a novel method of searching appropriate ranges of base isolation

STRUCTURAL CONTROL AND HEALTH MONITORING

Struct. Control Health Monit. (2008)Published online in Wiley InterScience(www.interscience.wiley.com). DOI: 10.1002/stc.259

A novel method of searching appropriate ranges of baseisolation design parameters through entropy-based

classification

Pei Chiung Huang1, Shiuan Wan2,�,y and Jia Yih Yen1

1Department of Civil Engineering, National Chung Hsing University, Taichung, Taiwan2Department of Information Management, Ling Tung University, Taichung, Taiwan

SUMMARY

This paper proposes an idea of selecting the proper ranges of design parameters for modifying the responsesof base-isolated systems. Parametric analysis of base isolation designs on seismic protection is generallyachieved through multiple adjustments of system parameters based on the observations of system responses,but applying such procedures is very time consuming. If the ranges of inputs (design parameters) could bereasonably estimated earlier to obtain suitable outputs (structural responses), then the computational effortscould be reduced. This study generates limited design cases to analyze the relationships between designs andsystem responses. Specifically, the Shannon entropy, a Data Mining method, can be used to classify theproper and improper ranges of design parameters. A case study on the lead rubber bearing base-isolatedmachinery located at Central Science Park in Taiwan subjected to several records of Chi-Chi earthquakefrom different stations is presented. Entropy-based classification is applied in the case study and thepracticability of the analyzed result is verified. Copyright r 2008 John Wiley & Sons, Ltd.

KEY WORDS: base isolation design; entropy-based classification; seismic protection

INTRODUCTION

Base isolation system has become a practical strategy for an earthquake-resistant design inrecent years. In the past literature, parametric studies on base-isolated structures are performedthrough iterations and adjustments of system parameters to produce acceptable design strategiesbased on the performance of base isolation devices and the reduction of structural responses toearthquakes. System parameters and structural responses become important concepts of baseisolation. Clearly speaking, on the one hand, the system parameters considered are usually

*Correspondence to: Shiuan Wan, Department of Information Management, Ling Tung University, Taichung, Taiwan.yE-mail: [email protected]

Contract/grant sponsor: National Science Council project; contract/grant numbers: 95-2415-H-275-001, 95-2221-E-005-121

Copyright r 2008 John Wiley & Sons, Ltd.

Received 2 August 2007Revised 9 January 2008

Accepted 22 February 2008

involved with basic properties of the structures and the isolators such as the mass, stiffness, yieldstrength, damping ratio and natural periods. On the other hand, structural responses are used intwo major aspects: (1) measurable indexes: acceleration, displacement, forces and ductility (suchas [1–6]) and (2) energy indexes (such as [7–9]).

A possible manner of finding the optimal design for the base isolation systems is usuallyachieved through adjustments of the system parameters. These adjustments are concerned withthe iterations in dynamic analyses, whereas the structural responses are ensured to protectstructural safety. However, in fact, it could be difficult to obtain fitting design strategies if (1)initial design parameters are selected inappropriately (insensitive or out of range) or (2) multipleearthquakes are considered for the excitations of a base isolation system. Base isolation designswith the above problems may cause tedious and repeated iteration calculations to obtainexpected structural responses.

Therefore, Wan and Yen [10] used the modified back propagation neural network (BPNN)as a predictor to substitute the step-by-step integration of the dynamic approach. Hundreds ofdesign cases (containing different combinations of design parameters and correspondingsystem responses) are first sampled and learned from BPNN. Then, the system responsesfor any combination of design parameters can be precisely predicted through the learnedsystem of BPNN. In their study, a convenient method is found to obtain relationshipsamong massive inputs and system responses. Unfortunately, many design cases are necessaryto construct a reliable learned system of BPNN. In addition, a learned system of BPNNfrom the design cases with the same background may be merely suitable for a specificcondition with a given structural system and a ground motion excitation. Thus, a conciseapproach, the response surface method (RSM) introduced by Box and Wilson [11], is thendiscussed.

Specifically, RSM has been extensively applied in many fields for parametric optimizationssuch as materials [12], food chemistry [13], structural reliability [14,15] and so on. In thismethod, the relationships between parameters (or variables, xi) and responses (y) in a givensystem can be described as an approximation surface. This approximation surface can beexpressed by common models such as

y ¼ b0 þXi

bixi þ E ð1Þ

y ¼ b0 þXi

bixi þXi

Xj

bijðiojÞxixj þ E ð2Þ

y ¼ b0 þXi

bixi þXi

biix2i þ

Xi

Xj

bijðiojÞxixj þ E ð3Þ

where e is the error component and b is the coefficient of weight. The optimal approximationsurface for suitable values of b can be obtained by minimizing the error component e throughregression. Then, the ideal design point or the best values of parameters will be located at thehighest point, lowest point or saddle point of the approximation surface. This approximationsurface can be constructed from a few samples of the relationships between xi and y. Higher ordersof Equations (1)–(3) simulate more complicated approximation surfaces than those of the lowerorders. Although RSM provides a concise manner of finding optimal design results, the quantityand the quality of the given samples (xi and y) are significant. Some experimental techniques areused to select suitable samples such as the central composite design and Box–Behnken design [16]

P. C. HUANG, S. WAN AND J. Y. YEN

Copyright r 2008 John Wiley & Sons, Ltd. Struct. Control Health Monit. (2008)

DOI: 10.1002/stc

and so on, but even then the optimal design results are sensitive to the samples especially for theproblems of complex approximate surfaces. Further, even if a suitable design result is chosen, thisresult may still be unreasonable if a local minimum (or maximum) is encountered.

In this study, the objective of a precise optimal design point (or the best combination ofparameter values) is replaced by a general concept of suitable ranges of design parameters todecrease the influence of the selecting samples on the optimal design results. Additionally, acommon random sampling procedure is adopted instead of sampling by experimentaltechniques. Then, an efficient data mining technique, entropy-based classification, is used toallocate the optimal boundaries of the suitable ranges from given initial ranges of designparameters. Also, the local minimum (or maximum) can be rationally avoided. This papershows that a few given selected (design) samples can attain good design strategies throughentropy-based classification. That is, our study relies on two basic concepts of entropy-basedclassification including (1) the main process of ‘classification’, and (2) the criterion of ‘entropy’.In the next section, these concepts are briefly introduced.

Classification is a task of grouping data with multiple attributes into relevant categories.Examples of classification [17] include detecting spam e-mail messages based on the messageheader and content, categorizing cells as malignant or benign based on the results of MRI scansand classifying galaxies based on their shapes. The objective of classification herein is to classifythe structural responses with multiple attributes of system (or design) parameters.

Many techniques were developed to handle classification problems, such as decision tree,rule-based and nearest-neighbor classifiers and support vector machine. Each of them includesrespective criteria in the analyzing procedures. This study utilizes the Shannon entropy [18] inthe decision tree as the analysis criterion. Briefly, the Shannon entropy is the level of disorderbetween causes (or attribute) and results (or decision) in a concerned subject. It can be employedto decide clearly whether the entropy in a subject is small. This investigation applies theShannon entropy with classification concept (namely entropy-based classification) to assessthe relationships between the decisions of the structural response categories and the attributes ofthe given design parameters.

The Shannon entropy is applied in many fields such as the data exploration task in theepidemiology of school children injuries by Vorko and Franjo [19], the potential availability ofwater resources in an area in terms of disorder in intensity and over-a-year apportionment ofmonthly rainfall by Maruyama et al. [20] and the possibility of adopting the class entropy of theoutput of a connectionist phoneme recognizer to predict time boundaries between phoneticclasses by Salvi [21].

The steps for the entire study are shown in Figure 1. A study data set is usually requiredbefore using entropy-based classification for knowledge extraction. This study used a well-developed study data set to describe the feasible designs of lead rubber bearing (LRB) base-isolated machinery systems. The data set was predetermined by randomly generating thepractical values of design parameters. The structural responses from the ground motion inputsof Chi-Chi earthquake (station TCU053) were calculated for an illustration example. That is,the given parameters and corresponding dynamic responses were used as the elements of thestudy data set. In the next procedure, entropy-based classification was applied in the study dataset to attain the knowledge rules on allocating the proper ranges of the design parameters forsatisfactory structural responses.

In the results, the knowledge rules obtained through entropy-based classification wereverified by a test data set with a series of design examples generated randomly based on these

SEARCHING PROPER RANGES OF BASE ISOLATION DESIGN PARAMETERS


DOI: 10.1002/stc

rules. In addition, 10 Chi-Chi earthquake records of surrounding stations (shown in Figure 2)were used as different input forces of the base isolation systems, respectively. These verificationsensured that the outcomes were informative.

DEVELOPMENT OF A STUDY DATA SET OF BASE-ISOLATED MACHINERYDESIGNS

An LRB base-isolated two degrees of freedom (DOF) machinery (Figure 3(a)) system is used asthe structural model for the construction of study data set. Figure 3(b) shows the masses m1, m2,equivalent stiffness k1, k2 and equivalent damping ratios x1 and x2.

Referring to the New Zealand Ministry of Works and Development [22], the stiffness of LRBdevices in the horizontal direction is assumed to be bilinear (Figure 4) and has the followingmechanical properties:

Kd ¼GrAr

lr1þ 12

Al

Ar

� �ð4Þ

Kd ¼ aKu ð5Þ

Qd ¼ ð1� arÞfylAl ð6Þ

Set up initial ranges of design parameters

Construct a study data set

1. Randomly generate design parameters2. Obtain structural responses through dynamic analysis

Generate a test data set

Attain knowledge rules of modified designranges through entropy-based classification

Verify the acceptable level of the knowledge rules

Plot time-histories andresponse spectra

Figure 1. Steps of the entire study.



DOI: 10.1002/stc

where a is a ratio of stiffness after yielding (Kd) to that before yielding (Ku) in LRB devices withyield strength Qd. The parameters of Gr, Ar, lr and ar denote the shear modulus of elasticity, areaunder pressure, net height and strain hardening coefficient of the rubber layers in LRB devices,respectively. The parameters Al and fyl represent the section area and yield stress of the leadplug-in LRB devices, respectively. The equivalent damping ratio (xe) of LRB devices can bederived as

xe ¼4Qdðdi � dyÞ2pKeffd

2i

¼4Qd di �

aQd

ð1�aÞKd

� �

2p QdþKddidi

� �d2i

ð7Þ

where di, dy and Keff indicate the design deformation, yield deformation and effective stiffness ofLRB devices, respectively.

Central Science Park

TCU105TCU104

TCU100TCU053

TCU048

TCU061

TCU057

TCU056

TCU051TCU050

4km

0 20 40 60 80 100

Time (sec)

-0.33

0

0.33A

ccel

erat

ion

(g) TCU048

0 20 40 60 80 100 100

Time (sec)

-0.33

0

0.33

Acc

eler

atio

n(g

) TCU050

0 20 40 60 80

Time (sec)

Time (sec)

Time (sec)

-0.33

0

0.33

Acc

eler

atio

n( g

) TCU051

0 20 40 60 80 100

Time (sec)

Time (sec)

-0.33

0

0.33

Acc

eler

atio

n (g

) TCU053

0 20 40 60 80 100

0 20 40 60 80 100

-0.33

0

0.33

Acc

eler

atio

n(g

) TCU056

0 20 40 60 80 100-0.33

0

0.33

Acc

eler

atio

n(g

) TCU057

-0.33

0

0.33

Acc

eler

atio

n(g

) TCU061

0 20 40 60 80 100Time (sec)

0 20 40 60 80 100Time (sec)

0 20 40 60 80 100Time (sec)

-0.33

0

0.33

Acc

eler

ati o

n(g

)

-0.33

0

0.33

Acc

eler

ati o

n(g

)

-0.33

0

0.33A

ccel

erat

i on

(g)TCU100 TCU104 TCU105

Figure 2. Stations and associate Chi-Chi earthquake records around Central Science Park in Taiwan.



DOI: 10.1002/stc

This study further assumes that all the designs have the following characteristics: (1) the LRBdevices are fixed on the base mounted below the precision machinery; (2) the horizontal stiffnessof the machinery is assumed to be elastic; (3) the damping ratio of the machinery system is 0.02and (4) rocking and vertical vibrations of the machinery system are prevented.

In an information system, in general, two parts (1) attributes and (2) decisions areincorporated into the information table. It can be presented as follows:

Attributes: Based on the above assumptions, four independent design parameters (m1/m2,k1/k2, a and Qd) are used to describe the mechanical properties of the LRB base-isolatedmachinery systems. These four parameters are generated randomly in practical ranges duringdesign procedures, and assigned to the attributes in the study data set.

Earthquake-induced ground motions

Mass ofmachinery

Mass of a base andbase isolation devices

Equivalentdamping ofmachinery

Equivalentstiffness ofmachinery

Equivalentdamping of

base isolation device

Equivalentstiffness of

base isolation device

Displacementof machinery

Displacement of base and base isolation device

m2

m1

u2

u1

k2 �2

k1 �1

üg

(a) (b)

Figure 3. (a) Precision machinery and (b) base-isolated structural model.

Displacement

Force

di

Keff

Qd

dy

Ku

Kdfyl

Figure 4. The force–displacement relationship of LRB devices.



DOI: 10.1002/stc

Decision: Three basic system responses are assigned to the decisions in the study data set:(A) Energy dissipation of LRB devices, (B) the maximum relative displacement of LRB devices(u1;max, m), and (C) the maximum acceleration response of machinery ( €u2;max, m/s2). Amongthese decisions, B and C can be attained directly through numerical analysis. Besides, a ratioof the hysteretic energy dissipated by LRB devices to the earthquake-induced input energy(ELRB/Ein) is defined as the decision of A. Specifically, decision A suggests the capacity of theLRB device to absorb earthquake-induced input energy of the structural system. The parameterof ELRB can be obtained by summing the area of the hysteresis loops of LRB devices, and Ein isdefined by the following equation:

Ein ¼ �Z u

0

M €ugðtÞ du ð8Þ

where M and u are the mass and relative displacement of the structural system, and €ugðtÞ is theground acceleration.

ENTROPY-BASED CLASSIFICATION CONCEPT

A classification approach based on the Shannon entropy is introduced to search therelationships between attributes (or design parameters) and decisions (or the structuralresponses) in the study data set. The basic concepts of classification are (1) quantifying thedisordered degree (Shannon entropy) between attributes and decisions and (2) obtaining theappropriate ranges of attributes to achieve the better category of decision based on the lowervalue of entropy.

An example is given to clarify the procedures of entropy-based classification. Table I shows aconcerned study data set of 10 data (x1 to x10 in column (a)) with an attribute (column (b)) and acorresponding decision (column (c)). The detailed calculation procedures of entropy-basedclassification techniques are shown in the following steps.

In the first step, the data attributes are sorted in an ascending order.Second, a fictitious cutting point (FCP(t)) is defined as the mean values of two different

attribute values from column (e) of Table I, and t represents the ID number of the fictitiouscutting point. The parameter FCP(t) divides the attributes into two different classes (attributeclasses): attribute-class 1 means that the attribute value is smaller than FCP(t) and attribute-class 2 indicates that it is larger than FCP(t). In a real database, it is possible that two data havethe same attribute classes with different decision classes. It can be considered that one of them isa noise data. Therefore, in this case, there may be different values of entropy with regard to thedata of the two attribute classes. The values of entropy can be calculated by the followingequation:

entropyðtÞj ¼ �X2i¼1

pðijtÞj log2 pðijtÞj ð9Þ

where i is the decision class (‘1’ or ‘2’ in this example), j the attribute class and pðijtÞj theprobability of data for decision-class i in the data for attribute-class j. For instance,pði ¼ 2jt ¼ 3Þj¼1 represents the probability of the data for decision-class 2 in the data for theattribute value smaller than FCP(3) (attribute-class 1). However, pðijtÞj ¼ 0 is assigned aspðijtÞj log2 pðijtÞj ¼ 0.



DOI: 10.1002/stc

Entropy is an index of disorderliness among decisions and attributes. The values of entropy,herein, vary from 1.0 to 0.0. For instance, there are one opaque box and 10 balls (containing fivewhite balls and five black balls). If these 10 balls are put into the box, the probability of selectinga black ball the first time is only 0.5. That is, it is difficult to ensure the color of the first selectedball. Mathematically, the entropy is then used to describe the disorder level of the first selectedblack ball:

entropyðtÞfirst selected black ball ¼ ð�0:5 log2 0:5Þblack balls þ ð�0:5 log2 0:5Þwhite balls ¼ 1:0

If only black balls are put into the box, the color of the first selected ball from the box isundoubtedly black, and the entropy of the first selected black ball is

entropyðtÞfirst selected black balll ¼ ð�1:0 log2 1:0Þblack balls þ ð�0:0 log2 0:0Þwhite balls ¼ 0:0

In other words, when entropyðtÞj is 1.0, it means the attribute class (the black-and-white balls inthe box) has only a 50% of chance of making the right decision (selecting a black ball). IfentropyðtÞj is 0.0, it signals that the attribute class is 100% related to the decision class.Therefore, it is expected that one obtains a smaller value of entropyðtÞj in each attribute class.

Another concept of information gain (IG) is then introduced following the concept ofentropy. IG can be computed by the following equation:

igðtÞj ¼ 1:0� entropyðtÞj ð10Þ

Table I. An example of entropy-based classification.

Data Attribute mi Decision t Attribute of FCP(t) IG(t)

(a) (b) (c) (d) (e) (f)

x1 1 11 2.0 0.108

x2 3 12� 5.0y 0.236z

x3 7 23 8.0 0.035

x4 9 14 10.0 0.125

x5 11 15 13.0 0.278

x6 15 26 19.0 0.125

x7 23 17 26.5 0.396

x8 30 28 33.0 0.236

x9 36 29 38.0 0.108

x10 40 2

�t5 2.yFCP(2)5 5.0. FCP(2) is the mean value of 3 and 7. That is, the cutting point is 5, which divides mio5 and mi45 intotwo parts.zThe entropy-based classification of FCP(2) is given in Table II.



DOI: 10.1002/stc

Referring to entropyðtÞj , the values of igðtÞj also vary from 0.0 to 1.0. It has to be noted that thevalues of igðtÞj are contrary to entropy and are in accordance with the tendency of the attribute-class proportion in the study data set. In the next step, a given FCP(t) can be derived to theproportion of each attribute class. It can be consequently multiplied by igðtÞj to obtain the resultof IG(t) as the following equation:

IGðtÞ ¼X2j¼1

PðjjtÞ � igðtÞj ð11Þ

where P(j|t) is the proportion of data for attribute-class j to all of the data.Therefore, the index of IG(t) used in this study can be considered as a measurable value of the

assessment for a given FCP(t). The larger IG(t) means the better choice of FCP(t). That is, incomputing a series of IG(t), we selected the largest IG(t) to display the best relationshipsbetween attributes and decisions.

For example, IG(2) (t5 2 and FCP(2)5 5.0 in columns (d) and (e) of Table I) is cited asfollows and is shown in Table II:

(1) In column (b) of Table I, there are two (x1 and x2) and eight (x3 to x10) data that fall intoattribute-class 1 (the data for attribute value smaller than FCP(2) with j5 1) andattribute-class 2 (for those larger than FCP(2) with j5 2), respectively. That is,Pðj ¼ 1jt ¼ 2Þ ¼ 0:2 and Pð2j2Þ ¼ 0:8, as shown in column (c) of Table II.

(2) A(i,j) (from column (d) of Table II) is the number of data for conditions of i and j. Boththe above-mentioned data (x1 and x2) for attribute-class 1 fall into decision-class 1, but nodata fall into decision-class 2. Thus, Aði ¼ 1; j ¼ 1Þ ¼ 2 and Að2; 1Þ ¼ 0. Moreover,Að1; 2Þ ¼ 3 represents the data of x4, x5, and x7 and Að2; 2Þ ¼ 5 represents the data of x3,x6, x8, x9 and x10.

(3) Attribute-class 1 (j5 1) has two data (x1 and x2). Both these data satisfy i5 1. Therefore,the ratio pði ¼ 1jt ¼ 2Þj¼1 ¼ 2=ð2þ 0Þ ¼ 1:0 in column (e) of Table II. Additionally,pð2j2Þ1 ¼ 0=ð2þ 0Þ ¼ 0. Using the same procedure, pð1j2Þ2 ¼ 3=ð3þ 5Þ ¼ 0:375 andpð2j2Þ2 ¼ 5=ð3þ 5Þ ¼ 0:625.

(4) Columns (f)–(h) are based on Equations (9)–(11).

Steps (1)–(4) are repeated to attain the rest of the values on IG(t) with respect to differentconditions of FCP(t). These results are shown in column (f) of Table I. Among these results,IG(7) is the biggest one; hence, FCP(7)5 26.5 denotes the optimal cutting point. Table I shows

Table II. Entropy-based classification of FCP(2) of Table I.

Description

j i P(j|t) A(i,j) p(i|t)j Entropy(t)j ig(t)j IG(t)

(a) (b) (c) (d) (e) (f) (g) (h)

1 1 0.2 2 1.0 0 1.0 0.2362 0 0

2 1 0.8 3 0.375 0.954 0.0462 5 0.625



DOI: 10.1002/stc

that the majority of decision falls into class 1 if the attribute value is located between 1 and 26.5.Additionally, the decision class is 2 when the attribute value is larger than 26.5. Accordingly,knowledge rules for the appropriate range of attributes to achieve the expected decision classesare obtained.

EXAMPLE OF CALCULATION

This calculation example has two steps: (1) the development of a study data set with the designparameters and the structural responses obtained through step-by-step integration of numericalanalysis and (2) the application on entropy-based classification techniques to obtain theknowledge rule of appropriate ranges of design parameters from the study data set.

A study data set was built initially by 25 design cases containing the four design parameters(m1/m2, k1/k2, a and Qd) generated randomly in practical ranges [10] (0:15om1=m2o0:3,0:15ok1=k2o0:3, 0:15oao0:3 and 15oQdo30 kN). Besides, the three system responses(ELRB=Ein, u1;max and €u2;max) are computed through the dynamic analysis of the two DOFsystem subjected to Chi-Chi earthquake excitation recorded by station TCU053. Table IIIshows an example of the study data set for the attributes (columns (b)–(e)) and the decision(column (f)). To simplify this example and to clarify the processes of entropy-basedclassification, only the response ELRB=Ein is demonstrated as the decision in this section. Acutting point of the decision is assumed to be the average value (0.288) of column (f). Thedecision class is then defined based on this cutting point with ‘1’ for the smaller value of decisionand ‘2’ for the larger value of decision (as shown in column (g)). The objectives of this study areto allocate the proper cutting point of attribute and to select the suitable attribute class (as theappropriate range), and then the results of decision class mostly fall on 2 (the larger energy ratioof ELRB=Ein).

To allocate the appropriate ranges of design parameters, the attributes of m1/m2, k1/k2, a andQd are first assumed to be independent. The attribute in the aforementioned example of entropy-based classification (column (b) of Table I) can then be replaced by each one of these fourattributes (columns (b)–(e) of Table III) and the decision (column (c) of Table I) of eachattribute is all referred to column (g) of Table III. Finally, the optimal cutting point of eachattribute can be obtained by an entropy-based classification. In Table III, the results of optimalcutting points for the attributes of m1/m2, k1/k2, a and Qd are 0.264, 0.249, 0.270 and 22.8 kN,respectively. Table IV presents the classified results based on these optimal cutting points, withattribute-class 1 for the smaller value of attribute and 2 for the larger value of attribute.

Table V presents a statistic result to assess the appropriate attribute class for the objective ofdecision-class 2 (the larger energy ratio of ELRB/Ein). In this table, column (a) shows theattributes; columns (b) and (c) list the classes of attribute and decision, individually; and column(d) presents the number of data for the conditions of columns (b) and (c). For instance, there areno data in Table IV for attribute-class 2 of m1/m2 and decision-class 1. Additionally, there arefour data (x14, x16, x19 and x21) for attribute-class 2 of m1/m2 and decision-class 2. The properchoice in column (e) on the attribute class is defined as the majority of the portion of decision-class 2. Finally, the modified ranges of attributes (where columns (f) and (g) also show the initialranges as the comparisons) can be obtained by considering the initial practical ranges, theoptimal cutting points and the proper choices of attribute class. These modified ranges are much



DOI: 10.1002/stc

smaller than the initial ranges. Further, the modified ranges could provide the effective designstrategy to increase the responses of ELRB/Ein.

A PRACTICAL CASE STUDY

In the previous example, in the first stage, a limited number of design examples in the study dataset were generated, and one of the system responses (ELRB/Ein) was sampled as the decision. Thestudy data set was then analyzed through an entropy-based classification for a series ofquestionable knowledge rules (modified ranges of design variables). In this section, fourquestions arise from the knowledge rules:

(1) Are the processes of entropy-based classification feasible for the different decisions (ELRB/Ein, u1;max and €u2;max) in the study data set?

(2) How much data in the study data set are adequate for gaining reliable knowledge rules?(3) How is the reliability of the knowledge rules assessed?(4) Is the entropy-based classification applicable to other different excitations?

Table III. An example of the study data set.

DataAttribute

Decision Decision classm1/m2 k1/k2 a Qd ELRB/Ein

(a) (b) (c) (d) (e) (f) (g)

x1 0.187 0.215 0.292 27.7 0.20 1x2 0.167 0.226 0.188 16.1 0.45 2x3 0.239 0.178 0.243 23.5 0.28 1x4 0.219 0.228 0.245 29.3 0.19 1x5 0.220 0.200 0.296 29.0 0.21 1x6 0.249 0.249 0.292 23.7 0.29 2x7 0.202 0.178 0.279 24.0 0.12 1x8 0.183 0.185 0.186 26.9 0.20 1x9 0.182 0.248 0.284 28.6 0.26 1x10 0.222 0.168 0.203 29.4 0.19 1x11 0.190 0.181 0.292 29.8 0.15 1x12 0.247 0.198 0.236 15.1 0.40 2x13 0.176 0.178 0.273 22.9 0.23 1x14 0.273 0.299 0.243 24.6 0.29 2x15 0.205 0.279 0.256 16.2 0.39 2x16 0.285 0.186 0.262 16.9 0.37 2x17 0.231 0.177 0.252 22.6 0.33 2x18 0.254 0.278 0.297 29.4 0.14 1x19 0.293 0.174 0.190 24.2 0.37 2x20 0.193 0.295 0.192 20.2 0.43 2x21 0.292 0.172 0.188 20.3 0.32 2x22 0.248 0.177 0.250 24.5 0.18 1x23 0.174 0.271 0.255 27.5 0.34 2x24 0.166 0.257 0.164 16.4 0.49 2x25 0.195 0.193 0.266 17.9 0.38 2



DOI: 10.1002/stc

In this section, all the three system responses will be used for the first question. Also, threedifferent numbers of study data sets (containing numbers 25, 50 and 100) with 10 repeatinggenerations in each number are used to consider the second question. Then, a test data setcontaining 200 design examples were randomly generated based on the knowledge rules (theproper ranges of design parameters obtained from analyzing the study data sets) to discuss thethird question. An artificial index of Pfailure is defined as the percentage of the data in the testdata set for the failure predictions on the decision classes. The proper decision classes for aneffective isolator of ELRB/Ein, u1;max and €u2;max are herein assumed to be 2, 2 and 1, respectively.That is, larger system responses of ELRB/Ein and u1;max and smaller responses of €u2;max areexpected. The lower Pfailure in the test data set indicates a better combination of design variables.

Table VI presents the results of Pfailure from the base-isolated system subjected to theexcitation of TCU053. The results of the three system responses are included in this table.Column (a) of Table VI shows the different numbers of study data sets. Row (a) shows the 10repeating generations in each number of the study data set.

In Table VI, the Pfailure values of ELRB/Ein show reliable results. About 90% of the decisionsin the test data set (generated from the proper ranges of the design parameters) exceed theaverage of the decisions in the study data set. In addition, most Pfailure results of the repeatinggenerations from the 25 data in the study data set are acceptable compared with those from the

Table IV. Classified results of Table III through the Shannon entropy.

DataAttribute class

Decision classm1/m2 k1/k2 a Qd

(a) (b) (c) (d) (e) (f)

x1 1 1 2 2 1x2 1 1 1 1 2x3 1 1 1 2 1x4 1 1 1 2 1x5 1 1 2 2 1x6 1 2 2 2 2x7 1 1 2 2 1x8 1 1 1 2 1x9 1 1 2 2 1x10 1 1 1 2 1x11 1 1 2 2 1x12 1 1 1 1 2x13 1 1 2 2 1x14 2 2 1 2 2x15 1 2 1 1 2x16 2 1 1 1 2x17 1 1 1 1 2x18 1 2 2 2 1x19 2 1 1 2 2x20 1 2 1 1 2x21 2 1 1 1 2x22 1 1 1 2 1x23 1 2 1 2 2x24 1 2 1 1 2x25 1 1 1 1 2



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numbers of 50 and 100 data. Although the Pfailure of u1;max and €u2;max are larger than ELRB/Ein

and, respectively, achieve about 50 and 30%, all these results indicate that 25 data in the studydata set are feasible compared with the results of the 50 and 100 data. Even if the first generationof the 25 data fails to meet this requirement, it is easy to ensure the acceptable results in solidnumbers of generations.

Another assessing index (Increment) to the test data is then defined and discussed as thefollowing equation:

Increment ¼Decavg;test �Decavg;study

Decavg;study� 100ð%Þ

Table V. Strategies of attribute classes for decision-class 2 in Table IV.

Conditions Modified range

Attribute Attribute class Decision classNumber of data for

the conditions Choice Min. Max.

(a) (b) (c) (d) (e) (f) (g)

m1/m2 1 1 12 —2 9

2 1 0 V 0.264 0.32 4 (0.15�) (0.3�)

k1/k2 1 1 11 —2 7

2 1 1 V 0.249 0.32 6 (0.15�) (0.3�)

a 1 1 5 V 0.15 0.272 12 (0.15�) (0.3�)

2 1 7 —2 1

Qd (kN) 1 1 0 V 15.0 22.82 9 (15.0�) (30.0�)

2 1 14 —2 2

�Initial range.

Table VI. The Pfailure from different data numbers of the study data set.

Decision (a) 1 2 3 4 5 6 7 8 9 10

ELRB/Ein 25 0.06 0.02 0.01 0.01 0.05 0.07 0.10 0.09 0.25 0.1350 0.05 0.09 0.09 0.01 0.06 0.23 0.02 0.15 0.05 0.20100 0.12 0.08 0.06 0.03 0.02 0.08 0.11 0.02 0.15 0.06

u1;max (m) 25 0.33 0.69 0.56 0.40 0.31 0.44 0.47 0.26 0.45 0.3850 0.33 0.41 0.44 0.43 0.32 0.31 0.58 0.41 0.56 0.40100 0.53 0.59 0.44 0.56 0.25 0.36 0.33 0.43 0.36 0.30

€u2;max (m/s2) 25 0.18 0.29 0.26 0.20 0.02 0.30 0.24 0.14 0.25 0.0850 0.38 0.05 0.02 0.03 0.18 0.10 0.08 0.03 0.13 0.04100 0.04 0.16 0.02 0.15 0.11 0.30 0.02 0.04 0.46 0.15

Pfailure is the percentage of the data in the test data set for the failure predictions on the decision classes.



DOI: 10.1002/stc

where Decavg;study and Decavg;test are the average values of decisions (the values of ELRB/Ein,u1;max or €u2;max) in the study data set and the test data set, respectively. The Increment points outthe increment percentage for the system responses in the isolation system designed by themodified ranges to the initial ranges. Referring to the assumed proper decision classes of eachdecision, it is considered that the values of Increment are positive for ELRB/Ein and u1;max andnegative for €u2;max. That is, the larger values of ELRB/Ein and u1;max (for decision-class 2) and thesmaller values of €u2;max (for decision-class 1) are expected to obtain a better design of an isolator.

Table VII presents the results of Increment. The design cases from the modified ranges ofdesign parameters could provide more than 30% of ELRB/Ein to the designs from the initialranges. Besides, the Increment for €u2;max is about �5%. The deviations in Increment of u1;max forthe different generations are not good enough; therefore, the detailed values of Decavg;study andDecavg;test for 25 data for each decision are further discussed and presented in Table VIII.

Table VIII presents the values of Decavg;study in u1;max for the different generations located at alarge range from about 0.09 to 0.11m, but the modified results of Decavg;test are almost largerthan 0.12m. The Decavg;study of €u2;max is located between about 4.5 and 4.6m/s2, but the values ofDecavg;test are almost less than 4.4m/s2. The Decavg;study of ELRB/Ein is located between about0.28 and 0.36, but the values of Decavg;test are nearly larger than 0.4. Therefore, even if the valuesof Pfailure in u1;max are not good enough, the results of Increment are still acceptable. That is, 25data generated randomly in a study data set is acceptable, the tests of the three differentdecisions are feasible and the modified ranges of design parameters obtained from entropy-based classification are applicable to the designs of base-isolated machinery systems.

The above processes discussed the front three questions. The last examination is thecondition for several different excitations of the isolation system. Thus, 10 records of the Chi-Chi earthquake (normalized to 0.33 g) are taken from seismology stations around a science parkin Taiwan. These input forces are used in the procedures to obtain the respective study data sets(containing 25 data) for the three system responses. The corresponding test data sets (including200 data) for the study data sets are then processed for the results of Pfailure and Increment aspresented in Table IX.

In Table IX, column (a) shows the IDs of the record stations. Columns (b)–(g) present theresults of Increment and Pfailure in the three decisions of system responses. Most of these resultsare obtained in the first generation of the study data set. Only a few results are obtained fromthe second or the third generations. These results show that the values of Increment are reliable

Table VII. The Increment (%) from different data numbers of the study data set.

Decision (a) 1 2 3 4 5 6 7 8 9 10

ELRB/Ein 25 32.0 41.0 43.5 32.4 37.9 25.7 17.0 28.3 10.2 30.350 34.6 25.4 23.0 41.2 25.0 20.9 30.3 23.9 27.4 18.5100 22.1 22.2 25.9 32.0 49.7 32.5 26.7 40.4 20.7 29.4

u1;max (m) 25 44.7 �5.6 7.2 18.8 38.1 25.8 22.1 51.4 24.8 30.450 35.0 18.5 24.9 18.4 43.1 47.6 1.1 24.8 4.7 28.0100 14.5 3.5 21.7 4.0 51.2 34.2 40.3 24.8 31.8 38.5

€u2;max (m/s2) 25 �7.2 �4.4 �3.7 �4.5 �13.0 �3.5 �4.9 �7.0 �4.8 �9.250 �3.1 �10.8 �11.7 �11.2 �7.1 �8.1 �7.3 �11.4 �8.1 �10.4

100 �12.1 �6.7 �10.6 �6.2 �8.6 �4.5 �10.9 �10.1 �1.5 �7.9

Increment is the increment percentage for the system responses from the modified ranges of design variables to the initialranges.



DOI: 10.1002/stc

in the three decisions, but the values of Pfailure for u1;max are not good enough. This issue wasdiscussed in the previous questions and the other index of Increment is then considered. TheIncrement of u1;max shows the ability of raising the displacement of an isolator. Table IXpresents the base-isolation designs by modified ranges through entropy-based classification,which can effectively increase or decrease system responses for the requirements when thestructural system is subjected to different excitations.

To concisely show the feasibility of entropy-based classification, three cases of the isolationsystem subjected to the excitation of TCU053 are compared to note the system responses: (1) anon-isolated machinery, (2) an improper design case: the base-isolated machinery designed bythe initial ranges of variables but not by the modified ranges and (3) a proper design case: thebase-isolated machinery designed by the modified ranges. The design ranges for the impropercases and the proper cases are analyzed and listed in Table X (where the ranges of ELRB/Ein werecarried out in columns (f) and (g) of Table V).

Figure 5 shows the variation in energy ratio (ELRB/Ein) of time histories from the base-isolated system subjected to the excitation of TCU053 while the natural period of the non-isolated machinery is 0.3 s. The x-axis represents the time history from 30 to 90 s. The y-axisrepresents the energy ratio of ELRB/Ein. This figure has two lines: (1) a thin line for an improper

Table VIII. The Decavg;study and Decavg;test for the case of 25 data in Table VII.

Decision (a) 1 2 3 4 5 6 7 8 9 10

ELRB/Ein Decavg;study 0.316 0.288 0.297 0.320 0.284 0.320 0.357 0.304 0.318 0.310Decavg;test 0.417 0.406 0.426 0.423 0.392 0.402 0.418 0.391 0.351 0.404

u1;max (m) Decavg;study 0.089 0.102 0.102 0.095 0.089 0.104 0.104 0.111 0.099 0.106Decavg;test 0.128 0.096 0.109 0.113 0.123 0.131 0.128 0.168 0.123 0.138

€u2;max (m/s2) Decavg;study 4.626 4.587 4.591 4.596 4.552 4.395 4.574 4.487 4.556 4.630Decavg;test 4.295 4.384 4.422 4.391 3.959 4.242 4.348 4.174 4.337 4.206

�Decavg;study and Decavg;test are the average values of the decisions in the study data set and the test data set.

Table IX. The Increment and Pfailure from different excitations.

ELRB/Ein u1;max (m) €u2;max (m/s2)

Station (TCU) Increment (%) Pfailure Increment (%) Pfailure Increment (%) Pfailure

(a) (b) (c) (d) (e) (f) (g)

048 15.7 0.05 27.6 0.37 �8.1 0.07050 22.3 0.04 20.7 0.40 �7.4 0.07051 21.6 0.03 22.1 0.26 �7.2 0.06053 32.9 0.08 36.8 0.37 �12.6 0.02056 18.0 0.06 18.8 0.44 �6.7 0.10057 9.9 0.06 9.8 0.44 �7.7 0.13061 11.5 0.13 23.0 0.29 �11.1 0.03100 17.7 0.02 27.3 0.28 �3.7 0.19104 14.5 0.03 20.5 0.34 �19.3 0.06105 21.3 0.02 32.3 0.31 �9.3 0.08



DOI: 10.1002/stc

case with m1/m2 5 0.184, k1/k2 5 0.211, a5 0.293 and Qd 5 27.2 kN and (2) a thick line for aproper case with m1/m2 5 0.288, k1/k2 5 0.275, a5 0.173 and Qd 5 18.1 kN. In Figure 5, the casein the thick line shows better performance of energy dissipation by the LRB devices than for thethin line.

Figures 6 and 7 show the time histories of isolator deformation and machinery accelerationfor the three cases (the responses of the non-isolated machinery are shown as a dash line inFigure 7). The natural period of case 1 (non-isolated machinery) used here is 0.3 s. It is foundthat the proper case (thick line) has larger maximum deformation (u1;max) and lower maximumacceleration ( €u2;max) than the improper case (thin line) following the objectives of this study. Inaddition, the machinery acceleration responses of isolated machinery are much lower than thenon-isolated machinery.

A more practical study on the response spectrum is further discussed for the cases ofTCU053. Figures 8–10 show the response spectra of ELRB/Ein, u1;max and €u2;max for the improper

30 35 40 45 50 55 60 65 70 75 80 85 90Time (s)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

EL

RB

/Ein

Isolated (an improper case)Isolated (a proper case)

Figure 5. The variation in energy ratio of time-history analysis (TCU053, T5 0.3).

Table X. The design ranges for the improper cases and the proper cases.

m1/m2 k1/k2 a Qd (kN)

Decision Improper Proper Improper Proper Improper Proper Improper Proper

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

ELRB/Ein Min. 0.150 0.264 0.150 0.249 0.270 0.150 22.8 15.0Max. 0.264 0.300 0.249 0.300 0.300 0.270 30.0 22.8

u1;max Min. 0.208 0.150 0.172 0.150 0.230 0.150 15.0 25.9Max. 0.300 0.208 0.300 0.172 0.300 0.230 25.9 30.0

€u2;max Min. 0.150 0.182 0.209 0.150 0.173 0.150 21.4 15.0Max. 0.182 0.300 0.300 0.209 0.300 0.173 30.0 21.4



DOI: 10.1002/stc

and the proper cases. Besides, the machinery deformation response spectra for the impropercase and the proper case are also drawn in Figure 9 (the thin dash line and the thick dash lines).In these figures, the x-axes are the natural periods of the non-isolated machinery. Owing to (1)the natural period of non-isolated machinery used herein is 0.3 and (2) the values of ELRB/Ein

and €u2;max between the proper and the improper cases at the period of 0.9 s are nearly the same,the values of x-axes are limited between 0.1 and 0.9 s.

Figures 8–10 clearly show that as the natural period of non-isolated machinery is lower than0.5 s, the proper case has better performances on the responses of ELRB/Ein, u1;max, €u2;max and the

40 42 44 46 48 50

Time (s)

-8

-4

0

4

8

Mac

hine

ryA

ccel

erat

ion

(m/s

2 )

Isolated (an improper case)Isolated (a proper case)Non-isolated

Figure 7. Time histories of machinery acceleration (TCU053, T5 0.3).

30 32 34 36 38 40 42 44 46 48 50Time (s)

-0.08

-0.04

0

0.04

0.08

0.12

Isol

ator

Dis

plac

emen

t(m

)


Figure 6. Time histories of isolator deformation (TCU053, T5 0.3).



DOI: 10.1002/stc

maximum deformation of machinery simultaneously. In addition, if the response spectra for theperiods (non-isolated machinery) are higher than 0.5 s, a series of new case studies must begenerated to attain better design ranges.

CONCLUSIONS

This study proposed an entropy-based classification procedure to allocate proper ranges ofdesign variables in the designs of isolation systems. A case study based on an LRB base-isolated

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Nature period of non-isolated machinery (s)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

EL

RB

/Ein


Figure 8. Energy ratio response spectra of TCU053.

0.1 0.3 0.5 0.7 0.90

0.02

0.04

0.06

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9


0

0.1

0.2

0.3

0.4

0.5

0.6

Max

imum

Def

orm

atio

n( m

)

Isolator defomation (an improper case)Isolator defomation (a proper case)Machinery defomation (an improper case)Machinery defomation (a proper case)

Figure 9. Maximum isolator deformation response spectra of TCU053.



DOI: 10.1002/stc

machinery system subjected to Chi-Chi earthquake nearby the Central Science Park isconsidered. This paper draws the following conclusions.

Three system responses of ELRB/Ein (the ratio of the energy dissipating by LRB to inputenergy), u1;max (the maximum relative displacement of LRB devices) and €u2;max (the maximumacceleration response of machinery) are used as the indexes of the system responses. Entropy-based classification can effectively decrease or increase these system responses by modifying theinitial ranges of design parameters and allocating the proper ranges.

The study shows that 25 data (design examples) randomly generated in the study data set forthe analysis of entropy-based classification attain a concise and feasible manner of achieving theacceptable results of the three system responses. It is also found that more than 25 data in theanalysis process (entropy-based classification) are surplus.

Ten different earthquake-induced excitations recorded around a science park are used for theinput forces of the isolation system. The applicable levels on the knowledge rules obtained fromthe design cases with different input forces are then verified.

The system responses of ELRB/Ein, u1;max and €u2;max in a base-isolated machinery caneffectively improve our understanding through the designs from the proper ranges of designparameters. In the results of this case study, ELRB/Ein shows the best outcomes for following thedesign philosophy. That is, ELRB/Ein plays a vital role in preliminary design range for theisolation system.

NOMENCLATURE

m1 mass of machinerym2 mass of a base and base isolation devicesk1 equivalent stiffness of base isolation devices

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9


3.2

3.6

4

4.4

4.8

5.2

5.6

6

6.4

6.8

7.2

Max

imum

Mac

hine

ryA

ccel

erat

ion

(m/s

2 )


Figure 10. Maximum machinery acceleration response spectra of TCU053.



DOI: 10.1002/stc

k2 equivalent stiffness of machineryx1 equivalent damping of base isolation devicesx2 equivalent damping of machineryKd post-elastic stiffness in the horizontal direction of LRB devicesGr shear modulus of elasticity of the rubber layers in LRB devicesAl section area of the lead plug in LRB deviceslr net height of the rubber layers in LRB devicesAr area under pressures of the rubber layers in LRB devicesa ratio of stiffness after yielding to that before yielding in LRB devicesKu elastic stiffness in the horizontal direction of LRB devicesQd yield strength of LRB devicesar strain harden coefficient of the rubber layers in LRB devicesfyl yield stress of the lead plug in LRB devicesxe equivalent damping ratio of LRB devicesdi design deformation of LRB devicesdy yield deformation of LRB devicesKeff effective stiffness of LRB devicesELRB hysteretic energy dissipated by LRB devicesEin earthquake-induced input energyM mass of the structureu relative displacementugðtÞ ground accelerationu1;max maximum relative displacement of LRB devicesu2;max maximum acceleration response of machineryFCP(t) fictitious cutting pointt ID number of the fictitious cutting pointi decision classj attribute classentropy(t)j Shannon entropy based on t and jIG(t) information gain based on tP(j|t) proportion of data for attribute-class j to all of the datap(i|t)j probability of data for decision-class i in the data for attribute-class jA(i,j) number of data for conditions of i and jPfailure percentage of the failure predictions on the target decision-classesIncrement increment percentage for the system responses from the modified ranges

of design variables to the initial rangesDecavg;study average values of the decisions in the study data setDecavg;test average values of the decisions in the test data set

ACKNOWLEDGEMENTS

The authors appreciate the financial support from the National Science Council project (95-2415-H-275-001) and (95-2221-E-005-121). In addition, the native English editor of K. T. Lee foundations helped inmodifying the paper.



DOI: 10.1002/stc

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DOI: 10.1002/stc

a novel method of searching appropriate ranges of base isolation

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