Pounding force response spectrum under earthquake excitation

Engineering Structures 28 (2006) 1149–1161www.elsevier.com/locate/engstruct

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Pounding force response spectrum under earthquake excitation

Robert Jankowski∗

Faculty of Civil and Environmental Engineering, Gdansk University of Technology, ul. Narutowicza 11/12, 80-952 Gdansk, Poland

Received 25 April 2005; received in revised form 22 December 2005; accepted 23 December 2005Available online 23 March 2006

Abstract

Earthquake-induced pounding between inadequately separated structures may cause considerable damage or even lead to a strcollapse. The assessment of the damage magnitude as well as the design of some pounding mitigation method requires the knowledge of thmaximum impact force value expected during the time of earthquake. The aim of the present paper is to propose the idea of impact forcspectrum for two adjacent structures, which shows the plot of the peak value of pounding force as a function of the natural structuralperiods. In the analysis, both structures have been modelled by single-degree-of-freedomsystems and pounding has been simulated by the nlinear viscoelastic model. The analysis has been conducted for elastic and inelastic (elastoplastic) structures under different ground motions.heexamples of response spectra show that the selection of the structural parameters, such as the gap size between structures, their natperiods, damping, mass andductility as well as the time lag of input ground motion records, might have a substantial influence on thepounding force value. The results of the study indicate that impact force response spectra might serve as a very useful tool for the desigof closely-spaced structures on seismic areas.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Structural pounding; Earthquakes; Response spectrum; Peak impact force

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1. Introduction

The problem of pounding between insufficiently separastructures, or structural members, during earthquakesattracted researchers’ attention for several years now.interest results from the fact that a growing amount of evidecan be found in reports after major earthquakes indicatingstructural pounding may cause considerable damage orlead to a structure’s collapse (see, for example, [1,2]). It hasbeen recognised that the main reason leading to collisbetween buildings is usually their out-of-phase vibratiocaused by the difference in their dynamic characteristics3,4]. On the other hand, in the case of longer bridge structupounding between adjacent superstructure segments isinduced due to the seismic wave propagation effect, whichresults in different seismic inputs acting on supports alongstructure [5,6]. The results of various numerical studiesusing different structural models and applying different modof collisions confirm that pounding, due to imposing additiona

∗ Tel.: +48 58 3472497; fax: +48 58 3471670.E-mail address: [email protected].

0141-0296/$ - see front matterc© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2005.12.005

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impact forces, may result in damage at contact points as wemay considerably increase the structural response.

The most natural way to prevent structural pounding isprovide sufficiently large spacing between adjacent structuror structural members. Thus, the minimum seismic gaspecified in most recent earthquake-resistant design codenewly constructed buildings (see, for example, [7]). However,due to the land shortage and high land prices in many clocated on seismic areas, this solution is usually very diffito be accepted by the land owners. It might be an especproblematic issue, for example, when one of the ownwants to build his house close to the existing one, whhas been constructed up to the property line. Moreover, tare many examples of old buildings with different dynamcharacteristics, which have been constructed in contacteach other, as this was not prohibited by the old earthquresistant design codes. Similar situations concern also bridThe earthquake design codes specify that the gap size bebridge segments should be large enough to avoid collis(see, for example, [8]). However, enlarging the space betweadjacent girders, or between a girder and an abutment, mighbe an expensive and problematic solution because of h

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1150 R. Jankowski / Engineering Structures 28 (2006) 1149–1161

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Because of the above reasons, intensive investigation hasbeen carried out on pounding mitigation techniques in orto enhance the seismic performance of structures withsufficient in-between space. Westermo [9] suggested, forexample, linking buildings by beams, which can transmit thforces between the structures and thus eliminating collisioThe connections between adjacent structures can alsosome energy dissipating properties and impacts can be parabsorbed [10]. The idea of filling the separation gap by aenergy absorbing material or providing strong collision waprotecting part of the structure has been studied [11]. Jankowskiet al. [12] considered the use of bumpers, crushable deviand shock transmission units to suppress the blows of impin bridges. The effectiveness of variable dampers as welrestrainers has also been analysed [13,14].

For the design purposes of pounding-prone structures,magnitude of the impact force,which can be expected duringthe time of earthquake, needs to be known in orderassess the potential damage due to collisions. Also the deof pounding reduction methods, such as collision wallsbumpers, for example, requires the knowledge of the peacollision forces. For these purposes, the idea of an impforce response spectrum for earthquake-induced poundbetween two adjacent structures is introduced in this paper. Thespectrum shows the maximum pounding force value, whican be expected during the earthquake, and thus might sas a very useful tool for the design purposes of closely-spastructures.

2. Pounding force response spectrum for elastic structures

2.1. Pounding force response spectrum concept

The displacement, velocity (pseudo-velocity) and acceler-ation (pseudo-acceleration) response spectra are well knowpractical means of characterising ground motions and theirfects on structures. The response spectrum for a particular qtity is defined as a plot of the peak value of a response quanas a function of the natural vibration periodT of the system, ora related parameter (circular or cyclic frequency) [15]. The plotshows the peak elastic response of the structure, modelledsingle-degree-of-freedom system, for a fixed value of structudamping,ξ . Among response spectra for different quantities,the displacement response spectrum is the most popular onis defined as a plot of the peak deformation responses,xmax,obtained for different values ofT under fixedξ [15]

xmax(T, ξ) = maxt

|x(t, T, ξ)|. (1)

In order to predict the maximum relative displacemebetween two adjacent structures with different natuperiods, the relative displacement response spectrum hasconsidered [16]. Ruangrassamee and Kawashima [17] proposedalso the concept of relative displacement response spectruwith pounding effect, which might be helpful in studying thinfluence of pounding on structural behaviour.

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Fig. 1. Model of interacting structures.

In this paper, the idea of impact force spectrumearthquake-induced pounding between two neighboustructures is introduced. This response spectrum can be deas a plot of the peak value of pounding force as a functof natural vibration periodsT1, T2 (or frequenciesf1, f2)of colliding structures modelled by single-degree-of-freedsystems (seeFig. 1). On the contrary to the response spectrufor an independently vibrating single structure, the impact fospectrum for pounding between two structures will dependonly on damping ratiosξ1, ξ2 but also on massesm1, m2 and thein-between gap size,d. Moreover, inthe case of long structuressuch as bridges, buildings with spatially extended foundatior life-line systems, the incorporation of the seismic wapropagation effect might be important [18]. In such a casethe influence of a time lag,τ , for the input ground motionrecords acting on two adjacent structures (or their parts) shoulalso be considered [19]. Taking all the above into account, thpounding force response spectrum can be defined as a plthe peak impact force,Fmax, obtained for different values ofT1andT2 under fixed values ofξ1, ξ2, d, m1, m2, τ

Fmax(T1, T2, ξ1, ξ2, d, m1, m2, τ )

= maxt

|F (t, T1, T2, ξ1, ξ2, d, m1, m2, τ )| . (2)

2.2. Numerical model of colliding structures and simulation ofpounding force during collision

The dynamic equation of motion for elastic poundininvolved response of two adjacent structures modelledsingle-degree-of-freedom systems (seeFig. 1) can be written inthe form[

m1 00 m2

] [x1(t)x2(t)

]+

[C1 00 C2

] [x1(t)x2(t)

]

+[

K1 00 K2

] [x1(t)x2(t)

]+

[F(t)

−F(t)

]

= −[

m1 00 m2

] [xg1(t)xg2(t)

](3)

where xi (t), xi (t), xi(t), Ci , Ki , xgi(t) are the horizontaldisplacement, velocity, acceleration, damping coefficientstiffness coefficient and acceleration input ground motionthe structure with massmi (i = 1, 2), respectively, andF(t)denotes the pounding force.

The appropriate numerical model of pounding force,F(t),during collision between structures is essential for the precdetermination of the impact force response spectrum. Pounitself is a complex phenomenon involving plastic deformatioat contact points, local cracking or crushing, fracturing due

R. Jankowski / Engineering Structures 28 (2006) 1149–1161 1151

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to impact, friction, etc. Forces generated by collisionsapplied and removed during a short interval of time initiatstress waves, which travel away from the region of contactThe process of energy transfer during impact is higcomplicated which makes the mathematical analysis of thitype of problem difficult. Several models have been usedsimulate pounding force during collision between structuresThe simplest model applies a linear elastic spring and dnot take into consideration the energy dissipation duimpact [4,6,20]. The more precise linear viscoelastic mod(see, for example, [3,5,21]) accounts for the energy loss duto pounding but the impact force–deformation relation is stilsimplified. In order to model this relation more realisticallynon-linear elastic model, following the Hertz law of contahas been adopted by a number of researchers [22–24]. Thismodel, however, is fully elastic and does not account forenergy dissipation during contact due to plastic deformatilocal cracking, friction, etc. Therefore, for the purposof determination of the pounding force response spectrpresented in this paper, the non-linear viscoelastic modebeen chosen [25]. In the model, a non-linear spring followinthe Hertz law of contact is applied together with an additionon-linear damper, which is activated during the approperiod of collision in order to simulate the process of eneloss taking place mainly during that period. It has been verifiethat, comparing to other models, the non-linear viscoelamodel is the most precise one in simulating the impact fotime history during impact as well as the pounding-involvstructural response (see [25]). In the model mentioned, thpounding force,F(t), between colliding structures with massm1, m2 is expressed as [25]

F(t) = 0 for δ(t) ≤ 0 (no contact)

F(t) = βδ32 (t) + c(t)δ(t) for δ(t) > 0 andδ(t) > 0

(contact-approach period)

F(t) = βδ32 (t) for δ(t) > 0 andδ(t) ≤ 0

(contact-restitution period)

(4a)

δ(t) = x1(t) − x2(t) − d (4b)

whered is the initial separation gap,β stands for the impacstiffness parameter depending on material propertiesgeometry of colliding bodies andc(t) is the impact element’sdamping, which at any instant of time can be obtained fromformula [25]

c(t) = 2ξ

√β√

δ(t)m1m2

m1 + m2(5)

where ξ denotes an impact damping ratio correlated with acoefficient of restitution,e, which accounts for the energdissipation during collision. The approximate relation betwξ ande in thenon-linear viscoelastic model is expressed byformula [26]

ξ = 9√

5

2

1 − e2

e (e(9π − 16) + 16). (6)

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d

e

If required, the value of impact damping ratio can be furtadjusted during the numerical simulations in order to satexactly the relation between the post-impact and the primpact velocities (see [25] for details).

The numerical study has been conducted in orderdetermine the elastic pounding force response spectradifferent parametric values, such as mass, damping ratio,size between structures and time lag of ground motion recoWhen the effect of one parameter has been investigated,values ofothers have been kept unchanged. In the analythe following basic values of the structural model’s parametehave been applied:m1 = m2 = 106 kg, ξ1 = ξ2 = 5%, d =0.05 m,τ = 0 s. Based on results of the experiments, the valuof the pounding force model’s parameters have been seleβ = 2.75× 109 N/m3/2 andξ = 0.35 (e = 0.65) [25,27]. Inthe analysis, the structural natural vibration periodsT1, T2 havebeen ranged from 0.05 s to 3 s with an increment of 0.05 s. Inthe case of constant mass value,the change of the natural periodhas been obtained by alteringthe structural stiffnessK1 or K2.For thebasic mass value of 106 kg, the stiffness has varied from4.39 × 106 N/m to 1.58 × 1010 N/m. In order to solve theequation of motion(3) numerically, atime-stepping integrationprocedure with constant time step of 0.005 s has been uThe analysis has been conducted for different ground morecords. In the following sections of this paper, the effectsof damping ratio, gap size between structures, mass andlag are investigated based on the results conducted forNS component of the El Centro earthquake (18 May 194The examples of pounding force response spectra for oearthquakes are also presented.

2.3. Effect of damping ratio

A standard displacement, velocity (pseudo-velocity) oracceleration (pseudo-acceleration) response spectrum for aindependently vibrating single structure is a simple 2D grshowing the peak structural response versus the natvibration periodT (or frequencyf ). Several plots for differenvalues of damping ratio are often presented on a sinfigure allowing for easy comparison between them. Howesince the pounding force response spectrum is a plot agtwo naturalvibration periodsT1, T2 (or frequenciesf1, f2)simultaneously, the spectrum needs to be presented in agraph (see also [17]) as obtained for a fixed pair of structurdamping ratios ξ1, ξ2. The examples of such spectra for thEl Centro earthquake are presented inFigs. 2 and 3. Fig. 2shows the peak pounding forces for different values of identidamping ratios(ξ1 = ξ2), whereasFig. 3 presents the spectrfor various damping ratios of one of the structures whilkeeping the value of the damping ratio of the other strucunchanged. It can be seen from both figures that the influof the structural damping on the impact force is substantiaThis is obviously due to the fact that increasing dampleads to the reduction of structural vibrations and therereduction in prior-impact velocities which, at the end, resultslower impact force values. It can be seen comparingFig. 2(a)with Fig. 2(b), for example, that the increase in damping ra


Fig. 2. Pounding force spectra for El Centro earthquakefor different values of identical damping ratios,ξ1 = ξ2, of both structures.

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for both structures from 0% to 2% leads to the decrein the peak pounding force by 22% on average. MoreoFig. 2 indicates that the increase in structural damping of bothstructures simultaneously leads to the extension of the regiothe spectrum, where the impactforce is equal to zero (a regiowhen pounding does not take place). This region concthe cases when the natural vibration periods are very smafor both structures (small deformations preventing impaand when the periods are equal or nearly equal (in-pvibrations). On the other hand,Fig. 3 shows that, when thdamping ratios of two structures are different, pounding fofor the cases of identical natural vibration periodsis not alwaysequal to zero. This is due to the fact that the differencestructural damping makes the structures vibrate with diffedeformation histories that may result in collisions.

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2.4. Effect of gap size between structures

The in-between gap size is a very important configuraparameter, which describes the spatial arrangementneighbouring structures. The easiest way to prevent strucpounding is to provide large spacing between adjacstructures. On the other hand, however, high land prices ogenerate pressure on engineers to construct as closepossible. The examples of pounding force response spectrdifferent values of gap size between structures are presentFig. 4. It can be clearly seen from the figure that the increin the separation gap allows usto prevent impacts for a widerange of the natural vibration periods of the structures. Incase of the El Centro earthquake, the gap size of 0.3 malready sufficiently large to avoid collisions for all consider


he
Fig. 3. Pounding force spectra for El Centro earthquake for different values of damping ratio of the right structure,ξ2 and constant value of damping ratio of tleft structure,ξ1 = 5%.
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combinations of the structural periods. It is worth notihowever, that for the cases when pounding cannot be prevethe peak impact force values arenearly at the same level apafrom the gap size value.

2.5. Effect of mass

The mass of colliding structures is an especially imporstructural parameter, which has a direct influence on structresponse and on pounding force during impact. The numeanalysis has been conducted for different values of idenmassesm1 = m2 as well as for various masses of one ofstructures while keeping the mass value of the other strucat constant level (different mass ratios). The examples oresults of the study in the form of the pounding force respo

,ed,

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eee

spectra for the El Centro earthquake are shown inFigs. 5and6. Both figures confirm that the increase in the mass valeads to the considerable increase in the peak impact forcFig. 5indicates that, in the case of identical mass values of botstructures, the shapes of the pounding force spectra aresimilar with nearly linear, relatively rapid increase in the peaforce values in the range of 200×103–5×106 kg. On the otherhand,Fig. 6 shows that, when the mass value of only one othe structures is being changed, the increase in the peak impforce is naturally not so rapid although also nearly linear inconsidered mass range. Moreover, the shapes of the specdifferent mass ratios (Fig. 6) arenot so similar one to anotheas in the case of the spectra obtained for identical mass vaof both structures (Fig. 5).


Fig. 4. Pounding force spectra for El Centro earthquake for different values of gap size between structures,d.

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2.6. Effect of time lag

Due to the seismic wave propagation effect, the input groumotion records acting on two adjacent long structures or theparts, such as superstructure segments of an elevated bmay be delayed by a time lag,τ [19]. This parameter dependon a mean apparent seismic wave velocity vector,v, andon adistance vector between the structural supports,r, determinedwith respect to a fixed coordinate system [18]. The value of thetime lag,τ , can be calculated as

τ = v ·rv2

(7)

wherev = |v| is a magnitude of velocity.The investigative study has been conducted for differ

values of time lag of the input ground motion records. T

d

ge,

t

examples of the results in the form of pounding force respospectra for the El Centro earthquake are presented inFig. 7.The results of the study show that the time lag of at le0.2 s results in the disappearance of the region in the spectrwhere the impact force is equal to zero, which is normalobserved for equal or nearly equal structural vibration perioThis is due to the fact that the seismic wave propagation einduces the out-of-phase vibrations even for structuresidentical dynamic properties. Moreover,Fig. 7(c)–(f) indicatethat the shapes of the pounding force response spectra doechange much for the range of time lag 0.2–0.5 s. It can beseen from the figure that the region with zero pounding foobserved in the case of very small natural vibration periodsbe found nearly unchanged in all spectra apart from the timevalue. This results from the fact that the initial separation g


Fig. 5. Pounding force spectra for El Centro earthquake for different values of identical masses,m1 = m2, of both structures.

er

beth

dsatunanhoiose

ak

r theringtbe

ural

rmsticablyring

(0.05 m) is sufficiently large to accommodate different but vsmall vibrations of adjacent structures.

2.7. Pounding force spectra for different earthquakes

The analysis of pounding force response spectrum hasconducted for different ground motions. The examples ofspectra for the earthquake records listed inTable 1are presentein Fig. 8. The results of the studyconfirm that the shapeof the spectra depend much on the peak ground acceler(PGA) and on the frequency contents of the input gromotion records. In the case of the Kobe and the San Fernearthquakes, for example, the pounding force spectrum sits peaks whenone of the structures has the natural vibratperiod of about 1 s. On the other hand, it is interesting tothat the spectra for the Kushiro and the Northridge earthqu

y

ene

ionddowsne

es

indicate that the maximum impact force can be expected fovibration period of about 0.5 s. It can be also seen compaFig. 8(b) with Fig. 8(d) that, in spite of similar peak impacforce levels, the region with zero pounding force canobserved for much different ranges of small natural structvibration periods.

3. Pounding force response spectrum for inelastic struc-tures

3.1. Numerical model of colliding structures

During intensive ground motions, structures often defointo their inelastic range. It can be expected that the inelastructural response of interacting structures may considerinfluence the maximum pounding force value observed du


re,
Fig. 6. Pounding force spectra for El Centro earthquakefor different values of mass of the right structure,m2 and constant value of mass of the left structum1 = 1000 000 kg.
Table 1Ground motion records used in the analysis

Earthquake Date Magnitude Station Component PGA (cm/s2)

Kobe (Hyogo-ken Nanbu) 17.01.1995 7.2 JMA NS 817.82Kocaeli (Izmit) 17.08.1999 7.4 Sakarya EW 369.28Kushiro 15.01.1993 7.8 JMA EW 919.13LomaPrieta 17.10.1989 6.9 Corralitos NS 631.51Northridge 17.01.1994 6.7 Tarzana, Cedar Hill EW 1745.54San Fernando 09.02.1971 6.6 Pacoima Dam N16◦W 1202.62

honntio

ingsticrce,

the time of earthquake. Because of the above, the studybeen extended to determination of the pounding force respspectra also for inelastic systems. For the inelastic respospectrum analysis, the elastic-perfectly plastic approxima

assesen

of a force–deformation relationship is ensured. FollowEq.(2), the pounding force response spectrum for elastoplastructures can be defined as a plot of the peak impact foFmax, obtained for different values ofT1 and T2 under fixed


Fig. 7. Pounding force spectra for El Centroearthquake for different values of time lag,τ .

ing

ndtic

e

,f

incegth

tofor

el’s

values ofξ1, ξ2, d, m1, m2, τ , µ1, µ2

Fmax(T1, T2, ξ1, ξ2, d, m1, m2, τ, µ1, µ2)

= maxt

|F (t, T1, T2, ξ1, ξ2, d, m1, m2, τ, µ1, µ2)| (8)

whereTi , ξi (i = 1, 2) are the natural period and the dampratio of the inelastic system with massmi vibrating within itslinearly elastic range, andµi is a ductility factor obtained for afreely vibrating structure.

The dynamic equation of motion(3) for elastic pounding-involved response of two adjacent structures has been exteto the more general form to account for the elastoplasbehaviour. Such equation can be written in the form[

m1 00 m2

] [x1(t)x2(t)

]+

[C1 00 C2

] [x1(t)x2(t)

]+

[FS1(t)FS2(t)

]

ed

+[

F(t)−F(t)

]= −

[m1 00 m2

] [xg1(t)xg2(t)

](9)

where FSi(t) is the inelastic restoring shear force for thstructure with massmi (i = 1, 2) which is equal to:FSi(t) = Ki xi (t) for the elastic range till the yield strengthFyi , and FSi (t) = ±Fyi for the plastic range. The value othe yield strength,Fyi , depends on the ductility factor,µi , andhas to be determined through the interpolative procedure sthe response of a system with arbitrarily selected yield strenseldom corresponds to the desired ductility value [15].

The numerical study has been conducted in orderdetermine the inelastic pounding force response spectradifferent structural properties. Similarly as for the elasticanalysis, the following basic values of the numerical mod


Fig. 8. Pounding force spectrafor different earthquakes.

enns(1

onftedin

e

resding

dingwith

ichres.ndsres

ureslts

parameters have been applied:m1 = m2 = 106 kg, ξ1 = ξ2 =5%, d = 0.05 m, τ = 0 s, β = 2.75 × 109 N/m3/2, ξ =0.35 (e = 0.65). The analysis has been conducted for differground motion records. The examples of inelastic respospectra for the NS component of the El Centro earthquakeMay 1940) are presented in the following sections.

3.2. Pounding force spectra for one elastic and oneelastoplastic structure

First, the study has been conducted for one elastic andinelastic system. It has been assumed in the analysis that the lestructure is the elastic one(µ1 = 1), whereas the right structurshows the elastoplastic behaviour. The examples of pounforce response spectra for different ductility factors of the rightstructure,µ2, are presented inFig. 9. It can be seen from th

te8

e

g

figure that the inelastic behaviour of only one of the structuleads to the substantial change of the shapes of pounforce spectra even for small ductility factors.Fig. 9 indicates,however, that the peak impact force is not significantly reduceas the value of the ductility factor increases. It is worth notthat the shapes of the spectra are much more asymmetricrelation to the line of equal natural vibration periods, whis in contrast to the spectra obtained for the elastic structuThis fact indicates that the peak impact force value depesubstantially on the consideration of which of the structu(left or right) is elastic and which is the inelastic one.

3.3. Pounding force spectra for two elastoplastic structures

The analysis has been further extended for both structshowing the inelasticbehaviour. The examples of the resu


Fig. 9. Pounding force spectra for one elastic (left) and one elastoplastic (right) structure under El Centro earthquake for different values of ductility factor, µ2.

ctr

inadca

60%(se

ionThe

atltin

eaytheto

gsiteblee isiourthed fore

of the study in the form of pounding force response spefor different values of identical ductility factors(µ1 = µ2)

are shown inFig. 10. The results indicate that the increasethe ductility factors for both structures simultaneously leto the substantial reduction in the peak impact forces. Itbe observed fromFig. 10(c), for example, that forµ1 =µ2 = 2, the decrease in the pounding force averagescomparing to the case of fully elastic structural behaviourFig. 10(a)). Moreover, the results show that with the increasein the values of ductility factors for both structures, the regin the spectra with zero pounding force is much enlarged.above observations are due to yielding, which takes placelower strength as the value of ductility factor increases resu

a

sn

e

ag

in the increased displacements ofthe structures during the timof earthquake. Therefore, already after the first hit, which minduce only moderate impact force, they often rebound inopposite directions so much that they might not be ablecome into contact again. Large displacements due to yieldinof one of the structures may also force it to shift in the oppodirection without impact, while the second structure is not ato approach it. In such a case, the zero pounding forcobtained. It should be underlined, however, that this behavmay not be similar when we switch the natural periods ofstructures, and that is why the response spectra obtainehigher ductility factors are asymmetric with relation to the linof equal natural vibration periods (seeFig. 10(e)–(f)).


Fig. 10. Pounding force spectra for two elastoplastic structures under El Centro earthquake for different values of identical ductility factors,µ1 = µ2.

rutwhete

thlnndia

oothing

nsturalratectthendforn be

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4. Conclusions

In the paper, the idea of impact force response spectfor earthquake-induced structural pounding betweeninsufficiently separated structures has been considered. Tanalysis has been conducted for elastic and inelastic sysunder different ground motion records.

The presented examples of response spectra showthe appropriate selection of theconfiguration and structuraparameters, such as the in-between gap size, natural vibratioperiod, damping coefficient, mass, time lag of input groumotion records or ductility factor, might have a substantinfluence on the peak pounding force value. The resultsthe study show that tuning the dynamic parameters of bstructures, so that they may vibrate in-phase, or provid

mo

ms

at

lf

sufficient separation between themwill minimize the negativeeffects of pounding or even prevent contact at all. Collisioare also avoided when both structures have very small naperiods (high frequency systems), which makes them vibwith very small deformations. The results indicate that impaforce response spectra provide valuable information onmaximum pounding force value expected during groumotions and thus they might serve as a very useful toolthe design purposes of closely-spaced structures. They casuccessfully used to assess the magnitude of pounding-inddamage, which can be expected during the design earthqThe spectra will be also very helpful during the design of sopounding reduction method, which requires the knowledgethe peak impact force.


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e

Since the pounding force response spectrum depon different combinations of dynamic parameters of bstructures, only the examples of the spectra have bpresented in this paper. In the case, however, whenof the structures has already been constructed (its dynproperties are known), the response spectrum analysis camuch simplified. Then, the spectra may be obtained onlyparameters of a new structure and simple 2D plots can beto show the results.

In order to determine the impact force response spectthe single-degree-of-freedom structural models of interacstructures have been used (such models are always used icase of response spectrum analysis). Therefore, the presresults concern mainly the structures, which can be successsimulated by such simplified models. In the case of mudegree-of-freedom systems, such as multi-storey buildingsexample, the more precise analysis would be required in oto verify the obtained results. The accuracy of the predictionapart from the model used, should be further validated throthe experimental study.

Acknowledgement

The study was supported by the European Commuunder the FP5 Programme, key-action “City of Tomorrow aCultural Heritage” (Contract No. EVK4-CT-2002-80005). Thsupport is greatly acknowledged.

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Pounding force response spectrum under earthquake excitation