Laying geogrids and compressible materials, such as expandable polystyrene (EPS), on the tops of culverts under high fill can improve the stability of the fill and reduce the earth pressure on the tops of culverts. The creep of geogrids and compressible materials under high fill affects the stress state of culvert–soil systems. However, the long-term behaviour of culverts with geogrids and EPS is not clear. This paper investigates the long-term behaviour of culvert–soil interactions under high fill, specifically focusing on the creep effects of geogrids and EPS boards. A novel mechanical model for determining the long-term stress characteristics of culverts is proposed based on limit equilibrium theory. A comparison with finite difference simulation results was conducted to validate the proposed model. The findings reveal that the vertical earth pressure on the culvert top gradually transfers to the culvert side, exhibiting a nonlinear decrease over time. Moreover, the horizontal earth pressure on the culvert sidewall exhibits nonlinear growth with time, decreasing incrementally with depth. The maximum horizontal earth pressure on the culvert sidewall increases by 12.5% postconstruction, emphasizing the importance of accounting for the creep effects of geogrids and EPS boards in engineering design to prevent structural issues.