Keywords: thin-walled beam, incremental formulation, finite element, non-linear displacement field, large rotations, external stiffness approach.

Thin-walled structures are widely used in the field of mechanical, naval, aerospace and civil engineering because of the product cost which is usually an objective function in an optimal procedure. Many of them can be modelled as thin- walled beams in static and dynamic analysis [1]. But, such weight-optimised structural members display a very complex structural behaviour because they are very susceptible to the loss of stable deformation forms and, in their design, an important consideration concerns the accurate prediction of their limit load-carrying capacity [2]. Thus, the development of effective non-linear numerical models as analysis tools proving indispensable. Furthermore, considering the response of frames involving three-dimensional large rotations, renders the analysis far more complicated. In this, beam-type finite elements are especially appealing to structural engineers because the use of shell-type finite elements is restricted only to studying the non-linear response of individual members or small assemblages [3].

In the finite element analysis a non-linear problem should be solved in an incremental manner where three main approaches being identified: total Lagrangian (TL), updated Lagrangian (UL) and Eulerian [4]. In the TL approach system quantities are referred to the initial configuration of the structural element, generally leading to complex strain-displacement relationship. In the UL approach, adopted in the present work, system quantities are referred to the last known equilibrium configuration, which results in simpler strain-displacement relationship. In the Eulerian approach system quantities are referred to the current unknown configuration, allowing linear strain-displacement relationship to be used in the local system, with geometric non-linearities introduced through transformations between the local and global system. The set of non-linear equilibrium equations of the structure obtained by each approach must be attempted by a combination of incremental and iterative procedures. These procedures consist of three main phases: predictor, corrector and checking phase, respectively. The first phase comprises evaluating the overall structural stiffness and solving for the displacement increments from the approximated incremental equilibrium equation for the structure; the second phase involves updating of the element geometry for each finite element and determining the corresponding nodal forces using a particular force recovery algorithm, while the third phase comprises checking if the adopted convergence criterion of iteration is achieved in the current increment by comparing with the pre-set tolerance value. In order to account for the effect of non- commutativity of the three-dimensional rotations, the updating of end orientations should be based on the large rotation theory [5].

As in this work the geometric stiffness matrix of a space beam element is developed by including the non-linear displacement field of a thin-walled cross- section, it cannot pass the rigid body test [6]. For this reason, the natural deformation approach, very often used in UL formulation, cannot serve as the force recovery algorithm. Instead, the external stiffness approach is applied by including the external stiffness matrix of the beam element in the force recovering and the purpose of which is to exclude the rigid-body effects from the calculation of element force increments.

The basic assumptions used in this analysis are: a thin-walled beam member is prismatic and straight; the cross-section is not deformed in its own plane, but is subjected to warping in the longitudinal direction; displacements and rotations are large but strains are small; the shear strain in the middle surface can be neglected; material is homogeneous, isotropic and linear-elastic; internal moments are represented as the resultants of stresses calculated by engineering theories; external load is static and conservative. The presented numerical is validated through the test problems.

1

Hu, Y., Jin, X., Chen, B., "A Finite Element Model for Static and Dynamic Analysis of Thin-Walled Beams with Asymmetric Cross-Section", Computers & Structures, 61, 897-908, 1996. doi:10.1016/0045-7949(96)00058-2

2

Trahair, N.S. "Flexural-Torsional Buckling of Structures", CRC Press, Boca Raton, 1993.

3

Chin, C.K., Al-Bermani, F.G.A., Kitipornchai, S. "Non-Linear Analysis of Thin-Walled Structures using Plate Elements", International Journal for Numerical Methods in Engineering, 37, 1697-1711, 1994. doi:10.1002/nme.1620371005

4

Belytschko, T., Liu, W.K., Moran, B., "Nonlinear Finite Elements for Continua and Structures", John Wiley & Sons, LTD, Chichester, 2000.

5

Crisfield, M.A., "Non-Linear Finite Element Analysis of Solids and Structures", Vol. 2, John Wiley & Sons, New York, 1997.

6

Yang, Y.B., Kuo, S.R., "Theory & Analysis of Nonlinear Framed Structures", Prentice Hall, New York, 1994.

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