Theoretical and Experimental Study of the Pounding Response for Adjacent Inelastic MDOF Structures Based on Dimensional Analysis

Dimensional analysis is applied to study the pounding response of two inelastic multidegree of freedom (MDOF) structures under simpliﬁed earthquake excitation. The forces and deformations of the collision processes are simulated by adopting the improved Kelvin pounding model. The inelastic characteristics of MDOF structures are described by the bilinear interstory resistance model, and the representations of dimensionless impact force and the dimensionless motion equation in the pounding process are derived. On the basis of the above-mentioned theoretical deduction, the superiority of the improved Kelvin model is veriﬁed by comparing the impact response of the improved Kelvin model and the Kelvin model. Finally, the validity of the proposed theoretical method is further proved by the comparison between results from the shaking table tests of adjacent four-story and three-story steel frame structures and the corresponding numerical results obtained by the MATLAB program.


Introduction
e past decades have witnessed numerous earthquake disasters, causing a great number of casualties and tremendous property losses [1][2][3][4]. During earthquakes, a great number of adjacent structures suffered severe damage owing to the induced structural pounding effects, which have been recorded in investigations of the Mexico earthquake of 1985 [5], the ChiChi earthquake of 1999 [6], the Wenchuan earthquake of 2008 [7], the earthquakes in Iraq of 2017 [8], and so on.
To this end, the pounding responses of adjacent multidegree of freedom (MDOF) structures have recently aroused the interests of many researchers including Maison and Kasai [9], Anagnostopoulos and Spiliopoulos [10], Mahmoud and Jankowshi [11], and Jankowshi [12]. It is found that the pounding between adjacent constructions beneath seismic excitation is associated with the characteristics of ground motions, which are also additionally affected by the characteristics of the adjacent structures. Structural collision is a higher-order nonlinear phenomenon affected by multiple parameters, and therefore the fundamental situation in analyzing the pounding responses of adjacent constructions underneath seismic excitations has to address plenty of parameters [13]. Dimensional analysis [14] with fewer dimensionless Π parameters can reduce the number of parameters while at the same time preserving the basic behaviors of the adjacent structural collision responses.
Makris and Black [15,16] were the first to study the response of structures under earthquake excitations using a dimensionless method. ereby, dimensional analysis was welcomed by many scholars in studying the impact of adjacent elastic constructions. Zhang and Tang [17] investigated the effect of the soil-structure interaction on the structure underneath earthquake ground motion constructions using dimensional analysis. Dimitrakopoulos et al. [18][19][20] analyzed the pounding response between multiple SDOF constructions via a dimensionless method. e pounding of structures with fewer dimensionless parameters Π is described by using Buckingham Π, in which the number of parameters that govern structural collision responses is significantly reduced. Zhai et al. [13] combined the dimensionless method with the contact element method to study the pounding responses between a single elastic oscillator and an inflexible barrier, in which the Kelvin model is employed to simulate the impact forces. Furthermore, descriptions of the forces and deformations of structures during the contact phase are written in the realtime domain. Wu et al. [21] chose a more accurate impact model, the modified Kelvin model [22], to study collisions between an elastic SDOF structure and a rigid body, overcoming the disadvantage of the Kelvin impact model and revealing the rules of structural pounding. In the abovementioned research, adjacent structures are mostly simplified into linear SDOF structures. Nevertheless, poundings of adjacent structures are nonlinear problems, which may occur at each story of adjacent structures under earthquake excitations. erefore, it is more realistic to model the adjacent structures as nonlinear MDOF structural models when studying the pounding response.
In this paper, the pounding response of two inelastic MDOF constructions undergoing simplified earthquake excitations is investigated by using the dimensional analysis and an improved Kelvin model. e inelastic characteristics of the MDOF structures are simulated by using the bilinear interstory resistance model [23], and MATLAB is used to deduce the expression for the dimensionless pounding force and the dimensionless equation of motion in the pounding process. e calculation results are compared to the corresponding results yielded by the Kelvin pounding model. Lastly, the shaking table tests of adjacent steel frame structures are performed, whose results are compared with the numerical simulations in order to verify the correctness and effectiveness of the proposed method.

Impact Analytical
Model. e pounding force generated during the collision of two adjacent inelastic MDOF structures is simulated by using an improved Kelvin pounding analytical model. e improved Kelvin analytical model is shown in Figure 1, and the pounding force can be expressed as where k is the contact unit stiffness coefficient and δ(t) and _ δ(t) are the relative penetrating displacement and velocity for two objects in pounding, respectively. e damping coefficient c can be expressed as follows: in which the damping constant ξ is Here, the coefficient of restitution is represented by e (an elastic pounding without energy loss is implied by e � 1 and complete plastic pounding is implied by e � 0). V 1 and V 2 indicate the velocities at which two objects collide.
Equations (1)- (3) show that the time-varying damping coefficients c have overcome the theoretical shortcomings of the Kelvin model, which involve the uniform energy loss in the contact and rebound stages of pounding, including the pounding force at the beginning of the collision and the pulling force during the rebound phase of the collision.

Equation of Motion.
e adjacent three-story structures are used to illustrate the calculation model of seismic pounding (see Figure 2). As can be seen in Figure 2, the mass, stiffness, and damping coefficient of the left structure for each story are m i , (i � 1, 2, 3), K i , (i � 1, 2, 3), and C i , (i � 1, 2, 3), respectively. e mass, stiffness, and damping coefficient of the right structure for each story are m j (j � 1, 2, 3), K j (j � 1, 2, 3), and C j (j � 1, 2, 3), respectively. e original interval in the middle of the adjacent structures is d. For the study of structural nonlinearity, the constitutive relationship of the structures is simulated by using the bilinear interstory resistance model [24].
Only the acceleration amplitude and angular frequency of ground motion are required to study the seismic pounding responses of adjacent inelastic MDOF structures by using the dimensional analysis method. Consequently, Figure 1: Improved Kelvin analytical model. Figure 2: Adjacent MDOF pounding models.
the seismic excitation can be simulated with a simplified model. Two models are used to simplify the simulations of the ground motion, which are the harmonic and pulse excitation models [25]. e ground motion excitation is assumed to be sinusoidal excitation in this paper. e acceleration amplitude is represented by a p , and the circular frequency is represented by ω p .
Under sinusoidal excitation, the equations of motion for the adjacent inelastic MDOF structures are where € X g (t) is the acceleration and € X g (t) � a p sin(ω p t). 6) are the displacement, velocity, and acceleration responses of each story for adjacent structures at different times t, respectively. F si (t) indicates the inelastic story shear force at different times t.
e incremental form of the inelastic resistance is ΔF si (t) � K i (t) · ΔX i (i � 1, . . . , 6), in which ΔX i is the story drift, K i is the stiffness of the structure, and it is related to the yielding displacement u yi of each structure (u ya for the left structure and u yb for the right structure). M is the matrix of the structural mass and F p (t) is the matrix of the pounding force. e damping matrix C is the Rayleigh damping matrix, expressed as

Shock and Vibration 3
where In equation (5c), ω 1a is the angular frequencies of the first and ω 2a is second modes for the left structure; ω 1b is the angular frequencies of the first and ω 2b is second modes for the right structure, respectively; and the angular frequency ω can be obtained by solving |K − Mω 2 | � 0. ξ � 0.05 is the structural damping ratio assigned to the first two modes.
In this paper, the mass of each floor for the left structure and right structure is assumed to be equal, that is, m i � m a (i � 1, 2, 3) and m j � m b (j � 4, 5, 6). In addition, the stiffness of each floor for the left and right structures is assumed to be equal, that is, K i (t) � K a (i � 1, 2, 3) and K j (t) � K b � μK a (j � 4, 5, 6) during the elastic stage. μ is the stiffness ratio of the right structure story to the left structure story (μ � (K a /K b )). During the plastic stage, the stiffness of each story for the left structure and right structure is K i (t) � αK a and K j (t) � αK b , respectively, where α is the postyielding stiffness ratio.
According to the above assumptions and the physical quantity characterizing the energy scale of excitation l e (l e � (a p /ω 2 p ) with the dimensionless expression of [L]) proposed by Makris and Black [15,16], the mass m b of each floor, the amplitude a p (m/s 2 ), and the circular frequency ω p (s −1 ) of the sinusoidal excitation of the right MDOF structure are selected as the basic quantity. And, the following quantities are rewritten: where τ is the dimensionless movement time.
are the relative displacement, relative velocity, and relative acceleration of each floor for the two structures, respectively.
Substituting equation (6) into equation (4a), the dimensionless equation of motion for adjacent inelastic MDOF structures can be obtained as By using the definition of the modified Kelvin pounding model [22] as mentioned in equation (1), the following conclusions are obtained.
When x i − x j > (d/l e ), pounding occurs, and the expression of the pounding force is (8a)

Shock and Vibration
When x 1 − x 2 < d/l e , no pounding occurs, and the pounding force can be expressed as Substituting equation (6) into equation (8a), when pounding occurs, the dimensionless pounding force can be obtained: where v 1 and v 2 are the dimensionless velocities (v � V · (ω p /a p )) when pounding occurs between the left and right structures, respectively. ω is the angular frequency of the contact element, and ω � ��������������� β(m a + m b )/m a m b . β is the stiffness of the contact element.
Substituting equation (6) into the damping matrix and the stiffness matrix as shown in equations (5a) and (5b), the dimensionless damping matrix and stiffness matrix can be obtained as Shock and Vibration 5 where e matrix of the dimensionless inelastic resistance F s /m b a p is related to the dimensionless story shear force of F si /m b a p . Because both adjacent structures are inelastic and ΔF si /m b a p � K i (t) · Δx i /m b a p , the dimensionless inelastic resisting force matrix F s /m b a p is relevant to the dimensionless story stiffness K i (t)/m b a p , dimensionless yield displacement u y /l e , and postyielding stiffness ratio α.

Dimensionless Equation of Motion Based on the Buckingham Π eorem.
On the basis of the Π theorem and the equation of motion of the pounding between two inelastic MDOF structures obtained above, the pounding reaction of two inelastic MDOF structures is characterized by the peak displacement X max and peak velocity _ X max of the stories of the structures. e parameters controlling the pounding response are the masses m a and m b , the stiffness K a of the left structure, the stiffness ratio μ(μ � (K a /K b )) measured from the right to left structure, the yield displacements u y1 and u y2 , the damping ratio ξ, the initial spacing d between the adjacent inelastic MDOF structures, the postyielding stiffness ratio α, the recovery coefficient e of the contact element, the angular frequency ω, the amplitude a p , and the angular frequency ω p of the sinusoidal excitation. e expression of the pounding response function of the two inelastic MDOF structures through the Π theorem: From equation ( (10) is Also, Equation (11) can be rewritten as where Π m � m a /m b is the mass ratio of the left to the right structure. Π k � (K a /m b ω 2 p ) is the dimensionless stiffness story of the left structure. Π uyi � u yi ω 2 p /a p (i � a, b) is the ratio of the yield displacement u yi to the excitation energy scale l e � a p /ω 2 p of each structure, that is, the dimensionless yield displacement. Π uyi and Π α are the parameters characterizing the structural inelasticity. Furthermore, Π ω con and Π d with Π e are the parameters that characterize the pounding properties; among them, the ratio of the angular frequency of the contact element to the sinusoidal excitation is represented by Π ωcon � (ω/ω p ).

Numerical Solution of the Pounding Response of Adjacent Inelastic MDOF Structures
Equation (7a) is solved by using the Newmark-β method [25], in which the parameters are taken as c � 1/2, β � 1/4. e time step is Δτ � 0.001. e following dimensionless Π-parameters are used rough the MATLAB program, the numerical procedure has been realized. e analysis results of the displacement, pounding force, and velocity histories of the adjacent inelastic MDOF structures are shown in Figures 3-5 for the first story, the second story, and the third story, respectively. e displacement histories, velocity histories, and pounding force contrast curve for the third story of the two adjacent structures with the improved Kelvin model and Kelvin model considering pounding and nonpounding are shown in Figure 6. erefore, it is unreasonable to consider the collision effect of the highest story if the collision model is simplified to ignore the collision effect of the remaining stories. In addition, the left structure with a small mass and stiffness is obviously suppressed in positive displacement after pounding and produces a larger negative displacement than the right structure; however, the peak displacement response of the left structure is significantly reduced compared to the nonpounding situation (as shown in Figures 3(a)-5(a)). e right structure with a larger mass and stiffness has a significantly increased displacement response after pounding.
Moreover, pounding has an obvious influence on the velocity response of adjacent structures, as shown in Figures 3(c)-5(c). e velocity histories of the two structures change dramatically after pounding, and the velocity response of the left structure changes from positive to negative while the velocity response of the right structure increases. erefore, the instantaneous sharp change in the velocity response is one of the basic features of the collision of the structure. e hysteresis curve of each story of the two adjacent inelastic structures under the parameters of this paper is given in Figures 3(d

)-5(d) and Figures 3(e)-5(e). e figures
show that the first and the second story of both structures are entering the plastic stage; however, the hysteretic curve of the third story structure is a straight line, indicating that it is already in the elastic stage.
In addition to the velocity histories, Figure 6 compares the collision force history of the third story that is obtained by the improved Kelvin model and the Kelvin model. From the velocity histories of both structures, the histories of the two pounding models basically coincide, which proves the correctness of the numerical solution method obtained by the improved Kelvin model. From the time history curves of pounding force obtained using the two pounding models shown in Figure 6(c), a negative pulling force occurs during the rebound phase when the pounding process is simulated by using the Kelvin model. However, there exists no negative pulling force when the modified Kelvin model is used to simulate the pounding process. is is also a good display of the velocity histories as shown in Figures 6(a) and 6(b). In the enlarged part of the two figures, due to the tension of the Kelvin model during the rebound phase, the velocity histories will decrease during the pounding rebound phase, but this is not the case with the velocity histories obtained by the improved Kelvin model. erefore, the improved Kelvin model can overcome the shortcomings of the Kelvin model, represent the real physical phenomena, and reflect the physical laws.

Shaking Table Pounding Test of Adjacent Multistory Structures
To further verify the correctness and effectiveness of the theoretical method proposed in this paper, the shaking table pounding test of adjacent four-story and three-story steel frame structure models is carried out, and the experimental model is numerically simulated such that the test results are dimensionless after being compared to the numerical simulation results.  Considering the convenience of material selection, the scale model structure of the test will be modeled with the same structural material (Q235 steel) as the original model structure. erefore, the scaled model has the same density and modulus of elasticity as the original model, where S E � S ρ � 1. According to the similarity criterion, the similarity relations for the parameters of the scale model can be determined, as shown in Table 1. e geometric similarity ratio can be determined according to Table 1, and the structural dimensions of the two adjacent multistory steel frame scaling models are determined as follows:  stories. A C20 prefabricated concrete slab is used as an additional mass of the story in this test. According to the calculation from the similarity theory, the detailed information of the various components of the scale model is shown in Table 2. Figure 7 shows detailed dimensions of the scale model, and Figure 8 shows the test model. Since the test model is bolted according to the fixed holes reserved on the shaking table, the distance between the adjacent steel frames is constant during the test; however, the neighboring structures cannot collide during the test because of the large distance between the two reserved holes. erefore, a scalable pounding element is designed that consists of an impacting and receiving end and is located on the second and third stories of the adjacent steel frames, respectively. To ensure that point-to-surface pounding occurs between the impacting end and the receiving end, a section of an 8 mm thick steel rod is welded at the center of the front end of the receiver. e detailed designs of the pounding end and the actual pounding end model are shown in Figure 9.

Scheme of Measurement.
e shaking table test is mainly used to study the pounding reaction between adjacent steel frames. e main recorded data include the displacement and acceleration responses of each story and the shaking table. erefore, the IMC and DH5922 dynamic acquisition instruments are selected for data acquisition, where the IMC dynamic acquisition instrument is used to collect the displacement and the DH5922 dynamic acquisition instrument is used to collect the acceleration. e arrangement of the displacement and acceleration measurement points is shown in Figure 10.    ere are two test conditions selected: no pounding and pounding with an adjacent spacing of 2 mm. Taking the case of no pounding as an example, the loading process of the whole test is as follows: first, the white noise is used to sweep the frequency of the two steel frames to check whether the dynamic characteristics of the structure have changed. en, a sinusoidal wave (the frequency of the sinusoidal wave is 20 rad/s) with a peak acceleration of 0.2 g being selected for loading, and finally, a white noise sweep is performed. To date, the first loading condition has been completed. Figure 11 shows the relative displacement response histories of the top and third stories of a four-story steel frame structure with or without pounding (spacing is 2 mm) (obtained by subtracting the measured displacement response histories data of the floor from the corresponding displacement response histories data of the shaking table). Figure 11, the actual displacement of the top and third stories are larger for the pounding case than that of the nonpounding case, which means that the pounding loading will amplify the displacement response of the structure. Figure 12 shows the acceleration response histories of the top and third stories of the four-story steel frame structure without and with pounding (spacing at 2 mm). e acceleration response of the top and third stories exhibits obvious pulses during pounding, and the acceleration of the top and third stories is much greater than that of the nonpounding case, indicating that the pounding loading also enlarges the acceleration response of the structure. 1.7 t and it is assumed that the lateral stiffness of each story is equal. rough analysis and calculation, the story stiffness is obtained, and the results of the white noise frequency sweep confirm that the structure is in an elastic state during the whole test process.

As shown in
Dimensionless parameters for the numerical calculation are obtained from the dimensionless test parameters, Figures 13 and 14 show that by numerically solving the problem using MATLAB, comparative analysis of the obtained results with the dimensionless measured results. Figure 13 shows the displacement and acceleration histories of the top and third floors of the four-story steel frame structure without pounding, and the comparison curves between the experimental and numerical results under the action of a sinusoidal wave with a peak acceleration of 0.2 g. e numerical results of the displacement response of the top and third floors are slightly larger than the experimental results, while the numerical results of the acceleration response are slightly smaller than the experimental results. is may be due to the fact that the actual structure is reduced to a centralized mass model, and the errors generated during the simplification process are caused. However, the frequency and waveform of the numerical results are consistent with the experimental results which can prove the validity and rationality of the mathematical model derived above. Figure 14 shows the comparison curves of the test results and numerical results of the displacement and acceleration histories of the top and third floors of the four-story steel frame structure with pounding. e numerical results of the displacement response of the top and third floors are broadly consistent with the results of the experiment, verifying that the method is correct. Due to the collision effect, the acceleration response of the structure has obvious pulses, but the number of collisions obtained by the numerical solution is more than the number recorded by the experiment as shown in Figure 4(d).
e numerical solution of the acceleration response of the top floor is smaller than the test result and the numerical solution of the acceleration response of the third floor is greater than the test result. e reason for this difference may be that the numerical model simplifies the experimental model into a lumped mass model, resulting in errors in the process of simplification.

Conclusion
In this paper, the dimensional analysis and the contact element method are combined to study the pounding response of adjacent nonlinear MDOF structures under simplified earthquake loadings. e improved Kelvin model is used to simulate the force and deformation of the two structures in the process of collision and contact, and the bilinear interstory resistance model is used to simulate the inelastic characteristics of the structure. Two inelastic dimensionless pounding force expressions and dimensionless motion equations of the MDOF structures are deduced. When dimensionless parameters Π are adopted, the number of variables affecting the pounding response of the two inelastic MDOF structures is reduced from 13 to 10. Moreover, the pounding law is clearly exhibited, and the contact deformation process is also presented. Under the selected parameters, the displacement, velocity, pounding force, and hysteretic curve of each story of the adjacent MDOF structures are predicted by the improved Kelvin model. e correctness of the proposed pounding response analysis method and the advantage of the improved Kelvin model are verified by comparing the pounding response obtained by the improved Kelvin model and that from the Kelvin model. Besides, the shaking table tests are carried out on adjacent four-story and three-story steel frame structures, and comparing the results from the experiment with the corresponding results from numerical simulations obtained by the MATLAB program further confirms the validity and effectiveness of the proposed pounding response analysis method.

Data Availability
Some data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.