Keywords: Monte-Carlo simulation, load incrementation, beam-colums, elasto-plastic FEA, stability analysis.

Simulation technics are already a possibility for design in structural engineering. In this paper we mean the term `simulation' as computer simulation, that is an experiment performed on a computer model. However these simulation results do not contain any explicit information about the behaviour of real systems, we want to regard those as real experiment results. Naturally from an engineering point of view this definition draws strict requirements for the computer model, but also we must make assumptions and simplifications. Nevertheless the experiment performed on a suitable model can have more appropriate result than the real experiment contains many uncertainties in equipment, measurements and evaluation of results. It can be stated that in simulation techniques the computer model is of primary importance.

In simulation techniques the most commonly used probabilistic approach is the Monte-Carlo method [3]. The Monte-Carlo method is the systematic use of samples of random numbers in order to estimate parameters of an unknown distribution by statistical simulation. Methods based on this principle of random sampling are indicated in cases where the dimensionality and/or complexity of a problem make straightforward numerical solutions impossible or impractical. When we apply the Monte-Carlo method in stability problems, the purpose is to take into consideration the probabilistic behaviour of the influential coefficients or initial parameters. A set of these random parameters - generated from a proper distribution - defines the initial state of structure. Applying a suitable deterministic procedure one can assess the stability resistance of this structure [5]. Repeating this process sufficiently many times one can obtain the probabilistic distribution of resistance. To create an efficient and applicable simulation process, there are several requirements for the deterministic procedure:

1. Accuracy. The calculations should reflect the real behaviour as close as possible, otherwise the sense of simulation fades away. Thus the mechanical model of the structure must contain all significant characteristics which influence the resistance.
2. Stability and reliability. The deterministic procedure must have very stable solution method, and should yield reliable results or else the simulation process has to stop prematurely.
3. Automatic working. To reach an efficient simulation process applicable for a wide range of problems it is essential to apply a highly automatic procedure. There can occur a lot of different stability problem during the simulation (e.g. according to the generated different load arrangement or amplitude of imperfections) which should be handled properly.
4. Rapidity. Because the great number of repetition the running of deterministic calculation should be very fast.

These requirements are generally in contradiction with each other, therefore it is important to choose and develop appropriate procedures. The first requirement is mostly connected with the mechanical model. In this paper a finite element model is applied [1] to elasto-plastic thin-walled beam-columns with taking a special cross-section treatment [4] into consideration, which can follow the real spread of yielding. The second, third and fourth requirements - and sometimes the first as well - are strongly connected with the solution technique of the non-linear set of equations. The most commonly used one is the Newton-Raphson incremental-iterative method, which advance the solution step by step following the equilibrium path of the structure [2]. The paper contains a fully automatic incremental technique, which does not require any preliminary knowledge of the structural problem, hence it is not necessary to set controlling parameters before every running. The main advantage of the method that such a scalar parameter was found - the maximum strain - what is an inner product of the calculation, and naturally contains all important information about the behaviour of the structure. Using this parameter the increments can reliably follow the softening of the non-linear system. An automatic selection of the size of first load increment is also possible according to the used approximation, which gives an effective result in case of imperfect beam-column structures.

1

W.F. Chen and T. Atsuta, "Theory of Beam-Columns: Space Behaviour and Design", Vol. 2., McGraw-Hill, 1977

2

M.J Clarke. and J. Hancock, "A Study of Incremental-Iterative Strategies for Non-Linear Analyses", Int. J. Numer. Methods Eng. 29, 1365-1391, 1990. doi:10.1002/nme.1620290702

3

P. Marek, M. Gustar, T. Anagnos, "Simulation-Based Reliability Assessment for Structural Engineers", CRC Press, 1996.

4

F. Papp, M. Iványi, K. Jármai, "Unified Object-Oriented Definition of Thin- Walled Steel Beam-Column Cross-Sections", Computer & Structures, 79, 839-852, 2001. doi:10.1016/S0045-7949(00)00183-8

5

J.Strating and H. Vos, "Computer Simulation of the E.C.C.S. Buckling Curves using a Monte-Carlo Method", HERON, 19(2), 1973.

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