Derivation of response spectrum compatible non-stationary stochastic processes relying on Monte Carlo-based peak factor estimation

  • Giaralis, Agathoklis (School of Engineering and Mathematical Sciences, City University London) ;
  • Spanos, Pol D. (Department of Mechanical Engineering and Material Sciences, Rice University)
  • Received : 2012.01.17
  • Accepted : 2012.02.26
  • Published : 2012.09.25


In this paper a novel approach is proposed to address the problem of deriving non-stationary stochastic processes which are compatible in the mean sense with a given (target) response (uniform hazard) spectrum (UHS) as commonly desired in the aseismic structural design regulated by contemporary codes of practice. The appealing feature of the approach is that it is non-iterative and "one-step". This is accomplished by solving a standard over-determined minimization problem in conjunction with appropriate median peak factors. These factors are determined by a plethora of reported new Monte Carlo studies which on their own possess considerable stochastic dynamics merit. In the proposed approach, generation and treatment of samples of the processes individually on a deterministic basis is not required as is the case with the various "two-step" approaches found in the literature addressing the herein considered task. The applicability and usefulness of the approach is demonstrated by furnishing extensive numerical data associated with the elastic design UHS of the current European (EC8) and the Chinese (GB 50011) aseismic code provisions. Purposely, simple and thus attractive from a practical viewpoint, uniformly modulated processes assuming either the Kanai-Tajimi (K-T) or the Clough-Penzien (C-P) spectral form are employed. The Monte Carlo studies yield damping and duration dependent median peak factor spectra, given in a polynomial form, associated with the first passage problem for UHS compatible K-T and C-P uniformly modulated stochastic processes. Hopefully, the herein derived stochastic processes and median peak factor spectra can be used to facilitate the aseismic design of structures regulated by contemporary code provisions in a Monte Carlo simulation-based or stochastic dynamics-based context of analysis.


  1. ASCE (2000), Seismic analysis of safety-related nuclear structures and commentary, ASCE standard no. 004-98, American Society of Civil Engineers.
  2. ASCE (2006), Minimum design loads for buildings and other structures, ASCE standard no. 007-05, American Society of Civil Engineers.
  3. Bogdanoff, J.L., Goldberg, J.E. and Bernard, M.C. (1961), "Response of a simple structure to a random earthquake-type disturbance", B. Seismol. Soc. Am., 51(2), 293-310.
  4. Bommer, J.J. and Martinez-Pereira, A. (1999), "The effective duration of earthquake strong motion", J. Earthq. Eng., 3(2), 127-172.
  5. Cacciola, P. (2010), "A stochastic approach for generating spectrum compatible fully nonstationary earthquakes", Comput. Struct., 88(15-16), 889-901.
  6. CEN (2004), Eurocode 8: Design of structures for earthquake resistance - Part 1: General rules, seismic actions and rules for buildings, EN 1998-1: 2004, Comite Europeen de Normalisation, Brussels.
  7. Chopra, A.K. (2007), "Elastic response spectrum: a historical note", Earthq. Eng. Struct. D., 36(1), 3-12.
  8. Clough, R.W. and Penzien, J. (1993), Dynamics of structures, Second Edition, Mc-Graw Hill, New York.
  9. Conte, J.P. and Peng, B.F. (1997), "Fully nonstationary analytical earthquake ground-motion model", J. Eng. Mech.-ASCE, 123(1), 15-24.
  10. Corotis, R.B., Vanmarcke, E.H. and Cornell, C.A. (1972), "First passage of nonstationary random processes", J. Eng. Mech.-ASCE, 98(2), 401-414.
  11. Crespi, P.G., Floris, C. and Paganini, P. (2002), "A probabilistic method for generating spectrum compatible earthquake time histories", Eur. Earthq. Eng., 16(3), 3-17.
  12. Faccioli, E., Paolucci, R. and Rey, J. (2004), "Displacement spectra for long periods," Earthquake Spectra, 20, 347-376.
  13. Falsone, G. and Neri, F. (2000), "Stochastic modeling of earthquake excitation following the EC8: power spectrum and filtering equations", Eur. Earthq. Eng., 14(1), 3-12.
  14. Fan, F.G. and Ahmadi, G. (1990), "Nonstationary Kanai-Tajimi models for El Centro 1940 and Mexico City 1985 earthquakes", Probabilist Eng. Mech., 5(4), 171-181.
  15. GB 50011 (2001), Code for seismic design of buildings, National Standard of the People's Republic of China, China Building Industry Press, Beijing.
  16. Giaralis, A. and Spanos, P.D. (2009), "Wavelets based response spectrum compatible synthesis of accelerograms- Eurocode application (EC8)", Soil Dyn. Earthq. Eng., 29(1), 219-235.
  17. Giaralis, A. and Spanos, P.D. (2010), "Effective linear damping and stiffness coefficients of nonlinear systems for design spectrum based analysis", Soil Dyn. Earthq. Eng., 30(9), 798-810.
  18. Gupta, I.D. and Trifunac, M.D. (1998), "Defining equivalent stationary PSDF to account for nonstationarity of earthquake ground motion", Soil Dyn. Earthq. Eng., 17(2), 89-99.
  19. Hancock, J. and Bommer, J.J. (2006), "A state-of-knowledge review of the influence of strong-motion duration on structural damage", Earthq. Spectra, 22, 827-845.
  20. Hancock, J. and Bommer, J.J. (2007), "Using spectral matched records to explore the influence of strong-motion duration on inelastic structural response", Soil Dyn. Earthq. Eng., 27(4), 291-299.
  21. Iervolino, I., Maddaloni, G. and Cosenza, E. (2008), "Eurocode 8 compliant real records sets for seismic analysis of structures", J. Earthq. Eng., 12(1), 54-90.
  22. Jayaram, N., Lin, T. and Baker, J.W. (2011), "A computationally efficient ground-motion selection algorithm for matching a target response spectrum mean and variance", Earthq. Spectra, 27, 797-815.
  23. Kanai, K. (1957), "Semi-empirical formula for the seismic characteristics of the ground", University of Tokyo, Bulletin of the Earthquake Research Institute, 35, 309-325.
  24. Katsanos, E.I., Sextos, A.G. and Manolis, G.D. (2010), "Selection of earthquake ground motion records: A stateof- the- art review from a structural engineering perspective", Soil Dyn. Earthq. Eng., 30(4), 157-169.
  25. Kaul, M.K. (1978), "Stochastic characterization of earthquakes through their response spectrum", Earthq. Eng. Struct. D., 6(5), 497-509.
  26. Kotz, S. and Nadarajah, S. (2000), Extreme value distribution.Theory and applications, Imperial College Press, London.
  27. Martinelli, L., Barbella, G. and Feriani, A. (2011), "A numerical procedure for simulating the multi-support seismic response of submerged floating tunnels anchored by cables", Eng. Struct., 33(10), 2850-2860.
  28. Mason, A.B. and Iwan, W.D. (1983), "An approach to the first passage problem in random vibration", J. Appl. Mech.-ASME, 50, 641-646.
  29. Mason, A.B. and Iwan, W.D. (1983), "An approach to the first passage problem in random vibration", J. Appl. Mech.-ASME, 50, 641-646.
  30. Morikawa, H. and Zerva, A. (2008), "Approximate representation of the statistics for extreme responses of single degree-of-freedom system excited by non-stationary processes", Probab. Eng. Mech., 23(2-3), 279-288.
  31. Newmark, N.M. and Hall, J.W. (1982), Earthquake spectra and design, Earthquake Engineering Research Institute, Oakland.
  32. Nigam, N.C. and Jennings, P.C. (1969), "Calculation of response spectra from strong-motion earthquake records", B. Seismol. Soc. Am., 59(2), 909-922.
  33. Nocedal, J. and Wright, S.J. (1999), Numerical optimization, Springer-Verlag, New York.
  34. Preumont, A. (1985a), "The generation of non-separable artificial earthquake accelerograms for the design of nuclear power plants", Nucl. Eng. Des., 88(1), 59-67.
  35. Preumont, A. (1985b), "On the peak factor of stationary Gaussian processes", J. Sound Vib., 100(1), 15-34.
  36. Priestley, M.B. (1965), "Evolutionary spectra and non-stationary processes", J. Roy. Statistic. Soc. Ser. B, 27(2), 204-237.
  37. Rezaeian, S. and Der Kiureghian, A. (2008), "A stochastic ground motion model with separable temporal and spectral nonstationarities", Earthq. Eng. Struct. D., 37(13), 1565-1584.
  38. Roberts, J.B. and Spanos, P.D. (2003), Random vibration and statistical linearization, Dover Publications, New York.
  39. Senthilnathan, A. and Lutes, L.D. (1991), "Nonstationary maximum response statistics for linear structures", J. Eng. Mech.-ASCE, 117(2), 294-311.
  40. Shrikhande, M. and Gupta, V.K. (1996), "On generating ensemble of design spectrum-compatible accelerograms", Eur. Earthq. Eng., 10(3), 49-56.
  41. Spanos, P.D. (1978), "Non-stationary random vibration of a linear structure", Int. J. Solids Struct., 14(10), 861- 867.
  42. Spanos, P.D. and Lutes, L.D. (1980), "Probability of response to evolutionary process", J. Eng. Mech.-ASCE, 106(2), 213-224.
  43. Spanos, P.D. and Vargas Loli, L.M. (1985), "A statistical approach to generation of design spectrum compatible earthquake time histories", Soil Dyn. Earthq. Eng., 4(1), 2-8.
  44. Spanos, P.D. and Zeldin, B.A. (1998), "Monte Carlo treatment of random fields: A broad perspective", Appl. Mech. Rev., 51(3), 219-237.
  45. Spanos, P.D., Giaralis, A. and Politis, N.P. (2007a), "Time- frequency representation of earthquake accelerograms and inelastic structural response records using the adaptive chirplet decomposition and empirical mode decomposition", Soil Dyn. Earthq. Eng., 27(7), 675- 689.
  46. Spanos, P.D., Giaralis, A. and Jie, L. (2009), "Synthesis of accelerograms compatible with the Chinese GB 50011-2001 design spectrum via harmonic wavelets: artificial and historic records", Earthq. Eng. Eng. Vib., 8(2), 189-206.
  47. Spanos, P.D., Giaralis, A., Politis, N.P. and Roessett, J.M. (2007b), "Numerical treatment of seismic accelerograms and of inelastic seismic structural responses using harmonic wavelets", Comput. Aided Civil Infrastruct. E., 22(4), 254-264.
  48. Taflanidis, A.A. and Jia, G. (2011), "A simulation-based framework for risk assessment and probabilistic sensitivity analysis of base-isolated structures", Earthq. Eng. Struct. D., 40(14), 1629-1651.
  49. Trifunac, M.D. and Brady, A.G. (1975), "A study on the duration of strong earthquake ground motion", B. Seismol. Soc. Am., 65(3), 581-626.
  50. Vanmarcke, E.H. (1976), Structural response to earthquakes, In C. Lomnitz & E. Rosenblueth, Eds., Seismic Risk and Engineering Decisions, Elsevier, Amsterdam, The Netherlands.
  51. Wang, J., Fan, L., Qian, S. and Zhou, J. (2002), "Simulations of non-stationary frequency content and its importance to seismic assessment of structures", Earthq. Eng. Struct. D., 31(4), 993-1005.
  52. Wen, Y.K. and Eliopoulos, D. (1994), "Method for nonstationary random vibration of inelastic structures", Probab. Eng. Mech., 9(1-2), 115-123.
  53. Zembaty, Z. (1988), "A note on non-sationary stochastic response and strong motion duration", Earthq. Eng. Struct. D., 16(8), 1189-1200.