24 Finite Element Model Updating Accounting for Modeling Uncertainty 213 Fig. 24.1 Framework for parameter estimation of nonlinear FE models using the UKF 24.2.2 Dual Approach Accounting for Modeling Uncertainty Compensation effects arise in the parameter-only estimation approach when moderate to high levels of modeling uncertainty exists, and biased estimates (reaching even unphysical values) of the unknown model parameters are obtained, implying large estimation errors for unobserved responses [11]. To alleviate this issue, a dual adaptive filtering approach is presented here. The aims is to estimateθand at the same time estimate the diagonal entries of Rk +1, i.e., the variances of the prediction error vector. To this end, a covariance-matching technique is employed, which goal is to make the innovations (vk+1|k) compatible with their expected covariance matrix [16]. Here, it is assumed that Rk +1 is diagonal and expressed as Rk+1 =diag (rk+1), i.e., that the simulation errors of the different response measurements are uncorrelated. Then, an UKF is used as master filter (MF) to estimate θ and a linear KF is employed as slave filter to estimate rk +1. Figure 24.2 shows the proposed dual filtering approach, where the highlighted portion corresponds to the additional calculations required to incorporate the estimation of rk +1. T and Uare the timeinvariant covariance matrices of the process and measurement noises, respectively, of the state-space model corresponding to the SF, both assumed Gaussian white with zero mean. Further details about the dual adaptive filtering approach can be found in [19]. 24.3 Validation Study The exterior north-south frame of a three-story steel moment-resisting frame building known as SAC-LA3 [20] under seismic base excitation is used as validation example. Columns and beams are made of A572 and A36 steel, respectively. Geometry of the frame is shown in Fig. 24.3a. A FE model is developed in the software OpenSees [21] using nonlinear force-based fiber-section beam-column elements. Each column and beam member is modeled with a single element and seven and six integration points, respectively. Rayleigh damping with mass- and tangent stiffness-proportional coefficients based on a critical damping ratio of 2% for the first two initial modes (T1 =1.06 [s] and T2 =0.35 [s]) is assumed. Element cross-
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