Sparse Bayesian learning with Gibbs sampling for structural health monitoring with noisy incomplete modal data

Most hidden damage that occurs in civil structures is in localized areas. In this paper, this information is exploited in a sparse Bayesian learning framework for inferring localized stiffness reductions as a proxy for structural damage that uses noisy incomplete modal data from before and after possible damage. The methodology proposes a new sparse Bayesian model that induces spatial sparseness in representations of the inferred stiffness reductions. To obtain not only the most plausible model of sparse stiffness reductions within a specified class of models, but also a quantification of its uncertainty, the method uses Gibbs sampling to generate samples from the posterior distribution for the structural stiffness parameters, system modal parameters and eigenequation-error precision parameter. The approach has five important benefits: (1) no matching of model and experimental modes is needed; (2) solving the eigenvalue problem of a structural model is not required; (3) all the uncertain parameters are sampled or estimated conditional on the modal data, and, therefore, no user-intervention is required; (4) the effective dimension for the Gibbs sampling only depends on the small number of parameter groups that are used for constructing the conditional PDFs for drawing samples; and (5) the inferred stiffness reductions are spatially sparse in a way that is consistent with a Bayesian Ockham's razor. A three-dimensional braced-frame model from the Phase II benchmark problem sponsored by the IASE-ASCE Task Group on Structural Health Monitoring is analyzed using the proposed method. The results show that the proposed approach reduces the occurrence of false and missed damage detections in the presence of modeling errors.