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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.

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References

Figures

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Fig. 1

Comparison of the spectral radius for μn=0.75

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Fig. 2

Comparison of the numerical damping ratio for μn=0.75

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Fig. 3

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

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Fig. 4

Comparison of the relative period error for μn=0.75

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Fig. 5

Two stories shear building of Example 1

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Fig. 6

Displacement responses for bottom story of two stories shear building

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Fig. 7

Displacement responses for top story of two stories shear building

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Fig. 8

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

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Fig. 9

Displacement responses of 50DOF system

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Fig. 10

Comparison of displacement errors of 50DOF system

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Fig. 11

Displacement responses of 100DOF system

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Fig. 12

Comparison of displacement errors of 100DOF system

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Fig. 13

Displacement responses of 200DOF system

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Fig. 14

Comparison of displacement errors of 200DOF system

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