The engineering merit of the "Effective Period" of bilinear isolation systems

Makris, Nicos;Kampas, Georgios

  • Received : 2012.07.05
  • Accepted : 2012.10.16
  • Published : 2013.04.25


This paper examines whether the "effective period" of bilinear isolation systems, as defined invariably in most current design codes, expresses in reality the period of vibration that appears in the horizontal axis of the design response spectrum. Starting with the free vibration response, the study proceeds with a comprehensive parametric analysis of the forced vibration response of a wide collection of bilinear isolation systems subjected to pulse and seismic excitations. The study employs Fourier and Wavelet analysis together with a powerful time domain identification method for linear systems known as the Prediction Error Method. When the response history of the bilinear system exhibits a coherent oscillatory trace with a narrow frequency band as in the case of free vibration or forced vibration response from most pulselike excitations, the paper shows that the "effective period" = $T_{eff}$ of the bilinear isolation system is a dependable estimate of its vibration period; nevertheless, the period associated with the second slope of the bilinear system = $T_2$ is an even better approximation regardless the value of the dimensionless strength,$Q/(K_2u_y)=1/{\alpha}-1$, of the system. As the frequency content of the excitation widens and the intensity of the acceleration response history fluctuates more randomly, the paper reveals that the computed vibration period of the systems exhibits appreciably scattering from the computed mean value. This suggests that for several earthquake excitations the mild nonlinearities of the bilinear isolation system dominate the response and the expectation of the design codes to identify a "linear" vibration period has a marginal engineering merit.


seismic isolation;equivalent linearization;bilinear behavior;system identification;health monitoring;earthquake protection


  1. Addison, P.S. (2002), The illustrated wavelet transform handbook, Institute of Physics Handbook.
  2. Anti-seismic devices (2009), European standard, FprEN 15129, Eurocode.
  3. Astrom, K.J. and Bohlin, T. (1965), "Numerical identification of linear dynamic systems from normal operating records", IFAC Symposium on Self-Adaptive Systems, Teddington, England.
  4. Buckle, I.G. and Mayes, R.L. (1990), "Seismic isolation: History, application and performance - A world view", Earthq. Spectra J., 6(2), 161-201.
  5. Caughey, T.K. (1960), "Random excitation of a system with bilinear hysteresis", J. Appl. Mech., 27(4), 649-652.
  6. Caughey, T.K. (1963), "Equivalent linearization techniques", J. Acoust. Soc. Am., 35(11), 1706-1711.
  7. Constantinou, M.C., Mokha, A.S. and Reinhorn, A.M. (1990), "Teflon bearing in base isolation. II: modeling", J. Struct. Eng.-ASCE, 116(2), 455-474.
  8. Crandall, S.H. (2006), "A half-century of stochastic equivalent linearization", J. Struct. Contr. Health Monitor., 13(1), 27-40.
  9. Design of lead-rubber bearings (1983), Civil division publication 818/A, New Zealand Ministry of Works and Development, Wellington, New Zealand.
  10. FEMA 310 (1998), Handbook for the seismic evaluation of buildings - A prestandard, ASCE.
  11. Garini, E., Gazetas, G. and Anastasopoulos, I. (2010), "Accumulated assymetric slip caused by motions containing severe 'Directivity' and 'Fling' pulses", Geotechnique, 61(9), 733-756.
  12. Guide specifications for seismic isolation design (1991), American association of state highway and transportation officials, Washington, D.C.
  13. Hwang, J.S. and Chiou, J.M. (1996), "An equivalent linear model of lead-rubber seismic isolation bearings", J. Eng. Struct., 18(7), 528-536.
  14. Hwang, J.S. and Sheng, L.H. (1993), "Equivalent elastic seismic analysis of base-isolated bridges with lead-rubber bearings", J. Eng. Struct., 16(3), 201-209.
  15. Hwang, J.S. and Sheng, L.H. (1994), "Effective stiffness and equivalent damping of base-isolated bridges", J. Struct. Eng., 119(10), 3094-3101.
  16. International Code Council (2000), International building code.
  17. Iwan, W.D. and Gates, N.C. (1979), "The effective period and damping of a class of hysteretic structures", J. Earthq. Eng. Struct. D., 7(3), 199-211.
  18. Iwan, W.D. (1980), "Estimating inelastic response spectra from response spectra", J. Earthq. Eng. Struct. D., 8(4), 375-388.
  19. Kelly, J.M., Eidinger, J.M. and Derham, C.J. (1977), A practical soft story system, Report No. UCB/EERC-77|27, Earthquake Engineering Research Center, University of California, Berkeley, CA, USA.
  20. Kelly, J.M. (1986), "A seismic base isolation: Review and bibliography", J. Soil Dyn. Earthq. Eng., 5(4), 202-216.
  21. Ljung, L. (1987), System identification-theory for the user, Prentice-Hall, New Jersey.
  22. Ljung, L. (1994), "State of the art in linear system identification: Time and frequency domain methods", Proceedings of '04 American Control Conference, 1, 650-660.
  23. Ljung, L. (2002), "Prediction error estimation methods", Circ. Syst. Signal Pr., 21(1), 11-21.
  24. Makris, N. and Black, C.J. (2004a), "Dimensional analysis of rigid-plastic and elastoplastic structures under pulse-type excitations", J. Eng. Mech., 130(9), 1006-1018.
  25. Makris, N. and Black, C.J. (2004b), "Dimensional analysis of bilinear oscillators under pulse-type excitations", J. Eng. Mech., 130(9), 1019-1031.
  26. Makris, N. and Black, C. (2004c), "Evaluation of peak ground velocity as a "good" intensity measure for near-source ground motions", J. Eng. Mech.-ASCE, 130(9), 1032-1044.
  27. Makris, N. and Chang, S. (2000), "Effect of viscous, viscoplastic and friction damping on the response of seismic isolated structures", J. Earthq. Eng. Struct. D., 29(1), 85-107.<85::AID-EQE902>3.0.CO;2-N
  28. Makris, N. and Vassiliou, M.F. (2011), "The existence of 'complete similarities' in the response of seismic isolated structures subjected to pulse-like ground motions and their implications in analysis", J. Earthq. Eng. Struct. D., 40(10), 1103-1121.
  29. Mallat, S.G. (1999), A wavelet tour of signal processing, Academic Press.
  30. MATLAB (2002), High-performance language software for technical computation, The MathWorks, Inc: Natick, MA, USA.
  31. Mayes, R.L., Buckle, I.G., Kelly, T.E. and Jones, L.R. (1991), "AASHTO seismic isolation design requirements for highway bridges", J. Struct. Eng., 118(1), 284-304.
  32. Mokha, A.S., Constantinou, M.C. and Reinhorn, A.M. (1990), "Teflon bearing in base isolation. I: Testing", J. Struct. Eng.-ASCE, 116(2), 438-454.
  33. Naeim, F. and Kelly, J.M. (1999), Design of seismic isolated structures, New York: Wiley Publications.
  34. Naeim, F. (2001), The seismic design handbook, Springer.
  35. Ricker, N. (1943), "Further developments in the wavelet theory of seismogram structure", B. Seismol. Soc. Am., 33(3), 197-228.
  36. Ricker, N. (1944), "Wavelet functions and their polynomials", Geophysics, 9(3), 314-323.
  37. Roberts, J.B. and Spanos, P.D. (2003), Random vibration and statistical linearization, New York, Dover Publications.
  38. Vassiliou, M.F. and Makris, N. (2011), "Estimating time scales and length scales in pulselike earthquake acceleration records with wavelet analysis", B. Seismol. Soc. Am., 101(2), 96-618.
  39. Veletsos, A.S. and Newmark, N.M. (1960), Effect of inelastic behavior on the response of simple systems to earthquake motions, University of Illinois, IL, USA.
  40. Veletsos, A.S., Newmark, N.M. and Chelepati, C.V. (1969), "Deformation spectra for elastic and elastoplastic systems subjected to ground shock and earthquake motions", Proceedings of the 3rd World Conference on Earthquake Engineering, II, Wellington, New Zealand: 663-682.
  41. Veletsos, A.S. and Vann, W.P. (1971), "Response of ground-excited elastoplastic systems", J. Struct. Div.-ASCE, 97(ST4), 1257-1281.
  42. Wen, Y.K. (1975), "Approximate method for nonlinear random vibration", J. Eng. Mech., 101(4), 389-401.
  43. Wen, Y.K. (1976), "Method for random vibration of hysteretic systems", J. Eng. Mech., 102(2), 249-263.

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