Structural Analysis of Historic Construction – D’Ayala & Fodde (eds)© 2008 Taylor & Francis Group, London, ISBN 978-0-415-46872-5

Integrated modeling method for dynamic behavior of ancient pagodas

Jianli YuanYangzhou Univ., Yangzhou, China

Ling Yao & Shengcai LiDepartment of Civil Engineering, Yangzhou Univ., Yangzhou, China

Donato AbruzzeseDepartment of Civil Engineering, Univ. of Rome, Tor Vergata, Rome, Italy

ABSTRACT: Dynamic behavior model is essential to the reliability evaluation and restoration scheme ofancient pagodas. In this paper, the identification techniques and the main influence factors on the dynamicbehavior of ancient pagodas are discussed, and the modeling method integrated the predominance of parameterforecast, sensitivity analysis, and model updating criteria is developed. The Huqiu Pagoda in Suzhou City,a famous leaning pavilion-style masonry pagoda, was selected as a research case to present the applicationcharacteristics of the method. A 3D finite element model of this pagoda has been constructed, and the mainstructural parameters were updated according to the model updating criteria to match with the measured dynamiccharacteristics of ambient vibration test to ensure the validity of the model.

Keywords: ancient pagoda; dynamic behavior; integrated modeling method; environmental random excitation;finite element analysis; sensitivity system; model updating criteria.

1 INTRODUCTION

Chinese pagodas are outstanding representation ofancient high-rise buildings. There are more than 3,000ancient pagodas in China, and the most of them need tobe maintained and repaired. Modeling of the dynamicbehavior is an important step in the structural relia-bility assessment and intervention scheme of ancientpagodas. Dynamic behavior model of a structure canbe obtained by computer analysis based on the dynam-ics theory or by testing the structure on site, the keyon the model is the analysis model is equivalent to thephysical system.

The development of modern test and numeri-cal modeling technology provide new approach forconstruction of dynamic behavior model of ancientpagodas. Combined with the application of ambi-ent vibration technology and finite element pro-gram ANSYS, this paper discusses the characteristicsof the integrated method and correlative influencefactors on modeling dynamic behavior of ancientpagodas. A study case, Huqiu Pagoda in SuzhouCity, was selected to introduce the essentials ofapplication.

2 PROCESS OF THE INTEGRATEDMODELING METHOD

The process of integrated modeling method fordynamic behavior of ancient pagodas is as follows(Figure 1): (1) Survey the structure, define structuralparameters and their variation degree, (2) Construct

N. Y.

1

2

3

4

5

6

7

8

9

Figure 1. Flow chart of the integrated modeling method.

393

the preliminary analysis model by ANSYS, (3) Esti-mate the dynamic eigenvalues of the structure for theselection of proper instrument parameters, (4)Acquirethe modal parameters by ambient vibration test,(5) Establish sensitivity system and optimize structuralparameters, (6) Select model updating criteria, updatethe analysis model, (7) Evaluate the error betweenthe analysis value and test value by Errors EvaluationProgram, (8) Select adjustment parameters for newupdating, (9) Obtain the perfect model.

3 THE MODELING TECHNIQUES ANDCORRELATIVE INFLUENCE FACTORS

3.1 Major factors influencing dynamic behaviorof ancient pagodas

Ancient pagodas were built long time ago, often suf-fered various natural disasters and damages, and mostof them had been repaired. Therefore most pago-das have following common characteristics: complexstructure and construction type, non-uniform mate-rials, as well as the coexistence of various damageconditions (Yuan Jianli, Shengcai Li, 2001). Sincethe structural parameters affect directly the dynamicbehavior of ancient pagodas, the dimension, construc-tion detail, damage condition and variation degreeshould be surveyed carefully and defined reasonablyfor a good analytical model to be developed.

According to the principle of dynamics, the motionequation of an ancient pagoda can be expressed asfollows:

where {x}, {x}, {x} are the structural nodal displace-ment, velocity and acceleration vectors, respectively;[K], [C], [M ] are, respectively, structural stiffness,damping, and mass matrices; xo is the ground acceler-ation. Taking [C] = α[M ] + β[K], if Rayleigh damp-ing is adopted, the mass M and stiffness K are the mainfactors to influence dynamic behavior of the structure.

The mass of an ancient pagoda can be obtained fromthe measurement of the geometrical dimensions andmaterial density of each component. The density ofmain materials in masonry pagodas is list in Table 1.

The masonry elastic modulus is one of importantinfluence factors to the stiffness of ancient pagodas.The test and statistical data indicate that: (1) For brickmasonry the elastic modulus in direct proportion tocompressive strength of the masonry, and is relatedto the strength degree of the mortar; (2) For stonemasonry the elastic modulus is mostly dependent onartifactitious precision of the stones and the strengthdegree of the mortar. When constructing the dynamicbehavior model of ancient pagodas, the elastic mod-ulus of the masonry can be determined on the basis

Table 1. Density of main materials.

Material Density

Brick masonry 18 (kN/m3)Granite masonry 26.4 (kN/m3)Limestone masonry 25.6 (kN/m3)Sandstone masonry 22.4 (kN/m3)China fir 4∼5 (kN/m3)Pine wood 5∼6 (kN/m3)Black tile roof 0.9∼1.1 (kN/m2)

Table 2. Fundamental period of pavilion style masonrypagoda.

T (s)H D d Doors/

Pagoda (m) (m) (m) storey Eq.2 Test

Dayan∗ 49.8 25.2 9.2 2 0.65 0.67Xiaoyan∗ 40.2 11.4 3.6 2 0.84 0.74Fawang∗ 27.8 8.4 2.7 2 0.55 0.59Liusheng∗∗ 36.1 14.3 2.0 4 0.56 0.58Zhengguo∗∗ 44.2 14.0 2.2 4 0.88 0.85Renshou∗∗ 40.2 13.8 2.2 4 0.73 0.80Huqiu∗∗∗ 47.5 13.7 1.9 4 0.84 0.83Wenfeng∗∗∗ 38.3 10.3 2.4 4 0.82 0.88Xingfusi∗∗∗ 52.1 9.0 1.9 4 1.68 1.61

Note: ∗: Brick pagoda in Shanxi province; ∗∗: Stone pagodain Fujian province; ∗∗∗: Brick pagoda in Jiangsu province.

of the material strength test and refer to the specifi-cation of code for design of masonry structures (GB50003-2001, 2002).

3.2 The fundamental period of ancient masonrypagodas

The fundamental period is a basic index of the dynamicbehavior of pagodas. It is useful for selection of theappropriate test parameter to estimate the fundamen-tal period before field test. Based on the comparisonof theoretical value with experimental value of ninepavilion-style masonry pagodas (Table 2), a simplifiedformula for fundamental period has been concluded:

where, T = fundamental period; α1 = influence factorof masonry elastic modulus, α1 equals to 1.0 for brickpagoda and 1.2 for stone pagoda; α2 = influence factorof door opening ratio of the wall, α2 equals to 1.05 fortwo door holes each storey and 1.1 for four door holeseach storey; H = high from the ground to top of thepagoda; D = diameter of inscribed circle of polygon

394

Figure 2. Sketch map of the parameters in Eq. (2).

foundation (Fig. 2); and d = thickness of the wall atfoundation.

3.3 Identification of modal parameters by ambientvibration testing

Ambient vibration technique uses natural ground andwind pulsations as excitation source, which is sim-ple to conduct and does not damage the structure.However, the excitation signal is generally faint forthe ancient pagoda with large stiffness. To obtain therequired input signal, the high sensitivity sensors andlow noise amplifiers should be selected. Furthermore,the pagoda is situated on the scenic spot usually, toreduce the exterior vibration disturbance the bettersampling time is that with lesser noise from touristsand vehicles.

The excitations from ground and wind usually occurat the same time, the output-only modal analysis isused in the ambient technique to avoid the difficultyof distinguishing the input signals.

According to the theory of random vibration, fre-quency response function H (ω) can be expressed bythe following formula:

where Gff (ω) and Gyy(ω) are the auto-power spectra ofexciting force f (t) and structure response y(t), respec-tively. When the frequency spectrum of input sourceis smooth and approximately close to white noise witha finite bandwidth, the power spectrum can be treatedas a constant C, which reduces Eq. (2) to:

It is now clear that the identification of natural fre-quencies of the structure can be deduced from theauto-power spectrum of structural response.

For output-only modal analysis, there exist classi-cal Peak Pick (PP) method (Bendat JS, Piersol AG.1993) and more recent Enhanced Frequency DomainDecomposition (EFDD) method (Brincker R., et al,2001) in the frequency domain. When the PP methodis used in analyzing testing data of ancient pagodas,the modal frequencies can be identified accordingto following criteria: (a) Peak value of auto-powerspectrum of structure response occurs with the samefrequency at each test point; (b) Coherence function islarger between test points at every modal frequency;(c) At every modal frequency, each test point hasapproximately identical or opposite phases.

Based on random vibration theory, components ofmode shape can be determined by the value of trans-fer function located at characteristic frequency. As tothe ambient vibration testing, transfer function is theratio of response at the test point to that at referencepoint. Taking reference point as input and test point asoutput, the mode shape can be analyzed using transferfunction between both points, which can be expressedas follows:

where Gff and Gfy, respectively, are the auto-powerspectrum function and cross-power spectrum functionof the response signal.

The calculation of transfer function can beexpressed in the form of complex numbers:

where |H (ω)| equals to the ratio of amplitude of thesignal at test point to that at reference point; and ϕ(ω)is the phase difference between the signal at test pointand that at reference point. The signs of mode shapecoordinate at each test point are correlated to the phaserelation of cross-power spectra: same signs for samephases, opposite signs for opposite phases.

For ancient pagodas with small damping ratios andwell-separated modal frequencies, when ω ≈ ω underthe arbitrary random excitation, the ratio betweenpeak value of auto-power spectrum and that of cross-power spectrum, that is its transfer function, can beapproximated by the ratio of mode shapes.

3.4 Model updating and the reference criteria

Due to the uncertainty of structural parameters ofthe ancient pagoda, dynamic characteristics predictedby the analytical model often differ from field mea-surements. The proper reference criteria should be

395

provided for the structural parameters identificationin the model updating.

From the field testing we can obtain the firstn order modal parameters of a structure with Ndegrees of freedom, i.e., natural frequencies [ω2

T ] =diag(ω2

1, ω22, . . . , ω2

n) and mode shapes [φT ] = diag[{φ1}, {φ2}, . . . , {φn}].The structural mass matrix [MA]and stiffness matrix [KA] can be determined throughstructural analysis (finite element analysis for exam-ple). Basically, there are two sets of reference criteriafor structural parameters identification (Li Guoqian,Li Jie, 2002): (1) [ω2] = [ω2

T ], [φ] = [φT ]; and (2)[ω2] = [ω2

T ], [M ] = [MA].Generally, [ω2

T ] measured from the field test is accu-rate while [φT ] is less accurate. On the other hand,[MA] obtained from structural analysis is compara-tively accurate but [KA] is less close to the actualvalues. For the ancient pagodas, because absence ofthe original constructing data, the field test data isvery valuable for the structural parameters identifi-cation. To take full advantage of the information fromfield test, the reference criteria (1) is usually selectedas the reference for the analysis model updating.

Many model updating techniques have been devel-oped in the past (Smith M J, Hutton S G. 1992,Farhat C, Hemez F M. 1993, Renken J A. 1995,Alvin K F. 1996, Chung Y T. 1997), however, for apagoda with complicated architectural details, the 3DF.E. model consists of tens of thousands of meshingunits and the degrees of freedom usually, it is still a bur-densome task on structural parameters identification.To improve the effect of the model updating, the sensi-tivity system should be constructed for selection of thestructural parameters firstly. Besides, taking the fastanalysis advantage of ANSYS program, the conven-tional trial-error method also can be used to simplifythe model updating procedure.

3.5 Sensitivity of dynamic behavior to structuralparameter adjustment

The sensitivity-based model updating procedurehas been recognized as an effective approach forimproving FE models and the correlative researches(Zhang Dewen, Zhang Lingmi, 1992, Jung H. 1992,Friswelli MI, and Mottershead JE. 1995, Dascotte E,et al. 1995) are helpful for the construction of thesensitivity system of ancient pagodas.

Suppose the structural parameters such as mass,stiffness, geometric dimensions and material char-acteristics described as pi(i = 1, 2,. . . , n), andthe eigenvalues or eigenvectors are consideredas derivative functions of structural parameters,the dynamic characteristics of the pagoda canbe expressed as F = F(p1, p2, . . . , pn). Then thesensitivity of F to structural parameter pi isSFpi = ∂F(p1, p2, . . . , pn)/∂pi, and the bigger the

absolute value of SFpi , the more the sensitivity of modelcharacteristic to structural parameter pi.A little adjust-ment to pi will often cause a big change in the dynamiccharacteristics.

According to the structural dynamics, the γ-ordereigenvalue λγ and eigenvector φ(γ), should satisfy:

By solving the above formula’s partial derivative tothe i−parameter, the sensitivity of the eigenvalue canbe obtained:

And the sensitivity of the eigenvector is:

Taking the stiffness parameter of structure as mainstudy object, the sensitivity of the γ-order eigenvalueand eigenvector to stiffness kij , respectively, are:

By assessing and analyzing the sensitivity ofdynamic response to the adjustment of structuralparameters, the sensitivity system of the model can beestablished, which will provide reference for optimiz-ing the structural parameter adjustment, distinguish-ing the influence of parameters, and improving theefficiency of model updating.

4 INTEGRATED MODELING OF DYNAMICBEHAVIOR OF HUQIU PAGODA

4.1 Prominent characteristics of Huqiu pagoda

Situated on the Huqiu hill of ancient city Suzhou,the Huqiu pagoda was built between A.D. 959 and961 and is considered as the one with the oldestage and the greatest inclination among existing Chi-nese pavilion style masonry pagodas. The pagoda isa seven-storey tube structure, with 47.68 meters inheight and 6100 tons in weight. Ever since its con-struction, the groundwork of the pagoda has started tosettle unevenly, causing the pagoda to incline north-ward. Under the erosion by winds and rains and with

396

Figure 3. Picture and plane of the Huqiu pagoda.

Table 3. Dimensions of the Huqiu pagoda (m).

Storey No. Storey height Outer length Inner length

1 7.80 5.47 2.492 6.37 5.21 2.593 6.06 4.90 2.494 5.95 4.57 2.485 5.48 4.23 2.166 5.20 3.79 1.907 10.60 3.36 1.99

the development of uneven foundation settlement, thepagoda has deteriorated severely. During the years of1981 to 1986, Chinese civil engineering communitycarried out an important intervention project to restorethis ancient pagoda which was then on the verge ofcollapse. At conclusion of the project, the foundationsettlement and the structure inclination have been wellstabilized (Yuan Jianli, et al. 2004). As present, thepagoda maintains a tip deflection of 2.325 meter withan angle of inclination of 2◦48′.

4.2 Primary parameters and preliminary model

Figure 3 shows the picture and plane of the Huqiupagoda and Table 3 lists the dimensions at each storey.Based on the actual survey data and documentedrecords (Suzhou Huqiu pagoda archives, 1988), mate-rial properties and their variation range were deter-mined (Table 4). Because various materials were usedduring the different repair projects, the compressivestrength of the brick masonry with larger variation atevery storey, and the mortar of brick masonry is mainlya mixture of clay-lime-sand, so the elastic modulusof the pagoda is a main uncertain factor on dynamicbehavior model.

First, a 3D physical model of the Huqiu pagodawas developed using AutoCAD software. The model

Table 4. Material properties of the Huqiu pagoda.

Elastic modulus Density PoissonLocation MPa kN/m3 ratio

7th storey wall 1694∼ 2197 18.0 0.152nd∼6th storey wall 1386∼1794 18.0 0.151st storey wall 1694∼2197 18.0 0.15RC foundation 25500 25.5 0.20Soil groundwork 30 21.4 0.22Clay loam 16 20.0 0.32

Figure 4. Physical and mathematical models.

is divided by floors and areas of known damage orretrofit of a large scale were incorporated into themodel. Next, the physical model was converted intoa 3D finite element model with the ANSYS finiteelement program. A preliminary model of the Huqiupagoda, as shown in Figure 4, was established by fur-ther defining element properties, element meshing,and boundary conditions. Structural flaws of a rela-tively smaller scale are not included in this preliminarymodel. Finer meshes were used for connection andtransition areas such as door holes and walls, floorslabs, joints of each floor. The number of elementsin the model added up to total of 73,241 (Yao Ling,2003), and the Power Dynamics technique is adoptedfor the analysis of modal parameters.

397

Figure 5. Test point locations.

4.3 Field testing of dynamic characteristics

The dynamic characteristics of the Huqiu Pagoda,prominently its natural frequencies and mode shapes,were tested by the ambient vibration method. Themain testing instrument used was an INV-306 intel-ligent signal collection, process and analysis systemmanufactured by Orient Vibration and Noise Technol-ogy Research Institute of China. It uses large scalesignal collection and analysis software DASP (DataAuto Sample and Process System), and the maxi-mum sampling frequency of the system is 100 KHz.The effective frequency scope of horizontal-velocitysensors (Type 891-II) is 0.01∼100 Hz.

For the appropriate test parameters selection, thedynamic characteristics of Huqiu pagoda were esti-mated before the field test. According to the Eq. (2),the natural period of Huqiu pagoda is about 0.84 sec-ond, namely the first frequency is 1.19 HZ. Besides,the truncated frequency of primitive signal was takenat 20 HZ and sampling frequency at 60 HZ during theambient vibration testing.

Seven testing points were instrumented on thepagoda with their relative elevations shown in Figure 5.Eight sensors were used for the testing, one is placedon the reference point and the other seven are placed onthe floors of the first to seventh storey. The reference

Figure 6. Auto-power spectrum of test point 3.

Figure 7. Transfer function of test point 3.

point was fixed on the second floor (i.e., the eighthpoint), which was also the location of the sixth point.The signals of eight sensors were collected at the sametime. All points were sampled in the X direction first,and were sampled in the Y direction again. To satisfythe request of higher orders frequencies to test time,each sampling time is more than 20 minutes.

The auto-power spectrum as well as the transferfunction at every test point (shown in Figs 6 and 7for testing-point 3, respectively) was analyzed. Thenatural frequencies and mode shapes were obtained bythe software DASP using the improved EFDD tech-nique (Ying Huaiqiao and Liu Jinming, 2002) to fittesting data (shown in Figure 8). Tables 5 and 6 list thenatural frequencies and mode shapes of the first fourorders respectively.

4.4 Modification of modal parameters

By comparing and analyzing on the sensitivity of mainstructural parameters such as the tower eaves, leaning

398

Figure 8. Modal parameters fitting.

Table 5. Natural frequencies from the test (Hz).

1st Freq. 2nd Freq. 3rd Freq. 4th Freq.

1.204 3.905 7.295 11.250

Table 6. Mode shapes from modal fitting.

Testpoint 1st Freq. 2nd Freq. 3rd Freq. 4th Freq.

1 1 1 −1 −12 0.719 −0.248 −7.268 −0.83 0.550 −1.098 −5.194 0.4984 0.415 −1.384 2.397 0.7295 0.183 −1.009 5.727 −0.3976 0.054 −0.482 3.575 −0.6027 0 0 0 0

gradient, interior connecting details, door openingratio of the wall, as well as the strength and elasticmodulus of the masonry, the following conclusions canbe drawn: (1) The wooden and tile tower eaves havelesser sensitivity to the dynamic modification, theirarchitectural details can be simplified as the battenswith the same mass and elastic modulus. In the caseof Huqiu pagoda, the most of eaves were destroyed byenvironment erosion long ago, and the influence of theremnant brick braces to the dynamic behavior can beignored. (2) Because the strengthened foundation canbe regarded as the fixed bearing of the analysis model,the sensitivity of the leaning gradient of Huqiu pagodais rather low and can be ignored; (3) The door openingratio of the wall and elastic modulus of the masonryhave evidence influence to the dynamic behavior, sothe great attention should be paid. To simplify the

Table 7. Comparison of the natural frequencies (Hz).

Frequency Analysis result Test result Error %

First 1.155 1.204 −4.05Second 3.963 3.905 1.46Third 7.778 7.295 6.21Fourth 11.876 11.250 5.27

updating process further, the masonry’s modulus isonly selected as the structural modification parame-ter on the basis that the accurate dimensions of doorson the wall have been measured.

The elastic modulus is depended mostly on the dam-age condition and material property of the brick andmortar. In this investigation, based on the field test dataof the materials and observation on the damage con-dition, the elementary elastic modulus of each storeyand their estimated adjusting range and positions weredetermined. To convenient for modeling and analysisof the computer, by adjusting the elementary elasticmodulus to simulate synthesis influence on the mate-rial, damage, and repaired conditions of the masonryto the structural stiffness parameter. Thus, the adjustedmodulus can be called as the synthesis modulus orcomposite modulus.

More than 40 elementary models that reflectingthe possible combination with various material prop-erties and damage conditions were selected for themodel updating. When the model is updated, the ref-erence updating criterion (1), that is, [ω2] = [ω2

T ] and[φ] = [φT ], is adopted. The modification to the synthe-sis modulus follows the rule: Predicted mode shapesare in general agreement with the measurements whilethe errors in natural frequencies are within allowablelimit.

The updating objective limit in this investigationis that: (1) The displacement of mode shape at everystorey is the same sign as the measured displace-ment, (2) The allowable error for natural frequenciesis ±10%.

The model updating was carried out by analysis pro-gram ANSYS. And an Errors Evaluation Program hasbeen compiled for the error evaluation and the newparameter selection. A refined model for the dynamicbehavior of the Huqiu Pagoda was obtained from the40 elementary models. The results show that whenthe synthesis modulus of the wall is 2029 MPa forthe first story and seventh story, and 1552 MPa for thesecond to sixth story, respectively, the errors in thefirst four order natural frequencies are within ±7%,and predicted mode shapes agree generally with themeasured ones.The comparison of natural frequenciesbetween the analysis model and test result is listed inTable 7. The corresponding mode shapes are showedin Figure 9.

399

a) The mode shapes from the test

b) The mode shapes from F.E.A.

Figure 9. Comparison of the first four order mode shapes.

5 CONCLUSIONS

The integrated modeling method for the dynamicbehavior takes advantage of ambient vibration tech-nique and finite element analysis, which can obtainnot only the synthesis dynamic response of the wholestructure but also the contributions of individual fac-tors such as connections, restraints, and damage con-ditions. In the construction of the dynamic behaviormodel of ancient pagodas, it is necessary to get holdof the main characteristics of the method to improvethe application effect.

(1) Mastery of the main influence factors on dynamicbehavior of ancient pagodas and adoption of theconcise and proper finite element model, whichwill redound to the parameter identification andmodification. The architectural details such as themast, eaves, and roofs of the pagoda can be simpli-fied to reduce the number of elements and degreesof freedom of the model. But the door holes, dif-ferent structural materials, damaged positions, andcomplicated connections, etc. should be reflected

in the model as much as possible to consider theinfluence to the dynamic behavior.

(2) Environmental random excitation test technologyuses natural ground and wind pulsations as theexcitation source and is non-destructive to ancientpagodas. But the excitation signals from groundand wind pulsations are generally faint, for theancient pagoda with large stiffness, only the lowerorder frequencies can be obtained distinctly, andthe identification effect on mode shapes is reduceobviously with the increase of the frequency. Inorder to obtain the satisfactory data, besides thesensors and amplifiers with high sensitivity andlow noise should be selected, it is important toensure enough sampling time for each signalrecord.

(3) The elastic modulus of the masonry has a remark-able influence to the dynamic behavior of theancient pagodas, which can be selected as the mainmodified parameter for model updating. It is con-venient for model updating and analysis of thecomputer by adjusting the elastic modulus to sim-ulate the synthesis influence on the material vari-ation, damage degree, and repaired conditions ofthe masonry to the structural stiffness parameter.

(4) For the ancient pagodas, because absence of theoriginal constructing data, the field test datais very valuable to identification of the modelparameters. To take full advantage of the informa-tion from field test, the model updating criteria,[ω2] = [ω2

T ], [φ] = [φT ], can be selected as prefer-ential reference for modification of the analysismodel. And the rule: “Predicted mode shapesare in general agreement with the measurementswhile the errors in natural frequencies are withinallowable limit” will be helpful for obtaining theappropriate analysis model.

(5) In order to enhance the effect of the integratedmodeling method, the link programs between themodel updating, the parameter optimizing system,as well as the error evaluation, should be improvedfurther.

ACKNOWLEDGMENTS

This research was supported by the CooperationProject granted by the 11th Sino-Italian S&T JointCommission (2002, N. 47).

Suzhou administration of cultural heritage is grate-fully acknowledged for the help in the investigation ofHuqiu pagoda.

REFERENCES

Alvin KF. 1996, Finite element model update via Bayesianestimation and minimization of dynamic residuals,

400

Proceedings of the 14th international modal analysisconference, society for experimental mechanics, Inc,561∼567.

Bendat JS, Piersol AG. 1993, Engineering applications ofcorrelation and spectral analysis. 2nd ed. USA, NewYork:Wiley Inter science.

Brincker R, Ventura CE, Andersen P. 2001, Damping estima-tion by frequency domain decomposition, Proceedings ofthe 19th international modal analysis conference (IMAC),Florida, USA, 441∼46.

Code for design of masonry structures (GB 50003-2001),China Construction Industry Press, 2002 (in Chinese).

ChungYT. 1997, Mass matrix updating for model validation,Proceedings of the 15th international modal analysis con-ference, society for experimental mechanics, Inc, Japan,818∼824.

Dascotte E, Strobbe J, Hua H. 1995, Sensitivity-basedmodel updating using multiple types of simultaneous statevariables. Proceedings of the 13th international modalanalysis conference, 1–5.

Farhat C, Hemez F M. 1993, Updating finite element dynamicmodels using an element-by-element sensitivity method-ology, AIAA Journal, 31(9): 1702∼1711.

Friswelli MI, Mottershead JE. 1995, Finite element modelupdating in structural dynamics, 2nd ed. The Netherlands:Kluwer Academic Publishers.

Jung H. 1992, Structural dynamic model updating usingeigensensitivity analysis, Ph.D. thesis. London (UK):Imperial College of Science, Technology and Medicine.

Li Guoqian, Li Jie. 2002, Dynamic Detection Theory andApplication for Engineering Structure, Science Press, (inChinese).

Renken JA. 1995, Optimal solution selection for sensitivity-based finite element model updating and structural dam-age detection. AD-A306053.

Smith MJ, Hutton SG. 1992, Frequency modification usingnewton’s method and inverse eigenvector updating, AIAAJournal, 30(7): 1866∼1891.

Suzhou Huqiu pagoda archives, 1988, the archives of nationalprotection unit of historical and cultural heritages, (inChinese).

Yao Ling, 2003, Research and application on modeling tech-nology of dynamic behavior of ancient masonry pagodas,Dissertation of Yangzhou University, April (in Chinese).

Ying Huaiqiao and Liu Jingming, 2002, DASP – data autosample and process system, China orient vibration andnoise institution, Beijing (in Chinese).

Yuan Jianli, Li Shengcai, 2001, Analysis and investigation ofseismic behavior for multistory pavilion ancient pagodasin China, Structural Studies, Repairs, and Maintenance ofHistorical Buildings, WIT Press, UK, 2001(5)129–137.

Yuan Jianli, et al., 2004, Inclination control and reinforce-ment technology applied to Huqiu Pagoda, China CivilEngineering Journal, 37(5): 44∼49(in Chinese).

Zhang Dewen, Zhang Lingmi, 1992, Matrix transforma-tion method for updating dynamic model. AIAA Journal,30(5): 1440∼1443.

401