Seismic Collapse Analysis of RC Highway Bridges Based on a Simplified Multiscale FE Modeling Approach

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 LS-DYNA software.The effectiveness of this method was verified by performing simulation experiments of both a single-column pier and a two-span 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.


Introduction
Highway bridges were severely damaged or collapsed during past strong earthquakes, which brought many difficulties to rehabilitation [1]. Figure 1 shows the unseating failure of bridge spans during the 2008 Wenchuan earthquake with a magnitude of ML = 8.0 [2].Therefore, the failure prediction of bridges, damage mitigation, and unseating prevention at expansion joints of multispan bridges are important issues of seismic design for highway bridges.There are three methods used for simulating the earthquake-induced failures of bridges: Discrete Element Method (DEM), Applied Element Method (AEM), and Finite Element Method (FEM).Currently, FEM is one of the most commonly used methods of simulating the collapse process of bridges [3][4][5].A detailed 3D FE model elaborately simulated the damage location and the local failure mechanism of highway bridges [6][7][8].
Although detailed 3D FE model provides good precision, the cost of computation is high.It is necessary to develop an appropriate method that not only accurately reflects the damage process but also improves the computational efficiency.To satisfy the demands of engineering practice, multiscale FE modeling uses different types of elements to simulate different parts of structures, which is an effective method to balance precision and efficiency [9].
In the past years, many researchers have applied multiscale FE modeling approach to study the failure mechanism of frame structures.Bin and Li [10] developed a suited concurrent multiscale approach to simulate the trans-scale failure of large concrete structures and performed tests to compare experimental observations with numerical analysis results.Tao and Nie [11] proposed a multiscale modeling method for evaluating the deformation mechanism of composite joint substructures in a CFST (concrete filled steel tubular) column-composite floor frame structural system.Margiacchi et al. [12] simulated the nonlinear static response of an infilled moment-resisting steel frame by using a multiscale FE model.It is important to study the coupling modes for realizing the incorporation of different dimensional elements in multiscale numerical model.McCune et al. [13] proposed a method that can correctly couple beam elements to shell elements and shell elements to solid elements, and Shim et al. [14] provided a novel relationship among nodal degrees of freedom between the different dimensional element types in elastic regions.In addition, the multiscale numerical simulation was also applied to bridge engineering to analyze the damage process of bridges [15][16][17].Wang [18] investigated the failure of a multispan simply supported girder bridge subjected to pounding and sliding under severe ground motion by numerical simulation.Chan et al. [19] studied the nonlinear physics-based modeling to simulate the dynamic characteristics and static response of civil infrastructure.Li et al. [20] proposed erosion criterion of multiscale FE model and numerical simulations of continuous RC bridge were conducted to study failure process under strong earthquake.Therefore, the multiscale FE modeling is an effective method to predict seismic collapse process of bridges.In this paper, the authors primarily focused on (1) the coupling method of rigid surface based on a simplified multiscale FE modeling approach, where the formula of displacement coordination was developed and verified for predicting the failure of bridge structures considering both computational precision and efficiency, and (2) the seismic collapse and the local failure process analysis of bridges using the LS-DYNA software for engineering applications.The numerical results were obtained to illustrate bridge dynamic failure due to extreme earthquake loading.

Rigid Surface Coupling Method
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 " * CONSTRAINED NODE RIGID BODY" was used in the simulations to couple beams and the rigid surface of solids in the LS-DYNA software, which provided good computational stability.The principle can be described as the nodes of beam elements on the coupling surface are master nodes and the nodes from the rigid surface of solid element are slave nodes, as shown in Figure 2. It is worth mentioning that the translation and rotation degrees of freedom of rigid surface are dominated by the master nodes.Figure 3 shows the displacement coordination principles of nodes on the coupling surface under axial force and moment.
In Figure 3, node  is the master node of beam element on the coupling surface, and   ( = 1, 2, 3, . . ., ) refers to the slave nodes of solid element.The nodes displacements of the detailed 3D solid element satisfy the following equations: where Δ  , Δ  , Δ   , and Δ   are the displacements that belong to node  of beam element and node   of solid element, respectively;  is the rotation angle of beam element at node ;   is the distance between node  and node   .Figure 4 shows the displacement coordination principles of nodes on the coupling surface under torque.
In Figure 4, the displacement coordination equation can be described as follows: where Δ   and Δ   are the displacements in the  and  directions of node   . is the rotation angle of the coupling surface about -axis.

Numerical Verification of Rigid Surface Coupling Method
To verify the efficiency of the rigid surface coupling method based on a simplified multiscale modeling method, a singlecolumn pier and a two-span simply supported beam bridge were simulated using the detailed 3D FE model and the multiscale FE model.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 7 shows the stress distribution of the five models at the same time.It can be seen that the rigid surface coupling method based on a simplified multiscale modeling method is effective.

Numerical Experiments of a
(2) Applying Torque on the Top of Bridge Pier.The contours of -coordinate of model 1 and model 3 when subjected to applied torque are shown in Figure 8. Obviously, there is a little difference between the solid model and the multiscale model under torque.
According to numerical experiment of the single-column 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.

Time History Analysis.
After the modal analysis and quasi-static analysis, time history analysis was subsequently conducted to verify the reliability of the multiscale models.Figure 9 shows the loading protocol of the numerical simulation.Figure 10 shows the comparison of von Mises stress at the same position between the solid model and multiscale models.Apparently, the von Mises stress of multiscale models is in good agreement with that of the solid model.
It is notable that the computation time of 3D detailed model was longer than multiscale models.According to the modal analysis, quasi-static 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.

Numerical Verification of a Two-Span Simply Supported
Girder Bridge

Bridge Model.
The time history analysis of a two-span simply supported girder bridge was conducted for validating the precision of multiscale models in structural level.Figure 11 shows the elevation view of the bridge (30 m + 30 m), the detailed elevation view of the pier, the cross-sectional view of the box girder, and the cross-sectional view of the pier.A 60 mm gap was set between the two girders as well as the girder and the abutment as expansion joint to allow for contraction and expansion of bridge girders.
The solid model and the multiscale model were established based on LS-DYNA software, as shown in Figure 12.The solid elements (Solid 164) were used in the detailed area.In order to increase the computational efficiency, some parts were modeled by beam elements (Beam 161) in 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 13.The spring elements only transform vertical force to girders and piers.Moreover, the vertical displacement of two ends of the spring is consistent.Mechanical behavior of the basin rubber bearing was modeled by nonlinear spring constitutive model as shown in Figure 14.The failure displacement can be defined by using keyword " * SECTION DISCRETE."HJC model proposed by Holmquist and Jonson [21] was utilized to describe the constitutive relation of concrete in the present study.Grade C50 concrete and grade C30 concrete (according to the Chinese concrete grade) were used for girders and piers, respectively.The parameters of grade C50 concrete and grade C30 concrete are listed in Table 3.The numbers in the bracket are the parameters of grade C30 concrete which are individually different from the parameters of grade C50 concrete.
The El Centro ground motion was adopted as the acceleration time history curves to conduct numerical analysis, as shown in Figure 15.In order to reduce the computational time, only the first 10 seconds of the El Centro ground motion were applied at the bottom of the piers in the longitudinal direction.

Modal Analysis.
Modal analysis is used to study the natural frequencies, which directly reflect the seismic responses of structures.Table 4 shows the values of the first eight frequencies.Obviously, the simulation results of the multiscale model are in agreement with the solid model, which validates the reliability of the simplified multiscale model.

Comparison of Support Reaction.
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 16. Figure 17 shows the time history curves of support reaction between the two models, including bearing 1, bearing 2, bearing 3, and bearing 4. Figure 17(a) presents a little difference of the failure time between the solid model and multiscale model.Figure 17(b) depicts that the peak of support reaction at 1.8 s has slight difference between the two models.Considering the efficiency of computation, these differences are accepted in practical engineering.Comparing the pounding responses between the solid model and the multiscale model, it can be seen that the pounding responses of the two models are identical in the first five seconds.However, there is a little difference between the two models in the last five seconds.The differences are accepted compared with the reduction of computational time.

Comparison of Responses at the Base of Bridge Piers.
The base shear forces of bridge piers directly represent the damage and failure of bridges.Figure 19 shows the comparisons of base shear force and moment of bridge piers with different models.The base shear forces and moments of the two models are essentially identical, especially in the first six seconds.The peak of base shear force and the peak of moment of the        multiscale model are 12.6% and 8.7% higher than those of the solid model, respectively.The main reason for the difference is that the elastic constitutive relation was utilized for beam element of multiscale model and the damage constitutive relation was used for concrete of solid model; therefore, the pier of multiscale model is stiffer than that of the solid model.The comparison of stress distribution at the base of piers under the peak moment is shown in Figure 20.It can be seen that the stress distribution of multiscale model is approximately in accordance with that of the solid model.

Comparison of Computational Efficiency.
According to the numerical experiment of single-column pier and a two-span simply supported girder bridge, the effectiveness of simplified multiscale modeling approach was verified.According to Table 8, it is apparent that the multiscale model is more efficient than the solid model.Although there is a little difference between the two models, the precision of the simplified multiscale model is accepted in practical engineering.

Collapse Analysis of a Multispan Simply Supported Girder
Bridge.A multispan simply supported girder bridge was collapsed in the Wenchuan earthquake; the collapse analysis of this bridge was conducted using LS-DYNA program based on simplified multiscale modeling approach.Figure 21 presents the elevation view of the bridge and the detailed view of the bridge pier, cap beam, and girder.The effect of structure-pier-soil interaction was not considered.
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 22.The material constitutive relation of concrete utilized the HJC model, as shown in Table 3.The rubber bearings were established by solid models, and " * MAT MOONEY RIVLIN RUBBER" was used as material constitutive relation of rubber.Seventeen seconds (36 s-53 s) of actual seismic record of Wenchuan earthquake through filtering was adopted as earthquake input [22].
Figure 23 shows the detailed damage and collapse process of the bridge.Obviously, the failure of A2 abutment results in the girders of the forth span falling down under pounding force.The collapse mode conducted by the numerical simulation in this paper is consistent with the real bridge collapse in Wenchuan earthquake.
Figures 24 and 25 show the pounding force and velocity of the girder on the forth span to present the demolishing process, respectively.Poundings result in cracks and failure of A2 abutment at 4 seconds.Poundings lead to most girders of the forth span successively falling down at 8 seconds.Finally, the pounding force and velocity of girder reach peak values of 2330 kN and 1.46 m/s, respectively, which induce to all girders of the forth span dropped down to the ground.

Collapse Analysis of RC Continuous Rigid Frame Bridge.
The length of the four-span 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 cross-sectional view of A-A and B-B are shown in Figure 26.
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 [21] was used to model the concrete response; the parameters of grade C50 concrete are listed in Table 3. " * MAT CSCM CONCRETE" model was used for shear keys and the backwall of abutments.The multiscale model of bridge is shown in Figure 27.

Selection of Ground Motion
Records. Figure 28 presents the first ten modes of bridge to display the dynamic characteristics.The first mode is in the longitudinal direction, and the primary mode in the transverse direction is the second mode, while the third and subsequent modes are a combination of longitudinal and transverse modes.
According to the dynamic characteristics of the bridge, actual earthquake records during the 1999 Chi-Chi earthquake were selected to simulate earthquake-induced 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 29.The uppermost 20 s of the acceleration history was utilized to simulate the failure process of the bridge structure to enhance computational efficiency.The peak ground acceleration is 1.12 m/s 2 ( = 9.3 s) in the longitudinal direction and 0.85 m/s 2 ( = 2.74 s) in the transverse direction.

Elastic Time History Analysis.
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 30 shows the relative displacement at the top of piers in the longitudinal and transverse directions.The peak displacement in the longitudinal direction is 189.6 mm at 8.83 s and the time is consistent with the time of the peak acceleration of ground motion in longitudinal direction.Obviously, the displacement tendencies of different bridge piers are the same in the longitudinal direction without consideration of traveling wave effect.Seismic response of the bridge in the longitudinal direction is primarily dominated by the first mode.
In the transverse direction, peak displacements at the top of the three piers are 75.8mm ( = 8.30 s), 136.9 mm ( = 8.25 s), and 87.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.

Simulation of Collapse Process.
In this bridge, contact problems were controlled by keyword " * CONTACT AUTOATIC SINGLE SURFACE" based on the penalty function method.The erosion criterion of elements was defined by keyword " * MAT ADD EROSION," which was used to rule out elements that have no contribution to resisting earthquake loading.Failure of concrete was controlled by the failure mode of tension and pressure.Failure parameter of the first principal strain is 0.002 and that of the third principal strain is −0.005.Besides, the failure strain of reinforcement is 0.12.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 31.
Figure 31 shows the seismic collapse process of the whole bridge at 0 s and 2 s.It can be seen that the cover concrete at the corner of piers (#1 and #3) got damaged in 2 s; stress concentration appeared between the start element and the pier.Moreover, #2 pier was intact because the stiffness of #2 pier is small.
The status of bridge at 4 s and 6 s is shown in Figure 32.At 4 seconds, the cover concrete cracked at the top and base of #2 pier.For #1 and #3 piers, the cover concrete in the plastic hinge regions mostly failed and the concrete core went into plastic state gradually.
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 33.For #3 pier, damage occurred on the top deck and base deck of start element and the lean of pier appeared.Lean of #1 pier and #2 pier was not observed and the first girder was still safe.
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.

Conclusions
The following conclusions can be drawn from the numerical simulation: (1) 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.
(2) The earthquake-induced 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.(3) The seismic collapse analysis of a four-span 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.

Figure 1 :Figure 2 :
Figure 1: The failure of bridges under earthquake.

Figure 3 :Figure 4 :
Figure 3: Displacement relationships of nodes on the coupling surface.

Figure 5 :
Figure 5: Five models of numerical simulation.

Figure 6 :
Figure 6: The comparison of von Mises stress between different models.

Figure 7 :Figure 8 :
Figure 7: Stress distribution of five models at the same time.
Figure 18  compares the pounding force time histories between the solid model and the multiscale model at different connection surfaces.Moreover, Tables 5-7 list the three main pounding forces and the corresponding time at the three connection surfaces.
Cross-sectional view of pier

Figure 11 :Figure 12 :Figure 13 :
Figure 11: Sketch map of a two-span simply supported girder bridge.

Figure 16 :Figure 17 :
Figure 16: Positions of bearings and connection surfaces.

Figure 18 :
Figure 18: Comparison of pounding process at the three connection surfaces.

Figure 19 :
Figure 19: Comparisons of responses at the base of piers with different models.

Figure 20 :
Figure 20: Stress distribution at the base of piers under the peak moment.
3 mm ( = 8.36 s), as shown in Figure 30(b).The times corresponding to the peak displacement are approximately equal for different bridge piers.Moreover, the values of the peak displacements are related to the height of piers.Seismic response of the bridge in the transverse direction is mainly dominated by the second mode.

t = 12 .
66 s; most of girders of the forth span abutment failed.t = 6.99 s; the backwall of A2 backwall of A2 abutment.t = 4.03 s; cracks appeared on the the third span, respectively, fell t = 10.35 s; girders of the forth span down.

Figure 23 :Figure 24 :
Figure 23: The collapse process of bridge.

Figure 28 :
Figure 28: The first ten modes of the bridge.

Figure 30 :
Figure 30: Relative displacement at the top of the piers.

Table 4 :
The values of the first eight frequencies (Hz).

Table 5 :
Pounding forces and the corresponding time at #1 connection surface.

Table 6 :
Pounding forces and the corresponding time at #2 connection surface.

Table 7 :
Pounding forces and the corresponding time at #3 connection surface.

Table 8 :
Computation of computational efficiency of different model.