Assessment of seismic responses of skewed bridges with bidirectional collision effect

Unbonded laminated elastomeric bearings have been used widely for skewed highway bridges in China. This type bearing is likely to slip during an earthquake, which will lead to large girder displacement and collision phenomenon. In order to discuss the effect of different collision models and design parameters on the seismic response of skewed bridges in China, 3D beam-stick models of three-span highway bridges with skew angles varying from 0°to 60°are developed in this paper. The results show that the collision effect is the principal contributor to girder rotation. The collision model considered nonlinear property in this paper can capture the irregular behavior of skewed bridges correctly. The uneven collision phenomenon will be more obvious for changing the initial gap of shear keys, and the overall seismic responses are likely to become complicated. Then, an appropriate gap is recommended to reduce the seismic damage of skewed bridges.

model was used. [9] studied the rotation mechanism of simply-support bridges. A double broken line model was used to simulate the pounding behavior in the study, but it did not include shear keys failure. From what has been discussed above, the skew angle and collision effect are the main factors to make the seismic response of skewed bridges differ from regular bridges. In order to investigate the seismic response of skewed bridges with unbonded laminated elastomeric bearings in China, threedimensional beam-stick models of three-span highway bridges with skew angles varying from 0°to 60°are developed. Subsequently, results of a comprehensive study on the effect of skewed angle, collision models and shear keys design parameters, are discussed.
2 Modeling methods

Modeling of skewed bridge
The benchmark bridge is a three-span skewed bridge with skew angle of 45° in Sichuan province of China. Each span length is 25m. The superstructure consists of four precast concrete box girders with height of 1.4m. Three-column bent is constructed with diameter of1.3m, the reinforcement ratios and stirrup ratios are 1.25% and 0.87%, respectively. The height of the columns from the top of the footing to the bottom of the cap is 5 m. The unbonded laminated elastomeric bearings are placed directly between superstructure and substructure without any anchoring. The site class of the bridge is Type II in Chinese Standard [10].
SAP2000 program was used to develop detailed nonlinear 3D model ( Figure 1). The linear elastic beam-column elements were selected to simulate the superstructure since it is not expected to damage during an earthquake. The plastic hinges were assumed to form at the top and bottom of the columns, the fiber P-M2-M3 (PMM) type hinge was used to define the characteristics of the plastic hinges. The cross section should be divided into enough fibers so that the properties difference between column cross section and fiber section are not more than 10%, and then the reasonable plastic hinge length and the position of the plastic hinges are determined [11]. The collision effects between girders and abutments as well as this between girders and shear keys were simulated using a combination of link elements and GAP element, which would be activated/inactivated through the gap element closing/opening. Soil-structure interaction was not taken into account. The benchmark bridge model was altered to develop models with various skew angles.

Shear keys behavior
Shear keys can restrict girder displacement effectively, but the collision effect between girders and shear keys may increase the seismic force in substructure. The commonly used shear keys mechanism models are linear elastic model and elastic-plastic model.

2.2.2
Elastic-plastic model. [12] conducted experimental research earlier on the nonlinear characteristics of shear keys. They found that the shear keys performed five working levels under collision effect and put forward two spring hysteresis model by separating the contribution of concrete and steel skeleton respectively. [13] modified the Megally model by considering with shear keys construction characteristics in China, the hysteresis rule is shown in Figure 2, and the related parameters definition can refer to [13].

Selection and input of ground motions.
According to site conditions and Chinese Standard [10], the target spectrum was established. Then three artificial ground motions were generated and four real ground motions were selected from PEER database. The match status between the average response spectrum of the seven ground motions and the target spectrum is shown in Figure 4. The PGA levels were amplified to 0.4g, and all ground motions were input along the longitudinal and transverse directions (X and Y) respectively.

The influence of difference collision models
Collision effect is the main factor influencing the seismic responses of the skew bridge. Three finite element models taking longitudinal and transverse collision effect into account were built to investigate the influence on the seismic response of the skewed bridge, so as to determine reasonable collision analysis model: (1) Model 1: Assuming that the gap is large enough so that no pounding effects happen.
(2) Model 2: It is supposed that shear keys would failure if its ultimate capacity has been reached, its function to restrict girders displacement would degrade, and abutment-soil interaction may perform nonlinear property under collision effect. The hysteresis rules are shown in Figure2 and Figure3. A combination of link elements considered nonlinear property and GAP element was used to simulate the collision effect.   Figure 5 presents the variation of girder rotation as skew angle increasing for different collision models. For the bridge with skew angle of 0° (regular bridge), there is no girder rotation occurring under any cases. For skewed bridges (skew angle larger than 0°), the girders perform small rotation without considering collision effect. While after taking collision effect into account, different levels of girder rotation occurs, and the rotation angle varies for different collision effect. The girder rotation angles of model 2, which considered nonlinear property of shear keys and abutments, were larger than that of model 3 considered linear property of shear keys and abutments. The ration between two models reached 3.02 times under the skew angle of 30°, which shows that collision effect plays an important role in the girder rotation of skew bridges. This can reasonably explain the common girder rotation of skewed bridges, which is often accompanied by the expansion joints and shear keys failure during the 2008 Wenchuan earthquake. In Figure 5, it also can be seen that the girder rotation angle increases as skewed angle increases, but does not increase linearly, which indicates skew angle complicates the seismic response of skewed bridges.  Figure 6 shows the impact force at 1# (obtuse corner) and 2# (acute corner) of right abutment. When skew angle less than 45°, the impact force at 1# (obtuse corner) and 2# (acute corner) are almost equal. Once skew angle reaches to 45°, the differences between the two impact forces get larger, especially for the model considered collision effect with nonlinear property of shear keys and abutments, the uneven collision phenomenon is obvious. To describe the uneven collision phenomenon between girders and abutment, impact coefficient k is defined as the ratio of the impact force at obtuse corner to the impact force at acute corner.   Figure 7: Impact coefficient k of right abutment The impact coefficient k of both side abutments changing along with skew angle increases is illustrated in Figure 7. The impact coefficient k increases obviously with the increases of skew angle. For model 2, which considered nonlinear property of shear keys and abutments, the impact coefficient k of right abutment is nearly 1.0 when the skew angle was 0°, and reached to 2.44 at skew angle of 60°, which shows that the uneven collision phenomenon between girders and abutments occurs significantly. The different collision models used in modeling skewed bridges will result in different collision effect, the degree of uneven collision of model 2 was greater than that of model 3. The differences between two models get larger with the increases of skew angle. In summary, the model 3 considered linear property of shear keys and abutments significantly overestimates the impact force between girders and abutment, and it weaken the uneven collision effect, which cannot accurately reflect the rotation features of the skewed bridges girders.

3.1.3
The moment of bents. Figure8 shows the base moment for bent 1-1. It can be observed that the base moments (Mx and My) of bent 1-1 increase with the skew angle under all cases. By comparison the base moments (Mx and My) of bent 1-1of the three model, it indicates that the base moments get larger under the collision effect, which suggests that the collision effect enlarges the seismic response of skewed bridges. The influence on seismic response of skewed bridges considered linear property of shear keys and abutments is more obvious, the base moment Mx of bent 1-1 of model 3 is larger than that of model 2 by a factor of 2.2 ( Figure 8(b) ) , which overestimated significantly the seismic responses at substructure.

Effect of the initial gap of shear keys
The initial gap of shear keys is an important parameter of the collision effect between girders and shear keys, which plays significant role in restricting the superstructure displacement and influencing bridges seismic response. Based on the finite model considered nonlinear property of shear keys and abutments (model 2), the influence of shear keys design parameters on seismic response of skewed bridges is investigated by changing the initial gap of 5cm, 8cm, 12cm and 15cm.  Figure 9: Girder rotation The girder rotation angles against with the initial gap are plotted in Figure 9. As can be seen from Figure 9, the trends of the rotation angles with the initial gap first decrease and then increase, especially for skew angle reaches to 45°. When the initial gap was 5cm, the impact force between girders and shear keys is large enough to make shear keys failure, then it loss the function of displacement restricting. While the initial gap was 15cm, the girders have enough space to move, meanwhile, the earthquake energy could be dissipated through unbonded laminated elastomeric bearings sliding and bents yielding, while the shear keys would remain elastic under collision effect, the displacement restricting function of shear keys could not work well, and the girder rotation angle is large enough. So, it is important to select an appropriate gap to assure shear keys restrict girder displacement effectively, but not increase the seismic force in substructure.  Figure 11: Impact coefficient k of right abutment The uneven collision phenomenon at right abutment under the application of different initial gap with 5cm, 8cm, 12cm and 15cm are investigated. As can be seen in figure10, both impact forces at 1# (obtuse corner) and 2# (acute corner) decrease with skew angles increase, and it decreases more obvious at 2# (acute corner). When skew angle is the same, the impact forces at 1# (obtuse corner) under different initial gap are essentially equal, the maximum difference happened at the skew angle of 30°with the value of 2.1%. While the impact forces at 2# (acute corner) under different initial gap decreased significantly with the increases of the initial gap, the impact force for 5cm gap was 1.96 times larger than that for 15cm at the skew angle of 60°. According to the previous analysis of the girder rotation, the girders is induced to rotate anticlockwise firstly by the impact abutment force at obtuse corner, while the shear keys in transverse direction limit this rotation, and push the girders rotate clockwise, which increases the impact force at acute corner. It can be concluded that the impact effects in longitudinal and transverse directions influence each other and change the girder rotation mechanism. The interaction of collision effects in two directions will weak with the increase of the initial gap, the impact force decreases gradually at acute corner and lead to impact coefficient k increase, the larger the skew angel is, the more impact coefficient k increases (Figure 11), which indicates the irregular behavior of skewed bridges increase.

Conclusions
(1) The skew angle will complicate the seismic response of skewed bridges. As the skewed angle increase, the uneven collision phenomenon at abutment become more obvious, and the interaction of collision effect in longitudinal and transverse directions may become closer, which means the irregular behavior of skewed bridges increased significantly.
(2) The collision effect between girders and abutment as well as this between girders and shear keys are the principal contributor to girder rotation. Different collision models lead to significant difference in seismic response. The collision effect considered linear property of shear keys and abutments could be better to restrict girder displacement, but it lead to overestimate the seismic force in substructure, especially for bridge with large skew angle. The collision effect considered nonlinear property of shear keys and abutments has the capability to capture the irregular behavior of skewed bridges. It is recommended that the elastic-plastic collision model should be used in the seismic design of skewed bridges.
(3) The initial gap of shear keys has great impact on the seismic response of skewed bridges. The girder rotation angles first reduce and then increase with the increase of the initial gap. The change of the initial gap will lead to suffer some degree of the uneven collision. An appropriate gap can decrease the seismic damage. (4) According to previous literature, the soil-structure interaction plays an important role in structural vibration, it will increase structural damping. And some studies also noted that the input angle of ground motion may have great influence on the seismic response of skewed bridges. The further re- search will be focused on the effect of soil-structure interaction and determining the most unfavorable input angle of ground motion of skewed bridges.

Funding
This study was supported by the NINGXIA Natural Science Foundation of China (Grant No.NZ1606).