使用高阻尼橡膠隔震支承之建築結構非線性地震力分析

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行政院國家科學委員會專題研究計畫成果報告

使用高阻尼橡膠隔震支承之建築結構非線性地震力分析研究成果報告(精簡版)

計畫類別：個別型

計畫編號： NSC 99-2221-E-011-036-

執行期間： 99 年 08 月 01 日至 100 年 07 月 31 日執行單位：國立臺灣科技大學營建工程系

計畫主持人：黃震興

計畫參與人員：碩士班研究生-兼任助理人員：Fazrin

處理方式：本計畫可公開查詢

中華民國 100 年 09 月 27 日

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行政院國家科學委員會專題研究計畫成果報告

使用高阻尼橡膠隔震支承之建築結構非線性地震力分析

計畫編號：NSC99-2211-E011-036

執行期限：99 年 08 月 01 日至 100 年 07 月 31 日主持人：黃震興教授國立台灣科技大學營建工程系

計畫參與人員：Mohammad Fazrin Assidiqy 國立台灣科技大學營建工程系

1. Abstract

The concept of base isolation is gaining widespread acceptance in the global earthquake engineering community due to the excellent performance of base-isolated structures during the 1994 Northridge and 1995 Kobe earthquakes. Some of the commonly used isolation systems are elastomeric bearings including lead-rubber bearings (LRB) and high damping rubber (HDR) bearings as well as sliding isolation systems. In recent years, a few mathematical models of high damping elastomeric isolation bearings have been proposed.

Hwang et al proposed a mathematical model modified from Pan and Yang’s model. The number of parameters used by Pan and Yang are reduced, and the model is extended to be capable of describing the Mullins effect and scragging effect of high damping rubber bearings.

A mathematical model of high damping elastomeric isolation bearings proposed by Hwang et al has been validated by the cyclic loading test of HDR materials. In this work, this model will be validated using cyclic loading tests of HDR bearings. The comparison between the predicted and experimental results indicates the proposed model is capable of predicting the dynamic hysteretic behavior of HDR bearings under different axial loads and different excitation frequencies. In addition, this proposed model of HDR isolation bearings will be used to predict the seismic responses of a base-isolated multistory building tested by a shaking table. A single set of 10 parameters

used in the proposed mathematical model is obtained by combined just two hysteresis loop of eight shaking table test. From the test result, it is concluded that the proposed model of HDR isolation bearing is capable of predicting the seismic response of a base-isolated multistory structure if the model is well calibrated by a set of dynamic tests.

Keywords ： seismic isolation, elastomeric bearing, high damping rubber bearing, cyclic loading test, shaking table test, analytical model

2. Background and Motivation

The concept of base isolation is gaining widespread acceptance in the global earthquake engineering community due to the excellent performance of base-isolated structures during the 1994 Northridge earthquake and the 1995 Kobe earthquake.

Some of the commonly used isolation systems are laminated rubber bearings (LRB and HDRB) and sliding bearings. Since the mechanical characteristics of different bearings may depend greatly on the compounds used in the high damping rubber [1], performance test are required in the current design practice to confirm that the bearing characteristics are approximately equal to the design parameters and satisfy the design requirements.

The behavior of a base-isolated structure is very similar to that of a single-degree-of-freedom system if the superstructure is rigid and displacement

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responses are concentrated in the isolation bearings. The force-displacement relationship of typical elastomeric isolation bearings is nonlinear as a result of their inherent damping properties. The accurate prediction of the seismic behavior of base-isolated structures depends strongly on the accuracy of the analytical model of the isolation bearings.

In recent years, a few mathematical models of high damping rubber (HDR) bearings have been proposed. Hwang et al [2]

proposed a mathematical model modified from Pan and Yang’s model [3]. The number of parameters used by Pan and Yang are reduced, and the model is extended to be capable of describing the Mullins effect and scragging effect of high damping rubber bearings.

A mathematical model of high damping elastomeric isolation bearings proposed by Hwang et al has been validated by the cyclic loading test of HDR materials. In this work, this model will be validated using cyclic loading tests of HDR bearings. In addition, this proposed model of HDR isolation bearings will be used to predict seismic responses of a base-isolated multistory building tested by a shaking table.

3. Analytical Model of HDR Isolation Bearings

In the mathematical model proposed by Hwang et al, the shear force experienced by the HDR bearing is attributed to the sum of the restoring force and the damping force.

The model is written in the form of

1 2

( , ) ( , ) ( , )

F u u =F u u +F u u (1) where F u u( , ) is the shear force transmitted by the bearing which is composed of the restoring force F u u₁( , ) and the damping force F u u₂( , ). Both the restoring force and damping force are functions of the relative displacement (u) and relative velocity (u) of the bearing.

( )

9 0

,

2 4 4

1 1 2 3 2

5

, cosh

t a F u u du

F u u a a u a u a e u

a u

⎡ ∫ ⎤

⎢ ⎥

= + + +

⎢ ⎥

⎣ ⎦

(2)

( ) ⁶ ⁷ ² ¹⁰⁰ ^{( )}^,

2 2 2

8

, 1

t a F u u du

a a u

F u u e u

a u

⎛ ∫ ⎞

+ ⎜ ⎟

= + ⎜⎜⎝ + ⎟⎟⎠

(3)

In this work, the model proposed by Hwang et al will be validated using cyclic loading tests of HDR bearings. The test setup is shown in Figures 1. In this test, each set of HDR bearings were deformed laterally with cyclic sinusoidal waves that deduced a maximum shear strain ranging from 50% to 200% with a step increment of 50%. Three excitation frequencies, 0.25Hz , 0.5 Hz , and 1Hz are selected for the test. Three axial loads, 25 kg cm/ ² , 50 kg cm/ ² , and 75kg cm are prescribed. Each test was / ² conducted for three cyclic reversals.

The 10 parameters of Eqs. (2) and (3) can be identified by the best fit to the cyclic loading test results. The best fit to the test results can be solved using nonlinear least square method [4] with the aid of the mathematical tools of MATLAB [5].

However, since the identification is performed in a 10 variable space, many local minima may exist in the identification process. Nevertheless, it is not the aim of this study to establish the relationship of these 10 parameters to the aforementioned influence factors, but to demonstrate the capability of the proposed model in predicting the force–displacement hysteresis of HDR bearings and the seismic response of base-isolated structures by HDR bearings.

The effect of frequency under an axial load of 43.38kN is summarized in Figure 2.

The parameters used for the analytical prediction are identified in Tables 1. From figure, it is seen that the measured and the calculated hysteresis loops have an excellent agreement. Therefore, the proposed model is validated for its capability of predicting the bearing behavior under different excitation frequencies.

The effect of frequency under different axial loads is also studied. The axial load was changed to 86.77kN. Figure 3 shows the measured and predicted hysteresis loops under different excitation frequencies. The parameters used for the analytical prediction

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are identified in Table 2. From Figure 3, one can see that the measured and calculated hysteresis loops have a good agreement.

Thus, it is concluded that the proposed model is capable of predicting the bearing behavior through different excitation frequencies and different axial loads.

In summary, an analytical hysteresis model of high damping elastomeric isolation bearings has been proposed by Hwang et al.

With extensive correlation between the experimental and predicted hysteresis behavior of the HDR bearings, the model has been proved to be capable of accurately predicting the behavior of bearings subject to dynamic shear loadings.

4. Prediction for Seismic Responses of A Base-Isolated Multistory Building For conducting shaking table tests, a three-story steel structure with one bay in each horizontal direction is designed to simulate a base-isolated structure. This structure model is illustrated in Figure 4.

The base isolation system is composed of four high damping rubber (HDR) bearings.

In this test, the structure is subjected to a series of earthquake ground motions as listed in Table 3.

The equation of motion for the superstructure and isolation system follows Eqs. (4) and (5), in which the high damping rubber (HDR) bearings are represented by the mathematical model given in Eqs. (2) and (3).

( ) ² ( ) ² ( ) ( ( ) ( ))

i i i i i i i b g

q t + ξ ωq t +ω q t = −L u t +u t

(4)

( ) ( ) ( ) ( ) ( )

tot b k b k b k b k b( )k

tot g k

M u t k t u t c t u t M u t

+ +

= −

(5)

where

T i

i T

i i

L Mr M φ

φ φ

= (6)

( ) ( ) ( ) ^{( )} ( )

0

sin

i i t

i t

i b g i

i

q t L u τ u τ e^{ω ξ} ^τ ω t τ τd ω

− −

⎡ ⎤

= − ∫⎣ + ⎦ − (7)

[ ]

{ }

*

1

1 ⁱ^{i k} sin cos

N

t

tot i i i k i k

i

M M M e⁻^{ω ξ} ξ ωt ωt

=

=⎡ + + − ⎤

⎢ ⎥

⎣ ∑ ⎦ (6)

Eq. (5) is the equation of motion of the base isolation layer. This type of equation is a nonlinear differential equation which may be solved by the B-spline collocation method

[6,7].

However, the 10 parameters identified in the cyclic loading test are not to be used in the analytical correlation. This is because the axial load exerting on HDR bearing during the ground shaking is not constant as that of cyclic loading tests. In addition, the natural frequency of the test structure is varying due to the fact that the hysteresis behavior of the HDR bearings is basically nonlinear.

Instead, the 10 parameters to be used in this analysis will be obtained by the hysteresis loops obtained from each of the seven shaking table tests. In another approach, a single set of 10 parameters is identified from combined hysteresis loops of two shaking table tests, 200% ELC270 and one new earthquake ground motion (200%

Sannomaru). The 10 parameters used in this analysis are shown in Table 4. The reason for so doing is that it is the attempt of this study to obtain only one set of 10 parameters rather than seven sets of 10 parameters from each shaking table test. If the identified single set of parameters can be successfully applied to predict the seismic responses of the test structure under each ground shaking, the single set of parameters may be used for the response prediction of the base-isolated structure subjected to other ground motions than the ground motions used in this test series.

In the first approach, a set of parameters from Table 4 is applied to the calculation procedure for each corresponding test. Each set of 10 parameters is identified from corresponding to each of the seven shaking table tests. However, only the 300%

TCU047 of the Chi-chi earthquake is used herein to illustrate the calculation procedure.

The comparison of measured and predicted relative displacement and absolute acceleration response histories of the base slab and superstructure are shown in Figures 5 and 6, respectively. From the Figures, it can be seen that the measured and predicted relative displacement and absolute acceleration responses under different ground excitation have shown a good agreement. It can then be concluded that the

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proposed model is capable of accurately predicting the seismic response of a base-isolated multistory building subjected to different ground excitations.

In the second approach, a single set of 10 parameters is obtained from the combined hysteresis loops of two shaking table tests.

The arbitrarily selected two tests are the 200% ELC270 and 200% Sannomaru tests.

This single set of 10 parameters will be used to predict the seismic response of structural model under the other two ground excitations (100% KJM000 and 300%

TCU047). However, only the 300%

TCU047 of the Chi-chi earthquake is used herein to illustrate the calculation procedure.

The comparisons of the measured and predicted relative displacement and absolute acceleration response histories of the base slab and superstructure under 300%

TCU047 ground excitation are shown in Figures 7 and 8. From figures can be seen that the measured and predicted relative displacements and absolute acceleration responses under differences seven grounds excitation have show a good agreement.

The significance of this result is that the identified single set of parameters from the combined hysteresis loops of two shaking table tests (200% ELC270 and 200%

Sannomaru tests) can accurately predict the seismic responses of the structure subjected to two different ground excitations (100%

KJM000 and 300% TCU047). Therefore, it can be concluded that the mathematic model with the 10 parameters identified from a number of appropriate ground shakings can be extended to accurately predict the seismic responses of base-isolated structures subjected to other ground motions than those used for the identification of the 10 parameters.

5. Conclusion

a. In this work, a mathematical model of HDR bearings proposed by Hwang et al has been validated by the cyclic loading tests of HDR bearings. The proposed model is proved to be capable of accurately predicting HDR bearing

behavior under different axial load and different excitation frequencies.

b. In addition, the proposed model is also capable of predicting the seismic responses of a base-isolated multistory building tested by a shaking table. The 10 parameters of the mathematical model to be used in analysis were obtained from the hysteresis loops obtained from each of the seven shaking table tests, and the prediction of the seismic responses of the isolated structure showed that the model can simulate the seismic behavior of HDR bearings with fidelity.

c. In another approach, a single set of 10 parameters was identified from the combined hysteresis loops of only two of all shaking table tests. The identified model was then used to predict test results including the tests whose measured hysteresis loops of the bearings were not used for the identification of the 10 parameters.

d. The proposed model of HDR isolation bearing is capable of accurately predicting the response of a base-isolated multistory structure to other ground motions than the ground motions from which the 10 parameters of the model are identified.

6. References

[1] Kawashima K, Okado M, and Horikawa M. Design example of a highway bridge based on the manual for Menshin design of highway bridges. Recent Selected Publications at Earthquake Engineering Division, Public Works Research Institute. Tsukuba: Ministry of Construction, 1993: 191-208.

[2] Hwang JS, Wu JD, Pan TC, and Yang G.

A Mathematical Hysteretic Model for Elastomeric Isolation Bearings.

Earthquake Engineering and Structural Dynamic. 2002; 31:771-789.

[3] Pan TC, Yang G. Nonlinear analysis of base-isolated MDOF structures.

Proceedings of the 11th World Conference on Earthquake Engineering.

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Mexico: 1996; 1534.

[4] Madsen K, Nielsen HB, and Tingleff O.

Methods for non-linear least square problems. 2nd Ed. Denmark: Informatics and Mathematical Modeling Technical University of Denmark. 2004.

[5] MATHWORKS, Inc. MATLAB Reference Guide. The Mathworks, Inc.

Natick, MA. 1992.

[6] Caglar H, Caglar N, Elfaituri K. B-spline

Interpolation Compared with Finite Difference, Finite Element and Finite Volume Methods Which Applied to Two-point Boundary Value Problems.

Applied Mathematics and Computation.

2006; 175:72-79.

[7] De Boor C. A practical guide to splines.

New York: Springer. 1978; 113-115.

Table 1 Identified parameters from cyclic loading tests with an axial load of 43.38 kN and various lateral excitation frequencies

Axial load 43.38 kN

Frequency 0.25 Hz 0.5 Hz 1 Hz

a1 912 867 846

a2 -9 -5 -4

a3 0.067 0.039 0.032

a4 758 543 535

a5 0.201 0.111 0.054 a6 2619 2771 3102

a7 13 13 18

a8 5 11 26

a9 0.0000094 -0.0000009 -0.0000198 a10 0.0000546 0.0000983 0.0001856

Table 2 Identified parameters from cyclic loading tests with an axial load of 86.77 kN and various lateral excitation frequencies

Axial load 86.77 kN

Frequency 0.25 Hz 0.5 Hz 1 Hz

a1 639 568 548

a2 -5 -2 -2

a3 0.039 0.019 0.015 a4 651 545 543 a5 0.205 0.112 0.054 a6 2636 3202 3976

a7 13 19 31

a8 6 14 34

a9 0.0000015 0.0000032 -0.0000145 a10 0.0000587 0.0001328 0.0002875 Table 3 Earthquake list for shaking table tests

Test PGA Test

Name Input Excitation Earthquake

Component Time Scale Shaking

Direction Original PGA

Test PGA Value (g)

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100% 0.40 TCU047

Chi-Chi/TCU047 Chi-Chi, Taiwan

1999/09/21 Real Earthquake

NS scale factor

=1/ 4 ^X _{300% 1.19}

100% 0.35 ELC

270

El Centro/I-ELC270 Imperial Valley, U.S.

NS scale factor

=1/ 4 ^X

200% 0.70

50% 0.42

80% 0.67 KJM

000

KJMA/KJM000 Kobe, Japan

NS scale factor

=1/ 4 ^X

100% 0.83

Table 4 Identified 10 parameters from each input ground motion and from combined two hysteresis loops

Parameter TCU047 100%

TCU047 300%

ELC270 100%

ELC270 200%

KJM000 50%

KJM000 80%

KJM000 100%

Combined 2 hysteresis

loops

a1 5566 3493 4714 3638 3850 3389 2916 2509 a2 -2523 -234 -1091 -170 -455 -231 -96 -88

a3 698 10 179 6 46 12 3 2

a4 904 18 1089 706 485 372 305 367 a5 0.30130 0.12474 0.09217 0.11115 -0.18359 0.09167 0.11653 0.11121 a6 1130 1762 1448 2202 1491 1883 2171 1703

a7 14 11 19 21 9 5 3 14

a8 -8 17 -10 22 12 18 22 17

a9 -0.00070 -0.00086 0.00006 -0.00012 0.00025 0.00057 0.00053 0.00035 a10 -0.00005 0.00005 -0.00003 -0.00001 -0.00002 -0.00008 -0.00005 0.00004

Figure 1 Set-up cyclic loading test

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-150 -100 -50 0 50 100 150 Displacement (mm)

-20 -10 0 10 20

Shear force (kN)

frequency 0.25 Hz, Axial load 43.38 kN Measured

Predicted

-150 -100 -50 0 50 100 150

Displacement (mm) -20

-10 0 10 20

Shear force (kN)

frequency 0.5 Hz, Axial load 43.38 kN Measured

Predicted

(a) 0.25 Hz (b) 0.5 Hz

-150 -100 -50 0 50 100 150

-10 0 10 20

Shear force (kN)

frequency 1 Hz, Axial load 43.38 kN Measured

Predicted

(c) 1 Hz

Figure 2 Comparison between measured and predicted hysteresis loops under different excitation frequencies (axial load 43.38 kN)

-150 -100 -50 0 50 100 150

-10 0 10 20

Shear force (kN)

Frequency 0.25 Hz, Axial load 86.77 kN Measured

Predicted

-150 -100 -50 0 50 100 150

-10 0 10 20

Shear force (kN)

Frequency 0.5 Hz, Axial load 86.77 kN Measured

Predicted

(a) 0.25 Hz (b) 0.5 Hz

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-150 -100 -50 0 50 100 150 Displacement (mm)

-20 -10 0 10 20

Shear force (kN)

Frequency 1 Hz, Axial load 86.77 kN Measured

Predicted

(c) 1 Hz

Figure 3 Comparison between measured and predicted hysteresis loops under different excitation frequencies (axial load 86.77 kN)

Figure 4 Plan dimensions of test structure: (a) x direction (b) y direction

-50 -25 0 25 50

Displacement (mm)

Relative Displacement of Base Isolation Layer Measured

Predicted

Relative Displacement of 2nd Floor Measured

Predicted

(a) base isolation layer (b) 2^nd floor

Unit : mm

(10)

10 15 20 25 30

Time (Sec)

-50 -25 0 25 50

Displacement (mm)

Relative Displacement of 3rd Floor Measured Predicted

10 15 20 25 30

Time (Sec)

Relative Displacement of Roof Floor Measured

Predicted

(a) 3^rd floor (b) roof

Figure 5 Comparison of measured and predicted relative displacements in X direction under TCU047 300% ground motion using a set of 10 parameters identified from each ground excitation

-4 -2 0 2 4

Acceleration (m/s2)

Absolute Acceleration of Base Isolation Layer Measured

Predicted

Absolute Acceleration of 2nd Floor Measured

Predicted

10 15 20 25 30

Time (Sec) -4

-2 0 2 4

Acceleration (m/s2)

Absolute Acceleration of 3rd Floor Measured

Predicted

10 15 20 25 30

Time (Sec)

Absolute Acceleration of Roof Floor Measured

Predicted

Figure 6 Comparison of measured and predicted absolute acceleration in X direction under TCU047 300% ground motion using a set of 10 parameters identified from each ground excitation

-50 -25 0 25 50

Displacement (mm)

Relative Displacement of Base Isolation Layer Measured

Predicted

Relative Displacement of 2nd Floor Measured

Predicted

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10 15 20 25 30

Time (Sec)

-50 -25 0 25 50

Displacement (mm)

Relative Displacement of 3rd Floor Measured

Predicted

10 15 20 25 30

Time (Sec)

Relative Displacement of Roof Floor Measured

Predicted

Figure 7 Comparison of measured and predicted relative displacements in X direction under TCU047 300% ground motion using a set of 10 parameters identified from combined hysteresis loops of two

ground excitations

-3.5 -1.75 0 1.75 3.5

Acceleration (m/s2)

Absolute Acceleration of Base Isolation Layer Measured

Predicted

Absolute Acceleration of 2nd Floor Measured Predicted

10 15 20 25 30

Time (Sec)

-3.5 -1.75 0 1.75 3.5

Acceleration (m/s2)

Absolute Acceleration of 3rd Floor Measured Predicted

10 15 20 25 30

Time (Sec)

Absolute Acceleration of Roof Floor Measured

Predicted

Figure 8 Comparison of measured and predicted absolute acceleration in X direction under TCU047 300% ground motion using a set of 10 parameters identified from combined hysteresis loops of two

ground excitations

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國科會補助計畫衍生研發成果推廣資料表

日期:2011/09/27

國科會補助計畫

計畫名稱: 使用高阻尼橡膠隔震支承之建築結構非線性地震力分析計畫主持人: 黃震興

計畫編號: 99-2221-E-011-036- 學門領域: 結構應力

無研發成果推廣資料

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99 年度專題研究計畫研究成果彙整表

計畫主持人：黃震興計畫編號：99-2221-E-011-036- 計畫名稱：使用高阻尼橡膠隔震支承之建築結構非線性地震力分析

量化

成果項目 ^{實際已達成}

數（被接受或已發表）

預期總達成數(含實際已

達成數)

本計畫實際貢獻百

分比

單位

備註（質化說明：如數個計畫共同成果、成果列為該期刊之封面故事 ...

等）

期刊論文 0 0 100%

研究報告/技術報告 1 2 100%

研討會論文 0 0 100%

論文著作篇

專書 0 0 100%

申請中件數 0 0 100%

專利已獲得件數 0 0 100% 件

件數 0 0 100% 件

技術移轉

權利金 0 0 100% 千元

碩士生 1 2 100%

博士生 0 0 100%

博士後研究員 0 0 100%

國內

參與計畫人力

（本國籍）

專任助理 0 0 100%

人次

期刊論文 0 1 100%

研究報告/技術報告 0 0 100%

研討會論文 0 0 100%

論文著作篇

專書 0 0 100% 章/本

申請中件數 0 0 100%

專利已獲得件數 0 0 100% 件

件數 0 0 100% 件

技術移轉

權利金 0 0 100% 千元

碩士生 0 0 100%

博士生 0 0 100%

博士後研究員 0 0 100%

國外

參與計畫人力

（外國籍）

專任助理 0 0 100%

人次

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其他成果

(

無法以量化表達之成

果如辦理學術活動、獲得獎項、重要國際合作、研究成果國際影響力及其他協助產業技術發展之具體效益事項等，請以文字敘述填列。)

無

成果項目量化 名稱或內容性質簡述

測驗工具(含質性與量性) 0

課程/模組 0

電腦及網路系統或工具 0

教材 0

舉辦之活動/競賽 0

研討會/工作坊 0

電子報、網站 0

科教處計畫加填項

目計畫成果推廣之參與（閱聽）人數 0

(15)

國科會補助專題研究計畫成果報告自評表

請就研究內容與原計畫相符程度、達成預期目標情況、研究成果之學術或應用價值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）、是否適合在學術期刊發表或申請專利、主要發現或其他有關價值等，作一綜合評估。

1. 請就研究內容與原計畫相符程度、達成預期目標情況作一綜合評估

■達成目標

□未達成目標（請說明，以 100 字為限）

□實驗失敗

□因故實驗中斷

□其他原因說明：

2. 研究成果在學術期刊發表或申請專利等情形：

論文：□已發表 □未發表之文稿 ■撰寫中 □無專利：□已獲得 □申請中 ■無

技轉：□已技轉 □洽談中 ■無其他：（以 100 字為限）

3. 請依學術成就、技術創新、社會影響等方面，評估研究成果之學術或應用價值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）（以 500 字為限）

The concept of base isolation is gaining widespread acceptance in the global earthquake engineering community due to the excellent performance of base-isolated structures during the 1994 Northridge and 1995 Kobe earthquakes. Some of the commonly used isolation systems are elastomeric bearings including lead-rubber bearings (LRB) and high damping rubber (HDR) bearings as well as sliding isolation systems. In recent years, a few mathematical models of high damping elastomeric isolation bearings have been proposed. Hwang et al proposed a mathematical model modified from Pan and Yang’s model. The number of parameters used by Pan and Yang are reduced, and the model is extended to be capable of describing the Mullins effect and scragging effect of high damping rubber bearings.

A mathematical model of high damping elastomeric isolation bearings proposed by Hwang et al has been validated by the cyclic loading test of HDR materials. In this work, this model will be validated using cyclic loading tests of HDR bearings.

The comparison between the predicted and experimental results indicates the proposed model is capable of predicting the dynamic hysteretic behavior of HDR bearings under different axial loads and different excitation frequencies. In addition, this proposed model of HDR isolation bearings will be used to predict

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table. A single set of 10 parameters used in the proposed mathematical model is obtained by combined just two hysteresis loop of eight shaking table test. From the test result, it is concluded that the proposed model of HDR isolation bearing is capable of predicting the seismic response of a base-isolated multistory structure if the model is well calibrated by a set of dynamic tests.