Assessment of robustness of hinged-bar systems

Abstract

Typically in structural design, foreseeable loads are assumed in a dimensioning exercise. Structures can, however, be exposed to largely unforeseeable events such as intense environmental phenomena, accidents, malicious acts, and planning or execution errors. This circumstance determines the interest in the problem of structural robustness, which has been the subject of many recent works.
 This paper focuses on methods for assessing the robustness of hinged bar systems, considering truss structures as an example. They are the simplest in terms of computation, but make it possible to fully illustrate the proposed approach.
 First, the differences between progressive collapse (description of the process) and the disproportionate propagation of local failures (description of the state) are analyzed. The generalizing nature of the concept of robustness and its differences from the concept of invulnerability are pointed out.
 The paper considers the problem of measuring robustness. The known quantitative estimates of robustness are analyzed focusing on estimates that are invariant with respect to the stress state, as more general ones. The paper considers estimates that use such properties of the stiffness matrix as the condition number, or based on a comparison of the determinants of the original and changed stiffness matrices. It is pointed out that the degree of static indeterminacy can serve only as a necessary, but insufficient measure of robustness.
 The paper considers a well-known method of robustness assessment using a redundancy matrix determined by the forces that must be applied to assemble the system from elements with the length different from the design one. This method is opposed to the use of a projection matrix, the main diagonal elements of which indicate the degree of participation of the bars in ensuring robustness. The main properties of the idempotent projection matrix are considered. The paper illustrates the possibility of recalculating the projection matrix for the changed system with the help of the Jordan elimination step. A simple example demonstrates assembling and changing the projection matrix.
 In addition to the failure of the bar, the case of its damage (partial failure) is also considered, it is shown how it affects the change in the projector and the redistribution of internal forces.