nonlinear pounding seismic

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EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2005; 34:595–611

Published online 3 March 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.434

Non-linear viscoelastic modelling of earthquake-inducedstructural pounding

Robert Jankowski∗;†

Faculty of Civil and Environmental Engineering; Gda nsk University of Technology;

ul. Narutowicza 11= 12; 80-952 Gda nsk ; Poland

SUMMARY

Past severe earthquakes indicate that structural pounding may cause considerable damage or even leadto collapse of colliding structures if the separation distance between them is not sucient. Because of its complexity, modelling of impact is an extremely dicult task, however, the precise numerical modelof pounding is essential if an accurate structural response is to be simulated. The aim of this paper isto analyse a non-linear viscoelastic model of collisions which allows more precise simulation of thestructural pounding during earthquakes. The eectiveness of the model is veried by comparing theresults of numerical analyses with the results of experiments conducted on pounding between dierenttypes of structures. The results of the study indicate that, compared to other models, the proposednon-linear viscoelastic model is the most precise one in simulating the pounding-involved structuralresponse. Copyright ? 2005 John Wiley & Sons, Ltd.

KEY WORDS: structural pounding; earthquakes; impact force; non-linear model, Hertz contact law

INTRODUCTION

During severe earthquakes, pounding between neighbouring, inadequately separated build-ing structures or bridge segments has been repeatedly observed. Rosenblueth and Meli [1]reported that in the Mexico City earthquake of 1985 about 40% of the damagedstructures experienced some level of pounding, 15% of them leading to structural collapse.Anagnostopoulos [2] re-assessed that statement to say that evidence of pounding was foundin 15% of buildings with major damage or collapse and in 20–30% of these cases poundingcould have been a signicant factor in the structural damage. During the 1989 Loma Prietaearthquake, over 200 pounding occurrences involving more than 500 buildings were observed

∗Correspondence to: Robert Jankowski, Faculty of Civil and Environmental Engineering, Gdansk University of Technology, ul. Narutowicza 11= 12, 80-952 Gdansk, Poland.†E-mail: [email protected]

Contract= grant sponsor: European Community FP5 Programme; contract= grant number: EVK4-CT-2002-80005

Received 16 May 2004

Revised 26 August 2004 and 11 October 2004Copyright ? 2005 John Wiley & Sons, Ltd. Accepted 11 October 2004


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at sites located over 90 km from the epicentre [3]. Signicant pounding damage was observedat expansion hinges and abutments of standing portions of a number of bridges during the1994 Northridge earthquake at the Interstate 5 and State Road 14 interchange [4]. The reportsof damage to highway bridges during the Kobe earthquake of 1995 have identied pounding

due to fracture of bearing supports as a reason for local damage and a potential contributionto the falling down of bridge decks [5]. The pounding-involved structural damage during other

past earthquakes has been also reported [6].The main factor recognized as a reason for the pounding of buildings is usually the dier-

ence in dynamic characteristics of adjacent structures [7–9]. This is due to the fact, that thedierence in mass or stiness induces out-of-phase vibrations which may lead to structuralinteraction during the time of earthquake. On the other hand, for the longer bridge structures,it is often the seismic wave propagation eect that is considered to be a dominant factor lead-ing to pounding of neighbouring superstructure segments [10–12]. This eect, due to timelag and spatial variation of seismic wave, results in dierent seismic input acting on supportsalong the structure [13].

Structural pounding is a complex phenomenon involving plastic deformations at contact points, local cracking or crushing, fracturing due to impact, friction, etc. Forces created bycollisions are applied and removed during a short interval of time initiating stress waves whichtravel away from the region of contact. The process of energy transfer during impact is highlycomplicated which makes the mathematical analysis of this type of problem dicult.

Structural pounding during earthquakes has been intensively studied recently by applyingvarious structural models and using dierent models of collisions. The fundamental studyon pounding between buildings in series using a linear viscoelastic model of collisions, has

been conducted by Anagnostopoulos [14]. Jankowski et al. [10] used the same model to study pounding of superstructure segments in bridges. Further analyses were carried out by applyingdiscrete multi-degree-of-freedom structural models and using FEM, though often incorporatingsimplied models of collisions (see, for example, References [8, 9, 11, 15]). In order to model

the impact force–deformation relation more realistically, a non-linear elastic model followingthe Hertz law of contact has been adopted by a number of researchers [16–18].Intensive study has also been carried out on mitigation of pounding hazards. One of the ob-

jectives is to develop procedures for evaluating adequate separation distance between structuresin order to prevent contact during earthquakes [19]. The minimum seismic gap is specied inmost recent seismic-resistant design codes. Another aim is to enhance the seismic performanceof existing structures without sucient in-between space and to develop proper approaches for reducing the pounding eects on structural members. Westermo suggested linking buildingswith beams which can transmit the forces between them eliminating dynamic contacts [20].The idea of lling the separation gap by energy absorbing material, providing bumpers or strong collision walls protecting parts of the structures, has been studied [2]. Jankowski et al.[15] considered also the use of dampers, crushable devices and shock transmission units which

provide sti linking of structural members during earthquakes and do not impose undesiredforces resulting from thermal elongation, creep or shrinkage eects.The aim of this paper is to analyse a non-linear viscoelastic model of collisions which

allows more precise simulation of the structural pounding. The model can be applied alsofor the assessment of the maximum pounding force value which is required for the design

purposes of dierent types of links, bumpers or crushable devices placed between structuresor structural members. The eectiveness of the model is veried by comparing the results of

Copyright ? 2005 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2005; 34:595–611


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case when the structures are simulated by multi-degree-of-freedom models or when the studyon the pounding of buildings in series or between several segments of a bridge is conducted,the structural response during the time when contact takes place is essential. This is due to thefact, that when the structural members rebound after collision they might come into contact

with other members. Moreover it may also happen that at the time of contact between twogiven structural members other members may collide with each other.

MODELLING OF POUNDING FORCE

The second approach to modelling of pounding is to simulate directly the pounding forceduring impact. The experimental results [21, 26, 27] have shown that pounding force historydepends substantially on a number of factors, such as masses of colliding structures, their rel-ative velocity before impact, structural material properties, contact surface geometry and pre-vious impact history. The examples of experimentally obtained load–time diagrams of impact

between concrete elements [26] are shown in Figure 1. Similar shapes of pounding force histo-ries have been also observed for steel-to-concrete as well as steel-to-steel impacts for dierentcontact surface geometries, mass values and impact velocities of colliding bodies [21, 27].

The pounding force time history during impact consists of two sub-intervals. The approach period extends from the beginning of contact up to the maximum deformation. It is followed by a restitution period lasting until the separation. The beginning of the approach periodis attributed to elastic material behaviour but soon plastic deformations, local cracking or crushing usually take place. In the restitution period, the accumulated elastic strain energyis released without considerable plastic eects. In addition, during the whole time of impact,friction between the colliding members takes place and this eect is especially important inthe case of rough surfaces. It has been shown that most of the energy which is dissipatedduring impact is lost during the approach period of collision and a comparably small amount

of energy is lost during the restitution period due to friction [21]. The experimental results

0

20

40

60

80

100

120

0 2 4 6 8 10 12

test 1

test 2

P o u n d i n g f o r c e ( k N )

Time (ms)

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12

test 1

test 2

P o u n d i n g f o r c e ( k N )

Time (ms)(b)(a)

Figure 1. Examples of experimentally obtained pounding force time histories for impact between concreteelements [26]: (a) impact velocity 0:5 m= s; and (b) impact velocity 2:5 m= s.



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(see Figure 1) indicate that during the approach period, the relatively rapid increase in the pounding force is observed. On the other hand, during the restitution period, the force de-creases with lower unloading rate which is reduced even more just before separation.

The pounding force between colliding structures is usually modelled by the use of elastic or

viscoelastic impact elements which become active when contact is detected. Several types of such elements, e.g., a linear spring, a bi-linear spring, a linear spring-damper and a non-linear spring, have been used for the simulation purposes. The simplest impact element consistsof a linear elastic spring and does not account for the energy dissipation during collision[8, 11, 28, 29]. The bi-linear spring element with dierent approaching and separation stinesshas been described by Valles and Reinhorn [30]. This model includes some energy dissipationdue to the hysteretic behaviour.

Linear viscoelastic model

The most frequently used type of an impact element is a linear spring-damper (Kelvin–Voigtmodel) [7, 10, 14, 30]. The pounding force during impact, F (t ), for this linear viscoelastic

model is expressed as

F (t ) = k(t ) + ċ(t ) (4)

where (t ) describes the deformation of colliding structural members, ̇(t ) denotes the relativevelocity between them, k is the impact element’s stiness simulating the local stiness atthe contact point and c is the impact element’s damping which can be obtained from theformula [14]:

c= 2

k m1m2m1 + m2

(5)

where m1 and m2 are masses of structural members and is a damping ratio correlated witha coecient of restitution, e, by the equation [14]:

= − ln e 2 + (ln e)2

(6)

The disadvantage of the linear spring-damper element is that its viscous component isactive with the same damping coecient during the whole time of collision. This results inthe uniform dissipation of energy during the approach and restitution periods which is notfully consistent with the reality [21, 30]. As was mentioned earlier, most of the energy isdissipated during the approach period and the restitution period is mainly attributed to elastic

behaviour where the accumulated elastic strain energy is released with minor energy loss. Nevertheless, owing to its simplicity, the linear viscoelastic model has been widely used for

the simulation purposes of structural pounding.

Non-linear elastic model

In order to model the pounding force–deformation relation more realistically, a non-linear elastic spring following the Hertz law of contact has been adopted by a number of researchers[16–18, 21]. The pounding force during impact, F (t ), for this type of impact element is



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expressed by the formula

F (t ) = 32 (t ) (7)

where is the impact stiness parameter which depends on material properties and the geom-etry of colliding bodies. Results of the experiments indicate that for impacts between concrete

elements, it ranges typically from 40 to 80 kN= mm3= 2 (1:2× 109 –2:6× 109 N= m3= 2

) dependingmainly on the contact surface geometry [26].

The impact stiness parameter for steel-to-steel impacts takes usually higher values [21, 31].The formulae to calculate values of for certain special impact cases, such as impacts be-tween two spheres or between a sphere and a massive plane surface, have been given byGoldsmith [21].

The disadvantage of the Hertz contact law model is that it is fully elastic and does notaccount for the energy dissipation during contact due to plastic deformations, local cracking,friction, etc.

Non-linear viscoelastic model

The aim of this paper is to analyse a non-linear viscoelastic impact element model whichovercomes the disadvantages of the linear viscoelastic and the non-linear elastic models andthus can be used to simulate structural pounding more precisely. In the model proposed, a non-linear spring following the Hertz law of contact is applied. Additionally, a non-linear damper is activated during the approach period of the collision in order to simulate the process of energy dissipation which takes place mainly during that period. For the simplicity reasons,the minor energy loss during the restitution period is neglected in the model. The poundingforce during impact, F (t ), for this type of impact element is expressed as

F (t ) = 32 (t ) + c(t )̇(t ) for ̇(t )¿0 (approach period)

F (t ) = 32 (t ) for ̇(t )60 (restitution period)

(8)

where is the impact stiness parameter and c(t ) is the impact element’s damping which atany instant of time can be obtained from the formula

c(t ) = 2

(t ) m1m2m1 + m2

(9)

where denotes a damping ratio correlated with a coecient of restitution, e.Expressing the impact element’s damping, c(t ), by Equation (9), which can be considered

as the extension of Equation (5) to the non-linear case, allows us to dene the non-linear

viscoelastic model by two constant parameters, and , where is independent from ,similarly to the case of the linear viscoelastic model. Values of both parameters should bedetermined based on the results of experiments. First, the damping ratio, , should be obtainedfor a given value of coecient of restitution, e, which is to be determined experimentally withthe help of Equation (2) or Equation (3). Since, however, both the spring and the damper are non-linear in the model, the relation between and e cannot be expressed by a relativelysimple formula (see Equation (6)) derived for the linear viscoelastic model by equating the



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m

h

y (t )

Figure 2. Model of a ball falling on a stationary rigid surface.

pounding between the falling ball and a stationary rigid surface, shown in Figure 2, is used.The dynamic equation of motion for such a model can be written in the form

m y(t ) + F (t ) =mg (12)

where m is the mass of a ball, y(t ) its vertical acceleration, g stands for the acceleration of gravity and F (t ) is the pounding force which is equal to zero when y(t )6h (h is a dropheight) and is dened by Equations (4), (7) or (8) when y(t )¿h, where deformation (t ) isexpressed as

(t ) =y(t ) − h (13)

The numerical analysis has been conducted for impacts of steel balls with dierent diametersdropped from various height levels applying three models of pounding force. In this paper,however, the results for a 5= 32inch diameter ball and 2inch drop height are presented. In theanalysis, the following values of parameters dening the dierent pounding force models have

been used: k = 2:07×107 N= m, = 0:14 (e= 0:65) for the linear viscoelastic model, = 4:66×

109 N= m3= 2 for the non-linear elastic model and = 1:03 × 1010 N= m3= 2, = 0:35 (e= 0:65)for the non-linear viscoelastic model. The above values of the impact stiness parameters: k , and have been determined through an iterative procedure in order to attain the maximum

pounding force of F max = 80:7 N since such a value was measured as the maximum oneduring the experiment [27]. A time-stepping integration procedure with constant time stept = 1 × 10−7 s has been applied to solve the equation of motion (12) numerically. The

pounding force time history measured during the experiment and the histories received fromthe numerical analysis for the rst impact are presented in Figure 3. Using Equation (10),the simulation errors for pounding force histories have been calculated as equal to: 15.8%for the linear viscoelastic model, 78.0% for the non-linear elastic model and 21.7% for thenon-linear viscoelastic model.

Concrete-to-concrete impact

In this example, the results of the numerical analysis are compared with the results of theexperiment conducted by van Mier et al. [26] on impacts between a pendulum concrete striker and a xed prestressed concrete pile for dierent contact surface geometries, striker mass andimpact velocity values. Neglecting the inuence of the xed pile, the numerical analysis can

be conducted using the model of one-sided pounding of a pendulum striker with a stationary



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0

20

40

60

80

100

0 5 10 15 20

experiment [27]

linear viscoelastic model

non-linear elastic model

non-linear viscoelastic model

P o u n d i n g f o r c e ( N )

Time (s)

Figure 3. Pounding force time histories during impact between falling steel balland steel hemisphere mounted on a beam.

m

x (0)

l

Figure 4. Model of a pendulum which strikes a stationary rigid barrier.

rigid barrier, as shown in Figure 4. The dynamic equation of motion for such a model can be expressed as

m x(t ) + mg

l x(t ) + F (t ) = 0 (14)

where x(t ), x(t ), m, and l are the horizontal displacement, acceleration, mass and length of the pendulum striker, respectively, F (t ) is the pounding force and g stands for the accelerationof gravity. In order to ensure a specied velocity at the rst impact, the pendulum striker hasto be initially displaced by an appropriate distance x(0)= x0, ( x0¡0). When contact between



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-2 104

0

2 104

4 104

6 104

8 104

1 105

1.2 105

0 2 4 6 8 10 12

experiment, test 1 [26]

experiment, test 2 [26]

linear viscoelastic model

non-linear elastic model

non-linear viscoelastic model

P o u n d i n g f o r c e ( N )

Time (ms)

Figure 5. Pounding force time histories during impact between concrete pendulum striker and reinforced concrete pile.

the pendulum and a rigid barrier is detected, i.e. when x(t )¿0, the pounding force, F (t ), inEquation (14) is dened using Equations (4), (7) or (8), in which the deformation (t ) isexpressed as

(t ) = x(t ) (15)

On the other hand, when x(t )60 (no contact), then the value of the pounding force is equalto zero.

The numerical simulations have been conducted for various values of striker mass andimpact velocity applying three dierent models of pounding force. In order to solve theequation of motion (14) numerically, a time-stepping integration procedure with constanttime step t = 0:0001 s has been used. An example of the results from the numerical anal-ysis and the experiment [26] for impact of a spherical concrete pendulum striker of mass

570 kg (concrete strength 38:2 N= mm2) with impact velocity 0:5 m= s is shown in Figure 5. In

the numerical analysis, the following values of parameters dening dierent pounding force

models have been used: k = 9:35 × 107

N= m, = 0:14 (e= 0:65) for the linear viscoelas-tic model, = 1:13 × 109 N= m

3= 2for the non-linear elastic model and = 2:75 × 109 N= m

3= 2,

= 0:35 (e= 0:65) for the non-linear viscoelastic model. The above values of the impactstiness parameters, k , and , have been determined through an iterative procedure inorder to attain the maximum pounding force of F max = 102:5 kN since such a value was mea-sured as the maximum one during the experiment [26]. The simulation errors for poundingforce histories presented in Figure 5 have been calculated as equal to 23.8% for the linear



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d

C

K

m x (t )

Figure 6. Model of a bridge girder which interacts with an abutment.

viscoelastic model, 97.6% for the non-linear elastic model and 22.9% for the non-linear vis-coelastic model.

Pounding between a bridge girder and an abutment

In the third example, the results of the numerical analysis are compared with the results of the experiment conducted by Zhu et al. [25] on pounding between a bridge girder and anabutment. In the experiment, the model of a bridge girder with mass of 2 kg placed with a0:5 cm gap close to the xed rigid barrier has been tested on the shaking table under a sinewave. A video camera has been employed to monitor the longitudinal displacement of themodel girder using image-processing techniques. In the numerical analysis, the bridge girder ismodelled as a single-degree-of-freedom system as shown in Figure 6. The dynamic equationof motion which simulates the pounding-involved response for such a model can be writtenin the form

m x(t ) + C ˙ x(t ) + K x(t ) + F (t ) = p(t ) (16)

where x(t ), ˙ x(t ), and x(t ) are the horizontal displacement, velocity and acceleration of a bridge girder, respectively, m stands for the girder’s mass, C and K are damping and stinesscoecients and p(t ) is an external excitation (sine wave). Moreover, in Equation (16), F (t )denotes the pounding force which is equal to zero when x(t )6d (d is an initial separationgap) and is dened by Equations (4), (7) or (8) when x(t )¿d, where deformation (t ) isexpressed as

(t ) = x(t ) − d (17)

The numerical analysis has been conducted for the structural model dened by Equa-tion (16) for: m=2kg, C = 4:1 kg= s, K = 210:125N= m (see Reference [25]), applying three

dierent models of pounding force. In the analysis, the following values of models’ parametershave been used: k = 1:5474 × 105 N= m, = 0:28 (e= 0:40) for the linear viscoelastic model

(values determined by Zhu et al. [25] based on experiments), = 4:15 × 106 N= m3= 2

for the

non-linear elastic model and = 2:47 × 106 N= m3= 2, = 0:99 (e= 0:40) for the non-linear viscoelastic model. The equation of motion (16) has been solved numerically using a time-stepping integration procedure with constant time step t = 0:001 s. The displacement timehistory of the bridge girder obtained from the experiment and the histories received from the



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C 2

K 2

m 2

d

x 2 (t )

K 1

m 1

x 1(t )

C 1

Figure 8. Model of interacting steel towers.

for such a model can be expressed as

m1 0

0 m2

x1(t )

x2(t ) + C 1 0

0 C 2

˙ x1(t )

˙ x2(t )+ K 1 0

0 K 2 x1(t )

x2(t ) + F (t )− F (t )

= −

m1 0

0 m2

xg(t )

xg(t )

(18)

where xi(t ), ˙ xi(t ), xi(t ), C i and K i are the horizontal displacement, velocity, acceleration,damping coecient and stiness coecient for Tower i (i= 1; 2), respectively, xg(t ) standsfor the acceleration input ground motion and F (t ) is the pounding force which is equal tozero when (t )60 and is dened by Equations (4), (7) or (8) when (t )¿0, where (t ) isdened as

(t ) = x1(t ) − x2(t ) − d (19)

The numerical analysis has been conducted by applying three dierent models of poundingforce. In the analysis, the following values of model parameters have been used: k = 1:40 ×

109 N= m, = 0:14 (e= 0:65) for the linear viscoelastic model, = 2:36 × 1010 N= m3= 2 for the

non-linear elastic model (value determined by Chau et al. [31]) and = 9:90 × 1010 N= m3= 2

,= 0:35 (e= 0:65) for the non-linear viscoelastic model. A time-stepping integration procedure

with constant time step t = 0:0001 s has been applied to solve the equation of motion (18)numerically. The velocity time history of Tower 1 obtained from the experiment and thehistories received from the numerical analysis are shown in Figure 9. The simulation errorsfor these histories have been calculated as equal to 39.0% for the linear viscoelastic model,30.4% for the non-linear elastic model and 28.3% for the non-linear viscoelastic model.

Results of experimental verication of pounding force models

It can be seen from Figures 3 and 5 that the linear viscoelastic and the non-linear viscoelasticmodels are the most precise ones in simulating the pounding force time histories duringimpact. Both models allow simulation of the relatively rapid increase in the pounding forceduring the approach period and the decrease in the force with lower unloading rate during therestitution period. However, it is only the non-linear viscoelastic model which allows the force



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(d) )s(emTi

s ) / m (

t y

l o c i

V e

00 5400 32010

1

50.

0

-0.5

-1

(c) )s(emTi

s ) / m (

t y

l o c i

V e

00 5400 32010

1

50.

0

-0.5

-1

(b) )s(emTi

s ) /

( m

y t i c o l

V e

00 5400 32010

1

50.

0

-0.5

-1

(a) )s(emTi

) / s

m

c i t y (

o l e V

00 50 40 32010

1

50.

0

5-0.

-1

Figure 9. Velocity time histories of Tower 1 for pounding of steel towers under the ElCentro earthquake: (a) experiment [31]; (b) linear viscoelastic model; (c) non-linear

elastic model; and (d) non-linear viscoelastic model.

to be even further reduced just before separation. On the other hand, the activated dashpotduring the restitution period of collision in the case of the linear viscoelastic model results inthe negative impact force just before the end of the contact which does not have a physicalexplanation. Figures 3 and 5 conrm also the disadvantage of the non-linear elastic model



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for which the simulation errors are the largest. In this case, owing to elastic behaviour, the pounding force history during the approach and the restitution periods is symmetric resultingin the longer time of contact.

The pounding force time histories for the non-linear viscoelastic model presented in Fig-

ures 3 and 5 show a change in curvature after passing the peak force value. This eect isrelated to the disengagement of the viscous damper in the restitution period of impact. Smooth-ing of the impact force diagram at this point is possible but it would require redening theformula (8) for the transition zone between the approach and restitution period. That wouldmake the model more complicated and therefore less practical for numerical simulations.

The results of the study presented in Figures 7 and 9 show that the application of the proposed non-linear viscoelastic model gives the smallest simulation errors in the response(displacement and velocity) time histories of the analysed examples of structural pounding. Itis worth noting that in the case of the analysis of pounding between a bridge girder and anabutment (see Figure 7), the use of the non-linear elastic model gives very poor results. Thisis due to the fact that the model does not account for the dissipation of energy during contactwhich is of great importance in this case since the relatively low value of the coecient of restitution (e= 0:4) has been determined based on experiments [25]. There is no doubt thatin such cases only the models that take into consideration the dissipation of energy should beused for the simulation purposes. On the other hand, the example of pounding of steel towers(see Figure 9) is quite a dierent case. Now, the impacts are much more elastic and theapplication of the non-linear elastic model (used also by Chau et al. [31]) gives very similar simulation errors as in the case of the non-linear viscoelastic model. It should be stressed alsothat the relatively high values of simulation errors obtained for all models in this exampleresult mainly from the fact that a less accurate single-degree-of-freedom structural model has

been used in the numerical analysis (used also by Chau et al. [31]). This fact, however, doesnot prevent us from drawing the comparative conclusions on the eectiveness of dierent

pounding force models.

CONCLUSIONS

In this paper, a non-linear viscoelastic model of pounding force, which is intended to en-hance the accuracy of the modelling of structural pounding during earthquakes, has beenanalysed. The comparison between the model proposed and two other commonly used pound-ing force models, i.e. the linear viscoelastic model and the non-linear elastic model followingthe Hertz contact law, has been conducted. In order to verify the accuracy of the mod-els, the results of the numerical analysis have been compared with the results of impactexperiments.

The results of the study show that the linear viscoelastic and the non-linear viscoelastic

models give the smallest simulation errors in the pounding force time histories during impact.In the case of the linear viscoelastic model, however, the negative impact force just beforeseparation, which does not have a physical explanation, has been observed. Further analysishas shown that the application of the proposed non-linear viscoelastic model results in thesmallest simulation errors in the response time histories of the analysed examples of structural

pounding. The results have also indicated that the non-linear elastic model should not be usedfor modelling of impacts with low values of coecient of restitution. However, it can be



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more accurate than the linear viscoelastic model when the loss of energy during collision isnot so signicant.

Among the considered pounding force models, the linear viscoelastic model is the simplestone, it can be easily applied in computer programs and it does not require numerical iterations

to obtain an appropriate value of the impact damping ratio as in the case of the non-linear viscoelastic model. It seems that the major shortcoming of the linear viscoelastic model, i.e.the negative impact force just before separation, can be eliminated by neglecting the negativeforce value when it occurs. This would require, however, the re-assessment of the relation

between the impact damping ratio and the coecient of restitution (see Equation (6)) in order to achieve the appropriate post-impact velocities during numerical simulations. Moreover, suchan improvement may be dicult to implement in the general-purpose computer programs.

The practical application of the proposed non-linear viscoelastic model for simulation purposes requires the knowledge of the model’s parameters. In this paper, the values of these parameters have been determined based on examples of results of experiments conducted onstructural pounding. It seems, however, that more extensive experimental studies are requiredin order to assess the range of the model’s parameters more precisely for dierent types of structures with various material and contact surface geometry properties.

ACKNOWLEDGEMENTS

This study was supported by the European Community under the FP5 Programme, key-action ‘Cityof Tomorrow and Cultural Heritage’ (Contract No. EVK4-CT-2002-80005). This support is greatlyacknowledged.

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Nanbu earthquake and the retrot of highway bridges in Japan. Third U.S.–Japan Workshop on Seismic Retrotof Bridges, Osaka, Japan, 10–11 December 1996.

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