## 5 Near-Field Foundation-Bridge Model and Numerical Verification  
In consideration of general vibration, research and testing using frequency domain methods can be achieved. But, considering the earthquake action, it is difficult to verify the test. It needs to be  
Figure 4: Relationship between wave velocity and horizontal apparent wave velocity.  
Figure 5: Schematic diagram of the free-field displacement expansion solution.  
verified by shaking table test or large-scale test like centrifuge. This paper mainly proposes a numerical calculation method that simultaneously considers multi-point seismic and soil-dynamic interactions. And for the curved beam bridge, there is no test to verify it. The new method proposed in this paper is mainly compared with the extended numerical method that considers all the integral soil under the bridge. The results show that the method can achieve the accuracy of the extended numerical solution.  
As shown in Figure 6, a pier model is located in a homogeneous semi-infinite space, regardless of the damping effect, and the pier parameters are shown. According to the characteristics of the soil-bridge structure system, a near-field foundation-bridge system model (right) shown in Figure 7 was established, and a direct solution model (left) was established for comparison to verify the model shown in Figure 7. In the direct solution model, a large space should be selected to eliminate the influence of the boundary, and the size of the selected soil region is \(1000\times 1000\times 200\) m. In the near-field foundation-structure system model, a small part of the soil area near the pier of dimensions \(10\times 10\times 60\) m was cut, and the equivalent viscoelastic artificial boundary element was introduced on the intercepting boundary. This is about 2 or 3 times by the dimension of the foundation.  
The displacement time history for input wave is shown in Figure 8. In the direct solution model, the horizontal displacement time history was applied at the bottom of the soil layer. In the substructure model, the input of ground motion was applied by transforming the displacement response of the  
Figure 6: Simplified model of the pier.  
Figure 7: Direct solution model (**Left**) and the substructure model (near-field.foundation-bridge model) (**Right**).  
intercepted area in the direct model into the equivalent load. The acceleration and displacement response time history at the media surfaces of the two models and those on the pier top are shown in Figure 9.  
Figure 9 compares the surface acceleration response and displacement response of the two models. The blue lines in (a) and (b) are the theoretical solutions of the free-field surface response of the without-pier model. The acceleration and displacement responses of the two models are highly fitted and are close to the free field theoretical solution of the without-pier structure. The impact of the pier on the free field response is negligible. Because of the kinematic interaction and the inertial interaction between the soil and foundation, the acceleration and displacement responses at the surfaces of the two models both oscillate after 1.17 s. Further, (c) and (d) show the acceleration and displacement response time history of the pier top, respectively. The results of the two models are very close. Therefore,  
Figure 8: Displacement loading function.  
Figure 9: Comparison of model responses.  
the seismic response analysis based on the near-field foundation-bridge model is correct and has sufficient accuracy.