Mode‐Superposition Methods for Elastoplastic Systems
Publication: Journal of Engineering Mechanics
Volume 115, Issue 10
Abstract
In the framework of modal analysis the dynamic correction method is applied to evaluate the response of elastic perfectly plastic systems having numerous degrees of freedom. According to this method, the nodal structural response is evaluated as the sum of a pseudo‐static response, which is the particular solution of the differential equations of motion in nodal coordinates, and a dynamic response evaluated using a reduced number of natural modes. Once the modal analysis is applied, the elastoplastic response is evaluated by using a stepwise approach in modal space and by solving at each step a linear complementarity problem or equivalent quadratic programming problems. In a numerical example it is shown that by using the dynamic correction method greater accuracy, especially for elastoplastic systems, is obtained with respect to the traditional mode‐displacement methods, without considerable increase in the computational effort.
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References
1.
Arnold, R. R., et al. (1985). “Application of Ritz vectors for dynamic analysis of large structures.” Computers and Structs., 21(5), 901–907.
2.
Bathe, K. J., and Wilson, E. L. (1973). “Stability and accuracy analysis of direct integration methods.” Earthquake Engrg. and Struct. Dynamics, 1, 283–291.
3.
Belytschko, T. (1980). “Explicit time integration of structure‐mechanical systems.” Advanced Structural Dynamics, by J. Donéa, ed., Applied Science Publs., London, U.K., 97–122.
4.
Borino, G., Caddemi, S., and Polizzotto, C. (1987). “Mathematical programming methods for evaluating dynamic deformation of elastoplasticity structures.” Computational Plasticity Models, Software and Applications, D. R. Owen, E. Hinton, and E. Oñate, eds., Pineridge Press, Swansea, U.K., 1231–1245.
5.
Borino, G., and Muscolino, G. (1986). “Mode‐superposition methods in dynamic analysis of classically and non‐classically damped linear systems.” Earthquake Engrg. and Struct. Dynamics, 14, 705–717.
6.
Borino, G., and Polizzotto, C. (1988). “A mathematical programming approach to dynamic elastoplastic structural analysis.” Meccanica, 11–19.
7.
Capurso, M., and Maier, G. (1970). “Incremental elastoplastic analysis and quadratic optimization.” Meccanica, 5(2), 107–116.
8.
Caughey, T. K., and O'Kelly, M. E. J. (1965). “Classical normal modes in damped linear dynamic systems.” J. Appl. Mech., 32, 583–588.
9.
Clough, R. W., and Mojtahedi, S. (1976). “Earthquake response analysis considering non‐proportional damping.” Earthquake Engrg. and Struct. Dynamics, 4, 489–456.
10.
Comwell, R. E., Craig, R. R., Jr., and Johnson, C. P. (1983). “On the application of the mode‐acceleration method to structural engineering problems.” Earthquake Engrg. and Struct. Dynamics, 11, 679–688.
11.
Cottle, R. W. (1979). “Fundamentals of quadratic programming and linear complementarity.” Engineering Plasticity by Mathematical Programming, M. Z. Cohn and G. Maier, eds., Pergamon Press, New York, N.Y., 293–323.
12.
Dungar, R. (1982). “An imposed force summation method for non‐linear dynamic analysis.” Earthquake Engrg. and Struct. Dynamics, 10, 165–170.
13.
Feijoo, R. A., and Zouain, N. (1987). “Variational formulations for rates and increments in plasticity.” Computational Plasticity Models, Software and Applications, D. R. Owen, E. Hinton, and E. Oñate, eds., Pineridge Press, Swansea, U.K., 33–57.
14.
Foss, K. A. (1958). “Co‐ordinates which uncouple the equations of motion of damped linear dynamic systems.” J. Appl. Mech., 25, 361–364.
15.
Geschwindner, L. F., Jr. (1981). “Nonlinear dynamic analysis by modal superposition.” J. Struct. Engrg., ASCE, 107(12), 2324–2336.
16.
Hansteen, O. E., and Bell, K. (1979). “On the accuracy of mode superposition analysis in structural dynamics.” Earthquake Engrg. and Struct. Dynamics, 7, 405–411.
17.
Hodge, P. G., Belytschko, T., and Herakovich, C. T. (1969). “Quadratic programming and plasticity.” Computational Approaches in Applied Mechanics, E. Sevin, ed., ASME, New York, N.Y., 73–97.
18.
Houbolt, J. C. (1950). “A recurrence matrix solution for the dynamic response of elastic aircraft.” J. Aeronaut. Sci., 17, 540–550.
19.
Maddox, N. R. (1975). “On the number of modes necessary for accurate response and resulting forces in dynamic analysis.” J. Appl. Mech., 42, 516–517.
20.
Maier, G. (1968). “A quadratic programming approach for certain classes of nonlinear structural problems.” Meccanica, 3(2), 121–130.
21.
Molnar, A. J., Vaschi, K. M., and Gay, C. W. (1976). “Application of normal mode theory and pseudo‐force methods to solve problems with nonlinearities.” J. Pressure Vessel Tech., 98, 151–156.
22.
Morris, N. F. (1977). “The use of modal superposition in nonlinear dynamics.” Computers and Structs., 7, 65–72.
23.
Muscolino, G., and Polizzotto, C. (1984). “A mathematical programming approach to dynamic elastoplasticity.” VII National Congress AIMETA, 5, 181–192 (in Italian).
24.
Newmark, N. M. (1959). “A method of computation for structural dynamics.” J. Engrg. Mech. Div., ASCE, 85(3), 67–94.
25.
Nickell, R. E. (1976). “Nonlinear dynamics by mode superposition.” Computer Methods in Appl. Mech. and Engrg., 7, 107–129.
26.
Nour‐Omid, B., and Clough, R. W. (1984). “Dynamic analysis of structures using Lanczos co‐ordinates.” Earthquake Engrg. and Struct. Dynamics, 12, 565–577.
27.
Owen, D. R. J. (1980). “Implicit finite methods for the dynamic transient analysis of solids with particular reference to non‐linear solutions.” Advanced Structural Dynamics, by J. Donéa, ed., Applied Science Publs., London, U.K., 123–152.
28.
Polizzotto, C. (1984). “Dynamic shakedown by modal analysis.” Meccanica, 19, 133–144.
29.
Polizzotto, C. (1985). “A solution method for elastoplastic structures under dynamic agencies.” Numerical Methods in Engineering: Theory and Applications, Middleton and G. N. Pande, eds., A. A. Balkema, Rotterdam, The Netherlands, 243–251.
30.
Polizzotto, C. (1986). “Elastoplastic analysis method for dynamic agencies.” J. Engrg. Mech., ASCE, 112(3), 293–310.
31.
Traill‐Nash, R. W. (1981). “Modal methods in the dynamics of systems with nonclassical damping.” Earthquake Engrg. and Struct. Dynamics, 9, 153–169.
32.
Wilson, E. L., and Itoh, T. (1983). “An eigensolution strategy for large systems.” Computers and Structs., 16, 259–265.
33.
Zienkievicz, O. C. (1977). The finite element methods. 3rd Ed., McGraw‐Hill, London, U.K.
34.
Zienkievicz, O. C., et al. (1984). “A unified set of single step algorithms.” Int. J. Numerical Methods in Engrg., 20, 1529–1552.
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Copyright © 1989 ASCE.
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Published online: Oct 1, 1989
Published in print: Oct 1989
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