## 5 Conclusions  
In this research, we analytically investigate the dynamic response of the pavement subjected to a moving load from a vehicle, modeled as a two-degree-of-freedom system. The pavement is simulated with both the E-B beam and the Timoshenko beam theories, while the foundation is approximated by a nonlinear viscoelastic Winkler foundation featuring cubic characteristics. Using the Adomian decomposition method, combined with the Fourier transform and Duhamel's integral, an integral form of the analytical solution for pavement deformation under the action of vehicles is obtained. Based on the analytical results, we conducted an in-depth study on the influence of the foundation's nonlinear stiffness on the pavement deformation. The conclusions are as follows:  
* When the excitation of the pavement takes the form of a sine wave, the force exerted by the tire upon contact is a harmonic load. * Regardless of whether it is the E-B beam model or the Timoshenko beam model, the presence of cubic nonlinear stiffness will lead to an increase in the higher harmonic terms in the pavement response. * Both the E-B beam model and the Timoshenko beam model have a threshold for the nonlinear stiffness of the foundation. When the nonlinear stiffness is below this threshold, its impact on the pavement deformation is negligible or minimal. However, when the nonlinear stiffness exceeds the threshold, the pavement deformation initially experiences a minor reduction, which is then followed by a substantial increase as the nonlinear stiffness rises. The magnitude of this threshold is not fixed and is primarily influenced by the vehicle velocity and the foundation damping coefficient, increasing with higher vehicle velocities and greater foundation damping. Under the same parameters, the threshold for the Timoshenko beam is significantly higher than that for the E-B beam. * For both beam models, when the nonlinear stiffness is relatively low, the position of the wheel is almost coincident with the location of maximum pavement deformation. As the nonlinear stiffness increases, a noticeable offset occurs between the two positions, with the wheel positioned ahead of the area where the pavement deforms the most. The greater the nonlinear stiffness, the larger the offset between them. Comparatively, the offset effect is more pronounced in the E-B beam model. In previous studies [46, 47, 48, 49, 50, 51], it was commonly presumed that the beam was subjected to a moving force of unvarying magnitude. When the magnitude of the force is small, the foundation's nonlinear stiffness has minimal impact on the beam deformation. However, when the force is substantial, the deformation of the beam increases with the growing nonlinear stiffness [8]. Under identical parameter conditions, the results derived from the E-B beam model are consistently higher than those from the Timoshenko beam model. Consequently, employing an E-B beam in design calculations for dynamic strength and fatigue is acceptable as it leads to an overestimation [52, 53]. In this paper, the force acting on the beam is harmonic, and the research indicates that, under the same parameter conditions, the maximum deformation calculated using the E-B beam model exceeds that of the Timoshenko beam model. When the nonlinear stiffness is not significant, it is feasible to disregard the nonlinearity and use a linear model. When the nonlinear stiffness is considerable, there is a marked difference in the deformation distribution calculated by the E-B beam model compared to that of the Timoshenko beam model. If the concern extends beyond just the maximum value of deformation and also includes the discussion of deformation distribution along the beam, careful consideration in model selection is necessary. Our research results notably deepen the comprehension of the sophisticated, nonlinear mechanics that govern the dynamics of interactions between vehicles and pavements. Conceptualization, B.Z. ; methodology, B.Z. ; software, Z.X. ; investigation, L.O. and Z.X.