Semi-Implicit Integration Algorithm for Stochastic Analysis of Multi-Degree-of-Freedom Structures
Publication: Journal of Engineering Mechanics
Volume 128, Issue 6
Abstract
This paper presents a semi-implicit integration algorithm for random vibration problems that is appropriate for analyzing large structures, nonlinear hysteretic systems, and structural control problems. This semi-implicit approach results in a recursive expression for the mean and covariance response. A state-space representation of the equations of motion is adopted for deriving the algorithm. The solution of the state-space equations is first obtained, after which the expected value of the resulting equations is taken so as to obtain the first two moments. A stability condition for the method is also derived. Three numerical examples, a linear oscillator, a Duffing oscillator, and a multi-degree-of-freedom system with hysteretic supplemental damping devices, are provided to illustrate the effectiveness of the proposed method. Results compare well with Monte Carlo simulation, indicating that the semi-implicit integration algorithm is accurate and stable.
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References
Chang, T.-P., Mochio, T., and Samaras, E.(1986). “Seismic response analysis of nonlinear structures.” Probab. Eng. Mech., 1(3), 157–166.
Eliopoulos, D. F., and Wen, Y. K. (1991). “Method of seismic reliability evaluation for moment resisting steel frames.” Civil Engineering Studies, Structural Research Series, No. 562, Univ. of Illinois, Urbana, IL.
Emam, H. H., Pradlwarter, H. J., and Schuëller, G. I. (1999). “On the computational implementation of EQL in FE analysis.” Stochastic structural dynamics, Spencer and Johnson, eds, Balkema, Rotterdam, The Netherlands, 85–91.
Emam, H. H., Pradlwarter, H. J., and Schuëller, G. I.(2000). “A computational procedure for the implementation of equivalent linearization in finite element analysis.” Earthquake Eng. Struct. Dyn., 29, 1–17.
Lutes, L. D. and Sarkani, S. (1997). Stochastic analysis of structural and mechanical vibrations, Prentice-Hall, Englewood Cliffs, N.J.
Miao, B.(1993). “Direct integration variance prediction of random response of nonlinear systems.” Comput. Struct., 46(6), 979–983.
Ohtori, Y., and Spencer, B. F. (2000). “Semi-implicit integration algorithm for solution of nonlinear stochastic vibration problems.” Proc., 8th ASCE Specialty Conf. on Probabilistic Mechanics and Structural Reliability (CD-ROM; PMC2000-323), p. 6.
Pradlwarter, H. J., and Schuëller, G. I. (1987). “Accuracy and Limitations of the Method of Equivalent Linearization for Hysteretic Multi-Story Structures.” Proc., Nonlinear Stochastic Dynamics Engineering Systems IUTAM Symposium, 3–21.
Pradlwarter, H. J., and Li, W.(1991). “On the computation of the stochastic response of highly nonlinear large MDOF-system modeled by finite elements.” Probab. Eng. Mech., 6(2), 109–116.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannerry, B. P. (1992). Numerical recipes in C, 2nd Ed., Cambridge University Press, Cambridge.
Roberts, J. B., and Spanos, P. D. (1990). Random vibration and statistical linearization, Wiley, New York.
Schuëller, G. I., Pandey, M. D., and Pradlwarter, H. J.(1994). “Equivalent linearization (EQL) in engineering practice for aseismic design.” Probab. Eng. Mech., 9, 95–102.
Schuëller, G. I., Pradlwarter, H. J., Vasta, M., and Harnpornchai, N. (1998). “Benchmark study on non-linear stochastic structural dynamics.” Structural safety and reliability, N. Shiraishi, M. Shinozuka, and Y. K. Wen, eds., Balkema, Rotterdam, The Netherlands, 355–362.
Shinozuka, M., and Sato, Y.(1967). “Simulation of nonstationary random process.” J. Eng. Mech., 93(EM1), 11–40.
Simulescu, I., Mochio, T., and Shinozuka, M.(1989). “Equivalent linearization method in nonlinear FEM.” J. Eng. Mech., 115(3), 475–492.
Soong, T. T., and Grigoriu, M. (1992). Random vibration of mechanical and structural systems, Prentice–Hall, Englewood Cliffs, N.J.
To, C. W. S.(1986). “The stochastic central difference method in structural dynamics.” Comput. Struct., 23(6), 813–818.
To, C. W. S.(1988a). “Random response of duffing oscillator by the stochastic central difference method.” J. Sound Vib., 124(3), 427–433.
To, C. W. S.(1988b). “Recursive expressions for random response of nonlinear systems.” Comput. Struct., 29(3), 451–457.
To, C. W. S.(1988c). “Direct integration operators and their stability for random response of multi-degree of freedom systems.” Comput. Struct., 30(4), 865–874.
To, C. W. S.(1992). “A stochastic version of the Newmark family of algorithms for discretized dynamic systems.” Comput. Struct., 44(3), 667–673.
Wen, Y. K.(1980). “Equivalent linearization for hysteretic systems under random excitation.” J. Appl. Mech., 47(1), 150–154.
Wen, Y. K., and Eliopoulos, D.(1994). “Method for nonstationary random vibration of inelastic structures.” Probab. Eng. Mech., 9, 115–123.
Zhang, L., Zu, J. W., and Zheng, Z.(1999). “The stochastic Newmark algorithm for random analysis of multi-degree-of-freedom nonlinear systems.” Comput. Struct., 70, 557–568.
Zhang, S. W., and Zhao, H. H.(1992). “Effects of time step in stochastic central difference method.” J. Sound Vib., 159(1), 182–188.
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Published In
Journal of Engineering Mechanics
Volume 128 • Issue 6 • June 2002
Pages: 635 - 643
Copyright
Copyright © 2002 American Society of Civil Engineers.
History
Received: Feb 23, 2001
Accepted: Nov 20, 2001
Published online: May 15, 2002
Published in print: Jun 2002
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