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EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 16,443-456 (1988)

POUNDING OF BUILDINGS IN SERIES DURING EARTHQUAKESSTAVROS A. ANAGNOSTOPOULOS*

Department of Civi l Engineering, University of Patras, (26110) Patras, Greece

S U MMA R YA simplified model of several adjacent buildings in a block is used to study the pounding of such buildings due to strongearthquakes. Considerable structural damage and even some collapses have sometimes been attributed to this effect. Eachstructure is mod elled as a S.D.O.F. ystem and pou nding is simulated using impact elements. A parametric investigation ofthis problem shows that the end structures experience almost always substantial increases in their response w hile for'interior' structures the o pposite often happens. This may explain why h igh percentages of co rner buildings have collapsedin some earthquakes.

I N T R O D U C T I O NPou ndin g between adjacent buildings or between parts of the same building du e to earthquakes has often beenrecorded as on e of the causes of significant or even severe structural dam age.'-6 This problem is particularlycom mon in many cities located in seismically active regions, where d ue to various socioeco nom ic factors andland usage requirements the codes permit contact between adjacent buildings. In man y parts of the w orld, theso-called 'continuous building system' co nstitutes by far the pred om inant practice for large areas in c ities ortowns, where every building in a block is in full or partial contact, typically at tw o o ppo site sides, with itsneighbouring bu ildings. This can be seen in Figure 1, which shows three actual b uilding blocks fro m the city ofThessaloniki, Greece. Th e numb ers in circles indicate different lots an d the s had ed area s mark the buildinglayout in each lot. D ue to differences in their d ynamic characteristics, adjacent buildings will vibrate o ut ofphase during an earth qua ke and pou ndin g will occur if there is not sufficient separation distance between them.When several buildings are next to each other forming a row in a block, then there is some evidence tha t theend or corne r buildings are mo re heavily penalized by p ~ u n d i n g . ~ . ~his can have a n intuitive explanation: abuilding at the end of a row p oun ds o n on e side only while being free to move tow ards th e oppo site side. Thesame happens with corner buildings in city blocks except that p oun ding in this case takes place along twoorthogonal directions. O n the other hand , if a building is between two o ther buildings, it will poun d on bothsides but at the same time it will not be free to m ove excessively in either direction.Work on the problem of pounding of adjacent buildings is limited. In R eference 7 the pou ndin g of the O liveView Hospital and its stairway tower during the 1971 San Fer na nd o earthq uak e was accounted for in ananalytical investigation of that famous collapse. The contribution of earthquake induced pounding to thecollapse of ano ther b uilding has been studied in Reference 8, while in Reference 9 the pounding of a nuclearreactor building and an adjacent auxiliary structure has been examined. The problem of pounding of twomasses under steady-state conditions is known a s a vibroimpact p roblem a nd has been stud ied analytically formechanical systems. Vibroimpact concepts have also been applied in References 11and 12 to investigate theproblem of two adjacent linear Single Degree of Freedom (S.D.O.F.) systems subjected to h arm onic grou ndmotion.The results of the aforementioned investigations indicate that poun ding, in addition to th e local dama ge itusually causes, increases structural respon se. It is not clear, however, whether th e same conclusion is app licableto all the buildings in a row, particularly those subjected t o two-sided impacts. In fact, there is a notion am ong

* Associate Professor.0098-8847/88/030443-14 07.00

988 by Jo hn Wiley & Sons, Ltd.Received 3 March 1987Revised 10 August 1987

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POUNDING OF BUILDINGS IN SERIES 445S I M P L I F IE D M O D E L O F S EV ER AL A D J A C EN T S T R U C T U R E S

The simplest possible idealization for studying the effects of earthquake induced pounding between severaladjacent buildings is shown in F igure 2 (a). Each struc ture is idealized as a S.D.O.F. system with m ass m i ,viscous damping constant c i , nitial stiffness K , , yield level RYi nd post-yield stiffness p K , [Figure 2(b)]. Fornumerical applications, these parameters can be taken as generalized properties of actual buildingscorresponding to so me assume d deflected shape (e.g. that of the first mode). Pou ndin g is simulated by m eans oflinear viscoelastic impact elements (sp ring-dashpots) that a re introduced between the m asses and act onlywhen the masses are in contact. These elements are characterized b y the linear spring constants s and thedashpot constants ci j . I t is assumed th at all systems are subjected to the sam e input grou nd motion u , t ) , i.e. theeffects of phase difference due to travelling w aves are not considered.

Equations of motiondifferentiation w ith respect to time, then the equation of motion for mass i isIf we denote by ui the displacement of mass i relative to the ground displacement u,and use do ts to indicate

(1)where R i is the non -linear resistance of the system, a function o f the relative displacement u , , and F i . i , F,. ,+are impact forces due to pound ing of m ass i with masses i 1 and i + 1, respectively. These last two forces willnot be always present but will act only when the corresponding masses are in contact. If we callmi ui +c, l i R i- F i- ,i + F,., = mi u ,

(2)where di, i , is the distance between systems i and i 1,then the condition for contact between masses i and i+ 1 is bi .>0. Thus, the exp ressions for the forces Fi- .i and Fi.i I are

6. = u .- . t -d i . i t 1

(3)i l . i = 0 for < 0

F i - l . i = s i . iS i - + c i - l . i S i - , for 6 , - > 0Fi it 1 = 0 for S i < 0Fi i 1 = si i 1 hi + ci.i+ 1 A i for i> 0

By writing equation 1) for every S.D.O.F. system in the row, we obta in the system of differential eq uation sdescribing the response of the configuration to the grou nd acceleration ti . This system is uncou pled if there isno pounding. Coupling between two or more equations is introduced whenever the corresponding massescome into contact. In matrix form, the system of equations can be written as[MI { 0 +CCl { i } + R } [ S ] { U }+ D } = - , {m } 4)

i

Figure 2. Idealization of several adjacent structures

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446 S. A. ANAGNOSTOPOULOSwhere[ M I = diagonal m ass matrix, with elements the masses m ,[C] = [ C , ] + [C,] = damping matrix[ C , ] = diagonal damping matrix, with elements the damping constants ci[C 3 = damping matrix corresponding to the impact elements{ R r = vector of structural resistances Ri unctions of the displacements ui{ U } = vector of unknown displacements u i{ m } = vector containing the masses m,[S] = stiffness matrix corresponding to the impact elements{ D } = vector including stiffness terms d ue to the im pact elements,

Matrices [ C , ] and [S] and the vector { D } are of the form

- c 2 3 c 2 3 + c 3 4 - c3 4-c34 etc.

c12 -c12- c 1 2 c 1 2 + c 2 3 -c2300 0. . . . . . . . . . . . . . . . . . . . . . . . . . . .

C,I =

- s 1 2 s 1 2 + s 2 3 -s23- s2 3 s23 + s 3 4

etc.. . . . . . . . . . . . . . . . . . . . . . . . . .

s 12 - s 1 2 0

0 0CSl =

In these matrices,[equation (2)]. and si.i+ are set equal to zero if the masses mi and m i + are n ot at im pact, i.e. if di d 0

Solution of the equations of motionThe equations of motion, equ ations 4), ere solved numerically using central d ifferences (constan t velocitymetho d) and the linear acceleration method to start the solution. For com putation al efficiency, two differenttime steps were employed: a large time step, equal to 0.01 sec, and a finer time step whose size depend ed o n thestiffness of the imp act springs. The typ ical value of the finer time s tep wa s O W 0 5 sec, which is less than abo ut1/30 of the lowest natural period of any system configuration with the g aps closed. Th e small time step wasapplied whenever one or more masses were at impact. If an impact was detected while the large time ste p was inuse, the integration was repeated with the finer time ste p from th e beginning o f the last time station. The finer

time step remained in effect for the duration of the impact or for the duration of all consecutive overlappingimpacts.PARAMETRIC INVESTIGATIONS

To obta in results of a more ge neral nature , i.e. covering a w ide range of parameters and independent of specificmotion characteristics, a large n umber of analyses was carried out. T he systems used for these analyses wereassigned masses increasing with the natural period T according to the expression m = mo (0.25+0.75 T).Correspon ding stiffnesseswere then determ ined from the natura l periods considered, while viscous dam ping 3per cent of critical was assigned to each system . Yield forces were comp uted on th e basis of the ATC -3 variationof the design base shear ~o eff ici en t , '~ssuming that R , = V 0 / 3 ,where Vo is the design shea r according toATC-3 for q = 1. As for the impact elemen ts, their stiffnesses were set equa l to twenty times the stiffness of thestiffer S.D.O.F.system in the p air to which each element was assigned. This value was estimated on the basis of

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448 S. A. ANAGNOSTOPOULOSThe most imp ortant param eters that characterize ou r problem and whose effects will be exam ined here are:(1) system configuration, i.e. number of systems in a row and their periods(2) gap size(3) streng th (yield level) of the systems4) relative size of system masses5 ) impact element damping6) impact element stiffness.

1. System configurationOne of the difficulties in studying the problem at hand is associated with the great number of possibleconfigurations, as determined by a variable numb er o f buildings in a row and by the different com binations oftheir natural periods. This was dealt with by considering two, three, four and five systems in a row and byintroducing the ratio of the period of the interior system to the period of the adjacent exterior system as aparameter. The five configurations exam ined are shown as insets in the upper right part of Figures 3 to 7, whilethe four com binations of periods used for the study are indicated in the u pper left part. In Figure 3,we have twosystems with periods T,and T,, where T , / T , = A. In Figures 4,5 and 6, the two exterior systems are identical

with period T ,,and so are the interior systems with period Tin.The last configuration, tha t in Figure 7, differsfrom the configuration in Figure 6 only in the middle system, which in Figure 7 is identical to the two exteriorsystems rather than the two interior. In all these cases the system masses were placed in practical contact byassuming a gap size equal to 0 1 cm. Results have been expressed in terms of displacement amplification factorsi.e. as ra tios of peak displacements, denoted by u L , u , , u c x ,u i n , o the corresponding peak displacementsdenoted by uo of the same systems responding independently (i.e. without pound ing). Mean a nd maximumvalues of these factors for the five earthq uake motions h ave been plotted versus the period of the systemexamined. They are both given in the figures, the mean values at left and the m aximum values at right. Thesystem periods T in all the configurations ranged from 0.125 sec to 4.0 sec.In Figure 3 the results are for a two-system configuration, in which we have only one-sided impacts. Thegraphs in the upper part are for the left structure and the grap hs in the lower part are for the right structure.

2

3 1 2 3T sec) T sec )

00 1 2 3 0 1 2 1

T sec) TR sec)Figure 3. Effects of pounding on seismic response: two-system configuration

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POUNDING OF BUILDINGS IN SERIES 449

07 2 3 0 1 2 3T sec) T sec)

Figure 4. Effects of pounding on seismic response: three-system configuration

Ok-?-?--3Tan 9 ~ ) Tm ( 5 ~ )

Figure 5 . Effects of pounding on seismic response: four-system configuration

Differences between responses of the left and right system are due to th e depend ence of the time a nd sequenceof impacts on the p osition of each system in the co nfiguration. Such differences, however, are of no practicalsignificance. We also o bserve that when the system examined isadjacent to a mo re flexible system A or A > l ,the mean amplification is always greater than one, increasing typically for higher values of the ra tios A or A . Inthis case, the mean amp lification fac tors vary between 1.0and 1-5,while their peak values vary from 1.15 to 2.5.

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450 S. A. ANAGNOSTOPOULOS

r

7 -

1

o t0

21 21

Figure 6. Effects of pounding on seismic response: five-system configuration

19 v v P P I

0 O L1 7 30 1 7 3T sec) T.. sec)

(5)1

0

+ *1

f t

0 1 7 31,set) T,. sec)Figure 7. Effects of pounding on seismic response: second five-system configuration

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POUNDING OF BUILDINGS IN SERIES 45 1When the system examined is adjacent to a stiffer system A or I 1) the mean amplification factors can beeither less or greater than one, depending upon the frequency of the system as well as upon the ratio i ori .The mean amplifications in this case vary from 0.80 to 1.25 and their peak values from 1.05 to 1.60.For more than two systems in a row, we can distinguish between exterior and interior systems, the firstsubjected to one-sided impacts and the second to two-sided impacts. Mean and maximum amplificationfactors are presented in Figures 4 to 7, for exterior systems in the upper part and for interior systems in thelower part. In computing mean values for the five motions, only the largest factor from each group ofsystems-xterior or interior-has been used, i.e. u,,is the larger peak displacement of the two exterior systemsand uin s the largest peak displacement of the two or three interior systems. Thus, the graphs depict the worsteffects of pounding in the row rather than effects on individual systems. In Figure 7, a third set of graphs showsamplification factors for the middle system which is identical to the two exterior systems) plotted versus itsperiod. Comparison of the graphs for exterior and interior structures indicates that the former are much moreheavily penalized than the latter. While the exterior structures in all configurations exhibit mean displacementamplifications greater than one practically over the full range of periods and for all the ratios Tin/Te,considered, the opposite is true for the interior structures which exhibit substantially lower amplifications.Moreover, for T e X / T i , 1.5 or 2.0, the effect of pounding is to reduce, on the average, the peak response of theinterior structures over almost the complete range of periods. The greatest increases in response due topounding, up to 3 times, appear to occur when stiff exterior systems collide with flexible interior systems.If the same group of plots i.e. mean or maximum amplifications-interior or exterior systems) from thevarious configurations is compared, an overall similarity, qualitative as well as quantitative, will becomeapparent. This similarity is even greater for the graphs of Figures 4 and 7, which correspond to twoconfigurations such that one is identical to a portion of the other. What this indicates is that the effects ofmultiple poundings on the response of any system in a given configuration are predominantly determined bythe properties of the system itself and in relation to the properties of the adjacent systems. The response andcollisions of systems in the configuration that are not adjacent to the system considered, and thus do notinteract with i t directly, do not influence this systems response appreciably.2. Effects of gap size

A lower limit of the gap size between two adjacent structures, if pounding is to be avoided, is obviously thesum of their absolute maximum displacements due to their independent responses to the earthquake motionconsidered. Because, however, it is highly unlikely that these two maximum displacements will both occur atthe same instant and with opposite signs, a smaller gap size will usually be sufficient to avoid pounding. As thegap size increases, the number of impacts decreases and typically the amplification of structural responsedecreases. This can be confirmed in Figure 8 where the response amplifications, mean and maximum, of theexterior systems in the four-system configuration are shown for two ratios Ti , /T , , (= 0*5,2.0 and for five gapsizes. The solid lines are for the case of practical contact between the systems (d = 0.1 cm) that has already beenpresented in Figure 5. The dotted lines correspond to the largest gap size d o ) onsidered, which for any twoadjacent systems was taken equal to the square root of the sum of the squares SRSS)of their design peakdisplacements. These peak displacements have been estimated as a (0.85 x 3.0)6,, following the ATC-3provisions3 for a response reduction factor q = 3.0, a ratio q / c , = 0.85 and a yield displacement 6 , = R , / K .The remaining three lines are for intermediate gap sizes, constant fractions of do .As expected, there is a ratherconsistent reduction of response amplification due to pounding with increasing gap sizes. Moreover, a gap sizeequal to do appears to be generally sufficient for avoiding impact problems.3. Effects ofstructural strength

To see how the amplification of structural response due to pounding is affected by the system strength RFigure l ) , the four-system configuration with Tin/TeX 0 5 and 2.0 was analysed for two additional structurafstrengths: a) for R, = very large elastic response) and b) for R, = V0/6, where Vo s the design base shearaccording to ATC-3 without any reduction i.e. q = 1). In Figure 9, the resulting amplifications, mean andmaximum, corresponding to three yield levels are presented for the exterior systems the yield level R, = V o 3

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452 S.A. ANAGNOSTOPOULOS-24

2

1

-1 . , yJ,0 1 2 3 0 1 2 3Figure 8. Effects of gap size on the seismic response of four adjacent systems

Figure 9. Effects of structural strength on the seismic response of four adjacent systems

was assumed for the basic designs). It can be seen that, w hile differences in amplification fac tors resulting fromthe different strengths can be significant, n o clear depend ence between yield level and response amplificationdue t o pou nding is obvious. Fo r some periods, higher strengths lead to higher amplifications, yet for others theopposite happens. Thu s, the only conclusion th at may be drawn is that pounding can be as bad for elastic as forinelastic responses.

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POUNDING OF BUILDINGS IN SERIES 4534. Eflects of relative mass size

It is intuitively obvious that when two bodies collide, the consequences of th e collision for o ne of thembecome greater when the mass of the other body increases. The extent to which the mass sizes affect theresponse amplifications due t o po unding was examined for the four-system co nfiguration, by varying the m assof the tw o interior systems. If mbis the mass of the interior systems for the basic design, the cases with minterior= O-2mb,2-Omband5-0m were also analysed, with the m asses of the e xterior systems kept un changed. Resultsare given in Figure 10 for two ratios of Tin/Tex:0 5nd 2.0. It is seen that, for bo th these ratios and practicallyfor all the periods co nsidered, the respo nse amplification of the ex terior systems increases, often substantially,as the interior systems become mo re m assive. In practice, large differences in m asses of adjacent structures th atmay experience pounding will be foun d in industrial facilities or in buildings with external stairway towers.5 . lmpact element damping

As stated earlier, the dam ping con stant of the impa ct element determines the am ou nt of energy dissipatedduring imp act, i.e. the degree to which the impact is plastic. There is obviously a g reat deal o f un certainty as t owhat m ight be a reasonable value for th e coefficient of restitution t o describe the earthqu ake induced co llisionsbetween real buildings. The value used in this study r = 0.65) is nothing mo re than a n educated guess made onthe basis of some data from impact experiments with spheres and ~ 1 a t e s . l ~his value is probably high(conservative assum ption). It turns ou t, however, tha t this uncertainty is not im portan t. Th is can be seen inFigure 11, where response amplifications for the exterior systems of the four-system configuration, withTin/Tex 0 5 and 2.0, are presented for four values of the impact damping ratio

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454 S. A. ANAGNOSTOPOULOS

1 - -0 1 2 3 0 1 2 3

Figure 1 1 . Effects of impact element damping on the seismic response of four adjacent systems

the same appro ach a s before, the four-system con figuration with Ti,/T,., = 0.5 and 2.0 was analysed for twoadditional impact element stiffnesses:K i / 1 0 and Ki/lOO, where K i is the basic stiffness used prev iously. Animpact element stiffnessof K i /100is actually very low but was included to simulate properties of som e materialthat could be placed in the gap between adjacent structures to act as an impact absorber. The results arepresented in Figure 12 and show that a ten-fold decrease of the im pact element stiffness produce s negligibleeffects (the response amplifications due t o po und ing have remained practically the sam e in all cases). O n theother h and, the reduction of K i y a factor of 100 caused a substantial decrease in response amplifications,though not below the value of one. Thus, it may be concluded that the use of some soft material to fill the gapbetween adjacent buildings can dam pen the imp acts significantly. Such practice, however, is not useful as avibration red uction m echanism i.e. for reducing b uilding response below the values reached withou t pou nding.It should be pointed o ut that the foregoing com men ts ab ou t insensitivity of response to imp act elementproperties apply to displacements only. Impact generated accelerations, on the other hand and to a lesserdegree the correspond ing velocities, are q uite sensitive to change s in the impact element prop erties, especiallyto changes in the spring stiffnesses. These accelerations can cause dama ge to the conten ts of the building, buthave little effect on the displacement response of the colliding masses, as Figure 12 indicates.

CONCLUSIONSOn the basis of the results obtained herein and subject to the limitations imposed by the underlyingassump tions the following conclusions can be summarized:

1) Th e effects of earthq uak e induced pound ing on the ov erall (global) response of a structu re in a row ofseveral adjacent structures depend primarily on (a) he prop erties of the structure itself and in relation tothe properties of the t w o other structures th at are next to it on either side, (b) whether th e structu re issubjected to one or two-sided impac ts (i.e. whe ther an exterior structure-at the en d of the row-or aninterior structure), and (c) the gap size.

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POUNDING OF BUILDINGS IN SERIES 455

0 1 21 (Se c )

0 0 1 2

Figure 12. Effects of impact element stiffness on the seismic response of four adjacent systems

(2) Exterior structures (end of the row ) are subjected to one-sided impacts and a s a rule experience responseamplifications tha t can be quite substantial. Interior structures, on the oth er han d, are subjected t o two-sided impacts tha t can produce either increases or redu ctions of response depend ing on the ratio of theirperiods to the pe riods of the adjacent structures. When this ratio is smaller tha n one , pound ing amp lifies,in most cases, the response of interior structures. This amplification, however, is lower than that ofexterior structures. W hen the ratio is greater than one, the response of interior s tructures is typicallyreduced du e to pound ing. The se results could explain why end buildings in city blocks appear to havesuffered more than interior buildings in past earthquakes.

3) Increasing the ga p size decreases the effects of p ounding . A gap size equal to the SRSSof the design peakdisplacements of the adjacent structures could be sufficient to avoid po und ing.4) Pou ndin g causes similar effects on elastic and on inelastic structures. Th e consequences, however, forinelastic structures will normally be more serious.5) Larger differences in the masses of two adjacent structures make the effect of pounding more

pronounced for the structure with the smaller mass.6) The computed displacement amplifications due to pounding are not very sensitive to changes in theparame ters of the impact elements sim ulating the collisions.(7) The use of a soft viscoelastic material filling the ga p between two adjacent structures can reduce theeffects of pound ing significantly. It is not effective, however, as a mo tion reduction mechanism.

The above suggest that althou gh p oun ding m ay sometimes reduce the overall structura l response and thu s beconsidered beneficial n such cases, m ore o ften it will amplify the resp ons e significantly. This is particu larly truefor end (o r corne r) buildings in city blocks. If one also takes into accoun t the local damage tha t is almost alwayscaused as a result of pound ing, it follows that p oun ding m ust be avoided by providing a sufficient seismic ga pbetween adjacent buildings.

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456 S. A. A N A G N O S T O P O U L O SR E F E R E N C E S

I . G . V. Berg and H. J. Degenkolb Engineeringlessons rom the M ana gua earthquake, American Iron an d Steel Institute Report, 1973.2. V. V. Bertero and R. G. Collins, Investigation of the failures of the Olive View stairtowers during the San Fern ando e arthquake andtheir imp lications on seismic design, Report No. EERC 73-26, Ea rthqu ake Engineering Research Center, University of California,Berkeley, CA, 1973,3. -The Roman ia, 4 March 1977, earth qua ke and its effects on structures, Short Monog raph prepared by ICC PDC , INCE RC andIPCT for C 0 . P . I . S . E . E .congress, earthquake protection of construction in seismic areas, Bucharest, 1978.4. ~ Thessaloniki,Greece, earthqua ke-June 20, 1978. EER I Reconn aissance Rep ort, 1978.5. ~ The Central Greece earthquakes of February-March, 1981, EE RI, Reconnaissance and E ngineering Report, 1982.6. -Impressions o f the Guerrero-Michoacan , Mexico earthquake of 19 September 1985. EE Rl Preliminary Rec onnaissance Report,1985.7. S. A. Mahin, V. V. Bertero, A. K. Chopra a nd R. G. Collins, Response of the Olive View H ospital main building durin g the SanFernand o earthquake, Report No. EERC 76-22, Earthq uak e Engineering Research Ce nter, U niversity of Ca lifornia, Berkeley, CA,1976.8. A. Wada, Y. Shinozaki and N . Nakam ura, Collapse ofbuilding with expansion joints throu gh collision caused by ear thqu ake motion,Proc. 8th world conJ earthquake eng. San Francisco. IV, 855-862 (1984).9. J. P. Wolf and P. E. Skrikerud, Mutual pounding of adjacent structures during earthquakes, Nucl. eng. des. 57, 253-275 (1980).10. A. E. Kobrinski, Dynamics ofMechanism s with Elastic Connections and Impact System s, English Translation: R. ennox-Napier, Iliffe,11. R. K. Miller, Steady vibroimpa ct at a seismic joint betwee n adjacent struc tures, Proc. 7thworldconf. earthquake eng. Istanbul, Turkey12. R. K. Miller and B. Fate mi, An efficient techniqu e for the approxim ate analysis of vibroimpact,J.mech. des. A S M E preprint, Design13. __Tentative provisions for the development of seismic regulations for buildings, Applied Tec hnology Co uncil, Publication ATC 3-14. W . Goldsmith, Impact: The Theory and Physical Behaoiour of Colliding Solids, Edward Arnold, London, 1960.15. A . Arias, A measure of eart hqua ke intensity, Seminar seismic des. nucl. power plants Department of Civil Engineering, M.I.T.,

London, 1969.6, 57-64 (1980).Engineering Technical Conference, Hartford, Connecticut, Paper No. 81-DET-16 (198 I).06, USA, 1978.

Cambridge, Mass., 1969.