Research Article Numerical Study on Pounding between Two ...downloads.hindawi.com/journals/sv/2016/1504783.pdf · Numerical Study on Pounding between Two Adjacent Buildings under

Research ArticleNumerical Study on Pounding between Two Adjacent Buildingsunder Earthquake Excitation

H Naderpour1 R C Barros2 S M Khatami1 and R Jankowski3

1Faculty of Civil Engineering Semnan University Semnan 3513119111 Iran2Faculty of Engineering University of Porto (FEUP) 4200-465 Porto Portugal3Faculty of Civil and Environmental Engineering Gdansk University of Technology 80-233 Gdansk Poland

Correspondence should be addressed to R Jankowski jankowrpggdapl

Received 30 July 2015 Accepted 15 October 2015

Academic Editor Georges Kouroussis

Copyright copy 2016 H Naderpour et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Seismic excitation which results in large horizontal relative displacements may cause collisions between two adjacent structuresdue to insufficient separation distance between them Such collisions known as earthquake-induced structural pounding mayinduce severe damage In this paper the case of pounding between two adjacent buildings is studied by the application of singledegree-of-freedom structural models Impact is numerically simulated with the use of a nonlinear viscoelastic model Specialattention is focused on calculating values of impact forces during collisions which have significant influence of pounding-involvedresponse under ground motions The results of the study indicate that the impact force time history is much dependent on theearthquake excitation analyzed Moreover the peak impact forces during collision depend substantially on such parameters as gapsize coefficient of restitution impact velocity and stiffness of impact spring element The nonlinear viscoelastic model of impactforce with the considered relation between the damping coefficient and the coefficient of restitution has also been found to beeffective in simulating earthquake-induced structural pounding

1 Introduction

During ground motions buildings often collide with eachother due to different dynamic characteristics insufficientgap between them and out-of-phase vibrations [1] Thephenomenon related to such collisions is often calledthe earthquake-induced structural pounding Pounding isexpressed by an instance of rapid strong pulsation whichmay cause severe damage [2] Consequently the probabil-ity of structural interactions during earthquakes must beassessed A number of researchers have studied the problemof pounding for different structural configurations undervarious ground motions

(i) Anagnostopoulos [3] was among the first scientistswho investigated the earthquake-induced structuralpounding by analyzing interactions between build-ings in a row He also described in detail the threatsrelated to such collisions during ground motions [4]

(ii) Maison and Kasai [5] presented a formulation andsimulated the multi degree-of-freedom equations ofmotion for floor-to-floor pounding between two 15-storey and 8-storey buildings The influence of build-ing separation relative mass and contact locationproperties was investigated

(iii) Jankowski [6 7] andMahmoud et al [8 9] carried outa number of studies concerning structural poundingrelated to both experimental and numerical aspects

(iv) Komodromos et al [10ndash12] studied poundingbetween seismically isolated buildings

(v) Barros and Khatami [13] examined a new model ofimpact to coordinate of results among numerical andexperimental studies They also suggested an approx-imate trend to select coefficient of restitution whichbecomes equal to impact velocity [14] Subsequentlya new equation of motion was suggested to simulateimpact and figure damping terms out of collision [15]

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 1504783 9 pageshttpdxdoiorg10115520161504783

2 Shock and Vibration

Furthermore some more recent numerical analyses havebeen carried out to study the influence of different parametersin pounding of buildings [16ndash20]

Nevertheless there is still a need to investigate differentmodels of structural pounding so as to verify their accu-racy in the case of different configurations under differentearthquake excitationsThis concerns especially the values ofimpact forces during collisions which are often not studied inthe analyses (or the analyses are simplified) since the inves-tigations are rather focused on pounding-involved responseunder ground motions

2 Contact Element Methods

Thecontact element is a special element (usually consisting ofa spring and damper) to model impact between two collidingstructures which is widely used to simulate impact forceImpact is parametrically modelled in this way that whenrelative displacement exceeds the separation distance thecontact element is activated The general formula for theimpact force during collision can be expressed as follows (see[21])

119865imp (119905) = 119896119904 sdot 120575119899

(119905) + 119888imp (119905) sdot (119905) (1)

where 119896119904is stiffness of spring 119888imp(119905) denotes damping of

dashpot and 120575(119905) and (119905) describe lateral displacement andvelocity respectively The power of 119899 has been recommendedto be 1 or 15 depending on the model considered Thedamping coefficient 119888imp(119905) is usually related to the coefficientof restitution CR which is defined as the ratio between thepostimpact velocity rebound and the prior-impact velocityimp [21]

0 lt CR =rebound

implt 1 (2)

Anagnostopoulos The linear viscoelastic model of impactforce (with 119899 = 1) was considered by Anagnostopoulos [3]The following formulae were suggested [3 22]

119888imp (119905) = 2 sdot 120577 sdot radic119896119904 sdot119898119894119898119895

119898119894+ 119898119895

120577 =ln (CR)

radic1205872 + (ln (CR))2

(3)

where 119898119894 119898119895are the masses of colliding structures The

impact force versus time and lateral displacement for thelinear viscoelastic model is shown in Figure 1

Jankowski Jankowski [6] considered the nonlinear viscoelas-tic model when 119899 = 15 He proposed activating the dampingterm only during the approach period in which most of

the energy is lost Therefore the following formulae wereconsidered [6]

119865imp (119905) = 119896119904 sdot 120575119899

(119905) + 119888imp (119905) sdot (119905) 997888rarr

(119905) gt 0 (approach period)

119865imp (119905) = 119896119904 sdot 120575119899

(119905) 997888rarr

(119905) le 0 (restitution period)

(4)

119888imp (119905) = 2 sdot 120577 sdot radic119896119904radic120575 (119905) sdot119898119894119898119895

119898119894+ 119898119895

(5)

It was suggested that the appropriate value of impact dampingratio 120577 for a specified value of CR can be obtained numericallythrough iterative simulations in order to satisfy the relationbetween the postimpact and the prior-impact velocitiesdefined by (2) [6] The following formula was also proposed[9]

120577 =9radic5

2sdot

(1 minus CR2)CR (CR (9120587 minus 16) + 16)

(6)

An alternative approach might also be needed to considerthe relation between the damping coefficient 119888imp(119905) and thecoefficient of restitution CR in the following form

119888imp (119905) = 120572 sdotCR120573imp (1 minus CR)

(119905)

sdot 119896119904sdot 120575119899

(119905) (7)

where 120572 and 120573 are the parameters of the model obtainedby fitting the experimental data using the method of theleast squares (determined in this way typical values are120572 = 001557 120573 = 02706) As it can be seen from(7) the impact damping coefficient depends directly on theprior-impact velocity value which can be obtained for eachimpact separately The impact force versus time and lateraldisplacement for the nonlinear viscoelastic model is shownin Figure 2

Muthukumar and DesRoches A different model (Hertzdampmodel) was considered byMuthukumar andDesRoches [23]Theypresented a formula for the impact force during collisionwith 119899 = 15 which depends on three parameters includingcoefficient of restitution stiffness of spring and prior-impactvelocity as

119888imp (119905) = 120577 sdot 120575119899

(119905) (8)

120577 =

3 sdot 119896119904(1 minus CR2)

4 sdot imp (9)

The impact force versus time and lateral displacement for theHertzdamp model is shown in Figure 3

Ye et al Ye et al [24] claimed that (9) is incorrect to calculatethe damping ratio for the pounding simulation in order to

Shock and Vibration 3

minus1000

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)

minus1 minus08 06040 02 08 1minus04 minus02minus06

Lateral displacement (mm)

minus1000

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)

5 6 7 8 9 104Time (s)

Figure 1 Impact force versus time and lateral displacement for the linear viscoelastic model

0

2000

4000

6000

8000

10000

12000

Impa

ct fo

rce (

kN)

55 6 65 7 75 8 85 9 95 105Time (s)

0604020 08 1minus04minus06minus08 minus02minus1


0

2000

4000

6000

8000

10000

12000Im

pact

forc

e (kN

)

Figure 2 Impact force versus time and lateral displacement for the nonlinear viscoelastic model

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)

5 6 7 8 9 104Time (s)

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)



Figure 3 Impact force versus time and lateral displacement for the Hertzdamp model


0

1000

2000

3000

4000

5000

6000

7000

8000Im

pact

forc

e (kN

)

5 6 7 8 9 104Time (s)

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)



Figure 4 Impact force versus time and lateral displacement for the modified Hertzdamp model

d

m1 m2

C1 C2

K1 K2

x1(t) x2(t)

Figure 5 Model of interacting structures

evaluate impact between two buildings They numericallyindicated that the appropriate relation is [24]

120577 =8 sdot 119896119904(1 minus CR)

5 sdot CR sdot imp (10)

The impact force versus time and lateral displacement for themodified Hertzdamp model is shown in Figure 4

3 Numerical Study

A parametric study has been conducted in order to ver-ify the effectiveness of the nonlinear viscoelastic modelof impact force during structural pounding described by(4) and (7) Two single degree-of-freedom systems (seeFigure 5) with the gap size of 119889 = 2 cm have been used tomodel the behaviour of adjacent structures under groundmotions

The dynamic equation of motion for such a model can beexpressed as [25]

[

1198981

0

0 1198982

][

1(119905)

2(119905)] + [

1198621

0

0 1198622

][

1(119905)

2(119905)]

+ [

1198701

0

0 1198702

][

1199091(119905)

1199092(119905)] + [

119865imp (119905)

minus119865imp (119905)]

= minus[

1198981

0

0 1198982

][

119892(119905)

119892(119905)]

(11)

where 119909119894(119905)

119894(119905)

119894(119905) 119862

119894 119870119894are the horizontal displace-

ment velocity acceleration damping coefficient and stiffnesscoefficient for structure 119894 (119894 = 1 2) respectively

119892(119905) stands

for the acceleration input ground motion and 119865imp(119905) is thepounding force which is equal to zero when 120575(119905) le 0 and isdefined by (4) when 120575(119905) gt 0 where 120575(119905) is defined as

120575 (119905) = 1199091(119905) minus 119909

2(119905) minus 119889 (12)

In the numerical analysis stiffness of each structure has beentaken to be 119870

1= 1198702= 740MNm and storey masses have


Right buildingLeft building

5 10 15 20 25 30 350Time (s)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4La

tera

l disp

lace

men

t (cm

)

(a) El Centro


minus3

minus2

minus1

0

1

2

3

4

5

Late

ral d

ispla

cem

ent (

cm)

5 10 15 20 25 30 350Time (s)

(b) Kobe


5 10 15 20 25 30 35 40 450Time (s)

minus3

minus2

minus1

0

1

2

3

Late

ral d

ispla

cem

ent (

cm)

(c) San Fernando


minus3

minus2

minus1

0

1

2

3La

tera

l disp

lace

men

t (cm

)

5 10 15 20 25 30 35 40 450Time (s)

(d) Parkfield

Figure 6 Lateral displacement time histories under different earthquakes

been assumed to be 1198981= 110 tons and 119898

2= 145 tons

respectively The structural damping ratio of 005 has beenconsidered in the analysis

Thedynamic analyses under the Parkfield (1966) San Fer-nando (1971) Kobe (1995) and El Centro (1940) earthquakerecords have been performed These records have differentcontents of the excitation frequencies different magnitudeof the accelerations and different time durations Besidestheir place of occurrence and geological conditions close tothe epicentre are distinct San Fernando earthquake had thehighest Peak Ground Acceleration (PGA) among the fourrecords discussed The PGA of the earthquake amounted to

1164 g with an epicentre distance less than 12 km The PGAof the Kobe earthquake was 07105 g and it was measured at adistance of 183 kmThe PGA of the Parkfield earthquake wasequal to 0462 g (measured at a distance of 32 km) Finally thePGA of the El Centro earthquake was equal to 0347 g Allmentioned records have been normalized to investigate theeffect of earthquake properties on pounding-involved struc-tural response The examples of the results of the numericalanalysis in the form of the lateral displacement time historiesunder different earthquakes are shown in Figure 6 Addition-ally the examples of the impact force time histories for the


KobeEl Centro

5 10 15 20 25 30 350Time (s)

0

1

2

3

4

5

6

Impa

ct fo

rce (

kN)

times102

Figure 7 Impact force time histories for the Kobe and El Centro earthquakes

Kobe and the El Centro earthquakes are presented inFigure 7

Using four different earthquake excitations the peaklateral displacements velocities and accelerations have alsobeen calculated for different structural periods of collid-ing structures The results of the analyses are presentedin Figure 8 They indicate that with the increase in thestructural period the peak lateral displacements show thenonuniform increase trend Among the ground motionsanalyzed the Kobe record gives the maximum lateral peakdisplacement equal to 387 cm while the minimum peakdisplacement of about 064 cm has been observed for theParkfield earthquake In the case of velocity when thestructural period is increased the peak velocities are nearlythe same in the range of 1ndash4 sec and after that the curvesshow a sudden decrease trend Slightly different trend isobserved for the El Centro record which demonstrates aslight increase from 51ms to 7ms and subsequently showsa sharp decline to 264ms when the structural periodchanges its value from 1 to 9 sec Finally the peak accelerationcurves are quite stable at the beginning of analyzed range ofstructural period and then they show a substantial increasetrend

31 Effect of Gap Size In order to investigate the effectof separation distance between structures a gap size hasbeen varied from 0 to 8 cm Figure 9 shows the effect ofseparation distance on the peak impact force under fourdifferent earthquake records It can be seen from the figurethat the curves follow an irregular decrease trend when thegap size increases In the case of the San Fernando and Kobeground motions a sudden decrease is observed after passinga specific gap size value while a slight declining tendency isvisible for two other earthquake records

32 Effect of Coefficient of Restitution Different values ofcoefficient of restitution CR have been considered to inves-tigate the impact forces between structures under differentearthquakesThe results of the investigation showing the peakvalues of impact forces are presented in Figure 10 Similartrend can be observed for all analyzed excitationsThe resultsshow a uniform decrease in the force when the coefficientof restitution increases For instance the peak impact forcefor the Kobe earthquake is equal to 341 kN and 92 kN forCR = 01 and CR = 09 respectively

33 Effect of Impact Velocity In order to obtain the responsesand compare the results of peak impact forces differentvalues of impact velocity have been considered from therange 1ndash25ms The relations between the peak impact forceand impact velocity values under different earthquakes arepresented in Figure 11 The results show a uniform increasein the peak impact forces with the increase in the impactvelocity The peak impact forces are nearly equal to zero forthe velocity of 1ms and are as large as 1725 kN 1578 kN1560 kN and 1405 kN for the 25ms impact velocity underthe El Centro Kobe San Fernando and Parkfield earthquakerespectively

34 Effect of Stiffness of Impact Spring Stiffness of impactspring element is considered to be one of the most impor-tant parameters when the impact force during collision iscalculated The results of the parametric study showing thepeak values of impact forces with respect to stiffness ofspring are presented in Figure 12 It can be seen from thefigure that the trend for all earthquakes analyzed is similarThe impact forces show nearly linear increase from 210 kN214 kN 273 kN and 352 kN to 2122 kN 2242 kN 2625 kN and


KobeEl Centro

San FernandoParkfield

KobeEl Centro


KobeEl Centro


2 3 4 5 6 7 8 91Period (s)

05

1

15

2

25

3

35

4La

tera

l disp

lace

men

t (cm

)

2 3 4 5 6 7 8 91Period (s)

25

3

35

4

45

5

55

6

65

7

Velo

city

(ms

)

2 3 4 5 6 7 8 91Period (s)

02

03

04

05

06

07

08

09

1

Acce

lera

tion

(ms2)

Figure 8 Peak lateral displacement velocity and acceleration with respect to structural period under different earthquakes

3342 kN for the Parkfield San Fernando Kobe and El Centroearthquake respectively

4 Conclusions

In this paper earthquake-induced pounding between twoadjacent buildings has been studied by the application ofsingle degree-of-freedom structural models Impact has beennumerically simulated with the use of a nonlinear viscoelasticmodel Special attention has been focused on calculatingvalues of impact forces during collisions which have signif-icant influence of pounding-involved response under groundmotions

The results of the study indicate that the impact force timehistory depends substantially on the earthquake excitation

analyzed Moreover the peak impact force during collision ismuch dependent on such parameters as gap size coefficientof restitution impact velocity and stiffness of impact springelement The nonlinear viscoelastic model of impact forcewith the considered relation between the damping coefficientand the coefficient of restitution has also been found to beeffective in simulating pounding between structures duringseismic excitations

The conclusions of the study can be very valuablefor the purposes of accurate modelling the phenomenonof earthquake-induced structural pounding This concernsespecially the issue of determination of the precise values ofimpact forces during collisions which are often not studied inthe analyses (or the analyses are simplified) since the inves-tigations are rather focused on pounding-involved response


KobeEl Centro


1 2 3 4 5 6 7 80Gap size (cm)

0

50

100

150

200

250

300

Impa

ct fo

rce (

kN)

Figure 9 Peak impact force with respect to gap size under differentearthquakes

KobeEl Centro

San FernadoData 4

0

50

100

150

200

250

300

350

400

Impa

ct fo

rce (

kN)

02 03 04 05 06 07 08 0901Coefficient of restitution (CR)

Figure 10 Peak impact forcewith respect to coefficient of restitutionunder different earthquakes

under ground motions It can be considered as the mostsignificant element of the analysis described in this paper ascompared to other relevant research studies

Conflict of Interests

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

KobeParkfield

San FernandoEl Centro

5 10 15 20 250Impact velocity (ms)

0

200

400

600

800

1000

1200

1400

1600

1800

Impa

ct fo

rce (

kN)

Figure 11 Peak impact force with respect to impact velocity underdifferent earthquakes

KobeEl Centro


2000 3000 4000 5000 6000 7000 8000 9000 100001000Stiffness of spring (Nmm)

0

500

1000

1500

2000

2500

3000

3500

Impa

ct fo

rce (

kN)

Figure 12 Peak impact force with respect to impact spring stiffnessunder different earthquakes

References

[1] K Kasai and B F Maison ldquoBuilding pounding damage duringthe 1989 Loma Prieta earthquakerdquo Engineering Structures vol19 no 3 pp 195ndash207 1997

[2] R Jankowski ldquoAssessment of damage due to earthquake-induced pounding between the main building and the stairwaytowerrdquo Key Engineering Materials vol 347 pp 339ndash344 2007

[3] S A Anagnostopoulos ldquoPounding of buildings in series duringearthquakesrdquo Earthquake Engineering amp Structural Dynamicsvol 16 no 3 pp 443ndash456 1988

[4] S A Anagnostopoulos ldquoBuilding pounding re-examined howserious a problem is itrdquo in Eleventh World Conference on


Earthquake Engineering Pergamon Elsevier Science OxfordUK 1996

[5] B F Maison and K Kasai ldquoDynamics of pounding when twobuildings colliderdquoEarthquake EngineeringampStructuralDynam-ics vol 21 no 9 pp 771ndash786 1992

[6] R Jankowski ldquoNon-linear viscoelastic modelling of earth-quake-induced structural poundingrdquo Earthquake Engineeringamp Structural Dynamics vol 34 no 6 pp 595ndash611 2005

[7] R Jankowski ldquoTheoretical and experimental assessment ofparameters for the non-linear viscoelastic model of structuralpoundingrdquo Journal of Theoretical and Applied Mechanics vol45 no 4 pp 931ndash942 2007

[8] S Mahmoud and R Jankowski ldquoModified linear viscoelasticmodel of earthquake-induced structural poundingrdquo IranianJournal of Science and Technology vol 35 no C1 pp 51ndash62 2011

[9] S Mahmoud A Abd-Elhamed and R Jankowski ldquoEarth-quake-induced pounding between equal height multi-storeybuildings considering soil-structure interactionrdquo Bulletin ofEarthquake Engineering vol 11 no 4 pp 1021ndash1048 2013

[10] P Komodromos P C Polycarpou L Papaloizou and M CPhocas ldquoResponse of seismically isolated buildings consideringpoundingsrdquo Earthquake Engineering amp Structural Dynamicsvol 36 no 12 pp 1605ndash1622 2007

[11] P C Polycarpou and P Komodromos ldquoNumerical investigationof potential mitigation measures for poundings of seismicallyisolated buildingsrdquo Earthquakes and Structures vol 2 no 1 pp1ndash24 2011

[12] P Komodromos and P Polycarpou ldquoA nonlinear impact modelfor simulating the use of rubber shock absorbers for mitigatingthe effect of structural pounding during earthquakerdquo Earth-quake Engineering amp Structural Dynamics vol 42 pp 81ndash1002012

[13] R C Barros and S M Khatami ldquoDamping ratios for poundingof adjacent building and their consequence on the evaluation ofimpact forces by numerical and experimental modelsrdquoMecanica Experimental vol 22 pp 119ndash131 2013

[14] R C BarrosHNaderpour SMKhatami andA RMortezaeildquoInfluence of seismic pounding on RC buildings with andwithout base isolation system subject to near-fault groundmotionsrdquo Journal of Rehabilitation in Civil Engineering vol 1no 1 pp 39ndash52 2013

[15] H Naderpour R C Barros and S M Khatami ldquoA new modelfor calculating the impact force and the energy dissipationbased on CR-factor and impact velocityrdquo Scientia Iranica vol22 no 1 pp 48ndash63 2014

[16] G Cole R Dhakal A Carr and D Bull ldquoAn investigation ofthe effects of mass distribution on pounding structuresrdquo Earth-quake Engineering amp Structural Dynamics vol 40 no 6 pp641ndash659 2011

[17] S Yaghmaei-Sabegh and N Jalali-Milani ldquoPounding forceresponse spectrum for near-field and far-field earthquakesrdquoScientia Iranica vol 19 no 5 pp 1236ndash1250 2012

[18] E Tubaldi M Barbato and S Ghazizadeh ldquoA probabilis-tic performance-based risk assessment approach for seismicpoundingwith efficient application to linear systemsrdquo StructuralSafety vol 36-37 pp 14ndash22 2012

[19] C Zhai S Jiang S Li and L Xie ldquoDimensional analysis ofearthquake-induced pounding between adjacent inelasticMDOF buildingsrdquo Earthquake Engineering amp EngineeringVibration vol 14 no 2 pp 295ndash313 2015

[20] B Madani F Behnamfar and H Tajmir Riahi ldquoDynamicresponse of structures subjected to pounding and structuremdashsoilmdashstructure interactionrdquo Soil Dynamics and EarthquakeEngineering vol 78 pp 46ndash60 2015

[21] W Goldsmith Impact The Theory and Physical Behavior ofColliding Solids Edward Arnold London UK 1st edition 1960

[22] S A Anagnostopoulos ldquoEquivalent viscous damping for mod-eling inelastic impacts in earthquake pounding problemsrdquoEarthquake Engineering amp Structural Dynamics vol 33 no 8pp 897ndash902 2004

[23] SMuthukumar andRDesRoches ldquoAHertz contactmodel withnon-linear damping for pounding simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 35 no 7 pp 811ndash8282006

[24] K Ye L Li and H Zhu ldquoA note on the Hertz contact modelwith nonlinear damping for pounding simulationrdquo EarthquakeEngineering amp Structural Dynamics vol 38 no 9 pp 1135ndash11422009

[25] R Jankowski ldquoImpact force spectrum for damage assessmentof earthquake-induced structural poundingrdquo Key EngineeringMaterials vol 293-294 pp 711ndash718 2005

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Furthermore some more recent numerical analyses havebeen carried out to study the influence of different parametersin pounding of buildings [16ndash20]

Nevertheless there is still a need to investigate differentmodels of structural pounding so as to verify their accu-racy in the case of different configurations under differentearthquake excitationsThis concerns especially the values ofimpact forces during collisions which are often not studied inthe analyses (or the analyses are simplified) since the inves-tigations are rather focused on pounding-involved responseunder ground motions

2 Contact Element Methods

Thecontact element is a special element (usually consisting ofa spring and damper) to model impact between two collidingstructures which is widely used to simulate impact forceImpact is parametrically modelled in this way that whenrelative displacement exceeds the separation distance thecontact element is activated The general formula for theimpact force during collision can be expressed as follows (see[21])

119865imp (119905) = 119896119904 sdot 120575119899

(119905) + 119888imp (119905) sdot (119905) (1)

where 119896119904is stiffness of spring 119888imp(119905) denotes damping of

dashpot and 120575(119905) and (119905) describe lateral displacement andvelocity respectively The power of 119899 has been recommendedto be 1 or 15 depending on the model considered Thedamping coefficient 119888imp(119905) is usually related to the coefficientof restitution CR which is defined as the ratio between thepostimpact velocity rebound and the prior-impact velocityimp [21]

0 lt CR =rebound

implt 1 (2)

Anagnostopoulos The linear viscoelastic model of impactforce (with 119899 = 1) was considered by Anagnostopoulos [3]The following formulae were suggested [3 22]

119888imp (119905) = 2 sdot 120577 sdot radic119896119904 sdot119898119894119898119895

119898119894+ 119898119895

120577 =ln (CR)

radic1205872 + (ln (CR))2

(3)

where 119898119894 119898119895are the masses of colliding structures The

impact force versus time and lateral displacement for thelinear viscoelastic model is shown in Figure 1

Jankowski Jankowski [6] considered the nonlinear viscoelas-tic model when 119899 = 15 He proposed activating the dampingterm only during the approach period in which most of

the energy is lost Therefore the following formulae wereconsidered [6]

119865imp (119905) = 119896119904 sdot 120575119899

(119905) + 119888imp (119905) sdot (119905) 997888rarr

(119905) gt 0 (approach period)

119865imp (119905) = 119896119904 sdot 120575119899

(119905) 997888rarr

(119905) le 0 (restitution period)

(4)

119888imp (119905) = 2 sdot 120577 sdot radic119896119904radic120575 (119905) sdot119898119894119898119895

119898119894+ 119898119895

(5)

It was suggested that the appropriate value of impact dampingratio 120577 for a specified value of CR can be obtained numericallythrough iterative simulations in order to satisfy the relationbetween the postimpact and the prior-impact velocitiesdefined by (2) [6] The following formula was also proposed[9]

120577 =9radic5

2sdot

(1 minus CR2)CR (CR (9120587 minus 16) + 16)

(6)

An alternative approach might also be needed to considerthe relation between the damping coefficient 119888imp(119905) and thecoefficient of restitution CR in the following form

119888imp (119905) = 120572 sdotCR120573imp (1 minus CR)

(119905)

sdot 119896119904sdot 120575119899

(119905) (7)

where 120572 and 120573 are the parameters of the model obtainedby fitting the experimental data using the method of theleast squares (determined in this way typical values are120572 = 001557 120573 = 02706) As it can be seen from(7) the impact damping coefficient depends directly on theprior-impact velocity value which can be obtained for eachimpact separately The impact force versus time and lateraldisplacement for the nonlinear viscoelastic model is shownin Figure 2

Muthukumar and DesRoches A different model (Hertzdampmodel) was considered byMuthukumar andDesRoches [23]Theypresented a formula for the impact force during collisionwith 119899 = 15 which depends on three parameters includingcoefficient of restitution stiffness of spring and prior-impactvelocity as

119888imp (119905) = 120577 sdot 120575119899

(119905) (8)

120577 =

3 sdot 119896119904(1 minus CR2)

4 sdot imp (9)

The impact force versus time and lateral displacement for theHertzdamp model is shown in Figure 3

Ye et al Ye et al [24] claimed that (9) is incorrect to calculatethe damping ratio for the pounding simulation in order to


minus1000

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)



minus1000

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)

5 6 7 8 9 104Time (s)


0

2000

4000

6000

8000

10000

12000

Impa

ct fo

rce (

kN)

55 6 65 7 75 8 85 9 95 105Time (s)



0

2000

4000

6000

8000

10000

12000Im

pact

forc

e (kN

)


0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)

5 6 7 8 9 104Time (s)

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)





0

1000

2000

3000

4000

5000

6000

7000

8000Im

pact

forc

e (kN

)

5 6 7 8 9 104Time (s)

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)




d

m1 m2

C1 C2

K1 K2

x1(t) x2(t)



120577 =8 sdot 119896119904(1 minus CR)



3 Numerical Study



[

1198981

0

0 1198982

][

1(119905)

2(119905)] + [

1198621

0

0 1198622

][

1(119905)

2(119905)]

+ [

1198701

0

0 1198702

][

1199091(119905)

1199092(119905)] + [

119865imp (119905)

minus119865imp (119905)]

= minus[

1198981

0

0 1198982

][

119892(119905)

119892(119905)]

(11)

where 119909119894(119905)

119894(119905)

119894(119905) 119862



119892(119905) stands


120575 (119905) = 1199091(119905) minus 119909

2(119905) minus 119889 (12)





5 10 15 20 25 30 350Time (s)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4La

tera

l disp

lace

men

t (cm

)

(a) El Centro


minus3

minus2

minus1

0

1

2

3

4

5

Late

ral d

ispla

cem

ent (

cm)

5 10 15 20 25 30 350Time (s)

(b) Kobe


5 10 15 20 25 30 35 40 450Time (s)

minus3

minus2

minus1

0

1

2

3

Late

ral d

ispla

cem

ent (

cm)

(c) San Fernando


minus3

minus2

minus1

0

1

2

3La

tera

l disp

lace

men

t (cm

)

5 10 15 20 25 30 35 40 450Time (s)

(d) Parkfield



2= 145 tons





KobeEl Centro

5 10 15 20 25 30 350Time (s)

0

1

2

3

4

5

6

Impa

ct fo

rce (

kN)

times102









KobeEl Centro


KobeEl Centro


KobeEl Centro


2 3 4 5 6 7 8 91Period (s)

05

1

15

2

25

3

35

4La

tera

l disp

lace

men

t (cm

)

2 3 4 5 6 7 8 91Period (s)

25

3

35

4

45

5

55

6

65

7

Velo

city

(ms

)

2 3 4 5 6 7 8 91Period (s)

02

03

04

05

06

07

08

09

1

Acce

lera

tion

(ms2)



4 Conclusions






KobeEl Centro


1 2 3 4 5 6 7 80Gap size (cm)

0

50

100

150

200

250

300

Impa

ct fo

rce (

kN)


KobeEl Centro

San FernadoData 4

0

50

100

150

200

250

300

350

400

Impa

ct fo

rce (

kN)






KobeParkfield



0

200

400

600

800

1000

1200

1400

1600

1800

Impa

ct fo

rce (

kN)


KobeEl Centro



0

500

1000

1500

2000

2500

3000

3500

Impa

ct fo

rce (

kN)


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus1000

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)



minus1000

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)

5 6 7 8 9 104Time (s)


0

2000

4000

6000

8000

10000

12000

Impa

ct fo

rce (

kN)

55 6 65 7 75 8 85 9 95 105Time (s)



0

2000

4000

6000

8000

10000

12000Im

pact

forc

e (kN

)


0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)

5 6 7 8 9 104Time (s)

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)





0

1000

2000

3000

4000

5000

6000

7000

8000Im

pact

forc

e (kN

)

5 6 7 8 9 104Time (s)

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)




d

m1 m2

C1 C2

K1 K2

x1(t) x2(t)



120577 =8 sdot 119896119904(1 minus CR)



3 Numerical Study



[

1198981

0

0 1198982

][

1(119905)

2(119905)] + [

1198621

0

0 1198622

][

1(119905)

2(119905)]

+ [

1198701

0

0 1198702

][

1199091(119905)

1199092(119905)] + [

119865imp (119905)

minus119865imp (119905)]

= minus[

1198981

0

0 1198982

][

119892(119905)

119892(119905)]

(11)

where 119909119894(119905)

119894(119905)

119894(119905) 119862



119892(119905) stands


120575 (119905) = 1199091(119905) minus 119909

2(119905) minus 119889 (12)





5 10 15 20 25 30 350Time (s)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4La

tera

l disp

lace

men

t (cm

)

(a) El Centro


minus3

minus2

minus1

0

1

2

3

4

5

Late

ral d

ispla

cem

ent (

cm)

5 10 15 20 25 30 350Time (s)

(b) Kobe


5 10 15 20 25 30 35 40 450Time (s)

minus3

minus2

minus1

0

1

2

3

Late

ral d

ispla

cem

ent (

cm)

(c) San Fernando


minus3

minus2

minus1

0

1

2

3La

tera

l disp

lace

men

t (cm

)

5 10 15 20 25 30 35 40 450Time (s)

(d) Parkfield



2= 145 tons





KobeEl Centro

5 10 15 20 25 30 350Time (s)

0

1

2

3

4

5

6

Impa

ct fo

rce (

kN)

times102









KobeEl Centro


KobeEl Centro


KobeEl Centro


2 3 4 5 6 7 8 91Period (s)

05

1

15

2

25

3

35

4La

tera

l disp

lace

men

t (cm

)

2 3 4 5 6 7 8 91Period (s)

25

3

35

4

45

5

55

6

65

7

Velo

city

(ms

)

2 3 4 5 6 7 8 91Period (s)

02

03

04

05

06

07

08

09

1

Acce

lera

tion

(ms2)



4 Conclusions






KobeEl Centro


1 2 3 4 5 6 7 80Gap size (cm)

0

50

100

150

200

250

300

Impa

ct fo

rce (

kN)


KobeEl Centro

San FernadoData 4

0

50

100

150

200

250

300

350

400

Impa

ct fo

rce (

kN)






KobeParkfield



0

200

400

600

800

1000

1200

1400

1600

1800

Impa

ct fo

rce (

kN)


KobeEl Centro



0

500

1000

1500

2000

2500

3000

3500

Impa

ct fo

rce (

kN)


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

1000

2000

3000

4000

5000

6000

7000

8000Im

pact

forc

e (kN

)

5 6 7 8 9 104Time (s)

0

1000

2000

3000

4000

5000

6000

7000

8000

Impa

ct fo

rce (

kN)




d

m1 m2

C1 C2

K1 K2

x1(t) x2(t)



120577 =8 sdot 119896119904(1 minus CR)



3 Numerical Study



[

1198981

0

0 1198982

][

1(119905)

2(119905)] + [

1198621

0

0 1198622

][

1(119905)

2(119905)]

+ [

1198701

0

0 1198702

][

1199091(119905)

1199092(119905)] + [

119865imp (119905)

minus119865imp (119905)]

= minus[

1198981

0

0 1198982

][

119892(119905)

119892(119905)]

(11)

where 119909119894(119905)

119894(119905)

119894(119905) 119862



119892(119905) stands


120575 (119905) = 1199091(119905) minus 119909

2(119905) minus 119889 (12)





5 10 15 20 25 30 350Time (s)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4La

tera

l disp

lace

men

t (cm

)

(a) El Centro


minus3

minus2

minus1

0

1

2

3

4

5

Late

ral d

ispla

cem

ent (

cm)

5 10 15 20 25 30 350Time (s)

(b) Kobe


5 10 15 20 25 30 35 40 450Time (s)

minus3

minus2

minus1

0

1

2

3

Late

ral d

ispla

cem

ent (

cm)

(c) San Fernando


minus3

minus2

minus1

0

1

2

3La

tera

l disp

lace

men

t (cm

)

5 10 15 20 25 30 35 40 450Time (s)

(d) Parkfield



2= 145 tons





KobeEl Centro

5 10 15 20 25 30 350Time (s)

0

1

2

3

4

5

6

Impa

ct fo

rce (

kN)

times102









KobeEl Centro


KobeEl Centro


KobeEl Centro


2 3 4 5 6 7 8 91Period (s)

05

1

15

2

25

3

35

4La

tera

l disp

lace

men

t (cm

)

2 3 4 5 6 7 8 91Period (s)

25

3

35

4

45

5

55

6

65

7

Velo

city

(ms

)

2 3 4 5 6 7 8 91Period (s)

02

03

04

05

06

07

08

09

1

Acce

lera

tion

(ms2)



4 Conclusions






KobeEl Centro


1 2 3 4 5 6 7 80Gap size (cm)

0

50

100

150

200

250

300

Impa

ct fo

rce (

kN)


KobeEl Centro

San FernadoData 4

0

50

100

150

200

250

300

350

400

Impa

ct fo

rce (

kN)






KobeParkfield



0

200

400

600

800

1000

1200

1400

1600

1800

Impa

ct fo

rce (

kN)


KobeEl Centro



0

500

1000

1500

2000

2500

3000

3500

Impa

ct fo

rce (

kN)


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











5 10 15 20 25 30 350Time (s)

minus5

minus4

minus3

minus2

minus1

0

1

2

3

4La

tera

l disp

lace

men

t (cm

)

(a) El Centro


minus3

minus2

minus1

0

1

2

3

4

5

Late

ral d

ispla

cem

ent (

cm)

5 10 15 20 25 30 350Time (s)

(b) Kobe


5 10 15 20 25 30 35 40 450Time (s)

minus3

minus2

minus1

0

1

2

3

Late

ral d

ispla

cem

ent (

cm)

(c) San Fernando


minus3

minus2

minus1

0

1

2

3La

tera

l disp

lace

men

t (cm

)

5 10 15 20 25 30 35 40 450Time (s)

(d) Parkfield



2= 145 tons





KobeEl Centro

5 10 15 20 25 30 350Time (s)

0

1

2

3

4

5

6

Impa

ct fo

rce (

kN)

times102









KobeEl Centro


KobeEl Centro


KobeEl Centro


2 3 4 5 6 7 8 91Period (s)

05

1

15

2

25

3

35

4La

tera

l disp

lace

men

t (cm

)

2 3 4 5 6 7 8 91Period (s)

25

3

35

4

45

5

55

6

65

7

Velo

city

(ms

)

2 3 4 5 6 7 8 91Period (s)

02

03

04

05

06

07

08

09

1

Acce

lera

tion

(ms2)



4 Conclusions






KobeEl Centro


1 2 3 4 5 6 7 80Gap size (cm)

0

50

100

150

200

250

300

Impa

ct fo

rce (

kN)


KobeEl Centro

San FernadoData 4

0

50

100

150

200

250

300

350

400

Impa

ct fo

rce (

kN)






KobeParkfield



0

200

400

600

800

1000

1200

1400

1600

1800

Impa

ct fo

rce (

kN)


KobeEl Centro



0

500

1000

1500

2000

2500

3000

3500

Impa

ct fo

rce (

kN)


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










KobeEl Centro

5 10 15 20 25 30 350Time (s)

0

1

2

3

4

5

6

Impa

ct fo

rce (

kN)

times102









KobeEl Centro


KobeEl Centro


KobeEl Centro


2 3 4 5 6 7 8 91Period (s)

05

1

15

2

25

3

35

4La

tera

l disp

lace

men

t (cm

)

2 3 4 5 6 7 8 91Period (s)

25

3

35

4

45

5

55

6

65

7

Velo

city

(ms

)

2 3 4 5 6 7 8 91Period (s)

02

03

04

05

06

07

08

09

1

Acce

lera

tion

(ms2)



4 Conclusions






KobeEl Centro


1 2 3 4 5 6 7 80Gap size (cm)

0

50

100

150

200

250

300

Impa

ct fo

rce (

kN)


KobeEl Centro

San FernadoData 4

0

50

100

150

200

250

300

350

400

Impa

ct fo

rce (

kN)






KobeParkfield



0

200

400

600

800

1000

1200

1400

1600

1800

Impa

ct fo

rce (

kN)


KobeEl Centro



0

500

1000

1500

2000

2500

3000

3500

Impa

ct fo

rce (

kN)


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










KobeEl Centro


KobeEl Centro


KobeEl Centro


2 3 4 5 6 7 8 91Period (s)

05

1

15

2

25

3

35

4La

tera

l disp

lace

men

t (cm

)

2 3 4 5 6 7 8 91Period (s)

25

3

35

4

45

5

55

6

65

7

Velo

city

(ms

)

2 3 4 5 6 7 8 91Period (s)

02

03

04

05

06

07

08

09

1

Acce

lera

tion

(ms2)



4 Conclusions






KobeEl Centro


1 2 3 4 5 6 7 80Gap size (cm)

0

50

100

150

200

250

300

Impa

ct fo

rce (

kN)


KobeEl Centro

San FernadoData 4

0

50

100

150

200

250

300

350

400

Impa

ct fo

rce (

kN)






KobeParkfield



0

200

400

600

800

1000

1200

1400

1600

1800

Impa

ct fo

rce (

kN)


KobeEl Centro



0

500

1000

1500

2000

2500

3000

3500

Impa

ct fo

rce (

kN)


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










KobeEl Centro


1 2 3 4 5 6 7 80Gap size (cm)

0

50

100

150

200

250

300

Impa

ct fo

rce (

kN)


KobeEl Centro

San FernadoData 4

0

50

100

150

200

250

300

350

400

Impa

ct fo

rce (

kN)






KobeParkfield



0

200

400

600

800

1000

1200

1400

1600

1800

Impa

ct fo

rce (

kN)


KobeEl Centro



0

500

1000

1500

2000

2500

3000

3500

Impa

ct fo

rce (

kN)


References






























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Numerical Study on Pounding between Two ...downloads.hindawi.com/journals/sv/2016/1504783.pdf · Numerical Study on Pounding between Two Adjacent Buildings under