A Novel Numerical Method for Nonlinear Time History Analysis of  MDOF Structures: Newton-Cotes-Hermite-4Point

بابائی قلعه جوق, مهدی; Alidoost, Mohammad Reza; Hanafi, Mohammad Reza

doi:10.22065/jsce.2023.400538.3134

A Novel Numerical Method for Nonlinear Time History Analysis of MDOF Structures: Newton-Cotes-Hermite-4Point

Document Type : Original Article

Authors

مهدی بابائی قلعه جوق ¹

Mohammad Reza Alidoost ²

Mohammad Reza Hanafi ³

¹ Assistant professor, Department of Civil Engineering, University of Bonab, Bonab, Iran

² M.S. of Structural Engineering, Department of Civil Engineering, University of Bonab, Bonab, Iran

³ M.S. Student of Construction Management and Engineering, Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran

10.22065/jsce.2023.400538.3134

Abstract

This paper presents an efficient numerical analysis method called the Newton-Cotes-Hermite-Four-Point (NCH-4P) method for the time history analysis of structural systems with one and multiple degrees of freedom under earthquake effect. The method combines the numerical integration formula of the Newton-Cotes four-point method with the Hermite interpolation formulas to form a new algorithm for solving the vibration equation. The method can handle both linear and nonlinear systems, as well as different types of loading, such as external forces and earthquake excitation. The method is developed for the first time for the analysis of linear and nonlinear damped and undamped structural systems with one and multiple degrees of freedom. The method shows remarkable performance superiority in terms of accuracy, speed, convergence and simplicity compared to the pseudo-analytical Newmark-beta method and the semi-analytical Duhamel integral method. The method modifies the differential equation of motion to have a suitable form for numerical integration. Unlike the nonlinear Newmark-beta method, the method does not require an independent process such as Newton’s iteration to account for nonlinear effects; instead, a series of simple repeated calculations leads to the convergence of the response in nonlinear behavior. The numerical results demonstrate the efficiency of the method in estimating the response of systems under the famous EL-Centro record.

Keywords

Time-History analysis

Seismic response

Non-Linear analysis

Multi-Degree-of-Freedom (MDOF)

Newton-Cotes quadrature

Hermite interpolation

Newton-Cotes-Hermite-4Point method

Subjects

Structural and Earthquake Engineering

[1] Brezinski, C. and L. Wuytack. (2012). Numerical Analysis: Historical Developments in the 20th Century. 64-238 Ed. New York, USA: Elsevier Science. 512.

[2] Stuart, A. M. (1994). Numerical analysis of dynamical systems. Acta numerica, 3(1), 467-572. https://doi.org/10.1017/S0962492900002488.

[3] Hochbruck, M. and A. Ostermann. (2010). Exponential integrators. Acta Numerica, 19(1), 209-286. https://doi.org/10.1017/S0962492910000048.

[4] Katsikadelis, J. T. (2020). Dynamic Analysis of Structures. 6th Ed. New York, USA: Elsevier Science. 512.

[5] Paz, M. and W. Leigh. (2004). Damped Single Degree-of-Freedom System. 5th Ed. Netherlands: Springer. 790.

[6] Chopra, A. K. (2007). Dynamics of Structures. 2nd Ed. New Delhi: Pearson Education. 844.

[7] Veletsos, A., N. Newmark and C. Chelapati. (1965). Deformation spectra for elastic and elastoplastic systems subjected to ground shock and earthquake motions. In: Proceedings of the 3rd world conference on earthquake engineering. New Zealand: American Society of Civil Engineers, 663-682. https://doi.org/10.13140/RG.2.1.4008.8167.

[8] Izadifard, R. A., S. Mollaei and M. E. N. Omran. (2016). Preparing pressure-impulse diagrams for reinforced concrete columns with constant axial load using single degree of freedom approach. International Journal of Advancements in Technology, 7(4), 9. http://dx.doi.org/10.4172/0976-4860.1000173.

[9] Newmark, N. M. (1959). A method of computation for structural dynamics. Journal of the engineering mechanics division, 85(3), 67-94. https://doi.org/10.1061/taceat.0008448.

[10] Ebeling R.M, Green R.A and French S.E. (1997). Accuracy of response of single-degree-of-freedom systems to ground motion. [online] Washington, D.C: U. S. A. C. o. Engineers, pp. 6-26. Available at: https://www.researchgate.net/publication/272682432_Accuracy_of_Response_of_Single-Degree-of-Freedom_Systems_to_Ground_Motion [1997.12.01].

[11] Li Q.S. (2002). Forced vibrations of single-degree-of-freedom systems with nonperiodically time-varying parameters. Journal of engineering mechanics, 128(12), 1267-1275. https://doi.org/10.1061/(ASCE)0733-9399(2002)128:12(1267).

[12] Chang, S. Y. (2004). Studies of Newmark method for solving nonlinear systems:(I) basic analysis. Journal of the Chinese Institute of Engineers, 27(5), 651-662. http://dx.doi.org/10.1080/02533839.2004.9670913.

[13] Li, P. and B. Wu. (2004). An iteration approach to nonlinear oscillations of conservative single-degree-of-freedom systems. Acta Mechanica, 170(1), 69-75. https://doi.org/10.1007/s00707-004-0112-3.

[14] Vamvatsikos, D. and C. A. Cornell. (2005). Direct estimation of seismic demand and capacity of multidegree-of-freedom systems through incremental dynamic analysis of single degree of freedom approximation. Journal of Structural Engineering, 131(4), 589-599. https://doi.org/10.1061/(ASCE)0733-9445(2005)131:4(589).

[15] Kurt, N. and M. Cevik. (2008). Polynomial solution of the single degree of freedom system by Taylor matrix method. Mechanics Research Communications, 35(8), 530-536. http://dx.doi.org/10.1016/j.mechrescom.2008.05.001.

[16] Bathe, K.-J. (2007). Conserving energy and momentum in nonlinear dynamics: a simple implicit time integration scheme. Computers & structures, 85(7-8), 437-445. https://doi.org/10.1016/j.compstruc.2006.09.004.

[17] Bathe K.J and N. G. (2012). Insight into an implicit time integration scheme for structural dynamics. Computers & Structures, 90(98), 1-6. https://doi.org/10.1016/j.compstruc.2012.01.009.

[18] Katsikadelis, J. T. (2014). A new direct time integration method for the equations of motion in structural dynamics. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 94(9), 757-774. https://doi.org/10.1002/zamm.201200245.

[19] Katsikadelis, J. T. (2021). A new method for numerical integration of higher-order ordinary differential equations without losing the periodic responses. Frontiers in Built Environment, 7(1), 621037. https://doi.org/10.3389/fbuil.2021.621037.

[20] Noh, G. and K.-J. Bathe. (2018). Further insights into an implicit time integration scheme for structural dynamics. Computers & Structures, 202(1), 15-24. https://doi.org/10.1016/j.compstruc.2018.02.007.

[21] Noh, G. and K.-J. Bathe. (2019). The Bathe time integration method with controllable spectral radius: The ρ∞-Bathe method. Computers & Structures, 212(1), 299-310. https://doi.org/10.1016/j.compstruc.2018.11.001.

[22] Zhang, J. (2020). A‐stable two‐step time integration methods with controllable numerical dissipation for structural dynamics. International Journal for Numerical Methods in Engineering, 121(1), 54-92. https://doi.org/10.1002/nme.6188.

[23] Zhang, J. (2021). A‐stable linear two‐step time integration methods with consistent starting and their equivalent single‐step methods in structural dynamics analysis. International Journal for Numerical Methods in Engineering, 122(9), 2312-2359. https://doi.org/10.1002/nme.6623.

[24] Zhang, J., Y. Liu and D. Liu. (2017). Accuracy of a composite implicit time integration scheme for structural dynamics. International Journal for Numerical Methods in Engineering, 109(3), 368-406. http://dx.doi.org/10.1002/nme.5291.

[25] Babaei M, Mollaei S, Moslemi Petrudi A, J. M. and Scurtu IC. (2021). Numerical and analytical study of seismic response of structural systems with new formulation using energy and impact methods. In: E3S Web of Conferences. Hong Kong: EDP Sciences, 040141-040148. https://doi.org/10.1051/e3sconf/202128604014.

[26] Babaei M. (2013). A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization. Applied Soft Computing, 13(7), 3354-3365. https://doi.org/10.1016/j.asoc.2013.02.005.

[27] Babaei M, Jalilkhani M, Ghasemi SH and Mollaei Somayeh. (2022). New Methods for Dynamic Analysis of Structural Systems under Earthquake Loads. Journal of Rehabilitation in Civil Engineering, 10(3), 81-99. https://doi.org/10.22075/jrce.2021.23323.1506.

[28] Hanafi M.R, Babaei M and Narjabadifam P. (2023). New Formulation for dynamic analysis of nonlinear time-history of vibrations of structures under earthquake loading. Journal of Civil and Environmental Engineering, [Online]. 28(2). Available at: https://ceej.tabrizu.ac.ir/article_16408.html [Accessed June 11 2023]. https://doi.org/10.22034/ceej.2023.54564.2209.

[29] Babaei, M., M. Jalilkhani, S. H. Ghasemi and S. Mollaei. (2022). New Methods for Dynamic Analysis of Structural Systems under Earthquake Loads. Journal of Rehabilitation in Civil Engineering, 10(3), 81-99. https://doi.org/10.22075/jrce.2021.23323.1506.

[30] Babaei, M. and J. Farzi. (2023). Derivation of weighting rules for developing a class of A-stable numerical integration scheme: α I-(2+ 3) P method. Journal of Difference Equations and Applications, [Online]. 29(4), 1-30. Available at: https://www.tandfonline.com/doi/citedby/10.1080/10236198.2023.2219785?scroll=top&needAccess=true [Accessed Jun 08 2023]. https://doi.org/10.1080/10236198.2023.2219785.

[31] Chopra, A. K. (2017). Dynamics of Structures: Theory and Applications to Earthquake Engineering. 4th Ed. Upper Saddle River, New jersey, USA: Pearson. 876.

[32] Slepyan, L. (2022). Phase shift in forced oscillations and waves. European Journal of Mechanics-A/Solids, 96(1), 104762. https://doi.org/10.1016/j.euromechsol.2022.104762.

[33] Rao S.S. (2017). Mechanical Vibrations, in SI Units, Global Edition. 6th Ed. London, UK: Pearson Education. 1152.