An investigation of the effects of mass distribution on pounding structures

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2011; 40:641–659Published online 13 September 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.1052

An investigation of the effects of mass distributionon pounding structures

Gregory Cole∗,†, Rajesh Dhakal, Athol Carr and Des Bull

Department of Civil and Natural Resources Engineering, University of Canterbury, Private Bag 4800,

Christchurch 8140, New Zealand

SUMMARY

The effects of diaphragm mass distribution are investigated for building pounding. Elastic diaphragm-to-diaphragm collisions are explained by considering the total momentum over the length of each diaphragmat three critical instants during collision. Expressions for collision force and collision duration are produced,providing additional information about the collision process. Equations for the post collision velocity ofeach diaphragm are produced and are found to appreciably differ from conventional impact—momentumequations under certain conditions. The change in post collision velocity is found to be dependent onthe ratio of the axial periods of free vibration of the two diaphragms and the ratio of their masses. Anequivalent lumped mass model is proposed and assessed against simplified distributed mass models withnumerical modelling of two two-storey buildings. Finally, a new parameter is introduced to represent theplasticity of an inelastic collision between the two distributed masses. This paper highlights the significantinfluence that diaphragm mass distribution may have on the analysis of pounding structures. Copyright� 2010 John Wiley & Sons, Ltd.

Received 14 April 2010; Revised 1 July 2010; Accepted 2 July 2010

KEY WORDS: distributed mass; pounding; equivalent lumped mass; coefficient of restitution; poundingforce

INTRODUCTION

Building pounding describes the collision of two or more adjacent buildings caused by externalloadings. Seismic forces are the leading cause of building collision and can result in severe structuraldamage or complete building collapse [1, 2]. Numerical modelling provides a relatively inexpensivemethod of predicting pounding-induced building damage and has been used by many researchersto further understand pounding [1, 3, 4]. Such modelling involves idealization of the diaphragms inthe affected buildings and development of a mechanism to recognize contact between the buildings(Figure 1). Typically, a contact mechanism is activated when the condition u1(t)−u2(t)�ugap ismet. The modelling of the contact itself has been subject to a multitude of approaches [5–8].

Idealization of a collision as an impact between two lumped masses leads to a method known asstereo mechanics [9]. Stereo mechanics uses conventional impact–momentum equations to predictthe velocities of two objects after they collide. In time history analyses, the collision is assumed to

∗Correspondence to: Gregory Cole, Department of Civil and Natural Resources Engineering, University of Canterbury,Private Bag 4800, Christchurch 8140, New Zealand.

†E-mail: [email protected]

Copyright � 2010 John Wiley & Sons, Ltd.

642 G. COLE ET AL.

m1 m2

u1(t) u2(t)

ugap

ugap = initial separationui

i

(t) = displacement of diaphragm i m = mass of diaphragm i i = 1, 2

Figure 1. Two SDOF oscillators at risk of pounding.

Figure 2. Common contact element hysteresis plots (compression is positive).

occur instantaneously and the velocities of each diaphragm are updated that the time step collisionis detected. The post collision velocities are calculated using [9]:

v′1 = v1 −(1+e)

m2

m1 +m2(v1 −v2) (1)

v′2 = v2 −(1+e)

m1

m1 +m2(v2 −v1) (2)

where vi = velocity of diaphragm i at impact; v′i = velocity of diaphragm i after impact; and

e= the coefficient of restitution. The coefficient of restitution describes the level of plasticity inthe collision. When e=1.0, the collision is fully elastic, while e=0.0 corresponds to a fullyplastic collision. Stereo mechanics cannot predict the size of the collision force or the duration ofthe collision, due to the method’s underlying assumptions. However, the biggest drawback arisesfrom the programming of the contact. Common numerical time history analysis programs generallyrequire changes to their core coding before stereo mechanics can be used. This drawback alsoexists for the Lagrange and Laplace methods, which use augmented matrices to apply poundingas additional geometric boundary conditions [5], or Laplace transforms to include soil–structureinteraction [8], respectively. While these methods have all been used for pounding research [4, 5, 8],the need for specifically developed software limits their adoption by others.

The more popular contact elements use combinations of springs and dashpots. These elementscan be characterized by their force–displacement hysteresis (Figure 2). Contact elements generallyrequire the determination of three parameters: initial separation, and once contact does occur,element stiffness and element damping. The collision stiffness is relatively arbitrarily assigneddue to its reported insensitivity on the structures’ displacement responses [7, 10]. The stiffness istypically assigned to be larger than the axial stiffness of either diaphragm [7].

The Kelvin Voight element has been commonly used to model contact [7, 11]. This elementcomprises a single spring and dashpot in parallel. Both the spring and the dashpot are activatedwhen u1(t)−u2(t)�ugap. This element is popular because it is simple and has a clearly definedmathematical relationship between the element’s damping ratio and the coefficient of restitu-tion [12]. The element does suffer a major drawback, which is shown in Figure 2. Towards the endof contact, tensile loading can occur in the element. Physically, this represents a ‘sticky’ contact,which does not occur in normal pounding scenarios [13]. The magnitude of the tensile loading

Copyright � 2010 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2011; 40:641–659DOI: 10.1002/eqe

AN INVESTIGATION OF THE EFFECTS OF MASS DISTRIBUTION 643

depends on both the level of element damping and the relative velocity of the collision. Variationson the Kelvin Voight element, such as the no tension variation by Shakya et al. [14], address thisinaccuracy (Figure 2). However, these variations invalidate the assumptions made when relating thecoefficient of restitution to the collision element damping ratio. The Kelvin Voight element has alsobeen used in three-dimensional analysis [15]. When used in this way, the contact element responsenormal to the contact surface follows the Kelvin Voight hysteresis, while transverse loading isgoverned by friction.

The Hertzdamp element [3, 13] has recently been promoted as a better alternative. The formu-lation produces a continuous expression for the contact stiffness, which provides more numericalstability in time history programs (Figure 2). The Hertzdamp element does not produce any tensileloadings, and the relationship between element damping and the coefficient of restitution is alsoknown. However, the Hertzdamp element shares drawbacks also present in the Kelvin Voightelement. The element stiffness lacks a rational method of quantification. Furthermore, neitherelement can accurately predict even approximate collision force values, or the duration of colli-sions. These drawbacks are inherent in the lumped mass assumption used when modelling thecolliding diaphragms.

Modelling diaphragms as a collision between two lumped masses is a major assumption thathas, until now, received little attention. In fact, incorporating distributed mass approximationsinto existing models is not overly complex. Furthermore, distributed mass models can provideadditional information that is unavailable in the abovementioned methods.

Modelling a diaphragm as a lumped mass implies that the diaphragm acts as a perfectly rigidobject. In reality all diaphragms have some degree of in-plane flexibility. This flexibility becomesespecially important when two diaphragms with similar stiffness’s collide, as each diaphragm’sflexibility affects the post collision velocity of both diaphragms. The distributed mass model isdescribed with a new general configuration (Figure 3).

Distributed masses were first used in pounding analyses by Watanabe and Kawashima [16] whenmodelling bridge decks. The primary goal of their work was to determine the optimum value ofthe contact element stiffness (kC). The analyses used an elastic Kelvin Voight element to modelthe contact (e=1.0). Both pounding decks had the same axial stiffness and mass per unit length.The optimized value for the contact element stiffness (kC) was determined as the adjacent elementstiffness (kE), as shown in Figure 4. Here E is the modulus of elasticity; A is the cross-sectionalarea of the element; and kD is the overall deck axial stiffness.

Figure 3. Two distributed masses at risk of pounding.

Figure 4. Optimal distributed mass collision element stiffness.


644 G. COLE ET AL.

Watanabe and Kawashima also used the one-dimensional wave equation to produce a formulafor the duration of contact and the internal stress in each deck. Finally, they compared a singlesituation where a collision between two distributed masses occurs. The distributed mass modelwas compared with a lumped mass model that possesses the same properties. Momentum andenergy was conserved in both models but the distributed mass’ final velocities differed from thatof the lumped masses. This is because the distributed mass model had some energy stored as strainenergy in one diaphragm. This strain energy causes the diaphragm to axially oscillate after thecollision has occurred.

To the authors’ knowledge, no research has been undertaken on the modelling of buildingpounding using distributed masses, other than conference papers by the authors [17, 18]. Thefirst of these papers presented the explicit behaviour of two colliding diaphragms using the one-dimensional wave equation [17]. A parameter, named �, was identified, which indicates whendistributed mass effects may affect the post collision velocities of each building. Three buildingconfigurations were also tested to compare lumped mass and distributed mass models. Subsequentwork refined the calculation of �, which was found to be dependent on only two factors: the ratioof the colliding diaphragms’ mass and the ratio of the diaphragms’ axial period [18].

This paper extends the work of both previous papers by calculating an effective coefficient ofrestitution, eeff, and relating this to the stiffness and damping of the contact element, assuminga Kelvin Voight hysteresis. These parameters produce the same post collision velocities as thosepredicted using distributed mass diaphragms, but remove the need to explicitly model the distri-bution of each mass. The new lumped mass modelling method is then compared to other lumpedand distributed mass models. The proposed method is then adapted to allow energy dissipationfrom other sources, such as those usually modelled by the coefficient of restitution, by introducinga distributed mass plasticity index. This method can be incorporated into any existing model thatuses either stereo mechanics or the Kelvin Voight hysteresis. The parameter � is also developedfurther and reinterpreted as an ‘influence coefficient’, which indicates the degree of velocity changecaused by collision.

THEORETICAL BACKGROUND

The collision of two distributed masses can be understood by consideration of a few critical instantsduring the collision. A more formal derivation can be obtained using the one-dimensional waveEquation [9]; however, the method presented herein allows a better physical understanding of thephenomenon.

When a collision occurs, a compression wave propagates from the collision interface in bothdiaphragms (Figure 5). As the compression wave passes, it changes the velocity of the diaphragmat that point. This velocity is present in both diaphragms and is termed vc (refer Figure 5(c)). Thecompression wave is reflected when it reaches the free end of a diaphragm. The reflected wavecauses a second change in velocity, termed v′

i . The collision ends when a reflected wave reachesthe collision interface. By this time the velocity in one diaphragm has completely changed fromthe beginning of collision. This velocity change causes the diaphragms to separate. Unless bothreflected waves reach the collision interface at the same time, one diaphragm will be partiallycompressed at the end of collision. This diaphragm will oscillate after the collision has finished.

The velocity at which the compression wave propagates through a diaphragm can be found fromone-dimensional wave theory [9]. The wave velocity (c), of diaphragm i , is;

ci =√

Ei

�i(3)

where Ei is the elastic modulus and �i is the mass density of the diaphragm. The time taken fora compression wave to propagate and then reflect back to the collision interface can be calculated



Diaphragm 1 Diaphragm 2

(v1 > v2)

Collision interface

v2 v1 v1 v2

v1 vc v2 vc v2

v1 vc v2 v1 vc v 2

(v 1 < vc)

(a)

(c)

(e) (f)

(d)

(b)

Figure 5. Snapshots of a distributed mass collision.

by considering the distance the wave must travel;

Ti = distance

velocity= 2Li

ci=2

√�i L2

i

Ei(4)

where Li is the length of the diaphragm and Ti is the ‘collision period’ for the considered diaphragm.The collision period of a diaphragm defines the time a collision would take between that diaphragmand a completely rigid wall. When two diaphragms collide, the collision duration is the lesser of thetwo diaphragms’ collision periods. Equation (4) is simplified with two substitutions; mi =�i Ai Liand ki = Ei Ai/Li , where Ai is the cross-sectional area of the considered diaphragm.

Ti =2

√mi

ki(5)

where mi is the total mass and ki is the total axial stiffness.To calculate the values of vc and v′

i , consider the momentum at the beginning, the midpoint andthe end of the collision (Table I). These three instants correspond to part b, d and f of Figure 5,respectively. The proportion of diaphragm 2 moving at a given velocity is governed by thediaphragms’ collision period ratio. For example, in Table I T1 = 3

4 T2. This means that when allof diaphragm 1 moves at vc, only three quarters of diaphragm 2 moves at velocity vc, while theremaining quarter travels at v2 (Table I(d)).

The internal sections of each diaphragm experience only up to three discrete velocities duringa collision. This is because the velocity within an element can only change when there is a stressimbalance, which occurs only at the front of the compression wave. Throughout the collision, thecollision interface moves at an intermediate velocity (vc), thus this intermediate velocity is labelledthe ‘collision velocity’. When the two diaphragms are in contact, the change in momentum ofdiaphragm one (�p1) is equal and opposite to the change in momentum of diaphragm two (�p2).Considering just the first half of the collision (p1d − p1b = p2b − p2d ) produces a relationship for vc.

vc =m1v1 +m2

T1

T2v2

m1 +m2T1

T2

(6)


646 G. COLE ET AL.

Table I. Diaphragm momentum at critical instants during collision.

Diaphragm velocity Diaphragm 1 momentum Diaphragm 2 momentum

b p1b =m1v1 p2b =m2v2

d p1d =m1vC p2d =m2

(T1T2

vC +(

1− T1T2

)v2

)

f p1 f =m1v′1 p2 f =m2

(2(

1− T1T2

)vC

+(

2 T1T2

−1)v′

2

)

Using just the second half of the collision, vc can also be found in terms of the final velocities v′1

and v′2. By equating the two expressions for vc, an expression directly relating final velocity to

initial velocity is found;

v′1 =v1 +2

1

1+ m1T2

m2T1

(v2 −v1) (7)

The expression for diaphragm 2’s ‘post collision’ velocity (v′2) is similar but has the subscripts

1 and 2 exchanged. However, as shown in Table I, (f), different segments of diaphragm 2 havedifferent velocities at the end of the collision. This means diaphragm 2 continues to oscillate afterthe collision is completed. The average velocity of diaphragm 2 (v′′

2 ) is defined as the averagevelocity the diaphragm moves at as a whole after the collision. In contrast, v′

2 is the velocity in asection of the diaphragm, which is caused by the propagation and reflection of the compressionwave. For example, in Table I, (f), v′

2 is the velocity of only the uncompressed diaphragm; whilev′′

2 is the average velocity found when considering all of diaphragm 2 (and is not shown in thefigure). The average velocity is found by considering conservation of momentum of the entirelength of diaphragm 2 after it separates from diaphragm 1:

m2v′′2 = m2

(2

(1− T1

T2

)vC +

(2

T1

T2−1

)v′

2

)

v′′2 = v2 +2

1T2

T1+ m2

m1

(v1 −v2)(8)

This problem does not occur in diaphragm 1 since at the end of collision all diaphragm sectionsmove at v′

1 (Table I, (f)). Thus, the overall diaphragm velocity is equal to v′1. For the purposes of

the rest of this paper, v′′2 shall be referred to as v′

2. This is because the original definition of v′2 is

only an intermediate step in the derivation, and relabeling provides consistency of nomenclaturein the following sections. Note the diaphragm labelling must be assigned using the criteria T2�T1,in order to make Equations (7) and (8) valid.

The one-dimensional wave equation also allows the calculation of the theoretical maximumcollision force. This is achieved by determining the compression strain in the diaphragm as a resultof collision. To find the strain consider Table II, which shows a section of diaphragm 1 during acollision with diaphragm 2.



Table II. Calculation of diaphragm strain as a result of collision.

Time Layout

t<t1

t = t1

t = t1 + �xc

Before time t1, both segment ends move at the same velocity and the segment is under nostrain. At time t1, the velocity at point (x +�x) changes to vc. This change creates a relativevelocity between points x and (x +�x), which lasts until the wavefront reaches point x . Sincethe wave velocity (c) is known, the time for the wave to travel between these two points is alsoknown (�t =�x/c). The strain is found from the ratio of the segment deformation and the segmentundeformed length. The segment deformation is found by

deformation= rel. velocity×time= (v1 −vC )× �x

c(9)

Thus, the equation for strain may be found. Generalizing, the strain in diaphragm i is

εi = vi −vC

ci(10)

The force is subsequently calculated using basic mechanics (F = Ai Eiεi ). After multiple substitu-tions, a simple relationship is found where

F =2v1 −v2

T1

m1+ T2

m2

(11)

To the authors’ knowledge, this is the only formula for collision force to be presented in buildingpounding research to date. It is acknowledged that the coefficient of restitution is likely to affect thecollision force away from the contact. However, the force at the collision interface is anticipatedto start at the calculated value and then decay over time. Note the collision force does not have asubscript. This is because the force is equal in both diaphragms. The derivation has no allowancefor three-dimensional effects, such as diaphragm torsion.

COMPARISON OF DISTRIBUTED VERSUS LUMPED MASS MODELLING

Stereo mechanics will produce the same post collision velocity as lumped mass contact modelsproviding four criteria are met:

1. There is no acceleration in either diaphragm at the onset of collision.2. No other external load is applied to either diaphragm during the collision.3. Any collision damping is calibrated to the coefficient of restitution [3, 12].4. The collision element stiffness is sufficiently large to cause a near instantaneous collision.


648 G. COLE ET AL.

Table III. Formulation of � based on mass distribution.

Lumped mass Distributed mass (T1�T2)

�11

1+ m1m2

1

1+ m1T2m2T1

�21

1+ m2m1

1T2T1

+ m2m1

Even if criteria 1 and 2 are not met, stereo mechanics provides a close approximation to the postcollision velocities resulting from other lumped mass models. The effects of mass distributioncan be assessed by comparing stereo mechanics with the equations derived above. To do this,Equations (7) and (8) are compared with Equations (1) and (2), when e=1. All four equationscan be expressed in terms of the general equation;

v′1 =v1 −2�1(v1 −v2) (12)

where the subscripts 1 and 2 are swapped when considering the post collision velocityof diaphragm 2. The value of � ranges between 0 and 1. When �=0, then v′

1 =v1, causing nochange in velocity. Thus, � may be considered as an ‘influence coefficient’, which indicates thedegree of velocity change caused by a collision. A greater value of � indicates a greater velocitychange in the diaphragm as a result of collision, and thus the diaphragm has a greater vulnerabilityto damage due to pounding.

Comparing Equations (1), (2), (7) and (8) with Equation (12), the formulations for � can bederived as presented in Table III. Note that T1/T2 is always less than or equal to one due to therestrictions resulting from Equation (8). The value of � is the same for both mass formulationsonly if T1 =T2. This corresponds to the case when diaphragm 2 does not oscillate after impact(refer to Table I, (f)). Figure 6 presents the likely range of �.

The distributed mass formulation cannot produce a greater value of � than the lumped massmodel. As the collision period ratio (T1/T2) is reduced, � is also reduced. This is because lessenergy is transferred between the diaphragms. The ‘lost’ energy is stored instead as strain energy indiaphragm 2, causing ongoing oscillation after contact. As the mass ratio increases, �1 reduces while�2 increases. This finding supports previous researchers’ comments stating building pounding cansignificantly damage the lighter building when one building is greatly heavier than the other [19].Note that mass ratio and collision period ratio are not independent parameters (refer Equation (5)).The figures can instead be presented using mass ratio and axial stiffness ratio. However, the presentformulation allows much more intuitive interpretation of results.

The influence coefficient, �, provides previously unrealized insight into the collision process.Specifically, it provides a means of identifying when distributed mass effects are likely to beimportant. In some circumstances reference to Figure 6 alone may be sufficient to show thatlumped mass modelling is suitable for the modelling of a specific building configuration. Using� as a measure of how much pounding influences the velocity of a structure is also informative.Building 1 is almost entirely unaffected by a collision if m1/m2>20 while building 2 is similarlyunaffected if m1/m2<0.05. However, local damage may still be present as a result of the force ofany collision. In such circumstances, pounding analysis may still be required if the collision forceor the performance of the other structure is a concern.

EQUIVALENT LUMPED MASS FORMULATION

Equation (12) can be generalized to also apply to non-linear lumped mass impact. This is achievedby incorporating the coefficient of restitution from stereo mechanics (refer Equations (1) and (2))

v′1 =v1 −(1+e)�1(v1 −v2) (13)



Figure 6. Influence of mass and period ratio on �.


650 G. COLE ET AL.

Figure 7. Calculation of the effective coefficient of restitution.

Using this formula an effective coefficient of restitution can be found. The effective coefficientenables lumped mass modelling that emulates distributed mass collisions. The effective coefficient(eeff) is found by relating Equation (12) to Equation (13). Note Equation (12) uses the distributedmass formulation for �1, while Equation (13) uses the lumped mass formulation.

(1+eeff)1

1+ m1

m2

= 2

1+ m1T2

m2T1

eeff = 2

⎛⎜⎜⎝

1+ m1

m2

1+ m1T2

m2T1

⎞⎟⎟⎠−1 (14)

The same effective coefficient is found if the equations for diaphragm 2 are used in the derivation.The effective coefficient produces the same post collision velocity as an elastic collision betweenthe two distributed masses. Equation (14) is plotted in Figure 7. It is possible for Equation (14) toproduce effective coefficients less than 0. However, this result is unlikely to occur for real buildingconfigurations, due to the usual range of mass and axial stiffness ratios. If a negative value foreeff is calculated, another modelling approach must be used. This is because the coefficient ofrestitution generally forms a logarithmic relationship to the collision element damping ratio (forexample, Equation (17)).

The duration of the collision for the lumped mass model can also be calibrated to the theoreticalcollision duration. This is achieved by modifying the contact element stiffness, which affectscontact duration. When using the Kelvin Voight element, the duration of a lumped mass collisionis calculated as [12];

T = �

�d= �√

kCm1 +m2

m1m2(1−�2)

= �√√√√√kCm1 +m2

m1m2

⎛⎝1−

(− ln(eeff)√

�2 +(ln(eeff))2

)2⎞⎠

(15)



The theoretical duration of the collision for two distributed masses is T1. Substituting and rear-ranging of the collision element stiffness (kC) produces

kC =m1m2

m1 +m2

(�

T1

)2

1−(

− ln(eeff)√�2 +(ln(eeff))2

)2(16)

The collision element damping is then calculated in the standard manner [12]:

C =2�

√kc

m1m2

m1 +m2where �= − ln(eeff)√

�2 +(ln(eeff))2(17)

A Kelvin Voight element using these values produces the same post collision velocity in eachdiaphragm and has the same collision duration as the theory presented in this paper. Note that theseformulae are subject to the four assumptions outlined in the previous section. The Hertzdamp modelcan have its parameters calibrated in a similar manner; however, its derivation is considerably morecomplicated due to the element’s non-linear stiffness and damping. To date, the Hertzdamp modelcalibration has not been attempted.

NUMERICAL MODELLING OF DISTRIBUTED MASS COLLISION

The accuracy of the ‘equivalent lumped mass’ model may be quantified using numerical analysis.This section focuses on four specific aspects when a number of models are subjected to multipleearthquake excitations:

1. Whether a significant difference in maximum drifts are recorded between the distributed andlumped mass model.

2. Whether the proposed ‘equivalent lumped mass’ model can sufficiently approximate thedistributed mass model.

3. Whether a simplified distributed mass representation with only a few elements can sufficientlymodel a distributed mass.

4. Whether Equation (11) accurately predicts the collision forces in each model.

Two simplified buildings, each with two storeys, are modelled for three excitations. The buildingsare modelled using Ruaumoko, a non-linear time history program developed at the University ofCanterbury [20]. General building dimensions and key properties are presented in Figure 8 andTable IV, respectively. The buildings are separated by 8 mm.

El Centro (East-West, 1940), Mexico City (East-West, 1985) and Loma Prieta (North-South,1989) earthquake components were adopted as input records (Figure 9). Ten seconds of eachrecord were used to reduce the overall computation time. The intensities of each record were alsomodified to adjust each building’s inelastic response. The scaling factors for El Centro, Mexico

50 m20 m

10 m

Figure 8. Building layout.


652 G. COLE ET AL.

Table IV. Key building parameters.

Building 1 Building 2 Ratio

Floor seismic weight 1600 kN 3000 kN 0.53Interstorey stiffness 420000kN/m 350000kN/m 1.2Diaphragm stiffness (kD) 1881300kN/m 752520kN/m 2.5Structure period 0.2 s 0.3 s 0.67Floor collision period (T ) 0.019 s 0.04 s 0.48Lumped mass � 0.652 0.348 —Distributed mass � 0.464 0.248 —Storey yield 2400 kN 1400 kN 1.7

Figure 9. Scaled input excitations.

City and Loma Prieta are 1.0, 2.0 and 1.0, respectively. After scaling, El Centro induces littleinelastic interstorey deformation, while Loma Prieta causes a significant inelastic displacementearly in the record. Mexico City causes moderate inelastic deformation.

Ten tests are run for each excitation. Four of these tests use lumped mass models, while theremaining six tests use distributed masses. The optimal contact element stiffness has already beendetermined for distributed mass modelling (Figure 4). However, a rational choice for the number ofelements in each diaphragm is still required. Additional floor elements add accuracy, but can alsoincrease computation time significantly. This is because the maximum time step is often sensitiveto the contact element’s stiffness, which is in turn affected by the number of diaphragm elements(Figure 4). The number of elements in each diaphragm affects the model configuration (Figure 10).The six distributed mass tests use 1, 2, 3, 4, 5 and 20 axial elements per diaphragm, respectively.The 20 element test is used as the benchmark for all other tests. This model was determined tosufficiently represent theoretical distributed mass behaviour by comparison with more accuratemodels in earlier testing [17]. The benchmark test uses a time step of 10−7 s while all other tests use10−6 s. These values were determined by reducing the time step until the energy calculated by theequation of dynamic equilibrium for a given configuration equalled that of the input energy fromeach input record. The time steps could have been increased incrementally for models with lesselements per diaphragm; however, in these tests the time step was held constant for consistency.

When considering lumped masses, the number of elements in each diaphragm is fixed at zero.However, traditional analysis uses only general rules to calculate the collision element stiffness (kC),which is usually described in terms of the diaphragm stiffness (kD). Three values of kC are testedfor the lumped mass models, in addition to the equivalent lumped mass formulation (Table V). Alllumped mass analysis is undertaken with the Kelvin Voight element. This is because the equivalentlumped mass formulation is not available for the Hertzdamp element, and the Hertzdamp elementis not presently available in the adopted software. Note the diaphragm axial stiffness (kD) used inTable V is taken as the larger value from Table IV.

The resulting drift envelopes for buildings 1 and 2 are presented in Figures 11 and 12, respec-tively. The drifts are presented as a percentage error from that of the benchmark solution, i.e. thesolution obtained from 20 elements per diaphragm model. The figures present two series for eachearthquake; the first series shows the displacement envelope for leftward building movement ineach inset diagram, while the second series shows the corresponding movement to the right. Anegative error value signifies that the test recorded a smaller magnitude than the benchmark test.



0 elements per diaphragm (lumped masses)

1 element per diaphragm

n elements per diaphragm 2 elements per diaphragm

Figure 10. Model configurations as defined by the number of axial elements per diaphragm.

Table V. Collision element stiffness’s for lumped mass tests.

Test name 0(10) 0(1) 0(0.1) 0(equiv)

kC kC =10kD kC = kD kC =0.1kD kC = Equation (16)

All traditional lumped mass models are found to have at least 10% error, regardless of the locationof the recorded drift. Surprisingly, the lumped masses frequently underestimate the response. Thisresult shows that a lower value of � (or equivalently, a lower value of e) does not necessarily reducethe global damage caused by pounding on a structure. The equivalent lumped mass formulation,test 0(equiv), provides a significant increase in accuracy and is approximately as accurate as theone element per diaphragm case. Drift error does exceed 10% for one excitation in the secondstorey of building 2, but all other drift recordings show marked improvement in accuracy. However,the equivalent lumped mass formulation cannot as accurately represent the opposite ends of eitherbuilding (refer to the inset diagrams in Figures 11 or 12). This is due to the diaphragm beingmodelled as perfectly rigid. The rigid diaphragm reduces the shear demands of the columns atthe collision interface as the contact loading is directly applied to all columns that are connectedto that diaphragm. The figures also illustrate that increasing the number of diaphragm elementsprogressively increases the displacement envelope accuracy. If two or more elements are used perdiaphragm, then all recorded errors are less than 5%.

The accuracy of the calculated collision forces are assessed in a different manner. The collisionforce is first evaluated for the 20 element model only (Figure 13). The vertical axis displays thepercentage error in the calculation of force using Equation (11), when compared to the contactforce determined from the recorded force history of the contact element.

All three excitations present similar collision force accuracy. El Centro was the only excitationto demonstrate collision at the first floor. First floor collisions appear to be not as accuratelypredicted by Equation (11). Figure 13 demonstrates that the accuracy of Equation (11) is affectedby the magnitude of the collision force. The drop in accuracy at lower force magnitudes can beattributed to secondary factors such as the acceleration in each floor at the onset of collision. Atsmaller collision velocities, the acceleration is much more likely to play an important role in thecontact. Overall, Equation (11) is shown to be very accurate when using the 20 element model.

The accuracy of Equation (11) is then assessed for all the test cases, as shown in Figure 14.Each test is compared to the recorded collision force from the 20 element model. The presentedresults show all collisions from the El Centro excitation only. If no collision was recorded, theerror has been reported as −100%.

Significantly more scatter is present in Figure 14. Accuracy of Equation (11) increases withincreasing model complexity. Ignoring the traditional models [0(10), 0(1) and 0(0.1)], all results>1000kN are within 15% of the accurate model recorded results, and all except two of theseresults are within 10%. Furthermore, the collision force is accurately calculated for major collisions


654 G. COLE ET AL.

Figure 11. Building 1 drift envelope error relative to the benchmark model.

(>2000kN) in all tests. Equation (11) is therefore a universally useful tool for predicting collisionforce magnitudes when two diaphragms collide.

INCORPORATING INELASTIC COLLISIONS INTO THE DISTRIBUTED MASS MODEL

This section proposes a method by which inelastic effects may be modelled in a distributed masscollision. When considering lumped masses, the coefficient of restitution represents the linearcombination of two extreme states; completely elastic collision and completely plastic collision.When two lumped masses undergo elastic collision, both momentum and kinetic energy areconserved. Using these rules, both mass’ post collision velocity can be calculated (Equation (12)).Similarly, a completely plastic collision between two lumped masses causes both masses to moveat the same velocity. This velocity can be calculated by considering conservation of momentumalone, since v′

1 =v′2;

v′1 = v1m1 +v2m2

m1 +m2=v1 − 1

1+ m1

m2

(v1 −v2) (18)



Figure 12. Building 2 drift envelope error relative to the benchmark model.

Figure 13. Calculated collision force accuracy for 20 element model. Note force error is calculated as thepercentage error between the predicted and the recorded collision force.


656 G. COLE ET AL.

Figure 14. Calculated collision force accuracy for all test cases.

Equations (12) and (18) can be combined so that when e=1, only Equation (12) contributes to thepost collision velocity. Similarly when e=0, only Equation (18) contributes to the post collisionvelocity. This is achieved by multiplying the two equations by the coefficients (e) and (1−e),respectively. Thus, the inelastic expression for the coefficient of restitution is calculated as:

v′1 = e

⎛⎜⎝v1 −2

1

1+ m1

m2

(v1 −v2)

⎞⎟⎠+(1−e)

⎛⎜⎝v1 − 1

1+ m1

m2

(v1 −v2)

⎞⎟⎠

= v1 − (1+e)

1+ m1

m2

(v1 −v2) (19)

This expression matches Equation (1) and is thus one of the equations of stereo mechanics. Nowconsider the distributed mass case. Inelasticity can be incorporated into the distributed mass modelby following the same method. A new parameter is introduced to describe the inelasticity ofthe distributed mass collision. The parameter is an index indicating the level of plasticity in adistributed mass collision. It is here termed the plasticity index, r . The fully elastic post collisionvelocities of both masses are already known (Equation (12)). The fully plastic collision causes thesame post collision velocities as the lumped mass model, by the same considerations as presentedin the previous paragraph (Equation (18)). The general distributed mass expression is thus:

v′1 = r

⎛⎜⎜⎝v1 − 2

1+ T2m1

T1m2

(v1 −v2)

⎞⎟⎟⎠+(1−r)

⎛⎜⎝v1 − 1

1+ m1

m2

(v1 −v2)

⎞⎟⎠

v′1 = v1 −

⎛⎜⎜⎝ 2r

1+ T2m1

T1m2

+ 1−r

1+ m1

m2

⎞⎟⎟⎠ (v1 −v2)

(20)



The expression for diaphragm 2 is similar:

v′2 =v2 −

⎛⎜⎜⎝ 2r

T2

T1+ m2

m1

+ 1−r

1+ m2

m1

⎞⎟⎟⎠ (v2 −v1) (21)

When r =1 the collision is completely elastic, and when r =0 the collision is completely plastic.Finally, the coefficient of restitution can be related to the plasticity index by equating Equations(19) and (20);

eeff =r

⎛⎜⎜⎝2

(m1

m2+1

)T2m1

T1m2+1

−1

⎞⎟⎟⎠=r ×[elastic eeff] (22)

Equation (22) can also be derived using the expressions for diaphragm 2. Comparison ofEquation (22) with Equation (14) shows that the plasticity index creates a linear scaling of theelastic effective coefficient of restitution. When r =0, eeff is also zero, regardless of the diaphragmproperties. The non-linear version of Figure 7 is thus the same as the existing version, but thevertical axis is scaled between r and −r .

When the level of plasticity increases, the duration of the collision also increases. This meansEquations (16) and (17) are no longer strictly valid for the equivalent lumped mass model. Whenthe plasticity index increases, collision stiffness must decrease to change the collision duration.However, the way contact duration increases with increasing plasticity is not known. In the absenceof further information, Equations (16) and (17) remain a reasonable approximation for inelasticcollision duration.

Note that the meaning of the plasticity index is similar, but not identical, to the coefficient ofrestitution. Appropriate values of r will not necessarily be the same as the commonly used valuesfor e. Recent studies have also highlighted major limitations of the coefficient of restitution [21].With these considerations in mind, values for r have not been investigated further here. Nevertheless,Equation (22) provides sufficient framework to incorporate inelastic effects into the equivalentlumped mass model.

CONCLUSIONS

Based on the works presented herein, the following conclusions are drawn:

1. A mathematical investigation is presented, which examines the differences between lumpedmass and distributed mass collisions. By making four basic assumptions, the post collisionvelocities of each approximation can be related. The differences in post collision velocityare found to be dependent on mass ratio and period ratio between the two diaphragms.Distributed mass collisions are found to always reduce the change in velocity resulting fromcollision, when compared to lumped mass models.

2. The differences in post collision velocity resulting from the choice of mass distribution can becharacterized by the change in the influence coefficient, �, between the lumped mass and thedistributed mass formulations. A larger change in � increases the significance of the choiceof mass distribution. The significance of mass distribution on a specific pounding scenariocan thus be qualitatively assessed by calculation of the change in � for each diaphragm. Thisevaluation must be performed for both buildings.

3. The influence coefficient can also be used to indicate the severity of damage that poundingmay have on a structure. As the value of � increases, the velocity change of that diaphragmdue to pounding also increases. If the value of � is equal to or near zero, then the velocitychange as a result of collision can be considered to be negligible.


658 G. COLE ET AL.

4. An ‘equivalent lumped mass model’ is derived, which produces the same collision durationand post collision velocities as the theory governing distributed mass collisions. The model canbe directly incorporated into any existing model, which employs either stereo mechanics orthe Kelvin Voight hysteresis. Using numerical analysis, this model is shown to be significantlymore accurate than existing lumped mass models at predicting the post collision velocity ofdistributed masses.

5. A numerical example is presented and found to support the mathematical findings describedin this paper. Lumped mass models produce significantly different results when compared tothe distributed mass models. Simplified distributed mass models are also tested and foundto produce agreeable drift results provided the model contains at least two elements perdiaphragm.

6. A formula for collision force is presented and numerically verified. The collision force isfound to be directly proportional to the relative velocity of the diaphragms at the onset ofcollision. The formula is found to be particularly accurate when a collision results from alarge relative velocity.

7. A framework is presented to incorporate inelastic distributed mass collisions in the equivalentlumped mass model. The method introduces a plasticity index into the calculation of theeffective coefficient of restitution. This method allows plasticity in distributed mass collisionsto be modelled using lumped mass models.

ACKNOWLEDGEMENTS

The first author acknowledges the Tertiary Education Commission and Beca, Carter, Hollings and FernerLtd. for personal financial assistance to conduct this research.

REFERENCES

1. Anagnostopoulos SA. Building pounding re-examined: how serious a problem is it? Eleventh World Conferenceon Earthquake Engineering. Elsevier Science Ltd.: Pergamon, 1996.

2. Kasai K, Jagiasi AR, Maison BF. Survey and analysis of building pounding during 1989 Loma Prieta earthquake.Eleventh World Conference on Earthquake Engineering. Elsevier Science Ltd.: Pergamon, 1996.

3. Muthukumar S, DesRoches R. A Hertz contact model with non-linear damping for pounding simulation.Earthquake Engineering and Structural Dynamics 2006; 35(7):811–828. DOI: 10.1002/ege.557.

4. Conoscente JP, Hamburger RO, Johnson JJ. Dynamic analysis of impacting structural systems. Proceedings ofthe Tenth World Conference on Earthquake Engineering. A. A. Balkema: Rotterdam, 1992; 3899–3903.

5. Papadrakakis M, Mouzakis H, Plevris N, Bitzarakis S. Lagrange multiplier solution method for pounding ofbuildings during earthquakes. Earthquake Engineering and Structural Dynamics 1991; 20(11):981–998.

6. Shakya K, Wijeyewickrema AC. Mid-column pounding of multi-story reinforced concrete buildings consideringsoil effects. Advances in Structural Engineering 2009; 12(1):71–85. DOI: 10.1260/136943309787522687.

7. Anagnostopoulos SA. Pounding of buildings in series during earthquakes. Earthquake Engineering and StructuralDynamics 1988; 16(3):443–456.

8. Chouw N. Influence of soil-structure interaction on pounding response of adjacent buildings due to near-sourceearthquakes. Japanese Society of Civil Engineers Journal of Applied Mechanics 2002; 5:543–553.

9. Goldsmith W. Impact: the Theory and Physical Behaviour of Colliding Solids. E. Arnold: London, 1960; 379.10. Anagnostopoulos SA, Spiliopoulos KV. Investigation of earthquake induced pounding between adjacent buildings.

Earthquake Engineering and Structural Dynamics 1992; 21(4):289–302.11. Jankowski R. Non-linear viscoelastic model of structural pounding. 13 WCEE: 13th World Conference on

Earthquake Engineering Conference Proceedings, Vancouver, BC, Canada, 2004.12. Anagnostopoulos SA. Equivalent viscous damping for modeling inelastic impacts in earthquake pounding problems.

Earthquake Engineering and Structural Dynamics 2004; 33(8):897–902. DOI: 10.1002/eqe.377.13. Muthukumar S, Desroches R. Evaluation of impact models for seismic pounding. 13 WCEE: 13th World

Conference on Earthquake Engineering Conference Proceedings, Vancouver, BC, Canada, 2004.14. Shakya K, Wijeyewickrema AC, Ohmachi T. Mid-column seismic pounding of reinforced concrete buildings in a

row considering effects of soil. 14th World Conference on Earthquake Engineering: Innovation Practice Safety.International Association for Earthquake Engineering, Beijing, China, 2008.

15. Zhu P, Abe M, Fujino Y. Modelling three-dimensional non-linear seismic performance of elevated bridges withemphasis on pounding of girders. Earthquake Engineering and Structural Dynamics 2002; 31(11):1891–1913.DOI: 10.1002/eqe.194.



16. Watanabe G, Kawashima K. Numereical simulation of pounding of bridge decks. 13 WCEE: 13th World Conferenceon Earthquake Engineering Conference Proceedings, Vancouver, BC, Canada, 2004.

17. Cole GL, Dhakal RP, Carr AJ, Bull DK. The effect of diaphragm wave propagation on the analysis of poundingstructures. Computational Methods in Structural Dynamics and Earthquake Engineering, Rhodes, Greece, 2009;CD200.

18. Cole GL, Dhakal RP, Carr AJ, Bull DK. The significance of lumped or distributed mass assumptions on theanalysis of pounding structures. 13th Asia Pacific Vibration Conference, Christchurch, New Zealand, 2009; Paper66.

19. Jeng V, Tzeng WL. Assessment of seismic pounding hazard for taipei city. Engineering Structures 2000;22(5):459–471.

20. Carr AJ. Volume 2: User Manual for the 2 Dimensional Version Ruaumoko 2D. Ruaumoko Manual, Vol. 2.University of Canterbury, Christchurch, 2007.

21. Jankowski R. Experimental study on earthquake-induced pounding between structural elements made of differentbuilding materials. Earthquake Engineering and Structural Dynamics 2010; 39(3):343–354. DOI: 10.1002/eqe.941.


An investigation of the effects of mass distribution on pounding structures

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