Uncertainty quantification in the damage assessment of a seven-story RC building slice through Bayesian FE model updating

Abstract

Bayesian finite element (FE) model updating is used for uncertainty quantification (UQ) purposes in the vibration-based damage assessment of a seven-story reinforced concrete building slice. This structure was built and tested at full scale on the USCD-NEES shake table: a progressive damage pattern was induced by subjecting the structure to a number of historical earthquake records. At each damage stage, modal characteristics (i.e. natural frequencies and mode shapes) were determined through vibration testing, these data are used in the Bayesian FE model updating schemes. In order to analyze the results of the Bayesian scheme and gain insight into what information is contained in the data, a comprehensive uncertainty and resolution analysis performed. It is shown that the Bayesian UQ approach and subsequent resolution analysis are effective in assessing uncertainty in FE model updating. Furthermore, it is demonstrated that insight into the Bayesian FE model updating approach provides a very natural way to regularize its often ill-posed deterministic counterpart. 1 THE SEVEN-STORY TEST STRUCTURE The seven-story test structure (Moaveni et al. 2007, Moaveni et al. 2009, Moaveni et al. 2010, Moaveni et al. 2011) represents a slice of a prototype reinforced concrete mid-rise residential building and is shown in Figure 1a. The structure consists of two perpendicular walls (i.e. a main wall and a back wall for transverse stability), seven concrete floor slabs, an auxiliary post-tensioned column for torsional stability, and four gravity columns to transfer the weight of the slabs to the ground level (Figure 1b). A progressive damage pattern was induced in the test structure through four historical earthquake records, leading to 5 damage states S0 to S4 (Table 1). After each seismic excitation sequence, vibration tests were performed to obtain modal characteristics of the structure. The experimentally identified natural frequencies of the first three longitudinal modes are listed in Table 2 for each damage state; it is clear that, as expected, the values decrease as the damage increases. Figure 2 shows the corresponding employed mode shapes obtained at damage state S0 using 28 sensors located along the main wall and on the floor slabs. For the damage identification, only the mode shape measurements obtained using 14 sensors located along the main wall are employed. In the following, experimental eigenvalues and mode shapes are denoted as λr = (2πfexp,r) 2 and φr ∈ Ro , respectively, where No represents the number of observed degrees of freedom. Both experimental data are collected in the vector d = {. . . , λr, . . . , φTr , . . .} . These modal data are used in five consecutive damage analyses: for each damage state, Bayesian FE Table 1: The five damage states and corresponding imposed historical earthquake records. Damage Earthquake record state Earthquake Comp. Recorded at M S0 None S1 1971 S. Fernando long. Van Nuys 6.6 S2 1971 S. Fernando trans. Van Nuys 6.6 S3 1994 Northridge long. Oxnard Blvd 6.7 S4 1994 Northridge 360◦ Oxnard Blvd 6.7 (a) (b) Figure 1: (a) Seven-story test structure and (b) elevation view. Table 2: Experimentally identified natural frequencies and damping ratios for the five damage states. Damage fexp [Hz] state Mode 1 Mode 2 Mode 3 S0 1.91 10.51 24.51 S1 1.88 10.21 24.31 S2 1.67 10.16 22.60 S3 1.44 9.23 21.82 S4 1.02 5.67 15.10 model updating is applied to quantify the uncertainties on the damage identification results. To this end, a detailed 3D linear elastic FE model was constructed with 322 shell and truss elements and Nd = 2418 degrees of freedom (Figure 3a), using the generalpurpose FE analysis program FEDEASLab (Filippou and Constantinides 2004). In order to model the damage, the structure is divided into 10 substructures, each consisting of part of the main wall (Figure 3b). It is assumed that each substructure has a uniform effective stiffness (Young’s modulus); these stiffness values will be the updating parameters θM in the Bayesian updating schemes. Initial values θ M of the 10 Young’s moduli are obtained trough concrete cylinder testing at various heights along the building (Moaveni et al. 2010): θ M = [ 24.5 24.5 26.0 26.0 34.8 34.8 30.2 28.9 32.1 33.5 ] T GPa (1) The FE model allows for the computation of the modal data as a function of the model parameters θM, where the modal data consist of Nm eigenvalues λr and corresponding mode shapes φr ∈ Rd , which are the solutions of the (undamped) eigenvalue equation K(θM)Φ = MΦΛ, where K(θM) is the FE model stiffness matrix and M the mass matrix. Φ collects the eigenvectors φr corresponding to the eigenvalues λr located on the diagonal of Λ. In the Bayesian updating scheme, these computed modes are paired to Figure 2: First three longitudinal mode shapes obtained at damage state S0. (a) (b) Figure 3: (a) FE model of the seven-story test structure and (b) definition of the substructures along the main wall. the experimentally identified modes by means of the Modal Assurance Criterion (or MAC); furthermore, a least squares scaling factor is introduced in order to ensure that paired modes are scaled equally. The set of computed data for a certain model parameter set θM is referred to as GM(θM) in the following. 2 BAYESIAN FE MODEL UPDATING 2.1 Model classes and uncertainties In general terms, a model MM(θM) belonging to the model classMM provides a mapping from the parameters θM to an output vector GM(θM) ∈ R through the transfer operatorGM. In the ideal case, the model outputGM(θM) corresponds perfectly to the true system output d, i.e. GM(θM) = d. This is the main starting point for deterministic parameter identification, where the objective is to determine the model parameters θM for a given set of observed system outputs d. However, the equality GM(θM) = d is only valid when it is assumed that the underlying fundamental physics of the system are fully known. This is of course never the case, as no model is capable of perfectly representing the behavior of the true physical system. A modeling error ηG is therefore always present, and can be described as the discrepancy between the model predictions GM(θM) and the true system output d, i.e. ηG = d−GM(θM). As the true system output has to be measured and processed experimentally, the data d are always subject to measurement error, resulting in a discrepancy between the true system output d and the actually observed data d. This difference is defined as the measurement error ηD = d−d. Eliminating the unknown true system output d from the error equations and collecting both errors on the right hand side of the equation yields: d−GM(θM) = ηG + ηD = η (2) The sum of both errors is equal to the total observed prediction error η, defined as the difference between the observed and predicted response quantities. This expression serves as a starting point for the Bayesian uncertainty quantification method. 2.2 Bayesian inference methodology The general principle behind Bayesian model updating is that the structural model parameters θM ∈ RM that parametrize model class MM are modeled as random variables, i.e. probability density functions (PDFs) are assigned to these parameters, which are then updated in the inference scheme based on the available information. Measurement and modeling uncertainty are taken into account by modeling the respective errors as random variables as well: PDFs are assigned to ηG and ηD, which are parametrized by parameters θG ∈ RG and θD ∈ RD . These parameters are added to the structural model parameters θM to form the general model parameter set θ = {θM,θG,θD} ∈ R . This in fact corresponds to adding two probabilistic model classes to the structural model class MM to form a joint model class M =MM ×MG ×MD, parametrized by θ. To express the updated joint PDF of the unknown parameters θ, given some observations d and a certain joint model classM, Bayes’ theorem is used: p(θ | d,M) = c p(d | θ,M) p(θ | M) (3) where p(θ|d,M) is the updated or posterior joint PDF of the model parameters given the measured data d and the assumed model class M; c is a normalizing constant (independent of θ) that ensures that the posterior PDF integrates to one; p(d|θ,M) is the PDF of the observed data given the parameters θ; and p(θ|M) is the initial or prior joint PDF of the parameters. In the following, the explicit dependence on the model classM is omitted in order to simplify the notations.