Abstract
In this research the tail equivalent linearization method (TELM) has been extended to study structures with degrading materials. The responses of such structures to excitations are non-stationary, even if the excitations are stationary. Non-stationary behavior of the system cannot be considered by conventional TELM. Applying the conventional TELM, the only distinction in the design point excitation for two stationary excitations with different durations is in the addition of a zero value part at the beginning of the design point of the longer excitation. This means that the failure probability is the same for the non-stationary systems under excitations with different durations. Therefore, this solution cannot be correct. In this study, in using TELM for systems with degrading materials, hysteretic energy is replaced by average hysteretic energy, calculated by averaging the obtained hysteretic energy of the structure subjected to a few random sample load realization. In this way, the degradation parameters under design point coincide with those under sample load realizations. Since the average of the hysteretic energy is converges very fast, the modified TELM only requires about tens to hundreds solutions of the response in addition to the ordinary calculations of conventional TELM.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baber TT and Noori MN (1986), “Modeling General Hysteretic Behavior and Random Vibration Application,” ASME J.Vib., Acoust., Stress, Reliab.Des., 108: 411–420.
Baber TT and Wen YK (1981), “Random Vibration of Hysteretic Degrading Systems”, Journal of Engineering Mechanics Division, Proceedings of ASCE, 107: 1069–1087.
Bouc R(1967), “Forced Vibration of Mechanical Systems With Hysteresis,” Proceedings of the 4th Conference of Nonlinear Oscillations, Prague, Czechoslovakia p.315.
Caughey TK (1963), “Equivalent Linearization Techniques,” Journal of the Acoustical Society of America, 35 (11): 1706–1711.
Der Kiureghian A and Fujimura K (2009), “Nonlinear Stochastic Dynamic Analysis for Performance-Based Earthquake Engineering,” Earthquake Engineering and Structural Dynamics, 38: 719–738.
Fujimura K and Der Kiureghian A (2007), “Tail Equivalent Linearization Method for Nonlinear Random Vibration,” Probabilistic Engineering Mechanics. 2: 63–76.
Greco R, Marano GC and Fiore A (2017), “Damage-Based Inelastic Seismic Spectra,” International Journal of Structural Stability and Dynamics, 17 (10): 1750115–11750115-23.
Haukaas T and Der Kiureghian A (2004), “Finite Element Reliability and Sensitivity Methods for Performance-Based Engineering,” Report No. PEER 2003/14, Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA.
Iourtchenko D, Mo E and Naess A (2008), “Reliability of Strongly Nonlinear Single Degree of Freedom Dynamic Systems by the Path Integration Method,” Journal of Applied Mechanics, 75 (6): 061016–1–061016–8.
Koo H, Der Kiureghian A and Fujimura K (2005), “Design-Point Excitation for Non-Linear Random Vibration,” Probabilistic Engineering Mechanics, 20 (2): 136–147.
Li J and Chen J (2009), “Stochastic Dynamics of Structures,” J. Wiley and Sons.
Liu P-L and Der Kiureghian A (1991) “Optimization Algorithms for Structural Reliability,” Structural Safety, 9(3): 161–177.
M Broccardo and A Der Kiureghian (2012), “Multi-Component Nonlinear Stochastic Dynamic Analysis Using Tail-Equivalent Linearization Method,” Proceeding of 15th World Conference on Earthquake Engineering, Lisbon, Portugal.
Marano GC, Acciani G, Fiore A and Abrescia A (2015), “Integration Algorithm for Covariance Non-Stationary Dynamic Analysis of SDOF Systems Using Equivalent Stochastic Linearization,” International Journal of Structural Stability and Dynamics, 15 (2): 1450044–1–1450044–17.
Marano GC and Greco R (2006), “Damage and Ductility Demand Spectra Assessment of Hysteretic Degrading Systems Subject to Stochastic Seismic Loads,” Journal of Earthquake Engineering, 10 (5): 615–640.
Marano GC, Greco R, Quaranta G, Fiore A, Avakian A and Cascella D (2013), “Parametric Identification of Nonlinear Devices for Seismic Protection Using Soft Computing Techniques,” Advanced Materials Research, 639–640 (1): 118–129.
PD Spanos, A Di Matteo, Y Cheng, A Pirrotta and J Li (2016), “Galerkin Scheme-Based Determination of Survival Probability of Oscillators With Fractional Derivative Elements,” Journal of Applied Mechanics, 83: 121003–121003.
Raoufi R and Ghafory-Ashtiany M (2016), “Nonlinear Biaxial Structural Vibration under Bidirectional Random Excitation with Incident Angle θ by Tail-Equivalent Linearization Method,” Journal of Engineering Mechanics, 142 (8): 04016050–1–04016050–16.
Rezaeian S and Der Kirureghian A (2011), “Simulation of Orthogonal Horizontal Ground Motion Components for Specified Earthquake and Site Characteristics,” Earthquake Engineering and Structural Dynamics, 41: 335–353.
Rice OC (1944), “Mathematical Analysis of Random Noise,” Bell System Technical Journal, 2446–2456.
Socha L (2008), “Linearization Methods for Stochastic Dynamic Systems, LectureNotes in Physics,” 730, Springer.
Song J and Der Kiureghian A (2006), “Generalized Bouc-Wen Model for Highly Asymmetric Hysteresis,” Journal of Engineering Mechanics, 132 (6): 610–618.
Song JK and Gavin HP (2011), “Effect of Hysteretic Smoothness on Inelastic Response Spectra with Constant-Ductility,” Earthquake Engineering and Structural Dynamics, 40: 771–788.
Spanos PD and Kougioumtzoglou IA (2014), “Galerkin Scheme Based Determination of First-Passage Probability of Nonlinear System Response,” Structure and Infrastructure Engineering, 10: 1285–1294.
Spanos PD and Kougioumtzoglou IA (2012), “Harmonic Wavelets Based Statistical Linearization for Response Evolutionary Power Spectrum Determination,” Probabilistic Engineering Mechanics, 27: 57–68.
Vanmarcke EH (1975), “On the Distribution of the First-Passage Time for Normal Stationary Random Processes,” Journal of Applied Mechanics (ASME) 42: 215–220.
Wen YK (1980), “Equivalent Linearization for Hysteretic Systems under Random Excitation,” Journal of Applied Mechanics (ASME), 47 (3): 150–154.
Wen YK (1976), “Method for Random Vibration of Hysteretic Systems,” Journal of Engineering Mechanics, 102: 249–263.
Wong CW, Ni YQ and Ko JM (1994), “Steady-State Oscillation of Hysteretic Differential Model I: Response Analysis,” Journal of Engineering Mechanics, 120: 2271–2298.
Zhang Y and Der Kiureghian A (1997), “Finite Element Reliability Methods for Inelastic Structures,” Report No.UCB/SEMM-97/05, Department of Civil and Environmental Engineering, University of California, Berkeley, CA.
Zhang Y and Der Kiureghian A (1994), “Two Improved Algorithms for Reliability Analysis,” Proceedings of the 6th IFIPWG 7.5 conference on optimization of structural systems, 297–304.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Raoufi, R., Ghafory-Ashtiany, M. Random vibration of nonlinear structures with stiffness and strength deterioration by modified tail equivalent linearization method. Earthq. Eng. Eng. Vib. 18, 597–610 (2019). https://doi.org/10.1007/s11803-019-0524-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11803-019-0524-7