Mechanical equivalent of Bayesian inference from monitoring data

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 inference. However, structural engineers are often unenthusiastic about Bayesian logic and prefer to make inference using heuristics. Herein we propose a quantitative method for logical inference based on a formal analogy between linear elastic mechanics and Bayesian inference with Gaussian variables. We start by discussing the estimation of a single parameter under the assumption that all of the uncertain quantities have a Gaussian distribution and that the relationship between the observations and the parameter is linear. With these assumptions, the analogy is stated as follows: The expected value of the considered parameter corresponds to the position of a bar with one degree of freedom and uncertain observations of the parameter are modelled as linear elastic springs placed in series or parallel. If we want to extend the analogy to multiple parameters, we simply have to express the potential energy of the mechanical system associated to the inference problem. The expected value of the parameters is then calculated by minimizing that potential energy. We conclude our contribution by presenting the application of mechanical equivalent to a real-life case study in which we seek the elongation trend of a cable belonging to Adige Bridge, a cable-stayed bridge located North of Trento, Italy.