## 5 Results and Discussion
### Comparison of Moving Load Models  
It is assumed that \(v_{\mathrm{load}}=100\) m/s, \(\bar{I}/L=0.3\), \(\lambda_{1}=\lambda_{2}=0.5\), \(n=2\). Figure 11 compares the displacement histories of the midpoint of the beam under the three kinds of loads in different situations. It can be seen that there are apparent differences between the  
Figure 10: Three models for the moving load. (**a**) Non-uniformly distributed moving load; (**b**) Uniformly distributed moving load; (**c**) Concentrated moving load. Figure 9: Variation of the foundation stiffness and damping with variation coefficients \(\lambda_{1}\) and \(\lambda_{2}\). (**a**) Foundation stiffness; (**b**) Damping coefficient. dynamic responses computed by the three load models in all cases. For example, in the case of power material gradient and a linear varying foundation, as Figure 11a shows, compared with the first peak displacement under the non-uniformly distributed load, that under the concentrated load is 4.82% higher, while that under the uniformly distributed load is 3.33% lower. For another example, in the case of exponential material gradient and parabolic varying foundation, the first peak displacement under the concentrated load is 4.70% higher than that under the non-uniformly distributed load, while the one under the uniformly distributed load is 3.17% lower than that. In the other two cases, as shown in Figure 11b,c, similar phenomena are also observed. The load models also affect the calculated results of the maximum displacement \(W_{\text{max}}=\max(|w(x,t)|)\) as the loads travel through the beam, as shown in Figure 12. Compared with the maximum displacement computed by the non-uniformly distributed load model, the results computed by the concentrated load model are greater, while the results calculated by the uniformly distributed load model are smaller, regardless of the travel speed. The maximum difference between the results obtained by the non-uniformly distributed load model and the concentrated load model is 5.41% in the case of the power-graded material and linearly varying foundation, as shown in Figure 12a. In the other three cases, this value is respectively 5.45%, 5.46%, and 5.39%. The maximum difference between the results obtained by the non-uniformly distributed load model and the uniformly distributed load model in the four cases are respectively 3.33%, 3.24%, 3.20%, and 3.17%. It should be pointed out that these differences increase as the ratio of the load span to the length of the beam (\(\tilde{l}/L\)) increases, as shown in Figure 13. For non-uniformly and uniformly distributed loads, the increase in the load span reduces the maximum displacement, which is more obvious for the uniform one. This can be concluded by comparing the downward trend of the green and red lines in Figure 13.  
Figure 11: Comparison of the displacement histories at the midpoint computed by the three load models. (**a**) Power material, linear foundation; (**b**) Power material, parabolic foundation; (**c**) Exponential material, linear foundation; (**d**) Exponential material, parabolic foundation. Although the total work of the concentrated moving load is equivalent to that of the non-uniformly distributed moving load, the concentrated load model makes the external  
Figure 12: Comparison of the maximum displacement when the moving loads travel through. (**a**) Power material, linear foundation; (**b**) Power material, parabolic foundation; (**c**) Exponential material, linear foundation; (**d**) Exponential material, parabolic foundation. Figure 13: Influence of the load span on the maximum displacement, exponential material and parabolic foundation. (**a**) \(v_{\text{load}}=25\) m/s; (**b**) \(v_{\text{load}}=50\) m/s; (**c**) \(v_{\text{load}}=100\) m/s; (**d**) \(v_{\text{load}}=200\) m/s. force act on the concentrated point, which is difficult to make appear in practice, and the concentrated load will increase the local displacement, resulting in the distortion of the calculation results.