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Elimination of the Stops Because of Failure of Nonlinear Solutions in Nonlinear Seismic Time History Analysis

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Abstract

Purpose

For analyzing structures’ nonlinear dynamic behaviors, it is broadly accepted to use time integration methods and model the nonlinearities using iterative methods. The iterations may however fail. With attention to the sources of computational errors, in 2015 and 2022, the authors proposed continuation of the analysis, even when the iteration fails, and briefly discussed the accuracy and efficiency. In this paper, the proposed change is discussed further, in application to analysis, according to the seismic code of New Zealand, NZS 1170.5:2004, the unique seismic code with established analysis procedure for nonlinear time history analysis. The purpose is to study how the proposed change affects the response accuracy, and the analysis efficiency and simplicity.

Method

Based on the analysis procedure in the seismic code of New Zealand, a new procedure that considers the proposed change is suggested. Using the new procedure, the effects of the change on the features of the analysis is discussed through theoretical discussions and piece-wisely linear examples. The limitations, conceivable generalizations, and challenges ahead are discussed, and a vision for the future is provided.

Results and Conclusion

As the result, the proposed changes lead to sufficiently accurate responses and analyses simpler and mostly more efficient compared to the ordinary analyses. Besides, not only the improvement in efficiency may be considerable, but also no negative effects on the efficiency are observed. The achievements are important, also because of indirectly encouraging the structural engineers to make more use of nonlinear time history analysis. It is finally recommended to use the new procedure for the analysis of piece-wisely linear systems, when the nonlinearities are modelled using fractional time step**.

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References

  1. Nayfeh AH, Mook DT (2008) Nonlinear oscillations. Wiley, Weinheim

    MATH  Google Scholar 

  2. Soroushian A (2003) New methods to maintain responses’ convergence and control responses’ errors in the analysis of nonlinear dynamic models of structural systems. Ph.D. Thesis, University of Tehran, Tehran (in Persian)

  3. Bathe KJ (1996) Finite element procedures. Prentice-Hall, Upper Saddle River

    MATH  Google Scholar 

  4. Belytschko T, Hughes TJR (1983) Computational methods for transient analysis. Elsevier, Amsterdam

    MATH  Google Scholar 

  5. Belytschko T, Liu WK, Moran B (2000) Non-linear finite elements for continua and structures. Wiley-Intersciences, Chichester

    MATH  Google Scholar 

  6. Cook RD, Malkus DS, Plesha ME, Witt RJ (2002) Concepts and applications of finite element analysis. Wiley, New York

    Google Scholar 

  7. De Borst R, Crisfield MA, Remmers JJ, Verhoosel CV (2012) Nonlinear finite element analysis of solids and structures. Wiley, Chichester

    Book  MATH  Google Scholar 

  8. Soroushian A (2017) Integration step size and its adequate selection in analysis of structural systems against earthquakes. In: Papadrakakis M, Plevris V, Lagaros ND (eds) Computational methods in earthquake engineering, vol 3. Springer, Cham, pp 285–329. https://doi.org/10.1007/978-3-319-47798-5_10

    Chapter  Google Scholar 

  9. Wriggers P (2008) Nonlinear finite element methods. Springer, Berlin

    MATH  Google Scholar 

  10. Clough RW, Penzien J (1993) Dynamics of structures. McGraw-Hill, Singapore

    MATH  Google Scholar 

  11. Wood WL (1990) Practical time step** schemes. Oxford University Press, New York

    MATH  Google Scholar 

  12. Hughes TJR (1987) The finite element method: linear static and dynamic finite element analysis. Prentice-Hall, Upper Saddle River

    MATH  Google Scholar 

  13. Krenk S (2006) State-space time integration with energy control and fourth-order accuracy for linear dynamic systems. Int J Numer Methods Eng 65:595–619. https://doi.org/10.1002/nme.1449

    Article  MathSciNet  MATH  Google Scholar 

  14. Soroushian A (2018) A general rule for the influence of physical dam** on the numerical stability of time integration analysis. J Appl Comput 4(5):467–481. https://doi.org/10.22055/jacm.2018.25161.1235

    Article  Google Scholar 

  15. Wen W, Li H, Liu T, Deng S, Duan S (2022) A novel hybrid sub-step explicit time integration method with cubic B-spline interpolation and momentum corrector technique for linear and nonlinear dynamics. Nonlinear Dyn 110:2685–2714. https://doi.org/10.1007/s11071-022-07740-9

    Article  Google Scholar 

  16. Hughes TJR (1976) Stability, convergence and growth and decay of energy of the average acceleration method in nonlinear structural dynamics. Comput Struct 6:313–324. https://doi.org/10.1016/0045-7949(76)90007-9

    Article  MathSciNet  MATH  Google Scholar 

  17. Krishnan S (2009) FRAME3D V2. 0—a program for the three-dimensional nonlinear time-history analysis of steel structures: user guide. Report No. EERL2009-04, California Institute of Technology, Pasadena

  18. Rashidi S, Saadeghvaziri MA (1997) Seismic modeling of multi-span simply-supported bridges using ADINA. Comput Struct 64:1025–1039. https://doi.org/10.1016/S0045-7949(97)00016-3

    Article  MATH  Google Scholar 

  19. Rezaiee-Pajand M, Hashemian M, Bohluly A (2017) A novel time integration formulation for nonlinear dynamic analysis. Aerosp Sci Technol 69:625–635. https://doi.org/10.1016/j.ast.2017.07.032

    Article  Google Scholar 

  20. Cao S, Li Z, Liu B (2019) Nonlinear time history analysis of a large-scale complex connected structure base on an explicit friction pendulum element. 工程力学 36:128–137. https://doi.org/10.6052/j.issn.1000-4750.2018.05.0259

  21. Soroushian A, Wriggers P, Farjoodi J (2013) Practical integration of semidiscretized nonlinear equations of motion: proper convergence for systems with piecewise linear behavior. J Eng Mech 139:114–145. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000434

    Article  Google Scholar 

  22. Vaiana N, Sessa S, Marmo F, Rosati L (2019) Nonlinear dynamic analysis of hysteretic mechanical systems by combining a novel rate-independent model and an explicit time integration method. Nonlinear Dyn 98:2879–2901. https://doi.org/10.1007/s11071-019-05022-5

    Article  MATH  Google Scholar 

  23. Trcala M, Němec I, Gálová A (2022) On the nonlinear transient analysis of structures. In: Earthquakes-recent advances, new perspectives and applications. IntechOpen

  24. ** schemes for non-linear dynamic equations. Commun Numer Methods Eng 10:393–401. https://doi.org/10.1002/cnm.1640100505

    Article  MATH  Google Scholar 

  25. Yang YS, Wang W, Lin JZ (2017) Direct-iterative hybrid solution in nonlinear dynamic structural analysis. Comput Aided Civ Infrastruct 32:397–411. https://doi.org/10.1111/mice.12259

    Article  Google Scholar 

  26. Allgower EL, Georg K (1990) Numerical continuation methods, an introduction. Springer, Berlin

    Book  MATH  Google Scholar 

  27. Bathe KJ, Cimento AP (1980) Some practical procedures for the solution of nonlinear finite element equations. Comput Methods Appl Mech Eng 22:59–85. https://doi.org/10.1016/0045-7825(80)90051-1

    Article  MATH  Google Scholar 

  28. Crisfield MA, Jelenic G, Mi Y, Zhong HJ, Fan Z (1997) Some aspects of the non-linear finite element method. Finite Elem Anal Des 27:19–40. https://doi.org/10.1016/S0168-874X(97)00004-8

    Article  MATH  Google Scholar 

  29. Geradin M, Idelsohn S, Hogge M (1980) Nonlinear structural dynamics via Newton and quasi-Newton methods. Nucl Eng Des 58:339–348. https://doi.org/10.1016/0029-5493(80)90147-8

    Article  Google Scholar 

  30. Nau JM (1983) Computation of inelastic spectra. J Eng Mech 109:279–288. https://doi.org/10.1061/(ASCE)0733-9399(1983)109:1(279)

    Article  Google Scholar 

  31. Soroushian A, Wriggers P, Farjoodi J (2015) From the notions of nonlinearity tolerances towards a deficiency in commercial Transient Analysis softwares and its solution. In: International conference on computational methods in structural dynamics and earthquake engineering Crete, Greece. National Technical University of Athens, Athens, pp 1899–1907. https://doi.org/10.7712/120115.3509.653

  32. Soroushian A, Wriggers (2022) Test of an idea for improving the efficiency of nonlinear time history analyses when implemented in seismic analysis according to NZS 1170.5:2004. In: Dimitrovová Z, Biswas P, Gonçalves R, Silva T (eds) Recent trends in wave mechanics and vibrations, WMVC 2022, mechanisms and machine science, vol 125. Springer, Cham, pp 107–114. https://doi.org/10.1007/978-3-031-15758-5_10

    Chapter  Google Scholar 

  33. NZS 1170 (2004) Structural design actions—part 5: earthquake actions. Standards New Zealand, Wellington

    Google Scholar 

  34. NZS 1170.5 Supp 1 (2004) Structural design actions—part 5: earthquake actions. Standards New-Zealand, Wellington

    Google Scholar 

  35. Courant R, Friedrichs K, Lewy H (1928) Über die partiellen differenzengleichungen der mathematischen physik. Math Ann 100:32–74. https://doi.org/10.1007/BF01448839

    Article  MathSciNet  MATH  Google Scholar 

  36. Chopra AK (1995) Dynamics of structures: theory and application to earthquake engineering. Prentice-Hall, Upper Saddle River

    MATH  Google Scholar 

  37. Basöz NI, Kiremidjian AS, King SA, Law KH (1999) Statistical analysis of bridge damage data from the 1994 Northridge, CA, earthquake. Earthq Spectra 15:25–54. https://doi.org/10.1193/1.1586027

    Article  Google Scholar 

  38. Jeng V, Tzeng WL (2000) Assessment of seismic pounding hazard for Taipei City. Eng Struct 22:459–471. https://doi.org/10.1016/S0141-0296(98)00123-0

    Article  Google Scholar 

  39. Babitsky VI, Krupenin VL (2001) Vibration of strongly nonlinear discontinuous systems. Springer, Berlin

    Book  MATH  Google Scholar 

  40. Soroushian A, Farjoodi J, Mehrazin H (2006) A new measure for the nonlinear behavior of piecewisely linear structural dynamic models. In: International Congress on Sound and Vibration Vienna, Austria. International Institute of Acoustics and Vibration, Vienna

  41. Wilson EL, Habibullah A (1995) SAP2000, structural analysis program. Computers & Structures, Inc., Berkeley

    Google Scholar 

  42. Kojic M, Bathe KJ (2005) Inelastic analysis of solids and structures, vol 2, no. 1. Springer, Berlin, pp 2–4

  43. Mahin SA, Lin J (1983) Construction of inelastic response spectra for single degree-of-freedom systems. Report No. UCB/EERC-83/17, Earthquake Engineering Research Center, University of California, Berkeley

  44. Soroushian A, Moghadam AS, Sabzei A, Amiri S, Saaed A, Yahyapour A (2023) An engineering comment for simply accelerating seismic response history analysis of mid-rise steel-structure buildings. J Archit Eng Res. https://doi.org/10.54338/27382656-2023.4-001

    Article  Google Scholar 

  45. Soroushian A (2022) Performance of a Time integration acceleration technique applied to seismic analysis of non-classically damped structural dynamics. Iran J Sci Technol Trans Civ Eng 46(2):1281–1300

    Article  Google Scholar 

  46. Newmark NM (1959) A method of computation for structural dynamics. J Eng Mech 85:67–94. https://doi.org/10.1061/JMCEA3.0000098

    Article  Google Scholar 

  47. Frontczak M, Wargocki P (2011) Literature survey on how different factors influence human comfort in indoor environments. Build Environ 46:922–937. https://doi.org/10.1016/j.buildenv.2010.10.021

    Article  Google Scholar 

  48. Chung J, Hulbert GM (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-a method. J Appl Mech 60:371–375. https://doi.org/10.1115/1.2900803

    Article  MathSciNet  MATH  Google Scholar 

  49. Clough RW (1973) Numerical integration of equations of motion. In: Lectures on finite element methods in continuum mechanics. Univ. of Alabama, Tuscaloosa, AL, pp 525–533

  50. Ralston A, Rabinowitz P (1978) First course in numerical analysis. McGraw Hill, New York

    MATH  Google Scholar 

  51. Soroushian A (2010) Pseudo convergence and its implementation in engineering approximate computations. In: International conference from scientific computing to computational engineering Athens, Greece. Laboratory of Fluid Mechanics and Energy, Athens, pp 621–626

  52. Noble B, Daniel JW (1977) Applied linear algebra. Prentice-Hall, Upper Saddle River

    MATH  Google Scholar 

  53. Fajfar P (2018) Analysis in seismic provisions for buildings: past, present and future. In: European conference on earthquake engineering Thessaloniki, Greece. Springer, Cham, pp 1–49. https://doi.org/10.1007/s10518-017-0290-8

  54. Fragiadakis M, Papadrakakis M (2008) Modeling, analysis and reliability of seismically excited structures: computational issues. Int J Comput Methods 5:483–511. https://doi.org/10.1142/S0219876208001674

    Article  MATH  Google Scholar 

  55. Taghinia A, Vasseghi A, Khanmohammadi M, Soroushian A (2022) Development of seismic fragility functions for typical Iranian multi-span RC bridges with deficient cap beam–column joints. Int J Civ Eng 20:305–321. https://doi.org/10.1007/s40999-021-00661-5

    Article  Google Scholar 

  56. Ahmad A, Plevris V, Khan QUZ (2020) Prediction of properties of FRP-confined concrete cylinders based on artificial neural networks. Crystals 10:811. https://doi.org/10.3390/cryst10090811

    Article  Google Scholar 

  57. Lagaros ND, Fragiadakis M, Papadrakakis M, Tsompanakis Y (2006) Structural optimization: a tool for evaluating seismic design procedures. Eng Struct 28:1623–1633. https://doi.org/10.1016/j.engstruct.2006.02.014

    Article  Google Scholar 

  58. Soroushian A (2023) Test of an idea for setting the nonlinearity tolerance in nonlinear response history analyses according to procedures originated in the seismic code of New Zealand NZS 1170.5:2004. In: International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering Athens, Greece. National Technical University of Athens, Athens

  59. Capuano R, Vaiana N, Pellecchia D, Rosati L (2022) A solution algorithm for a modified Bouc-Wen model capable of simulating cyclic softening and pinching phenomena. IFAC-PapersOnLine 55:319–324. https://doi.org/10.1016/j.ifacol.2022.09.115

    Article  Google Scholar 

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Acknowledgments

The authors are grateful for the reviewers’ comments, which have caused significant improvements in the paper. The role of the editorial team in quickly reviewing the paper and editing it nicely is appreciated, as well. The helps of Mr. Hamid Doostie in the random selections in Table 10 is also appreciated. Finally, the first author acknowledges the financial support of the International Institute of Earthquake Engineering and Seismology (IIEES) in relation to the Projects 7420, 7528, and 7711.

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Authors and Affiliations

  1. Structural Engineering Research Center, International Institute of Earthquake Engineering and Seismology, 19537, Tehran, Iran

    Aram Soroushian

  2. Institut für Kontinuumsmechanik, Leibniz Universität Hannover, 30167, Hannover, Germany

    Peter Wriggers

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  1. Aram Soroushian
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Correspondence to Aram Soroushian.

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Soroushian, A., Wriggers, P. Elimination of the Stops Because of Failure of Nonlinear Solutions in Nonlinear Seismic Time History Analysis. J. Vib. Eng. Technol. 11, 2831–2849 (2023). https://doi.org/10.1007/s42417-023-00968-8

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