Review Article Seismic Response of Base-Isolated High-Rise...

Review ArticleSeismic Response of Base-Isolated High-Rise Buildings underFully Nonstationary Excitation

C F Ma12 Y H Zhang1 P Tan3 and F L Zhou3

1 State Key Laboratory of Structural Analysis for Industrial Equipment Faculty of Vehicle Engineering and MechanicsDalian University of Technology Dalian 116023 China

2 Bridge Science Research Institute Co Ltd China Railway Major Bridge Engineering Group Wuhan 430034 China3 Earthquake Engineering Research and Test Center Guangzhou University Guangzhou 510405 China

Correspondence should be addressed to Y H Zhang zhangyhdluteducn

Received 26 November 2013 Revised 12 January 2014 Accepted 28 February 2014 Published 24 June 2014

Academic Editor Jeong-Hoi Koo

Copyright copy 2014 C F Ma et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Stochastic seismic responses of base-isolated high-rise buildings subjected to fully nonstationary earthquake ground motion arecomputed by combining the pseudoexcitation and the equivalent linearization methods and the accuracy of results obtained bythe pseudoexcitation method is verified by the Monte Carlo method The superstructure of a base-isolated high-rise building isrepresented by a finite element model and a shear-type multi-degree of freedom model respectively The influence of the modeltype and the number of the modes of the superstructure participating in the computation of the dynamic responses of the isolatedsystem has been investigatedThe results of a 20-storey 3D-framewith height to width ratio of 4 show that storey drifts and absoluteaccelerations of the superstructure for such a high-rise building will be substantially underestimated if the shear-type multi-degreeof freedom model is employed or the higher modes of the superstructure are neglected however this has nearly no influence onthe drift of the base slab

1 Introduction

Seismic isolation is a technology that decouples a buildingstructure from the damaging earthquake motion It is asimple structural design approach to mitigate or reducepotential earthquake damage In base-isolated structures theseismic protection is obtained by shifting the natural periodof the structure away from the range of the frequenciesfor which the maximum amplification effects of the groundmotion are expected thus the seismic input energy is signif-icantly reduced At the same time the reduction of the highdeformations attained at the base of the structure is possiblethanks to the energy dissipation caused by the damping andthe hysteretic properties of these devices further improvingthe reduction of responses of the structures [1] Since the1995 Hyogoken-nanbu earthquake the number of seismicisolated buildings has been increasing remarkably includingresidential buildings nuclear power plants office buildingshospitals and schools [2 3] Base isolation is also an attractive

retrofitting strategy to improve the seismic performance ofexisting bridges and monumental historic buildings [1 3 4]

It is believed that isolation technology is very effectivein improving the seismic performance of low- and medium-rise buildings but it is not envisaged for high-rise buildingsHowever a lot of base-isolated high-rise buildings have beenbuilt in recent decades Sendai MT building is an 18-storeyoffice building with a height of 849m in Sendai city whichwas the first base-isolated building with a height exceeding60m [5] Thousand Tower a 41-storey building with a heightof 135mwas constructed in 2002 [5] and another super high-rise building in Japan was built in 2006 with a height of1774m and height to width ratio of 57 which is the highestbase-isolated building in the world so far [6]

A substantial amount of work has been done on base-isolated high-rise buildings Since isolators are easily dam-aged by uplift when such high-rise buildings are subjectedto major earthquakes Roussis and Constantinou [7] pro-posed some new devices to avoid damage caused by uplift

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 401469 11 pageshttpdxdoiorg1011552014401469

2 Shock and Vibration

of the isolators Hino et al [8] studied the limitation ofthe height to width ratio of the base-isolated buildings bythe Monte Carlo method Ariga et al [9] investigated theresonant behavior of base-isolated high-rise buildings underlong-period ground motions Takewaki [10] investigated therobustness of base-isolated high-rise buildings under code-specified ground motions and concluded that base-isolatedhigh-rise buildings have lower robustness than base-isolatedlow-rise buildings Takewaki and Fujita [11] studied theearthquake input energy to base-isolated high-rise build-ings by both time-domain and frequency-domain methodsYamamoto et al [12] studied the input energy and its rate to abase-isolated building during an earthquake in the frequencydomain Pourzeynali and Zarif [13] optimized the parametersof the base isolation system using genetic algorithms tosimultaneously minimize the displacement of the top storeyand that of the base isolation system Recently Islam et al[14 15] studied the nonlinear performances of base-isolatedmultistorey buildings in both time domain and frequencydomain

However in most of the above-mentioned research thestructure was simplified as a shear-type multi-degree of free-dom (MDOF) model (ie each storey of the superstructurewas simplified as one degree of freedom in the horizontaldirection and the vertical deformation of the columns wasneglected) Such modeling is not suitable when the height towidth ratio (defined by the ratio of the building height to thebuilding width) of the superstructure exceeds 4 as by thenthe flexural effect of the column cannot be neglected In thepresent work the superstructure of a base-isolated high-risebuilding is represented by a finite element (FE) model and ashear-typeMDOFmodel respectively to study the modelingeffect The stochastic response of this building is evaluatedby combining the pseudoexcitation method (PEM) and theequivalent linearization method (ELM) since this combinedapproach does not require a huge storage of data and vastcomputational effort [16]

The main objectives of this paper are (i) to evaluate thedynamic responses of base-isolated high-rise buildings underfully nonstationary earthquake excitations by combining thePEM and ELM and verify the accuracy of the results by theMonte Carlo method (ii) to investigate the influence of theflexure of the beams and extensibility of the columns on theresponses of such structures and (iii) to study the influenceof the higher modes of the superstructure on the responses ofbase-isolated high-rise buildings

2 Governing Equations of Motion forthe Base-Isolated System

21 Governing Equations of Motion for the Superstructure andBase Slab Considering a structure with 119873 DOFs the gov-erning equation ofmotion for the superstructure subjected tohorizontal seismic ground acceleration 119892 in the horizontal 119909

direction is [1]

M119904u119904 + C119904u119904 + K119904u119904 = minusM119904r119904 (119892 + 119887) (1)

where M119904 C119904 and K119904 are the 119873 times 119873 mass damping andstiffness matrices of the superstructure respectively u119904 u119904and u119904 are the acceleration velocity and displacement vectorof order 119873 relative to the base slab 119909119887 is the displacement ofthe base slab relative to the ground displacement 119906119892 and r119904 isthe 119873-dimensional influence coefficient vector

The governing equation of motion of the base slab withrotation and vertical deflection neglected can be expressed as[1]

119898119887 (119887 + 119892) + 119888119887119887 + 1205720119896119906119909119887 + (1 minus 1205720) 119896119906119911

+ r119879119904M119904 (u119904 + r119904119887 + r119904119892) = 0

(2)

where 1205720 denotes the post- to preyielding stiffness ratio 119911

is a hysteretic component which is a function of the timehistory of 119909119887 and 119887 and 119898119887 119888119887 and 119896119906 are the masssupplemental damping and preyielding stiffness of the baseslab respectively One has

119888119887 = 2120589119887radic119896119889119898119905 119896119889 = 1205720119896119906 119898119905 = r119879119904M119904r119904 + 119898119887

(3)

where 120589119887 is the supplemental damping ratio of the base slab119896119889 is the postyielding stiffness of the base slab and 119898119905 is thetotal mass of the superstructure and base slab

In (2) 119911 is related to 119909119887 and 119887 through the followingnonlinear differential equation [17]

= 119860119887 minus (1205741003816100381610038161003816119887

1003816100381610038161003816 119911|119911|120578minus1

+ 120573119887|119911|120578) (4)

Note that in (4) 120574 and 120573 control the shape of the hystereticloop 119860 controls the restoring force amplitude and 120578 controlsthe smoothness of the transition from elastic to plasticresponse These parameters are related by 119863119910 =

120578radic119860(120573 + 120574)

where 119863119910 is the yield displacement of the isolators

22 Static Correction Procedure In properly designed base-isolated systems the superstructure remains elastic evenwhen subjected to a major earthquake ground motionTherefore the modal superposition method is used to reducethe number of degrees of freedom of the system with the first119899 (119899 ≪ 119873) modes participating in the dynamic computationThis improves the computational efficiency but introducestruncation errors because of neglecting the influence of thehigher modesThe static correction procedure is employed totake into consideration the contribution of the higher modesThe displacement of the superstructure corresponding to thehigher modes is obtained based on the fact that the highfrequency modes react essentially in a static manner whenexcited by low frequencies [18] Assume that the displacementof the superstructure u119904 consists of two parts that is thedynamic part u119889

119904and the static part u119904

119904[19 20]

u119904 = u119889119904

+ u119904119904 (5)

The dynamic part of the displacement can be expressed as

u119889119904

= Φq =

119899

sum

119894=1

Φ119894119902119894 (6)

Shock and Vibration 3

where Φ119894 is the mass normalised eigenvector correspondingto the 119894th eigenvalue 120596

2

119894 frequencies and modes satisfy

K119904Φ119894 = 1205962

119894M119904Φ119894 (119894 = 1 2 119899) Φ = [Φ1 Φ2 sdot sdot sdot Φ119899] is

the 119873 times 119899 eigenvector matrix and 119902 is the 119873 times 1 generalisedmodal displacement vector

Premultiplying (1) byΦ119879 gives

q + C119904q + K119904q = L (119892 + 119887) (7)

where

C119904 = Φ119879C119904Φ

K119904 = Φ119879K119904Φ = diag 120596

2

1 1205962

2 120596

2

119899

L = minusΦ119879M119904r119904

(8)

in which L is the mode participation factor of order 119899The flexibility matrix of the superstructure can be ex-

pressed as

Kminus1119904

=

119873

sum

119894=1

1

1205962

119894

Φ119894Φ119879

119894 (9)

The remaining static displacement corresponding to thehigher modes is

u119904119904

= minus (Kminus1119904

minus

119899

sum

119894=1

1

1205962

119894

Φ119894Φ119879

119894)M119904 [u119904 + r119904 (119887 + 119892)]

= minusBM119904 [u119904 + r119904 (119887 + 119892)]

(10)

where

B = Kminus1119904

minus

119899

sum

119894=1

1

1205962

119894

Φ119894Φ119879

119894 (11)

23 Equivalent LinearizationMethod (ELM) Using the ELM(4) can be rewritten as follows when 120578 = 1 [17]

+ 119888119890119887 + 119896119890119911 = 0 (12)

with

119888119890 = radic2

120587(120574

119864 [119887119911]

120590 119909119887

+ 120573120590119911) minus 119860

119896119890 = radic2

120587(120574120590 119909119887

+ 120573119864 [119887119911]

120590119911

)

(13)

where 119864[ ] denotes the expectation operator and 120590119911 and 120590 119909119887are the standard deviations of 119911 and 119887 respectively

24 State Space Method Equations (2) and (7) can be com-pacted together into the single equation below

M119887

q + C119887

q + K119909119887

q + (1 minus 1205720) 119896119906

0 119911 = minusMl119892

(14)

where

M = [119898119905 minusL119879

minusL I119899] C = [

119888119887 0

0 C119904]

K = [

1205720119896119906 0

0 K119904] l = 1 0 0 0

119879

(15)

in which I119899 is the 119899 times 119899 identity matrixThe second-order (14) can be replaced by the first-order

differential equations by the state space method

119887

q119887

q

= [0 I119899+1

minusMminus1K minusMminus1C]

119909119887

q119887

q

+ 0a 119911 minus

0l 119892

(16)

where I119899+1 is the (119899 + 1) times (119899 + 1) identity matrix and (119899 + 1)-dimensional vector a is given by

a = Mminus1minus (1 minus 1205720) 119896119906 0 0 0119879

(17)

The state variable vector v of order (2119899 + 3) times 1 is nowintroduced as

v = 119909119887 q119879

119887 q119879

119911119879

(18)

Thus (12) and (16) can be replaced by the first-orderdifferential equation

v = Hv + r119892 (19)

where

H =[[[

[

0 I119899+1 0

minusMminus1119870 minusMminus1C a0 b minus119896119890

]]]

]

r = minus 0 l119879 0

(20)

in which (119899 + 1)-dimensional vector b is given by

b = minus119888119890 0 0 0 (21)

3 Nonstationary Stochastic Analysis by PEM

Because of the uncertainty of earthquakes stochastic seismicanalysis is a powerful tool in earthquake engineering and hasexperienced extensive development in recent decades Jangid[21] studied the performance of the isolated buildings andbridges and the stochastic responses of the isolated structuressubjected to uniformly modulated nonstationary earthquakeexcitations were obtained by solving Lyapunov equation Asthe stochastic response of a nonlinear system is stronglyaffected by the nonstationary behaviour of an earthquake[22] the fully nonstationary earthquake model proposed byConte and Peng [23] is adopted in this paper


Initial value obtained

ce =

Obtain E[xbz] 120590z 120590x119887 by PEM

ce kece = ce ke = ke

|ce minus ce| |ke minus ke|)

Err le tolNo

No

Yes

Yes

tk = tk + Δt

End

Time end

c(0)e ke = k(0)e

exp(i120596tk)

(ce ke tk

Err = max(

v = v(0)

ug(120596 tk) =radicSu119892u119892 (120596 tk)

v = H )v+rug(120596j tk)

vkj = v(120596j tk)

uds uss

us = uds + uss

Figure 1 Flow chart of the computation procedure

31 Fully Nonstationary Earthquake Excitation Model Theearthquake excitation model is considered a sigma-oscillato-ry process [23] which is a sum of 119901 zero-mean independentuniformly modulated Gaussian processes Each uniformlymodulated process consists of the product of a deterministictime modulating function 119860119896(119905) and a stationary Gaussianprocess 119884119896(119905) Thus the earthquake ground motion 119892(119905) isdefined as [23]

119892 (119905) =

119901

sum

119896=1

119883119896 (119905) =

119901

sum

119896=1

119860119896 (119905) 119884119896 (119905) (22)

In (22) the modulating function 119860119896(119905) is defined as

119860119896 (119905) = 120572119896(119905 minus 120589119896)120573119896

119890minus120574119896(119905minus120589119896)119867 (119905 minus 120589119896) (23)

where120572119896 and 120574119896 are positive constants120573119896 is a positive integer120589119896 is the ldquoarrival timerdquo of the 119896th subprocess 119883119896(119905) and 119867(119905)

is a unit step functionThe 119896th stationary Gaussian process 119884119896(119905) is character-

ized by its autocorrelation function

119877119884119896119884119896(120591) = 119890

minus]119896|120591| cos (120578119896120591) (24)

and its power spectral density (PSD) function

119878119884119896119884119896(120596) =

]1198962120587

[1

]2119896

+ (120596 + 120578119896)2

+1

]2119896

+ (120596 minus 120578119896)2

] (25)

in which ]119896 and 120578119896 are two free parameters representingthe frequency bandwidth and predominant frequency of theprocess 119884119896(119905) respectively


x

y

z

(a)

x

y

(b)

A B C D1

2

3

4

6m

3m

6m

x

z

3 times 6 m

(c)

Figure 2 A base-isolated building with 20 storeys (a) 3D model (b) evaluation of the frame and (c) plan view

008

007

009

006

004

005

003

001

002

0

1020

30

80

6070

4050

0

510

1520

2530

35

Time (s)

Frequency (rads)

PSD

(m2s

3)

Figure 3 Time varying PSD function of sigma-oscillatory processmodel for Orion Boulevard 1971 earthquake ground acceleration

Finally the evolutionary PSD of the earthquake groundexcitation 119892(119905) is given

119878119892119892(120596 119905) =

119901

sum

119896=1

1003816100381610038161003816119860119896 (119905)1003816100381610038161003816

2119878119884119896119884119896

(120596) (26)

32 PEM Computational Procedure The procedure of com-puting stochastic responses of base-isolated systems is sum-marized below

Step 1 Constitute the pseudoexcitation at instant 119905119896 as in [16]

119906119892 (120596 119905119896) = radic119878119892119892(120596 119905119896) exp (119894120596119905119896) (27)

where 119894 = radicminus1 Substitute 119906119892(120596 119905119896) into (19) with 119888119890 and 119896119890

in (13) and (19) given an initial value at 119905 = 0

Step 2 Compute the pseudoresponse v(120596119895 119905119896) by Runge-Kutta method

The corresponding nonstationary random vibrationresponse analysis is transformed into an ordinary directdynamic analysisThus k(120596119895 119905119896) can be evaluated at a series ofequally spaced frequency points 120596119895 = 119895Δ120596 (119895 = 1 2 119873120596)at 119905119896 by Runge-Kutta method where Δ120596 is the frequency stepand 119873120596 is the total number of frequency steps

Step 3 Obtain the cross- and auto-PSD of the responses byPEM at 119905119896 and 120596119895

119878 119909119887119911(120596119895 119905119896) is the cross-PSD of 119887(120596119895 119905119896) and 119911(120596119895 119905119896)

119878119911 119909119887(120596119895 119905119896) are the cross-PSD of 119911(120596119895 119905119896) and 119887(120596119895 119905119896) and

119878119911119911(120596119895 119905119896) and 119878 119909119887 119909119887(120596119895 119905119896) are the auto-PSD of 119887(120596119895 119905119896) and

119911(120596119895 119905119896) respectively and they are given as [16]

119878 119909119887119911(120596119895 119905119896) = 119909

lowast

119887(120596119895 119905119896) (120596119895 119905119896)

119878119911 119909119887(120596119895 119905119896) =

lowast(120596119895 119905119896)

119909119887 (120596119895 119905119896)

119878119911119911 (120596119895 119905119896) = lowast

(120596119895 119905119896) (120596119895 119905119896)

119878 119909119887 119909119887(120596119895 119905119896) = 119909

lowast

119887(120596119895 119905119896)

119909119887 (120596119895 119905119896)

(28)

In (28) 119909119887(120596119895 119905119896) and (120596119895 119905119896) are the pseudoresponsesobtained by Step 3 119909

lowast

119887(120596119895 119905119896) and

lowast(120596119895 119905119896) are the complex

conjugate of 119909119887(120596119895 119905119896) and (120596119895 119905119896) respectively


010

008

006

004

002

000

RMS

drift

of t

he b

ase s

lab

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

(m)

(a)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00030

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he3

rd st

orey

(m)

(b)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00030

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he13

th st

orey

(m)

(c)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he19

th st

orey

(m)

(d)

Figure 4 Comparison of the RMS storey drifts (a) base slab (b) 3rd storey (c) 13th storey and (d) 19th storey evaluated by PEM with thoseobtained by Monte Carlo simulation

Step 4 Compute the covariance and variance of the respons-es by Wiener-Khintchine theorem at 119905119896

The quantities 119864[119887119911] 120590119911 and 120590 119909119887used in (13) are given

by

119864 [119887119911] = int

+infin

minusinfin

119878 119909119887119911(120596 119905119896) d120596

= Δ120596

119873120596

sum

119895=1

[119878 119909119887119911(120596119895 119905119896) + 119878119911 119909119887

(120596119895 119905119896)]

1205902

119911= 2 int

+infin

0

119878119911119911 (120596 119905119896) d120596 = 2Δ120596

119873120596

sum

119895=1

119878119911119911 (120596119895 119905119896)

1205902

119909119887= 2 int

+infin

0

119878 119909119887 119909119887(120596 119905119896) d120596 = 2Δ120596

119873120596

sum

119895=1

119878 119909119887 119909119887(120596119895 119905119896)

(29)Step 5 Evaluate 119888119890 and 119896119890 by (13)

When the corresponding responses become convergent119905119896 is replaced by 119905119896+1 and Steps 1ndash5 are repeated for thenext time step Equations (13) (19) (28) and (29) make


22

20

18

16

14

12

10

8

6

4

2

Isolation

0000 0002 0004 0006 0008 0010 0012

Peak RMS storey drifts (m)

Base-fixed buildingBase-fixed building

Stor

eys

(a)

Base-fixed buildingBase-isolated building

22

20

18

16

14

12

10

8

6

4

2

0

Stor

eys

00 01 02 03 04 05

Peak RMS storey absolute accelerations (g)

(b)

Figure 5 Comparison of (a) the peak RMS storey drifts and (b) the absolute storey accelerations of the base-fixed and base-isolated buildings

up the complete formulation of the isolated system Thecomputational procedure is shown in Figure 1

4 Numerical Study

In the base-isolated frame structure shown in Figure 2 eachof the 20 storeys is 36m in height so the total height ofthe frame structure is 72m with 18m in width and 15min depth in the 119909 and 119911 directions respectively Thus itsheight to width ratios are 4 and 48 in the 119909 and 119911 directionsrespectively The reinforced concrete beams are all identicalwith width of 06m and depth of 08m The reinforcedconcrete columns are all of square cross-sections with sidelength 119889 The column properties that is their side length 119889extensional rigidities 119864119860 and flexural rigidities 119864119868 for thein-plane behavior are in three different values dependingupon the storey number as shown in Table 1 The total massof each storey is distributed uniformly as a lumped mass ateach of its nodes as also shown in Table 1 The dampingratios of the superstructure and isolation slab are 003 and010 respectively the fundamental period of the base-fixedsuperstructure is 119879119904 = 166 s and the fundamental period ofthe base-isolated system is 119879119889 = 350 s Other values usedare the post- to preyielding stiffness ratio 1205720 = 01 and theyielding displacement 119863119910 = 001m

A versatile fully nonstationary earthquake ground-mo-tion model proposed by Conte and Peng [23] is employedhere and this stochastic earthquake model is applied toan actual earthquake N00W (N-S) component of the SanFernando earthquake of February 9 1971 recorded at theOrion Boulevard site The corresponding parameters of thesigma-oscillatory process estimated are given in Table 2 [23]The model parameters are determined by adaptively least-squares fitting the analytical time varying (or evolutionary)

Table 1 Properties of columns for the 20-storey building and massdistribution for each storey

Storeys 119889

(m)119864119860

(GN)119864119868

(GNm2)Node mass(103 kg)

Isolation layer mdash mdash mdash 3441sim8 09 243 1640 2769sim16 08 192 1024 24117sim20 07 147 0060 207

Table 2 Estimated parameters of the ground accelerationmodel forOrion Boulevard 1971 earthquake record

119896 120572119896

120573119896

120574119896

120589119896

]119896

120578119896

1 02358 7 13375 08672 12618 310222 01394 8 14116 55717 15856 489623 1309862 10 42028 124408 19182 417204 33724 9 25692 151802 19097 276795 10659 2 01612 minus15150 12000 345886 717647 2 08956 98679 12000 1108557 00044 11 17482 minus10149 20905 1413498 02012 11 24117 38821 29589 1640599 37529 11 30778 87310 13580 9411010 04901 11 26406 14716 24796 19259611 126339 3 07620 61032 12927 20207912 41843 5 11922 minus01155 27419 31474813 58917 5 13786 54048 13335 28792814 21934 5 12384 minus01895 18083 43085015 203968 5 17774 54490 43403 375139

PSD function of the proposedmodel to the evolutionary PSDfunction estimated from the actual earthquake accelerogram


22

20

18

16

14

12

10

8

6

4

2

Isolation


Stor

eys

0005 0010 0015 0020 0025 0030 0035

n = 3

n = 3 + correctionn = 40

Figure 6 Influence of static correction procedure on the storey drifts

20

18

16

14

12

10

8

6

4

2

0


Stor

eys

0000500000 00010 00015 00020 00025 00030 00035

Shear-type modelFE model

(a)


20

18

16

14

12

10

8

6

4

2

0

Stor

eys

002000 004 006 008 010 012

Peak RMS absolute accelerations (g)

(b)

Figure 7 Comparison of the (a) peak RMS storey drifts and (b) absolute storey accelerations of the superstructure given by FE analysis withthose given by the shear-type MDOF model

The PSD function of the earthquake excitation is shown inFigure 3 Obviously this earthquake model can capture thetime variation of both the intensity and the frequency contentof the earthquake record at the target

Figure 4 compares the root mean square (RMS) storeydrifts of the isolated structure evaluated by PEM with thosegiven byMonte Carlo simulation (500 samples are used)Therelationship between the restoring force and the drift of theisolators is described by the Bouc-Wenmodel so each sampleofMonteCarlo simulation is a nonlinear time history analysisof the isolated system Clearly both the drift of the base slaband the storey drifts of the superstructure agree well with thetwo sets of results so that the accuracy of the results by thePEM is demonstrated

The peak RMS of the storey drifts and absolute accel-erations of the base-fixed structure and those of the base-isolated one are shown in Figure 5 It demonstrates that theresponses decrease significantly after isolation so they revealthat the isolation technology can still protect the structuresfromdamage during earthquakes even for high-rise buildingswith a proper design of the isolators employed

The improvement of accuracy of the storey drifts by thestatic correction procedure is given by Figure 6 It shows thatthe accuracy of the response is improved with employment ofthis method

Figure 7 illustrates the influence of flexure of the super-structure on the peak RMS storey drifts and absolute accel-erations of the storeys The FE model and shear-type MDOF


22201816141210

8642

Isolation

Stor

eys

00005 00010 00015 00020 00025 00030 00035Peak RMS storey drifts (m)

n = 3

n = 10

n = 40

(a)

22201816141210

8642

Isolation

Stor

eys

004 005 006 007 008 009Peak RMS storey absolute accelerations (g)

n = 3

n = 10

n = 40

(b)

Figure 8 Influence of the number of the modes participating in the computation on the (a) storey drifts and (b) absolute accelerations of thesuperstructure

014

012

010

008

006

004

002

000

Peak

RM

S dr

ift o

f the

bas

e sla

b

2 5 10 20

Number of the modes participating in the dynamic computation


(m)

(a)

2 5 10 20


010

009

008

007

006

005

004

003

002

001

000

Peak

RM

S ab

solu

te ac

cele

ratio

n


(g)

(b)

Figure 9 Influence of the number of the modes participating in dynamic computation on the (a) drift and (b) absolute acceleration of thebase slab

model are each applied to the superstructure with the first 20modes participating in the dynamic computation It showsthat storey drifts and absolute accelerations increase whenextensions of the columns and flexure of the beams aresuitably taken into account in the FE model

Figure 8 shows the influence of the number of the modesparticipating in the computation of the response of thesuperstructureThe results are obtainedwith 119899 = 3 10 and 40modes respectively They reveal that the storey drifts of thesuperstructure will be substantially underestimated if only

the first few modes are included in the dynamic computationfrom Figure 8(a) while Figure 8(b) demonstrates that theabsolute accelerations vary significantly with the numberof modes participating in the dynamic computation Sothe influence of the higher modes on the responses of thesuperstructures should not be neglected for the base-isolatedhigh-rise buildings

Figure 9 shows the influence of the number of the modesparticipating in the computation of the drift and absoluteacceleration of the base slab The number of the modes


participating in the dynamic computation is found to have asmall influence on the drift of the base slab but a remarkableinfluence on the absolute acceleration of the base slab

5 Conclusions

The stochastic responses of a base-isolated high-rise buildingsubjected to fully nonstationary ground excitations are ana-lyzed by combining the PEM and the ELM The conclusionscan be drawn as follows

(1) The results obtained by the PEM agree well withthose obtained by the Monte Carlo method and theaccuracy of the results of such hysteretic systemsevaluated by the PEM is verified

(2) The static correction procedure is employed for con-sidering the contributions of the higher modes ofthe structure which causes almost no increase of thecomputational effort

(3) An FE model and a shear-type MDOF model areimplemented for the superstructure of such high-rise buildings It is found that the storey drifts andabsolute accelerations are underestimated if the flex-ural deformation of the beam components and theaxial deformations of the column components areneglected in the shear-type MDOF model used Thepeak RMS storey drifts of the superstructure couldbe underestimated by about 60 and the peak RMSabsolute accelerations of the superstructure could beunderestimated by about 7

(4) The storey drifts and the absolute accelerations couldbe underestimated if the higher modes of the super-structure are neglected in the FE model used and thenumber of the modes participating in the dynamiccomputation has a small influence on the responseof the base slab but a remarkable influence on itsabsolute acceleration and sometimes the absoluteacceleration of base slab could be overestimated byabout 40

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful for support from the NationalScience Foundation of China under Grant no 11172056and from the National Basic Research Program of China(2014CB046803)

References

[1] F Naeim and J M Kelly Design of Seismic Isolated StructuresFrom Theory to Practice John Wiley amp Sons Chichester UK1999

[2] L Di Sarno E Chioccarelli and E Cosenza ldquoSeismic responseanalysis of an irregular base isolated buildingrdquo Bulletin ofEarthquake Engineering vol 9 no 5 pp 1673ndash1702 2011

[3] L Di Sarno and A S Elnashai ldquoInnovative strategies forseismic retrofitting of steel and composite structuresrdquo Progressin Structural Engineering andMaterials vol 7 no 3 pp 115ndash1352005

[4] R A Ibrahim ldquoRecent advances in nonlinear passive vibrationisolatorsrdquo Journal of Sound and Vibration vol 314 no 3ndash5 pp371ndash452 2008

[5] T Komuro Y Nishikawa Y Kimura and Y Isshiki ldquoDevel-opment and realization of base isolation system for high-risebuildingsrdquo Journal of Advanced Concrete Technology vol 3 no2 pp 233ndash239 2005

[6] S G Wang D S Du and W Q Liu ldquoResearch on key issuesabout seismic isolation design of high-rise buildings structurerdquoin Proceedings of the 11th World Conference on Seismic IsolationEnergy Dissipation and Active Vibration Control of Structurespp 17ndash21 Guangzhou China 2009

[7] P C Roussis and M C Constantinou ldquoUplift-restrainingFriction Pendulum seismic isolation systemrdquo Earthquake Engi-neering and Structural Dynamics vol 35 no 5 pp 577ndash5932006

[8] J Hino S Yoshitomi M Tsuji and I Takewaki ldquoBound ofaspect ratio of base-isolated buildings considering nonlineartensile behavior of rubber bearingrdquo Structural Engineering andMechanics vol 30 no 3 pp 351ndash368 2008

[9] T Ariga Y Kanno and I Takewaki ldquoResonant behaviour ofbase-isolated high-rise buildings under long-period groundmotionsrdquo Structural Design of Tall and Special Buildings vol 15no 3 pp 325ndash338 2006

[10] I Takewaki ldquoRobustness of base-isolated high-rise buildingsunder code-specified groundmotionsrdquoThe Structural Design ofTall and Special Buildings vol 17 no 2 pp 257ndash271 2008

[11] I Takewaki and K Fujita ldquoEarthquake input energy to tall andbase-isolated buildings in time and frequency dual domainsrdquoThe Structural Design of Tall and Special Buildings vol 18 no 6pp 589ndash606 2009

[12] K Yamamoto K Fujita and I Takewaki ldquoInstantaneous earth-quake input energy and sensitivity in base-isolated buildingrdquoThe Structural Design of Tall and Special Buildings vol 20 no6 pp 631ndash648 2011

[13] S Pourzeynali and M Zarif ldquoMulti-objective optimization ofseismically isolated high-rise building structures using geneticalgorithmsrdquo Journal of Sound andVibration vol 311 no 3ndash5 pp1141ndash1160 2008

[14] A B M S Islam R R Hussain M Z Jumaat and M ARahman ldquoNonlinear dynamically automated excursions forrubber-steel bearing isolation in multi-storey constructionrdquoAutomation in Construction vol 30 pp 265ndash275 2013

[15] A B M S Islam R R Hussain M Jameel and M Z JumaatldquoNon-linear time domain analysis of base isolated multi-storeybuilding under site specific bi-directional seismic loadingrdquoAutomation in Construction vol 22 pp 554ndash566 2012

[16] J H Lin W P Shen and F W Williams ldquoAccurate high-speedcomputation of non-stationary random structural responserdquoEngineering Structures vol 19 no 7 pp 586ndash593 1997

[17] Y K Wen ldquoEquivalent linearization for hysteretic systemsunder random excitationrdquo Journal of AppliedMechanics vol 47no 1 pp 150ndash154 1980

[18] R W Clough and J Penzien Dynamics of Structures McGraw-Hill New York NY USA 1993


[19] P Cacciola N Maugeri and G Muscolino ldquoA modal correc-tion method for non-stationary random vibrations of linearsystemsrdquo Probabilistic Engineering Mechanics vol 22 no 2 pp170ndash180 2007

[20] S Benfratello and G Muscolino ldquoMode-superposition cor-rection method for deterministic and stochastic analysis ofstructural systemsrdquo Computers and Structures vol 79 no 26ndash28 pp 2471ndash2480 2001

[21] R S Jangid ldquoEquivalent linear stochastic seismic response ofisolated bridgesrdquo Journal of Sound and Vibration vol 309 no3ndash5 pp 805ndash822 2008

[22] C-H Yeh and Y K Wen ldquoModeling of nonstationary groundmotion and analysis of inelastic structural responserdquo StructuralSafety vol 8 no 1ndash4 pp 281ndash298 1990

[23] J P Conte and B F Peng ldquoFully nonstationary analyticalearthquake ground-motion modelrdquo Journal of EngineeringMechanics vol 123 no 1 pp 15ndash24 1997

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



of the isolators Hino et al [8] studied the limitation ofthe height to width ratio of the base-isolated buildings bythe Monte Carlo method Ariga et al [9] investigated theresonant behavior of base-isolated high-rise buildings underlong-period ground motions Takewaki [10] investigated therobustness of base-isolated high-rise buildings under code-specified ground motions and concluded that base-isolatedhigh-rise buildings have lower robustness than base-isolatedlow-rise buildings Takewaki and Fujita [11] studied theearthquake input energy to base-isolated high-rise build-ings by both time-domain and frequency-domain methodsYamamoto et al [12] studied the input energy and its rate to abase-isolated building during an earthquake in the frequencydomain Pourzeynali and Zarif [13] optimized the parametersof the base isolation system using genetic algorithms tosimultaneously minimize the displacement of the top storeyand that of the base isolation system Recently Islam et al[14 15] studied the nonlinear performances of base-isolatedmultistorey buildings in both time domain and frequencydomain

However in most of the above-mentioned research thestructure was simplified as a shear-type multi-degree of free-dom (MDOF) model (ie each storey of the superstructurewas simplified as one degree of freedom in the horizontaldirection and the vertical deformation of the columns wasneglected) Such modeling is not suitable when the height towidth ratio (defined by the ratio of the building height to thebuilding width) of the superstructure exceeds 4 as by thenthe flexural effect of the column cannot be neglected In thepresent work the superstructure of a base-isolated high-risebuilding is represented by a finite element (FE) model and ashear-typeMDOFmodel respectively to study the modelingeffect The stochastic response of this building is evaluatedby combining the pseudoexcitation method (PEM) and theequivalent linearization method (ELM) since this combinedapproach does not require a huge storage of data and vastcomputational effort [16]

The main objectives of this paper are (i) to evaluate thedynamic responses of base-isolated high-rise buildings underfully nonstationary earthquake excitations by combining thePEM and ELM and verify the accuracy of the results by theMonte Carlo method (ii) to investigate the influence of theflexure of the beams and extensibility of the columns on theresponses of such structures and (iii) to study the influenceof the higher modes of the superstructure on the responses ofbase-isolated high-rise buildings

2 Governing Equations of Motion forthe Base-Isolated System

21 Governing Equations of Motion for the Superstructure andBase Slab Considering a structure with 119873 DOFs the gov-erning equation ofmotion for the superstructure subjected tohorizontal seismic ground acceleration 119892 in the horizontal 119909

direction is [1]

M119904u119904 + C119904u119904 + K119904u119904 = minusM119904r119904 (119892 + 119887) (1)

where M119904 C119904 and K119904 are the 119873 times 119873 mass damping andstiffness matrices of the superstructure respectively u119904 u119904and u119904 are the acceleration velocity and displacement vectorof order 119873 relative to the base slab 119909119887 is the displacement ofthe base slab relative to the ground displacement 119906119892 and r119904 isthe 119873-dimensional influence coefficient vector

The governing equation of motion of the base slab withrotation and vertical deflection neglected can be expressed as[1]

119898119887 (119887 + 119892) + 119888119887119887 + 1205720119896119906119909119887 + (1 minus 1205720) 119896119906119911

+ r119879119904M119904 (u119904 + r119904119887 + r119904119892) = 0

(2)

where 1205720 denotes the post- to preyielding stiffness ratio 119911

is a hysteretic component which is a function of the timehistory of 119909119887 and 119887 and 119898119887 119888119887 and 119896119906 are the masssupplemental damping and preyielding stiffness of the baseslab respectively One has

119888119887 = 2120589119887radic119896119889119898119905 119896119889 = 1205720119896119906 119898119905 = r119879119904M119904r119904 + 119898119887

(3)

where 120589119887 is the supplemental damping ratio of the base slab119896119889 is the postyielding stiffness of the base slab and 119898119905 is thetotal mass of the superstructure and base slab

In (2) 119911 is related to 119909119887 and 119887 through the followingnonlinear differential equation [17]

= 119860119887 minus (1205741003816100381610038161003816119887

1003816100381610038161003816 119911|119911|120578minus1

+ 120573119887|119911|120578) (4)

Note that in (4) 120574 and 120573 control the shape of the hystereticloop 119860 controls the restoring force amplitude and 120578 controlsthe smoothness of the transition from elastic to plasticresponse These parameters are related by 119863119910 =

120578radic119860(120573 + 120574)

where 119863119910 is the yield displacement of the isolators

22 Static Correction Procedure In properly designed base-isolated systems the superstructure remains elastic evenwhen subjected to a major earthquake ground motionTherefore the modal superposition method is used to reducethe number of degrees of freedom of the system with the first119899 (119899 ≪ 119873) modes participating in the dynamic computationThis improves the computational efficiency but introducestruncation errors because of neglecting the influence of thehigher modesThe static correction procedure is employed totake into consideration the contribution of the higher modesThe displacement of the superstructure corresponding to thehigher modes is obtained based on the fact that the highfrequency modes react essentially in a static manner whenexcited by low frequencies [18] Assume that the displacementof the superstructure u119904 consists of two parts that is thedynamic part u119889

119904and the static part u119904

119904[19 20]

u119904 = u119889119904

+ u119904119904 (5)

The dynamic part of the displacement can be expressed as

u119889119904

= Φq =

119899

sum

119894=1

Φ119894119902119894 (6)



2


K119904Φ119894 = 1205962




q + C119904q + K119904q = L (119892 + 119887) (7)

where

C119904 = Φ119879C119904Φ

K119904 = Φ119879K119904Φ = diag 120596

2

1 1205962

2 120596

2

119899

L = minusΦ119879M119904r119904

(8)


pressed as

Kminus1119904

=

119873

sum

119894=1

1

1205962

119894

Φ119894Φ119879

119894 (9)


u119904119904


minus

119899

sum

119894=1

1

1205962

119894

Φ119894Φ119879

119894)M119904 [u119904 + r119904 (119887 + 119892)]

= minusBM119904 [u119904 + r119904 (119887 + 119892)]

(10)

where

B = Kminus1119904

minus

119899

sum

119894=1

1

1205962

119894

Φ119894Φ119879

119894 (11)


+ 119888119890119887 + 119896119890119911 = 0 (12)

with

119888119890 = radic2

120587(120574

119864 [119887119911]

120590 119909119887

+ 120573120590119911) minus 119860

119896119890 = radic2

120587(120574120590 119909119887

+ 120573119864 [119887119911]

120590119911

)

(13)



M119887

q + C119887

q + K119909119887

q + (1 minus 1205720) 119896119906

0 119911 = minusMl119892

(14)

where

M = [119898119905 minusL119879

minusL I119899] C = [

119888119887 0

0 C119904]

K = [

1205720119896119906 0

0 K119904] l = 1 0 0 0

119879

(15)



119887

q119887

q

= [0 I119899+1


119909119887

q119887

q

+ 0a 119911 minus

0l 119892

(16)



(17)


v = 119909119887 q119879

119887 q119879

119911119879

(18)


v = Hv + r119892 (19)

where

H =[[[

[

0 I119899+1 0


]]]

]

r = minus 0 l119879 0

(20)


b = minus119888119890 0 0 0 (21)





ce =




Err le tolNo

No

Yes

Yes

tk = tk + Δt

End

Time end

c(0)e ke = k(0)e

exp(i120596tk)

(ce ke tk

Err = max(

v = v(0)

ug(120596 tk) =radicSu119892u119892 (120596 tk)

v = H )v+rug(120596j tk)

vkj = v(120596j tk)

uds uss

us = uds + uss



119892 (119905) =

119901

sum

119896=1

119883119896 (119905) =

119901

sum

119896=1

119860119896 (119905) 119884119896 (119905) (22)


119860119896 (119905) = 120572119896(119905 minus 120589119896)120573119896





119877119884119896119884119896(120591) = 119890

minus]119896|120591| cos (120578119896120591) (24)


119878119884119896119884119896(120596) =

]1198962120587

[1

]2119896

+ (120596 + 120578119896)2

+1

]2119896

+ (120596 minus 120578119896)2

] (25)



x

y

z

(a)

x

y

(b)

A B C D1

2

3

4

6m

3m

6m

x

z

3 times 6 m

(c)


008

007

009

006

004

005

003

001

002

0

1020

30

80

6070

4050

0

510

1520

2530

35

Time (s)

Frequency (rads)

PSD

(m2s

3)



119878119892119892(120596 119905) =

119901

sum

119896=1

1003816100381610038161003816119860119896 (119905)1003816100381610038161003816

2119878119884119896119884119896

(120596) (26)



119906119892 (120596 119905119896) = radic119878119892119892(120596 119905119896) exp (119894120596119905119896) (27)










119878 119909119887119911(120596119895 119905119896) = 119909

lowast

119887(120596119895 119905119896) (120596119895 119905119896)

119878119911 119909119887(120596119895 119905119896) =

lowast(120596119895 119905119896)

119909119887 (120596119895 119905119896)

119878119911119911 (120596119895 119905119896) = lowast

(120596119895 119905119896) (120596119895 119905119896)

119878 119909119887 119909119887(120596119895 119905119896) = 119909

lowast

119887(120596119895 119905119896)

119909119887 (120596119895 119905119896)

(28)


lowast

119887(120596119895 119905119896) and




010

008

006

004

002

000

RMS

drift

of t

he b

ase s

lab

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

(m)

(a)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00030

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he3

rd st

orey

(m)

(b)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00030

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he13

th st

orey

(m)

(c)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he19

th st

orey

(m)

(d)




by

119864 [119887119911] = int

+infin

minusinfin

119878 119909119887119911(120596 119905119896) d120596

= Δ120596

119873120596

sum

119895=1

[119878 119909119887119911(120596119895 119905119896) + 119878119911 119909119887

(120596119895 119905119896)]

1205902

119911= 2 int

+infin

0

119878119911119911 (120596 119905119896) d120596 = 2Δ120596

119873120596

sum

119895=1

119878119911119911 (120596119895 119905119896)

1205902

119909119887= 2 int

+infin

0

119878 119909119887 119909119887(120596 119905119896) d120596 = 2Δ120596

119873120596

sum

119895=1

119878 119909119887 119909119887(120596119895 119905119896)




22

20

18

16

14

12

10

8

6

4

2

Isolation

0000 0002 0004 0006 0008 0010 0012



Stor

eys

(a)


22

20

18

16

14

12

10

8

6

4

2

0

Stor

eys

00 01 02 03 04 05


(b)



4 Numerical Study




Storeys 119889

(m)119864119860

(GN)119864119868




119896 120572119896

120573119896

120574119896

120589119896

]119896

120578119896




22

20

18

16

14

12

10

8

6

4

2

Isolation


Stor

eys

0005 0010 0015 0020 0025 0030 0035

n = 3



20

18

16

14

12

10

8

6

4

2

0


Stor

eys

0000500000 00010 00015 00020 00025 00030 00035


(a)


20

18

16

14

12

10

8

6

4

2

0

Stor

eys

002000 004 006 008 010 012


(b)








22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(a)

22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(b)


014

012

010

008

006

004

002

000

Peak

RM

S dr

ift o

f the

bas

e sla

b

2 5 10 20



(m)

(a)

2 5 10 20


010

009

008

007

006

005

004

003

002

001

000

Peak

RM

S ab

solu

te ac

cele

ratio

n


(g)

(b)








5 Conclusions








Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2


K119904Φ119894 = 1205962




q + C119904q + K119904q = L (119892 + 119887) (7)

where

C119904 = Φ119879C119904Φ

K119904 = Φ119879K119904Φ = diag 120596

2

1 1205962

2 120596

2

119899

L = minusΦ119879M119904r119904

(8)


pressed as

Kminus1119904

=

119873

sum

119894=1

1

1205962

119894

Φ119894Φ119879

119894 (9)


u119904119904


minus

119899

sum

119894=1

1

1205962

119894

Φ119894Φ119879

119894)M119904 [u119904 + r119904 (119887 + 119892)]

= minusBM119904 [u119904 + r119904 (119887 + 119892)]

(10)

where

B = Kminus1119904

minus

119899

sum

119894=1

1

1205962

119894

Φ119894Φ119879

119894 (11)


+ 119888119890119887 + 119896119890119911 = 0 (12)

with

119888119890 = radic2

120587(120574

119864 [119887119911]

120590 119909119887

+ 120573120590119911) minus 119860

119896119890 = radic2

120587(120574120590 119909119887

+ 120573119864 [119887119911]

120590119911

)

(13)



M119887

q + C119887

q + K119909119887

q + (1 minus 1205720) 119896119906

0 119911 = minusMl119892

(14)

where

M = [119898119905 minusL119879

minusL I119899] C = [

119888119887 0

0 C119904]

K = [

1205720119896119906 0

0 K119904] l = 1 0 0 0

119879

(15)



119887

q119887

q

= [0 I119899+1


119909119887

q119887

q

+ 0a 119911 minus

0l 119892

(16)



(17)


v = 119909119887 q119879

119887 q119879

119911119879

(18)


v = Hv + r119892 (19)

where

H =[[[

[

0 I119899+1 0


]]]

]

r = minus 0 l119879 0

(20)


b = minus119888119890 0 0 0 (21)





ce =




Err le tolNo

No

Yes

Yes

tk = tk + Δt

End

Time end

c(0)e ke = k(0)e

exp(i120596tk)

(ce ke tk

Err = max(

v = v(0)

ug(120596 tk) =radicSu119892u119892 (120596 tk)

v = H )v+rug(120596j tk)

vkj = v(120596j tk)

uds uss

us = uds + uss



119892 (119905) =

119901

sum

119896=1

119883119896 (119905) =

119901

sum

119896=1

119860119896 (119905) 119884119896 (119905) (22)


119860119896 (119905) = 120572119896(119905 minus 120589119896)120573119896





119877119884119896119884119896(120591) = 119890

minus]119896|120591| cos (120578119896120591) (24)


119878119884119896119884119896(120596) =

]1198962120587

[1

]2119896

+ (120596 + 120578119896)2

+1

]2119896

+ (120596 minus 120578119896)2

] (25)



x

y

z

(a)

x

y

(b)

A B C D1

2

3

4

6m

3m

6m

x

z

3 times 6 m

(c)


008

007

009

006

004

005

003

001

002

0

1020

30

80

6070

4050

0

510

1520

2530

35

Time (s)

Frequency (rads)

PSD

(m2s

3)



119878119892119892(120596 119905) =

119901

sum

119896=1

1003816100381610038161003816119860119896 (119905)1003816100381610038161003816

2119878119884119896119884119896

(120596) (26)



119906119892 (120596 119905119896) = radic119878119892119892(120596 119905119896) exp (119894120596119905119896) (27)










119878 119909119887119911(120596119895 119905119896) = 119909

lowast

119887(120596119895 119905119896) (120596119895 119905119896)

119878119911 119909119887(120596119895 119905119896) =

lowast(120596119895 119905119896)

119909119887 (120596119895 119905119896)

119878119911119911 (120596119895 119905119896) = lowast

(120596119895 119905119896) (120596119895 119905119896)

119878 119909119887 119909119887(120596119895 119905119896) = 119909

lowast

119887(120596119895 119905119896)

119909119887 (120596119895 119905119896)

(28)


lowast

119887(120596119895 119905119896) and




010

008

006

004

002

000

RMS

drift

of t

he b

ase s

lab

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

(m)

(a)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00030

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he3

rd st

orey

(m)

(b)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00030

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he13

th st

orey

(m)

(c)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he19

th st

orey

(m)

(d)




by

119864 [119887119911] = int

+infin

minusinfin

119878 119909119887119911(120596 119905119896) d120596

= Δ120596

119873120596

sum

119895=1

[119878 119909119887119911(120596119895 119905119896) + 119878119911 119909119887

(120596119895 119905119896)]

1205902

119911= 2 int

+infin

0

119878119911119911 (120596 119905119896) d120596 = 2Δ120596

119873120596

sum

119895=1

119878119911119911 (120596119895 119905119896)

1205902

119909119887= 2 int

+infin

0

119878 119909119887 119909119887(120596 119905119896) d120596 = 2Δ120596

119873120596

sum

119895=1

119878 119909119887 119909119887(120596119895 119905119896)




22

20

18

16

14

12

10

8

6

4

2

Isolation

0000 0002 0004 0006 0008 0010 0012



Stor

eys

(a)


22

20

18

16

14

12

10

8

6

4

2

0

Stor

eys

00 01 02 03 04 05


(b)



4 Numerical Study




Storeys 119889

(m)119864119860

(GN)119864119868




119896 120572119896

120573119896

120574119896

120589119896

]119896

120578119896




22

20

18

16

14

12

10

8

6

4

2

Isolation


Stor

eys

0005 0010 0015 0020 0025 0030 0035

n = 3



20

18

16

14

12

10

8

6

4

2

0


Stor

eys

0000500000 00010 00015 00020 00025 00030 00035


(a)


20

18

16

14

12

10

8

6

4

2

0

Stor

eys

002000 004 006 008 010 012


(b)








22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(a)

22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(b)


014

012

010

008

006

004

002

000

Peak

RM

S dr

ift o

f the

bas

e sla

b

2 5 10 20



(m)

(a)

2 5 10 20


010

009

008

007

006

005

004

003

002

001

000

Peak

RM

S ab

solu

te ac

cele

ratio

n


(g)

(b)








5 Conclusions








Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











ce =




Err le tolNo

No

Yes

Yes

tk = tk + Δt

End

Time end

c(0)e ke = k(0)e

exp(i120596tk)

(ce ke tk

Err = max(

v = v(0)

ug(120596 tk) =radicSu119892u119892 (120596 tk)

v = H )v+rug(120596j tk)

vkj = v(120596j tk)

uds uss

us = uds + uss



119892 (119905) =

119901

sum

119896=1

119883119896 (119905) =

119901

sum

119896=1

119860119896 (119905) 119884119896 (119905) (22)


119860119896 (119905) = 120572119896(119905 minus 120589119896)120573119896





119877119884119896119884119896(120591) = 119890

minus]119896|120591| cos (120578119896120591) (24)


119878119884119896119884119896(120596) =

]1198962120587

[1

]2119896

+ (120596 + 120578119896)2

+1

]2119896

+ (120596 minus 120578119896)2

] (25)



x

y

z

(a)

x

y

(b)

A B C D1

2

3

4

6m

3m

6m

x

z

3 times 6 m

(c)


008

007

009

006

004

005

003

001

002

0

1020

30

80

6070

4050

0

510

1520

2530

35

Time (s)

Frequency (rads)

PSD

(m2s

3)



119878119892119892(120596 119905) =

119901

sum

119896=1

1003816100381610038161003816119860119896 (119905)1003816100381610038161003816

2119878119884119896119884119896

(120596) (26)



119906119892 (120596 119905119896) = radic119878119892119892(120596 119905119896) exp (119894120596119905119896) (27)










119878 119909119887119911(120596119895 119905119896) = 119909

lowast

119887(120596119895 119905119896) (120596119895 119905119896)

119878119911 119909119887(120596119895 119905119896) =

lowast(120596119895 119905119896)

119909119887 (120596119895 119905119896)

119878119911119911 (120596119895 119905119896) = lowast

(120596119895 119905119896) (120596119895 119905119896)

119878 119909119887 119909119887(120596119895 119905119896) = 119909

lowast

119887(120596119895 119905119896)

119909119887 (120596119895 119905119896)

(28)


lowast

119887(120596119895 119905119896) and




010

008

006

004

002

000

RMS

drift

of t

he b

ase s

lab

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

(m)

(a)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00030

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he3

rd st

orey

(m)

(b)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00030

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he13

th st

orey

(m)

(c)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he19

th st

orey

(m)

(d)




by

119864 [119887119911] = int

+infin

minusinfin

119878 119909119887119911(120596 119905119896) d120596

= Δ120596

119873120596

sum

119895=1

[119878 119909119887119911(120596119895 119905119896) + 119878119911 119909119887

(120596119895 119905119896)]

1205902

119911= 2 int

+infin

0

119878119911119911 (120596 119905119896) d120596 = 2Δ120596

119873120596

sum

119895=1

119878119911119911 (120596119895 119905119896)

1205902

119909119887= 2 int

+infin

0

119878 119909119887 119909119887(120596 119905119896) d120596 = 2Δ120596

119873120596

sum

119895=1

119878 119909119887 119909119887(120596119895 119905119896)




22

20

18

16

14

12

10

8

6

4

2

Isolation

0000 0002 0004 0006 0008 0010 0012



Stor

eys

(a)


22

20

18

16

14

12

10

8

6

4

2

0

Stor

eys

00 01 02 03 04 05


(b)



4 Numerical Study




Storeys 119889

(m)119864119860

(GN)119864119868




119896 120572119896

120573119896

120574119896

120589119896

]119896

120578119896




22

20

18

16

14

12

10

8

6

4

2

Isolation


Stor

eys

0005 0010 0015 0020 0025 0030 0035

n = 3



20

18

16

14

12

10

8

6

4

2

0


Stor

eys

0000500000 00010 00015 00020 00025 00030 00035


(a)


20

18

16

14

12

10

8

6

4

2

0

Stor

eys

002000 004 006 008 010 012


(b)








22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(a)

22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(b)


014

012

010

008

006

004

002

000

Peak

RM

S dr

ift o

f the

bas

e sla

b

2 5 10 20



(m)

(a)

2 5 10 20


010

009

008

007

006

005

004

003

002

001

000

Peak

RM

S ab

solu

te ac

cele

ratio

n


(g)

(b)








5 Conclusions








Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x

y

z

(a)

x

y

(b)

A B C D1

2

3

4

6m

3m

6m

x

z

3 times 6 m

(c)


008

007

009

006

004

005

003

001

002

0

1020

30

80

6070

4050

0

510

1520

2530

35

Time (s)

Frequency (rads)

PSD

(m2s

3)



119878119892119892(120596 119905) =

119901

sum

119896=1

1003816100381610038161003816119860119896 (119905)1003816100381610038161003816

2119878119884119896119884119896

(120596) (26)



119906119892 (120596 119905119896) = radic119878119892119892(120596 119905119896) exp (119894120596119905119896) (27)










119878 119909119887119911(120596119895 119905119896) = 119909

lowast

119887(120596119895 119905119896) (120596119895 119905119896)

119878119911 119909119887(120596119895 119905119896) =

lowast(120596119895 119905119896)

119909119887 (120596119895 119905119896)

119878119911119911 (120596119895 119905119896) = lowast

(120596119895 119905119896) (120596119895 119905119896)

119878 119909119887 119909119887(120596119895 119905119896) = 119909

lowast

119887(120596119895 119905119896)

119909119887 (120596119895 119905119896)

(28)


lowast

119887(120596119895 119905119896) and




010

008

006

004

002

000

RMS

drift

of t

he b

ase s

lab

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

(m)

(a)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00030

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he3

rd st

orey

(m)

(b)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00030

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he13

th st

orey

(m)

(c)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he19

th st

orey

(m)

(d)




by

119864 [119887119911] = int

+infin

minusinfin

119878 119909119887119911(120596 119905119896) d120596

= Δ120596

119873120596

sum

119895=1

[119878 119909119887119911(120596119895 119905119896) + 119878119911 119909119887

(120596119895 119905119896)]

1205902

119911= 2 int

+infin

0

119878119911119911 (120596 119905119896) d120596 = 2Δ120596

119873120596

sum

119895=1

119878119911119911 (120596119895 119905119896)

1205902

119909119887= 2 int

+infin

0

119878 119909119887 119909119887(120596 119905119896) d120596 = 2Δ120596

119873120596

sum

119895=1

119878 119909119887 119909119887(120596119895 119905119896)




22

20

18

16

14

12

10

8

6

4

2

Isolation

0000 0002 0004 0006 0008 0010 0012



Stor

eys

(a)


22

20

18

16

14

12

10

8

6

4

2

0

Stor

eys

00 01 02 03 04 05


(b)



4 Numerical Study




Storeys 119889

(m)119864119860

(GN)119864119868




119896 120572119896

120573119896

120574119896

120589119896

]119896

120578119896




22

20

18

16

14

12

10

8

6

4

2

Isolation


Stor

eys

0005 0010 0015 0020 0025 0030 0035

n = 3



20

18

16

14

12

10

8

6

4

2

0


Stor

eys

0000500000 00010 00015 00020 00025 00030 00035


(a)


20

18

16

14

12

10

8

6

4

2

0

Stor

eys

002000 004 006 008 010 012


(b)








22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(a)

22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(b)


014

012

010

008

006

004

002

000

Peak

RM

S dr

ift o

f the

bas

e sla

b

2 5 10 20



(m)

(a)

2 5 10 20


010

009

008

007

006

005

004

003

002

001

000

Peak

RM

S ab

solu

te ac

cele

ratio

n


(g)

(b)








5 Conclusions








Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










010

008

006

004

002

000

RMS

drift

of t

he b

ase s

lab

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

(m)

(a)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00030

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he3

rd st

orey

(m)

(b)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00030

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he13

th st

orey

(m)

(c)

minus5 0 5 10 15 20 25 30 35 40

Time (s)

PEMMonte Carlo

00025

00020

00015

00010

00005

00000

RMS

drift

of t

he19

th st

orey

(m)

(d)




by

119864 [119887119911] = int

+infin

minusinfin

119878 119909119887119911(120596 119905119896) d120596

= Δ120596

119873120596

sum

119895=1

[119878 119909119887119911(120596119895 119905119896) + 119878119911 119909119887

(120596119895 119905119896)]

1205902

119911= 2 int

+infin

0

119878119911119911 (120596 119905119896) d120596 = 2Δ120596

119873120596

sum

119895=1

119878119911119911 (120596119895 119905119896)

1205902

119909119887= 2 int

+infin

0

119878 119909119887 119909119887(120596 119905119896) d120596 = 2Δ120596

119873120596

sum

119895=1

119878 119909119887 119909119887(120596119895 119905119896)




22

20

18

16

14

12

10

8

6

4

2

Isolation

0000 0002 0004 0006 0008 0010 0012



Stor

eys

(a)


22

20

18

16

14

12

10

8

6

4

2

0

Stor

eys

00 01 02 03 04 05


(b)



4 Numerical Study




Storeys 119889

(m)119864119860

(GN)119864119868




119896 120572119896

120573119896

120574119896

120589119896

]119896

120578119896




22

20

18

16

14

12

10

8

6

4

2

Isolation


Stor

eys

0005 0010 0015 0020 0025 0030 0035

n = 3



20

18

16

14

12

10

8

6

4

2

0


Stor

eys

0000500000 00010 00015 00020 00025 00030 00035


(a)


20

18

16

14

12

10

8

6

4

2

0

Stor

eys

002000 004 006 008 010 012


(b)








22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(a)

22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(b)


014

012

010

008

006

004

002

000

Peak

RM

S dr

ift o

f the

bas

e sla

b

2 5 10 20



(m)

(a)

2 5 10 20


010

009

008

007

006

005

004

003

002

001

000

Peak

RM

S ab

solu

te ac

cele

ratio

n


(g)

(b)








5 Conclusions








Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










22

20

18

16

14

12

10

8

6

4

2

Isolation

0000 0002 0004 0006 0008 0010 0012



Stor

eys

(a)


22

20

18

16

14

12

10

8

6

4

2

0

Stor

eys

00 01 02 03 04 05


(b)



4 Numerical Study




Storeys 119889

(m)119864119860

(GN)119864119868




119896 120572119896

120573119896

120574119896

120589119896

]119896

120578119896




22

20

18

16

14

12

10

8

6

4

2

Isolation


Stor

eys

0005 0010 0015 0020 0025 0030 0035

n = 3



20

18

16

14

12

10

8

6

4

2

0


Stor

eys

0000500000 00010 00015 00020 00025 00030 00035


(a)


20

18

16

14

12

10

8

6

4

2

0

Stor

eys

002000 004 006 008 010 012


(b)








22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(a)

22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(b)


014

012

010

008

006

004

002

000

Peak

RM

S dr

ift o

f the

bas

e sla

b

2 5 10 20



(m)

(a)

2 5 10 20


010

009

008

007

006

005

004

003

002

001

000

Peak

RM

S ab

solu

te ac

cele

ratio

n


(g)

(b)








5 Conclusions








Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










22

20

18

16

14

12

10

8

6

4

2

Isolation


Stor

eys

0005 0010 0015 0020 0025 0030 0035

n = 3



20

18

16

14

12

10

8

6

4

2

0


Stor

eys

0000500000 00010 00015 00020 00025 00030 00035


(a)


20

18

16

14

12

10

8

6

4

2

0

Stor

eys

002000 004 006 008 010 012


(b)








22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(a)

22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(b)


014

012

010

008

006

004

002

000

Peak

RM

S dr

ift o

f the

bas

e sla

b

2 5 10 20



(m)

(a)

2 5 10 20


010

009

008

007

006

005

004

003

002

001

000

Peak

RM

S ab

solu

te ac

cele

ratio

n


(g)

(b)








5 Conclusions








Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(a)

22201816141210

8642

Isolation

Stor

eys


n = 3

n = 10

n = 40

(b)


014

012

010

008

006

004

002

000

Peak

RM

S dr

ift o

f the

bas

e sla

b

2 5 10 20



(m)

(a)

2 5 10 20


010

009

008

007

006

005

004

003

002

001

000

Peak

RM

S ab

solu

te ac

cele

ratio

n


(g)

(b)








5 Conclusions








Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











5 Conclusions








Acknowledgments


References



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Review Article Seismic Response of Base-Isolated High-Rise...

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