During the past few years, several studies have been conducted in various fields of civil engineering in order to design structures that can withstand the forces and deformations that might occur during seismic events. However, more recently, building adjacent structures close to each other and more resistant to earthquakes, provided with coupling systems, has been an issue of major concern. The effects of some parameters, such as the characteristics of adjacent structures and those of the coupling system, on the choice of the separation distance, were investigated using a program that was developed using MATLAB. This article aims to present a study that is intended to determine the parameters characterizing the coupling system. Moreover, the influence of rigidity of the structure was also examined. For this, three examples were investigated: a flexible structure, a rigid structure, and a very rigid structure. The results obtained from the numerical study made it possible to show that knowing the characteristics, number, and arrangements of the coupling systems can be used to find the minimum separation distance between two adjacent buildings.
Keywords
 Adjacent building
 Coupling
 Passive
 Separation distance
Over the past few years, seismic protection of adjacent buildings has been a topic of major interest for researchers throughout the world [1, 2, 3, 4]. For such systems, the effects of seismic excitations can lead to negative interactions and collisions between adjoining buildings [1, 4, 5, 6], which can cause serious damage to these structures, even if they are welldesigned and appropriately constructed.
The easiest and most effective way to limit risks and reduce the seismic pounding of adjacent buildings is to provide a sufficiently wide separation to avoid contact between them. Indeed, several studies have been conducted to determine that parameter [7, 8, 9, 10, 11].
Seismic pounding between adjacent buildings is a very complex phenomenon that could lead to the collapse of infill walls, plastic deformation and shear rupture of a column, local crushing and even possible destruction on the entire structure. It is important to know that adjacent structures with different ground levels are more vulnerable when subjected to seismic excitations due to additional shear forces acting on the columns. These forces would certainly lead to greater damage and therefore contribute to the instability of the buildings [12]. The phenomenon of seismic pounding between adjacent buildings is a quite complex issue that has been the subject of a number of studies around the world. The main concern of researchers was to understand the physical aspects of the pounding effect between adjacent structures in order to develop a rational basis that can help to mitigate the seismic risks.
Today, the greatest challenge facing civil engineers throughout the world is to design structures that are capable of withstanding the forces applied to them and therefore prevent the deformations induced by a seismic event. Researchers have found that pounding between two adjacent buildings can cause largescale damage during strong earthquakes. Consequently, increasing the rigidity of structures using infill wall panels was one of several solutions that were proposed by several researchers working in this field. It was found that these wall panels can have a significant influence on the seismic behavior of structures during seismic excitations. Indeed, they can prevent and reduce the risk of pounding between adjacent structures [13]. However, the problem with infill wall panels is that it is difficult to predict and quantify the damage caused by the frameinfill wall interaction due to complications in modeling the interaction between these structures [4]. This leads us to ignore this solution that consists in modifying the dynamic behavior of adjacent structures during an earthquake.
Furthermore, some researchers examined the effectiveness of coupling adjacent buildings (passive, active, and semiactive) [14, 15, 16, 17, 18, 19, 20, 21].
In 1999, Xu
On the other hand, Jeng
On the other hand, due to the fact that the seismic performance of buildings connected with friction dampers has not been sufficiently investigated, researchers such as Bhaskarar
Similarly, in 2014, Abbas
It was found that when a structure is subjected to a severe earthquake, all constituent elements of the structure are subjected to large deformations. Then, if the elements do not have enough ductility, they will be severely damaged and the structure could collapse. It is worth recalling that ductility, otherwise known as the strain capacity, is often used in the field of seismic engineering; it is considered as one of the most critical parameters in evaluating the seismic performance of structures. The problem with adjacent structures is that if the separation distance is not sufficient, the ductility of these structures could be larger than that required by national and international regulations [13]. In 2019, Miari
Based on previous research, we decided to do a parametric study to calculate an optimal separation distance instead of proposing some values [2, 3], to avoid hammering between adjacent structures. First, a MATLABbased software program was developed using the fundamental equations of structural dynamics in the presence of control systems, and, second, to carry out an extensive parametric study to determine the influence of the characteristics of structures and coupling systems on the selection of the separation distance between adjacent structures as well as the elements connecting them.
This section aims to present the mathematical equations that allow calculating the dynamic response of two adjacent structures. To do this, the structures are assumed to be subjected to the same acceleration; the soil–structure interaction effect is neglected. In this case, two shear buildings were considered with NOA and NOB levels, such that NOA ≤ NOB. These two structures are coupled with Maxwell dampers of stiffness
The equation of motion of structure A can be written as:
That of structure B is written like:
Combining Eqs (1) and (2), and adding the coupling system (passive, active, etc.) to find the equation of motion of the two coupled adjacent structures, helps to obtain:
It should be noted that the vector {
{X (t)} of dimension (NO × 1) represents the displacement vector of floors of the system
The mass matrix for
The stiffness matrix for
The damping matrix for
Matrices [Cd] and [Kd], of dimensions (NO × NO), are the damping and stiffness matrix, respectively, of the coupling dampers of the system.
The vector {
For
Matrix [
Eq. (3) can be written in the form of the following equation of state (Reteri and Megnounif, [32]):
{
{E} is the external disturbance vector; its dimension is (2NO × 1):
{C} is the vector related to the acceleration of the structure base; its dimension is (2NO × 1):
Based on previous theoretical developments on the topic, the present work attempts to carry out a numerical simulation using MATLAB. A program was developed for the purpose of studying the dynamic behavior of adjacent structures (columns and beams) for any number of floors, structural characteristics, and coupling systems.
It is worth indicating that the issue can be treated with and without coupling. The validation of the program is performed by comparing the results obtained with those reported by Palacios
In this work, we calculated the separation distance by the absolute sum (ABS) of the maximum displacements with coupling:
The ABS:
In accordance with the previous mathematical development, a MATLABbased program was established. The general flowchart considered is given as follows (Figure 2a). In order to validate the program for a numerical application, the dynamic loading used for the excitation of the two structures is the ElCentro 1940 earthquake with N–S components, as shown in Figure 2b.
The results obtained from our program, in the case with coupling, were compared with the results found by Palacios
The data of the model under study are presented in Table 1 [17].
Data of the second case
1  1.29 × 10^{6}  4 × 10^{9}  10^{5}  1.29 × 10^{6}  2 × 10^{9}  10^{5} 
2  1.29 × 10^{6}  4 × 10^{9}  10^{5}  1.29 × 10^{6}  2 × 10^{9}  10^{5} 
3  1.29 × 10^{6}  4 × 10^{9}  10^{5} 
The characteristics of the coupling system are given as:
Table 2 gives the results found by our simulation and those of Ref. [17]:
Maximum absolute interstory drifts (cm)







Free  By Palacios 
2.71  2.13  1.17  3.16  1.95 
Simulation  2.67  2.09  1.17  3.20  1.99  
%  4  4  0  4  4  
Passive  By Palacios 
1.65  1.32  0.72  1.81  1.10 
Simulation  1.61  1.30  0.72  1.80  1.10  
%  4  2  0  1  0 
The comparison of the results showed differences ranging from 0% to 4%, indicating very good agreement.
For the purpose of studying the performance of adjacent buildings connected by passive dampers and determining the important characteristics relating to the selection of the best separation distance between adjacent structures, a parametric study was conducted in this article on four models, as shown in Figure 3, in order to show the influence of some specific parameters, such as the characteristics and position of the coupling systems, number of floors, and variation of stiffness from one building to another, in the two examples. The soil–structure interaction was neglected and was not taken into consideration in this study.
In both cases, and for the two examples of the parametric study regarding the three models, a number of iterations were performed for all values of the stiffness ratio (
The figures on the left give the results obtained from the parametric study in order to determine the optimal value of
The separation distance between buildings was calculated according to the stiffness values
From Figures 4–7, we note that when the value of the ratio (
The results in Figures 4–7 (left) show the sensitivity of the change in the stiffness of the impact elements in the reduction of the separation distance (i.e., the reduction of the structural response). Contrary to the conclusion found in the literature [34], it was concluded that structural responses are not sensitive to changes in the stiffness of impact elements.
It can also be easily noted that the effect of the stiffness ratio (
The results in Figures 4–7 (right) of the damping rate (
Regarding the position of the coupling system, it was revealed that it is generally more efficient when placed in the upper stories of the structure.
The curves plotted in Figures 4–7 clearly show the effect of introducing the coupling system between adjacent structures. The differences between the uncoupled and coupled curves can range from important to very important, depending on the cases under study. The stiffness of buildings and the position of dampers (coupling system), for Examples 1 and 2 in both cases 1 and 2, have a significant influence on the coupling efficiency. A simple comparison between the results obtained in the two cases showed that the second case gave smaller separation distance values, which leads us to conclude that the rigidity of the structure has a more positive impact with regard to the separation distance between adjacent buildings.
The urban land scarcity and the need to build more homes have pushed decisionmakers to build taller and higherdensity buildings. Many housing structures are currently built next to each other, forming a set of adjacent buildings. These structures are usually separated by a gap that civil engineers are trying to minimize as much as possible in order to save space. The present research aimed to investigate several parameters characterizing the buildings in order to find a way in which they can relate with the coupling system used, in order to determine the smallest separation distance. For this purpose, a MATLABbased program, which was founded on a mathematical formulation, was developed. Several studies were conducted for the purpose of better understanding the impact of control systems on the seismic joint. The most important conclusions that could be drawn from this study are as follows:
When the coupling parameters, such as stiffness and damping, are varied, the stiffness has a greater influence on the separation distance than damping. In this case, the optimum stiffness value
Increasing the stiffness of the structure has a direct impact on the separation distance between structures and, consequently, on the choice of the position of the coupling system, which in turn has an effect on the joint. In the case of a single damper, it is much more interesting to place it at the top than at the bottom of the building.
The position of the coupling system depends on the height of the structure. The numerical results obtained suggest that it is possible to have a tall building adjacent to a short one without increasing the separation distance. In this case, the separation distance can be decreased even more.
Increasing the stiffness of one structure with respect to another may result in using smaller seismic joints. Additional numerical studies could be carried out to better understand and control the seismic joint systems. In this case, if one considers adding control systems to the coupling devices, then the distance between buildings can be even smaller.
Data of the second case
1  1.29 × 10^{6}  4 × 10^{9}  10^{5}  1.29 × 10^{6}  2 × 10^{9}  10^{5} 
2  1.29 × 10^{6}  4 × 10^{9}  10^{5}  1.29 × 10^{6}  2 × 10^{9}  10^{5} 
3  1.29 × 10^{6}  4 × 10^{9}  10^{5} 
Maximum absolute interstory drifts (cm)







Free  By Palacios 
2.71  2.13  1.17  3.16  1.95 
Simulation  2.67  2.09  1.17  3.20  1.99  
%  4  4  0  4  4  
Passive  By Palacios 
1.65  1.32  0.72  1.81  1.10 
Simulation  1.61  1.30  0.72  1.80  1.10  
%  4  2  0  1  0 