Application of a maximum entropy method to estimate the probability density function of nonlinear or chaotic behavior in structural health monitoring data

Bridges and other civil structures can exhibit nonlinear and/or chaotic behavior under ambient traffic or wind loadings. The probability density function (pdf) of the observed structural responses thus plays an important role for long-term structural health monitoring, LRFR and fatigue life analysis. However, the actual pdf of such structural response data often has a very complicated shape due to its fractal nature. Various conventional methods to approximate it can often lead to biased estimates. This paper presents recent research progress at the Turner-Fairbank Highway Research Center of the FHWA in applying a novel probabilistic scaling scheme for enhanced maximum entropy evaluation to find the most unbiased pdf. The maximum entropy method is applied with a fractal interpolation formulation based on contraction mappings through an iterated function system (IFS). Based on a fractal dimension determined from the entire response data set by an algorithm involving the information dimension, a characteristic uncertainty parameter, called the probabilistic scaling factor, can be introduced. This allows significantly enhanced maximum entropy evaluation through the added inferences about the fine scale fluctuations in the response data. Case studies using the dynamic response data sets collected from a real world bridge (Commodore Barry Bridge, PA) and from the simulation of a classical nonlinear chaotic system (the Lorenz system) are presented in this paper. The results illustrate the advantages of the probabilistic scaling method over conventional approaches for finding the unbiased pdf especially in the critical tail region that contains the larger structural responses.