Enhanced Solution Scheme for Nonlinear Analysis of Force-Based Beam for Large Rotations, Multiple Critical Points, and Random Quasi-Static Loading Input
Publication: Journal of Structural Engineering
Volume 142, Issue 9
Abstract
This paper attempts to enhance (1) the flexibility-corotational beam formulation for simulating highly geometrically nonlinear problems, and (2) the arc length method of integration to develop a robust and economical solution scheme for random quasi-static loading input. The existing flexibility-corotational formulation of a three dimensional beam is improved by (1) updating the rotation matrices by using the quaternions, and (2) proposing novel methods for tracking the displacement compatibility. Further, the cylindrical arc length method is modified by (1) introducing capabilities for handling random quasi-static loading, and (2) proposing automatic adjustment of the arc length. By using these developments several classical geometric nonlinearity problems are solved and path independency for buckling and postbuckling path with the random quasi-static loading is verified. The proposed variation in the arc length solution scheme is also shown to be better than the implementation of the arc length method in the state of the art commercial software.
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References
ABAQUS [Computer software]. Dassault Systèmes Simulia, Providence, RI.
Algohary, T. A., Dong, L., Junkins, J. A., and Atluri, S. N. (2014). “Solution of post-buckling & limit load problems, without inverting the tangent stiffness matrix & without using arc-length methods.” Comput. Model. Eng. Sci., 98(6), 543–563.
Argyris, J., et al. (1979). “Finite element method—The natural approach.” Comput. Methods Appl. Mech. Eng., 17/18(1), 1–106.
Bathe, K. J., and Dvorkin, E. N. (1983). “On the automatic solution of nonlinear finite element equations.” Comput. Struct., 17(5–6), 871–879.
Belytschko, T., and Hsieh, B. J. (1973). “Non-linear transient finite element analysis with convected coordinates.” Int. J. Numer. Methods Eng., 7(3), 255–271.
Bouc, R. (1967). “Forced vibration of mechanical system with hysteresis.” Proc., 4th Conf. on Non-Linear Oscillations.
Crisfield, M. A. (1981). “A fast incremental/iterative solution procedure that handles “snap-through”.” Comput. Struct., 13(1–3), 55–62.
Crisfield, M. A. (1990). “A consistent co-rotational formulation for nonlinear, three-dimensional, beam-elements.” Comput. Methods Appl. Mech. Eng., 81(2), 131–150.
Crisfield, M. A. (1997). Nonlinear finite element analysis of solids and structures, Wiley, Chichester, U.K.
De Souza Neto, E. A., Peric, D., and Owen, D. R. J. (2008). Computational methods for plasticity theory and applications, Wiley, Chichester, U.K.
Felippa, C. A., and Haugen, B. (2005). “A unified formulation of small strain corotational finite elements: I. Theory.” Comput. Methods Appl. Mech. Eng., 194(21–24), 2285–2335.
Feng, Y. T., Perić, D., and Owen, D. R. J. (1996). “A new criterion for determination of initial loading parameter in arc-length methods.” Comput. Struct., 58(3), 479–485.
Ghosh, S., and Roy, D. (2008). “Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam.” Comput. Methods Appl. Mech. Eng., 198(3–4), 555–571.
Gierlinski, J. T., and Graves Smith, T. R. (1984). “The geometric non-linear analysis of thin-walled structures by finite strips.” Thin-Walled Struct., 2(1), 27–50.
Harrison, H. B. (1978). “Post-buckling behavior of elastic circular arches.” Proc. Inst. Civ. Eng., 65(2), 283–298.
Huddleston, J. V. (1968). “Effect of axial strain on buckling and postbuckling behavior of elastic columns.” Proc., Southeastern Conf. on Theoretical and Applied Mechanics, 263–273.
Ibrahimbegovic, A. (1997). “On the choice of finite rotation parameters.” Comput. Methods Appl. Mech. Eng., 149(1–4), 49–71.
Ibrahimbegovic, A., Frey, F., and Kozar, I. (1995). “Computational aspects of vector-like parametrization of three-dimensional finite rotations.” Int. J. Numer. Methods Eng., 38(21), 3653–3673.
Ibrahimbegovic, A., and Taylor, R. (2002). “On the role of frameinvariance in structural mechanics models at finite rotations.” Comput. Methods Appl. Mech. Eng., 191(45), 5159–5176.
Jelenic, G., and Crisfield, M. A. (1999). “Geometrically exact 3D beam theory: Implementation of a strain-invariant finite element for statics and dynamics.” Comput. Methods Appl. Mech. Eng., 171(1–2), 141–171.
Nukala, P. K. V., and White, D. W. (2004). “A mixed finite element for three-dimensional nonlinear analysis of steel frames.” Comput. Methods Appl. Mech. Eng., 193(23–26), 2507–2545.
Padovan, J., and Arechaga, T. (1982). “Formal convergence characteristics of elliptically constrained incremental Newton-Raphson algorithms.” Int. J. Eng. Sci., 20(10), 1077–1097.
Powell, G., and Simons, J. (1981). “Improved iteration strategy for nonlinear structures.” Int. J. Numer. Methods Eng., 17(10), 1455–1467.
Ray, T., and Reinhorn, A. (2014). “Enhanced smooth hysteretic model with degrading properties.” J. Struct. Eng., 04013028.
Ray, T., Schachter-Adaros, M., and Reinhorn, A. M. (2015). “Flexibility-corotational formulation of space frames with large elastic deformations and buckling.” Comput.-Aided Civ. Infrastruct. Eng., 30(1), 54–67.
Rezaiee-Pajand, M., and Alamatian, J. (2011). “Atomatic DR structural analysis of snap-through and snap-back using optimized load increment.” J. Struct. Eng., 109–116.
Riks, E. (1979). “An incremental approach to the solution of snapping and buckling problems.” Int. J. Solids Struct., 15(7), 529–551.
Rodrigues, O. (1840). “Des lois géométriques qui régissent les déplacements d’un système solide dans l’espace, et de la variation des coordonnées provenant de ces déplacement considérés indépendant des causes qui peuvent les produire.” J. Math. Pures Appl., 5, 380–440.
Santos, H. A. F. A., Pimenta, P. M., and Almeida, J. P. M. (2011). “A hybrid-mixed finite element formulation for the geometrically exact analysis of three-dimensional framed structures.” Comput. Mech., 48(5), 591–613.
Schachter, M., and Reinhorn, A M (2007). “Three-dimensional modeling of inelastic buckling in frame structures.”, Univ. at Buffalo, SUNY, Buffalo, NY.
Simo, J. C., and Vu-Quoc, L. (1986). “A three dimensional finite strain rod model. II: Computational aspects.” Comput. Method Appl. Mech. Eng., 58(1), 79–116.
Sivaselvan, M. V., and Reinhorn, A. M. (2002). “Collapse analysis: Large inelastic deformations analysis of planar frames.” J. Struct. Eng., 1575–1583.
Sivaselvan, M. V., and Reinhorn, A. M. (2004). “Nonlinear structural analysis towards collapse simulation: A dynamical systems approach.” Univ. at Buffalo, SUNY, Buffalo, NY.
Szyszkowski, W., and Husband, J. B. (1999). “Curvature controlled arc-length method.” Comput. Mech., 24(4), 245–257.
Timoshenko, S., and Gere, J. (1961). Theory of elastic stability, 2nd Ed., McGraw-Hill, New York.
Wempner, G. A. (1971). “Discrete approximations related to nonlinear theories of solids.” Int. J. Solids Struct., 7(11), 1581–1599.
Wen, Y. K. (1976). “Method for random vibration of hysteretic systems.” J. Eng. Mech. Div., 102(2), 249–263.
Widjaja, B. R. (1998). “Path-following technique based on residual energy suppression for nonlinear finite element analysis.” Comput. Struct., 66(2–3), 201–209.
Yang, Y. B., and Shieh, M. S. (1990). “Solution method for nonlinear problems with multiple critical points.” AIAA J., 28(12), 2110–2116.
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Published In
Journal of Structural Engineering
Volume 142 • Issue 9 • September 2016
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Aug 13, 2015
Accepted: Jan 19, 2016
Published online: Mar 24, 2016
Discussion open until: Aug 24, 2016
Published in print: Sep 1, 2016
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