24 Finite Element Model Updating Accounting for Modeling Uncertainty 217 0 2 4 6 8 10 12 Time [s] -0.5 -0.25 0 0.25 0.5 Acceleration [g] Los Gatos 0°- 1989 Loma Prieta EQ. Fig. 24.4 Ground acceleration time history recorded at the Los Gatos station during the 1989 Loma Prieta earthquake (parameter-only and dual) are employed to update different nonlinear FE models belonging to model classes Mj (j =1, 2, . . . , 28). FE Model Updating Results Dynamic Response Simulation The horizontal absolute acceleration responses of the three levels of the frame are considered as measured responses and denoted by y1, y2, and y3 in Fig. 24.3a. The true response of the structure (ytrue) is numerically simulated using model class M0 with the following vector of true model parameters θ true =[200 GPa,345 MPa, 20, 0.08, 200 GPa,250 MPa,18,0.05]T and ϕtrue = ⎡ ⎢⎢ ⎢⎢ ⎣ 0 9.14 18.28 27.43 3.96 7.92 11.88 T [m] 64.94 129.88 129.88 194.81 64.94 129.88 129.88 194.81 69 138 138 207 T kN s 2 m 26.01 25.99 23.09 T kN m 0.1782 0.00167 T ⎤ ⎥⎥ ⎥⎥ ⎦ T . In the estimation phase, each component of the true response vector of the structure (ytrue) is polluted with an independent white Gaussian noise with zero-mean and 0.5%g root-mean-square (RMS), to generate the measured response, y. Thus, the measurement noise exact covariance matrix is 0.24 ×10−2 I 3 [(m/s 2)2], where I i denotes the i ×i identity matrix. Estimation of Unknown Model Parameters The model parameters characterizing the steel material constitutive model are considered unknown and to be estimated. Random initial values for the unknown model parameters are considered and then both estimation approaches are applied to calibrate the FE models (belonging to different model classes Mj, j =1, . . . , 28) using the measured response (y). The unknown model parameters are initially assumed statistically uncorrelated, and thus the initial estimate of their covariance matrix, Pθθ 0|0 , is diagonal, with entries computed as p× θ i 0|0 2 , where i =1, . . . , nθ =8 and p denotes the initial coefficient of variation of the unknown model parameters. Then, the initial estimates of the mean vector and covariance matrix of the unknown model parameters are taken as θ0|0 = 1.3 Ecol s , 0.8 f col y , 0.7 Rcol 0 , 1.25 b col, 0.8 Ebeam s , 0.75 f beam y , 1.3 Rbeam 0 , 1.4 b beam T =[260 GPa, 276MPa, 14, 0.1,160 GPa, 187.5MPa, 23.4,0.07]T Pθθ 0|0 = diag 0.2· θ0|0 2 ∈R8×8 As proposed in [17, 18], a diagonal process noise covariance matrix Q = diag q × θ i 0|0 2 with q =1 ×10−5 is assumed. In the parameter-only estimation approach, the simulation error covariance matrix is assumed fixed and equal to R=0.87×10−3 I 3 [(m/s 2)2]. It is noteworthy that excellent estimation results were obtained in previous studies when using
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