A case study on the practicability of using linear analysis results in a Bayesian inference model to predict nonlinear responses in performance-based design methods

Document Type : Research


1 Ph.D. Candidate, Faculty of Civil Engineering, K.N. Toosi University of Technology, Tehran, Iran.

2 Associate Professor, Faculty of Civil Engineering, K. N. Toosi University of Technology, Tehran, Iran.


Compared to traditional methods based on mean response evaluation of seismic parameters with significant confidence margin, the growing use of the new generation of performance-based design methods, which are based on loss and financial assessment, necessitates an increase in accuracy and reliability in probabilistic evaluation of structural response for all values of seismic parameters. Even with the same limited number of common nonlinear analyses, utilizing the Bayesian approach, which allows the use of diverse and even inaccurate data to form beliefs, is a powerful method to predict and enhance seismic response results. In this paper, the practicability of using linear analysis data in a Bayesian inference model to predict nonlinear responses is evaluated. A 20-story reinforced concrete special moment resisting frame is being considered, and a Bayesian model for prediction of the maximum story drift and the peak floor acceleration has been investigated. The Bayesian model was developed on linear results and finally updated with a limited number of nonlinear results. The predictability power of predictors, Bayesian model comparison among different likelihood functions, and common diagnostics tools in numerical solution of the Bayesian model developed on linear results, have all been examined. The results demonstrate a significant improvement in the outcomes, while proving the practicability of developing a stable and reliable model based on linear analysis data.


1. FEMA P58-1, Seismic Performance Assessment of Buildings: Volume1-Methodology, Washington, D.C.,USA, 2018.
2. F. Guimaraes, «Research: anyone can do it,» PediaPress, Mainz, 2011.
3. S. Geman et D. Geman, «Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,» Journal of Applied Statistics, vol. 20, p. 25-62, 1993. [DOI:10.1080/02664769300000058]
4. A. Agrawal et K. Gopal, Biomonitoring of water and waste water, Springer Science & Business Media, 2013. [DOI:10.1007/978-81-322-0864-8]
5. S. E. Fienberg et others, «When did Bayesian inference become" Bayesian"?,» Bayesian analysis, vol. 1, p. 1-40, 2006. [DOI:10.1214/06-BA101]
6. I. Yildirim, Bayesian inference: Gibbs sampling., Technical Note, University of Rochester, 2012.
7. A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari et D. B. Rubin, Bayesian data analysis, CRC press, 2013. [DOI:10.1201/b16018]
8. R. McElreath, Statistical rethinking: A Bayesian course with examples in R and Stan, Chapman and Hall/CRC, 2018. [DOI:10.1201/9781315372495]
9. Esmaili, O., Grant Ludwig, L., & Zareian, F. (2016). Improved performanceā€based seismic assessment of buildings by utilizing Bayesian statistics. Earthquake Engineering & Structural Dynamics, 45(4), 581-597. [DOI:10.1002/eqe.2672]
10. S. Kwag, J. Oh, J.-M. Lee et J.-S. Ryu, «Bayesian-based seismic margin assessment approach: Application to research reactor,» Earthquakes Struct., vol. 12, p. 653-663, 2017.
11. K. Erazo, B. Moaveni et S. Nagarajaiah, «Bayesian seismic strong-motion response and damage estimation with application to a full-scale seven story shear wall structure,» Engineering Structures, vol. 186, p. 146-160, 2019. [DOI:10.1016/j.engstruct.2019.02.017]
12. H. Gholami, B. Asgarian et S. Asil Gharebaghi, «Practical Approach for Reliability-Based Inspection Planning of Jacket Platforms Using Bayesian Networks,» ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, vol. 6, p. 04020029, 2020. [DOI:10.1061/AJRUA6.0001071]
13. Sh. Taheri, Prediction of engineering demand parameters using bayesian inference approach, Phd thesis, Department of Civil Engineering, K. N. Toosi University of Technology., Tehran, 2021.
14. A. Gelman et B. Carpenter, «Bayesian analysis of tests with unknown specificity and sensitivity,» Journal of the Royal Statistical Society: Series C (Applied Statistics), vol. 69, n° %15, pp. 1269--1283, 2020. [DOI:10.1111/rssc.12435]
15. American Society of Civil Engineers. (2017, June). Minimum design loads and associated criteria for buildings and other structures. American Society of Civil Engineers.
16. R. E. Kass et A. E. Raftery, «Bayes factors,» Journal of the American Statistical Association, vol. 90, p. 773-795, 1995. [DOI:10.1080/01621459.1995.10476572]
17. B. p. Carlin et s. Chib, «Bayesian model choice via Markov chain Monte Carlo methods,» Journal of the Royal Statistical Society: Series B (Methodological), p. 473-484, 1995. [DOI:10.1111/j.2517-6161.1995.tb02042.x]
18. C. B. Haselton, A. B. Liel, G. G. Deierlein, B. S. Dean et J. H. Chou, «Seismic collapse safety of reinforced concrete buildings. I: Assessment of ductile moment frames,» Journal of Structural Engineering, vol. 137, p. 481-491, 6 2011. [DOI:10.1061/(ASCE)ST.1943-541X.0000318]
19. F. McKenna, M. H. Scott et G. L. Fenves, «Nonlinear finite-element analysis software architecture using object composition,» Journal of Computing in Civil Engineering, vol. 24, p. 95-107, 6 2010. [DOI:10.1061/(ASCE)CP.1943-5487.0000002]
20. T. D. Ancheta, R. B. Darragh, J. P. Stewart, E. Seyhan, W. J. Silva, B. S.-J. Chiou, K. E. Wooddell, R. W. Graves, A. R. Kottke, D. M. Boore et others, «NGA-West2 database,» Earthquake Spectra, vol. 30, p. 989-1005, 2014. [DOI:10.1193/070913EQS197M]
21. J. Moehle, Y. Bozorgnia, N. Jayaram, P. Jones, M. Rahnama, N. Shome, Z. Tuna, J. Wallace, T. Yang et F. Zareian, Case Studies of the Seismic Performance ofTall Buildings Designed by Alternative MeansTask 12 Report for the Tall Buildings Initiative, Pacific Earthquake Engineering Research Center, 2011.
22. S. Karamizadeh, S. M. Abdullah, A. A. Manaf, M. Zamani et A. Hooman, «An overview of principal component analysis,» Journal of Signal and Information Processing, vol. 4, p. 173, 2013. [DOI:10.4236/jsip.2013.43B031]
23. H. Abdi et L. J. Williams, «Principal component analysis,» Wiley interdisciplinary reviews: computational statistics, vol. 2, p. 433-459, 2010. [DOI:10.1002/wics.101]
24. K. Y. Hogarty, C. V. Hines, J. D. Kromrey, J. M. Ferron et K. R. Mumford, «The quality of factor solutions in exploratory factor analysis: The influence of sample size, communality, and overdetermination,» Educational and psychological measurement, vol. 65, p. 202-226, 2005. [DOI:10.1177/0013164404267287]
25. R. B. Cattell, «Fixing the number of factors: The most practicable psychometric procedures,» chez The Scientific Use of Factor Analysis in Behavioral and Life Sciences, Springer, 1978, p. 72-91. [DOI:10.1007/978-1-4684-2262-7_5]
26. K. K. Vasan et B. Surendiran, «Dimensionality reduction using principal component analysis for network intrusion detection,» Perspectives in Science, vol. 8, p. 510-512, 2016. [DOI:10.1016/j.pisc.2016.05.010]
27. B. A. Bradley, «A Generalized Conditional Intensity Measure Approach and Holistic Ground motion Selection,» Earthquake Engineering & Structural Dynamics, vol. 39, p. 1321-1342, 2010. [DOI:10.1002/eqe.995]
28. G. W. Housner, «Spectrum intensities of strong-motion earthquakes,» Bulletin of the Seismological Society of America, vol. 53, p. 403-417, 6 1952. [DOI:10.1785/BSSA0530020403]
29. J. L. Von Thun, «Earthquake Ground Motions for Design and Analysis of Dams,» Earthquake Engineering & Soil Dynamics II-Recent Advances in Ground-Motion Evaluation, 1988.
30. E. Bojórquez, R. Chávez, A. Reyes-Salazar, S. E. Ruiz et J. Bojórquez, «A new ground motion intensity measure IB,» Soil Dynamics and Earthquake Engineering, vol. 99, p. 97-107, 2017. [DOI:10.1016/j.soildyn.2017.05.011]
31. A. Arias, «Measure of Earthquake Intensity.,» 1970.
32. N. Su, X. Lu, Y. Zhou et T. Y. Yang, «Estimating the peak structural response of high-rise structures using spectral value-based intensity measures,» The Structural Design of Tall and Special Buildings, vol. 26, p. e1356, 2017. [DOI:10.1002/tal.1356]
33. N. Luco et C. A. Cornell, «Structure-specific scalar intensity measures for near-source and ordinary earthquake ground motions,» Earthquake Spectra, vol. 23, p. 357-392, 2007. [DOI:10.1193/1.2723158]
34. A. Azarbakht, M. Mousavi, M. Nourizadeh et M. Shahri, «Dependence of correlations between spectral accelerations at multiple periods on magnitude and distance.,» Earthquake Engineering and Structural Dynamics, vol. 43, n° %18, pp. 1193-1204, 2014. [DOI:10.1002/eqe.2393]
35. I. M. Chakravarty, J. D. Roy et R. G. Laha, «Handbook of methods of applied statistics,» 1967.
36. I. W. Burr, «Cumulative frequency functions,» The Annals of mathematical statistics, vol. 13, p. 215-232, 1942. [DOI:10.1214/aoms/1177731607]
37. C. Dagum, «A new model of personal income distribution: specification and estimation,» chez Modeling income distributions and Lorenz curves, Springer, 2008, p. 3-25. [DOI:10.1007/978-0-387-72796-7_1]
38. K. Pearson, «XI. Mathematical contributions to the theory of evolution.-X. Supplement to a memoir on skew variation,» Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, vol. 197, p. 443-459, 1901. [DOI:10.1098/rsta.1901.0023]
39. Cowles, M. K., & Carlin, B. P. (1996). Markov chain Monte Carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association, 91(434), 883-904. [DOI:10.1080/01621459.1996.10476956]
40. Flegal, J. M., Haran, M., & Jones, G. L. (2008). Markov chain Monte Carlo: Can we trust the third significant figure?. Statistical Science, 250-260. [DOI:10.1214/08-STS257]
41. R. Team, "R: A language and environment for statistical computing.," R Foundation for Statistical Computing, Vienna, Austria, 2020.