Abstract
Based on the assumption of finite deformation, the Hamilton variational principle is extended to a nonlinear elastic Euler-type beam-column structure located on a nonlinear elastic foundation. The corresponding three-dimensional (3D) mathematical model for analyzing the nonlinear mechanical behaviors of structures is established, in which the effects of the rotation inertia and the nonlinearity of material and geometry are considered. As an application, the nonlinear stability and the post-buckling for a linear elastic beam with the equal cross-section located on an elastic foundation are analyzed. One end of the beam is fully fixed, and the other end is partially fixed and subjected to an axial force. A new numerical technique is proposed to calculate the trivial solution, bifurcation points, and bifurcation solutions by the shooting method and the Newton-Raphson iterative method. The first and second bifurcation points and the corresponding bifurcation solutions are calculated successfully. The effects of the foundation resistances and the inertia moments on the bifurcation points are considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen, Z. D. Finite Deformation Theory for Rods, Plates and Shells (in Chinese), Science Press, Beijing (1994)
Antman, S. S. The Theory of Rods, Handbuch der Physik, Vol. VIa/2, Springer-Verlag, Berlin/Heidelberg (1972)
Zhu, Y. Y., Miura, F., and Zhu, Z. Y. Nonlinear stability analysis for HDAJ spliced piles in elastic ground (in Chinese). Chinese Quarterly of Mechanics, 26(2), 216–223 (2005)
Gao, S., Zhu, Y. Y., and Zhu, Z. Y. Analysis of stability of pile with initial bending (in Chinese). Journal of Shanghai University (Natural Science), 15(3), 290–295 (2009)
Hu, Y. J., Zhu, Y. Y., and Cheng, C. J. Differential-algebraic approach to large deformation analysis of frame structures subjected to dynamic loads. Applied Mathematics and Mechanics (English Edition), 29(4), 441–452 (2008) DOI 10.1007/s10483-008-0403-7
Zhu, Z. Y. and Cheng C. J. Numerical Methods for Bifurcation Problems (in Chinese), Press of Lanzhou University, Lanzhou (1989)
Arnold, V. I. Dynamical Systems VI, Singularity Theory I (Encyclopaedia of Mathematical Sciences), Springer-Verlag, Berlin (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
Contributed by Chang-jun CHENG
Project supported by the National Science Foundation for Distinguished Young Scholars of China (No. 11002084), the Shanghai Pujiang Program (No. 07pj14073), and the Scientific Research Project of Shanghai Normal University (No. SK201032)
Rights and permissions
About this article
Cite this article
Zhu, Yy., Hu, Yj. & Cheng, Cj. Analysis of nonlinear stability and post-buckling for Euler-type beam-column structure. Appl. Math. Mech.-Engl. Ed. 32, 719–728 (2011). https://doi.org/10.1007/s10483-011-1451-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-011-1451-x
Key words
- beam-column structure
- Hamilton variational principle
- nonlinear stability
- shooting method
- Newton-Raphson iterative method
- post-buckling