Keywords: reliability analysis, limit analysis, shakedown analysis, residual strain energy, Monte Carlo simulation, optimal design.

In structural plasticity the static and kinematic limit theorems provide appropriate tools to solve complex problems [1,2]. The main structural elements of steel framed multi-storey structures are the columns, the beams and their connections, which can be semi-rigid. This can significantly influence the behaviour of the structure and therefore has to be taken into account in the analysis and design. Since the shakedown and limit analysis provide no information about the magnitude of the plastic deformations and residual displacements accumulated before the adaptation of the structure, therefore for their determination several bounding theorems and approximate methods have been proposed. Among others Kaliszky and Lógó [1,3] suggested that the complementary strain energy of the residual forces could be considered as an overall measure of the plastic performance of the structure and the plastic deformations should be controlled by introducing a limit on the magnitude of this energy.

In engineering practice uncertainties play a very important role [2,4] and need intensive calculations. The aim of this paper is to take into consideration the influence of the limited load carrying capacity of the connections on the plastic limit state of elasto-plastic steel (or composite) frames under multi-parameter static loading and probabilistically given conditions during the design. In addition to the plastic limit design to control the plastic behaviour of the structure, bound on the complementary strain energy of the residual forces is also applied. This bound has a significant effect on the load parameter [3]. If the design uncertainties (manufacturing, strength, geometrical) are expressed by the calculation of the complementary strain energy of the residual forces then the reliability based extended plastic limit and shakedown design problems can be formed. Two numerical procedures are elaborated: the first one is based on a direct integration technique and the uncertainties are assumed to follow Gaussian distribution. The formulations of the problems yield to nonlinear mathematical programming which are solved by the use of a sequential quadratic algorithm. The nested optimization procedure is governed by the reliability index calculation. The second procedure is based on the crude Monte Carlo simulation. The multi-parameter static loads follow a Gumbel distribution and the "design" uncertainties are assumed the Gaussian distributed data. The response surface method is applied [2] to ensure high efficiency.

1

S. Kaliszky, J. Lógó, "Optimal Plastic Limit and Shakedown Design of Bar Structures with Constraints on Plastic Deformation", Engineering Structures, 19(1), 19-27, 1997. doi:10.1016/S0141-0296(96)00066-1

2

K. Dolinski, J. Knabel, "Reliability-oriented Shakedown Formulation", in ICOSSAR 2005, G. Augusti, G.I. Schueller, M. Ciampoli, (Editors), Millpress, Rotterdam, 2323-2330, 2005.

3

J. Lógó, M. Movahedi Rad, J. Knabel, Zs. Hortobágyi, "Plastic limit and shakedown analysis of elasto-plastic steel frames with semi-rigid connections", in Proceedings of the Ninth International Conference on Computational Structures Technology, B.H.V. Topping, M. Papadrakakis, (Editors), Civil-Comp Press, Stirlingshire, United Kingdom, paper 161, 2008. doi:10.4203/ccp.88.161

4

R.B. Corotis, A.M. Nafday, "Structural System Reliability Using Linear Programming and Simulation", Journal of Structural Engineering, 115(10), 2435-2447, 1989. doi:10.1061/(ASCE)0733-9445(1989)115:10(2435)

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