Keywords: Fourier p-elements, dynamic stability, ill-conditioning, follower tension, buckling, beam-column.

High-order polynomials are inherently ill-conditioned. To improve the ill-conditioning problem, Leung and Chan [1] used the C⁰ Fourier enriched shape functions to predict the axial vibration frequencies of a beam. The sine functions represent the internal DOF. It is proved that the Fourier p-mass matrix is much more stable than that of the Legendre p-element [2].

The Fourier p-element has been successfully applied to various structures of beams and plates of different shapes and many three dimensional problems [3,4,5,6,7,8,9,10]. In this paper the Fourier p-element has been extended to the dynamic stability analyses of beam-columns under conservative and non-conservative axial forces. The results compare very well to the exact method of dynamic stiffness [11]. The problem of ill-conditioning associated with polynomial p-elements is eliminated and the numerical stability associated with the dynamic stiffness method for very high frequency and axial load is avoided. New results of follower tension are given and follower tension buckling under uniformly distributed follower tension is originally reported. The readers are referred to Langthjem and Sugiyama [12] for a comprehensive review of follower compression. Effects of follower tension have not been reported in literature.

1

A.Y.T. Leung and J.K.W. Chan, "Fourier p-element for the analysis of beams and plates", Journal of Sound and Vibration, 212, 179-185, 1998. doi:10.1006/jsvi.1997.1423

2

R.D. Cook, D.S. Malkus, M.E. Plesha, "Concepts and Applications of Finite Element Analysis", 4th Edition, John Wiley & Sons, New York, 1989.

3

A. Houmat, "A sector Fourier p-element for free vibration analysis of sectorial membranes", Computers & Structures, 79, 1147-1152, 2001. doi:10.1016/S0045-7949(01)00013-X

4

A. Houmat, "A sector Fourier p-element applied to free vibration analysis of sectorial plates", Journal of Sound and vibration, 243(2), 269-282, 2001. doi:10.1006/jsvi.2000.3410

5

A. Houmat, "Three-dimensional hierarchical finite element free vibration analysis of annular sector plates", Journal of Sound and Vibration, 276, 181-193, 2004. doi:10.1016/j.jsv.2003.07.020

6

Y.Q. Li, "Free vibration analysis of plate using finite strip Fourier p-element", Journal of Sound and Vibration, 294(4-5), 1051-1059, 2006. doi:10.1016/j.jsv.2006.01.003

7

A.Y.T. Leung, B. Zhu, J.J. Zheng, H. Yang, "Analytic trapezoidal Fourier p-element for vibrating plane problems", Journal of Sound and Vibration, 271 (1-2), 67-81, 2004. doi:10.1016/S0022-460X(03)00263-3

8

A.Y.T. Leung, B. Zhu, "Transverse vibration of thick polygonal plates using analytically integrated trapezoidal Fourier p-element", Computers & Structures, 82 (2-3), 109-119, 2004. doi:10.1016/j.compstruc.2003.10.002

9

A.Y.T. Leung, B. Zhu, "Fourier p-elements for curved beam vibrations", Thin-Walled Structures, 42(1), 39-57, 2004. doi:10.1016/S0263-8231(03)00122-8

10

A.Y.T. Leung, B. Zhu, J.J. Zheng, H. Yang, " A trapezoidal Fourier p-element for membrane vibrations", Thin-Walled Structures, 41(5), 479-491, 2003. doi:10.1016/S0263-8231(02)00117-9

11

A.Y.T. Leung, "Dynamic stiffness and substructures", Springer-Verlag, 1993.

12

M.A. Langthjem, Y. Sugiyama, "Dynamic stability of columns subjected to follower loads: A Survey", Journal of Sound and Vibration, 238(5), 809-851, 2000. doi:10.1006/jsvi.2000.3137

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