Multiscale finite element (FE) modeling offers a balance between computational efficiency and accuracy in numerical simulations, which is appropriate for analysis of seismic collapse of RC highway bridges. Some parts of structures that need detailed analysis can be modeled by solid elements, while some subordinate parts can be simulated by beam elements or shell elements to increase the computational efficiency. In the present study, rigid surface coupling method was developed to couple beam elements with solid elements using the LSDYNA software. The effectiveness of this method was verified by performing simulation experiments of both a singlecolumn pier and a twospan simply supported beam bridge. Using simplified multiscale FE modeling, analyses of collapse and local failure of a multispan simply supported beam bridge and a continuous rigid frame bridge were conducted to illustrate the approach in this paper. The results demonstrate that the simplified multiscale model reasonably simulates the collapse process and local damage of complex bridges under seismic loading.
Highway bridges were severely damaged or collapsed during past strong earthquakes, which brought many difficulties to rehabilitation [
The failure of bridges under earthquake.
Gaoyuan bridge
Nanba bridge
In the past years, many researchers have applied multiscale FE modeling approach to study the failure mechanism of frame structures. Bin and Li [
In this paper, the authors primarily focused on
Three kinds of elements coupling modes are applied in the practical engineering: beams to solids, beams to shells, and shells to solids. In this paper, only the coupling method of beams to solids is presented and the other coupling principles of different elements are the same. The keyword “
Model of coupling surface.
Displacement relationships of nodes on the coupling surface.
Under axial force
Under moment
In Figure
Figure
Displacement coordinate principles of nodes under torque.
In Figure
To verify the efficiency of the rigid surface coupling method based on a simplified multiscale modeling method, a singlecolumn pier and a twospan simply supported beam bridge were simulated using the detailed 3D FE model and the multiscale FE model.
Five models were developed to illustrate the precision of the proposed method with the LSDYNA program, including solid model (Solid 164, model 1), multiscale models (model 2–model 4), and beam model (Beam 161, model 5), as shown in Figure
Five models of numerical simulation.
Natural frequencies are the inherent characteristic of structures, which directly affects their seismic response. Table
Natural frequencies of structures (Hz).
Frequency  Model 1  Model 2  Model 3  Model 4  Model 5 

1  3.92 × 10^{1}  4.03 × 10^{1}  3.95 × 10^{1}  3.97 × 10^{1}  4.06 × 10^{1} 
2  1.68 × 10^{2}  1.71 × 10^{2}  1.72 × 10^{2}  1.71 × 10^{2}  1.74 × 10^{2} 
3  2.37 × 10^{2}  2.42 × 10^{2}  2.43 × 10^{2}  2.43 × 10^{2}  2.48 × 10^{2} 
4  5.18 × 10^{2}  5.30 × 10^{2}  5.31 × 10^{2}  5.31 × 10^{2}  5.44 × 10^{2} 
5  6.28 × 10^{2}  6.31 × 10^{2}  6.32 × 10^{2}  6.32 × 10^{2}  6.32 × 10^{2} 
Error analysis of natural frequencies (%).
Frequency  Model 2  Model 3  Model 4  Model 5 

1  2.70  0.74  1.15  3.42 
2  1.66  1.87  1.76  3.50 
3  1.97  2.51  2.35  4.45 
4  2.27  2.53  2.48  4.89 
5  0.54  0.63  0.67  0.49 
The pier was fixed at the bottom and a displacement and a torque were, respectively, applied on the top of the bridge pier.
The comparison of von Mises stress between different models.
Model 1 and model 2
Model 1 and model 3
Model 1 and model 4
Compared with the von Mises stress of model 1, the maximum error of model 2–model 4 on the coupling surface is 3.43%, 0.79%, and 1.73%, which indicates that the von Mises stress of multiscale models is in good agreement with that of the solid model. Figure
Stress distribution of five models at the same time.
Contours of
According to numerical experiment of the singlecolumn pier, the displacement coordination principles on the coupling surface are verified under force, moment, and torque, respectively. Obviously, the rigid surface coupling method of simplified multiscale model guarantees the steadiness and precision of calculation.
After the modal analysis and quasistatic analysis, time history analysis was subsequently conducted to verify the reliability of the multiscale models. Figure
Loading protocol.
The comparison of von Mises stress.
Model 1 and model 2 (element A)
Model 1 and model 3 (element A)
Model 1 and model 4 (element A)
Model 1 and model 4 (element B)
It is notable that the computation time of 3D detailed model was longer than multiscale models. According to the modal analysis, quasistatic analysis, and time history analysis, it is obvious that the proposed simplified multiscale models show satisfactory precision and stability of computation in component level. Therefore, the proposed coupling method of simplified multiscale model can be used to predict the performance of structures.
The time history analysis of a twospan simply supported girder bridge was conducted for validating the precision of multiscale models in structural level. Figure
Sketch map of a twospan simply supported girder bridge.
The elevation view of bridge
Elevation view of pier
Crosssectional view of box girder
Crosssectional view of pier
The solid model and the multiscale model were established based on LSDYNA software, as shown in Figure
The model of bridge.
The solid model
The multiscale model
Basin rubber bearings were used in the prototype bridge, which were simulated by spring elements (Combi 165) in the simulation, as shown in Figure
Numerical simulation method of the bearing.
Nonlinear spring constitutive model.
HJC model proposed by Holmquist and Jonson [
The parameters of grade C50 (C30) concrete.
Parameters of strength  










0.79  1.60  0.61  0.007  7.00  48 (30)  1.0  4.00 (3.39)  12.5 
Parameters of damage  



EFMIN 
0.04  1.0  0.01 
Parameters of state equation  








0.001 (4.73 × 10^{−4})  0.1 (0.073)  16 (10)  800  85 

208 
The El Centro ground motion was adopted as the acceleration time history curves to conduct numerical analysis, as shown in Figure
El Centro ground motion.
Modal analysis is used to study the natural frequencies, which directly reflect the seismic responses of structures. Table
The values of the first eight frequencies (Hz).
Frequency  1  2  3  4  5  6  7  8 

Solid model  6.139  7.140  7.690  8.136  8.398  8.430  8.506  8.666 
Multiscale model  6.140  7.147  7.707  8.162  8.426  8.475  8.551  8.712 
The time history curves of support reaction present the positional change between the girder and the abutment with the increase of time. The bearing positions and connection surfaces are shown in Figure
Positions of bearings and connection surfaces.
Time history curves of support reaction.
Bearing 1
Bearing 2
Bearing 3
Bearing 4
Figure
Pounding forces and the corresponding time at #1 connection surface.
Model type  1  2  3  

Pounding force (kN)  Time (s)  Pounding force (kN)  Time (s)  Pounding force (kN)  Time (s)  
Solid model  15170  5.885  14470  2.205  7830  3.230 
Multiscale model  17120  5.680  14200  2.195  7190  3.135 
Pounding forces and the corresponding time at #2 connection surface.
Model type  1  2  3  

Pounding force (kN)  Time (s)  Pounding force (kN)  Time (s)  Pounding force (kN)  Time (s)  
Solid model  14540  2.535  9930  2.660  8060  6.140 
Multiscale model  16330  2.525  12670  3.550  10430  6.125 
Pounding forces and the corresponding time at #3 connection surface.
Model type  1  2  3  

Pounding force (kN)  Time (s)  Pounding force (kN)  Time (s)  Pounding force (kN)  Time (s)  
Solid model  21390  2.595  10190  5.105  9670  2.780 
Multiscale model  23550  2.640  9590  4.920  7270  5.510 
Comparison of pounding process at the three connection surfaces.
#1 connection surface
#2 connection surface
#3 connection surface
The base shear forces of bridge piers directly represent the damage and failure of bridges. Figure
Comparisons of responses at the base of piers with different models.
Shear force
Moment
Stress distribution at the base of piers under the peak moment.
According to the numerical experiment of singlecolumn pier and a twospan simply supported girder bridge, the effectiveness of simplified multiscale modeling approach was verified. According to Table
Computation of computational efficiency of different model.
The number of elements  Computational time (s)  

Solid model  20532  13028 
Multiscale model  13046  6707 
A multispan simply supported girder bridge was collapsed in the Wenchuan earthquake; the collapse analysis of this bridge was conducted using LSDYNA program based on simplified multiscale modeling approach. Figure
Sketch map of bridge.
The elevation view of the bridge
Detailed view of bridge
The vulnerable parts of the bridge were established by solid elements, including cap beams, base of piers, top of piers, collar beams, and the end of girders. Beam elements were used in other parts of the bridge to increase the efficiency of computation. The multiscale model of the bridge is shown in Figure
The multiscale model of the bridge.
The material constitutive relation of concrete utilized the HJC model, as shown in Table
Figure
The collapse process of bridge.
Figures
The pounding force of girder (4E).
The velocity of girder (4E).
The length of the fourspan RC continuous rigid frame bridge is 450 m. A 360 mm expansion joint was designed between the girders and the abutments. The elevation view of bridge structure and crosssectional view of AA and BB are shown in Figure
Sketch map of RC fourspan continuous rigid frame bridge.
The elevation view of bridge
The crosssectional view of AA and BB
The solid elements were used to simulate concrete in the detailed area, including the top and the bottom of bridge piers and the end of main girder and abutments. Other parts of the bridge were established by beam elements to reduce the cost of computation. For solid elements, the HJC material constitutive relation [
The multiscale model of the bridge.
Figure
The first ten modes of the bridge.
According to the dynamic characteristics of the bridge, actual earthquake records during the 1999 ChiChi earthquake were selected to simulate earthquakeinduced collapse of the continuous rigid frame bridge. The effect of bidirectional earthquake was taken into account by inputting seismic waves in both the longitudinal and the transverse directions, as shown in Figure
Ground motion record.
Longitudinal direction
Transverse direction
To verify the rationality of the simplified multiscale model under the selected seismic wave, the elastic time history analysis was conducted. Elastic material constitutive models were selected for the model and the pounding effects between the girders and abutments were not taken into consideration.
Figure
Relative displacement at the top of the piers.
In the longitudinal direction
In the transverse direction
In the transverse direction, peak displacements at the top of the three piers are 75.8 mm (
The elastic time history analysis has proven that seismic response of the bridge is mainly dominated by the first or second mode. Therefore, the multispan model is reliable to simulate the collapse process of bridge.
In this bridge, contact problems were controlled by keyword “
In order to simulate the collapse process, the peak ground acceleration in the longitudinal direction and transverse direction is 3.19 m/s^{2} and 2.42 m/s^{2}, respectively. The whole collapse process sustains 12.5 s; the mode of collapse and local failure are shown in Figure
Collapse process of the bridge (
Figure
The status of bridge at 4 s and 6 s is shown in Figure
Collapse process of the bridge (
At 6 seconds, plastic hinges appeared at the top and base of #3 pier and #3 pier leaned to A1 abutment at an angle of 1.8 degrees, which made the forth girder fell off from A2 abutment. #2 pier still possessed great capacity and good performance in resisting the transverse force because the stiffness of pier is small, which limited the lean of #1 pier.
At 8 seconds, the forth girder became cantilever beam due to the failure of A2 abutment, as shown in Figure
Collapse process of the bridge (
At 10 seconds, the forth girder broke off at start element. #3 pier lost its capacity completely and the third girder became cantilever beam gradually. Under the interaction of earthquake force and collapse effect, plastic hinges appeared at #1 pier and leaned to A2 abutment with an angle of 5 degrees. Moreover, plastic hinges occurred at the base of #2 pier at a space of 1.7 m and leaned to A2 abutment with an angle of 2.2 degrees. Apparently, the whole bridge presented unstable status.
At 11 second, the concrete at the base of three piers crushed and reinforcement was buckled, as shown in Figure
Collapse process of the bridge (
The following conclusions can be drawn from the numerical simulation:
Based on the theory of multiscale FE modeling, the coupling method of rigid surface and displacement coordinate formulas were developed in this paper and reliability of coupling method was verified in both the component and structural levels. The numerical experiments show that the simplified multiscale FE model offers a good balance between efficiency and accuracy.
The earthquakeinduced collapse analysis of a multispan simply supported girder bridge was conducted using simplified multiscale FE modeling developed in this paper. The results show that the pounding forces at expansion joints are an important cause for collapse of this bridge. The energy dissipating dampers as restrainers for expansion joints are significantly effective in limiting the relative opening displacements at expansion joints of highway bridges.
The seismic collapse analysis of a fourspan RC continuous rigid frame bridge was conducted under bidirectional earthquake excitation based on simplified multiscale FE modeling approach. The results demonstrate that the simplified multiscale FE model developed in this paper can simulate the collapse process and local damage of complex bridge structures accurately and efficiently. Although the structural integrity of RC continuous rigid frame bridge is good, the main bridge structure will be instable and will collapse when the shortest bridge pier is invalid. It is important to conduct reasonable stiffness distribution for bridge piers with different heights in seismic design of RC continuous rigid frame bridges.
The authors declare that they have no conflicts of interest.
The authors gratefully acknowledge the support for this research by the National Natural Science Foundation of China under Grants no. 51578022 and no. 51678013 and the research project of Beijing Municipal Commission of Education under Grant no. KZ201610005013.