*2.2. Contact Collision Simulation*

During an earthquake, the actual collision between adjacent structures of a curved bridge is not a single problem ofnormal frontal contact, but a complex problem ofspatial contact. The collision between adjacent components that is caused by the in-plane rotation of the curved bridge is arbitrary and asymmetrical. In order to consider anon-uniform collision, the master–slave surface method [28] is used. This method is mainly applied to deal with the contact between two surfaces, and could simulate the non-uniform collision between adjacent contact surfaces at the expansion joints with relative accuracy. The master–slave surface method involvestwo surfaces in contact with each other as master and slave surfaces, respectively, and the nodes on the master and slave surfaces are defined as master and slave nodes, respectively. Duringthe analysis process, the following instruction s should be followed: first, search for the contact pairs on the master and slave surfaces; next, calculate the effect of the contact pairs. The algorithms involved mainly include search algorithm and contact algorithm. The surface-to-surface contact automatic search algorithm [28] is used in this paper, and the penalty function method [28] is used as the contact algorithm. The penalty function method is used to obtain the contact force by multiplying a penalty parameter related to the element stiffness by the contact surface intrusion value [31]. The calculation of the contact force mainly includes the normal contact force and tangential friction force. In ABAQUS software, the "surf-to-surf" command adopted to simulate the normal contact force is set as "hard contact", and the penalty friction is adopted to simulate tangential force [28]. According to [32], the tangential friction coefficient is set as 0.5.

There are three types of contact pairs in the bridge model. One is formed by the adjacent structural surfaces at the expansion joints of the curved bridge. The expansion joint at the abutment position takes the abutment contact surface as the master surface, the middle expansion joint takes the contact surface of long unit girder as the master surface, and the corresponding surface is the slave surface, including the free surface of the rubber pad. Another is formed by the top surface of the sliding support and the bottom surface of the girder at the corresponding position, with the bottom surface of the girder as the master surface and the top surface of the support as the slave surface. The other is formed by the contact surfaces of the other components, including the upper and lower surfaces of the fixed support and its adjacent contact surfaces, the contact surfaces between cylindrical piers and cap beams, and the contact surfaces between rubber pads and the girder attached to it.The three-dimensional finite element model of the whole structure and its components is shown in Figure 4. In order to express the collision response at different expansion joints more clearly, they were named as shown in Figures 4 and 5. The three expansion joints

are respectively named as short unit expansion joints, middle expansion joints, and long unit expansion joints; the positions of the collision surface are respectively named as the short-unit abutment position, the short-unit middle position, the long-unit middle position, and the long-unit abutment position; the nodes at both ends of the expansion joint are numbered.

**Figure 4.** Finite element model of curved bridge. (**a**) Overall structure model. (**b**) Double-column pier. (**c**) Abutment and bearing.

**Figure 5.** The name and node number of expansion joints. (**a**) Short unit expansion joints. (**b**) Middle expansion joints. (**c**) Long unit expansion joints.

### *2.3. Seismic Mitigation and Unseating Prevention Devices*

In order to reduce the collision response of the curved girder bridge and improve the seismic performance during seismic shock, a relatively effective method is to install unseating restrainers or damping devices inside and outside the expansion joints. The restraint effect of the unseating restrainer and the energy consumption effect of the damping device can effectively reduce the torsion and collision reaction of the beam to achieve the goal of pounding mitigation and unseating prevention. The seismic mitigation and unseating prevention devices used in this paperinclude viscoelastic dampers, viscous dampers, steel strand cables-rubber pads, and lead-core rubber bearings.

Steel strand cable restrainers are usually used to prevent the girder from falling, but traditional steel strand cables can only bear axial tension. Therefore, they prevent unseating but offer no pounding mitigation function. In order to achieve the goal of seismic mitigation and unseating prevention function simultaneously, the two ends of the steel stranded cable are anchored at the webs of the beams on both sides of the expansion joint, and rubber pads are set between the expansion joints, as shown in Figure 6. The setting of rubber pads on the contact surfaces of each expansion joint isshown in Figure 7. The restrainer can prevent excessive relative displacement between the beams to perform the unseating prevention function. The rubber pad can buffer collisions with adjacent beams, and can also absorb the energy generated by the collision between the beams and avoid the direct collision of adjacent beams, thereby mitigating pounding.

**Figure 6.** Steel wire rope-rubber mat device.

**Figure 7.** The location of rubber pads.

The pounding mitigation and unseating prevention device is composed of the Strand Cable-Rubber Blanket (SCRB), which is mainly composed of steel strand cables and rubber pads. Among these, the rubber pad is made of general rubber, and its elastic modulus is taken as the elastic modulus of natural rubber. The cable restrainer used on the bridge

usually is the steel stranded cable. The mechanical model of the cable restrainer can be the bilinear model. Its pre-yield stiffness is *K*1, its post-yield stiffness is *K*2, and it generally takes *K*2 = 0.05 *K*1 [12]. The elongation of the steel cable, whose length is *L*, is Δ*L*, and the pre-yield stiffness can be calculated using the following formula:

$$K\_1 = \frac{E'A\_0}{L} \tag{1}$$

$$E' = \frac{TL}{A\_0 \Delta L} = E \frac{\left(6L^3 + 1.0671S^3\right)L}{\left(6S + 1.0671L\right)S^3} \tag{2}$$

where *E* is the elastic modulus of steel strand,

> *T* is the total tension of steel strand,

*E* is the elastic modulus of steel wire, generally taken as 2 × 10<sup>5</sup> N/mm2,

*S* is the length of inner and outer wire of a twist pitch, *S* = √<sup>4</sup>*π*2*R*<sup>2</sup> + *L*2, *R* = *d*1+*d*2 2= 1.0165*d*1,

*R* is the distance between the centerline of the outer wire and the centroid of the steel strand,

*d*1 is the cross-section diameter of outer wire,

*d*2 is the cross-section diameter of inner wire.

Consider the use of lead rubber bearings (LRB) and laminated rubber bearings (GJZ) for seismic isolators. The shear force-displacement hysteretic curve of laminated rubber bearings, which can be approximately linearized, is narrow and long, [33]. The stiffness of the lead rubber bearing includes vertical stiffness, horizontal stiffness, and horizontal equivalent stiffness. The horizontal stiffness includes two important parameters, namely the pre-yield stiffness and the post-yield stiffness. The horizontal restoring force model can be simulated by the bilinear model.The pre-yield stiffness of the support is used in the elastic stage, and the post-yield stiffness is used in the plastic stage. According to industry standards [34], the ratio of the pre-yield and post-yield stiffness is between 0.15 and 0.16; in this paper, it is taken as 0.154.

Viscoelastic Dampers (VEDs) arecomposed of viscoelastic material, which is usually polymer, constrained steel plate, and related parts. Viscoelastic dampers depend on the damping of some viscoelastic material, and they feature initial stiffness. Theircommon mechanical calculation models include the Kelvin model, the standard linear and equivalent standard solid model, etc. The most commonly used model to describe the properties of Viscoelastic Dampers is the Kelvin model. The Kelvin model [35] is composed of a spring and a damper in parallel. This model is used in this article, and its mechanical equation is as follows:

> *f*

$$
\dot{u} = \mathbf{k}u + c\dot{u} \tag{3}
$$

where *u* is the displacement acting on the model,

*f* is the output force of the damper,

*k* is the spring equivalent stiffness,

*c* is the equivalent damping coefficient.

Viscous Fluid Damper (VFDs) arecomposed of pistons, cylinders, orifices, and other related components. Viscous dampersare velocity-dependent and can dissipate seismic energy without adding any initial stiffness to the bridge structure. The mechanical model of VFDs adopt the Maxwellmodel, which is created by connecting the spring and dampers in series. Due to the series of viscous elements, deformation will increase infinitely under any small external force, so the Maxwell model is essentially viscous. According to [36], if the actual frequency is less than the cut-off frequency of 4 Hz, the effect of the spring stiffness on the displacement is negligible. Therefore, the relationship between the damping force and the relative speed can be expressed by the following formula:

$$f = -c\dot{\bar{u}}''\tag{4}$$

where *c* is the damping coefficient,

> *α* is the speed index.

In order to achieve a better effect of pounding mitigation and unseating prevention, the aforementioned damping devices and unseating restrainers were optimized in combination. The four cases obtained are shown in Table 2. In the table, Case A represents the original structure, in which all supports are GJZ, and there is no unseating restrainer at the expansion joints. All the supports in Case F are GJZ, and the VED and SCRB are set inside and outside the expansion joints. All the supports in case G are GJZ, and the VFD and SCRB are set inside and outside the expansion joints. All the supports in case H are LRB, and SCRB is set inside and outside all the expansion joints.

**Table 2.** The arrangemen<sup>t</sup> of various cases.


Note: • means setting, - means not setting.

The design parameters of the damping devices and unseating restrainers selected in the calculation conditions are shown in Table 3.


**Table 3.** The parameters of the devices.

### **3. Dynamic Analysis Method and Ground Motion Input**

Implicit and explicit algorithms can be used to solve dynamic equations, sincethe low frequency component is usually the main component in the dynamic response of the bridge structure. Considering the calculation accuracy, a larger time step is allowed; therefore, the unconditionally stable implicit algorithm, the Newmark method, is used to solve the dynamic equation. The contact collision process is the nonlinear problem. Nonlinear dynamic equations require iterative solutions that adopt the modified Newton-Raphson iterative method [37]. The structure in this study considers Rayleigh damping.

To determine the α and β factors, the radial frequency of the first mode and the highest mode to achieve a 90% modal mass participation wereconsidered, respectively. We tookthe damping coefficient as 0.05, the calculated massdamping coefficient α = 0.4158, and the stiffness damping coefficient β = 0.006.

The bridge is located in Renhe District, Panzhihua City, Sichuan Province, China, with site category I1 (i.e., with a shear wave velocity, VS, in the range of 250 to 500 m/s, and with a thickness of covering soil of less than 5 m). The seismic fortification intensity is determined to be 7 degrees (the peak acceleration of rare earthquakes is 3.1 m/s2). Through a modal analysis, it was determined that the basic period of the bridge structure is 0.7966 s. The wave selection software, compiled by the Structural Engineering and Disaster Prevention Center of Chongqing University, was used to input the basic information aboutthe bridge structure and site for wave selection, and MEX00003 was finally selected. The Fourier spectra and the acceleration waveform of the scaled ground motion are shown in Figure 8. Due to the far-field type of the selected motions, the vertical component of the motion was neglected; only the horizontal seismic ground motion was considered.

**Figure 8.** MEX00003 ground motion. (**a**) Fourier spectrum of ground motion. (**b**) Acceleration of the ground motion.

When analyzing the seismic response of curved bridges, the seismic input should be carried out separately along the connecting direction of two adjacent bridge piers to determine the most unfavorable seismic horizontal input direction. The curved bridge model established in this paper features three piers. According to the requirements of the specification, the ground motion should be input along the connection direction of piers 1 and 2 and the connection direction of piers 2 and 3 . Comparing the effect of collision response of curved bridges at different input angles of ground motion, it was found that the connecting direction of piers 2 and 3 is the most unfavorable direction.Therefore, the seismic response of the structure was mainly analyzed when the seismic wave was input along the connecting direction of piers 2 and 3 .
