Abstract
A conventional approach for nonlinear random vibration analysis is using equivalent linearization method. Tail-Equivalent Linearization Method (TELM) is one the best proposed methods in the recent decade for determination of equivalent linear model. In TELM, the design point is obtained using first-order reliability method. In the current research, a non-gradient-based method is applied for determination of the design point. One of the main advantages of this method is non-application of limit-state function gradient for calculation of the design point. In the implemented method, n arbitrary points in n-dimension standard normal space are selected and limit-state function in these points is estimated. Then, these points converge to the design point using a convergent algorithm. Since many random variables are produced in TELM for discretization of seismic excitation, iterative algorithms for determination of design point numerical instability would be encountered. By modification of step length for each iteration and application of a magnification coefficient for each step, an appropriate non-gradient method is proposed for analysis of the problems with many random variables. The efficiency of this method was investigated by solving numerical examples. Moreover, the convergence of this method for finding the design point was presented. It was also indicated that results obtained using this method are in good agreement with results obtained by gradient methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bong, T. and Son, Y. (2018). “A stochastic dual response surface method for reliability analysis considering the spatial variability.” KSCE Journal of Civil Engineering, Vol. 22, No. 9, pp. 3524–3533, DOI: https://doi.org/10.1007/s12205-018-0803-2.
Bouc, R. (1967). “Forced vibration of mechanical systems with hysteresis.” Proceedings of the Fourth Conference on Non-linear Oscillation, Prague, Czechoslovakia, p. 315.
Broccardo, M. and Der Kiureghian, A. (2015). “Multicomponent nonlinear stochastic dynamic analysis by tail-equivalent linearization.” Journal of Engineering Mechanics, Vol. 142, No. 3, pp. 1–15, DOI: https://doi.org/10.1061/(ASCE)EM.1943-7889.0001026.
Caughey, T. K. (1963). “Equivalent linearization techniques.” The Journal of the Acoustical Society of America, Vol. 35, No. 11, pp. 1706–1711.
Der Kiureghian, A. (2000). “The geometry of random vibrations and solutions by FORM and SORM.” Probabilistic Engineering Mechanics, Vol. 15, No. 1, pp. 81–90, DOI: https://doi.org/10.1016/S0266-8920(99)00011-9.
Ditlevsen, O. and Madsen, H. O. (1996). Structural reliability methods, John Wiley & Sons, New York.
Faravelli, L. (1989). “Response-surface approach for reliability analysis.” Journal of Engineering Mechanics, Vol. 115, No. 12, pp. 2763–2781, DOI: https://doi.org/10.1061/(ASCE)0733-9399(1989)115:12(2763).
Fujimura, K. (2006). Tail equivalent linearization method for nonlinear random vibration, PhD Thesis, University of California, Berkeley, CA, USA.
Fujimura, K. and Der Kiureghian, A. (2007). “Tail-equivalent linearization method for nonlinear random vibration.” Probabilistic Engineering Mechanics, Vol. 22, No. 1, pp. 63–76, DOI: https://doi.org/10.1016/j.probengmech.2006.08.001.
Gong, J. X., Yi, P., and Zhao, N. (2014). “Non-gradient-based algorithm for structural reliability analysis.” Journal of Engineering Mechanics, Vol. 140, No. 6, pp. 1–16, DOI: https://doi.org/10.1061/(ASCE)EM.1943-7889.0000722.
Haukaas, T. and Der Kiureghian, A. (2006). “Strategies for finding the design point in non-linear finite element reliability analysis.” Probabilistic Engineering Mechanics, Vol. 21, No. 2, pp. 133–147, DOI: https://doi.org/10.1016/j.probengmech.2005.07.005.
Haukaas, T. and Der Kiureghian, A. (2007). “Methods and object-oriented software for FE reliability and sensitivity analysis with application to a bridge structure.” Journal of Computing in Civil Engineering, Vol. 21, No. 3, pp. 151–163, DOI: https://doi.org/10.1061/(ASCE)0887-3801(2007)21:3(151).
Huh, J. (2000). “Reliability analysis of nonlinear structural systems using response surface method.” KSCE Journal of Civil Engineering, Vol. 4, No. 3, pp. 135–143, DOI: https://doi.org/10.1007/BF02830867.
Jahani, E., Shayanfar, M. A., and Barkhordari, M. A. (2013). “A new adaptive importance sampling Monte Carlo methods for structural reliability.” KSCE Journal of Civil Engineering, Vol. 17, No. 1, pp. 210–215, DOI: https://doi.org/10.1007/s12205-013-1779-6.
Keshtegar, B. and Meng, Z. (2017). “A hybrid relaxed first-order reliability method for efficient structural reliability analysis.” Structural Safety, Vol. 66, No. 1, pp. 84–93, DOI: https://doi.org/10.1016/j.strusafe.2017.02.005.
Kim, T. and Song, J. (2018). “Generalized Reliability Importance Measure (GRIM) using Gaussian mixture.” Reliability Engineering & System Safety, Vol. 173, No. 1, pp. 105–115, DOI: https://doi.org/10.1016/j.ress.2018.01.005.
Kleiber, M., Antunez, H., Hien T. D., and Kowalczyk, P. (1997). Parameter sensitivity in nonlinear mechanics: Theory and finite element computations, John Wiley & Sons, Hoboken, NJ, USA.
Koduru, S. D. and Haukaas, T. (2010). “Feasibility of FORM in finite element reliability analysis.” Structural Safety, Vol. 32, No. 2, pp. 145–153, DOI: https://doi.org/10.1016/j.strusafe.2009.10.001.
Li, J. and Chen, J. B. (2009). Stochastic dynamics of structures, John Wiley & Sons, Singapore.
Lin, Y. K. and Cai, G. Q. (2004). Probabilistic structural dynamic: Advanced theory and applications, McGraw-Hill, NY, USA.
Liu, P. L. and Der Kiureghian, A. (1991). “Optimization algorithms for structural reliability.” Structural Safety, Vol. 9, No. 3, pp. 161–177, DOI: https://doi.org/10.1016/0167-4730(91)90041-7.
Lutes, L. D. and Sarkani, S. (2004). Random vibrations: Analysis of structural and mechanical systems, Elsevier Butterworth-Heinemann, Burlington.
Polyanin, A. D. and Manzhirov, A. V. (2006). Handbook of mathematics for engineers and scientists, CRC Press, Boca Raton, FL, USA.
Roberts, J. B. and Spanos, P. D. (1990). Random vibration and statistical linearization, John Wiley & Sons, NY, USA.
Shayanfar, M. A., Barkhordari, M. A., and Roudak, M. A. (2017). “An efficient reliability algorithm for locating design point using the combination of importance sampling concepts and response surface method.” Communications in Nonlinear Science and Numerical Simulation, Vol. 47, No. 1, pp. 223–237, DOI: https://doi.org/10.1016/j.cnsns.2016.11.021.
Soong, T. T. and Grigoriu, M. (1993). Random vibration of mechanical and structural systems, Prentice Hall, Englewood Cliffs, NJ, USA.
Thoft-Christensen, P. and Baker, M. J. (1982). Structural reliability theory and its applications, Springer-Verlag, Berlin, Germany.
To, C. W. S. (2012). Nonlinear random vibrations: Analytical techniques and applications, Taylor & Francis Group, Boca Raton.
Wang, Z. and Der Kiureghian, A. (2016). “Tail-equivalent linearization of inelastic multi-support structures subjected to spatially varying stochastic ground motion.” Journal of Engineering Mechanics, Vol. 142, No. 8, pp. 1–14, DOI: https://doi.org/10.1061/(ASCE)EM.1943-7889.0001106.
Wang, Z. and Song, J. (2017). “Equivalent linearization method using Gaussian mixture (GM-ELM) for nonlinear random vibration analysis.” Structural Safety, Vol. 64, No. 1, pp. 9–19, DOI: https://doi.org/10.1016/j.strusafe.2016.08.005.
Wen, Y. K. (1976). “Method for random vibration of hysteretic systems.” Journal of the Engineering Mechanics Division, Vol. 102, No. 2, pp. 249–263.
Zhang, Y. and Der Kiureghian, A. (1993). “Dynamic response sensitivity of inelastic structures.” Computer Methods in Applied Mechanics and Engineering, Vol. 108, Nos. 1–2, pp. 23–36, DOI: https://doi.org/10.1016/0045-7825(93)90151-M.
Zhang, Y. and Der Kiureghian, A. (1995). “Two improved algorithms for reliability analysis.” Reliability and Optimization of Structural Systems, pp. 297–304, DOI: https://doi.org/10.1007/978-0-387-34866-7_32.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pourzeynali, S., Abbaszadeh, H. The Non-gradient-based Reliability Method in Equivalent Linear Systems for Nonlinear Random Vibration. KSCE J Civ Eng 23, 632–640 (2019). https://doi.org/10.1007/s12205-018-1598-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12205-018-1598-x