Keywords: torque buckling analysis, spline finite element, cubic B-spline, initial stress, framed structure, initial stress stiffness matrix.

This paper uses B-splines to analyze dynamic buckling problems of framed structures subject to coupled initial end torque and axial force. It was assumed [1] that the torque applied at the end of the beam element is equally shared in x and y axes as initial stresses, which is satisfactory in such a case that the bending rigidities in x and y axes are equal [2]. In the case where the bending rigidities are different, the torque is shared in the x and y axes as initial stresses in the ratio I_x:I_y.

After a brief introduction, assumptions and a basic description of the problem of interest are presented. Next the energy equations are derived for uniform cross-sections, and are non-dimensionalized with the rigidity ratio to define the geometry. Using cubic B-splines, displacement fields are constructed. A derivation of the equation of motion based on Hamilton's principle is presented. Thereafter, the idea of finite element method is used to formulate the system matrices. To simplify the calculation, a coordinate transformation transforms a typical element into a standardized element. Details of how to deal with boundary conditions are described. The solution is given, which is a linear eigenvalue problem to describe the relationship between natural frequency, axial buckling force, and end buckling torque. Numerical examples for a cantilever beam element are discussed, in which numerous interactive diagrams are constructed and analyzed. The paper is concluded with some remarks.

A comprehensive study on the interactive dynamic axial-torsional buckling problem of a doubly symmetric beam element with uniform cross-section has been presented. Extensive interactive diagrams are constructed for analysis purpose. The numerical results indicate that the method is efficient and accurate in comparison with the results in [2,3,4]. Although only a single cantilever beam element has been investigated in the numerical examples, structural frames can be further studied without difficulties since the method is based on the finite element formulation.

1

N.-I. Kim, C.C. Fu, M.-Y. Kim, "Stiffness matrices for flexural-torsional/lateral buckling and vibration analysis of thin-walled beam", Journal of Sound and Vibration, 299(4-5), 739-756, 2007. doi:10.1016/j.jsv.2006.06.062

2

H. Ziegler, "Principles of structural stability", 2nd ed, Birkhauser Verlag Basel, Germany, 1977.

3

S.P. Timoshenko, J. Gere, "Theory of Elastic Stability", 2nd ed, McGraw-Hill, New York, 1961.

4

A.Y.T. Leung, "Exact dynamic stiffness for axial-torsional buckling of structural frames", Thin-Walled Structures, 46(1), 1-10, 2008. doi:10.1016/j.tws.2007.08.012

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