Keywords: reliability based optimization, truss structures, geometry optimization, topology optimization.

Methods for the optimization of truss topology, where the members' cross-sectional areas are taken as design variables, are well established and there is a rich literature on this subject. However, the problem of truss geometry optimization, where the nodes positions are taken as design variables, was not studied as extensively as the topology one, and consequently references on this subject are not so vast. The truss geometry optimization problem is non linear by its nature, and therefore it needs to be solved by nonlinear optimization methods, which are in general more complex and computationally demanding than the linear programming techniques. Some strategies have been developed to deal with the truss geometry optimization problem [1], such as: simultaneous optimization of truss topology and geometry, alternating optimization and implicit programming optimization.

In deterministic optimization, however, the uncertainties involved in the design problem, such as those affecting material properties and loads, among others, are not considered. Robust optimization or reliability based optimization (RBDO) methods are usually employed to take such uncertainties into account. The former has as a main goal the minimization of the variability of some parameters related to system response due to its uncertainties. The latter's main goal is to optimize structures guaranteeing that its probability of failure is lower than a certain level chosen a priori by the designer.

Only a few papers, however, have dealt with the reliability based geometry and topology optimization of truss structures [2,3]. Thus, we propose in this paper a RBDO approach for solving the simultaneous optimization of geometry and topology for statically indeterminate trusses, taking into account the uncertainties on the applied forces as well as on the yielding stresses. In this approach the applied forces and the yielding stresses are modelled as random variables, and the failure constraints are expressed in probabilistic terms.

This study has three main contributions: (i) it is shown how to efficiently pursue the sensitivity analysis required by the gradient based method used in the optimization process; (ii) based on the assumptions of linear structural behaviour and, independent and normally distributed random variables, the RBDO problem is posed in such a way that its computational cost is similar to a standard deterministic optimization problem; (iii) it is shown that, when uncertainties are considered, the resulting optimum structure may not be a simply scaled version of the deterministic solution (e.g. higher member areas), but there may be a change in the structural geometry as well.

1

W. Achtziger, "On simultaneous optimization of truss geometry and topology", Structural and Multidisciplinary Optimization, 33, 285-305, 2007. doi:10.1007/s00158-006-0092-0

2

Y. Murotsu, S. Shao, "Optimum shape design of truss structures based on reliability", Structural Optimization, 2, 65-76, 1990. doi:10.1007/BF01745455

3

R. Stocki, K. Kolanek, S. Jendo, M. Kleiber, "Study on discrete optimization techniques in reliability-based optimization of truss structures", Computers and Structures, 79, 2235-2247, 2001. doi:10.1016/S0045-7949(01)00080-3

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