## 3 Parameters of Equivalent Force Control HS Method  
### Parameter of Numerical Substructure Model  
The EFC method was used for HS [47] with an implicit integration algorithm, during which the mathematical iteration procedure was improved by using a force feedback control. Numerical and experimental results of simple model verified that, as long as the equivalent force controller was properly designed, proposed EFC method was sufficient. This section introduces the detailed test procedure for the aforementioned sophisticated structures with the elastic non-linear numerical substructure. Considering the asymmetric stiffness in the tension and compression direction caused by the asymmetric structural arrangement of the test model, an asymmetric restoring force model was used in this paper, as shown in Figure 6. In this model, an inflection point exists at the location of the zero displacement for the asymmetric restoring force model of the numerical substructure. Traditionally, the subdivision technique should be used to reduce the computational error at the location of the displacement inflection point. However, the measured error will be inevitably involved because of the increase of the loading step caused by the subdivision technique for the EFC-based HS test. Furthermore, the structural responses were calculated using the feedback from the previous integration steps to gradually approach the exact response at the current step by using the EFC-based HS test. It is difficult to trace back to the previous finished test steps. To this end, no subdivision of integral time interval was taken into account at the location of the displacement inflection point. To make sure the subdivision influence upon the test results, the numerical simulation comparisons between the subdivision and non-subdivision cases were done. The model parameters of the numerical simulation were taken from the test measurements, as listed in Table 1. The structural damping was \(\mathbf{C}=0.05\)\(\mathbf{K}\); the restoring force model of each story can be shown in Figure 6. The flowchart of both the subdivision and non-subdivision method is illustrated in Figure 7. The difference between subdivision and non-subdivision method is the step after checking the inflection point of displacement. For the subdivision method, once variations in the inter-story shear force are observed, the subdivision of integral time interval is processed, meaning that the time integration interval is subdivided one step prior to the displacement inflection point. At the same time, the earthquake record was subdivided accordingly. The stiffness at current step is updated and used in the calculations for the remainder of this time interval. For the non-subdivision method, a re-assembly for the stiffness matrices is following in the next step. Figure 6: Restoring force model of numerical substructure. Figures 8 and 9 show the hysteric curves on the first and second floors under a peak ground acceleration of 110 gal using subdivision and non-subdivision techniques, respectively (other floors are similar to these two floors). Note that both results agree well with the restoring force model as shown in Figure 6. This agreement suggests that the above-mentioned techniques and the corresponding analysis procedures are reliable. Figure 10 shows the comparisons of the displacement time history curve between the subdivided and non-subdivided integral steps for the first story (the case of the other story is similar and not shown in this research). In order to observe the relationship more clearly, Figure 11 shows the results from 4 s to 8 s. From the Figures 10 and 11, we can see that both the subdivided and non-subdivided results nearly coincide. Figure 12 shows the time histories of the equivalent force (EF) command and feedback of the first story using the strategy of non-subdivided integral steps (the similar trend in the other story). It is easy to see that the EF feedback tracks the EF command well.