Unified Hermite Polynomial Model and Its Application in Estimating Non-Gaussian Processes
Publication: Journal of Engineering Mechanics
Volume 145, Issue 3
Abstract
Non-Gaussian processes beset many aspects of structural engineering analysis. To estimate non-Gaussian processes, various third-order Hermite polynomial models have been proposed and widely applied. Different forms of expressions have been proposed for hardening and softening processes in existing Hermite polynomial models, which makes them inconvenient to implement. Furthermore, these models are either too simple to ensure accurate results or too complicated to implement conveniently. Thus, a unified third-order Hermite polynomial model that achieves a good balance between accuracy and convenience for both hardening and softening processes is proposed in this study. Explicit expressions for translations of the marginal distributions between the non-Gaussian and Gaussian processes using the proposed Hermite polynomial model are deduced, and the applicable ranges are provided. The accuracy of the proposed model is demonstrated by comparing the coefficients and estimated moments with those obtained from the moment-matching method. Furthermore, the application of the proposed model in evaluating first passage probability, analyzing fatigue damage, and estimating peak factors of non-Gaussian wind pressure coefficient histories is demonstrated with numerical and practical examples.
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Acknowledgments
This study was partially supported by the National Natural Science Foundation of China (Grant Nos. 51820105014, 51738001, U1434204, and 51421005), China Scholarship Council (Grant No. 201706370095), and the Fundamental Research Funds for the Central Universities of Central South University (Grant No. 1053320180507). All of the sources of support are gratefully acknowledged.
References
Almalki, S. J., and S. Nadarajah. 2014. “Modifications of the Weibull distribution: A review.” Reliab. Eng. Syst. Saf. 124: 32–55. https://doi.org/10.1016/j.ress.2013.11.010.
Blaise, N., T. Canor, and V. Denoël. 2016. “Reconstruction of the envelope of non-Gaussian structural responses with principal static wind loads.” J. Wind Eng. Ind. Aerodyn. 149: 59–76. https://doi.org/10.1016/j.jweia.2015.12.001.
Chen, X. Z. 2014. “Extreme value distribution and peak factor of crosswind response of flexible structures with nonlinear aeroelastic effect.” J. Struct. Eng. 140 (12): 04014091. https://doi.org/10.1061/(ASCE)ST.1943-541X.0001017.
Choi, M., and B. Sweetman. 2010. “The Hermite moment model for highly skewed response with application to tension leg platforms.” J. Offshore Mech. Arctic Eng. 132 (2): 021602. https://doi.org/10.1115/1.4000398.
Crandall, S. H. 1958. Random vibration. Cambridge, UK: MIT Press.
Davenport, A. G. 1964. “Note on the distribution of the largest value of a random function with application to gust loading.” Proc. Inst. Civ. Eng. 28 (2): 187–196. https://doi.org/10.1680/iicep.1964.10112.
Ding, J., and X. Z. Chen. 2016. “Moment-based translation model for hardening non-Gaussian response processes.” J. Eng. Mech. 142 (2): 06015006. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000986.
Ditlevsen, O. 1986. “Duration of Gaussian process visit to critical set.” Prob. Eng. Mech. 1 (2): 82–93. https://doi.org/10.1016/0266-8920(86)90030-5.
Fisher, R. A., and E. A. Cornish. 1960. “The percentile points of distributions having known cumulants.” Technometrics 2 (2): 209–225. https://doi.org/10.1080/00401706.1960.10489895.
Grigoriu, M. 1984. “Crossings of non-Gaussian translation processes.” J. Eng. Mech. 110 (4): 610–620. https://doi.org/10.1061/(ASCE)0733-9399(1984)110:4(610).
Grigoriu, M. 1995. Applied non-Gaussian processes: Examples, theory, simulation, linear random vibration, and MATLAB solution. Englewood Cliffs, NJ: Prentice Hall.
Gurley, K. R., M. A. Tognarelli, and A. Kareem. 1997. “Analysis and simulation tools for wind engineering.” Probab. Eng. Mech. 12 (1): 9–31. https://doi.org/10.1016/S0266-8920(96)00010-0.
He, J., and Y. G. Zhao. 2007. “First passage times of stationary non-Gaussian structural responses.” Comput. Struct. 85 (7): 431–436. https://doi.org/10.1016/j.compstruc.2006.09.009.
Hohenbichler, M., and R. Rackwitz. 1981. “Non-normal dependent vectors in structural safety.” J. Eng. Mech. Div. 107 (6): 1227–1238.
Huang, G. Q., X. W. Ji, H. T. Zheng, Y. Luo, X. Y. Peng, and Q. S. Yang. 2018. “Uncertainty of peak value of non-Gaussian wind load effect: Analytical approach.” J. Eng. Mech. 144 (2): 04017172. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001402.
Huang, G. Q., Y. Luo, Q. S. Yang, and Y. J. Tian. 2017. “A semi-analytical formula for estimating peak wind load effects based on Hermite polynomial model.” Eng. Struct. 152: 856–864. https://doi.org/10.1016/j.engstruct.2017.09.062.
Huang, M. F., W. J. Lou, C. M. Chan, N. Lin, and X. T. Pan. 2013. “Peak distributions and peak factors of wind-induced pressure processes on tall buildings.” J. Eng. Mech. 139 (12): 1744–1756. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000616.
Johnson, N. L., and S. Kotz. 1970. Continuous univariate distributions—1, New York: Wiley.
Kleiber, C., and S. Kotz. 2003. Statistical size distributions in economics and actuarial sciences. Hoboken, NJ: Wiley.
Kwon, D. K., and A. Kareem. 2011. “Peak factors for non-Gaussian load effects revisited.” J. Struct. Eng. 137 (12): 1611–1619. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000412.
Langley, R. S. 1988. “A first passage approximation for normal stationary random processes.” J. Sound Vib. 122 (2): 261–275. https://doi.org/10.1016/S0022-460X(88)80353-5.
Low, Y. M. 2013. “A new distribution for fitting four moments and its applications to reliability analysis.” Struct. Saf. 42 (42): 12–25. https://doi.org/10.1016/j.strusafe.2013.01.007.
Moarefzadeh, M., and R. Melchers. 2006. “Nonlinear wave theory in reliability analysis of offshore structures.” Probab. Eng. Mech. 21 (2): 99–111. https://doi.org/10.1016/j.probengmech.2005.04.002.
Pearson, K. 1895. “Contributions to the mathematical theory of evolution. Part II: Skew variation in homogeneous material.” Philos. Trans. R. Soc. Lond. (A) 186: 343–414. https://doi.org/10.1098/rsta.1895.0010.
Puig, B., and J. Akian. 2004. “Non-Gaussian simulation using Hermite polynomials expansion and maximum entropy principle.” Probab. Eng. Mech. 19 (4): 293–305. https://doi.org/10.1016/j.probengmech.2003.09.002.
Ramberg, J. S., and B. W. Schmeiser. 1974. “An approximate method for generating asymmetric random variables.” Commun. ACM 17 (2): 78–82. https://doi.org/10.1145/360827.360840.
Saha, S. K., K. Sepahvand, V. A. Matsagar, A. K. Jain, and S. Marburg. 2013. “Stochastic analysis of base-isolated liquid storage tanks with uncertain isolator parameters under random excitation.” Eng. Struct. 57: 465–474. https://doi.org/10.1016/j.engstruct.2013.09.037.
SNAME (Society of Naval Architects and Marine Engineers). 2008. Recommended practice for site specific assessment of mobile jack-up units. Jersey City, NJ: SNAME.
Stuart, A., and K. Ord. 2010. Kendall’s advanced theory of statistics: Distribution theory. 6th ed. New York: Wiley.
Winterstein, S. R. 1987. Moment-based Hermite models of random vibration. Lyngby, Denmark: Dept of Structural Engineering, Technical Univ. of Denmark.
Winterstein, S. R. 1988. “Nonlinear vibration models for extremes and fatigue.” J. Eng. Mech. 114 (10): 1772–1790. https://doi.org/10.1061/(ASCE)0733-9399(1988)114:10(1772).
Winterstein, S. R., and T. Kashef. 2000. “Moment-based load and response models with wind engineering applications.” J. Solar Energy Eng. 122 (3): 122–128. https://doi.org/10.1115/1.1288028.
Winterstein, S. R., and C. A. MacKenzie. 2013. “Extremes of nonlinear vibration: Comparing models based on moments, L-moments, and maximum entropy.” J. Offshore Mech. Arct. Eng. 135 (2): 021602. https://doi.org/10.1115/1.4007050.
Yang, L., K. R. Gurley, and D. O. Prevatt. 2013. “Probabilistic modeling of wind pressure on lowrise buildings.” J. Wind Eng. Ind. Aerodyn. 114: 18–26. https://doi.org/10.1016/j.jweia.2012.12.014.
Yang, Q. S., and Y. J. Tian. 2015. “A model of probability density function of non-Gaussian wind pressure with multiple samples.” J. Wind Eng. Ind. Aerodyn. 140: 67–78. https://doi.org/10.1016/j.jweia.2014.11.005.
Zhao, Y. G., and Z. H. Lu. 2007. “Fourth-moment standardization for structural reliability assessment.” J. Struct. Eng. 133 (7): 916–924. https://doi.org/10.1061/(ASCE)0733-9445(2007)133:7(916).
Zhao, Y. G., X. Y. Zhang, and Z. H. Lu. 2018a. “A flexible distribution and its application in reliability engineering.” Reliab. Eng. Syst. Saf. 176: 1–12. https://doi.org/10.1016/j.ress.2018.03.026.
Zhao, Y. G., X. Y. Zhang, and Z. H. Lu. 2018b. “Complete monotonic expression of the fourth-moment normal transformation for structural reliability.” Comput. Struct. 196: 186–199. https://doi.org/10.1016/j.compstruc.2017.11.006.
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©2019 American Society of Civil Engineers.
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Received: May 16, 2018
Accepted: Sep 4, 2018
Published online: Jan 2, 2019
Published in print: Mar 1, 2019
Discussion open until: Jun 2, 2019
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