Keywords: thin-walled frame, elastic-plasticity, space beam element, non-linear displacement field, incremental formulation, external stiffness approach, plastic hinges.

Thin-walled frames, as weight-optimised structures, display very complex structural behaviour and thus the development of advanced non-linear analysis tools has been a major activity of many structural engineering researchers in the past years [1]. Although shell finite elements may provide the accurate prediction of the non- linear response of such structures, the significant computational efforts demanded by their two-dimensional nature, especially when large rotations as well as elastic- plastic deformations are likely to occur, restrict the use of this type of finite element to studying the non-linear response of individual members or small assemblages.

The non-linear response of a load-carrying structure should be solved using numerical methods, e.g. the finite element method, and some of incremental descriptions like the total and updated Lagrangian approaches, respectively or the Eulerian approach [2]. Each description utilizes a different structural configuration for system quantities referring and results in the form of a set of non-linear equilibrium equations of the structure. This set can further be linearized and should be solved using some incremental-iterative scheme, which consists of three main phases. The first or predictor phase comprises evaluating the overall structural stiffness and solving for the displacement increments from the approximated incremental equilibrium equations for the structure. Using the standard transformation process displacement increments of each finite element can be determined immediately. The second or corrector phase involves the geometry updating of each finite element and the determination of element nodal forces using some force recovery algorithm. Before choosing an appropriate force recovery algorithm, the rigid-body test should be performed [3]. While the updating of nodal co-ordinates is a trivial task, the updating of nodal space orientations should be based on the large rotation theory. The third or checking phase comprises checking if the adopted convergence criterion of iteration is achieved in the current increment by comparing with the pre-set tolerance value.

The first part of the present work comprises only geometrical non-linearities of an elastic, straight and prismatic thin-walled beam member. It is assumed that cross- section is not deformed in its own plane, but is subjected to warping in the longitudinal direction. Displacements and rotations are allowed to be large but strains are small. The shear strain in the middle surface can be neglected. Loading of a considering structure is static and conservative, while internal moments are represented as the resultants of stresses calculated by engineering theories. The element geometric stiffness is derived using the updated Lagrangian description and the non-linear displacement field of a thin walled cross-section, which includes non- linear displacement terms due to three-dimensional rotation effects. In such a way, the incremental geometric potential corresponding to the semitangential moment is obtained for all the internal moments, ensuring thus the moment equilibrium conditions at a joint of beam members with different space orientations to be preserved but preventing the beam element to pass the rigid-body test. Thus, the external stiffness approach (ESA) is applied as the force recovery algorithm [4].

Introducing the possibility of elastic-perfectly plastic material behaviour, the analysis is further enhanced to the material non-linearity. Although the plastic-zone model is usually considered the exact model because it explicitly accounts for the spread of plasticity throughout the beam member, in this work, because of the computational advantages, the plastic hinge model is applied. In such a model, it is assumed that all plastic effects, when occur, are concentrated in the zero-length plastic hinges at the finite element ends, while the element between hinges remains linear-elastic [5]. Supposing the existence of a single-function yield surface in terms of the beam stress-resultants obtained by the ESA and using the normality principle, a new plastic reduction matrix of the beam element is derived, the function of which is to keep the element incremental forces at a plastic hinge to move tangentially to the yield surface. As at the end of that increment the yield criterion is violated, the return of element nodal forces at a plastic hinge to the yield surface should be performed. The abovementioned numerical algorithm is implemented into a computer program and the effectiveness is validated through the test problems.

1

J.F. Doyle, "Nonlinear Analysis of Thin-Walled Structures", Springer-Verlag, New York, 2001.

2

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

3

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

4

G. Turkalj, J. Brnic, J., Prpic-Orsic, "Updated Lagrangian Formulation Using ESA Approach in Large Rotation Problems of Thin-Walled Beam-Type Structures", In: B.H.V. Topping (ed.), "Proceedings of the Eight International Conference on Civil & Structural Engineering Computing", Civil-Comp Press, Stirling, Scotland, 2001. doi:10.4203/ccp.73.75

5

W.F. Chen, I. Sohal, "Plastic Design and Second-Order Analysis of Steel Frames", Springer-Verlag, New York, 1995.

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