Many combinatorial optimality problems appear in the structural engineering, particularly, the structural design problems. Practically, these problems are approached by both high speed computers and the efficient algorithms with a result in the best solutions based on problem­oriented constraints or heuristic rules. Many optimal design problems including reliability­based or minimum weight design of structural system subjected to external loadings by the mode approach require some predominant failure modes from enormous space of combined modes among elementary failure modes. Consequently, the essence is equivalent substantially to develop efficient problem­oriented technique to prune futile combinations in relation to optimality. A simple combinatorial optimality closely connected with optimal design problems is the failure load analysis of masonry structures such as a single masonry wall with openings subjected to vertical and horizontal loadings, which can be modeled into an equivalent truss system. Hence, plausible failure modes consist of elementary failure modes, which are described by topology and geometry, and their linear combinations. Such a modeled equivalent truss system of a practical masonry wall has enormous number of combination, and without efficient constraints it is almost impossible to search for the lowest load factor. The genetic algorithm approach based on the stochastic schemata exploiter [1] is, herein, applied to the present limit analysis.

Based upon the experimental results at University of Pavia [2] and others, failure patterns of single masonry walls with openings are categorized into the shear diagonal crack opening, local panel rocking, bed joint sliding, full or partial overturning of an entire wall, which can be modeled by the equivalent shear truss members with appropriate geometry and strength parameters. A single masonry with rectangular openings is modeled as an equivalent shear truss system whose failure modes consist of elementary failure modes, which are systematically generated by geometry and topology of the system, and their linear combinations. This becomes a typical combinatorial optimality problem that the minimum load factor should be searched among possible failure modes including combined modes.

Main features of the present algorithm based upon the schema theorem are coding by means of tree graphs and selection of schemata. Numerical simulation of a single masonry wall with 4 rectangular openings subject to incremental horizontal and constant vertical loadings, which is described by the equivalent shear truss model, is carried out with discussion on advantages of this algorithm such as the smaller number of parameters and preference of local optima. Unfortunately, no guarantee exists of converge to the rigorous optimal solution by the present GA. However, from the engineering point of view, near optimal solutions are frequently required instead of the rigorous optimal solution. Among many approaches the genetic algorithm becomes applicable because of its computing efficiency. The present method seems powerful at identifying regions where the optimum exists, or selecting candidate solutions, which are exploited for further analysis such as a reliability analysis of masonry structures by the mode approach.

1

A. N. Aizawa, "Evolving SSE : a stochastic schemata exploiter", Proc. of the 1st IEEE Conference on Evolutionary Computation, 525�529, 1994. doi:10.1109/ICEC.1994.349895

2

G. Magenes, G. M. Calvi, R. Kingsley, "Seismic Testing of a full�scale, two�story masonry building : test procedure and measured experimental response", Experimental and Numerical Investigation on a Brick Masonry Building Prototype - Numerical Prediction of the Experiment, Report 3.0, G.N.D.T., Pavia, January 1995.

3

Y. Kohama, T. Takada, A. DeStefano, A. Miyamura, T. Aoki, "Limit analysis of masonry wall with openings by genetic algorithm", Proc. of the 5th Int. Conf. on Computational Structures Technology, Computational Engineering us� ing Metaphors from Nature, pp.135�141, 2000.

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