Contribution of system elements to its static indeterminacy

Abstract

The article contains an overview of the main ideas regarding the new direction of construction mechanics, which considers the issue of survivability and is intensively developing. From the point of view of redundancy, as the ability of the system to provide alternative ways of transferring the load, which is one of the main possible strategies for designing reliability and survivability, the static-kinematic analysis of rod systems is also considered. The fundamental measure of the level of redundancy of rod systems is the degree of static uncertainty. But this numerical sign does not contain information about the role of each element of the system in forming the degree of static uncertainty. This role is performed by a specially constructed distributed static indeterminacy matrix (DSI-matrix), which contains comprehensive information about the contribution of system elements to its static and kinematic properties. Using the fundamental provisions of linear algebra, the properties of the matrix of coefficients of the system of linear equilibrium equations as an operator over the vector spaces of forces and displacements are analyzed. The mechanical content of the four fundamental subspaces associated with this matrix is indicated. This analysis determines the mathematical properties of the RSN matrix and its mechanical interpretation. Methods of forming the DSI-matrix are considered both for the case of a geometrically constant system and for the analysis of geometrically variable systems, when it is necessary to resort to singular decomposition of the matrix of equilibrium equations. All theoretical explanations are accompanied by illustrative examples, although the issue of numerical implementation of the considered methods of analysis is not considered. It is obvious that they deserve independent consideration. Information is provided on the possibility of using the DSI -matrix to assess the reliability and survivability of the structural complex and its use to analyze the sensitivity of the system to the inaccuracy of manufacturing elements.