Lopez-Garcia, Soong - 2009

8/18/2019 Lopez-Garcia, Soong - 2009

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Probabilistic Engineering Mechanics 24 (2009) 210–223

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Probabilistic Engineering Mechanics

journal homepage: www.elsevier.com/locate/probengmech

Assessment of the separation necessary to prevent seismic pounding betweenlinear structural systems

D. Lopez-Garcia a,∗, T.T. Soong ba Departamento de Ingenieria Estructural y Geotecnica, Pontificia Universidad Catolica de Chile, Av. Vicuna Mackenna 4860, Macul, Santiago 782-0436, Chileb Department of Civil, Structural & Environmental Engineering, University at Buffalo, 212 Ketter Hall, Buffalo, NY 14260, USA

a r t i c l e i n f o

Article history:

Received 2 July 2007

Received in revised form

28 May 2008

Accepted 12 June 2008Available online 18 June 2008

Keywords:

Seismic pounding

Separation distance

a b s t r a c t

This study examines the accuracy of the Double Difference Combination (DDC) rule (also known simply

as the CQC rule) in predicting the separation necessary to prevent seismic pounding between linearstructural systems. Seismic excitations were modeled as modulated and filtered modulated Gaussian

whitenoise random processes, andadjacentstructureswere modeled as 5%-dampedSDOF systems havinga wide range of values of natural periods. Results obtained through Monte Carlo simulations indicate that

the accuracy of the DDC rule depends not only on the ratio of the natural periods of the structures, butalso on the relationship between the values of the natural periods and the value of the period associated

with the main frequency of the excitation.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Seismic pounding occurs when the separation between adjacent buildings is not large enough to accommodate the dis-

placement response of the structures relative to each other during

earthquake events. Depending on the characteristics of the collid-

ing buildings [1], pounding might cause severe structural dam-

age in some cases [2], and even collapse is possible in some

extreme situations [3]. Further, even in those cases where it does

not result in significant structural damage, pounding always in-

duces higher floor accelerations in the form of large magnitude,

short duration pulses [4], which in turn cause greater damage to

building contents [2]. For these reasons, it is widely accepted that

pounding is an undesirable phenomenon that should be prevented

or mitigated. This is recognized in seismic design codes and regula-

tions worldwide, which typically specify minimum separations to

be provided between adjacent buildings. For instance, accordingto the 2003 edition of the International Building Code, minimum

separations are given by:

S = X A + X B(adjacent buildings separated by a property line) (1)

S =

X 2 A + X 2B(adjacent buildings located on the same property) (2)

∗ Corresponding author. Tel.: +56 2 354 7684; fax: +56 2 354 4243.E-mail address: [email protected] (D. Lopez-Garcia).

where S = separation distance and X A, X B = displacement responseof the adjacent structures ‘‘ A’’ and ‘‘B’’, respectively, at the location

where pounding is expected to occur (i.e., at the level coincidingwith the roof level of the shorter building [5]).In Eqs. (1) and (2) the separation is obtained by combining

the quantities X A and X B according to the well known ABS andSRSS combination rules, respectively. Previous studies [5,6] haveshown that the ABS rule is always conservative, and that thedegree of conservatism increases as the periods of the adjacentstructures become closer to each other. The same studies havealso shown that, as the periods of the adjacent structures becomecloser to each other, results given by the SRSS rule evolve fromreasonably accurate (not always conservative) to very conservativeas well (but not as conservative as those given by the ABS rule).Qualitatively, these observations apply to structures behavingeither linearly or nonlinearly.

A more rational approach to calculate minimum separations

between linear structures was proposed by Jeng et al. [5], who,following a ‘‘spectral difference method’’ approach, derived theDouble Difference Combination (DDC) rule, i.e.:

S =

X 2 A + X 2B − 2ρ X A X B (3)whereρ is the well-known cross-correlation coefficient commonly

used in the Complete Quadratic Combination (CQC) rule of modal

responses of linear MDOF structures, and is given by [7,8]:

ρ =8√ ξ Aξ B

ξ A + ξ B T AT B

T AT B

1.5

1 −

T AT B

22+ 4ξ Aξ B

1 +

T AT B

2T AT B

+ 4 ξ 2 A + ξ 2B

T AT B

2 (4)

0266-8920/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.probengmech.2008.06.002

http://www.elsevier.com/locate/probengmechhttp://www.elsevier.com/locate/probengmechmailto:[email protected]://dx.doi.org/10.1016/j.probengmech.2008.06.002http://dx.doi.org/10.1016/j.probengmech.2008.06.002mailto:[email protected]://www.elsevier.com/locate/probengmechhttp://www.elsevier.com/locate/probengmech


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D. Lopez-Garcia, T.T. Soong / Probabilistic Engineering Mechanics 24 (2009) 210–223 211

where T A, T B and ξ A, ξ B are the natural periods and damping ratios,respectively, of the adjacent structures ‘‘ A’’ and ‘‘B’’. Because of the similarity between Eq. (3) and the equation giving the CQCcombination of modal responses of linear MDOF systems, Eq. (3) is

sometimes also referred to simplyas the CQCrule [9–11].As shownin several analytical studies [5,6,10,11], the DDC rule providesreasonably accurate results regardless of whether T A and T B areclose to each other or not. Although Eq. (3) is, in a strict sense,valid for SDOF systems only, it is also applicable to MDOF systemswhose first mode response accounts for a large portion of thetotal response [5] (a rigorous, but not practical, random vibrationapproach to calculate separations between linear MDOF systemsconsidering the contribution of higher modes can be seen in [12,13]). The DDC rule has also been used to calculate separationsbetween nonlinear building structures, either considering Eq. (4)to calculate ρ [14,15], or considering alternative expressions forρ intended to somehow take into account the nonlinear nature of the response [6,9,16]. The degree of accuracy of these applications

of the DDC rule to nonlinear structures, however, turned out to bemuch less than that corresponding to linear systems.

Eq. (4) gives the correlation between stationary displacementresponse processes of linear SDOF systems subjected to white

noise excitations [7,8], and Eq. (3) was derived assuming thatthe ratio of the mean peak displacement response of a linearSDOF system (over a finite duration) to the corresponding RMSdisplacement response value is independent of the parameters of the system (natural period and damping ratio) [5]. Actual seismicexcitations, however, neither have white-noise characteristicsnor are stationary, and the mean-peak-displacement-responseto RMS-displacement-response ratio is not independent of thesystem parameters, not even under stationary conditions [7]. Forthese reasons, results obtained using the DDC rule can be expectedto exhibit some degree of error. The evaluation performed by Jenget al. [5] found that, when the seismic excitation consists of a

set of recorded seismic ground acceleration histories, the DDCrule generally provides slightly conservative results, somewhat

more conservative when the periods of the structures are closeto each other. More recently, Hong et al. [10] found that, whenthe seismic excitation is modeled as a random process, the DDCrule is, under stationary conditions, slightly unconservative whenthe periods of the structures are not close to each other, andsomewhat conservative otherwise. Finally, essentially the sameresults were obtained by Wang and Hong [11], who modeledthe seismic excitation as a nonstationary random process. In thelatter studies [10,11], the ‘‘exact’’ relative displacement responseof adjacent linear systems (against which the estimates providedby the DDCrule were compared)was calculated using approximateanalytical expressions (no exact analytical solutions are available

at this time).The objective of this study is to provide further insight into the

accuracy of the DDC rule in predicting the separation necessary toprevent seismic pounding between linear structural systems. Thisstudy is similar to that by Wang and Hong [11] in that the seismicexcitation is modeled as a nonstationary random process, but isdifferent in that: (a) the ‘‘exact’’ relative displacement responseof adjacent linear systems is obtained through Monte Carlosimulation; and (b) possible influence of others factor (e.g., therelationship between the frequency content of the excitation andthe natural periods of the linear systems) is investigated.

2. Description of the evaluation procedure

Adjacent structural systems ‘‘ A’’ and ‘‘B’’ are modeled as linearSDOF systems (Fig. 1). Damping ratiosξ A and ξ B are assumed equal

to 5%, the value typically assumed in code-regulated proceduresfor the analysis of conventional building structures. The seismic

Fig. 1. Adjacent structural systems ‘‘ A’’ and ‘‘B’’.

Fig. 2. Modulating function f e (t ).

excitation is modeled as a Gaussian, zero-mean nonstationaryrandom process Ü g (t ) whose evolutionary power spectral densityfunction S ̈Ug (t , ω) is given by:

S ̈Ug (t , ω) = [ f e (t )]2 S g (ω) (5)where t denotes time, ω indicates circular frequency, f e (t )is a modulating (sometimes also referred to as ‘‘envelope’’ or

‘‘window’’) time function and S g (ω) is a stationary power spectraldensity function. The modulating function f e (t ) is assumed equalto that initially proposed by Saragoni and Hart [17] and calibrated

later by Boore [18], which is given by:

f e (t ) = at be−ct (6)where:

a =

e

εT D

b(7)

b = − ε ln (η)1 + ε [ln (ε − 1)] (8)

c =b

εT D (9)

where, in turn, T D is the duration of the excitation and η and εare constants that define the shape of f e (t ) (Fig. 2). In this study,constants η and ε are set equal to 0.05 and 0.20, respectively.

The displacement (relative to the ground) response processesof the SDOF systems ‘‘ A’’ and ‘‘B’’ are denoted by U A (t ) and U B (t ),respectively, and the relative displacement response process U REL(t ) is given by:

U REL (t ) = U A (t ) − U B (t ) . (10)It is emphasized that the expression ‘‘relative displacement’’ refersto the displacement response of the SDOF systems ‘‘ A’’ and ‘‘B’’relative to each other . Extreme values of processes U A (t ), U B (t ) and

U REL (t ) are denoted by U Amax , U Bmax and U RELmax, respectively, andtheir corresponding mean values are denoted by X A, X B and X REL.


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212 D. Lopez-Garcia, T.T. Soong / Probabilistic Engineering Mechanics 24 (2009) 210–223

Fig. 3. Sample realization of a modulated white noise excitation process (T D = 15 s).

Estimates of X REL obtained using the DDC rule (i.e., calculated usingEqs. (3) and (4)) are denoted by S .

Monte Carlo simulations (500 samples) are performed asfollows. Realizations ü g (t ) of the excitation process Ü g (t ) aregenerated according to standard simulation techniques [19,20].

Realizations u A (t ) and uB (t ) of response processes U A (t ) and U B (t )are obtained by numerically solving the corresponding equationsof motion, i.e.:

ü A (t ) +4πξ A

T Au̇ A (t ) +

4π2

T 2 Au A (t ) = ü g (t ) (11)

üB (t ) +4πξ B

T Bu̇B (t ) +

4π2

T 2BuB (t ) = ü g (t ) (12)

where the overdots indicate time derivatives. Realizations uREL (t )of response process U REL (t ) are obtained by:

uREL (t ) = u A (t ) − uB (t ) . (13)

Sample values of random variables U Amax , U Bmax and U RELmax areobtained by:

u A max = maxt |u A (t )| (14)uB max = maxt |uB (t )| (15)uREL max = maxt |uREL (t )| (16)and the corresponding mean values X A, X B and X REL are obtained byaveraging the sample values u Amax, uBmax and uRELmax, respectively.

Estimates S of the relative displacement response calculatedfollowing the DDC rule (Eqs. (3) and (4)) are then comparedwith the ‘‘exact’’ relative displacement response X REL. Results areexpressed in terms of the S / X REL ratio. Hence, values of the S / X RELratio that are greater than unity indicate that results provided bythe DDC rule are conservative, and the opposite is indicated by

values of the S / X REL ratio that are less than unity.It is implicit in Eqs. (1) and (2) that it is assumed that both

adjacent structures experience the same excitation at the same

time. In reality, the seismic excitation experienced by a givenstructural system is not exactly the same excitation acting onan adjacent structure due to ground motion spatial variation.However, since the distance between adjacent structures prone topounding is relatively small, ground motion spatial variation maybe ignored. It has been shown [21,22] that the influence of groundmotion spatial variation on the relative displacement responseof adjacent structures is relevant only in the case of very stiff structures having relatively large horizontal dimensions, and onlywhen the natural periods of the structures are very close to eachother. In any case, even if both adjacent structures do experience

the same excitation, they do not experience it exactly at the sametime due to the traveling nature of seismic waves. However, it has

also been shown [23] that the effects of traveling seismic waves arealso negligible when the distance between the adjacent structuresis relatively small. For these reasons, it was assumed that bothadjacent structures experience the same excitation at the sametime.

Finally, it must be noted that the extreme values of responseprocesses U A, U B and U REL calculated with Eqs. (14)–(16) aredouble-sided extreme values, while the values needed to correctlyestimate the separation necessary to avoid pounding are actuallyone-sided extreme values [10,11]. However, double-sided extremevalues are considered because, in practice, the quantities X A and

X B in Eq. (3) are always estimates of double-sided extreme values,and then, for consistency, values of S obtained using double-sided extreme values of processes U A and U B (i.e., X A and X B)are compared with double-sided extreme values of process U REL(i.e., X REL). Monte Carlo simulations performed considering all theexcitation processes and all the combinations of natural periods T Aand T B that will be described laterindicate that theratio of double-sided extreme values to one-sided extreme values of the response

process U REL ranges from 1.01 to 1.10, i.e., double-sided extremevalues of U REL are only slightly conservative (by 10% at most).

3. Response to modulated white noise excitation

If function S g (ω) is set equal to a constant value S 0, the

resulting excitation process Ü g (t ) is then a modulatedwhite noise.For illustration purposes, a sample realization ü g (t ), generatedconsidering S 0 = 200 cm2/s3, T D = 15 s and a constant time stepequal to 0.001 s, is shown in Fig. 3, and the corresponding samplerealizations u A (t ), uB (t ) and uREL (t ) are shown in Fig. 4.

Values of the S / X REL ratio obtained by considering modulatedwhite noise excitation processes are shown in Figs. 5 and 6. Therange of selected values of duration T D is very similar to the

range of expected values of actual seismic excitations. Each dotin Fig. 5 indicates the value of the S / X REL ratio for a particularpair of adjacent structures ‘‘ A’’ and ‘‘B’’, where T B = 0.10 s,0.15 s, . . . , 4.00 s, and T A = 0.05 s, 0.10 s, . . . , T B – 0.05 s(i.e., T A


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Fig. 4. Sample realization u A(t ), uB(t ) and uREL(t ) of displacement response processes U A(t ), U B(t ), U REL(t ). Excitation process: modulated white noise (T D = 15 s).

unconservative (by 10% at most), and the S / X REL ratio seems to bea function of the period ratio only, i.e., it is essentially independent

of the values of T A and T B. When the period ratio is greater than0.75, on the other hand, the DDC rule is always conservative (by asmuch as 75%), thedegreeof conservatismincreases with increasingvaluesof the period ratio,and the S / X REL ratio doesnotseem tobe afunction of the period ratio only. This last observation can be moreclearly appreciated in Fig. 6, where each line in each plot indicatesvalues of the S / X REL ratio for 99 pairs of adjacent structures ‘‘ A’’ and‘‘B’’, T B has a constant value and T A = 0.01T B, 0.02T B, . . . , 0.99T B(hence, T A < T B in allcases). Indeed, results shown in Fig. 6 confirmthat, when T A/T B > 0.75, the S / X REL ratio is a function not only of the period ratio but also of the values of T A and T B: the greater thevalues of T A and T B, the more conservative the DDC rule.

Finally, it is also observed in Figs. 5 and 6 that the accuracy of the DDC rule improves as the duration of the excitation increases.

This last observation was expected since, as the duration of theexcitation increases, the characteristics of response processes U A

(t ), U B (t ) and U REL (t ) become more similar to those correspondingto the stationary conditions under which the equation of the DDC

rule was derived.

4. Response to modulated filtered white noise

More realistic seismic excitation processes can be obtainedby ‘‘filtering’’, in the frequency domain, a process having white-noise characteristics. A widely used filtered white noise excitationprocess is that defined by the modified Kanai–Tajimi equation,sometimes also referred to as the Clough–Penzien equation, whichis given by [19]:

S g (ω) = H CP (ω)H KT (ω) S 0 (17)where:

H KT (ω) =ω4 g

+ 4ξ 2 g ω

2 g ω

2

ω2 g − ω2

2 + 4ξ 2 g ω2 g ω2 (18)


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Fig. 5. Values of the S / X REL ratio: modulated white noise excitation process.

is the Kanai–Tajimi filter (defined by parameters ω g and ξ g ), and:

H CP (ω) = ω4ω2 f − ω2

2 + 4ξ 2 f ω2 f ω2(19)

is the Clough–Penzien filter (defined by parameters ω f and ξ f ).The Kanai–Tajimi filter amplifies the white-noise intensity S 0 inthe vicinity of frequency ω g , and reduces the intensity S 0 atrelatively large values of ω. The size of the range of frequencies atwhich the intensity S 0 is amplified is controlled by the parameterξ g , which takes values between zero and unity. As the value of parameter ξ g increases, the range of frequencies at which theintensity S 0 is amplified increases as well, and the characteristicsof the frequency content of the process evolve from those typicalof narrow-band processes to those representative of wide-bandprocesses. TheClough–Penzien filteris introduced in order to make

S g (ω) tendto zeroasω tends to zero, as observed in power spectraldensity functions of actual earthquake records. This is achieved by

conveniently setting the values of parameters ω f and ξ f . It can beshown that, when S g (ω) is given by the modified Kanai–Tajimi

(Eqs. (17)–(19)), the corresponding main frequency ωm (i.e., thefrequency at which S g (ω) takes its maximum value) is given by:

ωm = ±

−1 +

1 + 8ξ 2 g

2ξ g ω g (20)

which is not a function of ω f and ξ f because, typically, H CP (ωm)= 1.

Fig. 7 shows the function S g (ω) given by Eq. (17) when ω g =12.50 rad/s, ξ g = 0.60, ω f = 2.00 rad/s, ξ f = 0.70 andS 0 = 200 cm2/s3. The figure also illustrates the effect of eachof the filters of the modified Kanai–Tajimi equation. The mainfrequency isωm = 10.26 rad/s Eq. (20). For illustration purposes, asample realization

¨u

g (t ) of the resulting modulated filtered white

noise excitation process Ü g (t ), generated considering T D = 30 s


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Fig. 6. Values of the S / X REL ratio: modulated white noise excitation process.

and a constant time step equal to 0.005 s, is shown in Fig. 8. Thecorresponding 5%-damped mean pseudo-acceleration response

spectrum (Fig. 9) is similar to average spectra of actual seismicaccelerations recorded on firm soil conditions. Note that the periodat which the mean spectrum reaches its maximum value is 0.40s, which does not coincide with the period T m associated with ωm(= 2π/10.26 rad/s = 0.60 s in this case). Figs. 7 and 8 indicatethat the process has wide-band characteristics, which is consistentwith the relatively large value of parameter ξ g .

Values of the S / X REL ratio obtained by considering themodulated filtered white noise excitation process described aboveare shown in Figs. 10 and 11. The sets of pairs of adjacentstructures ‘‘ A’’and‘‘B’’ considered in these figures arethe same setsconsidered in Figs. 5 and 6, respectively. A comparison betweenFigs. 10 and 11 and Figs. 5 and 6 (plot corresponding to T D = 30 s)reveals that, while most of the observations made when describing

the results shown in Figs. 5 and 6 are still valid for the resultsshown in Figs. 10 and 11, there are a few differences. Firstly, the

DDC rule is now always conservative when T A/T B > 0.80 ratherthan when T A/T B > 0.75. Secondly, when T A/T B 1 whenT A/T B < 0.80. It was found that periods T A and T B are, in all of these cases, relatively small. More insight into this last observationis provided by Fig. 12, which was obtained by considering severalpairs of adjacent structures ‘‘ A’’ and ‘‘B’’, where T A = 0.050 s,0.055 s, . . . , 0.750 s and T B = 0.050 s, 0.055 s, . . . , 0.750 s.When the excitation is a modulated white noise (Fig. 12, left),the contour lines separating the ‘‘conservative’’ (S / X REL > 1) and‘‘unconservative’’ (S / X REL < 1) regions essentially coincide with

the lines analytically expressed by T B = 0.75T A and T B = 1.33T A,an observation that is entirely consistent with what was observed


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Fig. 7. Modified Kanai–Tajimi modulated filtered white noise excitation process: power spectral density function S g (ω) (ω g = 12.50 rad/s, ξ g = 0.60, ω f = 2.00 rad/s,ξ f = 0.70 and S 0 = 200 cm2/s3).

Fig. 8. Modified Kanai–Tajimi modulated filtered white noise excitation process (ω g = 12.50 rad/s, ξ g = 0.60, ω f = 2.00 rad/s, ξ f = 0.70 and S 0 = 200 cm2/s3): samplerealization (T D = 30 s).

Fig. 9. Modified Kanai–Tajimi modulated filtered white noise excitation process (ω g = 12.50 rad/s, ξ g = 0.60, ω f = 2.00 rad/s, ξ f = 0.70 and S 0 = 200 cm2/s3 andT D = 30 s): mean pseudo-acceleration response spectrum (damping ratio = 0.05).

in Figs. 5 and 6. When the excitation is the modulated filtered

white noise described above, on the other hand ( Fig. 12, right), thecontour lines separating the ‘‘conservative’’ and ‘‘unconservative’’

regions exhibit a different pattern when the values of T A and

T B are relatively small. Recalling that T m = 0.60 s in thiscase, the DDC rule is now also conservative essentially when


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Fig. 10. Values of the S / X REL ratio: modified Kanai–Tajimi modulated filtered white noise excitation process (ω g = 12.50 rad/s, ξ g = 0.60, ω f = 2 .00 rad/s, ξ f = 0.70,S 0 = 200 cm2/s3 and T D = 30 s).

Fig. 11. Values of the S / X REL ratio: modified Kanai–Tajimi modulated filtered white noise excitation process (ω g = 12.50 rad/s, ξ g = 0.60, ω f = 2 .00 rad/s, ξ f = 0.70,S 0

= 200 cm2/s3 and T D

= 30 s).

Fig.12. Contour linesof values ofthe S / X REL ratio: [left] modulated white noise (T D = 30 s); [right] modified Kanai–Tajimimodulated filtered white noise excitationprocess(ω g = 12.50 rad/s, ξ g = 0.60, ω f = 2.00 rad/s, ξ f = 0.70, S 0 = 200 cm2/s3 and T D = 30 s).

T B < T m − T A regardless of the value of the period ratio. In passing,it is perhaps opportune to mention that contour lines for other

values of the S / X REL ratio were calculated (a single contour plot

was initially planned instead of Figs. 10–12), but, especially in the

‘‘conservative’’ region, they turned out to be very close to each

other, making the figure confusing and not adequate to draw clear,unambiguous conclusions.

In order to get more insight into the possible influence of the

frequency content of the seismic excitation on the accuracy of

the DDC rule, a second modulated filtered white noise excitation

process was obtained by setting ω g = 7.50 rad/s, ξ g = 0.30, ω f =2.00 rad/s, ξ f = 0.70 and S 0 = 200 cm2/s3. The characteristicsof the resulting process (Fig. 13) are similar to those typical of seismic excitations recorded on soft soil. The corresponding main


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Fig. 13. Modified Kanai–Tajimi modulated filtered white noise excitation process (ω g = 7.50 rad/s, ξ g = 0.30, ω f = 2.00 rad/s, ξ f = 0.70, S 0 = 200 cm2/s3 andT D = 30 s): [top] sample realization; [bottom] mean pseudo-acceleration response spectrum (damping ratio = 0.05).

frequency is ωm = 6.98 rad/s and the associated period isT m = 0.90 s, which again does not coincide with the period atwhich the mean pseudo-acceleration response spectrum reachesits maximum value (=0.80 s). Fig. 13 indicates that the processhas in this case characteristics that are intermediate betweenthose typical of wide-band excitations and those representative of narrow-band processes.

The resulting values of the S / X REL ratio are shown in Fig. 14.Fig. 14 (top) shows that, while the S / X REL vs. T A/T B relationship is, inmost cases, again the same main relationship described before, thenumber of cases not conforming to the main relationship is nowgreater. Results shown in Fig. 14 (bottom) seems to indicate that,again, the S / X REL vs. T A/T B relationship might be characterized interms of T m (=0.90 s inthiscase).When T B is definitely greater thanT m (i.e., the lines corresponding to T B

= 2.0 s and 4.0 s), the S / X REL

vs. T A/T B relationship follow exactly the main pattern describedbefore. When T B is close to, but still greater than, T m (i.e., the linescorresponding to T B = 1.0 s), the S / X REL vs. T A/T B relationshipstill follow essentially, but not exactly, the same main patternmentioned before. When T B is definitely less than T m, however(i.e., the line corresponding to T B = 0.50 s), the S / X REL vs. T A/T Brelationship has now different characteristics. In this latter case,the DDC rule turns out to be always conservative, the more so asthe value of the period ratio T A/T B increases. Fig. 15 empiricallyconfirms that, indeed, the accuracy of the DDC rule might becharacterized in terms of T m: in theregion roughly defined by T B T m, (i.e., the line corresponding to T B = 4.0 s) still followthe same main patterndescribed before, the lines corresponding to

cases where T B < T m, (i.e., the lines corresponding to T B = 0.50 s,1.00 s and 2.00 s,) have the characteristics mentioned before when

describing the results for the same cases (i.e., T B < T m) obtainedconsidering the second modulated filtered white noise excitation

process: the DDC rule is always conservative in these cases, the

more so as the value of the period ratio T A/T B increases. It can alsobe observed that, when T B < T m, the S / X REL ratio is a function not

onlyof theperiod ratiobut alsoof the actual values of T A and T B:thelesser the values of T A and T B, the more conservative the DDC rule.


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Fig. 14. Values of the S / X REL ratio: modified Kanai–Tajimi modulated filtered white noise excitation process (ω g = 7.50 rad/s, ξ g = 0.30, ω f = 2.00 rad/s, ξ f = 0.70,S 0 = 200 cm2/s3 and T D = 30 s).

Fig. 15. Contour lines of values of the S / X REL ratio: modified Kanai–Tajimi

modulated filtered white noise excitation process (ω g = 7.50 rad/s, ξ g = 0.30,ω f = 2.00 rad/s, ξ f = 0.70, S 0 = 200 cm2/s3 and T D = 30 s).

Fig. 18 confirms once again that the accuracy of the DDC rule maybe characterized in terms of T m: as before, in the region roughly

defined by T B <

T 2m − T 2 A , the DDC rule is always conservative

regardless of the period ratio.

5. Main frequency of design response spectra

Fig. 19 show examples of typical design response spectra(damping ratio = 5%) indicated in many seismic design codes,

guidelines and regulations worldwide. The particular examples

shown in Fig. 19 were obtained following the procedure describedin the ASCE 7-05 standard Minimum Design Loads for Buildings and

Other Structures [24]. They were obtained assuming that spectral

response acceleration parameters S S and S 1 are equal to 1.5 g

and 0.6 g, respectively, and considering soil types B (i.e., firm

rock) and E (i.e., soft soil). The spectral shapes of the spectra

shown in Fig. 19 are typical of design response spectra in the sense

that there is a range of periods at which the pseudo-acceleration

has a constant value (i.e., the constant acceleration region of the

spectrum), followed by a range of periods at which the value of

the pseudo-acceleration is inversely proportional to the value of

the period (i.e., the constant velocity region of the spectrum).

In order to investigate the frequency content associated with

design spectral shapes, modulated filtered white noise excitation

processes were defined in such a way that their corresponding

mean (5% damped) pseudo-acceleration response spectra match

the design spectra described above. When the modulating function

of the excitation processes is again given by Eq. (6) and T D is set

equal to 30 s, the resulting functions S g (ω), obtained numerically

through an iterative procedure, are those shown in Fig. 20. In both

cases, the main frequency of the excitation processes turned out

to be sharply defined, and equal to the frequency associated with

the period defining the limit between the constant-acceleration

and constant-velocity regions of the design spectra. The general

validity of this empirical finding was confirmed by examining

other examples found in the literature [25–27]. For illustration

purposes, sample realizations of the modulated filtered white

noise excitation processes compatible with design responsespectra areshownin Fig.21, andvalues of the correspondingS / X REL


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Fig. 16. Modified Kanai–Tajimi modulated filtered white noise excitation process (ω g = 2.50 rad/s, ξ g = 0.10, ω f = 0.30 rad/s, ξ f = 0.70, S 0 = 200 cm2/s3 andT D = 30 s): [top] sample realization; [bottom] mean pseudo-acceleration response spectrum (damping ratio = 0.05).

Fig. 17. Values of the S / X REL ratio: modified Kanai–Tajimi modulated filtered white noise excitation process (ω g = 2.50 rad/s, ξ g = 0.10, ω f = 0.30 rad/s, ξ f = 0.70,S 0 = 200 cm2/s3 and T D = 30 s).


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Fig. 18. Contour lines of values of the S / X REL ratio: modified Kanai–Tajimi

modulated filtered white noise excitation process (ω g = 2.50 rad/s, ξ g = 0.10,ω f = 0.30 rad/s, ξ f = 0.70, S 0 = 200 cm2/s3 and T D = 30 s).

ratios are shown in Fig. 22. The latter figure indicates again that

the accuracy of the DDC rule can be characterized, for practical

purposes, in terms of T m: in the region roughly defined by T B <

T m −T A, the DDC rule is alwaysconservative regardless of the valueof the period ratio.

6. Conclusions

In this study, the accuracy of the DDC rule in predictingthe separation necessary to prevent seismic pounding betweenlinear structural systems was examined. Adjacent structures weremodeled as 5%-damped SDOF systems, and the range of naturalperiods considered is essentially the same range of possible naturalperiods of actual building structures prone to seismic pounding.

Modulated and filtered modulated Gaussian white noise randomprocesseswere considered as seismic excitations,and the responseof the structural systems was evaluated through Monte Carlosimulations.It wasfoundthat theaccuracyof the DDCrule dependsnot only on the ratio between the natural periods T A and T Bof the adjacent structural systems ‘‘ A’’ and ‘‘B’’, as suggested informer studies, but also on the relationship between T A, T B andthe period T m associated with the main frequency of the excitationωm. Further, it was also found that, qualitatively, the relationshipbetween the accuracy of the DDC rule and the periods T A, T Band T m is, for practical purposes, essentially invariant, i.e., it doesnot depend on whether the excitation has wide- or narrow-bandcharacteristics, or on whether the value of T m is relatively large orsmall. If the natural periods T A and T B are defined in such a waythat T A < T B (thus, 0 < T A/T B < 1), then results shown in thisstudy lead to conclude that, for practical purposes, the accuracy of the DDC rule can be characterized in general terms as follows:

(1) When T B < T m − T A, the DDC rule is always conservative. Thedegree of conservatism increases with increasing values of theT A / T B ratio and, for a given value of the T A/T B ratio, increaseswith decreasing values of T A and T B.

Fig. 19. Design 5%-damped response spectra.


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Fig. 20. Modulated filtered white noise excitation processes compatible with design response spectra (T D = 30 s): power spectral density functions S g (ω).

Fig. 21. Modulated filtered white noise excitation processes compatible with design response spectra (T D = 30 s): sample realizations.

(2) When T B > T m − T A, the accuracy of the DDC rule depends onthe value of the T A/T B ratio:

• when the value of the T A/T B ratio is greater than 0.75,the DDC rule is again always conservative. The degree of

conservatism increases with increasing values of the T A/T Bratio and, for a given value of the T A/T B ratio, increases with

increasing values of T A and T B.

• when the value of the T A/T B ratio is less than 0.75, the DDCrule is always unconservative, at most by 13% in the case

of seismic excitations expected at most sites, and up to 20%in the extreme case of excitations having markedly narrow-band characteristics.

(3) When the seismic excitation is characterized in terms of a de-

sign response spectrum, the period T m associated withthe mainfrequency ωm is the period defining the boundary between


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Fig. 22. Contour lines of values of the S / X REL ratio: modulated filtered white noise excitation process compatible with design response spectra (T D = 30 s).

the constant-acceleration and constant-velocity regions of the

spectrum.The boundary between the above mentioned situations (1) and

(2) was actually found to evolve from T B = T m − T A when theexcitation has wide-band characteristics to T B =

T 2m − T 2 A when

the excitation has narrow-band characteristics. However, keepingin mind that, for practical purposes, design recommendations

should be expressed in somewhat simplified terms and should beon the conservative side, the boundary T B = T m − T A is deemedadequate for all kinds of excitations because it is conservative(i.e., when the excitation does have narrow-band characteristics,

it indicates that the DDC rule is unconservative in cases where

T m − T A < T B <

T 2m − T 2 A , whereas in reality the DDC rule isactually slightly conservative in these cases).

Acknowledgements

The research described in this paper was financially supported

by MCEER (Buffalo, USA) and by the Pontificia Universidad Catolicade Chile (Santiago, Chile). This support is gratefully acknowledged.

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