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CivilComp Proceedings
ISSN 17593433 CCP: 81
PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by: B.H.V. Topping
Paper 100
Suboptimal Cycle Bases for Analysis of Frames with Semirigid Joints A. Kaveh, H. Moez and M.A. Barkhordari
Iran University of Science and Technology, Tehran, Iran A. Kaveh, H. Moez, M.A. Barkhordari, "Suboptimal Cycle Bases for Analysis of Frames with Semirigid Joints", in B.H.V. Topping, (Editor), "Proceedings of the Tenth International Conference on Civil, Structural and Environmental Engineering Computing", CivilComp Press, Stirlingshire, UK, Paper 100, 2005. doi:10.4203/ccp.81.100
Keywords: cycle basis, graph theory, frames, force method, semirigid joints.
Summary
For an efficient force method of frame analysis, special cycle bases should be
generated for the formation of localized selfequilibrating systems, leading to
sparse flexibility matrices. In this paper, an algorithm is presented using a
fundamental cycle basis, where the selected cycles are improved via an algebraic
exchange method. Optimal analysis is performed for frames with semirigid
joints.
In this method, flexibility matrices are generated which are highly sparse (see reference [1,2] for definition). An ordering algorithm is also used for profile reduction to acquire an efficient method for the solution of the corresponding equations.Force method analysis of semirigid frames is formulated and a computer code is developed. Examples are analyzed with the present approach and the results are compared to those of SAP 2000. An efficient algorithm is presented for computing a suboptimal cycle basis of a simple graph . This algorithm is based on the fact that a cycle basis is minimal when no cycle in it can be replaced by a smaller cycle. Fundamental cycle basis of is as input of the algorithm and suboptimal cycle basis is the output. This algorithm exchanges a cycle of a given basis with a smaller one, if possible. Minimum length admissible cycles of can be found by a mapping to an auxiliary graph , and finding specific shortest paths in the [3]. The procedure of the algorithm is described in the following:
In order to control the independency of a new cycle, one can use the kernel of the cyclemember incidence matrix. The kernel of is required to find out the independency of a new cycle, with respect to the remaining cycles. Gaussian elimination can be used to compute the kernel. In each step, the kernel of similar matrices is required. There is an efficient method to construct the kernels with little effort [3]. Another important criterion is the length of the new cycle. In the following sections efficient approaches are presented to obtain the kernel of cyclemember incidence matrix and finding shortest admissible cycles. The present method is efficient, and makes a fast and economical generation of suboptimal cycle basis feasible, and a complete automated analysis of a semirigid frame can easily be performed. The cycle basis selection algorithm leads to the formation of minimal cycle bases for graphs, and for practical models, this algorithm leads to the formation of a suboptimal cycle basis. The formation of selfequilibrating stress systems on the element of the selected cycle basis leads to the formation of a highly sparse matrix, making an efficient flexibility analysis of semirigid frame structures feasible. References
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