Abstract
Seismic excitation, which results in large horizontal relative displacements, may cause collisions between two adjacent structures due to insufficient separation distance between them. Such collisions, known as earthquake-induced structural pounding, may induce severe damage. In this paper, the case of pounding between two adjacent buildings is studied by the application of single degree-of-freedom structural models. Impact is numerically simulated with the use of a nonlinear viscoelastic model. Special attention is focused on calculating values of impact forces during collisions which have significant influence of pounding-involved response under ground motions. The results of the study indicate that the impact force time history is much dependent on the earthquake excitation analyzed. Moreover, the peak impact forces during collision depend substantially on such parameters as gap size, coefficient of restitution, impact velocity, and stiffness of impact spring element. The nonlinear viscoelastic model of impact force with the considered relation between the damping coefficient and the coefficient of restitution has also been found to be effective in simulating earthquake-induced structural pounding.
1. Introduction
During ground motions, buildings often collide with each other due to different dynamic characteristics, insufficient gap between them, and out-of-phase vibrations [1]. The phenomenon related to such collisions is often called the earthquake-induced structural pounding. Pounding is expressed by an instance of rapid strong pulsation, which may cause severe damage [2]. Consequently, the probability of structural interactions during earthquakes must be assessed. A number of researchers have studied the problem of pounding for different structural configurations under various ground motions:(i)Anagnostopoulos [3] was among the first scientists who investigated the earthquake-induced structural pounding by analyzing interactions between buildings in a row. He also described in detail the threats related to such collisions during ground motions [4].(ii)Maison and Kasai [5] presented a formulation and simulated the multi degree-of-freedom equations of motion for floor-to-floor pounding between two 15-storey and 8-storey buildings. The influence of building separation, relative mass, and contact location properties was investigated.(iii)Jankowski [6, 7] and Mahmoud et al. [8, 9] carried out a number of studies concerning structural pounding, related to both experimental and numerical aspects.(iv)Komodromos et al. [10–12] studied pounding between seismically isolated buildings.(v)Barros and Khatami [13] examined a new model of impact to coordinate of results among numerical and experimental studies. They also suggested an approximate trend to select coefficient of restitution, which becomes equal to impact velocity [14]. Subsequently, a new equation of motion was suggested to simulate impact and figure damping terms out of collision [15].
Furthermore, some more recent numerical analyses have been carried out to study the influence of different parameters in pounding of buildings [16–20].
Nevertheless, there is still a need to investigate different models of structural pounding so as to verify their accuracy in the case of different configurations under different earthquake excitations. This concerns especially the values of impact forces during collisions which are often not studied in the analyses (or the analyses are simplified) since the investigations are rather focused on pounding-involved response under ground motions.
2. Contact Element Methods
The contact element is a special element (usually consisting of a spring and damper) to model impact between two colliding structures, which is widely used to simulate impact force. Impact is parametrically modelled in this way that when relative displacement exceeds the separation distance, the contact element is activated. The general formula for the impact force during collision can be expressed as follows (see [21]):where is stiffness of spring, denotes damping of dashpot, and and describe lateral displacement and velocity, respectively. The power of has been recommended to be 1 or 1.5 depending on the model considered. The damping coefficient is usually related to the coefficient of restitution which is defined as the ratio between the postimpact velocity and the prior-impact velocity [21]:
Anagnostopoulos. The linear viscoelastic model of impact force (with ) was considered by Anagnostopoulos [3]. The following formulae were suggested [3, 22]:where are the masses of colliding structures. The impact force versus time and lateral displacement for the linear viscoelastic model is shown in Figure 1.
Jankowski. Jankowski [6] considered the nonlinear viscoelastic model when . He proposed activating the damping term only during the approach period, in which most of the energy is lost. Therefore, the following formulae were considered [6]:It was suggested that the appropriate value of impact damping ratio for a specified value of can be obtained numerically through iterative simulations in order to satisfy the relation between the postimpact and the prior-impact velocities defined by (2) [6]. The following formula was also proposed [9]:An alternative approach might also be needed to consider the relation between the damping coefficient and the coefficient of restitution in the following form:where and are the parameters of the model obtained by fitting the experimental data using the method of the least squares (determined in this way typical values are ). As it can be seen from (7), the impact damping coefficient depends directly on the prior-impact velocity value, which can be obtained for each impact separately. The impact force versus time and lateral displacement for the nonlinear viscoelastic model is shown in Figure 2.
Muthukumar and DesRoches. A different model (Hertzdamp model) was considered by Muthukumar and DesRoches [23]. They presented a formula for the impact force during collision with , which depends on three parameters, including coefficient of restitution, stiffness of spring, and prior-impact velocity, as The impact force versus time and lateral displacement for the Hertzdamp model is shown in Figure 3.
Ye et al. Ye et al. [24] claimed that (9) is incorrect to calculate the damping ratio for the pounding simulation in order to evaluate impact between two buildings. They numerically indicated that the appropriate relation is [24]The impact force versus time and lateral displacement for the modified Hertzdamp model is shown in Figure 4.
3. Numerical Study
A parametric study has been conducted in order to verify the effectiveness of the nonlinear viscoelastic model of impact force during structural pounding described by (4) and (7). Two single degree-of-freedom systems (see Figure 5) with the gap size of cm have been used to model the behaviour of adjacent structures under ground motions.
The dynamic equation of motion for such a model can be expressed as [25]where are the horizontal displacement, velocity, acceleration, damping coefficient, and stiffness coefficient for structure , respectively, stands for the acceleration input ground motion, and is the pounding force which is equal to zero when and is defined by (4), when , where is defined asIn the numerical analysis, stiffness of each structure has been taken to be and storey masses have been assumed to be tons and tons, respectively. The structural damping ratio of 0.05 has been considered in the analysis.
The dynamic analyses under the Parkfield (1966), San Fernando (1971), Kobe (1995), and El Centro (1940) earthquake records have been performed. These records have different contents of the excitation frequencies, different magnitude of the accelerations, and different time durations. Besides, their place of occurrence and geological conditions close to the epicentre are distinct. San Fernando earthquake had the highest Peak Ground Acceleration (PGA) among the four records discussed. The PGA of the earthquake amounted to 1.164 g, with an epicentre distance less than 12 km. The PGA of the Kobe earthquake was 0.7105 g and it was measured at a distance of 18.3 km. The PGA of the Parkfield earthquake was equal to 0.462 g (measured at a distance of 32 km). Finally, the PGA of the El Centro earthquake was equal to 0.347 g. All mentioned records have been normalized to investigate the effect of earthquake properties on pounding-involved structural response. The examples of the results of the numerical analysis in the form of the lateral displacement time histories under different earthquakes are shown in Figure 6. Additionally, the examples of the impact force time histories for the Kobe and the El Centro earthquakes are presented in Figure 7.
(a) El Centro
(b) Kobe
(c) San Fernando
(d) Parkfield
Using four different earthquake excitations, the peak lateral displacements, velocities, and accelerations have also been calculated for different structural periods of colliding structures. The results of the analyses are presented in Figure 8. They indicate that, with the increase in the structural period, the peak lateral displacements show the nonuniform increase trend. Among the ground motions analyzed, the Kobe record gives the maximum lateral peak displacement equal to 3.87 cm, while the minimum peak displacement of about 0.64 cm has been observed for the Parkfield earthquake. In the case of velocity, when the structural period is increased, the peak velocities are nearly the same in the range of 1–4 sec and after that the curves show a sudden decrease trend. Slightly different trend is observed for the El Centro record which demonstrates a slight increase from 5.1 m/s to 7 m/s and, subsequently, shows a sharp decline to 2.64 m/s when the structural period changes its value from 1 to 9 sec. Finally, the peak acceleration curves are quite stable at the beginning of analyzed range of structural period and then they show a substantial increase trend.
3.1. Effect of Gap Size
In order to investigate the effect of separation distance between structures, a gap size has been varied from 0 to 8 cm. Figure 9 shows the effect of separation distance on the peak impact force under four different earthquake records. It can be seen from the figure that the curves follow an irregular decrease trend when the gap size increases. In the case of the San Fernando and Kobe ground motions, a sudden decrease is observed after passing a specific gap size value, while a slight declining tendency is visible for two other earthquake records.
3.2. Effect of Coefficient of Restitution
Different values of coefficient of restitution have been considered to investigate the impact forces between structures under different earthquakes. The results of the investigation showing the peak values of impact forces are presented in Figure 10. Similar trend can be observed for all analyzed excitations. The results show a uniform decrease in the force when the coefficient of restitution increases. For instance, the peak impact force for the Kobe earthquake is equal to 341 kN and 92 kN for and , respectively.
3.3. Effect of Impact Velocity
In order to obtain the responses and compare the results of peak impact forces, different values of impact velocity have been considered from the range 1–25 m/s. The relations between the peak impact force and impact velocity values under different earthquakes are presented in Figure 11. The results show a uniform increase in the peak impact forces with the increase in the impact velocity. The peak impact forces are nearly equal to zero for the velocity of 1 m/s and are as large as 1725 kN, 1578 kN, 1560 kN, and 1405 kN for the 25 m/s impact velocity under the El Centro, Kobe, San Fernando, and Parkfield earthquake, respectively.
3.4. Effect of Stiffness of Impact Spring
Stiffness of impact spring element is considered to be one of the most important parameters when the impact force during collision is calculated. The results of the parametric study showing the peak values of impact forces with respect to stiffness of spring are presented in Figure 12. It can be seen from the figure that the trend for all earthquakes analyzed is similar. The impact forces show nearly linear increase from 210 kN, 214 kN, 273 kN, and 352 kN to 2122 kN, 2242 kN, 2625 kN, and 3342 kN for the Parkfield, San Fernando, Kobe, and El Centro earthquake, respectively.
4. Conclusions
In this paper, earthquake-induced pounding between two adjacent buildings has been studied by the application of single degree-of-freedom structural models. Impact has been numerically simulated with the use of a nonlinear viscoelastic model. Special attention has been focused on calculating values of impact forces during collisions which have significant influence of pounding-involved response under ground motions.
The results of the study indicate that the impact force time history depends substantially on the earthquake excitation analyzed. Moreover, the peak impact force during collision is much dependent on such parameters as gap size, coefficient of restitution, impact velocity, and stiffness of impact spring element. The nonlinear viscoelastic model of impact force with the considered relation between the damping coefficient and the coefficient of restitution has also been found to be effective in simulating pounding between structures during seismic excitations.
The conclusions of the study can be very valuable for the purposes of accurate modelling the phenomenon of earthquake-induced structural pounding. This concerns especially the issue of determination of the precise values of impact forces during collisions which are often not studied in the analyses (or the analyses are simplified) since the investigations are rather focused on pounding-involved response under ground motions. It can be considered as the most significant element of the analysis described in this paper, as compared to other relevant research studies.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.