Keywords: thin-walled beam, stability analysis, non-linear displacement field, large displacements, large rotations, shear-flexible beam element.

This paper presents a numerical stability analysis of shear deformable thin-walled beam structures, based on the finite element method. The finite element equation is derived using the linearized virtual principle under assumptions of large displacements, small strains, isotropic and linear elastic material. Shear effects of the beam cross section are also taken into account. Internal moments are calculated by engineering theories. Linear shape functions are used for the axial displacement, while cubic shape functions are employed for the transverse displacements and angle of twist. Also, using a non-linear displacement field for the cross-section that includes the large rotation effects, the geometric stiffness matrix for the spatial thin-walled beam finite element is derived, in which internal semitangential moments are obtained. In this paper the stability problem is treated as eigenvalue one. The accuracy of presented numerical algorithm is examined through test problems [1,2].

Load-carrying structures composed of thin-walled sections are extensively used in engineering practice, both in stand-alone forms and as stiffeners for plate-like or shell-like structures. Unfortunately, such weight-optimised structural components, especially those with open profiles, are commonly weak in torsion and very susceptive to instability or buckling failure [3,4].

Stability analysis of such structures can be performed using two different approaches, both of them furnish two different kinds of valid and important information. In the first case, the stability analysis, known as the linear one, is performed in an eigenvalue manner and it attempts to determine the instability load in a direct manner without calculating the deformations. The lowest eigenvalue is recognised as a critical or buckling load and the corresponding eigenvector represents a corresponding critical or buckling mode. In this, ideal or perfect structures and loading conditions are considered and the pre-buckling deformations prior to the attainment of buckling load are neglected.

In the second case, stability problems are approached using a load-deflection manner, which attempts to solve a stability problem by predicting the structural behaviour through the entire range of loading, including the pre-buckling as well as the post-buckling phase. This approach belongs to so-called non-linear stability analysis and in contrast to the linear one, it is much more complex and computationally expensive, but gives valid information in the case of imperfect or real structures and loading conditions with or without material non-linearity included. The eigenvalue approach generally gives overestimated results for such problems.

In the region of large spatial rotation stability, analysis is a very complex problem because of the non vectoral nature of large rotations. When a standard linear displacement field is used, the torsional moment is of a semi-tangental character and the bending moments are of a quasi-tangential character, so their induced moments due to large spatial rotation are not compatible. To avoid this problem this work uses a non-linear displacement field of cross section in which large rotation effects are included [5]. The geometric stiffness matrix of thin-walled beam finite element is derived in which all internal moments are supposed to be of a semitangental character.

This work proposes the following model: isotropic material; prismatic and straight beam members; displacements are large but strains are small; cross sections are non-deformable in plane but it is possible to warp; constitutive equations are linear; external loads are conservative.

1

S.P. Chang, S.B. Kim, Y.M. Kim, "Stability of shear deformable thin-walled space frames and circular arches", Journal of Engineering Mechanics, 122(9), 844-854, 1996. doi:10.1061/(ASCE)0733-9399(1996)122:9(844)

2

M.Y. Kim, S.P. Chang, S.B. Kim, "Spatial stability and free vibration of shear flexible thin-walled elastic beams. II: Numerical approach", International Journal for Numerical Methods in Engineering, 37, 4117-4140, 1994. doi:10.1002/nme.1620372311

3

T.V. Galambos, "Guide to Stability Design Criteria for Metal Structures", John Wiley & Sons, New York, 1998.

4

N.A. Alfutov, "Stability of Elastic Structures", Springer-Verlag, Berlin, 2000.

5

G. Turkalj, J. Brnic, J. Prpic-Orsic, "Large rotation analysis of elastic thin-walled beam-type structures using ESA approach", Computers & Structures, 81(18-19), 1851-1864, 2003. doi:10.1016/S0045-7949(03)00206-2

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