#### 3.1. 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

$2.0\times {10}^{6}\mathrm{kN}/{\mathrm{m}}^{3/2}$ 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

$\xi $ is the damping constant. The derivation of

$\xi $ is by equating the energy loss during stereo-mechanical collision to the energy dissipated by the damper, then a formula for the damping constant (

$\xi $) 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 (

$\xi $) 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

$\xi $ and

e (i.e.,

e = 1 →

$\xi $ = 0 and

e = 0 →

$\xi $ = ∞) can’t be reached by Equation (3). Therefore, they derived a more theoretically rational approximating formula for the damping constant (

$\xi $), 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.

#### 3.2. 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.

where

${\left[{A}_{s}\right]}_{k}^{g}$ 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;

$DO{F}_{i}^{1,u}$ is the position of horizontal displacement component at node

i of structure 1 in the structural displacement vector in the global coordinate system;

$DO{F}_{m}^{2,u}$ is the position of horizontal displacement component at node

m of structure 2 in the structural nodal displacement vector in the global coordinate system.

The compression deformation

${\delta}_{k}^{c,g}$ at the

k-th contact nodal pair in the global coordinate system can be expressed as

where

${\left\{q\right\}}^{g,1}$ and

${\left\{q\right\}}^{g,2}$ 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

where

$\left\{{\delta}_{N}^{g,1}\right\}$ is the compression deformation at the

l-th contact nodal pair in the global coordinate system;

${u}_{cl}^{g,1}$ and

${\upsilon}_{cl}^{g,1}$ are the horizontal and vertical displacement components at pounding point

N in the global coordinate system.

The compression deformation

$\left\{{\delta}_{N}^{g,1}\right\}$ can be expressed in element coordinate system as follows

where

${\left[T\right]}_{{i}^{e}}^{1}$ 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;

$\left\{{\delta}_{N}^{g,1}\right\}$ is the compression deformation in element coordinate system. The element displacement interpolation functions (

${\left[N\right]}_{{i}^{e}}^{e,1}$) used in

Section 2 for the third-order beam-column element is adopted, therefore

where

${\left\{\delta \right\}}_{{i}^{e}}^{e,1}$ is the nodal displacement of element

${i}^{e}$ at point

N in structure 1 in element coordinate system. The

${\left\{\delta \right\}}_{{i}^{e}}^{e,1}$ can be represented in global coordinate system as

where

${\left[T\right]}_{{i}^{e}}^{ge,1}$ is the transfer matrix for nodal displacement vector of element

${i}^{e}$ in structure 1 from global coordinate system to element coordinate system;

${\left\{\delta \right\}}_{{i}^{e}}^{g,1}$ 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

Set

${\left[\overline{T}\right]}_{{i}^{e}}^{1}$ 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

${\left[\overline{T}\right]}_{{i}^{e}}^{1}$ is a

$DO{F}_{{i}^{e}}^{1}\times TDO{F}^{1}$ matrix consisting of 0 and 1;

$TDO{F}^{1}$ is the dimension of the structural displacement vector of structure 1;

$DO{F}_{{i}^{e}}^{1}$ is the dimension of the nodal displacement of element

i^{e} in structure 1, for planar beam-column element,

$DO{F}_{{i}^{e}}^{1}$ = 6.

According to Equations (12) and (13), the following equation can be obtained and expressed in a brief form

where

${\left[{B}_{s}\right]}_{l}^{g,1}$ is the transfer coefficient matrix from structural nodal displacement vector to the horizontal and vertical displacement components at pounding point

N in the global coordinate system

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

$DO{F}_{n}^{2,u}$, Equation (6) can be changed to

Similar to Equation (7), the compression deformation

${\delta}_{l}^{c,g}$ 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

${\left[A\right]}_{s}$ is the transfer coefficient matrix for contact nodal pair;

$\left\{q\right\}$ is nodal displacement vector of structure 1 and structure 2 in the global coordinate system;

${\left\{\delta \right\}}_{s}$ is compression deformation vector at contact nodal pairs in the global coordinate system.

#### 3.3. Pounding Force

The penetration velocity ${\dot{u}}_{1}-{\dot{u}}_{2}$ 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

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

where

${\left[{N}_{{i}^{e}}\right]|}_{N}$ is the matrix of elemental displacement interpolation functions of element

${i}^{e}$ at 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

${\left[\overline{T}\right]}_{{i}^{e}}^{1}$ 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

$DO{F}_{m}^{2,u}$, we can get

If there are

${N}_{collision}$ contact pairs in the two structures, a packing equation for the structural equivalent nodal load can be given as