight GD, Kowalsky MJ (2004) A direct displacement-based design approach for unbonded post-tensioned masonry walls. In: Proceedings of the 13th world conference on earthquake engineering–13WCEEGoogle Scholar
  52. Wijesundara KK, Nascimbene R, Sullivan TJ (2011) Equivalent viscous damping for steel concentrically braced frame structures. Bull Earthq Eng 9(5):1535–1558CrossRefGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Civil EngineeringBabol Noshirvani University of TechnologyBabolIran

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