Mechanical equivalent of logical inference: application to structural health monitoring problems

Structural health monitoring requires engineers to understand the state of a structure from its observed response. When this information is uncertain, Bayesian probability theory provides a consistent framework for making inferences. However, structural engineers are often unenthusiastic about Bayesian logic and prefer to make inference using heuristics. Here we propose a quantitative method for logical inference based on a formal analogy between linear elastic mechanics and Bayesian inference with linear Gaussian variables. We start by discussing the estimation of a single parameter, by assuming that all of the uncertain quantities have a Gaussian distribution and that the relationship between the information and the parameter is linear. With these assumptions, the analogy is stated as follows: The expected value of the parameter corresponds to the position of a bar with one degree of freedom; uncertain information on the parameter is modelled as linear elastic springs in series or parallel. If we want to extend the analogy to N parameters, we prove that we must simply express the potential energy of the mechanical system, with N degrees of freedom, associated to the inference problem in question. We apply the mechanical equivalent to a real-life case study: The elongation of a cable belonging to Adige Bridge, a cable-stayed bridge North ofTrento, Italy, in the case of two and three parameters to estimate.