# New Methods for Dynamic Analysis of Structural Systems under Earthquake Loads

Document Type : Regular Paper

Authors

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

2 Engineering Faculty of Khoy, Urmia University of Technology, Urmia, Iran

3 Department of Civil Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran

Abstract

Two numerical methods are proposed for dynamic analysis of single-degree-of-freedom systems. Basics of dynamics and elementary tools from numerical calculus are employed to formulate the methods. The energy conservation principles triggered the basic idea of the first method, so-called energy-based method (EBM). It is devised for dynamic analysis of linear damped system whose damping ratio is greater than 1%. The second method uses function approximation theory and integration scheme, and called simplified integration method (SIM). Several numerical examples are investigated through SIM. A detailed comparison is made between the proposed methods and the conventional ones. The results show that the proposed methods can estimate the dynamic response of linear damped systems with high accuracy. In the first example, the peak displacement is obtained 6.8747 cm and 6.8290 cm which closely approximate the highly exact response of Duhamel integral. Results show that Newmark-β method is the fastest one whose run-time is 0.0019 sec. EBM and SIM computational times are 0.0722 sec and 0.0021sec, respectively. SIM gives more accurate estimate and convergence rate than Newmark-β method. The difference of peak displacement obtained from two methods is almost less than 1%. Thus, SIM reliably estimates the dynamic response of systems with less computational cost.

Keywords

Main Subjects

#### References

[1] Anderson, J. C., Naeim, F. (2012). “Basic structural dynamics.” John Wiley & Sons.
[2] Babaei, M. (2013). “A general approach to approximate solutions of nonlinear differential equations using particle swarm optimization.” Applied Soft Computing, Vol. 13, Issue 7, pp. 3354–3365.
[3] Bathe, KJ. (1996). “Finite element procedures.” Prentice Hall, Englewood Cliffs, NJ.
[4] Bazzi, G., Anderheggen, E. (1982). “The r-family of algorithms for time-step integration with improved numerical dissipation.” Earthquake Engineering & Structural Dynamics, Vol. 10, pp. 537–550.
[5] Benaroya, H., Nagurka, M., Han, S. (2017). “Mechanical vibration: analysis, uncertainties, and control.” CRC Press.
[6] Chang, S. Y. (2004). “Studies of Newmark method for solving nonlinear systems: (I) basic analysis.” Journal of the Chinese Institute of Engineers, Vol. 27, Issue 5, pp. 651–662.
[7] Chopra, A. K., Goel, R. K., Chintanapakdee, C. (2003). “Statistics of single-degree-of-freedom estimate of displacement for pushover analysis of buildings.” Journal of structural engineering, Vol. 129, Issue 4, pp. 459–469.
[8] Chopra, A. K. (2017). Dynamics of structures. theory and applications to. Earthquake Engineering.
[9] Clough, RW., Penzien, J. (1995). “Dynamics of Structures.” Computers & Structure Inc., 3rd Ed., CA.
[10] Craig, R. R., Kurdila, A. J. (2006). “Fundamentals of structural dynamics.” John Wiley & Sons.
[11] Ebeling, RM., Green, RA., French, SE. (1997). “Accuracy of response of single-degree-of-freedom systems to ground motion.” US Army Corps of Engineers, Technical report ITL-97-7, Washington, DC.
[12] Gatti, P. L. (2014). “Applied Structural and Mechanical Vibrations: Theory and Methods.” CRC Press.
[13] Géradin, M., Rixen, D. J. (2014). “Mechanical vibrations: theory and application to structural dynamics.” John Wiley & Sons.
[14] Hall, KC., Thomas, JP., Clark, WS. (2002). “Computation of unsteady nonlinear flows in cascades using a harmonic balance technique.” American Institute of Aeronautics and Astronautics Journal, Vol. 40, pp. 879–886.
[15] He, L. (2008). “Harmonic solution of unsteady flow around blades with separation.” American Institute of Aeronautics and Astronautics Journal, Vol. 46, pp. 1299–1307.
[16] Hilber, HM., Hughes, TJR., Taylor, RL. (1977). “Improved numerical dissipation for time integration algorithms in structural dynamics.” Earthquake Engineering & Structural Dynamics, Vol. 5, pp. 283–292.
[17] Hoff, C., Pahl, PJ. (1988). “Development of an implicit method with numerical dissipation from a generalized single-step algorithm for structural dynamics.” Computer Methods in Applied Mechanics and Engineering, Vol. 67, pp. 367–385.
[18] Hoff, C., Pahl, PJ. (1988). “Practical performance of the θ1-method and comparison with other dissipative algorithms in structural dynamics.” Computer Methods in Applied Mechanics and Engineering, Vol. 67, pp. 87–110.
[19] JalilKhani, M., Babaei, M., Ghasemi, S. Evaluation of the Seismic Response of Single-Story RC Frames under Biaxial Earthquake Excitations. Journal of Rehabilitation in Civil Engineering, 2020; 8(3): 98-108.
[20] Kazakov, K. S. (2008). “Dynamic Response of a Single Degree of Freedom (SDOF) System in some Special Load Cases, based on the Duhamel Integral.” In International Conference on Engineering Optimization, pp. 01–05.
[21] Kurt, N., Çevik, M., (2008). “Polynomial solution of the single degree of freedom system by Taylor matrix method.” Mechanics Research Communications, Vol. 35, Issue 8, pp. 530–536.
[22] Li, P. S., Wu, B. S. (2004). “An iteration approach to nonlinear oscillations of conservative single-degree-of-freedom systems.” Acta Mechanica, Vol. 170, Issue 1-2, pp. 69-75.
[23] Mohammadzadeh, B., Noh, H. C. (2014). “Investigation into central-difference and Newmark’s beta methods in measuring dynamic responses.” In Advanced Materials Research, Vol. 831, pp. 95–99.
[24] Newmark, N. M. (1959). “A method of computation for structural dynamics.” Journal of the engineering mechanics division, Vol. 85, Issue 3, pp. 67–94.
[25] Paz, M. (2012). “International handbook of earthquake engineering: codes, programs, and examples.” Springer Science & Business Media.
[26] Paz, M. and Leigh, W. (2004). “Structural Dynamics: Theory and Computation.” 5th Ed., Springer Science & Business Media, N.Y.
[27] Rahmati, MT., He, L., Wells, RG. (2010). “Interface treatment for harmonic solution in multi-row aeromechanic analysis.” Proceedings of the ASME Turbo Expo: Power for Land, Sea and Air, GT2010-23376, Glasgow, UK, June.
[28] Rao, S. S. (2017). “Mechanical vibrations in SI units.” Pearson Higher Ed.
[29] Tedesco, J. W., McDougal, W. G., Ross, C. A. (1999). “Structural dynamics. Theory and Applications.”
[30] Thomson, W. (2018). “Theory of vibration with applications.” CrC Press.
[31] Vamvatsikos, D., Cornell, CA. (2005). “Direct estimation of seismic demand and capacity of multi-degree-of-freedom systems through incremental dynamic analysis of single degree of freedom approximation.” ASCE Journal of Structural Engineering, Vol. 131, pp. 589–599.
[32] Wilson EL. (1968). “A computer program for the dynamic stress analysis of underground structures.” SEL. Technical Report 68-1, University of California: Berkeley.
[33] Wood, WL., Bossak, M., Zienkiewicz, OC. (1980). “An alpha modification of Newmark’s method.” International Journal for Numerical Methods in Engineering, Vol. 15, pp. 1562–1566.
[34] Wu, B. S., Lim, C. W. (2004). “Large amplitude non-linear oscillations of a general conservative system.” International Journal of Non-Linear Mechanics, Vol. 39(5), pp. 859–870.

### History

• Receive Date: 02 May 2021
• Revise Date: 25 October 2021
• Accept Date: 31 October 2021