The equation of motion for the base-isolated structure can be written as

where

and u_i is the relative displacement of the ith floor to the ground; x_i is the relative displacement of the ith floor to the base slab; x_b is the relative displacement of the base floor to the ground; R_b is the restoring force measured from the base isolator; is the ground acceleration; and [C_s] is the damping matrix of the superstructure. Introducing Eq. 5 into Eq. 3 and Eq. 4, the following equation can be obtained in terms of {u} and x_b as

where {1} is a vector with all elements equal to one.

3.3. Experimental set-up for pseudodynamic tests

The pseudodynamic test was performed using a test apparatus which has two hydraulic actuators: one in the horizontal direction and the other in the vertical direction as shown in Fig. 12. Vertical load and horizontal displacement imposed on a pair of base isolators were controlled simultaneously, whereas the displacements and restoring forces of the deformed base isolators were measured as feedback signals.

The restoring force produced from the specimens (two base isolators) during the pseudodynamic test can be measured through both load cells 1 and 2. Load cell 1 is the one built-in inside of the horizontal actuator which is located above the specimens, and load cell 2 is attached to the horizontal support below the specimens. The hysteretic relationships between the restoring force and the displacement obtained using the data from two load cells are compared for sinusoidal loads with a loading rate of 0.5 Hz in Fig. 13, and similar results are shown for a different loading rate of 0.05 Hz in Fig. 14. From the figures, it can be clearly seen that error has been introduced into the data from load cell 1. The error source consists of the inertia force from the heavy loading apparatus above the specimens and the friction force from the rollers at both ends of the loading beam. Therefore, in this study, the data from load cell 2 were used as the restoring force from the specimens. The stiffness from the skeleton curves are found to be 1.05 and 1.09 kN/cm for two different loading rates. The general shapes of the hystereses and the stiffnesses indicate that the effect of the loading rate is not significant for the cases with 0.5 and 0.05 Hz.

3.4. Comparison between pseudodynamic test and shaking table test results

In order to verify the accuracy of the substructuring pseudodynamic test method for base-isolated structures, the pseudodynamic test results with the quarter-scale base isolators are compared with the shaking table test results. The results in Fig. 15 and Fig. 16 show that the acceleration responses for the El Centro earthquake obtained from two different tests agree reasonably well. The small discrepancies may be caused by the inaccuracy in the analytical model for the superstructure as well as the assumed viscous damping effect of the base isolator in the pseudodynamic test.

Generally, the viscous damping of the base isolator has been ignored because the seismic input energy transmitted to the base isolator would mainly be dissipated by the hysteretic damping [9]. During pseudodynamic test, the hysteretic damping effect can be automatically included through the hysteretic relationship between the displacement and the reaction force of the base isolator. On the other hand, viscous damping, which is dependent on the velocity, cannot be considered, unless the effect is added in the on-line numerical integration procedure. Fig. 17 presents the experimental results of the base floor with three different viscous damping ratios (i.e. 1, 3, and 6%) used in the numerical integration. It can be seen that the viscous damping of the base isolator should not be ignored because it played an important role in dissipating the seismic energy transmitted to the base-isolated structure. Based on the results, the pseudodynamic tests were carried out by taking the viscous damping ratio of the base isolator as 6%.

3.5. Comparison between quarter-scale and prototype structures

In order to examine the effect of scaling for the base isolator, the pseudodynamic test is also conducted using the prototype base isolator, and the results for the El Centro earthquake are compared with those using the quarter-scale base isolator in Fig. 18, Fig. 19 and Fig. 20. The scale factors for the displacement and the restoring force are L and L², as shown in Table 1, indicating that responses of the scaled model are less than those of the prototype structure. Some physical quantities, such as acceleration and strain, remain the same even after scaling. The hysteresis loops of two cases are compared in Fig. 18. It was reported that the scaled base isolators usually exhibit a smaller shear modulus than a full-scale model because of the scaling effect during the curing process. Fig. 18 shows that the general shapes of two hystereses agree reasonably well. The stiffnesses from two skeleton curves for the quarter- and full-scale specimens are 1.05 and 4.72 kN/cm, respectively. After considering the scaling factor for stiffness which is 4 as in Table 1, the effective stiffness of the quarter-scale model is 4.20 kN/cm, which is approximately 88% of the level of the stiffness of the prototype. The horizontal deformations of the base isolator and the accelerations of the base floor are compared in Fig. 19 and Fig. 20. It can be seen that the maximum horizontal displacement of the scaled model is greater that of the prototype structure by a factor of 1.1. On the other hand, the maximum acceleration is smaller that of the prototype structure by a factor of 0.9. The above results indicate that the scaling effect of the quarter-scale model is not too grave for the response prediction of the base-isolated structure.

4. Numerical simulation and comparison

Numerical simulations were also carried out to reproduce the results of the shaking table test for the base-isolated structure. The hysteretic behavior of the base isolator is modeled as a bi-linear curve based on the force–displacement relationship obtained from the preliminary quasi-static test on the base isolator. Fig. 21(a) shows the force–displacement relationship of the quarter-scale base isolator tested quasi-statically for a shear strain range of 20–120%, with a constant vertical load of 14 kN. Fig. 21(b) shows the bilinear curve obtained from the test results using a simple error minimization procedure between two curves. The same analytical model of the superstructure as used in the pseudodynamic test is used. The roof acceleration time history obtained from the numerical simulation is compared with the measurement record from the shaking table test in Fig. 22. The comparison between the two time histories is found to be not superb but acceptable. The maximum floor accelerations obtained from the shaking table test, the pseudodynamic test, and the numerical simulation are compared in Table 6. The maximum responses from the numerical analysis are found to be in reasonable agreement with two sets of the test results. The differences among the results are within 13%.

5. Conclusions

A series of the shaking table and pseudodynamic tests were conducted on a three-storey steel structure supported by base isolators subjected to various earthquake loadings. Numerical simulations were also carried out to reproduce the test results. Based on the test results, the following conclusions can be drawn:

1. Base isolation is a very effective way to reduce the seismic response of a structure, particularly floor acceleration, base shear, and overturning moment at rock or stiff-soil sites. However, at soft-soil sites, it is less effective and horizontal displacement may be severely increased.

2. The pseudodynamic test incorporating a substructuring technique is very effective for predicting the dynamic response of the base-isolated structure.

3. The viscous damping effect of the base isolator shall be considered in the pseudodynamic test in addition to hysteretic damping, because the former also plays an important role in dissipating the seismic energy transmitted to the base-isolated structure. In the present case, a viscous damping ratio of 6% is found to be a reasonable value.

4. The stiffness of the scaled base isolator trends underestimated due to the scaling effect during the curing process. The displacements of the superstructure with the quarter-scale base isolators are overestimated by approximately 10%, whereas the accelerations are underestimated by 10%.

5. Numerical analysis by employing an approximate bi-linear hysteretic model for the base isolator can reasonably simulate the earthquake responses of the base isolation system, particularly for the maximum responses.