The concept of a pair of impulses with inverse directions to each other, called a double impulse, was introduced by Kojima and Takewaki (2015a) to represent a major part of near-fault ground motions. Since the characteristic main part of near-fault ground motions is well known to be expressed by a pulse-type one-cycle or 1.5-cycle sine wave and is influential particularly for tall and base-isolated buildings, with a rather long natural period (Sasani and Bertero, 2000; Mavroeidis and Papageorgiou, 2003; Makris and Black, 2004; Mavroeidis et al., 2004; Kalkan and Kunnath, 2006). Figure 1A shows an example of a near-fault ground motion (Rinaldi FN component during the 1994 Northridge earthquake) and its modeling into a one-cycle sine wave and a double impulse. Figure 1B illustrates the velocity and displacement properties of the one-cycle sine wave and the double impulse. The red arrows indicate the Dirac delta function δ ( t ) . V is the given velocity (the input velocity level) and t 0 is the time interval of two impulses. Consequently, the double impulse can be expressed by u ¨ g ( t ) = V δ ( t ) − V δ ( t − t 0 ) . (1)

To make the double impulse a reliable substitute for near-fault ground motions, it is important to compare it with the one-cycle sine wave. The equivalent one-cycle sine wave with the circular frequency ω p (the period is T p ) and the velocity amplitude V p 1 can be expressed by u ¨ g SW ( t ) = 0.5 ω p V p 1 ⁡ sin ( ω p t ) ( 0 ≤ t ≤ T p = 2 t 0 ) , (2) where V p 1 = 1.2222 V (Kojima et al., 2018). This relation can be derived from the equivalence of the maximum Fourier amplitudes of both inputs.

Consider a 40-story base-isolated frame building model as shown in Figure 3. Unlike the base-isolation system used in Section 3, lead rubber isolators and oil dampers are employed as the constituent members of the base-isolated story. The total depth of these isolators is 200 [mm]. This base-isolated frame building model is reduced to an SDOF mass-spring model with a dashpot via a 2DOF model as illustrated in Figure 3. For the simplicity of presentation of the proposed theory, the super building frame has a constant floor mass m U and the free wall (sub building) has a constant floor mass m F . Let m I denote the mass of the base-isolation story and let c denote the constant damping coefficient of connecting oil dampers placed at several stories. The base-isolation story is assumed to have a bilinear hysteretic restoring-force characteristic as usually used for lead rubber isolators. k I 1 , f y , d y I indicate the initial stiffness in the elastic range, the yield force, and the yield deformation of the base-isolation story, respectively ( k I 1 = f y / d y I ) . The first and second stiffnesses of each lead rubber isolator are K 1 = 20.01 [kN/mm] and K 2 = 1.54 [kN/mm] and the force axis percept is Q d = 360.6 [kN]. These properties are multiplied by 36 (number of lead rubber bearings) to obtain the restoring-force characteristics of the base-isolation story.

T u p and T s u b denote the fundamental natural period of the main frame building for the fixed base-isolation story and the fundamental natural period of the sub building. Furthermore, c I indicates the total damping coefficient of oil dampers in the base-isolation story. The total damping coefficient c I of oil dampers is to be specified so as to attain the damping ratio 0.15 for the equivalent stiffness k I e q of the base-isolation story at the deformation u I = 0.4 ( m ) . The structural damping ratio of the super-structure (instantaneous stiffness-proportional damping) is set to 0.03. The strength of concrete is Fc60 (60 [N/mm²]) and the Young’s modulus is 33,500 [N/mm²]. The floor mass is 286.2 × 10³ [kg] and the mass of the base-isolation story is 858.6 × 10³ [kg]. The parameters of the base-isolated building are shown in Table 1.

Consider the transformation of a base-isolated building frame into a 2DOF model.

Since the fundamental natural period of the super building with a fixed base is 3.0 (s), the stiffness of the super building can be obtained as k U = M U ( 2 π / 3.0 ) 2 . (9)

The initial stiffness k I 1 and the second stiffness k I 2 of the base-isolation story are determined by considering the hysteretic characteristics and the number of isolators and dampers (Figure 4A). Let d y I and f y denote the yield deformation and the yield force of the base-isolation story. The damping coefficient of the super building as an SDOF model is determined for the fundamental damping ratio = 0.03 by c U = 2 × 0.03 × M U k U . (10)

The damping coefficient of the base-isolation story is determined by considering the number of oil dampers.

The 2DOF model set above is modeled into an SDOF model as shown in Figure 3. Since m I is rather small compared to M U in high-rise buildings, m I is neglected here. Then M e = M U . (11)

The stiffness and damping coefficient of the SDOF model are determined differently in the elastic region and the plastic region of the base-isolation story. While Hayashi (2018) determined the equivalent damping coefficient only for the initial stiffness, that is determined differently in the elastic region and the plastic region of the base-isolation story.

In the region before the yielding of the base-isolation story, the equivalent stiffness k e 1 and the equivalent damping coefficient c e 1 of the SDOF model can be evaluated by the following series modeling of complex springs. 1 k e 1 + i ω e 1 c e 1 = 1 k I 1 + i ω 1 c I + 1 k U + i ω 1 c U , (12) where ω 1 is the elastic fundamental natural circular frequency of the 2DOF model and the natural circular frequency of the SDOF model is obtained from ω e 1 = k e 1 / M e . From Eq. 12, k e 1 and c e 1 are obtained. While Hayashi (2018) obtained the elastic stiffness and the damping coefficient from Eqs 4, 5, the proposed method provides those in a unified manner from Eq. 12.

After the yielding of the steel damper in the base-isolation story, the stiffness reduces to k I 2 (see Figure 4A) and the equivalent stiffness in the base-isolation story changes gradually in time. By responding to this phenomenon, assume the deformation u I n = 1 of the base-isolation story and the equivalent stiffness of the base-isolation story by u I n = 1 = d y I , (13) k I e q n = 1 = f y / u I n = 1 , (14) where n indicates the iteration cycle number. Based on these values, the equivalent stiffness and the equivalent damping coefficient of the SDOF model are obtained from 1 k e e q + i ω e 2 c e 2 = 1 k I e q + i ω 2 c I + 1 k U + i ω 2 c U , (15) where k e e q denotes the equivalent stiffness of the SDOF model and is defined by k e e q n = 1 = f y / u e n = 1 , (16) u e is an assumed maximum deformation of the SDOF model and is determined by u I . In this computation, the lowest mode of the 2DOF model is used so that the equivalent stiffness of the base-isolation story in the 2DOF model attains k I e q and defines the deformation of the super building as u e for the deformation u I of the base-isolation story.

In Eq. 15, ω 2 is the fundamental natural circular frequency of the 2DOF model with the equivalent stiffness k I e q of the base-isolation story. In addition, ω e 2 = k e e q / M e . From Eq. 15, k e e q and c e 2 are consequently obtained.

Once u e and k e e q are obtained for the SDOF model, the second stiffness k e 2 of the SDOF model is determined by using the force-deformation relation of the 2DOF model. Since the equivalent damping coefficient of the SDOF model changes depending on the yielding of the base isolation story, c e 2 for the post-yield case is used as the equivalent damping coefficient c m a i n in place of c e 1 for the elastic case.

Through the above-mentioned procedure, the parameters M e , k e 1 , k e 2 , c m a i n for the SDOF model are obtained.

The maximum deformation of the bilinear hysteretic SDOF model to the critical double impulse was derived by Akehashi et al. (2018). In this paper, this closed-form expression is used. The relation between u max and u i s o is derived as follows. u i s o = { f max / k I 1 ( f max − b I ) / k I 2 ( u max < d y e ) ( u max ≥ d y e ) , (17) f max = { k e 1 u max k e 2 u max + b e ( u max < d y e ) ( u max ≥ d y e ) , (18) where d y e is the yield displacement of the SDOF model and b I , b e are the force-axis intercept in the 2DOF model and the SDOF model. Because the SDOF model is modeled from the 2DOF model, the maximum deformation u i s o of the base-isolation story has to be transformed from the maximum deformation u max of the SDOF model. In this process, the maximum story shear force of the SDOF model is defined by f max , and the deformation u i s o of the base-isolation story of the 2DOF model is computed for the same maximum story shear force f max (see Figure 4B).

In Eq. 14, the parameters of the SDOF model are determined for an assumed equivalent stiffness k I e q of the base-isolation story of the 2DOF model. The assumed maximum deformation u I of the base-isolation story is determined iteratively so that the difference from the value u i s o obtained from Eqs 17, 18 is minimized. The flowchart is shown in Figure 5.

The parameters of the 2DOF model and the SDOF model obtained by the above-mentioned procedure are shown in Table 2.

Since the second stiffness k e 2 and the damping coefficient c m a i n of the SDOF model after yielding in the base-isolation story are computed iteratively, those values change depending on the input velocity level of the critical double impulse. The relations of k e 2 , c m a i n with the input velocity V are shown in Figures 6A,B. It can be observed that, as V increases and the plastic deformation becomes larger, the second stiffness k e 2 and the damping coefficient c m a i n of the SDOF model become larger and those values converge to constant values finally. It should be remarked that, while the damping coefficient c m a i n is constant at the lower level around 0.7 × 10⁷ (N/(m/s)) in the previous method (Hayashi, 2018; Hayashi et al., 2018), the proposed method provides a larger value depending on the input velocity level. Figure 6C indicates the maximum deformation comparison of the base-isolation story among the proposed one, the 2DOF model under the one-cycle sine wave, the plane frame under the one-cycle sine wave, and the previous one by Hayashi (2018). It can be seen that, while the previous method (Hayashi, 2018) exhibits an inaccurate result, especially for a larger input level due to the factors explained in Figure 6B (the estimation of a smaller damping coefficient for the SDOF model), the proposed method for the SDOF model can evaluate the maximum deformation of the base-isolation story of the plane frame accurately. In addition, it is noted that the present model employs a bilinear hysteretic restoring-force model (second stiffness ratio = 0.077) for the base-isolation story while an elastic-perfectly plastic model was used in the previous method mentioned above. This may slightly affect the difference in Figure 6C.

The member parameters of this system are the same as those in Table 1. The connection dampers are placed on stories 4, 8, 12, 16, 18, 20, 22, 24, and 26. While Hayashi (2018) treated a shear building model, the present paper deals with a reinforced concrete plane frame. As in Section 4.1, the structural damping ratio of the super-structure (instantaneous stiffness-proportional damping) is set to 0.03. The strength of concrete is Fc60 (60 [N/mm²]) and the Young’s modulus is 33,500 [N/mm²]. In addition, the restoring-force characteristic of the base-isolation story is extended from the elastic-perfectly plastic model to the bilinear hysteretic model for the purpose of practicality.

The simplified SDOF model of a base-isolated building is further reduced to another SDOF model by connecting that SDOF model to a rigid wall as a representative of a stiff sub-building as shown in Figure 7. In this simplification procedure, a compensation coefficient β d for connecting dampers is introduced as in the method of Hayashi (2018) to include 1) the effect of rigid modeling of the sub building for simple treatment, 2) the effect of the location of connecting dampers, and 3) the effect of height of the sub building. The novel point is in the damping evaluation in the simplification of the base-isolated building explained in Section 4.

As in the method by Hayashi (2018), the RMDOF model with the compensation factor β d for the connecting dampers is considered as shown in Figure 8A to reflect the above three factors (rigid modeling of sub building, location of connecting dampers, height of the sub building).

The compensation factor β d for the connecting dampers is determined so that the lowest damping ratio h R (by complex eigenvalue analysis) of the RMDOF model coincides with that h M of the hybrid-controlled plane building frame. h M = h R (19)

Equation 19 requires repetition for obtaining β d . To avoid this repetition, it is assumed that h R is linear to the total damper quantity. Let h R ∗ denote the lowest damping ratio of the RMDOF model with the original total damper quantity c × n c as shown in Figure 8B.

In the complex eigenvalue analysis, it is necessary to determine the equivalent stiffness of the base-isolation story. In this paper, the equivalent stiffness is to be obtained from the analysis for the SDOF model with β d = 1 (initial assumption) where the equivalent stiffness of the base-isolation story is evaluated using the 2DOF model. While the Hayashi’s model and method (Hayashi, 2018) requires repetition for evaluating β d due to the inaccuracy in the evaluation of the equivalent damping coefficient, the proposed method highly enhances the reliability and accuracy because of the upgrade of the accuracy in the response evaluation in terms of the SDOF model.

Figure 9A shows the relation of the lowest-mode damping ratio of the MDOF model and the RMDOF model with the equivalent stiffness of the base-isolation story where five damping coefficients of the connecting damper are taken. The following observation can be drawn from Figure 9A

*The lowest-mode damping ratio of the MDOF model exhibits different values depending on the equivalent stiffness. The variation is large in the range of smaller equivalent stiffness.

Once such a relationship is obtained, the compensation factor β d can be determined by using the equivalent stiffness corresponding to the input velocity level. Figures 9B,C present the relation of the equivalent stiffness of the base-isolation story with the input velocity level and the relation of the compensation factor with the input velocity level. It can be observed from Figures 9B,C that, as the damping coefficients of connecting dampers becomes large, the compensation factor becomes smaller for a smaller input velocity level. It is noted that only eigenvalue analysis is needed for MDOF and RMDOF models and the response evaluation is conducted only for the SDOF model by using the closed-form expression (Akehashi et al., 2018).

Figure 10 presents the comparison of the maximum deformation of the base-isolation story by the proposed method for the SDOF model (with and without compensation factor) with that by time-history response analysis for the plane frame model for various connection damper levels. It can be observed that, while the proposed method exhibits a good correspondence to the plane frame model regardless of the compensation factor β d in the small connection damper level, the compensation factor plays an important role in the larger connection damper level. It seems that this results from the fact that, when the connection damper level becomes larger, the effect of the assumption of the rigid sub-building arises strongly. This fact can also be understood from the result that, as the connection damper level becomes larger, β d becomes smaller (see Figure 9C). This indicates the compensation effect clearly.

Recently, many multi-purpose structural analysis programs have become available. In this paper, a general-purpose structural analysis program SNAP (Kozo System Co., 2019) is used. In this program, the accuracy check is included automatically, e.g., the processing of unbalanced forces.

The double impulse u ¨ g D I is to be expressed by a one-point (triangular) acceleration input u ¨ g T R I where the triangle area of this input corresponds to the impulse V I , N . This relation can be expressed by A I , N Δ t = V I , N . (20)

Kawai et al. (2021) showed that Δ t = 0.01 [s] is sufficient enough for the accuracy. It is noted that, since a rapid change of input acceleration may cause the occurrence of unbalanced forces, division of the time interval into short durations is often conducted.

Furthermore, Akehashi and Takewaki (2019) made clear that the critical input of the second impulse can be characterized by the criterion that the critical timing of the second impulse is the time of the zero story shear in the first story. For the present model, the following relation holds. F = F M 1 + F F 1 = 0 , (21) where F M 1 is the base-isolation story shear force of the main building and F F 1 is the first-story shear force of the sub building. Figure 11 shows the critical timing of the second impulse and the simulation of the impulses by one-point (triangular) acceleration inputs.

Figure 12A shows the comparison of the maximum deformation of the base-isolation story of the plane frame model under the one-cycle sine wave equivalent to the critical double impulse for the SDOF model with that under the critical double impulse (a pair of inverse-direction one-point acceleration inputs) for the plane frame model. The connection damping coefficient is given by c = 0.417 [kN/(mm/s)] and the time increment is specified as Δ t = 0.01 [s]. It can be observed that both plots correspond well. This indicates the reliability and accuracy of using the SDOF model and the critical double impulse for the SDOF model.

Figure 12B illustrates the time histories of the above-mentioned two inputs. The phases of the time histories are adjusted. It can be understood that the responses under both inputs coincide well and the simple response evaluation method proposed in the paper is reliable.