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 ones, respectively or the Eulerian description. 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 [1]. 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. 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.

Columns under sustained loads are generally unstable in the regime of creep. That means that loss of stability may occur during a period of exploitation of structure even for loads lower than critical buckling load. Due to that reason stability is characterized by critical buckling time defined as load duration for which buckling deflections becomes infinitive. From that aspect any load can become a long time critical buckling load while of course the larger load supposes the shorter time to collapse.

The present work firstly comprises only geometrical non-linearities of an elastic, straight and prismatic beam member. It is assumed that cross-section is not deformed in its own plane. Displacements and rotations are allowed to be large but strains are small. The shear strain in the middle surface are neglected. Loading of a considering structure is static and conservative.

Standard Eulerian description is linear at an element level and all geometrically non-linear effects are introduce through the transformation from local to global coordinate system. The local Eulerian system follows the element chord during deformation and allows the use of simplified strain-displacement relations on the local element level [2]. Using such a standard procedure it is unable to model pure torsional buckling because no geometric non-linearties due to cross-sectional deformation are taken into account to initiate it. From that reason an additional nonlinear part of stiffness matrix needs to be included which accounts for the Wagner effect as a necessary condition for pure torsional buckling behaviour.

Introducing the possibility of non-constant material behaviour in time, analysis is further enhanced with respect to creep modelling. An explicit time integration scheme is implemented. According to the material creep law the stress state corresponding to the strain state is established incrementally from the values at the last equilibrium step. The above mentioned procedure is implemented into a computer program BMCA and the effectiveness is validated through test examples [3].

1

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

2

B.A. Izzuddin, D. Lloyd Smith, "Large displacement analysis of elastoplastic thin-walled frames. I: Formulation and implementation", Journal of Structural Engineering, 122(8), 905-914, 1996. doi:10.1061/(ASCE)0733-9445(1996)122:8(905)

3

J. Walczak, J. Sieniawski, K.J. Bathe, "On the analysis of creep stability and rupture", Computers & Structures, 17(5-6), 783-792, 1983. doi:10.1016/0045-7949(83)90093-7

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