Effects of Isolation Period Difference and Beam-Column Stiffness Ratio on the Dynamic Response of Reinforced Concrete Buildings

As suggested by Roehl (1972), in this study, the dynamic characteristics of an RC frame are defined as follows based on the fundamental vibration period (T) and the beam-column stiffness ratio (ρ) of a building. The RC frame is modeled as a single-span frame with a story height of h and span of 2 h. It is assumed that all the members of the RC building have identical cross-sections and that the distribution of the mass and stiffness for each story is constant. Figure 1 is a diagram of the study model. The major variables considered in this study and its scope are as follows:

The beam-column stiffness ratio can be defined as shown in Eq. (1). This ratio represents the different characteristics of the frame as the value changes from 0 to ∞.Here, \( E \) is the elastic modulus, \( I_{b} \) is the geometrical moment of the inertia of the beam, \( I_{c} \) is the geometrical moment of the inertia of the column, \( L_{b} \) is the length of the beam, and \( L_{c} \) is the length of the column. The value of \( \rho \) is taken from the middle story of the RC building. The limit value \( \rho \) = 0 represents a cantilever that consists of a beam with no restraint on its nodal rotation, and the limit value \( \rho \) = ∞ represents a shear building in which the nodal rotation is completely restrained. The value between the two limit values represents a frame that induces bending deflection depending on the degree of the nodal rotation in the beam and column. In this paper, the range of the \( \rho \) value is defined as between 0.05 and 2.0, which is sufficient to show the characteristics of building frames in Asia, based on existing studies. Analysis was carried out on seven values: 0.05, 0.1, 0.15, 0.2, 0.5, 1.0 and 2.0.

The fundamental vibration period of an RC building has the range shown in Eq. (2), which approximates the number of stories in the building.

Here, \( T = 0.1\,{N} \) and \( N \) = number of stories in the building.

In this study, the number of stories in the RC building is 15 stories. Seven stiffness ratio models and six period models were developed with consideration of the above variables and the scope of the study, thereby interpreting and analyzing a total of 42 models. For the hysteretic characteristics of the isolation device, the characteristics (bilinear model) of a hybrid-type isolation system applied to an actual isolation design were used. For seismic motion, time history analysis was performed using the most widely used earthquake data for the El Centro (1942), Taft (1952), and Hachinohe (1968) earthquakes (refer to Fig. 2). ETABS v8.48 was the program used for analysis. The modeling of the superstructure was designed as for a frame element, and Isolator1 element was used to perform local nonlinear modeling of the isolation device. Also, five percent viscous damping was assumed for the damping of the superstructure. For the isolation layer, any viscous damping other than hysteretic damping of the isolation device was not taken into account.

Figures 3 and 4 show the distribution of the maximum response accelerations for each story according to the change in the period of the isolated RC building when \( \rho \) = 0.05~\( \rho \) = 2.0, which are used to examine the isolation effect according to the period difference between the superstructure and the isolation layer of the isolated building. In this study, the isolation effect according to the change in the stiffness ratio between the superstructure and isolation layer was examined by changing the vibration period of the isolated building under the fixed vibration period of the superstructure.

As shown in Figs. 3 and 4, with an increase in the period of the isolated building when \( \rho \) = 0.05~\( \rho \) = 2.0, the distribution of the response acceleration in each story of the building changes from the form that includes the effect of a higher order mode to a constant distribution that is close to rigid body behavior. In terms of reduced acceleration, both buildings showed a reduction in acceleration, but it was difficult to anticipate the effective isolation, as acceleration tends to increase at the uppermost story of a building when the difference of the vibration period between the superstructure and the isolation layer is small. Considering the fact that the reduction in the acceleration of the superstructure refers to the reduced load in the member design, it is necessary to design the difference of the vibration period between the superstructure and the isolation layer to be above a certain level in order to achieve an effective isolation effect.

When the vibration period ratio between the superstructure and the isolation layer is low, the distribution of the response acceleration becomes inconsistent due to the effect of the higher order mode, which makes it difficult to achieve very much improved usability of the building. Also, a large load can act on a specific unpredicted layer to form a soft layer, which also causes difficulty in predicting the overall behavior of the RC building and assessing hazards. Because such careful design is required, it is desirable to design the model for above a certain level of the vibration period ratio between the superstructure and the isolation layer. From a practical viewpoint, it is necessary to set forth a quantitative standard for the level of the period ratio that must be achieved in order to obtain an effective isolation effect. However, no accurately established quantitative standards are currently available that can determine the isolation effect from the form of a specific response. Nevertheless, based on the comparison results shown in Figs. 3 and 4, and considering the fact that the aim of an isolation design is to achieve a consistent response acceleration distribution in order to decrease the response acceleration and increase usability, it is recommended that the subject RC building should be designed with a isolation period that is at least 2.5 times longer than that of superstructure in order to obtain an effective isolation effect for the building.

Another interesting point that can be observed from Figs. 3 and 4 is that no large difference is evident in the responses to the beam-column stiffness ratio (\( \rho \)) of the superstructure when the period ratios of the superstructure and isolation layer are the same. This finding implies that the target isolation period does not need to be considered differently in terms of the stiffness (or the frame characteristics) of the superstructure for the isolation design. It shows that the isolation effect can be differentiated only in terms of the vibration period ratio of the superstructure and isolation layer.

For the experiments in this study, an isolated RC building model and a non-isolated RC building model scaled to one-tenth of an actual building size were designed with consideration of the capacity of a shaking table. The effects of the model building foundations and site on the seismic behavior of the RC buildings were considered using seismic waves with different characteristics. The ground motion records used for the tests include those for the Central Chile earthquake S2A059 (059), San Fernando earthquake S3A103 (103), Eureka earthquake S2A105 (105), and an artificial earthquake. Earthquake 105 and the artificial earthquake represent a harmonic type of motion that is critical for long period structures. The spectrum characteristics of these seismic waves are compatible with the Korean Building Code (KBC 2009) and the US Building Code (ASCE/SEI7-10, 2010) design spectrum. The fabrication of the specimens and experimentation were carried out at the China Academy of Building Research (CABR) in Beijing, China. Figure 5 shows a typical floor plan of the subject buildings and the facade of the specimen installed on top of a shaking table. A full-scale model of the specimen, shown in Fig. 5a, is a flat-type medium sized apartment building with 15 stories and four units in each story.

The total height of the RC building is 45,800 mm including the water tank room. The plane dimensions are 50,720 mm × 12,270 mm. The story height of each story is 2800 mm, and the height of the water tank room is 1400 mm. The target isolation period of the full-scale model originally was to be configured as 4.0 s based on the criteria described in Sect. 2.2, but the isolation period was changed to 3.0 s based on the manufacturing limits of the scaled model isolation device and the effectiveness of the isolation effect. Eighteen lead rubber bearings (LRB 80) and 23 laminated rubber bearings (RB 80) were used to make the isolation layer in order to attain the target isolation period (JSSI 2006; SIVIC 2009). Table 1 presents the characteristics of each isolation device. The scaled specimen was modeled using a maximum manufacturing size that is one-tenth an actual building size to accommodate the capacity of the shaking table (6 m × 6 m, 800 KN, 6 DOF). It is a 15-story model with a long side and a short side of 5.07 m × 1.23 m respectively, and an overall height of 4.6 m. Table 2 summarizes the law of similarity for the mixed similarity model applied in this study.

The concrete strength values of the full-scale model were C30 (30 MPa), C35 (35 MPa) and C40 (40 MPa), and the corresponding concrete strength values of the scaled model were M5.5 (5.5 MPa), M6.0 (6.0 MPa) and M6.5 (6.5 MPa). For the rebar, annealed fine-drawn steel bars with diameters of 0.9~2.2 mm were used based on similarity conditions. Table 3 shows the test results for the small aggregate concrete and rebar. The specimens were manufactured by first making a base-plate of reinforced concrete, then installing and curing the scaled isolation device on top of the base-plate, and creating a base-plate and superstructure for manufacture of the superstructure. The concrete placement was performed story by story, and the concrete was cured for about 3 days after placement before sequentially carrying out the construction of the upper stories.

Figure 6 shows a complete view of the specimen installed on top of the shaking table. After fixing the specimen to the shaking table, the insufficient additional mass was determined according to the law of similarity, as shown in Fig. 6b.

For additional artificial mass, 0.1 kN iron ingot was distributed as uniformly as possible as a single layer on top of the floor slab of each story. Seismic motion was applied sequentially in the x, y, and x + y directions of the model, and the peak ground acceleration (PGA) was increased gradually to 0.07, 0.1, 0.22, 0.4 and 0.9 g. In this study, the tests were conducted with 0.22 g as the design seismic motion in order to assess the residential performance and 0.9 g as the seismic motion used to verify the safety performance. The time of the seismic motion was scaled down to the ratio of 1/4.472 for the recorded time according to the law of similarity.

The story acceleration and inter-story drift were measured to compare the capacity of the test buildings. Based on the test conditions, the story acceleration was measured every two floors, measuring four points in each direction and the center point, to find the lateral movement and warping histories of each floor, whereas the inter-story drift was measured every five floors by connecting the corners of the upper and lower sides of each floor diagonally. The ARJ-50A (Tokyo Sokki, Japan) was employed as the accelerometer, and a cable-extension displacement sensor, CDS-30 (Vishay Precision Group, Inc.), was employed as the displacement meter.

The isolated structure and non-isolated structure showed substantial differences in terms of scale and the type of seismic motion. First, with regard to cracking and the status of the failure, the isolated RC structure did not show any visually observed cracks in the superstructure until 0.9 g. No destructive symptom appeared until the end of the experiment. In contrast, the non-isolated RC structure was found to maintain an elastic state under small seismic loads, but cracks occurred in a portion of a structural member at under 0.22 g. Also, the structure clearly reached an inelastic state after experiencing 0.9 g, and many cracks and the destruction of some members were observed (refer to Fig. 7).

Figure 8 shows the response to inter-story drift, which is a phenomenon that causes structural damage. In comparison to the non-isolated structure, the isolated structure shows that the almost constant response reduced by about one-third, depending on the height. Considering the fact that inter-story drift is limited as a means to determine the seismic performance of buildings, and the reduced inter-story drift in an isolated structure implies improved seismic performance, then the safety of a superstructure can be achieved. Also, Fig. 9 shows that the isolated RC structure experienced a very large drift in the isolation layer, but the maximum response drift of the isolation layer under 0.22 g was 14.76 mm, which is much lower than the allowable drift, i.e., smaller value between 0.55 \( d \) (\( d \) = diameter of isolation device) and 3 \( t_{r} \) (\( t_{r} \) = total thickness of rubber layer in isolation device), in the Chinese standard GB50011-2010 (2010) and CECS126 (2001). The maximum response drift under 0.9 g was found to be 26.46 mm, confirming that the design is appropriate for the target performance.

Figure 10 presents a comparison of the peak acceleration response of each story for the isolated RC structure and non-isolated RC structure. While the response acceleration of the non-isolated structure shows a response distribution of a typical fixed-based structure in which the acceleration gradually increases as the response moves closer to the upper stories, the isolated structure shows that the almost constant response distribution and acceleration at the uppermost story is reduced by about two-thirds compared to the non-isolated structure. This improved behavior can be expected from the isolated RC structure in terms of usability, and amplification of the response acceleration in the superstructure did not appear until 0.9 g. This phenomenon clearly explains the isolation effect. In addition to increased usability, greater reduction of the shear force in the superstructure can be obtained in comparison to that of the non-isolated RC structure.

However, because the period ratio between the superstructure and the isolation layer in this case did not configure as larger than 2.5 times, the distribution of the response acceleration in the isolated structure is closer to a K-shape than complete rigid body behavior. The effect of the higher order mode seems to be included, and the acceleration value did not decrease as much as the input value.

The acceleration response of the full-scale RC building could be obtained from the following relational equation with consideration of the similarity relationship presented in Table 2.

Here, \( a_{i} \) is the maximum acceleration response (g) of the \( i \) story of the full-scale building, \( a_{g} \) is the maximum acceleration (g) of the input seismic motion, and \( K_{i} \) is the dynamic magnification factor of the \( i \) story of the scaled model that corresponds to the input seismic motion of a full-scaled RC structure.

Figure 11 presents a comparison of the maximum acceleration response of each story that was obtained from the test results and analytical results according to different seismic waves.

As illustrated in the figure, the experimental values are slightly higher than the values obtained from the analysis of seismic motion to assess residential performance. The figure shows an extremely large difference of about two to three times in the seismic motion values, which confirms the model’s safety performance. Also, in terms of the mode shape, although the analytical results are similar to those for rigid body behavior, the experimental results include some effects of the higher order mode. The main cause of this difference is probably a technical error associated with the fabrication of the scaled model. As this problem occurred not only in the scaled model experiments but also can occur in practice, strict experimental tests and a review of the characteristics of the isolation device are essential for on-site application. The problem becomes even greater in terms of the seismic motion that is used to confirm safety performance. The large difference in responses probably resulted from the large reduction in the isolation effect on the superstructure due to the smaller stiffness value difference between the superstructure and the isolation layer than the target value and from the deepening effects of the higher order mode and torsion under strong vibration due to the increased eccentricity between the superstructure and the isolation layer. However, as shown by the responses of the seismic motion that are used for the assessment of residential performance, a very similar response is evident in the one-way seismic motion, except for the slight difference in the response acceleration that is caused by the difference in stiffness value between the test results and analysis results. Based on these findings, a sufficiently reliable response can be obtained for the interpretive isolation model by paying close attention to the performance of the isolation device and appropriately modeling the stiffness of the isolation layer.

The drift response of a full-scale RC building can be obtained from the following relational equation with consideration of the similarity relationship presented in Table 2.

Here, \( D_{i} \) is the drift response of the \( i \) story in a full-scale building (mm), \( D_{mi} \) is the drift response of the \( i \) story in the scaled model (mm), \( a_{mg} \) is the maximum input acceleration (g) of the scaled model shaking table (g), \( a_{tg} \) is the maximum input acceleration (g) of the scaled model shaking table that corresponds to \( D_{mi} \), and \( S_{d} \) is the drift similarity factor of the scaled model.

Figures 12 and 13 respectively show the maximum acceleration response and maximum story displacement response that correspond to the design seismic motion level in order to assess residential performance by comparing the experimental and analytical results. As shown in the figures, the drift responses are extremely similar and constant except that the analytical results for the artificial seismic wave are slightly larger than the experimental results. The established analytical model can be used appropriately to confirm the limitation of the isolation device and to review the seismic performance of an isolated building.