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An extended two-lane lattice model of traffic flow with consideration of the slope effect is proposed. The slope effect is reflected in both the maximal velocity and the relative current. The stability condition of the model is derived by applying the linear stability method. By using the nonlinear analysis method, we obtain the Korteweg–de Vries (KdV) equation near the neutral stability line and the modified Korteweg–de Vries (mKdV) equation near the critical point. The analytical and numerical results demonstrate that the stability of traffic flow is enhanced on the uphill but is weakened on the downhill when the slope angle increases.

In this paper, we deduced a macroscopic traffic model on the uphill and downhill slopes by employing the transformation relation from microscopic variables to macroscopic ones based on a microscopic car-following model considering the velocity difference between adjacent vehicles. The angle $𝜃$ of the uphill and downhill and the gravitational force have a great impact upon the stability of traffic flow. The linear stability analysis for macroscopic traffic model yielded the stability condition. The Korteweg–de Vries (KdV) equation is derived by nonlinear analysis and the corresponding solution to the density wave near the neutral stability line is obtained. By using the upwind finite difference scheme for simulation, the spatiotemporal evolution patterns of traffic flow on the uphill and downhill are attained. The unstable region is shrunken with slope of the gradient increasing and backward-traveling density waves gradually decrease and even disappear on uphill. Conversely, the unstable region on downhill is extended and density waves propagate quickly backward to the whole road with slope of the gradient increasing.

The rigid pavements and rigid base supporting rails can be considered as a strip footing resting over an embankment. In addition, there can also be single or double-strip footings resting over an embankment. These strip footings are influenced by the properties of the soil, slope angle of an embankment, setback distance, and spacing between the footings and the height of embankment. In this study, the bearing capacity factors for the single- and double-strip footings resting over an embankment are evaluated using the finite element method (FEM). The bearing capacity factors $N_c^{\prime }$ and $N_{\gamma }^{\prime }$ are evaluated for different soil friction angles $(\varphi )$, slope angles of embankment $(\beta )$, setback distances $(S)$, embankment heights $(D)$ and spacing between the footings $(t)$. Considering the practical aspect, i.e., the surcharge is not present for footing on an embankment, the $N_q^{\prime }$ is not evaluated in the study. Based on the analysis, it is observed that the bearing capacity factors $N_c^{\prime }$ and $N_{\gamma }^{\prime }$ increase with increase in friction angle of the soil and setback distance and reduce with increase in the slope angle of the embankment. The effect of depth of embankment on the $N_c^{\prime }$ and $N_{\gamma }^{\prime }$ is negligible for smaller friction angles $(\varphi \le 20^{\circ })$, whereas it increases marginally for higher friction angles $(\varphi >20^{\circ })$. Furthermore, the $N_c^{\prime }$ and $N_{\gamma }^{\prime }$ are influenced by the spacing between the footings. This study’s results are compared with the results available in the literature. This study’s results are presented as design charts, and these could be adopted in the practice in routine designs of shallow foundations.