COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2005; 21:619–629 Published online 3 June 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.778 Subminimal cycle basis of a graph for ecient force method of frame analysis A. Kaveh ∗; † and H. Moez Department of Civil Engineering; Iran University of Science and Technology; Narmak; Tehran 16; Iran SUMMARY A new method is presented for the formation of subminimal cycle bases of graphs corresponding to sparse exibility matrices. In this approach, a contraction method is used, leading to the reduction of the size of the graph at each step. This reduction facilitates the selection of the smallest cycle at subsequent steps. Copyright ? 2005 John Wiley & Sons, Ltd. KEY WORDS: cycle basis; graph; contraction; rst Betti number; frame; exibility matrix; force method 1. INTRODUCTION Consider a frame structure S with M (S ) members and N (S ) nodes, which is (S ) times statically indeterminate. Select (S ) independent unknown forces as redundants. These un- known forces can be chosen from external reactions and= or internal forces of the structure. Denote these redundants by q = {q 1 ;q 2 ;:::;q (S ) } (1) In order to obtain a statically determinate structure, the constraints corresponding to redundants should be removed. Such a structure is known as the basic (primary or released) structure of S . Rigidity of this basic structure is assumed to hold. Consider the external joint loads as p = {p 1 ;p 2 ;:::;p n } (2) where n is the number of components for the applied nodal forces. The stress resultant distribution due to the given load p for a general linear analysis by the force method can be ∗ Correspondence to: A. Kaveh, Department of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran 16, Iran. † E-mail: alikaveh@iust.ac.ir Contract=grant sponsor: Iran Section of TWAS Received 23 October 2004 Revised 24 February 2005 Copyright ? 2005 John Wiley & Sons, Ltd. Accepted 21 March 2005