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Nowadays, strain and displacement can be measured using techniques such as electronic speckle pattern interferometry and digital image correlation. However, usually, only some part of the domain of interest is accessible for measurement devices. In order to assess also inaccessible areas of the structure, a fundamental task for structural health monitoring is the inference of full-field displacements from sparse measurements. Another important task is the inference of mechanical stress from displacement or strain data. In computational mechanics, the link between such data and stress is established via constitutive models. A well-established approach for inferring full displacement and stress fields from possibly sparse data is to calibrate the parameter of a given constitutive model using a Bayesian update. After calibration, a (stochastic) forward simulation is conducted with the identified model parameters to resolve physical fields in regions that were not accessible to the measurement device. A shortcoming of model calibration is that the model is deemed to best represent reality, which is only sometimes the case, especially in the context of the aging of structures and materials. While this issue is often addressed with repeated model calibration, a different approach is followed in the recently proposed statistical Finite Element Method (statFEM). Instead of using Bayes' theorem to update model parameters, the displacement is chosen as the stochastic prior and updated to fit the measurement data more closely. For this purpose, the statFEM framework introduces a so-called model-reality mismatch, parametrized by only three hyperparameters. This makes the inference of full-field data computationally efficient in an online stage: If the stochastic prior can be computed offline, solving the underlying partial differential equation (PDE) online is unnecessary. Compared to solving a PDE, identifying only three hyperparameters and conditioning the state on the sensor data requires much fewer computational resources. Computational efficiency is an essential requirement for online applications. It is noted that conditioning a model-based prior field on sensor data represents a variant of physics-based regression. This paper presents two contributions to the existing statFEM approach: First, we use a non-intrusive polynomial chaos method to compute the prior, enabling the use of complex mechanical models in deterministic formulations. Second, we examine the influence of prior material models (linear elastic and St.Venant Kirchhoff material with uncertain Young's modulus) on the updated solution. We present statFEM results for 1D and 2D examples, while an extension to 3D is straightforward. [ABSTRACT FROM AUTHOR]

• A novel Bayesian formulation for the uncertainty quantification in modal identification was developed. • The proposed formulation is tailor-made for modal identification using seismic data. • The proposed formulation is computational efficient, and it is applicable under both well-separated and closely-spaced modes situations. • The proposed formulation was verified using data collected from shaking table tests and field tests on a real building. Modal identification is an important step to evaluate the basic modal properties of structures using measured data. In this process, the existence of modeling error and measurement noise will inevitably lead to uncertainty in modal identification. A Bayesian framework was established to determine the optimal values of modal parameters efficiently based on the structural response under earthquake excitations. In this paper, in the same framework, a new method is presented to determine the analytical formulation for carrying out uncertainty evaluation of modal parameters utilizing seismic structural responses. Based on Bayes' Theorem, the covariance matrix can be calculated based on the Hessian matrix determined using the negative log-likelihood function (NLLF). In this work, for determining the Hessian matrix, a series of formulations were derived analytically. Simulated data of a six-story building were generated to investigate the new formulations. The noise effects on the posterior uncertainty were investigated. After the verification, applications were carried out using the data in a shaking table test model under laboratory conditions and a real building from a field test. The modal properties and their uncertainty obtained by the proposed method were studied under different earthquake excitations. [ABSTRACT FROM AUTHOR]