Estimating the seismic pounding force between adjacent buildings and study the effect of gap distance on seismic pounding

Insufficient separation distance between adjacent buildings may lead to their pounding during strong earthquakes causing damage or local failure. Accordingly, most of the design codes specify a minimum separation or gap distance between adjacent buildings to prevent their seismic pounding. Main objectives of this research are studying the seismic pounding between adjacent reinforced concrete (RC) buildings under various ground motions and studying the effect of gap distance on seismic pounding. The studied buildings are modeled as three-dimensional frames to idealize the adjacent buildings. This study considers adjacent frame buildings having the same number of floors as well as frames having a different number of floors. Nonlinear finite element time history analysis is performed using ETABS commercial software. Three different real earthquake records with different characteristics are scaled and used to simulate ground motion. Nonlinear gap element is used to model pounding between buildings at their interface points. The results of the study indicate that, the maximum pounding force was investigated for each of the studied cases. Pounding does not occur between buildings having equal height. Increasing separation distance between buildings reduces the pounding force between them. Separation distance, which is estimated by the Egyptian code, is conservative.


Introduction
Usually, a single structure is the focus of a seismic design, neglecting any nearby ones that could exist. In the meanwhile, numerous studies demonstrate that pounding between adjacent structures during earthquakes might provide an unexpectedly higher impact force (Elwardany et al., 2019;Miari et al., 2020;Rezaei et al., 2020).
During the earthquake, there are many types of failures and damage that may occur to the building. Some are due to design errors and others are due to external factors that have not be taken into account in design such as pounding between adjacent structures (Abdel Raheem et al., 2018;Elwardany et al., 2017;Naeej et al., 2019). Structures with large plan dimensions, as well as those with parts of a different number of floors, might be required to be separated into two or more parts by expansion joints or separation distances. If the separation distance, during an earthquake, is not sufficient to accommodate the relative lateral deformations of the adjacent buildings, they might come into contacts or collide with each other. This collision between adjacent buildings during earthquake is referred to as seismic pounding (Shrestha & Hao, 2018). This pounding may cause local damage, significant damage or even total collapse of the adjacent buildings depending on the earthquake strength such as buildings in Christchurch CBD as shown in Fig. 1 (Khatami et al., 2019).
There are several types of structural pounding, such as floor-to-floor (Cole et al., 2010;Kazemi et al., 2020), floor-to-column (Favvata, 2017;Karayannis & Favvata, 2005;Kazemi et al., 2018), unequal weight of adjacent pounding (Kazemi et al., 2020;Mohebi et al., 2018), and eccentric or non-eccentric (Polycarpou et al., 2014). In specific, Karayannis and Favvata (2005) studied interactions between the slab of the shorter building and the columns of the taller structure in adjacent reinforced concrete structures of varied story heights. The shear strength in the columns may be exceeded as a result of such collisions because the shear forces of the columns at and above the collision level increase. In addition, the building's available ductility may also be exceeded as a result of the pounding that the affected columns are experiencing.

Background
Many analytical and experimental research works have been carried out to study the seismic pounding between adjacent buildings. Adjacent buildings that have different dynamic properties, such as time period, mass, rigidity and geometry, vibrate out of phase. These vibrations can be strong enough to create forces that affect the structural response of the buildings. Observations after many earthquake indicates that local damage at the interaction point is due to insufficient separation between buildings. Therefore, major efforts have been directed toward the problem of seismic analysis of buildings using different modeling techniques (Mohamed et al., 2021).

Different modeling of pounding phenomenon
This section explains the different models used to simulate impact between adjacent buildings. The distance between buildings (gap) is the significant element in pounding model because it affects the pounding force and level of damage. Adjacent buildings are modeled using masses and the pounding between these mass is simulated by contact or gap element. This gap element is activated when masses collides and deactivated when masses are apart. For example of this model that discussed are a stereo-mechanical model, linear spring model, Kelvin-Voight element model, and Analytical solution as explained later in this section.
Three effective parameter differences between pounding models are: • Damper which is responsible for energy dissipation. • Effect of non-linearity.
• Stiffness of gap which is dependent on the collision material.

Stereo-mechanical model
Stereo-mechanical model is used the final velocity of impacting bodies which is based on their initial velocity. It also considered the effect of material properties of the masses through the coefficient of restitution as simulated by Goldsmith (1961). The coefficient of restitution value can be obtained from any material by dropping test which threw sphere formed by any material from height (h) then measure the rebound height in opposite direction to get (h*). The value of coefficient of restitution (e) ranges between 0 and 1. When the (e) convergences to (0) it indicates plastic collision and when it convergences to (1) it refers to elastic collision as shown in Eqs.
(1)-(3). Furthermore, for multiple degrees of freedom system when several colliding expected at various time, the application of the Stereo-mechanicsbased model is seen as infeasible (Jankowski, 2005).
where υ' 1 , υ' 2 are the velocities of the colliding bodies (m 1 , m 2 ) after impact and (υ 1 , υ 2 ) are the velocities before impact and (e) is the coefficient of restitution.

Linear spring model
A linear elastic spring was used to simulate the contact element between adjacent structures this contact element called gap element when its stiffness depends on the axial stiffness of the collided elements of the structures as used by Maison and Kasai (1990) as shown in Fig. 2. When buildings vibrate out of phase, the relative displacement change by their motion and the spring begins to observe the force when the initial gap between the structures is less than relative gap between them. Force on contact or gap element can be calculated according to Eqs. (4) and (5).
where u 1 and u 2 are the displacements of the impacting masses during oscillator (pounding), k 1 is the spring stiffness

Kelvin-Voight element model
The significant addition in this model is the damper working parallel to linear spring as show in Fig. 3. The importance of the damper is that, it takes the effect of energy dissipation as used by Anagnostopoulos (1988).
The second term, taking into consideration the energy dissipation during mass vibration is taken into consideration by the expressed Eqs. (6) and (7).
where u 1 , u 2 and its derivatives are the displacements and velocities of the impacting bodies, k k is the spring constant of the element, and g p is the initial separation distance between the structures. The damping coefficient can be related to the damping ratio (ζ) which is related to coefficient of restitution (e) that is expressed by Eqs. (8) and (9).

Analytical solution
To obtain pounding force in spring, it must be added to the general equation of motion (Ks) was expressed by Eq. (10).
where x is the displacement vector for all degree of freedom, ẋ is the velocity vector for all degree of freedom, ẍ is the acceleration vector for all degree of freedom, m is the mass matrix for structure degree of freedom, c is the damping matrix estimated by Rayleigh equation, k is the stiffness matrix for structure degree of freedom, ẍg is the input ground motion, and k s is the stiffness matrix for gap element.
When colliding occurs between two structures, each of them estimated by Eqs. (11) and (12) according to gap element under compression or tension force as shown in Fig. 4 (Mate et al., 2014).   where x 1 and its derivatives ẋ1 and ẍ1 are displacement, velocity and acceleration for structure (1) respectively while x 2 and it derivatives ẋ2 and ẍ2 are displacement, velocity and acceleration for structure (2) respectively, and k s is spring stiffness.

Code requirement to avoid pounding
Most of the code calculates the gap distance based on the maximum displacement of the adjacent structure. Then different techniques are used to calculate the gap distance, some of the techniques are square root of sum of squares (SRSS) and complete quadratic combination (CQC). Table 1 summarizes the different equation to calculate the gap according to different international codes.

Brief description of the study buildings
The buildings in this study were selected as two-dimensional building with multi-floor and multi-bay reinforced concrete (RC) structure. Figure 5 shows two adjacent buildings with length of 18 m and 36 m were used in this study. Floor height of the buildings is equal 3.0 m. In a building with an 18 m length, four models were constructed with 3, 6, 8, and 12 floors, while two models were constructed with 6 and 12 floors in a building with a 36 m length. Buildings are symmetric along its axes and thus torsional effect was not considered in analysis accordingly. Buildings were considered fixed in their connection with the foundation. A 3D finite element is constructed for each one of the studied structures separately. An available software package known as ETABS was utilized for the modeling and analysis. Table 2 shows the list of study cases for the buildings. Sum of structure displacement and multiple it by R (response reduction factor) National Building Code (NBC: E030, 2003) Take sufficient gap between structures as sum of max distance of two structure times to 2/3

Materials properties
Characteristic compressive strength of reinforced concrete (RC) is 25 N/mm 2 , Yield stress of reinforced steel is 360 N/ mm 2 . Table 3 summaries the different properties of the materials used in the design of selected buildings. Young's modulus estimated according to Egyptian code by Eq. (13).
where F cu is the characteristic compressive strength of reinforced concrete.

Properties of gap element
Separation distance between adjacent buildings was modeled with gap element using ETABS program as shown in Fig. 6. This element is a compressive force only transmitted when buildings are in contact while zero in tension or when the relative displacement between the adjacent elements is less than the initial separation distance. The gap element is non-linear element has two significant parameters, namely the stiffness and the opening. Gap element force deformation relation is given by the following Eq. (14).
where k g is the spring stiffness of the contact element, d is the building displacement, and open is the separation or initial gap which must be zero or positive, the following criteria was considered for the gap element • Stiffness: take gap stiffness as axial stiffness of colliding slabs for two adjacent buildings using Kelvin-Voight method K = 2 × 10 6 KN/m (Maison & Kasai, 1992). • Opening: is a separation distance between two adjacent structures. In this study, opening was assumed 5 mm, 10 mm, 20 mm, and 40 mm.

Vertical Loads
Vertical loads included own weight of the different elements which is calculated by program. It includes also flooring cover as well as live load of the occupancies. Table 4 summarizes the vertical loads value utilized through this study.

Seismic loads
To get the structural response of the structure during and after seismic load excitation time history need to be carried out. In this study, three seismic records are utilized. These records are Friuli, Newhall, Sylmar earthquakes which selected to present low, moderate and high ground motion. The maximum ground accelerations for each record are 0.31 g, 0.6 g, and 0.76 g, respectively. The acceleration records of the selected ground motion are shown in Figs. 7, 8, and 9 for Friuli, Newhall, and Sylmar, respectively. Input earthquake records have a relationship between ground motion and time. The records are obtained from seismologist    1 3 records. The condition of ECP codes the artificial less than or equal three earthquakes. If exposed, the structure to earthquake such as Friuli ground motion, in which max design acceleration is by 0.31 g, should be scaled to ECP code max acceleration 0.15 g. Table 5 shows modeling assumptions for seismic design.

Design of buildings
According to the requirements of the Egyptian code, studied buildings were designed. The following assumptions were considered: • Ritz vector is used in dynamic analysis. • Damping ratio was 5%.
• The p-delta effect was ignored. • All beams are well designed due to moment of max beam but each column design alone according to number of stories.  Type of soil C R Reduction factor 5 a g Ground motion 0.15 (a g ) γ Importance factor of building 1.0 C t coefficient based on structure system 0.05 • Effective inertia of beams and columns are 0.5I g and 0.7I c respectively. Table 6 summarized the dimensions and reinforcements of columns and beams from structure design.

Model analysis
Structures are simulated using finite element program (ETABS) to obtain straining action and check safety of section. This section exhibits properties of model, such as time period of structure and mass participation in models, because ECP mentioned as in seismic analysis mass participation must exceed 90% of total mass. Table 7 presents time periods for first four models for each buildings. Table 8 summarized the result for calculated mass participation factor.

Colliding model
Model in this study assumed collision between structures (slab to slab), which linked (tied) at each floor by gap element in structures with equal height. But in unequal height, building floor linked at the shorter floor. Another floor in the tall building is free as shown in Fig. 10. Colliding occurs between two adjacent structures when the stress on spring is a higher value than tensile spring resistance.

Pounding verification model
To make sure this model is valid to use, check under another program such as SEISMOSTRUCT software

Sufficient gap according to the Egyptian code
This section discusses calculation of sufficient according to Egyptian code which accommodates relative displacement and input Newhall and Sylmar earthquake ground motion for all cases. Tables 9 and 10 show the sufficient gap according to Egyptian code under Newhall and Sylmar ground motion, respectively. Figure 11 shows that, pounding force did not occur at the 50%, 75% and 100% of sufficient separation distance in all cases but at 25% of sufficient separation distance pounding occur at case 1 and case 4 as value of pounding force 279 KN and 181 KN, respectively. All cases where no collision occurred when the distance between buildings (separation distance) became 50% of the value of the sufficient distance according to the Egyptian code. So, separation distance which estimated by the Egyptian code is conservative.

Max pounding force
Pounding force is discussed for each study cases. The pounding force is estimated for adjacent structures under various gap distances, such as 5, 10, 20, and 40 mm, under three earthquake ground motions. Cases 2 and 5 present buildings with equal height, while other cases simulate buildings with  different heights. Results of these cases will be presented and discussed in the following sections.

Case 1
In this case, two adjacent buildings vibrate under three earthquake ground motions are Friuli, Newhall, Sylmar. Left building is 3 floors with 4 bays and right building is 6 floors with 8 bays as shown in Fig. 12. The maximum pounding force at each floor for both 5 mm and 10 mm separation distance under three earthquake ground motions is shown in Figs. 13 and 14, respectively.
Maximum pounding force was observed at a top floor (last connected floor) under three earthquake ground motions. Pounding force reduced when gap distance increased from 5 to 10 mm for three earthquake ground motions. This indicates that the pounding force is inversely proportional to the separation distance. No pounding force observed on the 1st floor for gap 5 mm in addition to no pounding force observed on the 1st and 2nd floor for gap 10 mm. When gap increased to 20 mm and 40 mm, no pounding force was observed for three earthquake ground motions.

Case 2
In this case, two adjacent buildings vibrate under three earthquake ground motions are Friuli, Newhall, Sylmar. Left building is 6 floors with 4 bays and right building is 6 floor with 8 bays as shown in Fig. 15. Two structures did not collide with each other because the two adjacent structures have same time period, two structures vibrate in phase (vibrate in the same direction), so the force of the spring is equal zero for gaps 5, 10, 20 and 40 mm.

Case 3
In this case, two adjacent buildings vibrate under three earthquake ground motions are Friuli, Newhall, Sylmar. Left building is 6 floors with 4 bays and right building is 12 floors with 8 bays as shown in Fig. 16.
The maximum pounding force at each floor for 5 mm, 10 mm, and 20 mm separation distance under three earthquake ground motions is shown in Figs. 17,18,and 19,respectively. Maximum pounding force was observed at a top floor (last connected floor) under three earthquake ground motions. Pounding force reduced when gap distance increased from 5 to 20 mm for three earthquake ground motions. This indicates that the pounding force is inversely proportional to the separation distance. For Friuli earthquake ground motion, no pounding force observed on the 1st, 2nd, 3rd, and 4th floor for gap 5 mm, while no pounding force was observed for all floors for gap 20 mm. For Newhall and Sylmar earthquake ground motion, no pounding force observed on the 1st, and 2nd floor for gap 5 mm, while no pounding force was observed on the 1st, 2nd, 3rd, 4th, and 5th floor for gap 20 mm. When gap increased to 40 mm, no pounding force was observed for three earthquake ground motions.

Case 4
In this case, two adjacent buildings vibrate under three earthquake ground motions are Friuli, Newhall, Sylmar. Left building is 8 floors with 4 bays and right building is 12 floors with 8 bays as shown in Fig. 20.
The maximum pounding force at each floor for 5 mm, 10 mm, and 20 mm separation distance under three earthquake ground motions is shown in Figs. 21, 22, and 23, respectively.
Maximum pounding force was observed at a top floor (last connected floor) under three earthquake ground motions. Pounding force reduced when gap distance increased from 5 to 20 mm for three earthquake ground motions. This indicates that the pounding force is inversely proportional to the separation distance. For Friuli earthquake ground motion, no pounding force observed on the 1st, 2nd, 3rd, 4th, 5th, 6th, and 7th floor for gap 5 mm, 10 mm, and 20 mm. For Newhall and Sylmar earthquake ground motion, no pounding force observed on the 1st, 2nd, and 3rd floor for gap 5 mm, while no pounding force was observed on the 1st, 2nd, 3rd, 4th, 5th, 6th, and 7th floor for gap 20 mm. When gap increased to 40 mm, no pounding force was observed for three earthquake ground motions. In this case, two adjacent buildings vibrate under three earthquake ground motions are Friuli, Newhall, Sylmar. Left building is 12 floors with 4 bays and right building is 12 floors with 8 bays as shown in Fig. 24.
Two structure did not collide with each other because the two adjacent structures have same time period, two structure vibrate in phase (vibrate in the same direction), so the force of the spring is equal zero for gaps 5, 10, 20 and 40 mm.

Effect of gap distance on pounding force
The effect of various gap distance on pounding force is shown in Figs. 25, 26, and 27 for all study cases. Under three earthquake ground motions, increasing gap distance between adjacent buildings causes decrease in pounding force.

Conclusion
Upon the results and the parametric study of this research work, it may be concluded that: • Separation distance which is estimated by the Egyptian code is conservative. Pounding starts to occur between buildings at a distance equal to half that required by the code. • In case 2 and case 5, two adjacent buildings vibrate under three earthquake ground motions are Friuli, Newhall, and Sylmar. The analysis did not record any pounding force at all gaps sizes. The main reason is the two adjacent structures have same time period, two structures vibrate in phase (vibrate in the same direction), so the force of the spring is equal zero for all gaps sizes. • Maximum pounding force was observed on the top floor (last connected floor) under three earthquake ground motions in all studied cases. • The previous result shows generally for reinforced concrete (RC) buildings under ground motion, as increas- ing gap between adjacent buildings causes decrease in max pounding force. This has been observed in all studied cases. • When gap increased to 40 mm, no pounding force was observed under three earthquake ground motions in all studied cases. • Increased mass for two adjacent buildings under three earthquake ground motions leads to increase max pounding force for all gaps sizes.

Competing interests
The authors declare no competing interests.