0
Research Articles

A New Unconditionally Stable Time Integration Method for Analysis of Nonlinear Structural Dynamics

[+] Author and Article Information
Ali Akbar Gholampour

Graduate Student
e-mail: aagholampour@ut.ac.ir

Mehdi Ghassemieh

Associate Professor
e-mail: mghassem@ut. ac.ir

Mahdi Karimi-Rad

Graduate Student
e-mail: hk_civil@yahoo.com
School of Civil Engineering,
University of Tehran,
Tehran, 14174 Iran

The generalized-α method is unconditionally stable when the following conditions are provided: αmαf1/2,β0.25+0.5(αf-αm). The parameters β and γ for the method can be produced as follows (αm, αf, β, and γ are algorithmic parameters): γ=0.5-αm+αf,β=0.25(1-αm+αf)2.

1Corresponding author.

Manuscript received November 28, 2011; final manuscript received September 12, 2012; accepted manuscript posted September 25, 2012; published online January 30, 2013. Assoc. Editor: Marc Geers.

J. Appl. Mech 80(2), 021024 (Jan 30, 2013) (12 pages) Paper No: JAM-11-1451; doi: 10.1115/1.4007682 History: Received November 28, 2011; Revised September 12, 2012; Accepted September 25, 2012

A new time integration scheme is presented for solving the differential equation of motion with nonlinear stiffness. In this new implicit method, it is assumed that the acceleration varies quadratically within each time step. By increasing the order of acceleration, more terms of the Taylor series are used, which are expected to have responses with better accuracy than the classical methods. By considering this assumption and employing two parameters δ and α, a new family of unconditionally stable schemes is obtained. The order of accuracy, numerical dissipation, and numerical dispersion are used to measure the accuracy of the proposed method. Second order accuracy is achieved for all values of δ and α. The proposed method presents less dissipation at the lower modes in comparison with Newmark's average acceleration, Wilson-θ, and generalized-α methods. Moreover, this second order accurate method can control numerical damping in the higher modes. The numerical dispersion of the proposed method is compared with three unconditionally stable methods, namely, Newmark's average acceleration, Wilson-θ, and generalized-α methods. Furthermore, the overshooting effect of the proposed method is compared with these methods. By evaluating the computational time for analysis with similar time step duration, the proposed method is shown to be faster in comparison with the other methods.

FIGURES IN THIS ARTICLE
<>
Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

Chopra, A., 2007, Dynamics of Structures: Theory and Applications to Earthquake Engineering, 3rd ed., Prentice-Hall, Upper Saddle River, NJ.
Paz, M., and Leigh, W., 2003, Structural Dynamics: Theory and Computation, 5th ed., Springer, Netherlands.
Park, K. C., 1977, “Practical Aspects of Numerical Time Integration,” Comput. Struct., 7(3), pp. 343–353. [CrossRef]
Bathe, K. J., and Wilson, E. L., 1973, “Stability and Accuracy Analysis of Direct Time Integration Methods,” Earthquake Eng. Struct. Dyn., 1, pp. 283–291. [CrossRef]
Keierleber, C. W., and Rosson, B. T., 2005, “Higher-Order Implicit Dynamic Time Integration Method,” J. Struct. Eng., 131(8), pp. 1267–1276. [CrossRef]
Chung, J., and Hulbert, G. M., 1993, “A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method,” ASME J. Appl. Mech., 60, pp. 371–375. [CrossRef]
Kontoe, S., Zdravkovic, L., and Potts, D. M., 2008, “An Assessment of Time Integration Schemes for Dynamic Geotechnical Problems,” Comput. Geotech., 35(2), pp. 253–264. [CrossRef]
Hughes, T. J. R., 1987, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ.
Dokainish, M. A., and Subbaraj, K., 1989, “A Survey of Direct Time Integration Methods in Computational Structural Dynamics. I. Explicit Methods,” Comput. Struct., 32(6), pp. 1371–1386. [CrossRef]
Subbaraj, K., and Dokainish, M. A., 1989, “A Survey of Direct Time Integration Methods in Computational Structural Dynamics. II. Implicit Methods,” Comput. Struct., 32(6), pp. 1387–1401. [CrossRef]
Chen, C., and Ricles, J. M., 2008, “Stability Analysis of Direct Integration Algorithms Applied to Nonlinear Structural Dynamics,” ASME J. Eng. Mech., 134(9), pp. 703–711. [CrossRef]
Bathe, K. J., 1996, Finite Element Procedures, Prentice-Hall, Englewood Cliffs, NJ.
Hughes, T. J. R., and Belytschko, T., 1983, “A Precis of Developments in Computational Methods for Transient Analysis,” ASME J. Appl. Mech., 50(4b), pp. 1033–1041. [CrossRef]
Belytschko, T., and Lu, Y., 1993, “Explicit Multi-Time Step Integration for First and Second Order Finite Element Semi-Discretizations,” Comput. Methods Appl. Mech. Eng., 108(3–4), pp. 353–383. [CrossRef]
Hahn, G. D., 1991, “A Modified Euler Method for Dynamic Analysis,” Int. J. Numer. Methods Eng., 32(5), pp. 943–955. [CrossRef]
Rezaiee-Pajand, M., and Alamatian, J., 2008, “Implicit Higher-Order Accuracy Method for Numerical Integration in Dynamic Analysis,” J. Struct. Eng., 134(6), pp. 973–985. [CrossRef]
Razavi, S. H., Abolmaali, A., and Ghassemieh, M., 2007, “A Weighted Residual Parabolic Acceleration Time Integration Method for Problems in Structural Dynamics,” Comput. Methods Appl. Math., 7(3), pp. 227–238. [CrossRef]
Chang, S. Y., 2007, “Improved Explicit Method for Structural Dynamics,” J. Eng. Mech., 133(7), pp. 748–760. [CrossRef]
Hilber, H., and Hughes, T. J. R., 1978, “Collocation, Dissipation and Overshoot for Time Integration Schemes in Structural Dynamics,” Earthquake Eng. Struct. Dyn., 6(1), pp. 99–117. [CrossRef]
Kavetski, D., Binning, P., and Sloan, S. W., 2004, “Truncation Error and Stability Analysis of Iterative and Non-Iterative Thomas-Gladwell Methods for First-Order Non-Linear Differential Equations,” Int. J. Numer. Methods Eng., 60(12), pp. 2031–2043. [CrossRef]
Goudreau, G. L., and Taylor, R. L., 1972, “Evaluation of Numerical Integration Methods in Elastodynamics,” Comput. Methods Appl. Mech. Eng., 2, pp. 69–97. [CrossRef]
Chang, S. Y., 2009, “Accurate Integration of Nonlinear Systems Using Newmark Explicit Method,” J. Mech., 25(3), pp. 289–297. [CrossRef]

Figures

Grahic Jump Location
Fig. 2

Comparison of the numerical damping ratio for μn=0.75

Grahic Jump Location
Fig. 3

Comparison of the numerical damping ratio for μn=0.75 and δ=0.35

Grahic Jump Location
Fig. 1

Comparison of the spectral radius for μn=0.75

Grahic Jump Location
Fig. 4

Comparison of the relative period error for μn=0.75

Grahic Jump Location
Fig. 12

Comparison of displacement errors of 100DOF system

Grahic Jump Location
Fig. 13

Displacement responses of 200DOF system

Grahic Jump Location
Fig. 14

Comparison of displacement errors of 200DOF system

Grahic Jump Location
Fig. 5

Two stories shear building of Example 1

Grahic Jump Location
Fig. 6

Displacement responses for bottom story of two stories shear building

Grahic Jump Location
Fig. 7

Displacement responses for top story of two stories shear building

Grahic Jump Location
Fig. 8

n-DOF spring-mass system for Example 2 [18]

Grahic Jump Location
Fig. 9

Displacement responses of 50DOF system

Grahic Jump Location
Fig. 10

Comparison of displacement errors of 50DOF system

Grahic Jump Location
Fig. 11

Displacement responses of 100DOF system

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In