New Formulation for Dynamic Analysis of Nonlinear Time-History of Vibrations of Structures under Earthquake Loading

Authors

1 Bachelor of Civil Engineering, Department of Civil Engineering, University of Bonab, Bonab, Iran

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

Abstract

A fast and efficient numerical scheme is presented for time-history analysis of single-degree-of-freedom (SDOF) structural systems undergoing seismic excitation (Chopra, 2003). The new method is called Newton-Cotes-4P-θ Method. It uses the most known 4-point Newton-Cotes quadrature in its body to solve the vibration equation. Nonlinear analysis is covered as well as linear analysis. Any arbitrary external loadings of type force or seismic signals are welcome. The significant advantages of the new formulation are its great simplicity, running speed, and appropriate precision level compared with its counterparts such as Duhamel integral and Newmark-β methods. The accuracy level of the Newton-Cotes-4P-θ is close to the semi-analytical method of Duhamel integration and its speed is similar to the Newmark-β algorithm. Notably, against the nonlinear Newmark-β method, the new method does not require a standalone procedure to handle nonlinear analysis; instead, it simply triggers iteration of the same computation used in its first processing round. Moreover, the Newmark-β method loses its performance dealing with stiff and near-conservative () systems; however, the Newton-Cotes-4P-θ method does not loos its accuracy and keeps its well-performed analysis in this case. Numerical results reveal the superiority of the Newton-Cotes- 4P-θ method against its counterparts such as the Duhamel integral, Newmark-β, and Wilson-θ methods (Babaei et al., 2021; Babaei et al., 2022; Babaei et al., 2023).

Keywords

Main Subjects

Al-Subari L, Hanafi M, Ekinci A, “Effect of geosynthetic reinforcement on the bearing capacity of strip footing on sandy soil”, SN Applied Sciences, 2020, 2 (9), 1484. https://doi.org/10.1007/s42452-020-03261-5

Babaei M, Jalilkhani M, Ghasemi SH, Mollaei S, “New methods for dynamic analysis of structural systems under earthquake loads”, Journal of Rehabilitation in Civil Engineering, 2022, 10 (3), 81-99. https://doi.org/10.22075/jrce.2021.23323.1506

Babaei M, Jalilkhani M, Mollaei S, “A numerical method for estimating the dynamic response of structures”, Journal of Civil and Environmental Engineering, 2021, 29 (1), 1-19.

      https://doi.org/10.22034/jcee.2021.41770.1963

Babaei M, Hanafi MR, “A novel method for nonlinear time-history analysis of structural systems: improved newton-cotes-hermite-5p method”, Iranian Journal of Science and Technology, Transactions of Civil Engineering, 2022. https://doi.org/10.1007/s40996-024-01345-5

Babaei M, Alidoost M R & Hanafi M R, "A Novel Numerical Method for Nonlinear Time History Analysis of MDOF Structures: Newton-Cotes-Hermite-4Point", Journal of Structural and Construction Engineering, 2023, 1 (1), 1-20. https://doi.org/10.22065/jsce.2023.400538.3134

Bathe K-J, “Finite element procedures Prentice-Hall”, New Jersey, 1996, 1037, 1.

Bathe K, Finite Element Procedures, Watertown, MA: KJ Bathe, Beijing, China: Higher Education Press, 2016.

Chang SY, “Studies of newmark method for solving nonlinear systems:(ii) verification and guideline”, Journal of the Chinese institute of engineers, 2004, 27 (5), 663-675.

      http://doi.org/10.1080/02533839.2004.9670914

Chopra AK, “Dynamics of structures: theory and applications to earthquake engineering”, Prentice Hall, Upper Saddle River, NJ, 2012, 134-143.

Chopra AK, Goel RK, Chintanapakdee C, “Statistics of single-degree-of-freedom estimate of displacement for pushover analysis of buildings”, Journal of Structural Engineering, 2003, 129 (4), 459.

      https://doi.org/10.1061/(ASCE)0733-9445(2003)129:4(459)

Clough R, Penzien J, “Dynamics of structures, 3rd Ed”, McGraw-Hill, US, 1975, 186-190.

Ekinci A, Hanafi M, Aydin E, “Strength, stiffness, and microstructure of wood-ash stabilized marine clay”, Minerals, 2020, 10 (9), 796.

      https://doi.org/10.3390/min10090796

Ekinci A, Hanafi M, Ferreira PMV, “Influence of initial void ratio on critical state behaviour of poorly graded fine sands”, Indian Geotechnical Journal, 2020, 50 (5), 689-699.

      https://doi.org/10.1007/s40098-020-00416-4

Fahmi A, Zavaragh SR, Hanafi MR, Rahimpour H, Zinatloo-Ajabshir S, Asghari A, “Facile preparation, characterization, and investigation of mechanical strength of Starchy NaCl-binder as a lightweight construction material”, Scientific Reports, 2022, 13 (1), 19042. https://doi.org/10.1038/s41598-023-46536-8

Ebeling RM, Green RA, French SE, “Technical report ITL-97-7: accuracy of response of single-degree-of-freedom systems to ground motion”, Prepared by Army Engineer Waterways Experiment Station Vicksburg Ms Information …, Washington, D.C, 1997. https://www.researchgate.net/publication/272682432_Accuracy_of_Response_of_Single-Degree-of-Freedom_Systems_to_Ground_Motion

Evtushenko YG, Stoer J, “Numerical optimization techniques”, Springer, US, 1985, 436-437.

Fombrun CJ, “Structural dynamics within and between organizations”, Administrative Science Quarterly, 1986, 31 (3), 403-421.

      https://doi.org/10.2307/2392830

Ge X, Gong J, Zhao C, Azim I, Yang X, Li C, “Structural dynamic responses of building structures with non-viscous dampers under Kanai-Tajimi spectrum excitation”, Journal of Sound and Vibration, 2022, 517, 116556.

      https://doi.org/10.1016/j.jsv.2021.116556

Hanafi M, Abki A, Ekinci A, Baldovino JDJA, “Mechanical properties of alluvium clay treated with cement and carbon fiber: relationships among strength, stiffness, and durability”, International Journal of Pavement Engineering, 2022, 24 (1), 2094928. https://doi.org/10.1080/10298436.2022.2094928.

Hanafi M, Aydin E, Ekinci A, “Engineering properties of basalt fiber-reinforced bottom ash cement paste composites”, Materials, 2020, 13 (8), 1952. https://doi.org/10.3390/ma13081952

Hanafi M, Ekinci A, Aydin E, “Triple-Binder-Stabilized marine deposit clay for better sustainability", Sustainability, 2020, 12 (11), 4633. https://doi.org/10.3390/su12114633

Hanafi M, Ekinci A, Aydin E, “Engineering and microstructural properties of alluvium clay stabilized with portland cement and coal bottom ash for sustainable future”, KSCE Journal of Civil Engineering, 2022, 26 (12), 5049-5066. https://doi.org/10.1007/s12205-022-2388-z

Izadifard RA, Mollaei S, Omran MEN, “Preparing pressure-impulse diagrams for reinforced concrete columns with constant axial load using single degree of freedom approach”, International Journal of Advancements o ISSN: 0976-4860 Research Article Research Article e i in Technology, 2016, 7 (173), 2-6. https://doi.org/10.4172/0976-4860.1000173

Kazakov KS, “Dynamic response of a single degree of freedom (SDOF) system in some special load cases, based on the Duhamel integral”, International Conference on Engineering Optimization (EngOpt 2008), Rio de Janeiro, Brazil, 1-5 June 2008.

Kurt N, Cevik M, “Polynomial solution of the single degree of freedom system by Taylor matrix method”, Mechanics Research Communications, 2008, 35 (8), 530-536.

      https://doi.org/10.1016/j.mechrescom.2008.05.001

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

Mollaei S, Fahmi A, Jahani D, Babaei Golsefidi Z, Babaei R, Hanafi MR, “A predictive model for the strength of a novel geopolymer construction material produced by autoclaved aerated concrete waste”, International Journal of Sustainable Construction Engineering and Technology, 2022, 14 (1), 148-167.

https://doi.org/10.30880/ijscet.2023.14.01.015.

Malakiyeh MM, Shojaee S, Bathe K-J, “The bathe time integration method revisited for prescribing desired numerical dissipation", Computers and Structures, 2019, 212, 289-298.

      https://doi.org/10.1016/j.compstruc.2018.10.008

Meirovitch L, “Elements of vibration analysis((Book))”, New York, McGraw-Hill Book Co., 1986, 574, 1986.

Nguyen QH, Hanafi M, Merkl J-P, d'Espinose de Lacaillerie J-B, “Evolution of the microstructure of unconsolidated geopolymers by thermoporometry”, Journal of the American Ceramic Society, 2021, 104 (3), 1581-1591. https://doi.org/10.1111/jace.17543

Newmark N, “A method of computation for structural dynamics”, Journal of the Engineering Mechanics Division, 1959, 85 (3), 67-94.

Noh G, Bathe K-J, “The Bathe time integration method with controllable spectral radius: The ρ∞-Bathe method”, Computers and Structures, 2019, 212, 299-310. https://doi.org/10.1016/j.compstruc.2018.11.001

Paz M, Leigh W, “Structural dynamics: theory and computation”, Springer, US, 1997, 220-264.

Paz M, Leigh W, “Damped single degree-of-freedom system”, Structural dynamics: theory and computation, 2004, 1, 31-48.

Veletsos A, Newmark N, Chelapati C, “Deformation spectra for elastic and elastoplastic systems subjected to ground shock and earthquake motions”, Proceedings of the 3rd World Conference On Earthquake Engineering, Wellington, New Zealand, 10 January 1965.

Wen W, Wei K, Lei H, Duan S, Fang D, “A novel sub-step composite implicit time integration scheme for structural dynamics”, Computers and Structures, 2017, 182, 176-186.

      https://doi.org/10.1016/j.compstruc.2016.11.018

Wu J-S, “Analytical and numerical methods for vibration analyses”, John Wiley and Sons, US, 2013, 350-356.

Zhang J, Liu D, Liu Y, “Degenerated shell element with composite implicit time integration scheme for geometric nonlinear analysis”, International Journal for Numerical Methods in Engineering, 2016, 105 (7), 483-513.

https://doi.org/10.1002/nme.4975

Zhang J, Liu Y, Liu D, “Accuracy of a composite implicit time integration scheme for structural dynamics”, International Journal for Numerical Methods in Engineering, 2017, 109 (3), 368-406.

      https://doi.org/10.1002/nme.5291

References

Al-Subari L, Hanafi M, Ekinci A, “Effect of geosynthetic reinforcement on the bearing capacity of strip footing on sandy soil”, SN Applied Sciences, 2020, 2 (9), 1484. https://doi.org/10.1007/s42452-020-03261-5
Babaei M, Jalilkhani M, Ghasemi SH, Mollaei S, “New methods for dynamic analysis of structural systems under earthquake loads”, Journal of Rehabilitation in Civil Engineering, 2022, 10 (3), 81-99. https://doi.org/10.22075/jrce.2021.23323.1506
Babaei M, Jalilkhani M, Mollaei S, “A numerical method for estimating the dynamic response of structures”, Journal of Civil and Environmental Engineering, 2021, 29 (1), 1-19.
      https://doi.org/10.22034/jcee.2021.41770.1963
Babaei M, Hanafi MR, “A novel method for nonlinear time-history analysis of structural systems: improved newton-cotes-hermite-5p method”, Iranian Journal of Science and Technology, Transactions of Civil Engineering, 2022. https://doi.org/10.1007/s40996-024-01345-5
Babaei M, Alidoost M R & Hanafi M R, "A Novel Numerical Method for Nonlinear Time History Analysis of MDOF Structures: Newton-Cotes-Hermite-4Point", Journal of Structural and Construction Engineering, 2023, 1 (1), 1-20. https://doi.org/10.22065/jsce.2023.400538.3134
Bathe K-J, “Finite element procedures Prentice-Hall”, New Jersey, 1996, 1037, 1.
Bathe K, Finite Element Procedures, Watertown, MA: KJ Bathe, Beijing, China: Higher Education Press, 2016.
Chang SY, “Studies of newmark method for solving nonlinear systems:(ii) verification and guideline”, Journal of the Chinese institute of engineers, 2004, 27 (5), 663-675.
      http://doi.org/10.1080/02533839.2004.9670914
Chopra AK, “Dynamics of structures: theory and applications to earthquake engineering”, Prentice Hall, Upper Saddle River, NJ, 2012, 134-143.
Chopra AK, Goel RK, Chintanapakdee C, “Statistics of single-degree-of-freedom estimate of displacement for pushover analysis of buildings”, Journal of Structural Engineering, 2003, 129 (4), 459.
      https://doi.org/10.1061/(ASCE)0733-9445(2003)129:4(459)
Clough R, Penzien J, “Dynamics of structures, 3rd Ed”, McGraw-Hill, US, 1975, 186-190.
Ekinci A, Hanafi M, Aydin E, “Strength, stiffness, and microstructure of wood-ash stabilized marine clay”, Minerals, 2020, 10 (9), 796.
      https://doi.org/10.3390/min10090796
Ekinci A, Hanafi M, Ferreira PMV, “Influence of initial void ratio on critical state behaviour of poorly graded fine sands”, Indian Geotechnical Journal, 2020, 50 (5), 689-699.
      https://doi.org/10.1007/s40098-020-00416-4
Fahmi A, Zavaragh SR, Hanafi MR, Rahimpour H, Zinatloo-Ajabshir S, Asghari A, “Facile preparation, characterization, and investigation of mechanical strength of Starchy NaCl-binder as a lightweight construction material”, Scientific Reports, 2022, 13 (1), 19042. https://doi.org/10.1038/s41598-023-46536-8
Ebeling RM, Green RA, French SE, “Technical report ITL-97-7: accuracy of response of single-degree-of-freedom systems to ground motion”, Prepared by Army Engineer Waterways Experiment Station Vicksburg Ms Information …, Washington, D.C, 1997. https://www.researchgate.net/publication/272682432_Accuracy_of_Response_of_Single-Degree-of-Freedom_Systems_to_Ground_Motion
Evtushenko YG, Stoer J, “Numerical optimization techniques”, Springer, US, 1985, 436-437.
Fombrun CJ, “Structural dynamics within and between organizations”, Administrative Science Quarterly, 1986, 31 (3), 403-421.
      https://doi.org/10.2307/2392830
Ge X, Gong J, Zhao C, Azim I, Yang X, Li C, “Structural dynamic responses of building structures with non-viscous dampers under Kanai-Tajimi spectrum excitation”, Journal of Sound and Vibration, 2022, 517, 116556.
      https://doi.org/10.1016/j.jsv.2021.116556
Hanafi M, Abki A, Ekinci A, Baldovino JDJA, “Mechanical properties of alluvium clay treated with cement and carbon fiber: relationships among strength, stiffness, and durability”, International Journal of Pavement Engineering, 2022, 24 (1), 2094928. https://doi.org/10.1080/10298436.2022.2094928.
Hanafi M, Aydin E, Ekinci A, “Engineering properties of basalt fiber-reinforced bottom ash cement paste composites”, Materials, 2020, 13 (8), 1952. https://doi.org/10.3390/ma13081952
Hanafi M, Ekinci A, Aydin E, “Triple-Binder-Stabilized marine deposit clay for better sustainability", Sustainability, 2020, 12 (11), 4633. https://doi.org/10.3390/su12114633
Hanafi M, Ekinci A, Aydin E, “Engineering and microstructural properties of alluvium clay stabilized with portland cement and coal bottom ash for sustainable future”, KSCE Journal of Civil Engineering, 2022, 26 (12), 5049-5066. https://doi.org/10.1007/s12205-022-2388-z
Izadifard RA, Mollaei S, Omran MEN, “Preparing pressure-impulse diagrams for reinforced concrete columns with constant axial load using single degree of freedom approach”, International Journal of Advancements o ISSN: 0976-4860 Research Article Research Article e i in Technology, 2016, 7 (173), 2-6. https://doi.org/10.4172/0976-4860.1000173
Kazakov KS, “Dynamic response of a single degree of freedom (SDOF) system in some special load cases, based on the Duhamel integral”, International Conference on Engineering Optimization (EngOpt 2008), Rio de Janeiro, Brazil, 1-5 June 2008.
Kurt N, Cevik M, “Polynomial solution of the single degree of freedom system by Taylor matrix method”, Mechanics Research Communications, 2008, 35 (8), 530-536.
      https://doi.org/10.1016/j.mechrescom.2008.05.001
Li P, Wu B, “An iteration approach to nonlinear oscillations of conservative single-degree-of-freedom systems”, Acta Mechanica, 2004, 170 (1), 69-75. https://doi.org/10.1007/s00707-004-0112-3
Mollaei S, Fahmi A, Jahani D, Babaei Golsefidi Z, Babaei R, Hanafi MR, “A predictive model for the strength of a novel geopolymer construction material produced by autoclaved aerated concrete waste”, International Journal of Sustainable Construction Engineering and Technology, 2022, 14 (1), 148-167.
https://doi.org/10.30880/ijscet.2023.14.01.015.
Malakiyeh MM, Shojaee S, Bathe K-J, “The bathe time integration method revisited for prescribing desired numerical dissipation", Computers and Structures, 2019, 212, 289-298.
      https://doi.org/10.1016/j.compstruc.2018.10.008
Meirovitch L, “Elements of vibration analysis((Book))”, New York, McGraw-Hill Book Co., 1986, 574, 1986.
Nguyen QH, Hanafi M, Merkl J-P, d'Espinose de Lacaillerie J-B, “Evolution of the microstructure of unconsolidated geopolymers by thermoporometry”, Journal of the American Ceramic Society, 2021, 104 (3), 1581-1591. https://doi.org/10.1111/jace.17543
Newmark N, “A method of computation for structural dynamics”, Journal of the Engineering Mechanics Division, 1959, 85 (3), 67-94.
Noh G, Bathe K-J, “The Bathe time integration method with controllable spectral radius: The ρ∞-Bathe method”, Computers and Structures, 2019, 212, 299-310. https://doi.org/10.1016/j.compstruc.2018.11.001
Paz M, Leigh W, “Structural dynamics: theory and computation”, Springer, US, 1997, 220-264.
Paz M, Leigh W, “Damped single degree-of-freedom system”, Structural dynamics: theory and computation, 2004, 1, 31-48.
Veletsos A, Newmark N, Chelapati C, “Deformation spectra for elastic and elastoplastic systems subjected to ground shock and earthquake motions”, Proceedings of the 3rd World Conference On Earthquake Engineering, Wellington, New Zealand, 10 January 1965.
Wen W, Wei K, Lei H, Duan S, Fang D, “A novel sub-step composite implicit time integration scheme for structural dynamics”, Computers and Structures, 2017, 182, 176-186.
      https://doi.org/10.1016/j.compstruc.2016.11.018
Wu J-S, “Analytical and numerical methods for vibration analyses”, John Wiley and Sons, US, 2013, 350-356.
Zhang J, Liu D, Liu Y, “Degenerated shell element with composite implicit time integration scheme for geometric nonlinear analysis”, International Journal for Numerical Methods in Engineering, 2016, 105 (7), 483-513.
https://doi.org/10.1002/nme.4975
Zhang J, Liu Y, Liu D, “Accuracy of a composite implicit time integration scheme for structural dynamics”, International Journal for Numerical Methods in Engineering, 2017, 109 (3), 368-406.
      https://doi.org/10.1002/nme.5291
Journal of Civil and Environmental Engineering
Volume 54.1, Issue 114 - Serial Number 114
Spring 2024
June 2024
Pages 22-35

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