Development of a Reﬁned Analysis Method for Earthquake-Induced Pounding between Adjacent RC Frame Structures

: Pounding of two adjacent structures is one of the factors that cause damage and hinder sustainable use of reinforced concrete (RC) frame structures under strong ground motion excitations. This study developed a pounding analysis method with a reﬁned beam-column element in order to solve the pounding problem between two RC frame structures. The analysis method combines the ﬁber beam-column element model with the element sections discretized into concrete and longitudinal rebar ﬁbers, the Hertz-damp contact element model to describe the pounding between beam-column elements, and the method to integrate the pounding force into the system dynamic equilibrium equation. The pounding can be considered either at the level between the story slab to slab or at the level between story slab to mid-column. The application of the proposed method in pounding analyses to provide a rational seismic separation gap between two adjacent RC frame structures is ﬁnally conducted to increase their safety and sustainability under strong earthquakes. and the damage indices of other members may increase or decrease. Therefore, the structural damage is complex by including poundings. Although local damage is important in pounding but the global damage is more important when evaluation of a structure under earthquake loads. A structural damage index D s shown as Equation (28) is used here to describe the global damage of the structure. Considering the correlation between the structural global damage and each beam is generally same, i.e., the contribution to structural global damage by beams in different stories may not have large differences, the weights of all beams are set with a same value. Assuming that the correlation between the structural global damage and each column generally decreases from lower stories to upper stories, the weights among different stories is set with the reciprocal of its story number and the weights of columns within a story are proportional to its axial stiffness.


Introduction
The damage of structures in strong earthquakes causes economic loss, casualties and downtime. Pounding between adjacent structures is the one of the reasons that causes such damage and is commonly observed after strong earthquakes, e.g., 1971 San Fernando earthquake [1], 1985 Mexico earthquake [2], 1989 Loma Prieta earthquake [3], 1999 Taiwan Chi-Chi earthquake [4], 2008 Wenchuan earthquake [5], 2011 Christchurch earthquake [6,7], and 2015 Gorkha Earthquake [8]. Usually, the pounding becomes the source of local damages and in some cases it may even lead to collapse of the structures. Pounding occurs in adjacent structures due to the difference of their dynamic properties and the insufficient seismic separation gap between them.
Studies on structural pounding during last two decades have been extensively conducted. Besides some experimental works [9][10][11], the researchers used various numerical structural models and different models to describe poundings. Structures are generally idealized as lumped mass story models with single degree of freedom (SDOF) systems or multi-degree of freedom (MDOF) systems. Pantelides and Ma [12] studied the dynamic behaviors of a SDOF system with one-sided pounding during an earthquake. Muthukumar and DesRoches [13] studied the cogency of various pounding models in capturing the seismic pounding response of adjacent structures. Dimitrakopoulos et al. [14]   Using damage indices with a range from 0 to 1 is a straightforward way to describe the damage degree of an element. For the beam-column element, the damage indices for axial force and bending moment at section level are firstly calculated by using the damage index formulas proposed by Talaat [40] (Equations (3.42)- (3.43) in the reference). Then, the elemental damage index is represented as the maximum value of the section damage indices among all integration points. In the study, the Gauss-Lobatto integration scheme is used to solve the integrals relevant to the element formulas and five integration points are used for each element. The material behavior of steel rebar is described by the constitutive model proposed by Menegotto and Pinto [38] as shown in Figure 1b. In the figure, σ y and ε y are the stress and strain at the point of modulus change, E s and E sh are the initial modulus and slope of a straight-line asymptote of the stress. Detailed formulas and rules governing hysteretic behaviors are based on the study by Spacone [39] (Equations (3.9)- (3.12) in this reference). In this model, the parameters R 0 = 20, a 1 = 18.5, and a 2 = 0.15 are used in the analyses. The above steel rebar constitutive model is extended to include the prediction of the rebar buckling and description of the post-buckling behavior. The implementation is a complex procedure and the formulas and rules are based on the study by Talaat [40] (Equations (2.12)-(2.38) in the reference).
Using damage indices with a range from 0 to 1 is a straightforward way to describe the damage degree of an element. For the beam-column element, the damage indices for axial force and bending moment at section level are firstly calculated by using the damage index formulas proposed by Talaat [40] (Equations (3.42)- (3.43) in the reference). Then, the elemental damage index is represented as the maximum value of the section damage indices among all integration points. In the study, the Gauss-Lobatto integration scheme is used to solve the integrals relevant to the element formulas and five integration points are used for each element.

Verification of Element Model
The developed beam-column element involves a series of components, e.g., the third-order beam-column element, uniaxial material models of modified Kent-Park and Menegotto-Pinto for the fiber section, the rebar buckling behavior, and the element damage index to describe the damage degree.
Three verification examples for the developed material and element are conducted, in which the first and second examples verify the rebar material behavior with buckling and the third example verifies the cyclic loading performance of a cantilever RC column. The monotonic and cyclic loading rebar buckling tests are used, respectively. Figure 2 shows the comparison results between the numerical predictions and the monotonic compression rebar buckling tests by Bae et al. [41] for nine levels of rebar slenderness ratio L/d = 4, 5, 6 . . . . . . , 12. Figure 3 shows the comparison results between the numerical prediction and the cyclic loading rebar buckling test with slenderness ratio L/d = 11 conducted by Monti and Nuti [42]. These two figures show that the method for predicting the rebar material behavior in this study can capture the tested rebar stress-strain curves and the bulking behavior of rebars. Figure 4 shows the prediction results on a cyclic loading test of a RC column (side column specimen) performed by Lu [43]. The comparison shows a good match between the numerical prediction and the test result. The beam-column elemental mesh is one in the numerical analysis and Figure 5 provides the development of elemental damage index of the RC column during the cyclic loading.

Verification of Element Model
The developed beam-column element involves a series of components, e.g., the third-order beam-column element, uniaxial material models of modified Kent-Park and Menegotto-Pinto for the fiber section, the rebar buckling behavior, and the element damage index to describe the damage degree. Three verification examples for the developed material and element are conducted, in which the first and second examples verify the rebar material behavior with buckling and the third example verifies the cyclic loading performance of a cantilever RC column. The monotonic and cyclic loading rebar buckling tests are used, respectively. Figure 2 shows the comparison results between the numerical predictions and the monotonic compression rebar buckling tests by Bae et al. [41] for nine levels of rebar slenderness ratio L/d = 4, 5, 6……, 12. Figure 3 shows the comparison results between the numerical prediction and the cyclic loading rebar buckling test with slenderness ratio L/d = 11 conducted by Monti and Nuti [42]. These two figures show that the method for predicting the rebar material behavior in this study can capture the tested rebar stress-strain curves and the bulking behavior of rebars. Figure 4 shows the prediction results on a cyclic loading test of a RC column (side column specimen) performed by Lu [43]. The comparison shows a good match between the numerical prediction and the test result. The beam-column elemental mesh is one in the numerical analysis and Figure 5 provides the development of elemental damage index of the RC column during the cyclic loading.

Verification of Element Model
The developed beam-column element involves a series of components, e.g., the third-order beam-column element, uniaxial material models of modified Kent- Park and Menegotto-Pinto for the  fiber section, the rebar buckling behavior, and the element damage index to describe the damage  degree. Three verification examples for the developed material and element are conducted, in which  the first and second examples verify the rebar material behavior with buckling and the third example verifies the cyclic loading performance of a cantilever RC column. The monotonic and cyclic loading rebar buckling tests are used, respectively. Figure 2 shows the comparison results between the numerical predictions and the monotonic compression rebar buckling tests by Bae et al. [41] for nine levels of rebar slenderness ratio L/d = 4, 5, 6……, 12. Figure 3 shows the comparison results between the numerical prediction and the cyclic loading rebar buckling test with slenderness ratio L/d = 11 conducted by Monti and Nuti [42]. These two figures show that the method for predicting the rebar material behavior in this study can capture the tested rebar stress-strain curves and the bulking behavior of rebars. Figure 4 shows the prediction results on a cyclic loading test of a RC column (side column specimen) performed by Lu [43]. The comparison shows a good match between the numerical prediction and the test result. The beam-column elemental mesh is one in the numerical analysis and Figure 5 provides the development of elemental damage index of the RC column during the cyclic loading.

Contact Element
Two modeling techniques have been primarily used: the stereo-mechanical method and the contact element method. The stereo-mechanical method is not easy to implement into the existing finite element method and software for analysis of MDOF structures [13]. Therefore, the contact element method is used in this study. This method is a force-based approach, which uses an activate contact element once contact occurs and delete the element when there is no contact. The pounding force can be seen as a complementary loading restricts to the original system dynamic equilibrium equation. Commonly there are four pounding models, i.e., the linear-spring model (this model is regarded as impractical as it cannot account for the energy loss during collision), the Kelvin model (this model is regarded as impractical as it results in tensile forces acting on the bodies just before separation), the Hertz model (this model is regarded as impractical as it fails to include the energy dissipation during collision), and the Hertz-damp model [13]. The Hertz-damp model can overcome the shortages of the above-mentioned models and it is used in this study. The following contact force (Fc) formulas of Hertz-damp model is originally used in areas of robotics and multi-body systems, and firstly introduced into the structural engineering area by Muthukumar and DesRoches [13], this model is based on the Hertz law and using a non-linear damper

Contact Element
Two modeling techniques have been primarily used: the stereo-mechanical method and the contact element method. The stereo-mechanical method is not easy to implement into the existing finite element method and software for analysis of MDOF structures [13]. Therefore, the contact element method is used in this study. This method is a force-based approach, which uses an activate contact element once contact occurs and delete the element when there is no contact. The pounding force can be seen as a complementary loading restricts to the original system dynamic equilibrium equation. Commonly there are four pounding models, i.e., the linear-spring model (this model is regarded as impractical as it cannot account for the energy loss during collision), the Kelvin model (this model is regarded as impractical as it results in tensile forces acting on the bodies just before separation), the Hertz model (this model is regarded as impractical as it fails to include the energy dissipation during collision), and the Hertz-damp model [13]. The Hertz-damp model can overcome the shortages of the above-mentioned models and it is used in this study. The following contact force (Fc) formulas of Hertz-damp model is originally used in areas of robotics and multi-body systems, and firstly introduced into the structural engineering area by Muthukumar and DesRoches [13], this model is based on the Hertz law and using a non-linear damper

Contact Element
Two modeling techniques have been primarily used: the stereo-mechanical method and the contact element method. The stereo-mechanical method is not easy to implement into the existing finite element method and software for analysis of MDOF structures [13]. Therefore, the contact element method is used in this study. This method is a force-based approach, which uses an activate contact element once contact occurs and delete the element when there is no contact. The pounding force can be seen as a complementary loading restricts to the original system dynamic equilibrium equation. Commonly there are four pounding models, i.e., the linear-spring model (this model is regarded as impractical as it cannot account for the energy loss during collision), the Kelvin model (this model is regarded as impractical as it results in tensile forces acting on the bodies just before separation), the Hertz model (this model is regarded as impractical as it fails to include the energy dissipation during collision), and the Hertz-damp model [13]. The Hertz-damp model can overcome the shortages of the above-mentioned models and it is used in this study. The following contact force (F c ) formulas of Hertz-damp model is originally used in areas of robotics and multi-body systems, and firstly introduced into the structural engineering area by Muthukumar and DesRoches [13], this model is based on the Hertz law and using a non-linear damper where k h is the contact stiffness, generally set as 2.0 × 10 6 kN/m 3/2 for concrete structures [44]; n is the Hertz coefficient for describing a nonlinear contact force-deformation relationship, which is typically taken as 3/2; u 1 − u 2 − g p is relative penetration (compression deformation), in which u 1 , u 2 , and g p are the displacements of the contact pairs along the contact axis and the gap between contact points; the dot denotes differentiation with respect to time; c h is the damping coefficient, which can be calculated as where ξ is the damping constant. The derivation of ξ is by equating the energy loss during stereo-mechanical collision to the energy dissipated by the damper, then a formula for the damping constant (ξ) can be expressed in terms of the spring stiffness (k h ), the restitution coefficient (e) and the relative approaching velocity (v 0 ), as follows [45] Other methods to determine the damping constant (ξ) were also proposed for a better accuracy [46,47]. In the study by Ye et al. [46], who thought that the Equation (3) is valid only for the case of restitution coefficient (e) approximating one and pointed out the logical relationship between ξ and e (i.e., e = 1 → ξ = 0 and e = 0 → ξ = ∞) can't be reached by Equation (3). Therefore, they derived a more theoretically rational approximating formula for the damping constant (ξ), as follows Hence, the force during pounding in Equation (1) can be expressed as Equation (5). Because the theoretically rational of Equation (4), Equation (5) for calculating the collision force is finally used in this study. The restitution coefficient (e) used to simulate real collision in structural engineering commonly varies in the range of 0.5 and 0.75, and 0.65 is adopted in the analyses.

Compression Deformation
The compression deformation of the Hertz-damp model needs to be calculated firstly and then calculation of the pounding force. There are two typical pounding situations between adjacent frame structures as shown in Figure 6, including (1) When the heights corresponding to story slabs of the two adjacent structures at which pounding occurs are same, pounding will be between the beam-column joint to joint (i.e., the pounding occurs at the level of story slab to slab. See Figure 6a); (2) Otherwise, the pounding may occur between the story slab and a point within the column (i.e., story slab to mid-column. See Figure 6b). In the finite element model to simulate the first situation, the pounding occurs between two elemental nodes; and for the second situation, the pounding may occur between two elemental nodes or between the elemental node and the mid-element.
For the first situation, the compression deformation of the Hertz-damp model can be directly expressed by using the structural nodal displacement vector. A vector is defined as follows to locate the pounding point.
is the position transfer coefficient vector for the k-th contact nodal pair; TDOF is total degree of freedoms of the structure, the superscript "1" and "2" denotes structure 1 and structure 2; where [A s ] g k is the position transfer coefficient vector for the k-th contact nodal pair; TDOF is total degree of freedoms of the structure, the superscript "1" and "2" denotes structure 1 and structure 2; DOF 1,u i is is the position transfer coefficient vector for the k-th contact nodal pair; TDOF is total degree of freedoms of the structure, the superscript "1" and "2" denotes structure 1 and structure 2; where { } ,1 g q and { } ,2 g q are nodal displacement vector of structure 1 and structure 2 in the global coordinate system. Note that the gap is not considered in Equation (7) for brief but it is easily to be added.
For the second situation, the horizontal displacement at pounding point N (see Figure 6b) needs to be firstly derived. Because in the finite element analysis procedure, only the elemental nodal displacement is recorded, the horizontal displacement at point N is certainly relevant to the element nodal displacement which the element includes the point N. As shown in Figure 6b, the displacement vector of point N in the global coordinate system is taken as The compression deformation δ c,g k at the k-th contact nodal pair in the global coordinate system can be expressed as where q g,1 and q g,2 are nodal displacement vector of structure 1 and structure 2 in the global coordinate system. Note that the gap is not considered in Equation (7) for brief but it is easily to be added. For the second situation, the horizontal displacement at pounding point N (see Figure 6b) needs to be firstly derived. Because in the finite element analysis procedure, only the elemental nodal displacement is recorded, the horizontal displacement at point N is certainly relevant to the element nodal displacement which the element includes the point N. As shown in Figure 6b, the displacement vector of point N in the global coordinate system is taken as N is the compression deformation at the l-th contact nodal pair in the global coordinate system; u g,1 cl and υ g,1 cl are the horizontal and vertical displacement components at pounding point N in the global coordinate system.
The compression deformation δ g,1 N can be expressed in element coordinate system as follows where [T] 1 i e is the transfer matrix for displacement vector of element i e at point N in structure 1 from element coordinate system to global coordinate system; δ g,1 N is the compression deformation in element coordinate system. The element displacement interpolation functions ([N] e,1 i e ) used in Section 2 for the third-order beam-column element is adopted, therefore  i e is the transfer matrix for nodal displacement vector of element i e in structure 1 from global coordinate system to element coordinate system; {δ} g,1 i e is the nodal displacement vector of element i e in the global coordinate system.
According to Equations (8)- (11), the following equation can be obtained ge,1 Set T 1 i e as the transfer coefficient matrix from the nodal displacement vector of structure 1 in the global coordinate system to the nodal displacement vector of element i e in the global coordinate system. Thus where T 1 i e is a DOF 1 i e × TDOF 1 matrix consisting of 0 and 1; TDOF 1 is the dimension of the structural displacement vector of structure 1; DOF 1 i e is the dimension of the nodal displacement of element i e in structure 1, for planar beam-column element, DOF 1 i e = 6. According to Equations (12) and (13) where the subscripts s1 and s2 correspond to the horizontal and vertical components. If denotes the position of horizontal nodal displacement of node N in structure 2 in structural nodal displacement vector 2 as DOF 2,u n , Equation (6) can be changed to  (16) Similar to Equation (7), the compression deformation The above derivations correspond to the collision occurs within the column in structure 1 and at the story slab in structure 2. If the collision occurs at the story slab in structure 1 and at within the column in structure 2, same derivation procedure can be adopted and will not be provided here again in the study for brief. In the calculation, it is easy to distinguish the pounding cases and use the formulas.
If there are collision N contact pairs in the two structures, a packing equation for the compression deformations can be given as Similar to Equation (7), the compression deformation δ c,g l at contact pair l can be expressed in the following Equation The above derivations correspond to the collision occurs within the column in structure 1 and at the story slab in structure 2. If the collision occurs at the story slab in structure 1 and at within the column in structure 2, same derivation procedure can be adopted and will not be provided here again in the study for brief. In the calculation, it is easy to distinguish the pounding cases and use the formulas.
If there are N collision contact pairs in the two structures, a packing equation for the compression deformations can be given as A brief expression of Equation (18) can be given as where [A] s is the transfer coefficient matrix for contact nodal pair; q is nodal displacement vector of structure 1 and structure 2 in the global coordinate system; {δ} s is compression deformation vector at contact nodal pairs in the global coordinate system.

Pounding Force
The penetration velocity . u 1 − . u 2 at the contact nodal pair used in Equation (5) can be calculated by differentiate the penetration displacements within the time step. Therefore, the pounding force at the contact nodal pair can be calculated. Then, the pounding force will be expressed by the format of external equivalent nodal loads.
Assuming the pounding forces on structure 1 and structure 2 at k-th contact nodal pair are expressed as the following Equation where k will change to l if it is at l-th contact nodal pair. For the first situation, the equivalent nodal load vector of structure in global coordinate system can be easily obtained by the Equation The above derivations correspond to the collision occurs within the column in structure 1 and at the story slab in structure 2. If the collision occurs at the story slab in structure 1 and at within the column in structure 2, same derivation procedure can be adopted and will not be provided here again in the study for brief. In the calculation, it is easy to distinguish the pounding cases and use the formulas.
A brief expression of Equation (18) can be given as where [ ] s A is the transfer coefficient matrix for contact nodal pair; { } q is nodal displacement vector of structure 1 and structure 2 in the global coordinate system; { } s δ is compression deformation vector at contact nodal pairs in the global coordinate system.

Pounding Force
The penetration velocity 1 2 u u −   at the contact nodal pair used in Equation (5) can be calculated by differentiate the penetration displacements within the time step. Therefore, the pounding force at the contact nodal pair can be calculated. Then, the pounding force will be expressed by the format of external equivalent nodal loads.
Assuming the pounding forces on structure 1 and structure 2 at k-th contact nodal pair are expressed as the following Equation where k will change to l if it is at l-th contact nodal pair. For the first situation, the equivalent nodal load vector of structure in global coordinate system can be easily obtained by the Equation For the second situation, the contact force at pounding point N needs to be firstly transferred to the elemental coordinate system, and then expressed by the equivalent elemental nodal load induced by the pounding as The matrix T 1 i e is used to get the elemental equivalent elemental nodal load vector for structure 1 in global coordinate system Therefore, if denotes the position of pounding force of node m in structure 2 in structural nodal load vector as DOF 2,u m , we can get pounding point N in structure 1; the superscript T denotes matrix transpose. Transfer the equivalent elemental nodal load from the elemental coordinate system to the global coordinate system The matrix Therefore, if denotes the position of pounding force of node m in structure 2 in structural nodal load vector as

Integration into System Equation
The Equation (27) can be seen as external nodal loads on the structures, which can be integrated into the system dynamic equilibrium equations of motion of the two adjacent structures. Newmark implicit numerical integration method is used with a default incremental time step of 5 × 10 −3 s. The time step can be automatically interpolated into a small value by checking whether the pounding occurs.

Building Information and Selected Ground Motions
Two RC frames with 5 stories (J-1 frame) and 11 stories (J-2 frame) are modeled to illustrate the collision analysis and discuss on the rational seismic separation gap. The two frames are with regular configuration, as shown in Figure 7. The fundamental periods are 1.11 s and 2.23 s for the 5-story frame and 11-story frame, respectively. For detail information of these RC frames, such as cross-If there are N collision contact pairs in the two structures, a packing equation for the structural equivalent nodal load can be given as

Integration into System Equation
The Equation (27) can be seen as external nodal loads on the structures, which can be integrated into the system dynamic equilibrium equations of motion of the two adjacent structures. Newmark implicit numerical integration method is used with a default incremental time step of 5 × 10 −3 s. The time step can be automatically interpolated into a small value by checking whether the pounding occurs.

Building Information and Selected Ground Motions
Two RC frames with 5 stories (J-1 frame) and 11 stories (J-2 frame) are modeled to illustrate the collision analysis and discuss on the rational seismic separation gap. The two frames are with regular configuration, as shown in Figure 7. The fundamental periods are 1.11 s and 2.23 s for the 5-story frame and 11-story frame, respectively. For detail information of these RC frames, such as cross-section dimensions of beams and columns, and materials of concrete and steel rebars, are provided in Tables 1-3 shows a total of 24 records of horizontal components of ground motion records selected for the pounding analyses and determination of the rational separation gap between the two frame structures. These ground motion records are recommended by the standard of general rule for performance-based seismic design of buildings [48]. section dimensions of beams and columns, and materials of concrete and steel rebars, are provided in Table 1 Table 2 Table 3 shows a total of 24 records of horizontal components of ground motion records selected for the pounding analyses and determination of the rational separation gap between the two frame structures. These ground motion records are recommended by the standard of general rule for performance-based seismic design of buildings [48].
(a) 5-story, J-1 frame (b) 11-story, J-2 frame  Note: As shown in Figure 7, AB, BC, and CD are first, second, and third bays; A, B, C, and D are axes; the compressive strength is 30MPa for concrete; the yielding strength is 400MPa for longitudinal rebars; the yielding strength is 235MPa for stirrups with Ф8@100/200; The rebars are continuing along the beam.   Note: As shown in Figure 7

Pounding Analysis and the Rational Separation Gap
The first ground motion record from Table 3, as shown in Figure 8, from 1979 Imperial Valley earthquake at station of El Centro, Array #5 (140 component) is selected to illustrate the pounding analyses. The 24 of ground motion records are then used for analyze the rational separation gap. The ground motion intensity is scaled to PGA = 0.4 g, which corresponds to the rare earthquake intensity at the building site (seismic hazard of 2-3% probability of exceedance in the 50 years).

Pounding Analysis and the Rational Separation Gap
The first ground motion record from Table 3, as shown in Figure 8, from 1979 Imperial Valley earthquake at station of El Centro, Array #5 (140 component) is selected to illustrate the pounding analyses. The 24 of ground motion records are then used for analyze the rational separation gap. The ground motion intensity is scaled to PGA = 0.4 g, which corresponds to the rare earthquake intensity at the building site (seismic hazard of 2-3% probability of exceedance in the 50 years).  Figure 9 shows the pounding forces at positions of J-1-D1 (1st story in axis D in J-1 RC frame) and J-1-D5 (5th story in axis D in J-1 RC frame). The figure illustrates that the pounding force and pounding frequency generally increase as the separation gap decreases. The pounding force and pounding frequency of upper stories (e.g., 5th story) are more serious than those of lower stories (e.g., 1st story). Figure 10 shows the horizontal acceleration and displacement responses at positions of J-1-D1 and J-1-D5. It seems that the pounding increases acceleration and displacement responses in most cases. The final states of beam and column damage indices are shown in Figure 11.  Figure 9 shows the pounding forces at positions of J-1-D1 (1st story in axis D in J-1 RC frame) and J-1-D5 (5th story in axis D in J-1 RC frame). The figure illustrates that the pounding force and pounding frequency generally increase as the separation gap decreases. The pounding force and pounding frequency of upper stories (e.g., 5th story) are more serious than those of lower stories (e.g., 1st story). Figure 10 shows the horizontal acceleration and displacement responses at positions of J-1-D1 and J-1-D5. It seems that the pounding increases acceleration and displacement responses in most cases. The final states of beam and column damage indices are shown in Figure 11.   Pounding aggravates member damage indices around the pounding point, and the damage indices of other members may increase or decrease. Therefore, the structural damage is complex by including poundings. Although local damage is important in pounding but the global damage is more important when evaluation of a structure under earthquake loads. A structural damage index Ds shown as Equation (28) is used here to describe the global damage of the structure. Considering the correlation between the structural global damage and each beam is generally same, i.e., the contribution to structural global damage by beams in different stories may not have large differences, the weights of all beams are set with a same value. Assuming that the correlation between the structural global damage and each column generally decreases from lower stories to upper stories, the weights among different stories is set with the reciprocal of its story number and the weights of columns within a story are proportional to its axial stiffness. Pounding aggravates member damage indices around the pounding point, and the damage indices of other members may increase or decrease. Therefore, the structural damage is complex by including poundings. Although local damage is important in pounding but the global damage is more important when evaluation of a structure under earthquake loads. A structural damage index Ds shown as Equation (28) is used here to describe the global damage of the structure. Considering the correlation between the structural global damage and each beam is generally same, i.e., the contribution to structural global damage by beams in different stories may not have large differences, the weights of all beams are set with a same value. Assuming that the correlation between the structural global damage and each column generally decreases from lower stories to upper stories, the weights among different stories is set with the reciprocal of its story number and the weights of columns within a story are proportional to its axial stiffness. Pounding aggravates member damage indices around the pounding point, and the damage indices of other members may increase or decrease. Therefore, the structural damage is complex by including poundings. Although local damage is important in pounding but the global damage is more important when evaluation of a structure under earthquake loads. A structural damage index D s shown as Equation (28) is used here to describe the global damage of the structure. Considering the correlation between the structural global damage and each beam is generally same, i.e., the contribution to structural global damage by beams in different stories may not have large differences, the weights of all beams are set with a same value. Assuming that the correlation between the structural global damage and each column generally decreases from lower stories to upper stories, the weights among different stories is set with the reciprocal of its story number and the weights of columns within a story are proportional to its axial stiffness. In which iλ Beam , iD Beam , jλ Column and jD Column are the weight and damage index of beam and column, respectively; N Beam is the number of beams; jλ Column Story is the sum of weights of columns in j-th story; jEA Column is the axial stiffness of columns in j-th story. The calculation of elemental damage index of a beam or column is same as that in Section 2.1. Figure 12 shows the structural damage index varied with the separation gap between the two RC frames. The figure illustrates the structural damage index is not linearly relevant to the separation gap. However, it seems that the structural damage will become to a constant value, i.e., it does not increase to larger values, when the separation gap is larger than a threshold value. In addition, the threshold value can use a value smaller than the required separation gap so no pounding occurs (see the damage index variation between the line for advised gap and line for no pounding in Figure 12). In this study, the advised threshold value is set as the separation gap, which after it the variation of the structural damage index is no larger than 2% of the maximum structural damage index. Follow this definition of a threshold value, more spaces between two adjacent structures can be saved to create additional economic profits in practical engineering projects. The 24 ground motion records in Table 3 are used to calculate the advised separation gap and required separation gap with no pounding. Figure 13 shows the calculated results and the mean ratio of the advised separation gap and the required separation gap with no pounding is about 0.9. In general, if the separation gap is set as 500 mm, there are over 85% probabilities that the damages are no more than that the damage without pounding. Column jEA is the axial stiffness of columns in j-th story. The calculation of elemental damage index of a beam or column is same as that in Section 2.1. Figure 12 shows the structural damage index varied with the separation gap between the two RC frames. The figure illustrates the structural damage index is not linearly relevant to the separation gap. However, it seems that the structural damage will become to a constant value, i.e., it does not increase to larger values, when the separation gap is larger than a threshold value. In addition, the threshold value can use a value smaller than the required separation gap so no pounding occurs (see the damage index variation between the line for advised gap and line for no pounding in Figure 12). In this study, the advised threshold value is set as the separation gap, which after it the variation of the structural damage index is no larger than 2% of the maximum structural damage index. Follow this definition of a threshold value, more spaces between two adjacent structures can be saved to create additional economic profits in practical engineering projects. The 24 ground motion records in Table 3 are used to calculate the advised separation gap and required separation gap with no pounding. Figure 13 shows the calculated results and the mean ratio of the advised separation gap and the required separation gap with no pounding is about 0.9. In general, if the separation gap is set as 500 mm, there are over 85% probabilities that the damages are no more than that the damage without pounding.   Column jEA is the axial stiffness of columns in j-th story. The calculation of elemental damage index of a beam or column is same as that in Section 2.1. Figure 12 shows the structural damage index varied with the separation gap between the two RC frames. The figure illustrates the structural damage index is not linearly relevant to the separation gap. However, it seems that the structural damage will become to a constant value, i.e., it does not increase to larger values, when the separation gap is larger than a threshold value. In addition, the threshold value can use a value smaller than the required separation gap so no pounding occurs (see the damage index variation between the line for advised gap and line for no pounding in Figure 12). In this study, the advised threshold value is set as the separation gap, which after it the variation of the structural damage index is no larger than 2% of the maximum structural damage index. Follow this definition of a threshold value, more spaces between two adjacent structures can be saved to create additional economic profits in practical engineering projects. The 24 ground motion records in Table 3 are used to calculate the advised separation gap and required separation gap with no pounding. Figure 13 shows the calculated results and the mean ratio of the advised separation gap and the required separation gap with no pounding is about 0.9. In general, if the separation gap is set as 500 mm, there are over 85% probabilities that the damages are no more than that the damage without pounding.

Conclusions
This study proposed a finite element analysis method with a refined beam-column element in order to solve the pounding problem between two reinforced concrete (RC) frame structures under strong ground motion excitations.
(1) The fiber beam-column element model is adopted and the element section discretized into concrete and reinforcement fibers with hysteretic models while considering bucking behaviors. The formulas that integration of the beam-column element and Hertz-damp contact element are proposed. The pounding can be considered either at the level between the story slab to slab or at the level between story slab to mid-column. (2) The application of the proposed method in pounding analyses to provide a rational seismic separation gap between two adjacent RC frame structures is conducted. It seems that the structural damage may not linearly vary as the separation gap reduces, but it may remain nearly unchanged when the separation gap is larger than a threshold value. The threshold value is smaller than the required separation gap meaning no pounding occurs. This value can be set as the rational seismic separation gap. (3) The proposed method can be used in damage analysis for structures which may be influenced by pounding or used to provides a rational seismic separation gap in a probabilistic way by analyses of a series of ground motion records at a specific earthquake intensity, in order to increase the safety and sustainability of a structure in earthquakes.