Bulletin of Earthquake Engineering

, Volume 17, Issue 6, pp 3495–3516 | Cite as

Direct estimation of the P-delta effect through the “stability-coefficient-response-spectra” by introducing the “first-storey-single-degree-of-freedom” system

  • Seyed Mohammad Fard MousaviEmail author
  • Serhan Sensoy
Original Research


Present study introduces two concepts for direct estimation of P-delta effect in both, strength based, and displacement based design methods. Although various previously conducted studies focused on inclusion of P-delta effect into the aforementioned design methods, development of reliable procedures is still attractive. The major argument of present study is that: treatment of P-delta effect can be enhanced by using an alternative response/design spectrum. To this end, and based on period-dependence feature of stability coefficient (SC), the “stability coefficient response spectra” (SCRS) is introduced. The SCRS, plots spectral acceleration versus SC, instead of period, for a pendulum with known height. To facilitate implementation of SCRS on multi-degree-of-freedom systems, and by considering some special features of the first-storey, the concept of “first-storey-single-degree-of-freedom” (FSSDOF) system is introduced. The FSSDOF system permits setting the minimum necessary lateral stiffness, conforming to a pre-selected SC limit, and a given ductility level, at very early stages of design process. Moreover, it is shown that implementation of SCRS and FSSDOF system can be extended to account for drift limits. This is done by introducing a modified version of the “yield-point-spectra” method in which period-dependence feature of SC is recognized. Several numerical examples are included as part of the presentation.


Stability-coefficient response spectra First-storey single-degree-of-freedom-system P-delta effect Pendulum-based spectra Inverted-pendulum 



  1. Adam C, Ibarra L (2015) Seismic collapse assessment. Earthq Eng encycl 3:2729–2752Google Scholar
  2. Adam C, Jäger C (2012a) Seismic collapse capacity of basic inelastic structures vulnerable to the P-delta effect. Earthq Eng Struct Dyn 41(4):775–793CrossRefGoogle Scholar
  3. Adam C, Jäger C (2012b) Simplified collapse capacity assessment of earthquake excited regular frame structures vulnerable to P-delta. Eng Struct 44:159–173CrossRefGoogle Scholar
  4. Adam C, Ibarra LF, Krawinkler H (2004) Evaluation of P-delta effects in non-deteriorating MDOF structures from equivalent SDOF systems. In: 13th world conference on earthquake engineering, Vancouver, BC, CanadaGoogle Scholar
  5. 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
  6. Andrews A (1977) Slenderness effects in earthquake resisting frames. Bull N Z Natl Soc Earthq Eng 10(3):71–75Google Scholar
  7. ASCE, SEI (2010) Minimum design loads for buidings and other structures. American Society of Civil Engineers/Structural Engineering Institute, RestonGoogle Scholar
  8. Aschheim M, Black EF (2000) Yield point spectra for seismic design and rehabilitation. Earthq Spectra 16(2):317–336CrossRefGoogle Scholar
  9. Aschheim M, Montes EH (2003) The representation of P-Δ effects using yield point spectra. Eng Struct 25(11):1387–1396CrossRefGoogle Scholar
  10. Asimakopoulos AV, Karabalis DL, Beskos DE (2007) Inclusion of P-Δ effect in displacement-based seismic design of steel moment resisting frames. Earthq Eng Struct Dyn 36(14):2171–2188CrossRefGoogle Scholar
  11. Aydınoğlu M, Fahjan Y (2003) A unified formulation of the piecewise exact method for inelastic seismic demand analysis including the P-delta effect. Earthq Eng Struct Dyn 32(6):871–890CrossRefGoogle Scholar
  12. Bernal D (1987) Amplification factors for inelastic dynamic p–Δ effects in earthquake analysis. Earthq Eng Struct Dyn 15(5):635–651CrossRefGoogle Scholar
  13. Bernal D (1992) Instability of buildings subjected to earthquakes. J Struct Eng 118(8):2239–2260CrossRefGoogle Scholar
  14. Bernal D (1998) Instability of buildings during seismic response. Eng Struct 20(4):496–502CrossRefGoogle Scholar
  15. Black E (2011) Use of stability coefficients for evaluating the P-Δ effect in regular steel moment resisting frames. Eng Struct 33(4):1205–1216CrossRefGoogle Scholar
  16. CEN (2005) Design of structures for earthquake resistance-part 1: general rules, seismic actions and rules for buildings. European Committee for Standardization, BrusselsGoogle Scholar
  17. Chopra AK (2007) Dynamics of structures: theory and applications to earthquake engineering. Prentice-Hall, Upper Saddle RiverGoogle Scholar
  18. Graizer V, Kalkan E (2008) Response of pendulums to complex input ground motion. Soil Dyn Earthq Eng 28(8):621–631CrossRefGoogle Scholar
  19. Gupta A, Krawinkler H (2000) Dynamic P-delta effects for flexible inelastic steel structures. J Struct Eng 126(1):145–154CrossRefGoogle Scholar
  20. Hjelmstad K, Williamson E (1998) Dynamic stability of structural systems subjected to base excitation. Eng Struct 20(4):425–432CrossRefGoogle Scholar
  21. Ibarra LF, Krawinkler H (2004) Global collapse of deteriorating MDOF systems. In: 13th world conference on earthquake engineering, Vancouver, BC, CanadaGoogle Scholar
  22. Ibarra LF, Medina RA, Krawinkler H (2005) Hysteretic models that incorporate strength and stiffness deterioration. Earthq Eng Struct Dyn 34(12):1489–1511CrossRefGoogle Scholar
  23. Jäger C, Adam C (2013) Influence of collapse definition and near-field effects on collapse capacity spectra. J Earthq Eng 17(6):859–878CrossRefGoogle Scholar
  24. Jennings PC, Husid R (1968) Collapse of yielding structures during earthquakes. J Eng Mech 94:1045–1065Google Scholar
  25. Kalkan E, Graizer V (2007) Coupled tilt and translational ground motion response spectra. J Struct Eng 133(5):609–619CrossRefGoogle Scholar
  26. 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
  27. MacRae GA (1994) P-Δ effects on single-degree-of-freedom structures in earthquakes. Earthq Spectra 10(3):539–568CrossRefGoogle Scholar
  28. Miranda E, Akkar SD (2003) Dynamic instability of simple structural systems. J Struct Eng 129(12):1722–1726CrossRefGoogle Scholar
  29. 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
  30. Priestley M (2000) Performance based seismic design. Bull N Z Soc Earthq Eng 33(3):325–346Google Scholar
  31. Priestley M, Calvi G, Kowalsky M (2007) Direct displacement-based seismic design of structures. In: Proceedings of the 2007 NZSEE conferenceGoogle Scholar
  32. Rahimi E, Estekanchi H (2015) Collapse assessment of steel moment frames using endurance time method. Earthq Eng Eng Vib 14(2):347–360CrossRefGoogle Scholar
  33. Rosenblueth E (1965) Slenderness effects in buildings. J Struct Div 91(1):229–252Google Scholar
  34. Sivaselvan MV, Reinhorn AM (2000) Hysteretic models for deteriorating inelastic structures. J Eng Mech 126(6):633–640CrossRefGoogle Scholar
  35. Sun C-K, Berg GV, Hanson RD (1973) Gravity effect on single-degree inelastic system. J Eng Mech Div 99(1):183–200Google Scholar
  36. TEC (2007) Specification for structures to be built in disaster areas. Ministry of Public Works and Settlement, TurkeyGoogle Scholar
  37. 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
  38. Williamson EB (2003) Evaluation of damage and P-Δ effects for systems under earthquake excitation. J Struct Eng 129(8):1036–1046CrossRefGoogle Scholar
  39. Williamson EB, Hjelmstad KD (2001) Nonlinear dynamics of a harmonically-excited inelastic inverted pendulum. J Eng Mech 127(1):52–57CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Civil Engineering DepartmentEastern Mediterranean UniversityFamagustaTurkey

Personalised recommendations