Kinematic interaction transfer functions and models

The procedure proposed by Kim and Stewart (2003) was implemented with some modifications to estimate kinematic transfer functions and minimize noise effects from the experimental data. The procedure is summarized below:

1. Lateral and rocking transfer functions were estimated using Equation (2-8). For both transfer functions, Sxx denotes the smoothed power spectral density function of FFM. For a lateral transfer function, Sxy denotes the smoothed cross-power spectral density function of FFM and FML. For a rocking transfer function, Sxy denotes the smoothed cross-power spectral density function of FFM and FMR.

2. Coherence functions were computed for the transfer functions, estimated in Step 1, using Equation (2-9). In this equation, Syy denotes the smoothed power spectral density function of FML and FMR, for lateral and rocking transfer functions, respectively.

3. “High coherence” data points in frequencies with coherency values more than 0.8 were

identified.

4. Kim and Stewart (2003) recommended removing transfer function points around a flexible- base natural frequency of a structure in order to decrease the inertial interaction effect on the experimental kinematic transfer functions. Thus, data points around the flexible-base natural frequencies of SFSM with a range of 0.5 Hz were extracted from high coherence data points. It is worth mentioning that this step is not applicable for SFM and AFM experiments.

5. Non-linear regression analyses were conducted to fit the analytical transfer functions, shown in Figure 2-9, with the calculated experimental transfer functions.

6. The lateral and rocking transfer function matching would be used to estimate incoherence parameters, κL, and κR.

Figure 4-14 compares the lateral and rocking transfer functions in the three physical models, while they were subjected to the SCZ01, A motion. In this figure, experimental transfer functions are shown with variations of dash-lines, while analytical transfer functions are shown with solid lines. In addition, high coherence data points are depicted with markers. According to Figure 4-13, the lateral and rocking flexible-base natural frequencies of SFSM were both about 2.2 Hz, when the physical model was subjected to the SCZ01, A motion. As shown in Figure 4-14, the general trends in the lateral and rocking transfer functions of SFSM around the flexible-base natural frequencies were significantly different from those of the two other physical models. This difference is mainly due to the inertial interaction effect. Figure 4-14 also demonstrates that while amplitudes of the lateral transfer functions in SFSM at the high-frequency range were generally smaller than those of the two other physical models, amplitudes of the rocking transfer function were larger. This behavior can also be demonstrated by comparing the incoherence parameters of the three physical models. For example, the lateral incoherence parameter of SFSM under this motion (𝜅𝐿,𝑆𝐹𝑆𝑀 = 5.02) was considerably larger than the incoherence parameter of the two other physical models under the same motion (𝜅_{𝐿,𝑆𝐹𝑀} = 1.17; 𝜅_{𝐿,𝐴𝐹𝑀}= 1.28 ). Moreover, as shown in Figure 4-14(b), the rocking incoherence parameter of SFSM (𝜅_{𝑅,𝑆𝐹𝑆𝑀} = 6.88) is also considerably larger than the incoherence parameter of the two other physical models (𝜅𝑅,𝑆𝐹𝑀 = 0.81; 𝜅𝑅,𝐴𝐹𝑀 = 0.53 ). It

increases, the amplitude of the lateral transfer function decreases starting from 1 at 0 Hz frequency. However, the amplitude of the rocking transfer function starts from zero, increases to a peak value, and then decreases, as shown in Figure 4-1. Moreover, as the incoherence parameter increases, the amplitude of the lateral transfer function decreases for all frequencies, however, the frequency corresponding to the peak value in the rocking transfer function decreases. Therefore, it is possible that the amplitude of the rocking transfer function with a larger incoherence parameter be smaller in some frequencies than the amplitude of the rocking transfer function with a smaller incoherence parameter.

Furthermore, Figure 4-14 illustrates that even by removing data points around the flexile-base natural frequencies, the incoherence parameters for SFSM are significantly different from those of the two other physical models. By comparing transfer functions of SFM and AFM, it can be seen that although the mass of the foundation in AFM, shown in Table 4-1, was significantly smaller than those of SFM (529% decrease), amplitudes of the lateral and rocking transfer functions were relatively close. This observation shows that the mass of the foundation had small effects on kinematic transfer functions. The figure also demonstrates that while the amplitudes of the lateral transfer function in SFM are smaller or larger than those of AFM depending on the frequency ranges, the amplitudes of the rocking transfer function in SFM are slightly larger than those of AFM in most frequencies. This behavior is intuitively expected since the effect of foundation mass on the lateral transfer function is less than the effects on the rocking transfer function.

Figure 4-14. Comparison of experimental and analytical foundation transfer functions of the three physical models, when excited by the SCZ01,A motion: (a) transfer function of the lateral component of FM to FFM; (b)

transfer function of the rocking component of FM to FFM. (Theoretical matches are shown with bold lines.)

Trends, shown in Figure 4-14, were also observed when the physical models were excited with other motions. As another example, the lateral and rocking transfer functions of the three physical models, when subjected to the JOS01 motion, are compared in Figure 4-15. By comparing Figure 4-14 and Figure 4-15 it can be seen that while the lateral incoherence parameter for SFM is smaller than those for AFM in Figure 4-14, it is larger in Figure 4-15. This comparison further shows that the effect of the foundation mass is small on the lateral transfer function. It is worth noting that the transfer functions, shown in Figure 4-14 and Figure 4-15, were repeated well in their conjugate repeatability check experiments.

Figure 4-15. Comparison of experimental and analytical foundation transfer functions of the three physical models, when excited by the JOS01 motion: (a) transfer function of the lateral component of FM to FFM; (b)

transfer function of the rocking component of FM to FFM. (Theoretical matches are shown with bold lines.)

Figure 4-16 compares the lateral and rocking incoherence parameters of SFSM experiments with those of SFM and AFM in all the tests. According to Figure 4-16(a) and (c), κL values in SFSM for all motions are larger than those in SFM and AFM. Thus, amplitudes of the lateral transfer functions of SFSM during all experiments were generally smaller than those of SFM and AFM. A practical conclusion of this observation is that using a semi-empirical model for the incoherence parameter, developed based on measurement of foundation motions, may lead to over-reduction of the lateral component of the foundation input motion. Furthermore, Figure 4-16(b) and (d) demonstrate that κR values in SFSM for all motions were larger than those in SFM and AFM. Therefore, amplitudes of the rocking transfer functions of SFSM are generally larger in most

frequencies than those of SFM and AFM, and a practical conclusion is that a semi-empirical model may lead to over introduction of the rocking component of the foundation input motion.

Figure 4-16. Comparison of the lateral and rocking incoherence parameters in SFSM experiments with those in SFM and AFM experiments.

Figure 4-17 compares κL and κR values for SFM with those from AFM experiments. According to Figure 4-17(a), while for low-intensity BMs, applied in Test-ASFM and Test-AAFM, κL values show relatively good agreements between the two experiments; however, when the physical models were excited with high-intensity BMs in Test-BSFM and Test-BAFM, κL decreases more in SFM experiments compared to AFM experiments. The bearing pressure of AFM is smaller than those of SFM; therefore, friction force, resisting against slippage of the physical model relative to the soil, is weaker than those of SFM. Consequently, slippage of the physical models could be different, specifically for high-intensity motions. It can be concluded that the effect of the foundation mass on the lateral transfer function is minimal, for low-intensity seismic motions. Figure 4-17(b) illustrates that while κR values were close in the two sets of the experiments, κR values in SFM experiments were slightly larger than those of AFM experiments. As discussed, it

those of AFM experiments. This observation is intuitively expected because the foundation mass may increase rocking.

Figure 4-17. Comparison of the lateral and rocking incoherence parameter in SFM experiments with those in AFM experiments.