#### Abstract

Collisions often happen between adjacent buildings without a sufficient gap under seismic loadings. Installing polymer bumpers may decrease the pounding forces between colliding buildings. This paper focuses on the analysis of suppressing pounding effects by using of polymer bumpers. In order to investigate pounding mitigation effectiveness in the condition with and without bumpers, different types of pounding force models are utilized to simulate and calculate frame vibration responses under seismic loads. Firstly, time-history results for pounding force, displacements, and shearing forces in selected floors are calculated in the condition with and without polymer bumpers. Secondly, differences between viscoelastic and elastic pounding force models are compared and discussed. And finally, parametric studies are carried out to investigate influences of different pounding parameters. The results show that polymer bumpers can reduce maximum values of pounding force but increase pounding times. And viscoelasticity of bumper materials has a certain range of influences on pounding force responses. Choosing improper material types and pounding parameters may lead to opposite effects.

#### 1. Introduction

Insufficient gaps between adjacent buildings of unequal heights may lead to poundings under seismic effects. This can result in severe damage and even loss of lives, which have been observed in all major earthquakes [1–4]. Large plastic deformations, splintering, and the collapse of the main components of buildings, as well as the dislocation or falling of bridge girders [5], are the most common functionally disabling phenomena of structures during earthquake.

Extensive literatures about earthquake-induced pounding have appeared in the past decade. Jankowski [6] analysed the parametric pounding of two buildings with the same number of stories. Based on Jankowski’s nonlinear viscoelastic analytical model, Pratesi [7] developed a special multilink viscoelastic finite element contact model and analysed seismic pounding of a slender reinforced concrete bell tower. Shakya [8] dealt with the seismic pounding between three typical reinforced concrete moment-resisting frame buildings considering the effects of the underlying soil on the structural response.

Reducing pounding effects between adjacent structures has been studied by many means. A sheared tuned mass damper was attached to two structures to reduce both the vibration and the probability of pounding effects by Abdullah [9]. Polycarpou [10] gave an efficient methodology to illustrate the performance of soft material (like rubber) bumpers during structural poundings.

Despite the abovementioned earthquake-induced pounding studies, a suitable pounding force model to illustrate the relationship between force and displacement is a key foundation to study the influence of pounding on structures. Many pounding force models or analysis methods have been proposed to simulate interactions between colliding bodies. The linear spring model, which is based on the classical theory of impact, considers the interaction between two colliding bodies as a linear spring. This analysis method is the simplest way to avoid ignoring energy dissipation. Ruangrassamee [11] and Wang [5] utilized the linear spring model to calculate the pounding of bridge decks and adjacent components.

The nonlinear spring model, also called Hertz spring model, was initially introduced into analysis for collision processes by Goodsmith [12]. This model is extremely effective for the analysis of elastic poundings and is suitable for curved surface contacts with different curvatures. Chau [13] simplified the modelling of structures as two single-degree-of-freedom oscillators and modelled poundings between two adjacent structures by utilizing a nonlinear Hertz impact model. Pantelides [14] and Cui [15] adopted the Hertz pounding model to study the pounding effects of bridges.

The Kelvin model is a type of linear viscoelastic model that can analyse energy dissipation during collision deformation. It can describe energy dissipation, with the exception of tension stress, at the end of a pounding process. This method was initially proposed and discussed by Anagnostopoulos [16, 17]. Shakya [8], Madani [18], and Komodromos [19, 20] utilized the Kelvin model to calculate poundings between adjacent buildings.

To get more precise results in simulating pounding processes, a nonlinear viscoelastic model was first developed by Jankowski based on the Kelvin model [21], which was used to calculate poundings of multistorey buildings [6] and isolated elevated bridges. At the same time, Muthukumar [22] combined the Kelvin model and the Hertz spring model to obtain the “Hertzdamp model.” Xue [23] also proposed a viscoelastic pounding force analysis method and utilized it in a PTMD (pounding tuned mass damper) simulation.

Rubber and other similar polymers perform efficiently in shock absorption. The viscoelastic behaviour of polymer shock absorbers under impact loading needs to be sufficiently modelled to obtain accurate numerical simulations and parametric studies. This paper presents a discussion of how polymer bumpers suppress vibration-induced pounding between adjacent structures. Different sizes, mechanical properties, and gaps between two buildings are calculated. The viscoelasticity of bumpers is analysed by using an updated pounding force method proposed by our previous work. Pounding forces, accelerations, and energy dissipation responses are compared to reveal the suppressing mechanism of polymer bumpers.

#### 2. Model of Adjacent Buildings

The study in this paper focuses on pounding between a high-rise building and a low-rise building, which is a very common form of construction for commercial and residential use. Figure 1 is the dynamic behaviour analysis model with which each storey’s mass is lumped on the floor level. The left building is a fifteen-storey concrete frame, and the right building is an eight-storey building. They have the same storey height, stiffness, and damping. Masses, stiffness, damping ratios, and the gap between the buildings are listed in Table 1.

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#### 3. Modelling of Pounding Force

The pounding force between colliding structures is usually modelled by using elastic or viscoelastic impact elements which become active when contact is detected. Here are three types of pounding force models to construct the interaction force relationship between collision bodies or bumpers. For structure pounding between two buildings without bumpers, Jankowski’s model is used to calculate pounding responses. This is because a number of studies have verified its effectiveness. For the polymer bumper’s viscoelastic properties, elastic and viscoelastic analysis models are introduced to verify effectiveness of the bumper and reveal influences of viscoelasticity.

##### 3.1. Concrete Pounding Force Model

To remove nonexisting tension stress in the Kelvin model, Jankowski [6] constructed a segmental function to avoid the tension stress at the end of the restitution period. According to the model, the value of pounding force during contact between the th () storey of two buildings can be calculated aswhere is the relative displacement, is the initial separation gap between buildings, is the relative velocity between colliding th storeys, is the impact stiffness parameter, and is the impact element’s damping. at any instant of time can be obtained from the following formula:where and are the mass of the th storey of the left and the right buildings, respectively, and is the impact damping ratio related to the coefficient of restitution, , which can be defined as [22]

The interaction between components during collision is a process of energy dissipation and momentum exchange, a process which may induce large pounding forces when collision occurs in a very short time interval. In this present study, pounding forces between two pounding concrete frames will make use of Jankowski’s model, whose effectiveness has been proven by Raheem [24] and Mahmoud [25]. The following values of the nonlinear viscoelastic pounding force model’s parameters have been applied: and [6].

##### 3.2. Polymer Pounding Force Model

Pounding force, as an additional effect, can be utilized as a controlling force to reduce structure vibrations. Xue [23] validated that the energy dissipation can be enhanced by viscoelastic materials, which are squeezed between collision components.

Two types of energy dissipation devices with elastic and viscoelastic materials, respectively, were used in the simulation between two adjacent buildings during an earthquake (see Figure 2).

In the analysis, the pounding force in the elastic model can be calculated aswhere is the height of the polymer block, is the surface area of the polymer block, is the relative displacement, and is the modulus of elasticity for the polymer block, for 3-parameters solid, . In the constitute model of viscoelasticity which is shown in Figure 1(b), and are spring part parameters of elasticity with unit Pa, and is dash-port part of the viscous parameter with unit . represents two springs and in series relation, in which damper is ignored and is zero.

The pounding force with the viscoelastic model can be calculated aswhere and are no longer differential operators, but instead polynomials with parameter through the Laplace transformation. is Poisson’s ratio of the viscoelastic material. The original parameters of the elastic and viscoelastic models of pounding force have been incorporated into the study: , , , and . The initial separation gap between buildings has been set to .

#### 4. Dynamic Equations of Motion

As an example, the study presented in this paper is focused on pounding between a fifteen-storey building on the left and an eight-storey building on the right. The dynamic equation of motion for such a structural model, including pounding between buildings at each floor level, can be written aswhere , and , represent the matrices of masses and damping coefficients for the left and right buildings, respectively; and are vectors consisting of the system inelastic resisting forces; , , and , , denote the displacement, velocity, and acceleration vectors for the left and right structures, respectively; and are the pounding force vectors for the left and right structures, respectively; is the vector of ground motion acceleration.

Let , , and , , be the masses, the viscous damping coefficients, and the inelastic storey shear forces for the left and the right buildings, respectively. Then, the matrices and vectors of Equation (6) can be defined as

Plastic effects of columns bearing shear forces are also considered. During the elastic stage, and take the forms and ; during the plastic stage, and , where , and , are the storey initial stiffness coefficients and yield forces for the left and the right buildings, respectively; , , and , , are the displacement, velocity, and acceleration of the left and right structures, respectively.

The KOBE earthquake wave, which was recorded in 1995, was utilized as seismic excitation. The numerical model has been built using the MATLAB SIMULINK software.

#### 5. Analysis and Simulation Results

Three types of analysis are calculated: structure without bumper, structure with bumper analysed by the elastic model, and structure with bumper analysed by the viscoelastic model. In the first part of the results, time-history responses are simulated to show influences of pounding suppression. Pounding forces, displacements, accelerations, and energy dissipations are illustrated separately. In the second part, parametric studies are carried out to reveal influences of bumpers. In last part, comparisons between the viscoelastic model and the elastic model are made.

##### 5.1. Time-History Response under KOBE Seismic Waves

Figure 3 shows pounding force time-histories with three types of analysis for pounding force under KOBE wave ground motions. Consequently, the maximum pounding force with polymer bumpers using the viscoelastic and elastic models has a significant decrease compared to the model without the polymer bumper. For an example, maximum pounding force in the 8th floor changes from 7.87×10^{6 }N to 4.78×10^{6 }N after installing bumpers. The reduction magnitude is 39.3%. On the other side, pounding times increase after installing bumpers, which is from 8 times to 42 times. Moreover, it has been observed that the pounding forces at lower storeys show smaller values compared to the value at higher storey levels. However, the polymer block between structures reduces the distance of the initial separation gap between buildings. This leads to the increase in the number of impacts when compared to the number of impacts for the case without the polymer block between buildings.

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The displacement time-histories with three types of analysis for collision under the KOBE earthquake record are shown in Figure 4. Compared with the displacement responses of the elastic model with polymer bumpers, similar results have been obtained for the viscoelastic model in the left and right buildings. The displacement responses with polymer bumpers are considerably lower than cases without bumpers in the left building, but they are higher than the pounding force models without polymer bumpers in the right building.

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Shearing force time-histories with three types of analysis for colliding buildings are shown in Figure 5. It can be seen from the figure that interactions between adjacent structures incorporating polymer bumpers result in a decrease in the observed shearing force for the storeys in the left building. Moreover, the results shown in Figure 5 indicate that collisions may lead to yielding in the case for the lower storeys of both the left and right buildings. It can also be observed that reaching threshold times of shearing forces are reduced for cases installing bumpers. For an example, after installing bumpers, times of shearing force researching the limit reduce from 6 to 1 in the first storey, which means polymer bumpers can decrease column damages and protect structure under seismic loads.

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##### 5.2. Comparison Study between Elastic and Viscoelastic Pounding Models

In the previous calculation results, the elastic pounding force model and the viscoelastic model have similar responses during the pounding process. This is because viscoelasticity is not significant for previously selected viscoelasticity parameters. The calculation in this section will illustrate differences between the viscoelastic model and elastic model.

###### 5.2.1. Pounding Force Time-History Results

Figure 6 shows the pounding force time-histories during one pounding process between buildings using polymer bumpers with elastic and viscoelastic models, in which the value of dash-pot parameter is and , respectively. The following parameters for the elastic and viscoelastic models of pounding force have been incorporated in this section: Pa, Pa, and Pa. The results indicate that the pounding force using the polymer bumper with the elastic model is smaller than the viscoelastic model. However, the duration of the collision process with the viscoelastic model is shorter than the duration of the collision process with the elastic model. Moreover, using the polymer bumper with the viscoelastic model decreases the pounding time between buildings during the whole earthquake, compared to using the polymer bumper with the elastic model. For example, the times of pounding between structures using polymer bumpers with the elastic and viscoelastic models during the whole earthquake are 51 and 47 times, respectively.

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###### 5.2.2. Displacement Time-History

The comparison of displacement results between the elastic and viscoelastic models with different dash-pot parameters between buildings is presented in Figure 7, where the values of , , , and are , Pa, Pa, and Pa, respectively. Different from Figure 4, considering viscoelasticity of bumper materials shows certain differences with results obtained by the elastic analysis model. These suggest that structural collision responses during earthquakes, including the collision displacements and the collision duration time, are affected by viscoelastic properties of bumpers. This means that although the elastic model can obtain enough precise results in most cases, viscoelasticity of bumper materials will affect analysis results in certain circumstances. The effect of the various parameters on the collision response behaviour is investigated in following sections.

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##### 5.3. Parametric Study for Viscoelastic Material Properties

In this section, different material properties between structures are applied to conduct the parametric analysis. This is done in order to study the influences of the parameters and to obtain an optimal viscoelastic parameter setting for the pounding force model.

To obtain the results of the influence of different viscoelastic materials between adjacent buildings, a series of materials with different values for equivalent elastic modulus, defined as in Equation (4), is presented in Table 2 [26].

The pounding force time-histories for five material types between two unequal height buildings under the KOBE earthquake record are presented in Figure 8. It can be seen from Figure 9 that the pounding force increases with the increasing of the equivalent elastic modulus for materials for all storeys in the buildings. However, the pounding times decrease with the increase of the equivalent elastic modulus for materials during the collision. For example, the collision times between adjacent buildings at the storey with the viscoelastic material of types A, B, C, D, and E are 71, 54, 53, 48, and 46, respectively.

Figure 10 shows the pounding force influenced by the parameter for the different material types presented in Table 2. For the polymer bumper with material types A, B, and C, the peak values of pounding force decrease during one collision process when improving the value of . In contrast, for the polymer bumper with type D material, which has a larger value, the peak values of pounding force increase during one collision process when improving the value of . The value of pounding force also has a sudden increase at the beginning of the collision.

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##### 5.4. Parametric Study for Pounding Geometries Sizes

###### 5.4.1. Different Gaps between Adjacent Structures

To obtain the results of the influence of gaps between adjacent buildings, four different values of seismic gap are used to calculate the responses of structures under the earthquake-induced record. The following parameters for the viscoelastic model of pounding force have been incorporated in this section: , Pa, and Pa.

The pounding force time-histories between buildings with different seismic gaps are shown in Figure 11. It is clear that the pounding times decrease with the increase of the seismic gap values. The peak values of pounding force between structures with different gaps in Figure 11 for the storey of the building are , , , and , respectively. This means that larger gaps cannot always decrease the pounding force.

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Figure 12 shows the pounding force time-history with different gaps between buildings during one collision process at about 40 s in the 4th storey and the 8th storey. Among the four different gaps in Figure 12, the pounding force between structures with is higher than the others. Moreover, the duration of pounding also changes with seismic gaps. Figure 13 is displacement time-history with different gaps between buildings. It can be seen that there is a vibration reduction effect on the displacement response for the storeys in the right building.

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###### 5.4.2. Different Sizes of Bumpers

The performance of polymer bumpers with different radii is researched in the paper. Four different radii of polymer bumpers are used in the simulation. Figure 9 illustrated that the value of pounding force increases with increasing radius of the polymer bumper. This means the larger area polymer bumper has the better carrying capacity. On the other hand, pounding times has no significant differences as size changes.

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The effect of polymer bumpers with different types of materials and radii is presented in Figure 14. All four groups of curves show that the larger polymer bumper can reduce the collision time during one collision process.

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#### 6. Conclusions

In this paper, pounding mitigation effectiveness for polymer bumpers is investigated in adjacent buildings, and the viscoelastic and elastic pounding force models are used to calculate the pounding responses. The following conclusions can be obtained:(1)During the pounding process, polymer bumpers can reduce pounding forces and shearing forces between two buildings. For the 8th storey of building, maximum pounding force reduces from 7.87 × 10^{6} N to 4.78 × 10^{6} N after installing bumpers. For the 1st storey, times of shearing force researching the limit reduce from 6 to 1. This can decrease column damages and protect structure under seismic loads.(2)Dynamic responses of adjacent building under Kobe seismic wave are calculated by using of the viscoelastic pounding force model and the elastic pounding force model. The comparison result shows that only a small difference (less than 1%) appears in pounding simulation results between these two types of analysis pounding force models. The variation degree of influence depends on the pounding force model parameter values.(3)The viscoelastic properties of polymer bumpers are a primary factor in influencing the performance of pounding suppression. A larger equivalent modulus elasticity value will induce the opposite effect on pounding reduction.(4)Gaps between buildings and sizes of bumpers are also key factors that influence the pounding responses between buildings. Small gaps may increase the pounding force but decrease the displacement. Larger bumpers can decrease the pounding time but increase pounding forces.

In a word, installing polymer bumpers is an effective way to reduce the pounding responses between adjacent buildings with insufficient gaps. It is necessary to choose the proper parameters for the polymer bumpers to ensure their effectiveness. Simulation results will vary with the viscoelasticity of the bumpers.

#### Data Availability

The properties of bumpers (shown in Table 2) used to support the findings of this study have been deposited on Wiley online library (https://doi.org/10.1002/eqe.2194). Constitutive parameter values of viscoelastic material used to support the findings of this study have been deposited to Hindawi repository (https://doi.org/10.1155/2016/2596923). No other raw data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The work was supported by the National Natural Science Foundation of China (Project nos. 51409056 and 51678322), the Fundamental Research Funds for the Central Universities (Project no. HEUCF180204), and the Taishan Scholar Priority Discipline Talent Group Program funded by the Shandong Province.