# Predicting Maximum and Cumulative Response of A Base-isolated Building Using Pushover Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Proposed Procedure

#### 2.1. Basis Procedure

- The considered base-isolated building predominantly oscillates in the first mode.
- The superstructure behavior is linearly elastic, while nonlinear behavior is assumed only in the isolated layer.
- The local response (e.g., floor displacement and acceleration) can be approximated by a combination of the first and second modal responses.
- The cumulative response in the isolated layer (e.g., the cumulative strain energy of the damper) can be approximated by the first modal contribution.

#### 2.2. Outline of the Procedure

#### 2.2.1. Step 1: Pushover Analysis of the Building Model (First Mode)

_{1}

^{*}-D

_{1}

^{*}) relationship of the equivalent SDOF model is determined based on the pushover analysis results. The equivalent displacement and acceleration at step n (namely,

_{n}D

_{1}

^{*}and

_{n}A

_{1}

^{*}) are determined using Equations (1) and (2), respectively, assuming that the displacement vector

**is proportional to the first mode vector ${}_{n}\mathsf{\Gamma}_{1}{}_{n}\mathsf{\phi}_{\mathbf{1}}$ at each loading step as follows:**

_{n}d_{j}is the mass of the jth floor;

_{n}M

_{1}

^{*}is the effective first modal mass at step n. The A

_{1}

^{*}-D

_{1}

^{*}relationship obtained from the pushover analysis result is idealized as a bilinear curve, and the normal bilinear hysteresis rule is adopted to model the nonlinear behavior of the first mode. Viscous damping is not considered for the equivalent SDOF model, because the energy absorption of the first mode is already included in the hysteresis energy.

#### 2.2.2. Step 2: Nonlinear Analysis of the Equivalent SDOF Model (First Mode)

_{1}

^{*}

_{max}and maximum equivalent acceleration A

_{1}

^{*}

_{max}) and the cumulative response (the equivalent velocity of the cumulative input energy [29] is calculated for the first modal response) as follows:

_{g}(t) is the ground acceleration, defined within the range [0, t

_{d}].

#### 2.2.3. Step 3: Prediction of Local Cumulative Response from First Mode

_{1}

^{*}

_{max}is determined from the results of step 1. Then, the first mode vector and the effective modal mass of the first mode corresponding to D

_{1}

^{*}

_{max}, namely, ${\mathsf{\Gamma}}_{1ie}{\mathsf{\phi}}_{1ie}$ and M

_{1ie}

^{*}, are determined, respectively, and the maximum horizontal deformation of the isolators and dampers at the isolated layer δ

_{Dk}(= x

_{01max}: the relative displacement at the 0th floor corresponding to D

_{1}

^{*}

_{max}) is also determined. The cumulative strain energy of the kth damper, E

_{SDk}, is calculated as follows:

_{SDk}is the plastic strain energy of the kth damper under monotonic loading calculated from the pushover analysis results. While in Equation (7), Q

_{yDk}and δ

_{yDk}denote the yield strength and displacement of the kth damper, respectively.

#### 2.2.4. Step 4: Calculation of Equivalent SDOF Model Properties (Second Mode)

_{2e}. The viscous damping ratio of the elastic SDOF model, h

_{2e}, is calculated as follows:

_{1fix}and h

_{1fix}are the natural period and viscous damping ratio of the first mode of the non-base-isolated frame building model, respectively. In this study, the damping ratio h

_{2e}is proportional to the ratio of the natural frequency of the natural modes because the damping of the superstructure is assumed to be proportional to the stiffness matrix of the superstructure, as described later.

#### 2.2.5. Step 5: Linear Analysis of Equivalent SDOF Model (Second Mode)

_{2e}

^{*}

_{max}). Then, the maximum equivalent acceleration of the second mode is calculated by taking into account the change in the effective modal mass as follows:

_{2e}

^{*}is the effective second modal mass in the elastic range. The derivation of Equation (12) is discussed in the Appendix A.

#### 2.2.6. Step 6: Prediction of Local Maximum Response Considering the Contribution of the Second Mode

**p**and

^{+}**p**is calculated in terms of ${\mathsf{\Gamma}}_{1ie}{\mathsf{\phi}}_{1ie}$ and ${\mathsf{\Gamma}}_{2ie}{\mathsf{\phi}}_{2ie}$ as follows:

^{−}**p**and

^{+}**p**(referred to as Pushovers 1 and 2, respectively) until the equivalent displacement

^{−}_{n}D

^{*}reaches D

_{1}

^{*}

_{max}, using the following calculation formula:

**a**is calculated as follows:

_{max}**f**is the maximum restoring force vector obtained from the Pushovers 1 and 2 envelopes.

_{Rmax}## 3. Building and Ground Motion Data

#### 3.1. Building Data

_{2}Y

_{2}in the X-direction and columns X

_{1}Y

_{2}and X

_{3}Y

_{2}in the Y-direction. This study only considered horizontal earthquake excitation in the X-direction. Figure 4 shows some details of column X

_{2}Y

_{2}with the spandrel wall. The thickness of the spandrel wall is assumed as 150 mm. The floor weights per unit area above level Z

_{1}and at level Z

_{0}are assumed as 14 and 32 kN/m

^{2}, respectively. The floor masses above level Z

_{1}and at level Z

_{0}are 504.9 ton and 1154 ton, respectively. Therefore, the total mass above the isolated layer M is 8223 ton (total weight W = 80581 kN). The behavior of all beams and columns is assumed to be linearly elastic.

_{f}and the shear force coefficient of dampers α

_{s}, are defined, respectively, as follows:

_{f}is the total stiffness of the isolated layer without dampers (calculated as the sum of the initial stiffness K

_{1}of the NRBs and the post yielding stiffness K

_{2}of the LRBs). In Equation (17),

_{s}Q

_{y}is the sum of the yield strength of the damper Q

_{yDk}(including the LRBs). The calculated values of T

_{f}and α

_{s}are T

_{f}= 4.84 s and α

_{s}= 0.048. The modelling scheme for the frame building is the same as that in the previous study by the authors [26].

_{ke}is the kth natural period in the elastic range (k = 1,2) and m

_{ke}

^{*}is the effective modal mass ratio of the kth mode.

_{f}= 4.84 s and the ratio T

_{f}/T

_{1fix}= 4.84/0.854 = 5.67.

#### 3.2. Ground Motion Data

_{I}, is calculated from the following equation for the measurement of the total input energy:

#### 3.3. Nonlinear Time-History Analysis Cases

## 4. Analysis Results

_{1}

^{*}

_{max}, A

_{1}

^{*}

_{max}, A

_{2}

^{*}

_{max}, and V

_{I}

_{1}

^{*}used in the following prediction are considered as the mean of the results obtained from the nonlinear and linear time-history analyses for the equivalent SDOF models by considering 12 waves in each series.

#### 4.1. Pushover Analysis Results

_{1}

^{*}-D

_{1}

^{*}relationship obtained from the pushover analysis result. In this study, the bi-linear idealization of the A

_{1}

^{*}-D

_{1}

^{*}relationship is made according to the initial natural period T

_{1e}and the two points on the A

_{1}

^{*}-D

_{1}

^{*}curve (D

_{1}

^{*}

_{0.10}, A

_{1}

^{*}

_{0.10}) and (D

_{1}

^{*}

_{0.40}, A

_{1}

^{*}

_{0.40}), respectively, where A

_{1}

^{*}

_{0.10}is the equivalent acceleration at D

_{1}

^{*}

_{0.10}(= 0.10 m) and A

_{1}

^{*}

_{0.40}is the equivalent acceleration at D

_{1}

^{*}

_{0.40}(= 0.40 m).

_{1}

^{*}increases, and the components of all floors are approximately equal to unity when D

_{1}

^{*}= 0.4 m. This implies that, in the first mode response, the superstructure behaves as a rigid body. Figure 11b shows the variation of the effective modal mass for the first and second modes. As shown in this figure, the change of m

_{1}

^{*}is relatively stable and closer to unity as D

_{1}

^{*}increases. The change of m

_{1}

^{*}occurs from 0.971 at first to 0.999 at D

_{1}

^{*}= 0.4 m. In contrast, the change of m

_{2}

^{*}is significant; m

_{2}

^{*}is 0.028 at first and drops to 9.340 × 10

^{-4}when D

_{1}

^{*}= 0.4 m. Figure 11c shows the change of the m

_{2}

^{*}/m

_{2e}

^{*}ratio. As can be seen, the m

_{2}

^{*}/m

_{2e}

^{*}ratio drastically drops as D

_{1}

^{*}increases; the m

_{2}

^{*}/m

_{2e}

^{*}ratio is 0.161 when D

_{1}

^{*}= 0.1 m, and 0.0329 when D

_{1}

^{*}= 0.4 m. Therefore, the change in the effective modal mass ratio of the second mode is significant for properly predicting the response of base-isolated buildings.

#### 4.2. Validation of the Predicted Results

#### 4.2.1. Maximum Floor Response

_{0}and Z

_{14}levels, that is, the lowest and highest floors. These figures also show that the maximum floor acceleration predicted by the envelope slightly underestimates the time-history analysis results when the ground motion intensity is 50% and overestimates it when the ground motion intensity is 100%.

#### 4.2.2. Maximum Shear Forces of Vertical Members

_{2}Y

_{1}and (b) column X

_{2}Y

_{2}(with the spandrel wall) are provided in the figures. As can be seen, the results predicted by considering only the first mode significantly underestimate the average time-history analysis results. However, the predicted results presented as envelopes are in good agreement with the average time-history analysis results when the ground motion intensity is 50% or 75%, and conservative when the ground motion intensity is 100%.

#### 4.2.3. Maximum Response of Isolators

_{1}Y

_{1}, LRB at X

_{1}Y

_{2}). The relationship between the maximum shear strain γ and the maximum and minimum compression stress σ is shown. In this figure, the “ultimate compressive stress” is the final ultimate property obtained from a catalog provided by the Bridgestone Corporation [31]. Notably, the peak shear strain γ and the compressive stress σ obtained from the time-history analysis results are the same as those obtained from the average nonlinear time-history analysis results for 12 waves in each series, without consideration to the simultaneity of the peak responses.

_{1}Y

_{1}obtained from the envelope when the ground motion intensity is 100%. However, the nominal stress of isolator X

_{1}Y

_{1}(and isolator X

_{1}Y

_{2}) obtained from all time-history analysis results is within the compression range. Therefore, the behavior of all isolators satisfies the given ultimate properties under the given ground motion intensities.

#### 4.2.4. Cumulative Strain Energy of Isolators and Dampers

## 5. Discussion

#### 5.1. Accuracy in Predicting Maximum Modal Responses

_{1}

^{*}

_{max}and A

_{1}

^{*}

_{max}) by the equivalent SDOF model is in very good agreement with that of the MDOF model. However, according to the second mode, the predicted maximum acceleration of the second modal response by the equivalent SDOF model is conservative. The results obtained by the equivalent SDOF model are in good agreement with those obtained by the MDOF model in the case wherein the ground motion intensity is 50%. However, the difference between the results obtained by the equivalent SDOF and MDOF models is larger in the cases wherein the ground motion intensities are 75% and 100%.

_{1}

^{*}-D

_{1}* relationship) and the time histories obtained by the equivalent SDOF model are in very good agreement with those obtained by the MDOF model.

_{2}

^{*}(t) for two different ground motion intensities (Art-S06). As shown in Figure 20a, the time-history obtained by the equivalent SDOF model is in very good agreement with that by the MDOF model when the ground motion intensity is 50%. However, the difference of the time-history becomes significant when the ground motion intensity is 100%, as shown in Figure 20b.

_{2}

^{*}(t) obtained from the equivalent SDOF and MDOF models, Fourier transform analysis is carried out for A

_{2}

^{*}(t). Figure 21 presents the comparisons between the Fourier amplitudes of the second modal acceleration for two different ground motion intensities. In this figure, the natural frequency of the second modal response (f

_{2e}= 1.957 Hz) is also shown. Moreover, as shown in Figure 21a, the Fourier amplitude distributions in the results obtained by each of the two models are in good agreement when the ground motion intensity is 50%. Additionally, when the ground motion intensity is 100%, as shown in Figure 21b, the results are similar, although the predominant frequency is approximately 1.8 Hz. Therefore, the validity of the assumption of the second modal response being approximated as a linearly elastic response with consideration to the change in the effective second modal mass is confirmed. The conservative evaluation of A

_{2}

^{*}

_{max}may have arisen from the energy absorption of the second modal response, owing to the hysteresis response of the isolation layer. In the linear analysis of the equivalent SDOF model, only the elastic viscous damping is considered.

#### 5.2. Accuracy in Predicting Cumulative Responses

_{I}

_{1}

^{*}, obtained by the equivalent SDOF and MDOF models. In this figure, both results were obtained from the average of 12 waves in each series. The results of the equivalent SDOF model agree with the results of the MDOF model. Therefore, the accuracy of the predicted V

_{I}

_{1}

^{*}is satisfactory.

_{d}is defined as follows:

_{Dk}

_{max}(= x

_{0max}) is the maximum deformation of the kth dampers.

_{d}values of the LRBs and the dampers. The time-history analysis results presented in this figure were calculated according to the average E

_{SDk}and δ

_{D}

_{max}. As shown in Figure 23a, the difference between the predicted n

_{d}and the time-history analysis results is small. In contrast, the difference between the two results is significant in the case of the Art-L series, as shown in Figure 23b. The n

_{d}values predicted for the LRBs and dampers have the same value, while those obtained from the time-history analysis are different. For the LRBs, the n

_{d}prediction is underestimated, while n

_{d}prediction for the dampers is overestimated. The n

_{d}difference between the LRBs and the dampers observed in the time-history analysis results is attributed to the difference of their yield deformation. Notably, the yield deformation of the LRB was smaller (δ

_{yD}= 1.95 × 10

^{−2}m) than that of the steel damper (δ

_{yD}= 3.17 × 10

^{−2}m).

## 6. Conclusions

- The consideration of the second modal response’s contribution is important to better predict the maximum floor acceleration and maximum shear forces of the vertical members in the superstructure. The predicted maximum response obtained by the proposed procedure is in good agreement with the nonlinear time-history analysis results.
- The cumulative strain energy of the isolators (LRBs) and dampers in the isolation layer can be satisfactorily predicted by considering only the first modal response. The reason for this is that the effective first modal mass is approximately 100% of the total mass.
- The maximum and cumulative response of the first mode is satisfactorily predicted using the equivalent SDOF model.
- The maximum equivalent acceleration of the second modal response is conservatively predicted by magnifying the linear analysis results of the equivalent SDOF model and considering the change in the equivalent second modal mass.

_{f}to T

_{1fix}(the fundamental period of the superstructure, assuming it is non-base-isolated) is an important factor to investigate. In the case wherein T

_{f}/T

_{1fix}is small, or T

_{f}is not sufficiently separated from T

_{1fix}, the proposed procedure may be difficult to apply. The evaluation of both the maximum and cumulative energy of each modal response without carrying out time-history analysis for the equivalent SDOF model is another important issue. For design purposes, it is very useful that both responses are evaluated based on the response spectrums. Notably, the maximum response can be easily estimated from the predetermined design response spectrum. However, evaluating the cumulative response is more difficult because, thus far, a design cumulative energy spectrum has not been implemented in the design code. Therefore, the authors think that a procedure should be developed to estimate both the maximum and cumulative responses from the ground motion characteristics, which highlights another important issue in this study.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Magnification Factor for Equivalent Second Modal Acceleration

_{1}

^{*}

_{max}. It is assumed that the restoring force vector of the second and higher modal response,${f}_{R2h}\left(t\right)$, can be approximated as follows:

_{1}

^{*}

_{max}. Additionally, it is assumed that the equivalent acceleration of the second mode, in terms of the second mode vector in the elastic range, A

_{2e}

^{*}(t), can be approximated as follows:

_{2e}

^{*}is the effective modal mass of the second mode in the elastic range.

_{2e}

^{*}(t) and A

_{2}

^{*}(t), and the A

_{2}

^{*}(t)/A

_{2e}

^{*}(t) ratio is the magnification factor resulting from the change in the second mode shape.

_{2}

^{*}(t) and A

_{2e}

^{*}(t) can be derived. From the orthogonal condition of mode vectors ${\mathsf{\phi}}_{1ie}$ and ${\mathsf{\phi}}_{2ie}$, the ratio of constant c

_{1}to constant c

_{2}can be expressed as follows:

_{2}can be expressed as follows:

_{2e}

^{*}(t) can be expressed as follows:

_{12}(Equation (A11)) is negligibly small, a simpler relationship between A

_{2e}

^{*}(t) and A

_{2}

^{*}(t) can be obtained as follows:

_{2e}

^{*}on the left side of Equation (A18) is calculated from the linear analysis of the equivalent SDOF model by assuming the natural period as T

_{2e}and viscous damping ratio as h

_{2e}. Therefore, the change of the natural period and energy absorbing effect caused by the nonlinearity of the base isolation layer are not considered in the calculation of A

_{2e}

^{*}.

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**Figure 1.**Concept of two equivalent single-degree-of-freedom (SDOF) models representing the first and second modal responses used in the proposed procedure.

**Figure 2.**Concept of horizontal force distribution obtained from combination of two modal responses.

**Figure 3.**Simplified structural plan and elevation of the base-isolated building considered in this study: (

**a**) plan of levels Z

_{1}to Z

_{14}; (

**b**) plan of level Z

_{0}; (

**c**) elevation of frame Y

_{2}.

**Figure 5.**Force-deformation relationships for isolators and damper: (

**a**) NRB, (

**b**) LRB, and (

**c**) steel damper.

**Figure 6.**Shapes of the first and second natural mode vectors of the building models in the elastic range: (

**a**) non-base-isolated; (

**b**) base-isolated.

**Figure 7.**Elastic pseudo-acceleration target spectra of the generated artificial ground motions: (

**a**) Art-S Series; (

**b**) Art-L Series.

**Figure 9.**Elastic V

_{I}spectrum of the generated artificial ground motions: (

**a**) Art-S Series; (

**b**) Art-L Series.

**Figure 10.**Bi-linear idealization of A

_{1}

^{*}-D

_{1}

^{*}for the equivalent SDOF model representing the first modal response.

**Figure 11.**Influence of the mode shape change at each step: (

**a**) variation of the first mode shape; (

**b**) variation of the effective modal mass ratio for two modes; (

**c**) variation of m

_{2}

^{*}/m

_{2e}

^{*}.

**Figure 12.**Comparison of maximum floor response (Art-S series): (

**a**) relative displacement; (

**b**) absolute acceleration.

**Figure 13.**Comparisons of maximum floor response (Art-L series): (

**a**) relative displacement; (

**b**) absolute acceleration.

**Figure 14.**Comparisons between the maximum shear forces of vertical members (Art-S series): (

**a**) column X

_{2}Y

_{1}and (

**b**) column X

_{2}Y

_{2}(with spandrel walls).

**Figure 15.**Comparisons between maximum shear forces of vertical members (Art-L series): (

**a**) column X

_{2}Y

_{1}and (

**b**) column X

_{2}Y

_{2}(with spandrel walls).

**Figure 17.**Comparisons of cumulative strain energy between isolators and dampers: (

**a**) Art-S series; (

**b**) Art-L series.

**Figure 18.**Comparison of maximum modal responses: (

**a**) equivalent displacement of first mode; (

**b**) equivalent acceleration of first mode; (

**c**) equivalent acceleration of second mode.

**Figure 19.**Comparisons between first modal response calculated from time-history analysis of multi-degree-of freedom (MDOF) and SDOF models (Art-L00, 100%): (

**a**) A

_{1}

^{*}-D

_{1}

^{*}relationship comparisons; (

**b**) time-history of D

_{1}

^{*}; (

**c**) time-history of A

_{1}

^{*}.

**Figure 20.**Time-history comparison of second modal acceleration: (

**a**) Art-S06, 50%; (

**b**) Art-S06, 100%.

**Figure 21.**Comparisons of Fourier amplitudes of second modal acceleration: (

**a**) Art-S06, 50%; (

**b**) Art-S06, 100%.

**Figure 22.**Comparisons of equivalent velocities of cumulative energy input for first mode: (

**a**) Art-S series; (

**b**) Art-L series.

**Figure 23.**Comparisons between equivalent numbers of cycles for LRBs and dampers: (

**a**) Art-S series; (

**b**) Art-L series.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fujii, K.; Mogi, Y.; Noguchi, T.
Predicting Maximum and Cumulative Response of A Base-isolated Building Using Pushover Analysis. *Buildings* **2020**, *10*, 91.
https://doi.org/10.3390/buildings10050091

**AMA Style**

Fujii K, Mogi Y, Noguchi T.
Predicting Maximum and Cumulative Response of A Base-isolated Building Using Pushover Analysis. *Buildings*. 2020; 10(5):91.
https://doi.org/10.3390/buildings10050091

**Chicago/Turabian Style**

Fujii, Kenji, Yoshiyuki Mogi, and Takumi Noguchi.
2020. "Predicting Maximum and Cumulative Response of A Base-isolated Building Using Pushover Analysis" *Buildings* 10, no. 5: 91.
https://doi.org/10.3390/buildings10050091